Systems Self-Assembly: Multidisciplinary Snapshots

Series Dedication Studies in Multidisciplinarity is dedicated to the memory of Ray Paton. Sure, he that made us with s...

Author: Natalio Krasnogor | Steve Gustafson | David A. Pelta | Jose L. Verdegay

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Series Dedication

Studies in Multidisciplinarity is dedicated to the memory of Ray Paton. Sure, he that made us with such large discourse, Looking before and after, gave us not That capability and god-like reason To fust in us unused. – William Shakespeare, Hamlet

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A deeper understanding of the fundamentals of molecular self-assembly all the way up to the self-organisation of biological processes is set to profoundly affect the way we build systems, and more specifically, computational or informational ones. That is, self-assembly and self-organisation, ubiquitous natural processes, are likely to become key engineering tools in the quest for faster, cheaper, smaller and pervasive information processing. Self-assembly is a potentially advantageous fabrication and control process because, with an appropriate set of components and associated interactions, these components will autonomously, robustly and efficiently assemble into a desired system without recurring to a master plan or centralised (e.g., top down) control. Robustness and versatility are some of the most important properties of self-assembling natural systems and this makes self-assembly an even more appealing engineering methodology. The idea for compiling this book arose out of necessity: although there are many conference proceedings and journals that deal in details with, for example, DNA-based computation and DNA-based self-assembly, there was no one book that would overview self-assembly research from a wider, more systemic, perspective. This book is an attempt to, at least partially, cover this gap. Although systems where self-assembly takes place, or which are created by a selfassembling process, are remarkably varied, some common principles for selfassembly are starting to be discerned. This book is a showcase of self-assembly in systems that arise from the physical, chemical, biological, computational and engineering disciplines. The “Ariadna’s Thread” that unifies the various case studies described within the book is the computational nature of self-assembling systems. That is, this book proposes that computational tools might give us and advantageous perspective from where to study self-assembling systems as to be able to better design, program and control them. The contributions in this book cover a range of different research themes dealing with self-assembly and self-organisation either as the focus of research itself or as a mechanism to achieve some other technological goal, e.g., new computational architectures. The book includes thirteen chapters that were written by renown scientists. These chapters cover five main themes, namely, the language of self-assembly (Chapters 3, 7 and 9), optimal design of self-assembling systems (Chapters 4, 10 and 13), self-assembled computing architectures (Chapters 5, 6 and 8), self-assembly principles (Chapters 2 and 12) and molecular self-assembly (Chapters 1 and 11). The chapters are not clustered around these five themes as we feel that each and every chapter is a fascinating and complementary account of the state-of-the-art (and current challenges) in self-assembly and selforganisation research. We thus invite the reader to reach and read every chapter,

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regardless of the theme it belongs to and to do so even if a given chapter does not fall exactly within the reader’s own research domain. The first chapter, by C.P. Martin, M.O. Blunt, E. Vaujour, A. Fahmi, A. D’Aléo, L. De Cola, F. Vögtle and P. Moriarty, is entitled Self-Organised Nanoparticle Assemblies: A Panoply of Patterns. This chapter provides a well-balanced account of recent progress made in relation to the understanding of self-organisation in nanoparticles ensembles. The paper provides a general introduction to the field and its terminology and presents detailed experiments performed by the authors. They also review some of the main analytical tools that can be used to understand the processes behind self-assembly and self-organisation, e.g., information theoretic analysis, proximity graphs analysis (e.g., Voronoi tessellations), and morphological image analysis. They then proceed to explain the relevance and impact of these types of analysis and how they can be used to go back to the chemistry and physics of these systems with new and improved understanding. Biomimetic Design of Dynamic Self-Assembling Systems by K.J.M. Bishop, C.J. Campbell, G. Mahmud and B.A. Grzybowski, the second chapter in the book, presents a very detailed account of nature’s strategies for achieving highly sophisticated self-assembly structures and processes. The authors argue that, although remarkable progress has been made in what relates to self-organisation and self-assembly, many lessons remain to be uncovered by closely studying biologically-based systems. They provide a series of study cases and propose a heuristic design methodology for self-assembly systems that is based on a detailed understanding of how nature utilises competitive interactions, energy dissipation, the interplay between medium and length scale and the hierarchical organisation of complexity for the purpose of creating stunning dynamical selfassembling systems. M. Gheorghe and G. P˘aun’s Computing by Self-Assembly: DNA Molecules, Polyominoes, Cells, the third chapter in this book, provides a language theoretical characterisation of Turing computability by self-assembly through the use of the so-called “self-assembly universality” (SAU) lemma. The authors present three distinct paradigms: DNA self-assembly through Sticker systems, 2D “pixelated” tiles of arbitrary shapes and self-assembling membrane systems (i.e., P-systems). They demonstrate that, while the details that make up each one of these computing-by-self-assembly paradigms are different, they retain some essential common features that can best be understood through the tools provided by theoretical computer science. The chapter Evolutionary Design of a Model of Self-Assembling Chemical Structures, by A. Buchanan, G. Gazzola and M.A. Bedau, is the fourth in the book. The authors present a modified version of the well-known Dissipative Particle Dynamics simulation method that can cope with the formation and dissociation of strong bonds. They couple their DPD simulation environment to an

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evolutionary algorithm and they demonstrate how the latter can be used to automatically fine tune the parameters of the simulation as to produce specific emergent behaviour and, more specifically, prototype versions of templated ligation. The fifth chapter by S. Stauth, C.J. Morris and B. Parviz entitled Self-Assembly as an Engineering Concept across Size Scales surveys the limitations of current technology vis-a-vis self-assembly processes that could span many length scales and be able to handle heterogeneous fabrication processes. The authors focus on computing devices built through self-assembly that use micron-scale parts and provides two illustrating examples. The first example investigates the incorporation of single crystal silicon components on flexible plastic substrates for the construction of large area electronic systems. The second example explores an avenue for future construction of three-dimensional silicon computing structures. Limitations, potentials and future avenues of research are outlined from both case studies. In Probabilistic Analysis of Self-Assembled Molecular Networks, the sixth chapter, D. Bhaduri and S.K. Shukla propose the first hierarchical technique to introduce structural redundancy in systems built from nanoscale devices and networks. Structural redundancy is highly relevant since it may help to mitigate operational faults. As the devices studied are bound to have errors due to the stochastic nature of the self-assembly fabrication process (and natural wear), this chapter demonstrates that it is possible to produce an accurate probabilistic map of the fabrication errors and then, to use that map to insert structural redundancy in the reconfigurable molecular nanofabric. In this way, the combination of defect mapping and structural redundancy may help to circumvent both natural hard and soft faults. The seventh chapter, by A.M. Bouchard, C.E. Warrender and G.C. Osbourn, is entitled The “Programming Language” of Dynamic Self-Assembly. In this chapter the authors pose two very suggestive questions: (1) if self-assembly processes in living organisms are performing computations, what algorithms are they executing? and (2) if it was possible to understand the “programming language” of dynamic self-assembly, would it be possible to design programs in that language as to assemble novel structures and materials? The authors answer both questions positively through theoretical and simulation arguments. C. Dwyer and A.R. Lebeck, in Self-Assembled Computer Architectures, explore the potential and challenges behind the self-assembling of computing systems, that is, how to use current nanoscience technology to build from the bottom–up novel unconventional computer architectures. This chapter, the eight in the book, provides a general introduction to the field and its terminology, and then presents specific experiments and simulations performed by the authors. The chapter characterises the main features and limitations of state-of-the-art selfassembly technology, e.g., short-range control only, large scale randomness, high defect rates, etc. Novel computing architectures capable of coping and exploiting (some) of these features are presented.

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The chapter Simulation of Self-Assembly Processes using Abstract Reduction Systems by J.-L. Giavitto and A. Spicher is the ninth chapter on this book. The authors explore an abstract modelling language, MGS, for self-organising processes. MGS is a declarative and rule-based formalism that relies on local rewriting rules. The local rewriting rules are adequate to capture natural processes that exhibit emergent self-organisation. The authors explain in detail the motivation behind their work and show several examples that mimics accretive growth and carving processes. P. Moisset de Espanés’ chapter, the tenth of the book, Computer Aided Search for Optimal Self-Assembly Systems, presents a fascinating account of the research behind the optimal design of self-assembling tiles. The author reviews current theoretical results on the complexity of “counters” design, i.e., self-assembling tiles that form rectangles, and details the pivotal role that computer aided proofs had on finding some of the key milestones for DNA-based self-assembled counters. He finishes the work by providing a summary of both solved and open problems. In the eleventh chapter, Programmable Self-Assembly—Theoretical Aspects and DNA-Linked Nanoparticles by B. Högberg, J. Boo, J. Liu-Helmersson, L. Glans and H. Olin, the concept of “bond uniqueness” and its impact on programmable self-assembly is detailed. The authors present a mathematical tool called “sticky graphs” for modelling self-assembly systems. Some key properties of the needed building blocks are derived from formal models. Then, the authors propose a way to physically obtain such building blocks. Lastly, the programmability of DNA modified nanoparticles for the self-assembly of nanostructures is shown. The twelfth chapter in this book, From Microscopic Rules to Emergent Cooperativity in Large-Scale Patterns by B. Tadić focuses on the self-organisation of complex spin-networks. The author shows that, as the microscopic rules that govern the assembly of nodes into connected components change so do the macroscopic structural features of the resulting network. In turn the “function” of the network, e.g., its hysteresis, memory, etc., is strongly correlated with the macrostructure, thus the link between microscopic rules and emergent function at the macroscopic level is established. In the last chapter of the book, Automated Self-Assembling Programming, L. Li, P. Siepmann, J. Smaldon, G. Terrazas and N. Krasnogor review a series of computational studies of self-assembling systems. They explore various selfassembling and self-organising processes that range from the evolvability of abstract Wang tile-like systems and a nanoparticle simulator, to the self-assembly of Java programs and the use of self-assembly of artificial vesicles as a potentially realistic implementation of a novel computing paradigm. The editors are immensely indebted with the authors of the various chapters and with the series editors R. Paton, Anne Rusum and Laura McNamara for their

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support during the preparation of this book. If it was not for the hard work of chapters’ contributors and the patience of reviewers and of the series editors, this book would have not seen the light of days. Very sadly, Ray Paton passed away soon after the proposal that preceded this book was accepted by Elsevier. We will remain forever grateful to him for his initial encouragement and support with this project. D. Pelta and J.L Verdegay acknowledge the support of Projects TIN2005-08404-C04-01 (Co-HeuriFuzzy) and TIC-00129-JA (MINAS). N. Krasnogor acknowledges the support of the United Kingdom’s Engineering and Physical Sciences Research Council (EPSRC) for the funding of projects GR/T07534/01, EP/D021847/1 and EP/E017215/1. Natalio Krasnogor, Steven Gustafson, David A. Pelta, José L. Verdegay

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Self-Organised Nanoparticle Assemblies: A Panoply of Patterns Christopher P. Martina,1 , Matthew O. Blunta,1 , Emmanuelle Vaujoura , Amir Fahmib , Anthony D’Aléoc , Luisa De Colad , Fritz Vögtlee , Philip Moriartya,* a School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD, UK b School of Mechanical, Materials, and Manufacturing Engineering, Faculty of Engineering,

University of Nottingham, Nottingham NG7 2RD, UK c Universiteit van Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands d Physikalisches Institut, Mendelstrasse 7, D-48149 Münster, Germany e Kekulé-Institut für Organische Chemie & Biochemie, Universität Bonn, Gerhard-Domagk-Str. 1, D-53121 Bonn, Germany Abstract. An overview of self-organisation in an archetypal nanostructured system—2D nanoparticle assemblies—is given. We first focus on the parallels that may be drawn for pattern formation in nanoscopic, microscopic, and macroscopic systems (spanning, for example, nanoparticle arrays, phase-separated polymers, diatom microskeletons, and binary fluid separation) before discussing the quantification of morphology and topology in nanostructured matter. The question of quantification is of key importance for the development of programmable or directed assembly and we highlight the central role that image morphometry can play in the software control of matter. The nanostructured systems we describe are, in very many cases, far from their ground state and we show that Monte Carlo simulations (based on the approach pioneered by Rabani et al. [Nature 426 (2003) 271]) provide important insights into the coarsening (i.e. approach to equilibrium) of nanoparticle arrays. We conclude with a consideration of the near-term prospects for programmable matter.

1. Introduction At the time of writing (Spring 2007), the field of complexity science is arguably rivalled only by nanoscience/nanotechnology when it comes to general expectation and hype. With this in mind, in this chapter we will attempt to consider the connections between these areas, focussing specifically on self-assembly, selforganisation, and non-linear dynamics in a prototypical nanostructured system: colloidal nanoparticle assemblies. Our aim is to provide a useful overview of * Corresponding author. E-mail: [email protected], url: http://www.nottingham. ac.uk/physics/research/nano. 1 The authors contributed equally to the work described in this chapter.

STUDIES IN MULTIDISCIPLINARITY VOLUME 5 ISSN 1571-0831/DOI: 10.1016/S1571-0831(07)00001-9

© 2008 Elsevier B.V. All rights reserved.

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pattern formation—both near to, and far from, thermodynamic equilibrium—in nanoparticle systems and to review current strategies to exploit self-organisation as a mechanism for pre-defined nanostructure fabrication. The emphasis is on the patterns formed in extremely thin (∼ a few nm) films of nanoparticles on solid substrates. As this contribution necessarily represents a rather brief consideration of self-organised nanosystems, the particular subset of nanostructured materials we shall discuss will largely be informed by the authors’ research interests and, thus (from a materials science perspective), will be relatively narrow in scope. Nevertheless, the system chosen for discussion (colloidal nanoparticle assemblies) exemplifies many key aspects of the physical properties and behaviour associated with the self-organisation of nanoscale units. Perhaps ‘complexity’—a nebulous term in many contexts—is most simply defined on the basis of Aristotle’s observation in 150 BC that a system can be very much more than just the sum of its parts. That is, there is a particular “added value” arising from the interactions of the units (or “agents”) comprising the ensemble. In terms of nanostructured systems, to illustrate many of the fundamental physical (and physicochemical) phenomena stemming from the interactions of large numbers of virtually identical units, we shall use what we consider the archetypal exemplar: nanoparticles dissolved in an organic solvent, forming a colloidal solution. Colloidal nanoparticles deposited from a solution onto solid surfaces form a rich variety of intricate patterns and the focus in the following discussion is to highlight morphological (and topological) parallels between nanostructured, microstructured, mesoscale, and macroscale systems formed via self-assembly and self-organisation. To avoid potential confusion in later sections, we stress that we draw a distinction between the terms self-assembly and self-organisation where, throughout this chapter, the former is used to describe structures formed close-to-equilibrium whereas the latter refers to far-from-equilibrium dissipative processes involving energy/matter flow. Although this distinction is not always made in the literature, and in some cases the terms are used interchangeably, “self-organisation” has traditionally been reserved to describe processes occurring away from equilibrium. Perhaps the most important example, returning to the question of parallels with complex systems and complexity theory, is self-organised criticality. Although it is a moot point as to whether the systems we describe in the following sections can truly be described as complex systems (exhibiting emergent behaviour), there is no question that correlations in the spatial distribution of the nanoparticles occur on length scales far exceeding those associated with interparticle interactions. The development of correlations on mesoscopic length scales is an important signature of a self-organised system; in self-assembly, interparticle interactions (perhaps mediated by the underlying surface) define the length scales of the observed patterns. We note that other authors, most notably Whitesides and Grzybowski [1], have instead adopted somewhat different definitions. Whitesides and Grzybowski draw

Self-Organised Nanoparticle Assemblies

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a distinction between static and dynamic self-assembly, where they define the latter as arising only in dissipative systems. There is then an equivalence between the Whitesides and Grzybowski dynamic self-assembly process and selforganisation as described above. The colloidal nanoparticle assemblies described in the following sections could be seen to fall into yet a third class involving ‘arrested’ self-organisation (or, depending on the reader’s preference, “arrested dynamic self-assembly”!). By “arrested”, we mean that although the patterns may be defined by far-from-equilibrium processes such as convective flow, the system is strongly kinetically hindered at some point in its evolution so that even when the pattern-forming process is switched off, the associated structure is “frozen in”.

2. Pattern Formation: Spanning the Nanoscopic to the Macroscopic The striking array of patterns shown in Fig. 1.1 was created via a straight-forward experiment involving the deposition of a droplet of a nanoparticle/solvent solution onto a solid substrate (with subsequent evaporation of the solvent). In this case, the particles in question are alkylthiol-passivated Au nanoclusters of ∼2 nm diameter synthesised using the technique pioneered by Brust and co-workers [2]. Other types of nanoparticle, including, for example, CdSe [3] and PbSe [4], also produce a broad variety of complex patterns. The physics underlying the appearance of the types of pattern seen in Fig. 1.1 and in similar systems is rather complex and can involve very many coexisting

Fig. 1.1. A subset of the wide variety of patterns observed in colloidal nanoparticle assemblies formed via solvent evaporation. In each case tapping mode atomic force microscopy has been used to image the distribution of nanoparticles on a native oxide (SiO2 ) covered Si(111) sample. The images shown in (a)–(e) and (h) are of single layers of nanoparticles whereas those in (f) and (g) are bilayer samples. For colour version, see Colour Plate Section.

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phenomena. These include (but are not limited to): the (de)wetting properties of the solvent–nanoparticle solution (on a given substrate); hydrodynamics (including the Marangoni effect [5]); phase transitions related to variations in solvent and nanoparticle density; instabilities in the solvent–nanoparticle fluid front; and nucleation/growth at the solid–liquid and/or liquid–air interface. Although some of these effects have previously been elucidated through experimentation and/or computer simulations [3–10], there remain very many open questions regarding the dynamics of pattern formation in colloidal nanoparticle assemblies. The patterns shown in Fig. 1.1 range from isolated droplets (Fig. 1.1(a)) through dendritic/branched structures (Fig. 1.1(b)), worm-like and interconnected domains (Figs. 1.1(c) and (d)), and cellular networks (Figs. 1.1(e) and (f)), to relatively well-developed fractal morphologies (Figs. 1.1(g) and (h)). The interconnected and labyrinthine structures of Figs. 1.1(c) and (d) are, as shall be discussed below, strikingly similar to the self-organised patterns which form in binary fluid, polymer, and ferromagnetic systems. Importantly, Fig. 1.1 represents only a subset of the wide variety of patterns possible in colloidal nanoparticle systems: the parameter space associated with this—at first glance, rather simple—system is extremely wide. A significant point to realise is that many of the structures shown in Fig. 1.1 are formed far from equilibrium because the solvent (or the vast majority of the solvent) is rapidly driven off by spinning the sample at speeds of a few thousand rpm. As the nanoparticles are restricted from diffusing on the substrate in the absence of solvent [4,11], the system can be kinetically trapped very far away from its equilibrium state. In the limit of a perfectly wetting solvent–nanoparticle film, this equilibrium state is a single close-packed ‘island’ of nanoparticles covering a large surface area. Bigioni et al. [10] have, however, also shown that highly ordered assemblies comprising ∼108 close-packed nanoparticles can be formed by seeding growth of the assembly at the liquid–air interface. One might now enquire as to the “usefulness” of the various morphologies shown in Fig. 1.1 when it comes to developing novel nanostructured systems. Importantly, the majority of the different patterns shown in the figures are each associated with a particular length scale (or set of length scales). It is this presence of well-defined correlation lengths which makes the far-from-perfectly-ordered structures of Fig. 1.1 so interesting. First, the correlation lengths themselves (and other morphological/topological metrics) betray the influence of substantial longranged cross-talk between the elements of the system. These correlations in turn can represent a signature of a particular physical/physicochemical process. In the context of complex system dynamics, therefore, images of the type shown in Fig. 1.1 prompt questions related to the collective interaction of the nanoparticle units. The patterns shown have typical correlation lengths ranging from of order 100 nanometres to a few microns. This should be compared with the typical length scale associated with interparticle interactions which is of order a


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Fig. 1.2. A comparison of patterns formed in (a) phase-separating polymer and (c) binary fluid systems with those observed in colloidal nanoparticle assemblies ((b) and (d)). Figures (a) and (c) are reproduced from [13] and [14] respectively.

nanoparticle diameter (∼3 nm). Second, it is gaining control of this long-ranged collective behaviour that underpins much of the interest in pattern formation in far-from-equilibrium nanostructured systems. Certain types of pattern are ubiquitous in nature and appear in a remarkably wide range of materials and across length scales differing by many orders of magnitude. (We refer the reader to a comprehensive and immensely readable account of pattern formation in nature written within the last few years by Ball [12].) To illustrate the ubiquity of the patterns which appear in nanostructured assemblies, we shall focus on the “worm-like”, labyrinthine, and network structures shown in Figs. 1.1(c)–(f). Figure 1.2 places images of patterns formed in dewetting polymer films and phase separating binary fluids (Figs. 1.2(a) and (c) respectively) alongside atomic force micrographs of structures formed in nanoparticle assemblies (Figs. 1.2(b) and (d)). The qualitative similarity between Figs. 1.2(a) and (b) and between Figs. 1.2(c) and (d) is striking. Although close similarities in pattern formation of course do not necessarily arise from parallels in underlying physical/chemical behaviour, it is nevertheless important to note that polymer, binary fluid, and colloidal nanoparticle systems indeed share a number of common features. These include the influence of spinodal phase separation [3,4], nucleation and growth, and convective fluid flow on the evolution of the system. (We return to the question of the evolution/dynamics of pattern formation in Section 4 below.) A structural motif which appears repeatedly in nanoparticle assemblies is the cellular network [7,8,15]. (Cellular in this case refers to the geometric cells (polygons) comprising the network and does not allude to biological cells.) We

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Fig. 1.3. Cellular networks in nature: (a) the microstructure of a cork [16]; (b) the cellular pattern formed on the hide of a giraffe; (c) the Giant’s Causeway in Co. Antrim, Northern Ireland [17]; and (d) a frame taken from the Virgo Consortium Millennium simulation of the evolution of a structure in the Universe [18]. For colour version, see Colour Plate Section.

have observed a cellular morphology ranging from relatively simple networks which are associated with a single correlation length (see, for example, Martin et al. [8]) to the rather more complex structures shown in Figs. 1.1(e) and (f) which comprise multi-level hierarchies of holes. Although the most widely recognised example is perhaps the foam formed by soap bubbles [15,19], cellular structure of the type shown in Figs. 1.1(e) and (f) is ubiquitous in nature (as pointed out by Weaire and Rivier [15]). We show in Fig. 1.3 four examples of cellular networks ranging from the microstructure of a cork from a wine bottle to the large scale structure of the Universe. What is particularly intriguing about the morphology/topology of cellular networks is that it is possible, using the mathematics of statistical mechanics, to write down an equation of state for an ideal random cellular network which is analogous to the ideal gas law in classical thermodynamics [15,19]. Just as deviations from the ideal gas law betray the influence of new physics and chemistry, a statistical study of the structure of a cellular network (using Voronoi tessellations (see the following section) coupled with, for example, Lewis’ law and the Aboav–Weaire law [15]) provides key insights into the relative importance of physics and chemistry vs. the mathematics of space filling in determining the structure of the system. A particularly interesting example of multi-level network formation in nature (involving hierarchical distributions of holes/pores of different sizes) is found in the diverse set of complex microskeletons formed by the family of singlecelled algae known as the diatoms. These fascinating structures were first described in detail by Haeckel in the late 19th century [20]. Diatoms—of which there are estimated to be ∼105 distinct species—are capable of fabricating exquisitely microstructured silica shells which have, ever since the seminal work of


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Fig. 1.4. Mimicking diatom microskeletons in colloidal (inorganic) nanoparticle assemblies. (a) A SEM image of the valve architecture of Coscinodiscus wailesii (taken from [26]). The scale bar in this image and all other images represents 1 µm. (b) A tapping mode AFM image of a 1.5 mg/ml solution of C5 thiol-passivated Au nanoparticles (2.2 nm mean diameter) spin cast (at 4 krpm) onto a native oxide terminated silicon substrate. (c)–(f) AFM images of nanoparticle array morphologies resulting from spin-casting 1.0, 1.1, 1.25, and 1.5 mg/ml solutions, respectively, onto silicon.

both Haeckel and D’Arcy Thompson [21], fascinated scientist and layman alike. While many aspects of pattern formation in diatom frustules remain to be elucidated [22–24], the potential for exploitation of this class of algae as a “nanofabricator” technology has garnered a remarkable amount of cross-disciplinary attention [25]. Although forming an impressively broad assortment of different patterns [20,23], a distinctive structural theme common to very many species of diatom is the hierarchical nature of the shell architecture. A striking example of this is shown in Fig. 1.4(a), a scanning electron microscope (SEM) image of the morphology of the valve of Coscinodiscus wailesii [26]. A similar type of hierarchical patterning is also observed in the (entirely abiotic) gold nanoparticle assembly shown in Fig. 1.4(b). The ability to pattern matter via selfassembly/self-organisation across a range of different length scales, as exemplified by Figs. 1.4(a) and (b), is an increasingly important goal in nanoscience [27] and a substantial amount of the intense recent interest in the potential nanotech applications of diatoms stems from this morphological characteristic. (Moreover, ground-breaking efforts by Oliver et al. [22] have resulted in the fabrication of artificial diatom microskeleton forms via advanced inorganic morphosynthesis.)

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The nanoparticle assembly of Fig. 1.4(b) clearly comprises cellular networks— “foams”—having two very distinct length scales. Of particular significance to multiple length scale patterning, a short wavelength network formed in the first monolayer of particles is embedded within a larger cellular structure in the second layer, mimicking the hierarchical structuring (at the micron and sub-micron levels) of the diatom frustules shown in Fig. 1.4(a). We tentatively suggest that two distinct processes with well-defined correlation lengths—namely Marangoni convection [6] and evaporative nucleation of holes in the solvent–nanoparticle film [8]—drive the generation of the micron and sub-micron scale nanoparticle foams observed in Fig. 1.4(b). (As initially shown by Ge and Brus [3] and confirmed in recent simulations by Rabani et al. [4], spinodal decomposition can also play a role in forming spatially correlated nanoparticle patterns.) Although in previous works [7,8] we have discussed the ability of far-from-equilibrium assembly to generate nanostructured cellular networks, the diatom-like, ‘foam within foam’ patterning observed in Fig. 1.4(b) is an entirely new and unexpected nanoparticle array morphology. Its origins may be ascertained from an examination of Figs. 1.4(c)–(f). Figures 1.4(c)–(f) show the nanoparticle array morphology arising from spincoating progressively more concentrated nanoparticle solutions onto the silicon substrate. Note the absence of hierarchical patterning in Fig. 1.4(c) and its gradual emergence as the solution concentration is gradually increased. A plausible origin of the longer length scale network present in Figs. 1.4(d)–(f) is the Marangoni effect: a convective flow driven by the temperature gradient imposed due to the evaporation of the volatile solvent (in this case, toluene) [6,28]. In Fig. 1.4(d), a polygonal network (whose cells are spaced by approximately 1 micron) of densely packed nanocrystals in the first monolayer is observed and, as shown in Figs. 1.4(e) and (f), acts as a template for the adsorption of the second layer of nanoparticles (Figs. 1.1(e) and (f)). We are confident that there is a particular scope to program both the degree of network order and the associated correlation lengths (via, for example, the establishment and control of spatially well-defined temperature fields during solvent evaporation).

3.

Quantifying Morphology and Topology

Although qualitatively striking, it is obviously important to consider metrics that might be used to classify quantitatively the morphology and topology of the various nanostructured patterns shown in the previous sections. Not only is this necessary from the perspective of determining, for example, the connectivity and correlations within a given structure but quantitative metrics can be applied to ascertain what parallels might exist between apparently disparate systems. Although Fourier (and wavelet) analysis can be used to determine correlation lengths and


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Fig. 1.5. The ‘grain growth’ approach to image analysis using Minkowski measures [30]. The decoration of germs (light grey) with grains (dark grey) that are discrete approximations to circles (left) and squares (right).

degrees of orientational order, sophisticated complementary methods such as statistical crystallography [15] and Minkowski morphometry [30] are increasingly being used to characterise the morphology and topology of nanostructured systems. We have previously [7,8] used Voronoi tessellations [15] to elucidate the degree of spatial correlation (i.e. deviations from Poisson statistics) in nanoparticle assemblies. Mecke et al. [29] pioneered the use of Minkowski functionals in the analysis of spatial correlations in dewetting polymer films. In 2D, the Minkowski functionals [30] are reduced to three relatively straight-forward geometric measures: the total covered area, the total perimeter length, and the Euler characteristic. This latter measure effectively accounts for the number of interconnected regions in an image [30] and is of obvious significance if the connectivity of a nanostructured system is of interest (in, for example, electron transport via percolation). From an image where the centres of mass of the morphological features may be readily identified, Minkowski analysis may be applied using the method described in the review article by de Raedt et al. [30] and schematically shown in Fig. 1.5. From each initial “germ” (of edge length r = 1) representing the centre of mass of a feature in the original image (e.g. a hole in an otherwise continuous film or an island of material on a substrate), a square “grain” of edge length 2r +1 is grown and the Minkowski measures are calculated for each grain size or, more correctly, as a function of the normalised quantity x = r/L where L is the mean germ separation. Different spatial distributions of the germs will produce marked differences in plots of the Minkowski measures (perimeter, area, or Euler characteristic) against x. A number of groups have discussed the application of Minkowski measures to the analysis of structure formation in polymer thin films [29,31,32]. Cer-

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Fig. 1.6. A simple schematic illustration of an Au nanoparticle terminated by a 3,5-Bis(benzyloxy)benzylbromide S6G1 dendrimer. For colour version, see Colour Plate Section.

tain classes of pattern, however, specifically those associated with the nucleation and growth of holes or islands, give rise to effectively system-independent Minkowski characteristics. That is, despite having dramatically different materials characteristics, the same types of spatial correlations are seen in many systems. What is the origin of these similarities? To attempt to address this question we will consider a nanoparticle system which is markedly different from that discussed thus far. In Fig. 1.6 we show a schematic diagram of a gold nanoparticle where the surface-terminating species is not a simple thiol molecule but a 3,5Bis(benzyloxy)benzylbromide S6G1 dendrimer [33,34]—a complex branched polymer. Dendrimers [35] are increasingly used as stabilising agents for nanoparticle surfaces as they have a well-defined molecular weight, size, and structure and they can encapsulate both organic and inorganic hosts. We show in the following that despite these significant differences in nanoparticle size and structure, morphological measures based on Voronoi tessellations and Minkowski functionals show that there are striking parallels in the structure of assemblies of thioland dendrimer-terminated nanoparticles. Indeed, these parallels extend to entirely unrelated materials systems such as polymers [8] and small organometallic molecules [36]. Figure 1.7(a) shows the adsorbed “film” formed by S6G1-terminated Au nanoparticles following spin-coating of a colloidal solution of the particles on a silicon substrate. The dendrimer-stabilised nanoparticles aggregate to form globular structures with heights and diameters of order 50 and 250 nm respectively. In common with previous AFM measurements of spin-coated dendrimer films [37], we observe a relatively high density of ‘droplets’. What is intriguing, however, is that even from an initial qualitative appraisal of the images, the positions of the dendrimer-functionalised nanoparticle droplets appear to be spatially correlated to some degree (i.e. there are strong deviations from a Poisson-distributed point set). (This is true also of the images reported by Sano et al. [37].) To put our


11

Fig. 1.7. (a) A tapping mode atomic force microscope (AFM) image (40 × 40 µm2 ) of the film morphology formed by spin-coating a solution of dendrimer-functionalised Au nanoparticles in toluene onto a native oxide-terminated Si(111) substrate. Note that the nanoparticles aggregate to form droplets with a relatively narrow size distribution and with an apparent ordering in their spatial distribution. (b) A Voronoi tessellation constructed from the centres of the droplets shown in (a). (c) A histogram of the probability, p(n), of finding an n-sided cell. For colour version, see Colour Plate Section.

discussion on a firmer quantitative footing, we have carried out Voronoi tessellation and Minkowski functional analyses for the dendrimer–nanoparticle aggregate distributions (see Figs. 1.7(b) and 1.8). A Voronoi tessellation is the statistical crystallography analog of the Wigner– Seitz unit cell used extensively in conventional crystallography and condensed matter science. Briefly, the tessellation shown in Fig. 1.7(b) was constructed by: (i) identifying the centre of mass of each of the nanoparticle droplets shown in Fig. 1.7(a), (ii) connecting each centre of mass to its nearest neighbours, and (iii) finding the perpendicular bisectors of those connecting lines. This produces, for each droplet, a Voronoi cell (a polygon) which represents the smallest surface

12

Chapter 1

area associated with that droplet. The subsequent analysis of the distribution of polygon sidedness (i.e. the number of polygon sides), area, and perimeter length can yield substantial insights into the physics underlying the formation of a particular surface morphology. From the distribution of polygon sidedness shown in Fig. 1.7(b), it is straightforward to calculate a ‘tessellation entropy’ (a quantity that is obviously very much distinct from the overall thermodynamic entropy of the system) which is defined in an analogous fashion to the conventional statistical mechanics form: S = − pn ln pn , where pn is the probability associated with finding an n-sided polygon. For a Poisson point set, the tessellation entropy is ∼1.75. For the image shown in Fig. 1.7(b), and in common with a number of other nanostructured systems [7,8], we find an entropy of ∼1.4 (to two significant figures). That the value of S for the distribution of aggregates shown in Fig. 1.7(a) falls substantially below that expected for a Poisson distribution is indicative of the presence of spatial correlations in the nanoparticle arrangement. Moreover, it is intriguing that a value of S = 1.40(±0.05) is associated with a rather broad range of nanostructured thin films: could this be indicative of a common origin for the deviation from Poisson statistics? To address this question, we have calculated the variation of the Minkowski measures with ‘grain’ size, as described above. The deviation from a Poisson point set is also apparent from the Minkowski measure analysis. We show in Fig. 1.8 the variation of the Euler characteristic, χ, as a function of x for the distribution of nanoparticle aggregates shown in Fig. 1.7(a). (Note that although in general it is important to consider variations in all three Minkowski measures, we focus here on the Euler characteristic as a representative example of the analysis technique.) Figure 1.8(a) is a comparison of the variation in χ for the image shown in Fig. 1.7(a) with the variation expected for a Poisson point set. The difference between these quantities is shown in Fig. 1.8(b) where the deviation from Poisson statistics is clear, in agreement with the Voronoi tessellation analysis of Fig. 1.7. The Minkowski measures, however, provide substantially more insight into the morphology and topology of a system. Perhaps more importantly, the Euler characteristics shown in Fig. 1.8 overlap almost perfectly with those measured in very different systems including dewetting organometallic thin films [36] and thiol-passivated Au nanoparticles [8]. How does this similarity in morphological characteristics arise and, in particular, why are strong deviations from Poisson statistics observed in each of these systems? Although phenomena such as spinodal decomposition/dewetting or Marangoni convection are inherently associated with well-defined correlation lengths, in the systems discussed above and, in particular, that shown in Fig. 1.7, we have proposed [8] that the deviations from spatially uncorrelated morphological features arise simply from coalescence events which wipe out the clustering that is the signature of a Poisson distribution of points. Brinkmann et al. [38,39] have previously observed similar strong mesoscopic correlations in the positions


13

Fig. 1.8. (a) The variation of the Euler characteristic as a function of normalised grain size (see text for a discussion) for a point set derived from the centres of the droplets shown in Fig. 1.7(a). The graph in (b) shows the deviation of the Euler characteristic from that expected for a Poisson (spatially uncorrelated) distribution of points. For colour version, see Colour Plate Section.

of islands of organic molecules deposited by vacuum sublimation onto silicon substrates. They attribute the emergence of a cut-off distance in the distribution of nearest neighbour separations to the merging of neighbouring aggregates either via direct coalescence (touching) of two islands or via a ripening phenomenon involving overlapping island diffusion fields. Regardless of the coalescence mechanism, it is from some perspectives remarkable that an undirected assembly process (occurring on a homogeneous substrate) can give rise to such striking spatial correlations in disparate systems (colloidal nanoparticles vs. vacuumdeposited organic molecules). An open question relates to the degree to which the correlations can be tuned or programmed—an issue of obvious importance for self-organised nanostructured systems.

4.

Evolving to Equilibrium

Returning to a consideration of the thiol-passivated nanoparticle patterns shown in Fig. 1.1, Rabani et al. [4] have put forward an Ising model description of drying-mediated pattern formation in nanoparticle–solvent films where the Hamiltonian of Eq. (1) is used within a Monte Carlo algorithm to model the system dynamics. H = −l (1) li lj − n ni nj − n l ni lj − μ li . ij

ij

ij

i

This model reproduces many (though certainly not all—see below) of the types of patterns we [7,8] and others [3] observe experimentally and has also been shown to yield good agreement with experimental data on the dynamics of nanoparticle aggregate (i.e. cluster) growth [4]. Equation (1) comprises (from left

14

Chapter 1

to right) terms describing nanoparticle–nanoparticle, nanoparticle–solvent, and solvent–solvent interactions, with the final term related to the chemical potential of the solvent (which accounts for the solvent evaporation rate/vapour pressure). Changes in the value of the total (global) energy of the system (represented by the Hamiltonian in Eq. (1)) are used to assign probabilities (via a Boltzmann factor, exp(− H kT )) for a number of processes to occur (i.e. the standard Metropolis algorithm Monte Carlo strategy [40] is implemented). In the nanoparticle–solvent system of interest here, the dynamic processes are limited by Rabani et al. [4] to solvent evaporation/condensation and nanoparticle hopping on the substrate. Importantly, nanoparticles do not diffuse on dry substrate: the model restricts nanoparticle hopping into sites which are occupied with the solvent. As highlighted by Rabani et al., it is this coupling between the solvent and nanoparticle dynamics which gives rise to much of the interesting behaviour of the system. For certain classes of pattern, the Rabani et al. approach produces remarkable agreement with experiment [4,8]. The process by which a far-from-equilibrium system approaches its equilibrium state is known as coarsening or ripening. Just as the snapshots of patterns shown in the figures thus far (where each image shows the system at a particular point in time) exhibit striking system-independent morphologies, coarsening is a ubiquitous phenomenon which displays a key universal feature: a temporal selfsimilarity of the sample morphology. (For an excellent review of coarsening in 2D systems, see Zinke-Allmang [41].) The self-similar nature of the coarsening process means that the evolution of the system towards equilibrium needs only one parameter—the fundamental length scale—for its description. This length scale, r, follows a power-law dependence on time, i.e. r ∼ t γ , where γ is the growth exponent. Different diffusion mechanisms (which underlie the coarsening process) give rise to different values of γ (although it is important to note that knowledge of γ alone is not always sufficient to correctly characterise the coarsening mechanism). Perhaps of most relevance to the central theme of this book, the self-similarity inherent in coarsening systems enables a pattern initially present at the nanometre scale to be preserved and “expanded”, in principle, up to micron (or larger) length scales. In recent experiments we have observed a striking coarsening of nanoparticle assemblies driven by a scanning probe [42]. In this coerced coarsening phenomenon, the degree of “pattern expansion” is tunable simply by interrupting the scan process at a pre-defined time. As the work on probe-induced coarsening is detailed elsewhere [42], we shall instead focus here on an analysis of the evolution of a nanoparticle assembly (formed in the spinodal regime of solvent evaporation) using the Monte Carlo simulation described above. Figure 1.9 shows three snapshots from a simulation of the evolution of a “spinodal” assembly. It is clear that, as time progresses, while the average size and spacing of the domains increase, the general form of the pattern does not change. This is clear from an analysis


15

Fig. 1.9. Three frames showing coarsening of a nanoparticle assembly formed by solvent evaporation in the spinodal regime. (Simulation parameters: 1008 × 1008 pixel system with kB T = l /2, MR = 30, and a nanoparticle coverage of 30%.) (a) After 39 Monte Carlo steps, the majority of the solvent has already evaporated leaving only a thin wetting layer around the domains; (b) after 299 MC steps, a visibly increased length scale is present, and (c) after 999 MC steps where the length scale has clearly increased further. For colour version, see Colour Plate Section.

of the radially averaged Fourier transforms of the images (not shown). With appropriate scaling so that the magnitude of the transform is plotted not against q (where q represents wavevector) but against q/qmax (where qmax is the value of q at which the Fourier transform peaks, representing the “signature” length scale of the system), the Fourier transforms collapse onto a time-independent master curve [42,47]. As the position of the peak in the radially averaged Fourier transform (qmax ) represents the correlation length of the coarsening system at a given point in time, by plotting qmax versus MC step number (i.e. time) on a log–log scale, it is possible to extract the value of the growth exponent, γ . Rabani et al. [4] have shown that for nanoparticle assemblies comprising ensembles of isolated islands, the mean island radius scales so that γ = 0.25. An interesting question is whether the interconnected morphology of the assemblies shown in Fig. 1.9 might produce a different scaling exponent (i.e. are different diffusion pathways active?). Both experimental data [42] and the results of MC simulations (see Fig. 1.10(a)) indicate that the exponent of 0.25 is preserved in the spinodal regime. Figure 1.10(a) is a log–log plot of qmax vs. number of MC steps and it is clear that the slope of the graph asymptotically approaches −0.25 (the negative sign arises because in this case we have plotted wavevector rather than wavelength against time). We have recently verified that the 0.25 exponent derived from Fig. 1.10(a) for a relatively short simulation time does not change for simulation times up to a few hundred thousand MC steps. That is, in Fig. 1.10(a) the system has reached the appropriate scaling regime. We have also examined the evolution of the various Minkowski measures for the nanoparticle assembly shown in Fig. 1.9. Intuitively, the boundary length, U , of this structure will decrease over time. This can be used as a measure of

16

Chapter 1

Fig. 1.10. Analysis of the scaling regime in the evolution of a ‘spinodal’ nanoparticle assembly showing (a) the evolution of the peak in the 2D FFT, and (b) the evolution of the length scale L = Atotal /U of the pattern (where U is the total perimeter length—a 2D Minkowski measure). Both reveal an approach to an exponent of 1/4. For colour version, see Colour Plate Section.

the length scale of the pattern. We can define a length scale, L = Atotal /U , where Atotal is the total area of the simulation grid. L is then a measure of the mean size of features in the image and, as shown in Fig. 1.10(b), this quantity follows the same growth law (γ = 0.25) as the peak in the Fourier transform. More importantly, Minkowski measures can be used to demonstrate that despite an increasing length scale, the pattern retains the same morphology. If the Euler characteristic scales in the same manner as the perimeter, then the pattern can be said to be morphologically stable. The value of χ /U should tend to a constant, ) should therefore tend to zero. This is indeed the case: and the value of d(χ/U dt such a plot drops to within 10−4 of zero after 100 MC steps, and after 400 MC steps there is an even clustering around zero. Finally, despite its ability to accurately (and impressively) reproduce many of the patterns observed experimentally in colloidal nanoparticle assemblies (and their associated dynamics), there are a number of very important limitations of the Monte Carlo model described above. Many of the key limitations stem from the fact that only solvent evaporation or condensation is incorporated in the algorithm—there is no flow of solvent possible. In other words, although the nanoparticles hop on the substrate, the solvent molecules do not diffuse from site to site. This restriction in turn means that the effects of convective solvent dewetting cannot be reproduced in the Monte Carlo code. To build in convective dewetting processes requires not only a consideration of solvent diffusion but the modelling of a solvent film which can have local 2D or 3D character in different regions of the surface. (We are currently developing code which incorporates dewetting phenomena of this type. Note added in proof: This work was subsequently published


17

in September 2007 [43].) Important additional effects not built into the model— and pointed out by Rabani et al. [4] in the conclusions of their paper—are hydrodynamics driven by Marangoni convection [6,7] and front instabilities (which the results of our recent experiments [44] suggest are strongly mediated by solvent dewetting). These front instabilities give rise to structures very reminiscent of the “viscous fingering” patterns observed in a range of systems including mixtures of viscous and inviscid fluids and solidification from a melt. What is of particular importance is that each of these effects can be, in essence, “programmed” by varying the substrate chemistry, opening new avenues of research in directed assembly of nanostructured systems [42,43].

5.

Conclusions

Colloidal nanoparticle assemblies represent a fascinating archetype for the study of self-assembly, self-organisation, and pattern formation in nanostructured systems. Many complex and intricate patterns result from a rather simple experiment where droplets of colloidal solution are deposited or spun onto solid substrates. Despite the simplicity of the experiment, a variety of complex physicochemical effects give rise to intricate (but generally spatially correlated) patterns ranging from isolated nanoscale droplets through labyrinthine structures to fractal branches. In this chapter some of the phenomena underlying the origin and evolution of patterns in nanoparticle assemblies have been described but in many ways we (i.e. the nanoscience research community) have barely “scratched the surface” both in terms of our understanding of self-assembly and, of key importance, in developing protocols to reliably tune or, more excitingly, program the assembly/organisation process. Pioneering steps in controlling the assembly of nanoparticle arrays have been made by a number of groups (including Refs. [3,4,10,45,46] amongst others) but it is clear that even in arguably the most basic colloidal nanoparticle system (thiol-passivated Au clusters) there is a very wide parameter space associated with pattern formation. When one considers the broad variety of functional groups that can be added to either the nanoparticle or the substrate surface, the parameter space associated with pattern formation becomes not just wide but vast! As such, we can expect to continue to see (for quite some time) a wealth of exciting fundamental and applied science on self-assembly and self-organisation in colloidal nanoparticle arrays. Arguably, however, the most exciting element of the self-assembly/selforganisation processes described above lies in their potential to lead to programmable assembly of matter. Software compilation of matter from individual molecules forms the core of the molecular manufacturing concept pioneered by K. Eric Drexler in his book Nanosystems [48]. Although molecular manufacturing is an extremely controversial subject within the nanoscience community

18

Chapter 1

(see, for example, [49–52]), at its core lies an important and demonstrably valid idea: computer-controlled single molecule chemistry. In the Drexler scheme, the position of each atom in a macroscopic structure is completely defined and the structure is assembled via entirely deterministic molecular trajectories. An alternative approach to the realisation of programmable matter is to tune the pattern-forming processes described above. This could be achieved via a variety of physicochemical parameters including electric and magnetic field strengths (and/or frequencies, phases, etc.), pH differences, and optical wavefields. Of course, the exploitation of one or more of these parameters in directed selforganisation necessitates a careful consideration of the thermodynamic and kinetic constraints on a particular system and thus, although a number of groups are currently pursuing research programmes involving directed assembly, progress is relatively slow. Nevertheless, a number of approaches—including the use of genetic algorithms to tune a nanostructured system towards a particular type of non-equilibrium state [53]—appear viable and we are confident that in the next decade important advances will be made in the area perhaps best described as “matter compilation”.

Acknowledgements The authors gratefully acknowledge funding from the European Commission under Sixth Framework contract HMT-RTN-200450728 (the PATTERNS Marie Curie Research Training Network (RTN)). This work was also supported by a Fifth Framework RTN, Supramolecular Self-Assembly of Interfacial Nanostructures (SUSANA) (HPRN-CT-2002-00185) and by the UK Engineering & Physical Sciences Research Council (EPSRC). We thank members of the PATTERNS network (in particular, Uwe Thiele, Ulli Steiner, Mathias Brust, and Bosa Tadic) for extremely helpful and enlightening discussions on pattern formation in thin film and nanostructured systems.

References [1] George M. Whitesides, Bartosz Grzybowski, Science 295 (2002) 2418. [2] M. Brust, M. Walker, D. Bethell, D.J. Schiffrin, R. Whyman, J. Chem. Soc. Chem. Comm. 1994 (1994) 801. [3] G. Ge, L. Brus, J. Phys. Chem. B 104 (2000) 9573. [4] E. Rabani, D.R. Reichman, P.L. Geissler, L.E. Brus, Nature 426 (2003) 271. [5] M.P. Pileni, J. Phys. Chem. B 105 (2001) 3358. [6] M. Maillard, L. Motte, M.P. Pileni, Adv. Mat. 13 (2001) 200. [7] P. Moriarty, M.D.R. Taylor, M. Brust, Phys. Rev. Lett. 89 (2002) 248303. [8] C.P. Martin, M.O. Blunt, P. Moriarty, Nano Lett. 4 (2004) 2389. [9] X.M. Lin, H.M. Jaeger, C.M. Sorensen, K.J. Klabunde, J. Phys. Chem. B 105 (2001) 3353.


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[10] T.P. Bigioni, X.M. Lin, T.T. Nguyen, E.I. Corwin, T.A. Witten, H.M. Jaeger, Nature Materials 5 (2006) 265. [11] J.N. O’Shea, M.A. Phillips, M.D.R. Taylor, P. Moriarty, M. Brust, V.R. Dhanak, Appl. Phys. Lett. 81 (2002) 5039. [12] Philip Ball, The Self-Made Tapestry: Pattern Formation in Nature, Oxford University Press, 2001. [13] K. Fukunaga, H. Elbs, G. Krausch, Langmuir 16 (2000) 3474. [14] H. Tanaka, T. Araki, Phys. Rev. Lett. 81 (1998) 389. [15] D. Weaire, N. Rivier, Contemp. Phys. 25 (1984) 59. [16] Sidney Perkowitz, Universal Foam, Walker & Co., New York, 2000. [17] Image taken from The National Trust website: www.nationaltrust.org.uk. [18] www.virgo.dur.ac.uk. [19] N. Rivier, Philos. Mag. B 52 (1985) 795. [20] E. Haeckel, Challenger Monograph (1887). [21] D. Thompson, On Growth and Form, Cambridge University Press, Cambridge, 1917. [22] O. Oliver, et al., Nature 378 (1995) 47. [23] G.A. Ozin, Acc. Chem. Res. 30 (1997) 17. [24] M. Sumper, Science 295 (2002) 2430. [25] Journal of Nanoscience and Nanotechnology (January 2005) devoted entirely to diatom nanotechnology. [26] A.-M.M. Schmid, B.E. Volcani, J. Phycol. 19 (1983) 387. [27] M.D. Morariu, N.E. Voicu, E. Schaffer, Z.Q. Lin, T.P. Russell, U. Steiner, Nature Materials 2 (2003) 48. [28] H. Bénard, Rev. Gen. Sci. Pures Appl. Bull. 11 (1900) 1261. [29] K. Jacobs, S. Herminghaus, K.R. Mecke, Langmuir 14 (1998) 965. [30] K. Michielsen, H. De Raedt, Phys. Rep. 347 (2001) 461. [31] J. Becker, G. Grun, R. Seemann, H. Mantz, K. Jacobs, K.R. Mecke, R. Blossey, Nature Materials 2 (2003) 59. [32] J. Rysz, Polymer 46 (2005) 977. [33] S.W. Chen, R.W. Murray, Langmuir 15 (1999) 682. [34] A. D’Aléo, R.M. Williams, F. Osswald, P. Edamana, U. Hahn, J. van Heyst, F.D. Tichelaar, F. Vögtle, L. De Cola, Adv. Funct. Mat. 14 (2004) 1167. [35] F. Vögtle, S. Gestermann, R. Hesse, H. Schwierz, B. Windisch, Prof. Poly. Sci. 25 (2000) 987. [36] F. Frehill, K.H.G. Schulte, C.P. Martin, L. Wang, S. Patel, J.A. Purton, J.G. Vos, P. Moriarty, Langmuir 20 (2004) 6421. [37] M. Sano, J. Okamura, A. Ikeda, S. Shinkai, Langmuir 17 (2001) 1807. [38] M. Brinkmann, F. Biscarini, C. Taliani, I. Aiello, M. Ghedini, Phys. Rev. B 61 (2000) R16339. [39] M. Brinkmann, S. Graff, F. Biscarini, Phys. Rev. B 66 (2002) 165430. [40] G. Casella, C.P. Robert, Monte Carlo Statistical Methods, Springer-Verlag, New York, 2004. [41] M. Zinke-Allmang, Thin Solid Films 346 (1999) 1. [42] M. Blunt, et al., Nature Nanotech. 2 (2007) 167. [43] C.P. Martin, M.O. Blunt, E. Pauliac-Vaujour, A. Stannard, P.J. Moriarty, I. Vancea, U. Thiele, Phys. Rev. Lett. 99 (2007) 116103. [44] E. Pauliac-Vaujour, P. Moriarty, J. Phys. Chem. C, published online 9 October 2007, doi: 10.1021/jp074152t. [45] R.P. Andres, J.D. Bielefeld, J.I. Henderson, D.B. Janes, V.R. Kolagunta, C.P.V. Kubiak, W.J. Mahoney, R.G. Osifchin, Science 273 (1996) 1690. [46] C.J. Kiely, J. Fink, M. Brust, D. Bethell, D.J. Schiffrin, Nature 396 (1998) 444. [47] H.-J. Ernst, F. Fabre, J. Lapujoulade, Phys. Rev. Lett. 69 (1992) 458.

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[48] K. Eric Drexler, Nanosystems: Molecular Machinery, Manufacturing, and Computation, Wiley Interscience, Chichester, 1992. [49] K. Eric Drexler, R.E. Smalley, Chem. Engng. News 81 (2003) 37. [50] http://www.softmachines.org/wordpress/index.php?p=70. [51] P. Moriarty, Nanotechnology Perceptions 1 (2005) 115. [52] http://video.google.com/videoplay?docid=-6086023528639836493. [53] P. Siepmann, C.P. Martin, I. Vancea, P.J. Moriarty, N. Krasnogor, Nano Lett. 7 (2007) 1985.

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Chapter 2

Biomimetic Design of Dynamic Self-Assembling Systems Kyle J.M. Bishop, Christopher J. Campbell, Goher Mahmud, Bartosz A. Grzybowski* Department of Chemical & Biological Engineering and Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA Abstract. Biological systems at all scales self-assemble and maintain themselves in dynamic states away from thermodynamic equilibrium. Examination of various biological examples of dynamic self-assembly (DySA) enables one to extract general principles that govern it. These principles, in turn, can be applied to the rational design of artificial DySA systems by combining biological inspiration with the tools of chemistry, physics, and engineering. Such artificial DySA ensembles can be studied in quantitative detail and will ultimately lead to novel applications in biomimetic “smart” structures and materials.

1. Introduction Self-assembly (Figs. 2.1 and 2.2)—that is, the process by which isolated components organize autonomously and spontaneously into ordered and/or functional structures [1]—is nature’s preferred way of building its animate creations on various length scales. Science and technology have long recognized the promise of the “bioinspired” SA approach for building large structures from small objects. Nevertheless, despite several impressive demonstrations, such as SA of 2D and 3D structures via capillary forces [2,3], DNA templating [4,5], fluidic machines [6], microspheres with complex internal structures [7], 2D electrostatic crystals [8,9], and several more, we are still very far from using SA as flexibly and creatively as nature does, especially in dynamic systems operating far from thermodynamic equilibrium. Therefore, to better understand dynamic self-assembly (DySA), it is instructive to first examine representative biological ensembles and extract from them at least qualitative governing principles. This effort will, in turn, further our ability to create artificial DySA systems that can be studied in quantitative detail and will ultimately lead to novel applications in biomimetic “smart” structures and materials. * E-mail: [email protected].



22

Chapter 2

The present contribution is intended to provide a basis for a systematic study of DySA and to illustrate some early experimental developments in the field. We begin by defining and characterizing various modes of self-assembly with emphasis on dynamic systems. We then examine several examples of DySA in biological systems operating on length scales from molecular to macroscopic. Based on these examples, we develop a heuristic approach that allows for the rational design of simple artificial DySA systems and combines biological inspiration with the “tools” of physics, chemistry, and engineering. A case study of magnetohydrodynamic DySA illustrates how this approach can be used in practice to construct dynamic ensembles of increasing complexities. Finally, we discuss two artificial DySA systems that closely mimic biological phenomena of bacterial colonization and insect swarming, respectively.

2.

Definitions

Before exploring specific examples of SA, it is first necessary to define precisely what is meant by self-assembly in general. While we note that any realistic definition of SA is not without some ambiguity, it can nevertheless provide a useful guide with which to navigate a diverse and growing set of phenomena, ranging from supramolecular chemistry [10–12] to the rings of Saturn [13]. Here, we define self-assembly as the spontaneous formation of organized structures from many discrete components that interact with one another directly (e.g., by electrostatic repulsions [14]) and/or indirectly, through their environment (e.g., magnetohydrodynamic SA [15,16], cf. Section 6). The term “organized” refers to some form of spatial and/or temporal ordering (e.g., an ionic crystal) and distinguishes SA proper from less ordered aggregation processes such as precipitation [17] and ballistic deposition [18]. Furthermore, because the building blocks of SA structures are specified as “discrete”, our definition excludes pattern formation in continuous media such as Bénard convection [19,20], chemical waves [21–23], and Turing structures [24,25]. In general, SA cannot be described by continuum modeling approaches; instead, its description relies on discrete methods such as particle dynamics simulations [26] or agent based modeling [27]. Within the definition outlined above, we make two further distinctions between static self-assembly (SSA) and dynamic self-assembly (DySA) by considering the thermodynamic description of the resulting assemblies (cf. Fig. 2.1). SSA refers to stable equilibrium structures characterized by a maximum (local or global) in the system’s entropy and no systematic energy flows. Examples include organic and inorganic crystals, block copolymer assemblies [28,29], supramolecular structures [10–12], and capillary SA [30,31]; however, due to the constraints of thermodynamic equilibrium, the control and flexibility of artificial SSA systems is somewhat limited. In contrast, DySA refers to ordered

Biomimetic Design of Dynamic Self-Assembling Systems

23

Fig. 2.1. Schematic representations of (a) static and (b–d) dynamic self-assembly. (a) In static self-assembly, components form an ordered, thermodynamically isolated aggregate whose structure does not change in time. (b–d) Dynamic self-assembly evolves a disordered collection of components into an ordered structure through input of energy from an external source and dissipates this energy (e.g., as heat) into the environment. (b) The system can adapt to changes in the rate of energy input by realizing different configurations. (c) When no energy is delivered to “drive” the system, it “falls apart”. When the supply of energy is restored, the system self-heals into a dynamic, stable configuration. (d) Self-replication can be realized through creating daughter assemblies from parental templates with the input of energy. In this hypothetical example, individual components dock onto the templates through capillary interactions. Once two components dock on one template, external energy (here, light of a specified wavelength) is supplied to initiate a permanent bond between the two components to form a daughter assembly. Another form of external energy (e.g., light of different wavelength) is inputted to overcome the capillary bond between the template, and the new daughter assembly is then ready to act as a template to create additional replicas. The described idea has been recently realized in practice in our laboratory [32] and will be reported in the near future.

24

Chapter 2

Fig. 2.2. Examples of dynamic self-assembly found in nature across various length scales. (a) A helicase driven by ATP unwinds DNA for DNA replication. (b) Chlorophyll-containing chloroplasts transform the external energy of light into chemical energy used by the cell to power its DySA processes. (c) Flagella power autonomous motions of bacteria. (d) Dynamic cytoskeleton endows a micropatterned cell its structural integrity. In the picture, the cell is confined to a 40 µm triangle surrounded by a SAM-patterned surface of gold (staining scheme: green = microtubules, red = focal adhesions, blue = actin filaments). (e) Bacterial colony directs its growth in response to the scarcity of nutrients. (f) Fish school to minimize the chances of being attacked by a predator. [Credits: (a) T. Ha, University of Illinois, Urbana–Champaign; (b) L. Lin, SUNY–Buffalo; (c) Reprinted with permission from [33]; (d) K. Kandere-Grzybowska, Northwestern University. (e) E. Ben-Jacob, Tel Aviv University; (f) Reprinted with permission from [1].] For colour version, see Colour Plate Section.


25

nonequilibrium structures, which are maintained far from equilibrium by a supply and subsequent dissipation (e.g., into heat) of useful energy. Free from the constraints of entropy maximization, these systems can “reside” in low entropy states often characterized by complex spatial or coherent spatio-temporal organization. Furthermore, the variety of dissipative interactions (cf. Table 2.1) and the dependence of DySA systems on an external energy supply (e.g., the angular velocity of a rotating magnetic field [15]) provide a basis for the systematic design and control of synthetic DySA systems. Despite its prevalence in nature and several demonstrations in artificial systems [14,15], relatively little is known about DySA—both in the theory and in the experimental practice. From the perspective of thermodynamics, DySA systems belong to a larger class of nonequilibrium steady-state systems [34,35] also known as dissipative structures [36]. In these systems, thermodynamic gradients feed a variety of dissipative (i.e., irreversible) processes, which produce entropy and direct the emergence of order. While these concepts are well established in linear regime of nonequilibrium thermodynamics [34,35],1 they are of little use in understanding DySA, which occurs only under far-from-equilibrium conditions. Here, one must seek alternative theoretical tools, of which the theory of dynamical systems appears most relevant [37,38]. In this context, we [39] and others [40–42] have recently suggested that concepts such as phase space compression and Liapunov exponent spectra can provide a basis for a quantitative study and formulation of DySA. Of course, validation of these theoretical conjectures [39–41] will rely heavily on the knowledge obtained from careful experimental investigation of both natural and synthetic DySA systems.

3. Advantages Even in the absence of a complete theoretical description, DySA offers exciting practical opportunities that go well beyond the capabilities of equilibrium SA. In particular, systems based on DySA can exhibit adaptive/smart behaviors, can be robust in the face of large perturbations, are capable of self-healing, and can potentially self-replicate.

3.1. Adaptability Living organisms, which are paradigmatic DySA systems, have developed a range of flexible strategies that allow them to adapt to environmental changes by adjusting their mode of assembly.2 For example, individual cells respond to 1 In the linear regime, Onsager’s reciprocal relations hold, and the assumption of local thermodynamic equilibrium is valid. 2 This form of adaptation should be distinguished from evolutionary (i.e., genetic) adaptation.

26

Chapter 2

unfavorable increases in temperature by producing various heat shock proteins, which function as molecular chaperones to guide dynamic cellular processes such as synthesis, folding, translocation, assembly, and degradation of other proteins [43]. At the population level, chemotactic bacteria (cf. Section 4.5 and Fig. 2.2) respond to a scarcity of nutrients by clustering and forming metabolicallyefficient colonies [44]. Such adaptability can also be realized in artificial DySA, where changes in the rate of energy delivered to a system can alter its mode of organization, and enable it to toggle controllably between different metastable states (cf. Section 6.2 and Fig. 2.4). The ability to respond to environmental stimuli is a direct result of the nonequilibrium nature of DySA systems and also their common trait since these systems represent stable, dissipative structures that organize in response to various nonequilibrium constraints (e.g., energy flows or thermodynamic gradients).

3.2.

Self-Healing

In addition to adaptability, DySA systems—both natural and artificial—are robust in the face of perturbations and have the ability to heal themselves when wounded. Unlike typical manmade systems (e.g., a Boeing 747), where the loss of one critical part can lead to system failure, DySA systems are capable of rebuilding and maintaining themselves despite the loss or exchange of their underlying components—e.g., on average, the cells in the human body turnover in ∼2 years. From the perspective of dynamical systems theory, DySA structures represent stable attractors in the phase space (specifically, “strange” attractors [37,38]), towards which all neighboring configurations converge. Furthermore, depending on the specific DySA system and its environment, there may exist multiple attractors, each surrounded in the phase space by a corresponding “basin” of attraction.3 Therefore, while smaller perturbations that remain in the current basin are “healed”, larger perturbations force the system to adapt by transitioning to another mode of assembly characterized by a different attractor in the phase space.

3.3.

Self-Replication

Finally, DySA systems have all the necessary ingredients for self-replication. For example, cells undergo a complex self-replication process in which various pieces of cellular machinery first self-assemble and/or migrate to appropriate locations [45,46]. Subsequently, in the processes of cell mitosis and cytokinesis, microtubules and later microfilaments do mechanical work to separate the cell into two 3 In the dynamical systems theory, a basin of attraction is defined as the set of points in the phase

space such that initial conditions chosen from this set evolve to a particular attractor.


27

progenies [47]. Characteristic of all DySA processes, cell replication requires energy, which is delivered to the cell in the form of nutrients and is later dissipated into the environment as heat and chemical “waste” [48]. Despite the prevalence of self-replication in natural systems, there are very few examples of self-replication in artificial DySA ensembles (cf. one example from magnetohydrodynamic SA in Section 6.3).

4.

Biological Examples—Molecular to Macroscopic

The many advantages of dynamic self-assembly manifest themselves most clearly in biological systems. In this section, we illustrate how nature uses DySA across length scales from molecular to macroscopic (cf. Fig. 2.2). Through these examples we gain not only a deeper appreciation of DySA’s potential, but also abstract general characteristics that can then guide the design of artificial DySA systems.

4.1.

DNA Manipulation—Stitching Together and Ripping Apart (Fig. 2.3)

At the molecular level, the genetic code of life—persistent over eons through replication of DNA, transcription to RNA, and translation into functional proteins—represents a finely tuned symphony of DySA processes (Fig. 2.3a). For example, the replication of DNA is energetically driven by the hydrolysis of deoxyribonucleotide triphosphate precursors, coupled to the unwinding of the double helix [49]. The DNA replication requires that discrete proteins called helicase, DNA polymerase, and primase organize spontaneously in the nucleus of the cell (Fig. 2.3b). Helicase physically separates the antiparallel strands at the origin of replication [50], DNA polymerase powered by nucleoside triphosphates synthesizes new Watson–Crick base pairs [51], and ligase seals the new strands together [52] before returning the system to the final equilibrium state (Fig. 2.3c). The transcription of DNA into RNA also requires a concerted action of different self-assembling components, including RNA polymerase, transcription factors, and promoter regions. Proteins called transcription factors help RNA polymerase find promoter regions in the double-stranded DNA, and free energy liberated from a favorable conformation change is used to elongate the strand [53]. During strand elongation, localized melting forms a transcription bubble in the DNA, and RNA polymerase is allowed to base pair (A-U and G-C). At a transcription stop site, the DNA strands hybridize back together, and the RNA polymerase spontaneously releases the newly created primary transcript. In the nucleus, this precursor messenger RNA is then further processed and spliced to remove non-gene coding segments before finally being exported to the cytoplasm for translation.

28

Chapter 2

Fig. 2.3. (a) Powered by the energy “stored” in RNA polymerase, DNA is transcribed into mRNA in the nucleus of the cell. After being exported to the cytoplasm, mRNA is translated into polypeptides. With the physical aid of ribosomes, energy from GTP is used to synthesize a growing amino acid chain. (b) Using energy delivered by dNTP, parental DNA strands are physically split apart at the origin of replication by helicase proteins. After the replication process is complete, the two new daughter strands seal back together. (c) Energy from ATP hydrolysis is used to make new DNA from antiparallel, “parent” DNA strands. The assembly of nucleotides, DNA polymerase and ligase replicate the DNA strand from the 3’ to the 5’ direction. For colour version, see Colour Plate Section.


29

Energy-dependent self-assembly also underlies the translation of DNA. Released from the hydrolysis of guanosine triphosphate, energy is harnessed to aid in the assembly of polypeptides. A ribosome molecule “clamps down” the newly created mRNA, and tRNA is then allowed to interpret the instruction manual. Transfer RNA (tRNA) is the “Rosetta Stone” of translation, because it is the key link in decoding the hieroglyphic mRNA into amino acids. Once the amino acids are sequenced into a correct polypeptide, the ribosome, tRNA, and mRNA complex is released. With the aid of molecular chaperons, the newly manufactured polypeptide chain folds and coils until it reaches a final equilibrium, functional form. We learn two things from the above examples. Firstly, biological DySA usually involves multiple agents interacting synchronously. Secondly, not all of the interactions that take place are of the same “character”—some are designed to bring components together, and some to separate them. DySA occurs as the result of a fine-tuned balance between these two types of processes—this observation provides the first heuristic rule (“competing interactions”, cf. Section 5.1) for the design of artificial DySA.

4.2. Photosynthesis—Powering DySA Biological examples above illustrate how DySA can be driven by chemical energy. In addition, it can also be powered by energy harnessed from external fields. For example, in the photosynthesis process, the energy that plants use to build and maintain themselves is derived primarily from sunlight [54]. Here, the self-assembled system of molecules in the chloroplast (Fig. 2.2b) transduces the energy of impinging photons through a series of synchronized electron transfer steps. In photosystems I and II, the energy from a photon of light excites chlorophyll to initiate an electron cascade, generating high energy ATP and NADPH [55]. The energy stored in NADPH and ATP is then used to drive the synthesis (assembly) of macromolecules, such as sugars and starches, from simple CO2 . Overall, plants power their DySA processes by harnessing and processing the energy of light. As we will see shortly, similar—albeit infinitely simpler— harnessing mechanisms can be built into artificial ensembles when responsive materials (magnetic, photoactive, etc.) are used to transduce external fields into mechanical work necessary to develop and maintain nonequilibrium order.

4.3.

Cytoskeleton—Well Balanced Tension

On the cellular scale, dynamic self-assembly endows cells structural integrity and mechanical resilience. The so-called cytoskeleton responsible for these functions is not a rigid structure but rather a highly dynamic maze of microscopic fibers that

30

Chapter 2

constantly grow and shrink by utilizing chemical energy contained in high-freeenergy molecules (e.g., GTP [56]). The two major components of cytoskeleton are microtubules and actin filaments (Fig. 2.2d). The former are made of multiple repeats of a protein called tubulin and, from a mechanical point of view, serve as compression resisting elements. In contrast, actin filaments (made of actin proteins) act as tension-bearing, rubber-like strings. The interplay between the compression and tension keeps the cells structurally compact while allowing them to adjust to external forces and environmental cues. While the intricacies of these mechanisms are not important here, we note that what makes this fabulous DySA system work is the proper balance between the magnitudes of compression and tension—had one of these “tendencies” been much stronger than the other, the cell would simply either collapse or explode. The same is true for all other “competing” effects that one might want to use to engineer DySA systems—in order to achieve DySA, these tendencies should be of comparable magnitudes. The proper balance is the key to DySA.

4.4.

Flagella—DySA in Motion

The ability to move autonomously is another consequence of DySA in animate systems. To propel itself, however, an organism must first transduce molecular scale, energy generating DySA events into a micro/macroscopic mechanical deformation of its entire body or of its part. For instance, bacteria utilize the energy from protons moving down a gradient of chemical potential to power and synchronize nine outer and two inner microtubule-associated motor proteins—these processes ultimately generate enough torque to cause periodic beating of microscopic membrane extensions called flagella (Fig. 2.2c). Interestingly, this complex machinery is very sensitive to environmental changes. Receptors on the flagella continuously monitor and respond to chemical gradients in the surrounding medium and by having a temporary “memory” of the previous chemical concentration are able to bias their movements. If the chemical response was deemed favorable, the flagella will move counterclockwise and propel the bacterium towards the stimulus; if the response was unfavorable, the flagella will move clockwise, and the motion will be reversed [57]. We make two practical observations on the possibility of autonomous motion in artificial DySA systems. First, such systems should be based on materials that respond to environmental changes (e.g., thermoresponsive hydrogels [58,59], pH responsive polymers [60,61], or light-responsive polymers [62]) and/or should be able to cause mechanical disturbances to the environment (either by moving their parts or by emitting chemical substances). In addition, if we are to achieve some level of motion control, the system should be able to communicate with the environment and possibly include feedback elements.


4.5.

31

Cellular and Organismal Populations—Collaborating with Neighbors

When the communication “system” spans the entire DySA ensemble, new collective phenomena may arise. In bacterial colonies, communication mediated by chemicals each member secretes into the surrounding medium allows the group to adjust to changing nutrient conditions. When nutrients are abundant, bacteria form circular aggregates; when food is scarce, they organize into fractal or vortex-like structures to maximize the efficiency of food consumption [63,64] (cf. Fig. 2.2). The underlying assembly mechanism—similar to diffusion-limited aggregation (DLA) in inanimate systems [65]—relies on the coupling of the local reproduction rate to the diffusive delivery and consumption of nutrients [64]. Even more complex colonial assemblies emerge when group members are responsive to environmental stimuli that they themselves create or influence (e.g., chemical signaling in ant colonies [66] or chemotactic bacteria [44]). In this case, swarms, flocks and schools form that—depending on a particular species— improve chances of finding food (e.g., ants [67] and birds [67,68]) or minimize the risks of becoming a prey (e.g., fish schools [69]). In all cases, the collective behaviors lead to modes of dynamic organization that optimize one or more characteristics of an ensemble. For an engineer of artificial DySA systems, these examples are probably the ultimate inspiration since they illustrate how almost “intelligent” behaviors can be achieved by putting together relatively unintelligent but dynamic components. These examples have already stimulated development of widely used computational algorithms for resource allocation [70,71]. In the future, they should underlie development of actual physical systems with potential uses in sensors, smart materials, and reconfigurable devices. In the following, we outline the first steps towards these visionary but, as we argue, achievable goals.

5.

Heuristics of Rational DySA Design

Before turning to specific systems, we briefly summarize the general principles extracted from our short survey of biological DySA. We emphasize that these principles are not quantitative but only heuristic. Nevertheless, they provide a viable basis for the rational design of artificial DySA systems, which, in turn, can be used as test-beds for the development of rigorous theories. Nature provides many interactions that could potentially be used in dynamic self-assembly. To successfully engineer a prototype DySA system from the bottom–up, one must first identify a set of competitive interactions and then determine the mode of energy dissipation, appropriate medium, and length scale for these interactions. Once this prototype is understood, one can then increase its complexity and consider applications.

32

5.1.

Chapter 2

Competitive Interactions

As in the DNA or cytoskeleton examples, interactions between assembling components should not all be of the same nature (i.e., not all attractive or all repulsive) but, rather, should “compete” with one another to achieve a well-ordered balance. If, instead, all forces were purely attractive or purely repulsive, the assembly would clump together into an equilibrium structure or simply disintegrate— neither of which constitutes DySA. Furthermore, at least one of the interactions should depend on the amount of energy delivered to the system; otherwise, the system would be energetically isolated and by definition not a dynamic one.

5.2.

Energy Dissipation

In order for the system to be responsive to external stimuli, it must be displaced from equilibrium by an externally supplied energy flow, which sets up and/or modifies the interactions between the assembling components. Possible energy sources include time-varying magnetic, electrical, or optical fields, chemical reactions, or mechanical agitation. Once delivered, energy is subsequently dissipated—usually as heat—through the component’s interactions with each other and/or their environment. Energy dissipation is a necessary consequence of the nonequilibrium conditions under which DySA occurs and is essential to the emergence of order far from equilibrium.

5.3.

Medium and Length Scale

The nature of competitive interactions dictates the medium and length scale at which a DySA system can operate. To illustrate this, consider the following example, in which a collection of charged spheres is to assemble in two dimensions by balancing attractive capillary and repulsive electrostatic forces. An obvious choice of the medium/location is a liquid/air interface at which the capillary forces between the objects can be engineered [31]. In such a configuration, attractive (lateral) capillary forces “compete” against repulsive forces due to charge on the particles. Assuming the radii of the spheres, R,√and the interparticle distances, L, are smaller than the capillary length,4 κ −1 = γ /ρg, where γ is the surface tension, ρ is the density of the liquid, and g is the acceleration due to gravity, the attractive capillary force can be well approximated as FC ∼ γ R 2 /L [72]. Similarly, the repulsive electrostatic force between like-charged particles is given by FE ∼ kQ2 /L2 , where k is the Coulomb constant, and Q is the charge of an individual particle. Therefore, in order for these interactions to balance one another, we must satisfy the condition FC ∼ FE , which determines the length scale 4 Above this length gravity dominates capillarity, and an assembly via capillarity becomes prob-

lematic or impossible.


33

at which the system should operate. Specifically, it follows that the radii of the spheres and the characteristic distance between them should obey the relation: LR 2 = kQ2 /γ . For example, if the system is composed of particles of radius R = 100 µm and charge Q ∼ 10−11 C floating on a surface of oil (γ ∼ 20 mJ/m2 and ρ ∼ 1 g/cm3 ), the characteristic distance should be of the order of 1 mm. Importantly, we note that the characteristic lengths can be modified by adjusting the properties of the particles (e.g., charge) or the properties of the fluid (e.g., density or surface tension). Furthermore, to make this system truly a dynamic one, charge could be induced in dielectric particles by UV light [73] and subsequently dissipated through the surrounding moisture, thereby necessitating constant irradiation.

5.4.

Increasing Complexity

Finally, the biomimetic design of DySA systems should follow a bottom–up approach, in which one starts with relatively simple ensembles that can be understood in quantitative detail and then gradually increases their complexity. Following the example of biological DySA systems, complexity should be increased in a “hierarchical” manner by sequential and systematic addition of new types of components or interactions. Ultimately, entire DySA systems may be used as components of even larger ones, just as individual cells contribute to the organization of the entire colony.

6.

DySA Design in Practice: Magnetohydrodynamic Systems

In this section we illustrate how the proposed principles were applied to develop a class of DySA systems—the first rationally designed ones—based on competitive magnetic and hydrodynamic interactions. We begin by describing a basic configuration in which identical components are confined by a centrosymmetric magnetic potential, repel one another by vortex–vortex interactions, and organize into two-dimensional, metastable lattices. We then discuss how the forces acting within the system can be tailored in two and three dimensions, and how these modifications lead to unexpected behaviors such as symmetry breaking, chiral recognition, and pattern replication. We conclude by describing two real-world applications originating from our model systems: a microfluidic mixer and a microfluidic machine.

6.1.

The Origin of Competitive Interactions (Fig. 2.4)

Consider a collection of small (tens of microns to millimeters) magneticallydoped particles (usually, polymeric disks containing magnetite powder) floating

34

Chapter 2

at a liquid–air interface and subject to a field produced by a permanent magnet rotating below this interface. Under the influence of the rotating magnetic field, all disks experience a centrosymmetric force, Fm , directed towards the axis of the magnet’s rotation [74]. At the same time, their induced magnetic moments interact with the magnetic moment of the external magnet, and the disks spin around their axes at an angular velocity ω equal to that of the magnet. The fluid motion associated with spinning results in repulsive hydrodynamic interactions Fh between the disks (cf. Fig. 2.4a). The origin of these repulsions can be explained using ideas from low (but not zero) Reynolds number hydrodynamics [75], and their magnitude is proportional to ω2 and inversely proportional to the cube of the distance between the particles, d 3 .

6.2.

Two-Dimensional Assemblies Exhibit the Principles and Advantages of DySA

The balance between attractive magnetic and repulsive hydrodynamic forces evolves collections of randomly distributed spinners into ordered open-lattice assemblies capable of adaptive and self-healing behaviors (cf. Fig. 2.4b). Because the magnitude of the vortex–vortex interaction increases with ω, one can control the assembly by tuning the rate of energy supplied to the system. This control parameter enables one to modify the interparticle spacing, effectively expanding or contracting the lattice, or even to toggle the system between different stable configurations or “modes” of the assembly (Fig. 2.4b). In addition to this primitive adaptability, the system is capable of rebuilding itself when perturbed, provided that energy is continuously supplied by the rotating magnetic field. Characteristic of DySA, the assembly “dies”, when the flow of energy ceases, and the disks collapse into a disordered cluster. Building on this basic arrangement, more complex systems were constructed using differently shaped particles or mixtures of component types. For example, when circular particles are replaced by chiral, comma-shaped components, the magnitude and nature of the interparticle interactions change dramatically [16]. While they repel each other at low and high values of ω, particles of a specific chirality combine selectively to form dimers at intermediate speeds (cf. Fig. 2.4c). Interestingly, dimerization only proceeds in the presence of a third particle, which “catalyzes” the interaction. Further complexity can be built into the system if one allows different types of particles to co-assemble simultaneously. For example, collections of large and small particles create assembles that mimic classical “atoms”, which can “react” to form macroscopic DySA “molecules” (Fig. 2.4d) and can break system’s symmetry [76].


35

Fig. 2.4. Artificial DySA systems engineered from magnetic and vortex–vortex interactions. (a) The simplest experimental arrangement uses circular, polymer discs (0.02–2 mm in diameter) doped with magnetite. The discs are placed at a liquid–air interface, below which a permanent bar magnet rotates with angular velocity ω (200–1100 rpm). A magnetic force, Fm , attracts all discs towards the axis of the magnet’s rotation, while the spinning discs repel one another via pairwise hydrodynamic forces, Fh . (b) Dynamic patterns formed by 5 (top left), 37 (top right), and 10 disks (bottom). The assembly of 10 disks can be toggled between two different configurations by changing the rotational speed. (c) Comma-shaped spinners with in-plane chirality aggregate into dimers. Once formed, the dimers repel one another and organize into extended dynamic arrays. Disks of opposite chiralities (not shown) do not dimerize. (d) Fusion of two macroscopic “artificial atoms” into an “artificial molecule”; the analogy is based on the quantitative similarity of potentials acting in this system and in the so-called classical artificial atoms [74]. (e) Illustration of pattern replication using vortex–vortex interactions. Here, an assembly of magnetically doped squares (black) spinning at a PFD/EG-H2 O interface was used as template to assemble nonmagnetic PDMS rectangles (white) on the EG-H2 O/air interface above [77]. Objects rotating on different interfaces interact and assemble via Taylor vortices (right). [Reprinted with permission from (a) [78], (b) [15], (c) [16], (d) [76], (e) [79].] For colour version, see Colour Plate Section.

36

6.3.

Chapter 2

Pattern Replication via Three-Dimensional Magnetohydrodynamic DySA

Having explored the possibilities of magnetohydrodynamic DySA in 2D, the system was extended to three dimensions by assembling particles at multiple interfaces prepared by layering immiscible fluids of different densities [79]. In addition to “communicating” with their neighbors on the same 2D interface, particles on different interfaces influence one another through Taylor vortices [80] extending in the vertical direction. While this 3D system exhibits a variety of interesting new phenomena, most notable is its ability to spontaneously replicate dynamic assemblies onto neighboring interfaces (Fig. 2.4e). In one example [77], a set of square, magnetic particles makes a copy of itself by pairing up paramagnetic half-squares floating above them. The paramagnetic particles are first attracted towards the Taylor vortices created by the magnetic squares, and then pair-up to produce an exact replica of the magnetic pattern below. Through careful investigation of the basic magnetohydrodynamic DySA system and its more complex derivatives, one realizes the potential advantages of DySA—adaptability, self-healing, and self-replication—and gains the understanding necessary to apply these advantages to functional DySA applications. Here, we illustrate two such applications in the fields of microfluidics and micromechanical systems.

6.4.

Microfluidic Mixer (Fig. 2.5a)

Efficient mixing [81,82] in microfluidic devices [83,84] is a fundamental technical problem that can be overcome using the tools of magnetohydrodynamic DySA [78]. While both passive (i.e., no moving parts) [85,86] and active mixers [87–90] have been developed to address the challenge of microscale mixing, these methods often involve complicated microfabrication processes and/or have limited control. In contrast, we have developed an effective active micromixer [78] that builds itself spontaneously through dynamic self-assembly and can be carefully controlled through an external energy supply. Briefly, the device consists of a sealed microfluidic channel, whose floor is patterned with magnetic islands (25–100 µm in diameter). In the presence of an external, rotating magnetic field, these islands act as field concentrators, onto which small magnetic particles assemble to form needle like structures. Importantly, as the needles build themselves, they repel one another by vortex–vortex (cf. above) forces and redistribute their components such that ultimately all needles are of similar sizes and rotate in synchrony. Rotation of an array of such microscopic mixing arms, creates turbulent flows that mix fluids flowing laminarly above them. As in previous examples of DySA, this system is responsive to external control and heals itself spontaneously when damaged.


37

Fig. 2.5. Applications of magnetohydrodynamic self-assembly, (a) self-assembling microfluidic mixers and (b) self-assembling fluidic machines. (a) The mixers are formed when magnetic particles aggregate from an initially uniform dispersion in the fluid into needle-like aggregates above micropatterned field concentrators. Each needle rotates with angular velocity ω equal to that of the external rotating magnet. The top-right picture shows an inefficient oval-shaped micromixer operating at 100 rpm and at a flow rate of 50 µL h−1 . A more efficient pear-shaped design (lower right) operating at 700 rpm at the same flow rate achieves complete mixing of laminarly flowing streams. (b) Self-assembled carousel system. Seven gears (pink) are powered by magnetic spinners (black) interacting with external, rotating magnetic field (cf. Fig. 2.4). When the field profile is modified by electromagnetic field concentrators (i), the gears organize above “activated” electromagnets and can be used to manipulate nonmagnetic containers floating at the interface. Here, the carousel first incorporates an empty container (ii), turns it around (iii) until the filling point (iv), completes the revolution (v), expels the filled container (vi), and finally returns to its initial state. The motions of the carousel are caused by synchronous activation of the electromagnets. When the carousel opens or closes, three electromagnets are simultaneously turned on; when the carousel rotates, six electromagnets are activated at a time. [Reprinted with permission from (a) [78] and (b) [6].] For colour version, see Colour Plate Section.

38

6.5.

Chapter 2

Fluidic Machines (Fig. 2.5b)

Vortex–vortex interactions from magnetohydrodynamic self-assembly provide a basis for creating unconventional mechanical machines operating at fluidic interfaces [6,91]. The most illustrative system here consists of two-component rotors with a magnetic core and a non-magnetic, gear-shaped ring around it. When an external rotating magnetic field is applied, the cores begin to rotate and center inside the gears, generating a hydrodynamic flow that sets the gears into motion. Rotors can be manipulated at the interface by localized magnetic fields generated by electromagnetic actuators placed below the liquid’s surface. Importantly, because of vortex–vortex repulsions between any two rotors, only one rotor resides in one local minimum of magnetic field intensity thereby creating a unique configuration for a given “pattern” of magnetic field. In other words, configuration and function of these fluidic machines are directed by the self-assembly “template” of magnetic field concentrators. Figure 2.5b demonstrates how the rotors can act as a carousel system to move floating containers, fill them with analyte solutions, and deliver them to a specific end location.

7.

Examples of Bioinspired DySA

While they allow for many versatile designs, the magnetic and vortex–vortex forces are certainly not the only ones from which artificial DySA can be engineered. Within our heuristic engineering approach, one can flexibly use a variety of existing dynamic (i.e., energy driven and/or dissipative) interactions (cf. Table 2.1) and even discover new ones. To illustrate this, we describe two more artificial DySA systems whose components interact by, respectively, dynamic surface tension and by electrostatic forces mediated by contact electrification. In addition to reinforcing the rational design approach, these examples illustrate how inspiration gained from various biological systems can be translated into working DySA prototypes.

7.1.

DySA via Dynamic Surface Tension (Figs. 2.6a–c)

Inspired by chemotactic bacteria that move in response to chemical gradients which they, themselves, create, we developed a macroscopic DySA system designed to mimic basic communication and response behaviors observed in biology. In this artificial system (Figs. 2.6a–c) a collection of millimeter-sized, hydrogel particles of a specified geometry are doped with a surface-active agent (e.g., solid camphor) and are placed at a liquid–air interface. As the camphor diffuses from the particles, it creates dynamic gradients in the surface tension, which cause the particles to move about autonomously, further modifying the


39

Table 2.1. Examples of dynamic interactions suitable for designing artificial DySA systems Interaction

Particle size/type

Applications and references

Induced magnetic Colloidal SA microfluidic mixers [78]; Applications of dipole–dipole [92] 0.1–100 µm magnetorheological fluids: automotive clutches [93], polishing ferromagnetic fluids [94], smart knee prosthetics [95], anti-lock brakes [96], vibration control of washing machines [97]; 2D and 3D arrays of electro-magnetorheological colloids [98,99] Optical binding

Colloidal 0.1–100 µm

Formation of ordered 2D crystalline arrays of colloidal particles which are held together by light [100,101]

2D vortex–vortex µm–mm sized DySA of magnetized disks at a liquid–air interface objects [15,16,74–76]; Self-assembled gear systems and fluidic machines [6,91]; Cell membranes with SA arrays of rotating bio-motors [102]; Vortex crystals in magnetized electron columns [103,104] Flow-induced µm–mm sized Flow-induced structures in sheared colloidal suspensions [105] viscous interaction objects 3D vortex columns µm–mm sized DySA of 3D vortex crystals [77,79] objects Dynamic capillary µm–mm sized Dynamic self-assembly of camphor-doped polymeric particles objects at the fluid–air interface [106]; Dynamic behavior of camphor particles at the fluid–air interface: spontaneous movement [107], oscillations and symmetry-breaking [108], and mode selection [109] Induced electric dipole–dipole [110]

0.1–100 µm Applications of electrorheological fluids: Ordered arrays of semiconductor colloidal particles [111]; Smart inks [112–114]; Mechanical particles polishing [115,116]; Tactile displays [117]; Actuators [118]; Dynamic self-assembly in electrostatically driven granular media: crystals, rotating vortices, and pulsating rings [119,120]

Charge–charge via contact electrification

Sub-micron to millimeter; polymeric or metal

Scattered light

Electrostatic self-assembly of macroscopic crystals using contact electrification [8]; Control of contact charging by adjusting humidity [121]; DySA of rings of charged metal spheres [14]; Swarm dynamics of charged particles in air [122] 3T3x cell aggregation—lysosome-rich cells attract others by their enhanced ability to scatter near-IR light; The cells “feel” each other at distances as large as 200 µm [123]

Light-switchable 10 nm parti- Fundamental understanding of nanoparticle interactions aggregation of cles aggregate [124,125] magnetic colloids in clusters up to 100 µm

40

Chapter 2

Fig. 2.6. (a) Schematic explanation of the origin of autonomous propulsion caused by dynamic surface tension. A surface-active agent (e.g., camphor) diffuses from a gel particle onto a fluid interface lowering its surface tension. Because the stationary state is unstable, small fluctuations cause the particle to break its cylindrical symmetry and accelerate up the local surface tension gradient. (b) At low particle densities, particles move about autonomously and interact through repulsive surface tension interactions. (c) At high densities, the particles assemble into an open lattice, which is strongly influenced by the geometry of the boundary. (d) Schematic representation of DySA via contact electrification. (e) Time evolution of the system as it organizes from a random initial configuration into well-defined “trails”. For colour version, see Colour Plate Section.


41

interface. Importantly, the camphor is removed from the interface via sublimation/dissolution into the surrounding environment, thereby maintaining the nonequilibrium conditions. The dynamic feedback between the particles and their environment (here, interface on which they “live”) is analogous—although certainly not identical—to that of the biological system described in Section 4.5. Although the behavior of isolated camphor particles is relatively well understood [107,126], far more interesting dynamics arise when multiple particles begin to interact at an interface. At low densities ( r0 or rij > rIbJ , where rIbJ is the bond-breaking radius for types I and J . The strong bond strength parameter, k, governs the strength of all strong bonds, whether or not they were present in the initial conditions. These bonding rules allow emergent chemical reaction networks, including emergent side-reactions, to interact with self-assembly processes [15]. Note that the temperature of the system changes when bonds form and break. However, the momentum in the system is constant, since the changes in the momentum of individual elements due to bonding events are always symmetrical with respect to the bonded particles.

3.

Genetic Algorithm for Chemical Structures

We now describe an evolutionary algorithm (EA) for designing chemical systems. We use the EA to optimize the parameters of dbDPD models, where the parameters are given a wide range of possible values. Specifically, a genome, g, is a vector of chemical systems parameters: g = (g1 , . . . , gN ). The evolutionary algorithm we use is simple, but differs from a standard genetic algorithm in important respects. In deference to the expense of testing fitness, all previously tested genomes are candidate parents, rather than only those of the last generation, and children are not allowed to repeat previously tested genomes. In an attempt to better mimic real evolutionary processes, mutation, the sole genetic operator, is limited to a subset of the space near the parent’s value, rather than the whole range of the gene. Obvious possibilities for future work include extending our results with various more complicated evolutionary algorithms [14,17,20], such as those using crossover, and comparison with a standard genetic algorithm. The EA proceeds with discrete generations of size Pg . Parents are selected from the entire population of genomes tested in any previous generation, not

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just those tested in the immediately preceding generation. Selection is done via truncation: at the end of each generation the least fit members of the population are culled until the group of new parents reaches a pre-determined size, Pt . For all experiments discussed below, Pt = Pg = 10. Each parent produces one child genome gc by mutation, with the limitation that gc not duplicate a previous g tested by the EA. The mutation rate per locus is governed by a global parameter, μ (set to 0.5 for all experiments reported here). The mutation operator is given by: p p g gi p c + i gi = u 2gi − 2 2 where u is a uniform random number chosen from the interval (0, 1), and gic p and gi are the ith child and parent gene. Mutation thus has an asymmetric Hamp p ming distance around gp such that gi /2 < gic < 2gi . The mutation rate is relatively high to balance other differences between ours and a standard EA. Because mutations are limited to the range defined by the Hamming distance, even high mutation rates do not lead to mutant children being overly distant in genomic space from their parents. Additionally, because children may not duplicate earlier genomes, much of the genome may be effectively immune to mutation due to previous sampling, particularly in combination with the limit imposed by the Hamming distance.

4.

Results

We optimized dbDPD parameters for a handful of different kinds of selfassembled structures. Some involve only weak associative forces, while others also involve strong bonds. This section reports the typical and notable results from a large sample of optimizations of these structures by evolutionary algorithms.

4.1.

Optimization of Micelle Size

One set of experiments focussed on systems consisting of amphiphile aggregations in water. Amphiphiles in aqueous solution spontaneously assemble into a number of structures, depending on the type of amphiphile and the pH and temperature of the solution. Amphiphilic self-assembled structures typically arise solely because of weak associative forces. The smallest amphiphile self-assembled structures are micelles. In micelles, amphiphiles aggregate with their hydrophobic tails at the center and the hydrophilic heads around the outside (see Fig. 4.1, right). The size of a micelle is affected by a number of factors, but can be considered in the abstract to depend

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Fig. 4.1. The equilibrium state of the dbDPD system before (left) and after (right) evolutionary design of dbDPD parameters for the micelle size. Amphiphile heads are shown as light-colored spheres, tails as darker columns; water is present but not shown. The system on the left includes amphiphile tails, but most are too short to be seen. In the system on the right, note the self-assembly of ten different micelles of a certain characteristic size. Two micelles wrap around the toroidal edge of the space. For colour version, see Colour Plate Section.

primarily on the length of tail and the “width,” in terms of inter-molecular forces, of the head. If the tail is short and the head wide, small micelles composed of low numbers of amphiphiles will be formed. The EA tested dbDPD parameters in systems consisting of 1000 particles: half water (W) and half tail–head (T–H) dimers. The initial system parameters were such that T–H pairs were strongly bonded and attracted each other, T–H bonds could break but could not reform, Hs repelled each other, and Ts repelled each other. The genome in this EA consists of genes for the bond strength, k, and for the repulsion forces, αI J : g = (k, αWT , αWH , αHH , αTH , αTT ) where all are integers in [1, 400]. In this experiment, we use an EA to tune dbDPD parameters so that micelles spontaneously self-assemble and their size is maximized. The existence and size of micelles is determined by the spatial distribution of amphiphiles. So, the fitness function needs to detect micelles and to measure their size. In the fitness function used here, a micelle-detecting algorithm locates all the amphiphile structures made up of a core of tails surrounded by heads. The algorithm proceeds by (1) listing all the tail particles in the system, (2) choosing a tail particle from the list at random, (3) drawing a circle whose radius is the distance between that tail particle and the nearest head or water particle, (4) removing from the list all the tail particles located inside that circle, and then repeating steps (2)–(4) until the list of tail particles is empty. The algorithm then groups any overlapping circles (where overlap is transitive), and considers each group of overlapping circles as

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a single micelle. A micelle’s radius is considered to be the radius of the largest circle in the group corresponding to that micelle. The fitness of a given genome (vector of dbDPD parameters) is dependent on the size of the micelles (if any) that form after a set number of model updates when the dbDPD system is started from those dbDPD parameters. More precisely, the fitness, F , of a given genome, g, is the mean size of the n micelles it generates, F =

n

fi

i=1

where fi is the size of the ith micelle. An individual micelle’s size is given by ri si fi = ri where si is the number of tails in the micelle core and ri is the micelle’s radius. The fitness function thus increases with both the number of particles in the micelle and its radius. The evolutionary algorithm typically had no difficulty creating large micelles. Different runs of the EA would often take different routes to similar final solutions. In some cases, a good solution is found in less than 20 generations; see Fig. 4.2. But other times the process is more gradual, as in Fig. 4.3, which illustrates a series of discrete adaptations. The fixed dbDPD parameters in these two contexts were similar but not exactly the same. In both cases, αWW = 1, βTT = 3 and β = 1 for all other particle pairs, and for all particle pairs r f = 0 (no bonds form). In the EA shown in Fig. 4.2, k = 250, l = 0.05, and r b = 0.7. In the EA shown in Fig. 4.3, l = 0.01, r b = 0.9, k was added as a gene and allowed to evolve. These separate adaptations can be identified in the corresponding evolutionary activity plot. Evolutionary activity statistics are a method for visualizing and quantifying the dynamics of evolutionary adaptations [2,4,5,33]. Here, we focus on the activity of only the parents, for those genotypes will contain the significant adaptations. In the present context, the evolutionary activity A(a, t) at time t of a given allele a of a given parental gene gi is the sum of a’s concentration in past parental generations: A(a, t) =

n

c(a, t)

t=1

where n is the total number of generations and c(a, t) is the concentration of the a allele in the parental population at generation t if the allele is present, and 0 otherwise. A discrete number of different allele types are formed from continuous

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Fig. 4.2. Time series of fitness scatter plot of a micelle size EA, in which a good solution is found in less than 20 generations. Each point represents the fitness of a genome in the EA with all the genomes in a single generation at the same point on the x-axis.

Fig. 4.3. Time series of fitness scatter plot for a micelle size EA, illustrating successive adaptations.

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Fig. 4.4. Evolutionary activity waves of all parental loci in the EA shown in Fig. 4.3. Note the rise of a number of adaptations such as the dominant gene in k, αTT and αWT . Also notable is the quiescence of one αWT gene only to see a revival towards the end of the study. A gene at locus αTH can be seen dying out around generation 60 to be replaced by a number of dominant neutral variants. For colour version, see Colour Plate Section.

parameters by dividing the range of legal allele values into some finite number of bins (40 bins, in the present study). We can see the effects of two main adaptations in the evolutionary activity waves in Fig. 4.4. The first adaptation around generation 20 is lower amphiphile tail repulsion. This allows amphiphiles to become packed into micelles. One can see the creation of new types of αTT genes, one of which quickly dominates the population (the big αTT wave in the middle of the diagram). This first adaptation also involves increasing the repulsion between heads, but the activity waves for these innovations appear among those in the lower half of the wave diagram. The second main adaptation around generation 50 is increased repulsion between heads and tails. This increases the amphiphile’s length, and so increases the size of the micelles. One can see the creation of new activity waves for αTH genes at around this time. (The persisting allele waves involving αWH and k illustrate that these genes are not undergoing significant innovations.) If we compare the micelles produced by the parents at successive stages in the evolutionary algorithm in Fig. 4.2, we observe a clear directional change. Figure 4.5 compares the distribution of mean core size and mean radius of the micelles made by the parents in generation 10, 15, and 30. One can see that the values of both variables increase over the course of the EA.

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Fig. 4.5. Box plots of the mean values of core size (left) and radius (right) of micelles formed by parents of generation 10, 15, and 30, in the EA shown in Fig. 4.2. A clear directional change is evident. The bottom of the box marks the first quartile (x0.25 ), the top, the third (x0.75 ) and the thicker line, the median. The height of the box defines the interquartile range (IQR). A data point is considered an outlier, and shown as an empty dot, if it is smaller than the lower fence (LF = x0.25 − 1.5IQR) or bigger than the upper fence (UF = x0.75 + 1.5IQR). The bottom line marks the smallest data point bigger than LF, and the top line the biggest data point smaller than UF.

Figure 4.6 compares the most fit genomes in the EAs shown in Figs. 4.2 and 4.3. It is evident that both EAs produce roughly similar genomes, though they differ in some details. The fittest genomes from the EA shown in Fig. 4.2 produce micelles with a more regular shape than those from the EA shown in Fig. 4.3.

Fig. 4.6. Logarithmically scaled distribution of the most fit genomes in the EAs shown in Figs. 4.2 and 4.3. The bond strength, k, was a gene in the EA shown on the right, but not in the EA shown on the left. Note that the two sets of evolved genomes are similar, though not the same.

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Fig. 4.7. Three slices of the fitness landscape for the micelle size EA. Each slice is a series of scatter plots of twenty fitness measurements in dbDPD parameter space, showing how fitness varies as a single parameter is varied over its whole range of possible values, while all other parameters are held constant. The parameters varied are αTT (left), αTH (middle), and αHH (right). Some significant topographical variation is evident in each slice.

We also investigated the structure of the fitness landscape in the micelle size task by making a number of “fitness slices.” In these fitness slices, all parameters were kept constant except for one parameter which was smoothly varied between its minimum and maximum values, and we measured fitness of a number (typically 20) of replicate dbDPD systems. The fitness slices (Fig. 4.7) reveal that the fitness landscape has an interesting, sometimes multi-modal, structure in at least three of its dimensions (αTT , αTH and αHH ). The presence of outliers which we observe on the fitness slices for αTH is due to a chaotic point in the parameter space which yielded large but extremely unstable micelles that produced widely differing fitness measurements depending on the momentary state of the system in which the measurement was made. These fitness slices corroborate our explanation above of the adaptations seen in the activity wave diagram (Fig. 4.4). The first significant adaptation involves lowering the TT repulsion and increasing the HH repulsion, and Fig. 4.7 reveals that for at least one slice through parameter space fitness increases as TT repulsion is lowered (left) and increases as HH repulsion increases (right). The second significant adaptation involves increasing the TH repulsion, and we see that fitness increases as TH repulsion increases from its minimal value (Fig. 4.7, middle).

4.2.

Template-Directed Ligation of Uniform Oligomers

Chemical amplification via templating is the basic mechanism of DNA replication, and also of simpler replicator systems such as von Kiedrowski’s autocatalytic replicator system [44] and peptide replicators [40]. Monomers of a given type may participate in a weak interaction with monomers of a complementary type, and each may form strong bonds with a monomer of any type if the two

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are in the correct proximity and orientation. Given a template oligomer made up of different types of monomers and a reservoir of free floating monomers, each monomer of the template oligomer can associate weakly with a complementary free monomer. If the weak forces of the template oligomer bring the free monomers into the correct orientation and proximity, strong bonds form between the monomers producing a complementary oligomer through the process of ligation. In this way, the self-assembly process of template-directed ligation involves both weak and strong chemical bonds. If the paired complementary oligomers are separated by a mechanism such as duplex melting due to temperature change or protein action, then each oligomer may repeat the process, creating more templates and complements. By this means, the overall number of oligomers in the population increases. Although this process results in the chemical amplification of oligomers, the focus of the present work is simply ligation, and the optimization of parameters that result in the organization and ligation of monomers into oligomers. Unlike a DNA system we attempt to simulate a system in which template oligomers directly catalyze the formation of their complements without any facilitating proteins consistent with RNA world theories of the origin of life. The ligation systems we subject to evolutionary design are analogous to the chemical system of non-enzymatic template-directed synthesis [1,7,19,24,25,29, 50]. Like template-directed systems of ligation in vivo and in vitro, our system is supplied with a template molecule and an excess of monomers. It then evolves so as to optimize the assembly of monomers on the template to produce a ligated copy of the template. The simplest form of template-directed ligation involves only uniform oligomers, i.e., oligomers composed of a single species of monomer. The template for a uniform oligomer is another uniform oligomer composed of the complementary monomer. Pairs of opposite type units attract each other, while like type units are unlikely to become associated by weak forces. This is roughly analogous to complementary base pairing in the context of nucleotides. In this section we present results of evolutionary design of ligation of uniform oligomers. Results on evolutionary design of the more general form of ligation involving non-uniform oligomers are reported in Section 4.3. It is possible to measure the catalytic efficiency of the templating process by fitting rate constants and comparing with background rates, but this is not done here. When designing ligation of uniform oligomers, we augmented dbDPD with angular stiffness of covalent bonds, following Shillcock and Lipowsky [36]. The preferred angle was 0, tending towards parallel bonds, and the bending constant was 200. Our genes in this EA are three dbDPD parameters: the bond strength, the relaxed length of the bond, and the strength of the attractive conservative force between A and B particles: g = (k, l, βAB )

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Fig. 4.8. The dbDPD system before (left) and after (right) evolutionary design of uniform oligomer ligation. Template oligomers are dark, monomers and target oligomers lighter. Water is present but not shown. After evolutionary design, it is evident that the template successfully catalyzes the production of complementary oligomers. For colour version, see Colour Plate Section.

where k is an integer in [1, 1000], l is a real number in [0.01, 0.75], and βAB is a real number in [1, 2]. The other dbDPD parameters were fixed as follows: αWW = αWT = αWH = 3 and all other α values were set to 30, β values for all f f = 0.25 and r f values for all particle pairs except AB were set to 1, rAA = rBB other particle pairs were set to 0, and r b = −1 for all particle pairs (bonds do not break). A genome’s fitness is measured by starting the dbDPD with the genome’s parameters and seeding the system with one kind of free monomers and a template that is a uniform oligomer formed from the complementary monomer. The fitness of a genome is given by: F =

lmax

nl × k l

l=lmin

where l is the length of the oligomer, lmin is the length of the shortest oligomer which counts towards fitness, lmax is the length of the longest oligomers allowed in the system, nl is the number of oligomers of a given length l, and k is a scaling factor that governs the relative importance of forming longer oligomers. For all our experiments k = 1.5. Many generalizations and modifications of our search algorithm and fitness function could be explored. As can be seen in Fig. 4.8 the EA was successful at templating oligomers in the uniform case. Fitness improved from zero to over eighty percent of the maximum possible. The twenty percent shortfall from perfect performance is from incomplete ligation due in large part to insufficient diffusion of monomers in the time allotted for evaluating the system.

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4.3. Template-Directed Ligation of Non-Uniform Trimers Ligation of uniform oligomers is a limiting case of ligation of non-uniform oligomers. In this section, we report results on evolutionary design of the more general form of ligation. To simplify the chemistry as much as possible, we focus solely on the ligation of trimers composed of two types of monomers. Our genes are eight chemical system parameters: the bond strength, the relaxed length of the bond, the strength of the repulsive conservative force between A particles, the strength of the repulsive conservative force between B particles, the strength of the attractive conservative force between A and B particles, the bondforming radius for two A particles, the bond-forming radius for two B particles, and the bond-forming radius for A and B particles: f f f g = k, l, αAA , αBB , βAB , rAA , rBB , rAB f , where k, αAA and αBB are integers in [5, 200], βAB is an integer in [5, 100], rAA f and r f are real numbers in [0.05, 0.75] and l is a real number in [0.01, 0.75]. rBB AB The other dbDPD parameters were fixed as follows: α and β values for all particle pairs not controlled by genes were set to 1, and r b for all particle pairs were set to 0.8. Here, a genome’s fitness has three components, corresponding to the ability to ligate three different classes of trimers. Each fitness component is measured by starting the dbDPD with the genome’s parameters and seeding the system solely with free monomers and one of three kinds of template trimers—AAA, AAB, and ABA—and letting dbDPD run for a fixed number of model updates. These templates and their complements cover all possible trimers that can be formed from the monomers A and B. The components of a genome’s fitness are the number of correct template and complementary trimers formed when seeded with each kind of trimer template. Formally, a genome’s fitness is:

F = nAAA × nAAB × nABA where nX is the number of X trimers and their complements produced after seeding the system with templates of type X. Figure 4.9 shows a scatterplot of the fitness of each genome over a typical EA run. In addition to the fitness of the genomes in each generation, the fitness of the parents of each generation is also plotted. This makes more apparent the mechanism of the EA as well as allowing more detailed investigation of performance in terms of the component fitnesses. We observed a steady increase in fitness over the course of one hundred generations. Figure 4.10 shows the evolutionary activity of the parents in the EA shown in Fig. 4.9. (Recall definitions from Section 4.1.) We can see significant adaptations f and in the genes for different types of bonding rules: βAB first adapts, then rAA

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Fig. 4.9. Measured fitness of each parent system (small closed circles) used by a typical EA designing template-directed ligation systems, and of child systems (open circles) produced by the EA.

Fig. 4.10. Evolutionary activity waves of types of alleles at all parental loci in the EA shown in Fig. 4.9. One βAB activity wave ends around generation 50 when βAB waves for alleles with higher values originate; these new waves are mixed with others in the bottom of the diagram. For colour version, see Colour Plate Section.

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Fig. 4.11. Time series of scatter plots of the three components of fitness—nABA above, nAAA in the middle, and nAAB below—in conditions started with the three corresponding kinds of templates in the EA shown in Figs. 4.9 and 4.10. As in Fig. 4.9, the parents at each generation are shown with small closed circles and those of the children with open circles. Note the initial rise in nABA , followed by rises in nAAA and finally nAAB . f . (We also see nearly constant domination of the parent population by allerBB les for genes k and l, presumably because they are not undergoing significant change.) Examining the changes in each component of the fitness function underlines the implications of the activity wave data. Figure 4.11 shows a scatterplot of

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each component of fitness of the genomes in the EA in Fig. 4.9. The EA first increases nABA ; this is the simplest fitness component to optimize because it involves only the AB bond. This is followed by increasing nAAA ; this is somewhat more complex because it requires forming both AA and BB bonds for the template and target respectively, but always in isolation from each other. The fitness component that increases the slowest is nAAB . This involves correctly forming each possible type of bond. Forming both like-type and complement-type bonds within the same oligomer proves very difficult in dbDPD. Note also that the parents shift somewhat in their rank in nAAA and nABA , but are always near the top of nAAB —more evidence of the latter’s key importance.

5.

Discussion

The behavior of probabilistic models like dbDPD is stochastic. This makes measurement of their expected behavior noisy, presenting an extra hurdle for our EA. One simple solution is to do enough replicates to average away the noise [3], but this is expensive in terms of fitness measurements. EAs can compensate for this by rewarding high measured fitness and punishing low measured fitness more gently. But this lessens the EA’s ability to quickly abandon dead ends and capitalize on new breakthroughs. The stochasticity of the dbDPD simulations also explains in part why different EAs sometimes find different solutions to the same optimization problem, the remaining variation due to stochasticity within the EA. We ran many EAs identical in all but the initial seed for the random number generator and saw significant differences in the time it takes to reach optimal solutions, as well the variance in the fitness of the child genomes produced. The differences are analogous to the differences between the EAs shown in Figs. 4.2 and 4.3, though those EAs dealt with slightly different problems. The first finds an optimal solution faster than the second, but the greater variance of children fitness reveals that the solution is less robust. The heart of a EA is its fitness function. Our non-uniform ligation fitness function multiplies three components. The fitness component graphs (Fig. 4.11) reveal a significant gap between desired and observed dbDPD behavior. The EA achieves high values for one fitness component (ligating ABA trimers, nABA ), but the resulting systems are poor at ligating other kinds of trimers. A better fitness function might improve results. For example, our fitness functions for ligation merely counts the oligomers of the right types, but it ignored the actual process that produced the oligomers. One could use a fitness function that explicitly assays to what extent the process of template-directed ligation is actually happening. This would appropriately devalue systems that produce the right oligomers by some other process than ligation. This fitness function should produce general, non-uniform template-directed ligation, providing it is possible to realize this process in the dbDPD framework.

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But evolutionary design methods can succeed only if an adequate solution exists somewhere in the search space. We concluded from the present study that the dbDPD framework can produce only limited forms of templated ligation. Certain simple oligomers, such as the ABA case, were readily produced, but the mechanism for this was not solely template-directed ligation. Furthermore, the yield of ligation with more complex non-uniform oligomers, such as AAB, was poor. This limitation appears to stem from the radial symmetry and specific bond rules in the dbDPD framework. In real templated ligation, a free monomer is held in a specific spatial orientation by weak bonds with another monomer in the template. If this happens at two adjacent locations in the template, then the template can hold two free monomers in the correct spatial orientation long enough that a strong bond forms between them. Without the template, free monomers are unlikely to bond because of the low probability of them coming into contact in the correct spatial orientation and remaining there long enough for a bond to form. In dbDPD, on the other hand, weak associative forces attract any free monomers within r0 so a cluster of free monomers can be attracted to a given location in the template. But strong bonds may form between any two monomers which are within r f for their respective types. So in dbDPD, while strong bonds will form between two free monomers attracted to adjacent locations in the template (as in real templated ligation), strong bonds are just as likely to form between two free monomers attracted to the same location in the template. Thus, the sequence of strong bonds that form is not controlled specifically by the sequence of monomers in the template. For these reasons, it appears that the present dbDPD framework is insufficient to achieve templated ligation of non-uniform oligomers. Appropriate changes to dbDPD may resolve this problem, but the consequent increase in computational costs is presently unknown.

6. Conclusion We here achieve evolutionary design of a model chemistry involving selfassembling amphiphiles in water. This demonstrates how evolutionary algorithms and similar indirect design methods can be used to “program” or optimize selfassembling chemical structures. This method works in a number of contexts, including those illustrated by micelle size and templated ligation of uniform oligomers. The majority of dbDPD simulation parameters optimized here can be effectively varied in actual chemical systems. However, in some cases they cannot be varied independently—i.e., there may not exist an amphiphile which has a particular repulsion with other head groups as well as a particular length tail—but otherwise they are analogous to real chemical variables. Thus, it is possible that

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optimal dbDPD systems correspond to real chemical systems with the same desired characteristics. The evolutionary algorithm used here was intentionally kept very simple. EA performance can typically be boosted by a variety of methods [16]. So we expect that the performance shown here can be significantly improved. The evolutionary design method shown here can be used to optimize other chemical functionalities in other kinds of chemical systems, both real and simulated. It has already been applied with high-throughput screening to optimize real amphiphile systems in the wet lab [39]. The resulting marriage of machine learning with chemical screening yields an automated, intelligent screening method for finding solutions in very sparse samples of huge and high-dimensional search spaces.

Acknowledgements Thanks to the European Center for Living Technology (ECLT) for hospitality during some of the work reported here. Thanks to Rüdi Fuchslin and Anders Eriksson for helpful discussion of DPD, and to Thomas Maeke and John McCaskill for the DPD code from which the present model was developed. Thanks for helpful discussion to Norman Packard, Martin Hanczyc, and Morgan Theis. Thanks also to audiences at Dublin City University, the University of Nottingham, and EA’05 where some of this work was reported. This work was funded in part by PACE (Programmable Artificial Cell Evolution), a European Integrated Project in the EU FP6 IST FET Complex Systems Initiative.

References [1] O.L. Acevedo, L.E. Orgel, Nonenzymatic transcription of an oligodeoxynucleotide 14 residues long, Journal of Molecular Biology 197 (1987) 187–193. [2] M.A. Bedau, C.T. Brown, Visualizing evolutionary activity of genotypes, Artificial Life 5 (1999) 17–35. [3] M.A. Bedau, A.J. Buchanan, G. Gazzola, M. Hanczyc, T. Maeke, J.S. McCaskill, I. Poli, N.H. Packard, Evolutionary design of a DDPD model of ligation, in: Lecture Notes in Computer Science, vol. 3871, 2005, pp. 201–212. [4] M.A. Bedau, N.H. Packard, Measurement of evolutionary activity, teleology, and life, in: C. Langton, C. Taylor, D. Farmer, S. Rasmussen (Eds.), Artificial Life II, Addison–Wesley, Redwood City, 1991, pp. 431–461. [5] M.A. Bedau, E. Snyder, N.H. Packard, A classification of long-term evolutionary dynamics, in: C. Adami, R. Belew, H. Kitano, C. Taylor (Eds.), Artificial Life VI, MIT Press, Cambridge, 1998, pp. 228–237. [6] G. Besold, I. Vattulainen, M. Karttunen, J.M. Polson, Towards better integrators for dissipative particle dynamics simulations, Phys. Rev. E Rapid Comm. 62 (2000) 7611–7614. [7] C. Bohler, P.E. Nielsen, L.E. Orgel, Template switching between PNA and RNA oligonucleotides, Nature 376 (1995) 578–581.

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[8] S. Brenner, R.A. Lerner, Encoded combinatorial chemistry, Proceedings of the National Academy of Science USA 89 (1992) 5381–5383. [9] K.B. Chapman, J.W. Szostak, In vitro selection of catalytic RNAs, Current Opinions in Structural Biology 4 (1994) 618–622. [10] D. Cliff, I. Harvey, P. Husbands, Explorations in evolutionary robotics, Adaptive Behavior 2 (1993) 71–104. [11] A.D. Ellington, J.W. Szostak, In vitro selection of RNA molecules that bind specific ligands, Nature 346 (1990) 818–822. [12] J. Ellman, B. Stoddard, J. Wells, Combinatorial thinking in chemistry and biology, Proceedings of the National Academy of Science USA 94 (1997) 2779–2782. [13] S. Nolfi, D. Floreano, Evolutionary Robotics: The Biology, Intelligence, and Technology of Self-Organizing Machines, MIT Press, Cambridge, 2004. [14] S. Forrest, Genetic algorithms: Principles of natural selection applied to computation, Science 261 (1993) 872–878. [15] G. Gazzola, A.J. Buchanan, N.H. Packard, M.A. Bedau, Catalysis by self-assembled structures in emergent reaction networks, in: F. Almeida e Costa, L. Mateus Rocha, I. Harvey, A. Coutinho (Eds.), Advances in Artificial Life, 9th European Conference ECAL 2007, in: Lecture Notes in Artificial Intelligence, vol. 4648, Springer, Berlin, 2007. [16] D.E. Goldberg, The Design of Innovation: Lessons from and for Competent Genetic Algorithms, Kluwer Academic Publishers, Boston, MA, 2002. [17] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison– Wesley, Reading, MA, 1989. [18] R. Groot, P. Warren, Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulations, Journal of Chemical Physics 107 (1997) 4423–4435. [19] A.R. Hill Jr, L.E. Orgel, T. Wu, The limits of template-directed synthesis with nucleoside-5’phosphoro(2-methyl)imidazolides, Origins of Life and Evolution of the Biosphere 23 (1993) 285–290. [20] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975. Reprinted by MIT Press, Cambridge, MA, 1992. [21] P. Hoogerbrugge, J. Koelman, Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, Europhysics Letters 19 (1992) 155–160. [22] D. Irvine, C. Tuerk, L. Gold, SELEXION. Systematic evolution of ligands by exponential enrichment with integrated optimization by non-linear analysis, Journal of Molecular Biology 222 (1991) 739–761. [23] G. Joyce, Directed evolution of nucleic acid enzymes, Annual Review of Biochemistry 73 (2004) 791–836. [24] G.F. Joyce, T. Inoue, L.E. Orgel, Non-enzymatic template-directed synthesis on RNA random copolymers. Poly(C, U) templates, Journal of Molecular Biology 176 (1984) 279–306. [25] G.F. Joyce, L.E. Orgel, Non-enzymatic template-directed synthesis on RNA random copolymers. Poly(C, A) templates, Journal of Molecular Biology 202 (1988) 677–681. [26] S. Jury, P. Bladon, M. Cates, S. Krishna, M. Hagen, N. Ruddock, P. Warren, Simulation of amphiphilic mesophases using dissipative particle dynamics, Physical Chemistry and Chemical Physics 1 (1999) 2051–2056. [27] M. Kranenburg, M. Venturoli, B. Smit, Phase behavior and induced interdigitation in bilayers studied with dissipative particle dynamics, Journal of Physical Chemistry 107 (2003) 11491– 11501. [28] H. Lipson, J.B. Pollack, Automatic design and manufacture of robotic lifeforms, Nature 406 (2000) 974–978. [29] R. Liu, L.E. Orgel, Enzymatic synthesis of polymers containing nicotinamide mononucleotide, Nucleic Acids Research 23 (1995) 3742–3749.

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[30] C. Marsh, Theoretical aspects of dissipative particle dynamics, PhD Thesis, University of Oxford, 1998. [31] J.B. Pollack, H. Lipson, G. Hornby, P. Funes, Three generations of automatically designed robots, Artificial Life 7 (2001) 215–223. [32] S. Rasmussen, L. Chen, D. Deamer, D. Krakauer, N. Packard, P. Stadler, M. Bedau, Transitions from nonliving to living matter, Science 303 (2004) 963–965. [33] M.J. Raven, M.A. Bedau, General framework for evolutionary activity, in: Lecture Notes in Artificial Intelligence, vol. 2801, 2003, pp. 676–685. [34] E. Reddington, A. Sapienza, B. Gurau, R. Viswanathan, S. Sarangapani, E.S. Smotkin, T.E. Mallouk, Combinatorial electrochemistry: A highly parallel, optical screening method for discovery of better electrocatalysts, Science 280 (1998) 1735–1737. [35] R. Rohatgi, D.P. Bartel, J.W. Szostak, Nonenzymatic, template-directed ligation of oligoribonucleotides is highly regioselective for the formation of 3’-5’ phosphodiester bonds, Journal of the American Chemical Society 118 (1996) 3340–3344. [36] J. Shillcock, R. Lipowsky, Equilibrium structure and lateral stress distribution from dissipative particle dynamics simulations, Journal of Chemical Physics 117 (2002) 5048–5061. [37] J. Singh, M.A. Ator, E.P. Jaeger, M.P. Allen, D.A. Whipple, J.E. Soloweij, S. Chowdhary, A.M. Treasurywala, Application of genetic algorithms to combinatorial synthesis: A computational approach to lead compound identification and lead optimization, Journal of the American Chemical Society 118 (1996) 1669–1676. [38] J.W. Szostak, D.P. Bartel, P.L. Luisi, Synthesizing life, Nature 409 (2001) 387–390. [39] M. Theis, G. Gazzola, M. Forlin, I. Poli, M.M. Hanczyc, M.A. Bedau, Optimal formulation of complex chemical systems with a genetic algorithm, ComPlexUs (2006), in press. [40] T. Tjivikua, P. Ballester, J. Rebek Jr, A self-replicating system, Journal of the American Chemical Society 112 (1990) 1249–1250. [41] S. Trofimov, E. Nies, M. Michels, Thermodynamic consistency in dissipative particle dynamics simulations of strongly nonideal liquids and liquid mixtures, Journal of Chemical Physics 117 (2002) 9383–9394. [42] C. Tuerk, L. Gold, Systematic evolution of ligands by exponential enrichment: RNA ligands to bacteriophage T4 DNA polymerase, Science 249 (1990) 505–510. [43] I. Vattulainen, M. Karttunen, G. Besold, J. Polson, Integration schemes for dissipative particle dynamics simulations: From softly interacting systems towards hybrid models, Journal of Chemical Physics 116 (2002) 3967–3979. [44] G. von Kiedrowski, A self-replicating hexadeoxynucleotide, Angewandte Chemie International Edition in English 25 (1986) 932–935. [45] R.A. Watson, S.G. Ficici, J.B. Pollack, Embodied evolution: Distributing an evolutionary algorithm in a population of robots, Robotics and Autonomous Systems 39 (2002) 1–18. [46] L. Weber, S. Wallbaum, C. Broger, K. Gubernator, Optimization of the biological activity of combinatorial compound libraries by a genetic algorithm, Angewandte Chemie International Edition in English 34 (20) (1995) 2280–2282. [47] M. Wright, G. Joyce, Continuous in vitro evolution of catalytic function, Science 276 (1997) 614–617. [48] S. Yamamoto, S. Hyodo, Budding and fission dynamics of two-component vesicles, Journal of Chemical Physics 118 (2003) 7937–7943. [49] S. Yamamoto, Y. Maruyama, S. Hyodo, Dissipative particle dynamics study of spontaneous vesicle formation of amphiphilic molecules, Journal of Chemical Physics 116 (2002) 5842– 5849. [50] W.S. Zielinski, L.E. Orgel, Oligoaminonucleoside phosphoramidates. Oligomerization of dimers of 3’-amino-3’-deoxy-nucleotides (GC and CG) in aqueous solution, Nucleic Acids Research 15 (1987) 1699–1715.

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Self-Assembly as an Engineering Concept across Size Scales Sean Stauth, Christopher J. Morris, Babak Parviz* University of Washington, Electrical Engineering, Campus Box 352500, Seattle, WA 98195, USA Abstract. In this chapter we focus on systems made via self-assembly in the micrometer range or larger. Self-assembly at these size scales is important because although technological advances in the electronics industry continue to enable smaller, faster, cheaper, and more integrated systems, efforts in the area of integration have faced significant challenges in constructing 3D structures or building heterogeneous systems made of parts from incompatible microfabrication processes. Robotic “pick-and-place” techniques are currently used to integrate devices from many types of processes, but the ability to efficiently handle individual parts diminishes as the size scale of the components decreases below about 300 µm. Self-assembly is an attractive solution to overcome current integration limitations. Two micrometer-scale self-assembly systems are reviewed here: the first investigates the incorporation of single crystal silicon components, including single crystal silicon field effect transistors (FETs), on flexible plastic substrates for the construction of large area electronic systems, and the second system explores an avenue for future construction of threedimensional silicon computing structures. The results from the first system include the electrical characterization of a self-assembled logic inverter, and the self-assembly of approximately 10,000 single crystal silicon elements in 25 minutes, each element measuring 100 µm in diameter.

1. Introduction The aggressive progress of the semiconductor industry during the past four decades has enabled many new technologies. Currently, microfabrication lines routinely produce chips that incorporate billions of sub-100 nm devices that are interconnected in a pre-determined way. If progress in the area is to maintain its pace, the next generation of manufacturing methods must: (a) yield devices with nm-scale precision, (b) produce large numbers (109 ) of these devices in a parallel and cost-effective fashion, (c) enable interconnection of the devices to create systems and finally (d) enable the integration of devices that are made in different and potentially incompatible fabrication technologies. The latter item is increasingly important in modern systems that perform multiple functions. These systems use not only electronic interfaces with the outside world for data manipulation, but also photonic, chemical, and mechanical interfaces. * E-mail: [email protected].



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Fig. 5.1. Approximate speed (parts per hour) vs. approximate range of part size for a variety of serial assembly methods. The enclosed zone identifies the area in which self-assembly can have significant contributions. Sources are: (a) individual atom placement by STM [1]; (b) polymer memory storage device [2]; (c) optical tweezers assembly [3–5]; (d) 3-D microassembly [6]; (e) robotic pick-and-place printed wiring board assembly (model HLX8100, Essemtec, Switzerland); (f) robotic assembly of a grease gun coupler (Innovation, Inc., Birmingham, AL); (g) robotic assembly of automobile wheel bearings (Capitol Technologies, Inc., Niles, MI); (h) robotic assembly of an automobile body and chassis (Profibus International Case Study: OPEL, Germany). For colour version, see Colour Plate Section.

The extension of current microfabrication technologies to sub-100 nm size scales for device fabrication is likely to yield devices and interconnected systems in the 20 nm size range; however, the successful extension of these technologies to yet smaller size scales is in doubt. A greater challenge for the extension of conventional techniques is system integration. These technologies have been markedly less successful in the integration of multi-functional systems with many parts and three-dimensional architectures. Figure 5.1 summarizes the state of the art for serial, robotic assembly techniques, by plotting assembly throughput vs. characteristic size of the parts being assembled. At larger scales, for example in an automotive robotic assembly plant, the assembly speed of meter-sized components is limited by the difficulty in applying large forces to overcome inertia. As component sizes shrink, the assembly speed increases, peaking for robotic pick-and-place systems that are used for electronic circuit board manufacturing. A state-of-the-art robot can assemble more than 26,000 parts/hour, each measuring 300 µm. Serial robotic assembly can be employed down to the atomic scale. However, as shown in Fig. 5.1 these techniques rapidly lose their efficiency and throughput when handling smaller components. It is unlikely that these techniques can meet the future challenges

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of producing and integrating components with nano-scale futures into larger systems. New manufacturing technologies must be explored. An excellent example of a manufacturing technology that can meet all requirements outlined above, both in terms of producing large numbers of parts with nano-scale precision, and integrating them to form larger systems, is provided by biology. A cursory look at an insect such as a fruit fly (Drosophila) reveals that biology has already met the challenges that are forthcoming for the semiconductor industry. From an engineering standpoint, a Drosophila is a multifunctional system with numerous parts each having nano-scale structural precision, each part being interconnected to provide system functionality. It also becomes clear that nature constructs this system via self-assembly—a process that is fundamentally different from what is used today to manufacture engineered products. The immediate lesson from observation of nature is that it is possible to construct systems via self-assembly and fundamental laws of physics are not violated during the process. It is also important to note that self-assembly processes in nature are not limited to producing small scale systems. Through examples such as trees and large mammals, nature offers an inspiration to use self-assembly as a manufacturing process for systems that span many meters in size or more. In order to realize the true potential of self-assembly as a method of manufacturing, one needs to design self-assembled structures not only in the nano-scale but also in the micron, millimeter, or even larger size scales. Nature constructs large scale structures through a hierarchical self-assembly scheme in which molecules with increasing levels of complexity form the final structure. Rarely, if ever, does nature use micron-scale parts to construct a millimeter-scale or larger structure. Although following nature’s approach may yield a methodology that will eventually enable the construction of large scale structures, current technological challenges call for nearer-term solutions. Nearer-term solutions are needed because the methodology to construct electronically functional, micron-scale components (possibly with many interconnected transistors) via self-assembly starting at the atomic scale does not exist. It is possible to produce such micron-scale parts without self-assembly, effectively bypassing some of hierarchical steps that normally would occur in a biological self-assembly process. By skipping to the step of having functional micron-scale components, one may design and experiment with self-assembly processes to construct interconnected systems from the functional parts. This self-assembly procedure, albeit different from the biological one, allows investigation of selfassembly processes in the micron-scale and identification of suitable programming methods, agitation mechanisms, and binding forces. Other chapters of this book discuss issues regarding the construction and design of nano-scale self-assembling systems. Here, we focus exclusively on systems that are made via self-assembly in the millimeter-range or larger. It should also be noted that current approaches to the construction of larger scale systems

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via engineered self-assembly are different from the path taken by nature in another important aspect. All naturally self-assembling biological systems involve self-replication. Self-replication, a process through which the parts produce new parts for the continuation of the assembly process, possesses a level of complexity that is beyond the state of the art in engineering. In the engineered selfassembly processes considered here, a collection of parts are induced to bind and constitute a new system with all parts either being present at the start or gradually introduced from outside the system, because construction of self-replicating components is not currently possible. The absence of the exponential efficiency of self-replication poses the question of how large can the number of parts be? There are technologies available that can generate large numbers of micron-scale (or larger) parts including the printing industry, the solid-state microfabrication industry, and agriculture. To create truly functional systems, the components need to perform tasks with a complexity beyond that of passive, structural materials. Therefore the technology used to generate the components must be able to incorporate basic functionality in their structure. The most mature technology today for generating large numbers of micron-scale parts that incorporate some function is solid-state microfabrication. Given the preceeding observations and constraints, the task of self-assembling a large scale system converts to answering the questions of how can one (a) generate a large number of micron-scale parts that carry basic functionality, and (b) induce the parts to self-assemble and form a large interconnected structure. In a typical micron-scale self-assembly procedure, parts that participate in the process are fabricated separately and then introduced in a common medium. The parts should have the ability to recognize (the self-assembly programming mechanism) and connect (self-assembly binding/driving force) to the proper adjacent part or template. Binding events between the components drive the system toward a lower energy state, and occur with a finite error rate due to the statistical nature of the process. In order to drive the system from local energy minima to a global minimum that corresponds to the correct self-assembly, an external agitation mechanism is needed. The design and implementation of such a self-assembly procedure involves not only the construction of parts incorporating recognition and binding mechanisms, but also the proper design of external agitation to arrive at a high yield process. In the next two sections, we consider two examples of simple self-assembly procedures that use micron-scale parts. The first example investigates the incorporation of single-crystal silicon components on flexible plastic substrates for the construction of large area electronic systems. The second one explores an avenue for future construction of three-dimensional silicon computing structures. The motivating factor for both of these examples has been an immediate engineering need. Our current approach for advancing self-assembly is the pursuit of a series of individually justifiable steps that bring us closer to the final goal of designing and engineering an arbitrary large scale functional system.

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2. Two-Dimensional Templated Self-Assembly The incorporation of single crystal silicon electronic devices onto flexible plastic is an avenue to greatly increase the power and speed of plastic electronics. With electronics embedded onto flexible substrates, previously rigid electronic devices such as computers, intelligent notebooks and e-books may become as flexible as a sheet of paper [7]. In addition, the ability to incorporate active circuitry into large area and non-conventional substrates can extend the application and range of computing into new domains. Applications currently utilizing flexible substrates include liquid crystal displays, smart cards, price and inventory tags, and large area sensor arrays [8,9]. However, these technologies are limited by the relatively low performance of the organic or amorphous silicon electronics employed. Low electron mobility in organic semiconductors (charge carrier mobility < 1 cm2 /Volt-second in organic field effect transistors) and amorphous silicon (electron mobility ∼ 1 cm2 /Volt-second in an amorphous silicon thin film transistor) leads to higher power consumption and lower switching speeds than their equivalent single crystal silicon counterparts (electron mobility ∼ 1000 cm2 /Volt-second) [10]. The low performance of these semiconductors have prevented the direct manufacturing of computing or radio frequency devices on large area plastic substrates. Standard semiconductor process requirements such as high temperatures and the use of chemicals that readily degrade organic materials do not allow for direct fabrication of single crystal silicon devices on plastic substrates. The silicon devices must be microfabricated prior to integration; therefore an integration technique that is both fast and accurate is needed to embed silicon devices into plastic substrates, while providing both mechanical and electrical connectivity. A promising candidate to meet these requirements is fluidic self-assembly. Fluidic self-assembly is the process whereby freestanding microcomponents spontaneously assemble at prescribed locations on a substrate, without utilizing external control mechanisms such as a robotic arm. Massively parallel by nature, selfassembly processes have demonstrated the ability to accurately assemble large numbers of microcomponents with minimal time and effort. In other self-assembly schemes, gravitational [11] and capillary [12–14] forces have been used to drive assembly processes. In [11], components slide over a template and fall into wells at assembly sites. In [12–14], components assemble after making contact with an adhesive or solder. Solder-based assembly methods offer an additional advantage, in that the solder can provide both binding forces for assembly as well as electrical connections between the assembled part and substrate. Jacobs et al. [12] assembled Light Emitting Diodes (LEDs) onto a flexible substrate to create a cylindrical display. In all these examples, effective part-to-substrate self-assembly was achieved, although post-possessing was required to complete electrical connections between the parts and substrate.

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Fig. 5.2. Illustration of the self-assembly process. Microfabricated silicon components assemble at complementary shaped prescribed binding sites on the plastic substrate. (A) Side and top views of the substrate with components. The molten alloy establishes both electrical and mechanical connection between the part and substrate. (B) The fluidic self-assembly process. The substrate is tilted and immersed in a heated aqueous solution. Only one shape of components (in this case rectangles) is assembled at a time. For colour version, see Colour Plate Section.

Fluidic self-assembly can be employed for the low temperature integration of freestanding single crystal silicon devices onto clear, flexible, plastic substrates in a single step which provides both mechanical and electrical connections. The mild self-assembly conditions (temp 10% and 30% and b, then do x,” where a represents the population of kinases, b the population of phosphatases, and “do x” is the reaction catalyzed by the target protein. However, because the kinase and phosphatase molecules diffuse around in the “soup” of the cytosol, reacting at random with the target proteins, there is a “race” between the two species to react with the target protein. Sometimes when there are fewer kinases than phosphatases, a phosphorylated target protein will have the opportunity to perform its catalytic function before

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it becomes dephosphorylated. Similarly, occasionally when there are more kinases than phosphatases, the target protein will not get a chance to catalyze a reaction before being dephosphorylated. Statistically, when a b, “do x” will almost always occur, and when a b, “do x” will almost never occur, but when a ≈ b, there is a smooth sigmoidal transition where the probability of “doing x” goes from 0 to 1. Thus, the stochastic nature of such races leads to “errors” in computation, particularly when the race is close. Decision making or branching can also be accomplished with only one or two molecules. The only requirement is two possible outcomes. For example, a new tubulin dimer could be added to the end of a microtubule (increment), or the last tubulin dimer might dissociate (decrement). A motor protein may take another step along the cytoskeletal fiber to which it is currently bound, or it might fall off. In both of these cases, there is a “race” between two stochastic events. There are two possible outcomes; the decision is made by whichever one happens first. In general, the decision-making/branching we observe in Nature involves some sort of race or competition. The decision is made to follow the “winner”—usually (or most likely) the faster, stronger, or more numerous.

4.

Hierarchical RAM Computing

The RAM computing model is applicable to living systems not only at the level of protein networks, but at other hierarchy levels as well. By encapsulating proteins within membranes (the nuclear membrane, organelle membranes, or the cell’s plasma membrane), RAM algorithms can be carried out at the level of cellular networks. Then, by encapsulating cells within another “membrane” (e.g., a skin or exoskeleton), algorithms can again be carried out at the level of organism networks. In this way, RAM programs can be scaled up to greater spatial and temporal scales. Since cells and organisms are readily observed to make decisions and change their behavior, the branching and sequencing aspects of the RAM model are obvious at these levels, so we will not discuss them further. What may not be obvious is how the registers and their operations are represented at the cellular or organism levels. Here, again, living systems embody registers in a unary representation, which can perhaps account for their lack of previous recognition. Figure 7.3 shows some examples of similar registers and operations at the levels of proteins, cells, and multi-cellular organisms. In Fig. 7.3a, the number of proteins is the register, and synthesis and degradation of the proteins are the increment and decrement operations, respectively. Compare this to Fig. 7.3b, in which the number of cells is a register. Cell division increments the register; apoptosis (or other mechanisms of cell death) decrements the register. Figure 7.3c shows a population of organisms as a register. Birth is an increment; death is a decrement.

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Fig. 7.3. Illustration of registers and their operators at different hierarchy levels. In (a)–(c), the register is represented by the number of objects, (a) molecules, (b) cells, and (c) organisms. The increment (decrement) operations are shown in the lower left (lower right) corners of each panel. In (d)–(f), the register is represented by a localization of objects in the center, (d) intracellular cargoes, (e) bacteria, and (f) ants. Increments (decrements), illustrated by + (−) arrows, are implemented by (d) motor proteins adding (removing) cargoes from the center, (e) bacteria moving by chemotaxis to (from) the food source, and (f) ants walking to (from) the food pile.

We mentioned above (and illustrated in Figs. 7.1e and f) that a localized collection of objects can represent a register. Figure 7.3d illustrates a cell in which motor proteins carry cargo by “walking” on microtubules (lines radiating outward from the center). Microtubules are cytoskeletal filaments that nucleate on a protein complex, the centrosome, usually located near the nucleus of a eukaryotic cell. The number of cargoes at the centrosome represents a register. Motor proteins carrying cargo to the centrosome increment the register; those carrying cargo away decrement it. This is analogous to Fig. 7.3e, which illustrates bacterial chemotaxis. The number of bacteria that have reached the attractant (e.g., food source) in the center represents a register. Each individual bacterium’s arrival at the food source increments the register. A bacterium that leaves the attractant serves as a decrement. Figure 7.3f shows ants arriving at a food source, incrementing the register (number of organisms at the source). When an ant leaves the source, it decrements the register.


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Although these examples are far from exhaustive, they illustrate a key point. Just as, at the molecular level, the same algorithms can be implemented by different hardware, the same algorithms can even be implemented at different hierarchy levels, again with different hardware. We hypothesize that the algorithms that life does implement are the ones that can be reproduced at the next level, so that life can scale up and evolve more complex forms. Indeed, there may be survival advantages for algorithms that can be implemented in many different hardware forms. That is, the algorithms that survive evolution may be the very ones that can be re-used robustly in many different physical contexts at different scales.

5.

RAM Programs in Living Systems

Having established that proteins in general have all the properties necessary to carry out RAM programs, we now turn to specific proteins in real living systems and examine what algorithms they actually implement. That is, by scrutinizing a specific biological process, we identify unary registers and the mechanisms of incrementing and decrementing those registers. Then we track the sequencing of those operations in order to capture explicitly (in pseudocode or a flow diagram) the RAM program corresponding to that biological process. Although the examples we present are relatively simple, they do demonstrate how RAM programs can be implemented in real systems and give us the first steps toward understanding this different kind of programming language.

5.1.

Melanophore Programs Using Active Transport

In color-changing fish skin cells (melanophores), pigment granules (cargoes) with both inward- and outward-walking microtubule-associated motor proteins enable the cell to change color. One cellular signal drives the inward motors to carry the pigment to the centrosome, leaving the body of the cell free of pigment, thereby making the cell transparent. A different cellular signal stimulates the outward motors to disperse the pigment, making the cell opaque [16]. We identify a collection of pigment particles at the centrosome (as in Fig. 7.3d) as register A. An accumulation of pigment particles anywhere else within the cell represents registers B, C, D, . . . , as many registers as necessary to account for all of the collections of pigment particles at any time. Anywhere a single pigment granule sits by itself is a register of value 1. A motor protein that carries a pigment granule to a pile increments that register, while one that carries a pigment granule away from a pile decrements the register. So, with these definitions, if a pigment granule is sitting by itself (B = 1) and a motor protein moves it somewhere else, this is equivalent to (decrement B, increment C) so that (B = 0; C = 1). The two registers’ operations are coupled by the single physical process.

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First we examine the algorithm to disperse the pigment. From the perspective of an individual cargo/motor-protein “agent,” the algorithm is extremely simple: decrement reg, increment reg (reg = reg ). Combining a large number of agents, all executing such a simple algorithm, results in a complex collective behavior. If we were to capture this collective behavior as if it were a single global algorithm, the dispersal algorithm could be viewed as loop: for reg = A, B, C, D, . . . for i = n(reg) down to 1 decrement reg increment reg next i next reg until n(reg) = 1 for all reg where n(reg) is the value in register reg. Although this is written as if there are three nested loops, nested in a particular order, in reality, there is no real ordering. All of the registers can be decremented simultaneously, and multiple decrements can occur on a single register at the same time. Because when a register is decremented, it might inadvertently increment another register, we add the outer loop to split up piles that accidentally got piled together. In a deterministic sense, the process continues until every pigment granule sits alone. That would be perfect dispersal. However, because of the stochastic nature of the system, there may be places (including at the centrosome) where a small number of pigment granules stay together, so that the registers are not exactly one. Nevertheless, the end result is close enough to have the physical effect of making the cell opaque. The condensation algorithm can be written as for reg = B, C, D, . . . for i = n(reg) down to 1 decrement reg increment A next i next reg Again, the loops need not be performed in order. The important part is “decrement reg; increment A.” All pigment particles can be moved to the centrosome simultaneously. The outcome of a deterministic version of this algorithm would be every granule residing at the centrosome (register A). However, again because of the stochastic nature of the system, perhaps not every particle will be moved to the centrosome, but it will be close enough that the cell is transparent.


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These programs are so simple that, from the agent’s perspective, no branching is required. Since all registers can run in parallel (i.e., the innermost loop of both programs is like a subroutine that can be run on each register simultaneously), there is very little sequencing to enforce either. It just needs a signal to start either the dispersal program or the condensation program on each register.

5.2. Scaling Algorithms to Higher Hierarchy Levels As mentioned previously, the same RAM algorithms can be implemented by hardware at different levels of hierarchy (and therefore at different scales), such as populations of molecules, cells, or organisms. We just described the condensation algorithm for populations of molecules in a melanophore. This same algorithm is applicable also to bacteria in chemotaxis. The motility of each bacterium itself acts as a combination of decrementing another register and incrementing register A, the number of bacteria at the focus of the attractant. Similarly, one could view the number of ants (or food particles) in an ant colony as register A. The ants gather food from other locations (other registers) to the colony, decrementing the other registers and incrementing register A. Hence, this one condensation, or “gathering,” algorithm is implemented in Nature on at least three different scales or levels of hierarchy.

5.3.

Search-and-Stabilize Program Using Structural Assembly

The second example program we describe has much more complicated sequencing, with numerous branches and loops. In contrast to the previous example, where active transport was used to operate on registers, in this example registers and their operations are implemented by the structural assembly of microtubules (MTs). In order to put this example in context, we briefly review a current wellaccepted model of MT growth and dynamics [12,17]. In a living animal cell, tubulin dimers bind to nucleating sites on the centrosome and then stack together to grow a cylindrical MT from each of these nucleating sites. Each free tubulin dimer is tightly bound to a guanosine triphosphate (GTP) molecule that is hydrolyzed to guanosine diphosphate (GDP) shortly after the tubulin is added to the growing MT. Whereas the GTP-tubulin packs efficiently together, GDPtubulin molecules have a different conformation and bind less strongly to each other. If GTP-tubulin is added faster than the GTP in the MT is hydrolyzed, a cap of GTP-tubulin holds together the growing end, and the MT continues to grow for some time. However, if, due to the randomness of chemical processes, the GTP is hydrolyzed all the way to the end of the MT before new GTP-tubulin is added, the weakly interacting GDP-tubulin at the end will unravel the MT, often in a catastrophic manner. If new GTP-tubulin attaches before the MT has collapsed completely, it can be “rescued” and resume growth. This alternating MT

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Fig. 7.4. Schematic illustration of registers represented by a microtubule. When a new GTP-tubulin dimer is added, register A is incremented. When a GTP-tubulin is converted to GDP-tubulin by GTP hydrolysis, register A is decremented and register B is incremented.

growth and collapse is termed dynamic instability. If a MT collapses completely, a new one is quickly nucleated in its place, and its growth is, in general, in a different direction from the previous MT. In living cells, dynamic instability is a natural and common behavior of microtubules. Dynamic instability can be suppressed, however, by stabilizing proteins that bind to the ends of MTs (or along their length), stabilizing them against disassembly. Such stabilized MTs serve as tracks for the transport of intracellular cargo by motor proteins, and help position organelles where needed by the cell. Therefore, dynamic instability, with growth in a new direction after complete MT collapse, provides a mechanism for searching the space of the cell until the stabilizers are located and bound. The result is a long-term physical structure of stabilized MTs in support of the cell’s activities. Now let us examine this search-and-stabilize behavior as a RAM computer program. Each MT in the cell is executing the same program, so here we discuss the algorithm for a single MT. We have identified two registers embodied by the MT, as illustrated in Fig. 7.4. Register A consists of the GTP-tubulin capping the MT; register B is made up of the GDP-tubulin closer to the centrosome. When a new GTP-tubulin dimer is added, register A is incremented. When a GTPtubulin is converted to GDP-tubulin by GTP hydrolysis, register A is decremented and register B is incremented. If there is no GTP cap, GDP-tubulin dimers can dissociate and decrement register B. Figure 7.5 shows a flow diagram of the search-and-stabilize program. STABILIZED may be viewed as a register with a Boolean value. This register is implemented by a bond between the end of the MT and a stabilizer, as in Fig. 7.1i. If a bond exists, STABILIZED is true, otherwise false. (Alternatively, if a bond exists, STABILIZED is 1, otherwise 0.) At the beginning of the program, STABILIZED is false. DIRECTION can also be viewed as a register that represents the direction of MT growth. A DIRECTION is selected at random (rand) by physical processes. Then the program enters a network of operations that can branch and loop, all running in parallel, with each operation occurring at a stochastic time, depending on diffusion of molecules, collisions, and binding rates. Adding a GTP-tubulin to


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Fig. 7.5. Flow diagram of the microtubule search-and-stabilize program. Each box represents a RAM computer operation. A and B are registers as shown in Fig. 7.4. STABILIZED and DIRECTION are registers that hold a Boolean value and a direction, respectively; a and b are the values of registers A (number of GTP-tubulin dimers) and B (number of GDP-tubulin dimers), respectively. Arrows show the sequencing of operations, with labels indicating branch conditions; there are many parallel sequences in this program. Growth of the microtubule occurs through the [+]A operation; collapse occurs via the [−]B/jump operation; paths that lead to rescue and re-nucleation are labeled.

the MT increments register A (operation [+]A). After this operation, two operations follow. Note that this is the only fork in the flow diagram where both paths are executed; all others are conditional branches. On the left branch, if the MT has reached a stabilizer, it binds to it, setting the STABILIZED register to true. If not, it loops back to adding another tubulin dimer ([+]A). The right branch from [+]A implements GTP-hydrolysis. When this occurs, at some stochastic time later, this decrements register A and increments register B. Both occur simultaneously, so we have shown it as one operation ([−]A; [+]B). Note that we used [−]A rather than [−]A/jump here. The jump occurs when the value of the register is already zero when the decrement is attempted. In this program, there is no way to reach this instruction if the value of register A is already zero, therefore no jump can occur. After GTP-hydrolysis of one tubulin dimer, what happens next depends on whether all of the GTP has been hydrolyzed. If there are still GTP-tubulin dimers left in the MT, i.e., the value of register A, a, is not zero, and the MT is not stabilized, then it loops back to [+]A, adding another GTP-tubulin. Thus there are two loops in this program (so far) that represent MT growth, both originating and

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returning to [+]A. The left loop implements the situation where multiple tubulin dimers are added without any GTP-hydrolysis occurring in between. The right loop includes GTP-hydrolysis, but at a slow enough rate that the GTP-tubulin cap remains (a = 0) and growth continues. If, after the GTP hydrolysis, there are no GTP-tubulin dimers left (a = 0) and the MT is not stabilized, GDP-tubulin can dissociate from the MT, decrementing register B ([−]B/jump). As mentioned before, branching (decision making) is one of the mechanisms that make this type of computing stochastic and nondeterministic. The GDP-tubulin could begin dissociating before a is exactly zero. There may be a few dimers of GTP-tubulin left, but not enough to effectively cap the MT. However, such stochastic errors are irrelevant to the overall MT behavior, just as the stochastic errors in the dispersal and condensation programs of the melanophore were irrelevant (i.e., even with errors it was close enough to serve the function of making the cell transparent or opaque). The [–]B/jump operation branches depending on whether the value of register B, b, is zero or non-zero, i.e., whether or not the MT has completely collapsed. If b = 0 (or, with stochasticity, reasonably close to it), the program loops all the way back to picking a new random direction and nucleating a new MT. If the MT has not completely collapsed (b > 0), then the program loops back to two different places in the program. Another tubulin can be added ([+]A), or as long as a is still 0 and the MT is still not stabilized, another GDP-tubulin can dissociate ([–]B/jump). The next operation in both of these loops occurs at some stochastic time later, so essentially they are in a race. If GDP-tubulin dissociates first ([–]B/jump), then the MT continues to collapse. If GTP-tubulin is added first ([+]A), the MT is rescued and the growth mode resumed. So far, we have examined all of the paths in the program involved in the “search” part of “search-and-stabilize.” This fully describes the algorithm for MT dynamic instability, in which paths that lead to the growth of the MT ([+]A) race against the paths leading to GTP-hydrolysis ([–]A; [+]B) and GDP-tubulin dissociation ([–]B/jump). It also includes nucleating a MT in a new growth direction after the complete collapse of a previous one. When the MT encounters a stabilizer and is stabilized (STABILIZED = T), that is the end of the left branch, but not the end of the program. Recall that different paths through the program are running in parallel, so there are many paths down the right branch that have already been launched. Specifically, GTPtubulin can still be hydrolyzed ([–]A; [+]B). As long as there are still GTPtubulin dimers in the MT (a = 0), the program keeps looping back to hydrolyzing it. Finally, when all the GTP is hydrolyzed (a = 0), the program goes into a “sleep” mode in which it simply keeps “checking” if the MT is still stabilized. If this loop were programmed on a modern computer, it would be an infinite loop, and just checking the flag over and over would be a waste of processing resources. However, in reality, the MT does not have to do anything active to


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check. It waits passively until something occurs to release the stabilizer from the MT. For example, suppose a signal molecule reaches the stabilizer, stimulating it to unbind from the MT. At that point STABILIZED becomes false, and the program immediately switches over to [–]B/jump, re-launching the race between collapse and growth, and the MT automatically resumes the dynamic instability, “search” part of the program. This behavior will appear below in our examples of programmed dynamic self-assembly.

6.

Programmed Dynamic Self-Assembly

In the previous section, we examined dynamic self-assembly from the perspective of understanding algorithms that are already implemented in living systems. We now investigate how to “program” these processes to build novel structures or materials. In this section, we describe our first dynamic self-assembly programs, designed to build nanostructures, and show their implementation in computer simulations. As mentioned before, “writing” a dynamic self-assembly program requires cleverly determining an appropriate set of hardware components that, when allowed to interact, executes the desired program (including the operations on appropriate registers, and the proper sequencing of these operations) by building, reconfiguring, or disassembling physical structures. Rather than start from scratch designing hardware components, we take advantage of hardware components that already perform some of the functions we desire—specifically, MTs, motor proteins, and the proteins associated with them. Many behaviors of these proteins have been shown to work outside of cells. For example, under suitable conditions, and as long as they are supplied with energy, MTs can form asters on centrosomes [18,19] and undergo dynamic instability [19–22]; motor proteins can carry cargo, such as pigment granules [23,24], drive stabilized microtubules [25], and help self-organize MTs into asters and spirals [26]. The idea of harnessing these molecular machines to build nanostructures bottom-up is attractive. We are beginning to learn what it is possible to program these proteins to do. We design a dynamic self-assembly program by prescribing specific combinations of these proteins plus others specifically engineered to sequence and direct them, and by also specifying the initial spatial distributions of some of these components. To test the program, we include models of these proteins in the prescribed configuration in computer simulations, and watch their collective behavior. Details of the protein models and computer simulations are described elsewhere [27,28].

6.1.

Reconfiguring Network Program

Our first example program uses only MTs, centrosomes, stabilizers, and a few specifically engineered molecules to block and unblock the stabilizers. Figure 7.6

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Fig. 7.6. Screen grabs of a simulation demonstrating a reconfigurable MT network. A detailed description of each panel is provided in the text. For colour version, see Colour Plate Section.

shows screen grabs from a computer simulation of this system. We assume an initial spatial pattern of centrosomes and two types of stabilizers (Fig. 7.6a). These stabilizers are identical in their interaction with MTs, but their blocker binding sites differ, so that they bind two different types of blockers. This will enable us to turn on and off the two different types of stabilizers independently. In Fig. 7.6, centrosomes are shown in green, type-1 stabilizers are yellow, and type-2 stabilizers are red. The two sets of stabilizers are arranged in an alternating pattern around the grid of centrosomes. Such an initial pattern could be achieved with currently available nanopatterning technologies. When we introduce a concentration of GTP-tubulin into this system, MTs automatically execute the search-and-stabilize algorithm—they are already programmed to do so—and self-assemble a network. If neither type of stabilizer is blocked (i.e., the MTs can bind to either type of stabilizer), the result is a random network of MTs (green lines), as shown in Fig. 7.6b. This random network is perhaps uninteresting from the standpoint of building nanostructures. However, when we include type-1 blocker molecules, these bind to the type-1 stabilizers (yellow) and change their conformation so that MTs cannot bind to them (blocked stabilizers are shown grey in Fig. 7.6). In RAM computing jargon, the blocker acts as a decrement operator on the STABILIZED register, changing the value from 1 to 0. Any MTs that were bound to those stabilizers are released, and they naturally resume the dynamic instability, “search” part of the search-andstabilize algorithm, as discussed at the end of the previous section. When they


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are stabilized by the only available stabilizers, which are all type-2, the resulting pattern, after a stochastic amount of time, is a square grid network (Fig. 7.6c), a more interesting structure. This structure could be used, for example, as a template for mineralization [29,30], to make a network of nano-wires. Furthermore, if we introduce releasers for the type-1 blockers and also introduce type-2 blockers, once again the MTs are released and resume the searchand-stabilize algorithm. Again they bind to the only available stabilizers, but this time they are type-1 stabilizers. The result (again, after a stochastic “wait”) is a diagonal grid pattern (Fig. 7.6d). This new grid could also be used as a template for another mineralization step, if desired. As such, it may be possible to fabricate two interpenetrating grids of two different materials, by using first one grid as a template, then the other. This simple dynamic self-assembly program is essentially the search-andstabilize “subroutine” of Fig. 7.5 triggered a number of times. In this example, we explicitly control the sequencing and the geometry of the desired structure by introducing the blockers and releasers at prescribed times.

6.2.

Directed Transport Program

We now discuss a second example, which uses not only MTs, but also motor proteins carrying cargo, to build a nanostructure. In this case, we assume a symmetry-breaking mechanism can be employed to drive all of the stabilizers to one side. There are numerous mechanisms employed by cells to guide proteins to a localized region within the cell or on its membrane [12]. While it may be relatively easy to localize proteins using these mechanisms, we can then use the pattern of localized proteins as a template to build a nano “rock wall” of any kind of cargo the motor proteins can carry. More importantly for our purposes, this example illustrates how we can program a sequence of dynamic self-assembly subroutines to execute either in parallel or sequentially, in order to collect randomly distributed cargo and deliver it to the desired location. Figure 7.7a shows the initial configuration. A single centrosome (green) is located in the center of the environment (black). Stabilizers (grey) are located on the edge of the environment to the right. Cargoes (red), with both inward and outward motor proteins attached, are randomly distributed throughout the environment. Initially, the conditions are such that the dynein (inward walking) motor protein is more likely to bind to the MT and carry the cargo. So, if there are MTs to walk on, the cargo/motor-protein complexes will execute the condensation algorithm described in Section 5.1. When GTP-tubulin is added, MTs automatically execute their search-andstabilize algorithm (Fig. 7.7b), which serves two purposes. First, when a MT grows out to (or past) a cargo/motor-protein complex, the MT acts as a temporary

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Fig. 7.7. Screen grabs of a simulation demonstrating a program to harvest randomly distributed nano-particles and deliver them to a localized region to build a nano “rock wall.” A detailed description of each panel is provided in the text. For colour version, see Colour Plate Section.

bridge. The dynein hops onto the MT and carries the cargo inward to the centrosome (executing the condensation algorithm). When the MT collapses, a new one is nucleated in a different direction, and can act as another temporary bridge. As many MTs rapidly grow and collapse, many directions are sampled, so most of the cargo is condensed to the center in a reasonably short time (Fig. 7.7c). In parallel with the cargo condensation process, the MT search-and-stabilize algorithm achieves another purpose: building a solid bridge to the stabilizers to the right (Fig. 7.7c). So far, the program has involved the search-and-stabilize algorithm of the MTs and the condensation algorithm of the cargo/motor-protein complexes running in parallel. Also in parallel, the concentration of cargoes at the centrosome is compared against a threshold value. When the concentration of cargo at the center is sufficiently high, this stimulates the release of a population of centrosome blockers. When a MT collapses, the centrosome blockers compete with GTP-tubulin to bind to sites on the centrosome. This eventually shuts off MT growth from the centrosome, because all available MT nucleating sites are bound by blockers. Eventually, only the MT bridge to the stabilizers remains. When the concentration of centrosome blockers bound to the centrosome exceeds a threshold, this triggers the next subroutine, by releasing the signal to enhance the activity of the kinesin (outward walking) motor proteins and suppress


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the dynein. These motor proteins cross the bridge to deliver the cargo to the stabilizers (Fig. 7.7d), building a “rock wall” of cargo over the original template of stabilizers. If one considers the cargo at the centrosome and stabilizers as registers A and B, respectively, then the motors carrying the cargo from the centrosome to the stabilizers are decrementing A and incrementing B. Like the condensation algorithm, the “loop” over the operations (decrement A; increment B) need not be sequential. Rather all the motors can perform these operations in parallel with other motors. A sufficiently high concentration of cargo at the stabilizers triggers the next subroutine, releasing the MTs and suppressing their growth dynamics, so that collapse is far more likely than growth (Fig. 7.7e). When the last MTs have collapsed, the remaining centrosome blockers block the centrosome entirely, so no more MT growth can occur. We end up with a nano version of a rock-wall (Fig. 7.7f). The cargo itself could be the nanostructure desired. Or again, it could be used as a template for some other structure. We should note that in the system just described, the molecules and complexes themselves carried or released the signaling molecules for the next step in the sequence of assembly and disassembly processes. An alternative approach would be to manually include signaling molecules at the appropriate times. This was the approach taken in the previous example, when specific blockers and releasers were introduced at desired times. These examples highlight the striking difference between programmed static and dynamic self-assembly. In programmed static self-assembly, the components are objects designed to settle into a low-energy, equilibrium configuration (the self-assembled structure of interest) that can remain stable for some time. The predominant approach exploits the self-assembly of DNA base pairs. See, for example, papers in Ref. [31]. In contrast, programmed dynamic self-assembly involves components that are animate—molecular machines, for example—that are energy-driven, far-from-equilibrium, and dynamic. They work as little “construction crews” to perform a sequence of steps (a multi-line program, so to speak) to self-assemble (and possibly reconfigure) the structure of interest. For additional examples of programmed dynamic self-assembly, see Ref. [28].

6.3.

Feasibility of Experimental Realization

We selected motor proteins and microtubules as the hardware for our first dynamic self-assembly programs because many of the required features have been observed in vivo or in vitro, and therefore experimental realization seems quite feasible. The reconfigurable network simulation requires microtubules to nucleate on centrosomes, undergo dynamic instability, and become stabilized when they encounter stabilizing molecules. Microtubules have been seen to form asters on centrosomes [18,19] and undergo dynamic instability [19–22]. Microtubule-capping

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proteins [32] may serve as stabilizers. That simulation also requires molecules that control the activity of the stabilizers. Molecules that naturally regulate microtubule capping proteins may act in that role in experiments. The directed transport simulation additionally requires two types of motor proteins that can carry cargo on microtubules, one walking in the plus and the other in the minus direction. Conventional kinesin, e.g., Drosophila melanogaster KHC, walks in the plus direction. The kinesin Saccharomyces cerevisiae Kar3 walks in the minus direction. Motor proteins have been seen to carry cargo, such as pigment granules, in experiments. [23,24,33]. This simulation also requires a means of reducing the activity of each type of motor as desired. Tomishige and Vale [34] showed that it is possible to reversibly lock kinesin motors. Centrosome blockers were also included in the directed transport simulation. In experiments, anti-γ tubulin antibodies [35] might function effectively as centrosome blockers. Many of the key ingredients of these simulations have already been seen experimentally. We hope to see experimental realization of the full simulated systems in the near future.

7.

Conclusion

In this chapter, we have presented the notion that dynamic self-assembly processes of biological systems can be viewed as carrying out RAM computing. A key aspect of the RAM model in living systems is the unary representation of registers (numbers) by the count or magnitude of physical objects (like molecules, particles, or polymers). Increasing a number is achieved via self-assembly (synthesis of new molecules, transporting particles to a pile, adding a subunit to a polymer), while decreasing a number is achieved by disassembly (degradation of a molecule, transporting away from the pile, removing a subunit from the polymer). Thus, when viewed from the RAM perspective, even activities that at first blush appear to have nothing to do with information processing—activities like assembly, movement, synthesis, disassembly, and degradation—do carry out computational operations. We discussed how a dynamic self-assembly program is essentially a set of cleverly determined hardware components that, when allowed to interact, executes the desired program, by building, reconfiguring, or disassembling physical structures. Living systems have already “cleverly determined” (through evolution) appropriate sets of hardware to program the construction of countless structures and drive complex processes. Indeed, many different sets of hardware, at different hierarchy levels (e.g., molecules, cells, and organisms) can run similar “software.” We do not believe that the purpose of such programs is to perform numeric computations, but rather, to build and organize structures and maintain the system in a far-from-equilibrium (ordered) state. We have begun to understand the “programming language” of dynamic self-assembly by examining a few specific biological


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processes in terms of the RAM computing model. That is, we identified registers and their increment and decrement/jump operators, and followed the sequence of these operations to capture explicitly, in pseudocode or a flow diagram, the RAM algorithm that describes each biological process. Specifically, we examined the search-and-stabilize algorithm executed by MTs and the condensation and dispersal algorithms executed by motor proteins carrying pigment granules in melanophores. We can say that these proteins, MTs and motor proteins, are programmed to execute these behaviors (subroutines). We then took advantage of these subroutines to design our first dynamic selfassembly programs to build and reconfigure nanostructures. We combined these pre-programmed proteins with others specifically engineered to direct, control, or sequence the subroutines and demonstrated (at least in computer simulation) one program to build and reconfigure a network of MTs and another to harvest randomly distributed particles (cargoes) and then localize them to build a nano rock wall. One of the motivations for understanding the computational nature of dynamic self-assembly was to be able to program novel structures and materials. We hope to report experimental verification of these simulations in the near future. This would confirm that we really are learning to program in this very different way. In addition, if we can solve the problem of manufacturing small machines that have all the properties of protein dynamic self-assembly, then we envision being able to program life-like systems that can self-assemble, self-heal, and self-reconfigure, from totally non-biological components. The other motivation for examining dynamic self-assembly from a computational perspective was to enable us to identify larger, more complicated algorithms that living systems are executing. What algorithms enable living systems to develop, persist, reproduce, and evolve? What enabled higher levels of hierarchy, such as multi-cellular organisms and colonies of organisms, to emerge? Indeed, is the ability to execute algorithms central to the emergence of life itself? In this chapter, we identified a few common themes. (1) Living systems represent quantities directly, using unary registers. This representation can be implemented physically in a wide variety of ways, and the fundamental operations on these registers are easy to achieve physically. (2) Living systems can implement the same software with different hardware, and (3) can scale algorithms up by applying the same algorithms at different levels of hierarchy (molecules, cells, organisms). (4) Sequencing mechanisms wire together fundamental operations in a modular, evolvable manner. (5) Decisions are often made by races or competitions; the decision follows the winner—usually the faster, stronger, or more numerous. As we continue to examine biological systems in the RAM computing framework, we will continue to test and refine these themes, and search for additional themes. Over time, being able to tease out common algorithms from diverse and seemingly unrelated biophysical processes may enable us to make progress toward some aspects of ambitious questions regarding the algorithms of life.

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Acknowledgements We thank Alex Slepoy, George Bachand, and Matt Glickman for useful comments and suggestions. This work was supported in part by the Division of Materials Sciences and Engineering, Office of Basic Energy Science, U.S. Department of Energy. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

References [1] M. Conrad, K.-P. Zauner, Conformation-driven computing: A comparison of designs based on DNA, RNA, and protein, Supramolecular Science 5 (5–6) (1998) 787–790. [2] M. Conrad, Molecular and evolutionary computation: The tug of war between context freedom and context sensitivity, BioSystems 52 (1999) 99–110. [3] D. Bray, Protein molecules as computational elements in living cells, Nature 376 (1995) 307– 312. [4] A. Ben-Hur, H.T. Siegelmann, Computation in gene networks, Chaos 14 (2004) 145–151. [5] M. Conrad, Cross-scale interactions in biomolecular information processing, BioSystems 35 (1995) 157–160. [6] M.O. Magnasco, Chemical kinetics is Turing Universal, Physical Review Letters 78 (1997) 1190–1193. [7] A.P. Arkin, J. Ross, Computational functions in biochemical reaction networks, Biophysical Journal 67 (1994) 560–578. [8] P.S. Agutter, D.N. Wheatley, Information processing and intracellular ‘neural’ (protein) networks: Considerations regarding the diffusion-based hypothesis of Bray, Biology of the Cell 89 (1997) 13–18. [9] M.L. Minsky, Computation: Finite and Infinite Machines, Prentice–Hall, Englewood Cliffs, NJ, 1967. [10] A.K. Dewdney, The New Turing Omnibus, Computer Science Press, New York, 1993. [11] A.M. Bouchard, G.C. Osbourn, Dynamic self-assembly in living systems as computation, Natural Computing 5 (4) (2006) 321–362. [12] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, fourth edition, Garland Science, New York, 2002. [13] D.G. King, M. Soller, Y. Kashi, Evolutionary tuning knobs, Endeavour 21 (1997) 36–40. [14] S.-H. Park, A. Zarrinpar, W.A. Lim, Rewiring MAP kinase pathways using alternative scaffold assembly mechanisms, Science 299 (2003) 1061–1064. [15] M. Kirschner, J. Gerhart, Evolvability, Proceedings of the National Academy of Sciences USA 95 (1998) 8420–8427. [16] H.N. Sköld, S. Aspengren, M. Wallin, The cytoskeleton in fish melanophore melanosome positioning, Microscopy Research and Technique 58 (2002) 464–469. [17] B. Alberts, D. Bray, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Essential Cell Biology: An Introduction to the Molecular Biology of the Cell, Garland Publishing, New York, 1998. [18] R.E. Palazzo, J.B. Brawley, L.I. Rebhun, Spontaneous aster formation in cytoplasmic extracts from eggs of the surf clam, Zoological Science 5 (3) (1988) 603–612. [19] T. Mitchison, M. Kirschner, Microtubule assembly nucleated by isolated centrosomes, Nature 312 (5991) (1984) 232–237.


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[20] T. Mitchison, M. Kirschner, Dynamic instability of microtubule growth, Nature 312 (5991) (1984) 237–242. [21] T. Horio, H. Hotani, Visualization of the dynamic instability of individual microtubules by dark-field microscopy, Nature 321 (6070) (1986) 605–607. [22] T.J. Mitchison, Polewards microtubule flux in the mitotic spindle evidence from photoactivation of fluorescence, Journal of Cell Biology 109 (2) (1989) 637–652. [23] S.L. Rogers, I.S. Tint, P.C. Fanapour, V.I. Gelfand, Regulated bidirectional motility of melanophore pigment granules along microtubules in vitro, Proceedings of the National Academy of Sciences USA 94 (8) (1997) 3720–3725. [24] H. Nilsson, W. Steffen, R.E. Palazzo, In vitro reconstitution of fish melanophore pigment aggregation, Cell Motility and the Cytoskeleton 48 (1) (2001) 1–10. [25] J. Howard, A. Hunt, S. Baek, Assay of microtubule movement driven by single kinesin molecules, Methods in Cell Biology 39 (1993) 137–147. [26] T. Surrey, F. Nédélec, S. Leibler, E. Karsenti, Physical properties determining selforganization of motors and microtubules, Science 292 (5519) (2001) 1167–1171. [27] A.M. Bouchard, G.C. Osbourn, Dynamic self-assembly and computation: From biological to information systems, in: Biologically Inspired Approaches to Advanced Information Technology, Springer, Berlin, 2004, pp. 95–110. [28] A.M. Bouchard, C.E. Warrender, G.C. Osbourn, Harnessing microtubule dynamic instability for nanostructure assembly, Physical Review E 74 (4) (2006) 041902-1–16. [29] R. Kirsch, M. Mertig, W. Pompe, R. Wahl, G. Sadowski, K.J. Böhm, E. Unger, Threedimensional metallization of microtubules, Thin Solid Films 305 (1–2) (1997) 248–253. [30] A.K. Boal, T.J. Headley, R.G. Tissot, B.C. Bunker, Microtubule-templated biomimetic mineralization of lepidocrocite, Advanced Functional Materials 14 (1) (2004) 19–24. [31] J. Reif (Ed.), Foundations of Nanoscience: Self-Assembled Architectures and Devices, ScienceTechnica, 2004. [32] N. Galjart, CLIPs and CLASPs and cellular dynamics, Nature Reviews Molecular Cell Biology 6 (6) (2005) 487–498. [33] G. Muthukrishnan, B.M. Hutchins, M.E. Williams, W.O. Hancock, Transport of semiconductor nanocrystals by kinesin molecular motors, Small 2 (2006) 626–630. [34] M. Tomishige, R.D. Vale, Controlling kinesin by reversible disulfide cross-linking: Identifying the motility-producing conformational change, Journal of Cellular Biology 151 (2000) 1081– 1092. [35] B.R. Oakley, C.E. Oakley, Y. Yoon, M.K. Junk, γ -tubulin is a component of the spindle pole body that is essential for microtubule function in Aspergillus nidulans, Cell 61 (7) (1990) 1289–1301.

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Self-Assembled Computer Architectures Chris Dwyera,* , Alvin R. Lebeckb,** a Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA b Department of Computer Science, Duke University, Durham, NC 27708, USA

Abstract. This chapter summarizes current work on DNA-based self-assembly of computing systems. Section 2 presents a technology overview, specifically a discussion of nanoelectronic devices and desirable characteristics. It also describes two forms of DNA-based self-assembly (DNAguided and DNA-scaffolded). Section 3 then discusses the limitations of the technology and the challenges this presents to the design of computer systems. Three case studies of computer architectures are presented in Section 4. A particular emphasis is placed on articulating how the architectures overcome the challenges of the self-assembly technology. We conclude in Section 5.

1. Introduction The rising cost of conventional lithographic fabrication facilities and the predicted limits of complementary metal-oxide semiconductors (CMOS) motivate research on replacement technologies. Self-assembly of nanoelectronic components is a potential alternative to top–down lithography for the fabrication of computing systems. However, a move to self-assembly requires system designers to re-examine many of the assumptions with respect to fabrication capabilities. Further, manufacturing defects, power density, process variability, transient faults, bulk silicon limits, rising test costs and multibillion dollar fabrication facilities are some of the challenges facing the continued scaling of CMOS. While architectural modifications (e.g., multicore) can provide some short-term relief, the semiconductor industry recognizes the importance of these issues and the need to explore long term alternatives to CMOS devices and fabrication techniques. One promising alternative is deoxyribonucleic acid (DNA) based self-assembly of nanoscale components using inexpensive laboratory equipment to achieve terato peta-scale integration. Although much of this technology is in its infancy (i.e., demonstrated in research lab experiments), by studying its potential uses for building computing systems, architects can gain a deeper understanding of its * E-mail: [email protected]. ** E-mail: [email protected].



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limitations and opportunities while providing important feedback to the scientists developing the new technologies.

2.

Technology for Self-Assembled Computers

The application of nanoscale self-assembly to computer fabrication relies on two important design factors: (i) the devices, and (ii) the self-assembly process. The specific devices and self-assembly process impact the architecture of the system because of technological limitations with self-assembly. The extent and impact of these limitations is discussed in Section 3.

2.1.

Nanoscale Devices

There are many potential nanoscale devices that may eventually replace CMOS as the foundation for computer systems. In this section we focus on two types of devices that are particularly well suited for interaction with DNA-based selfassembly: crossed carbon nanotube field effect transistors and ring-gated field effect transistors. 2.1.1. Crossed Carbon Nanotube FET Studies of individual carbon nanotubes [22] have demonstrated that they have excellent electrical properties, including high electron mobility. Experiments with CNFETs [10,28,31] have further demonstrated that these devices can have large transconductances, which indicates a great potential for nanoelectronic circuits. The crossed carbon nanotube FET (CNFET) is implemented by two crossed carbon nanotubes, one metallic and the other semiconducting. The metallic nanotube acts as a gate electrode to create a local electric field that modulates the conductance through the nearby semiconducting nanotube. CNFETs behave like field effect transistors and can be used in logic circuitry [1,17,19]. Although the demonstrations of CNFET logic rely on conventional patterning to organize the carbon nanotubes (or in some cases a template for the nanotubes), this work has established the fundamental possibility of using CNFET-based circuitry in computer systems. The availability of chemical functionalization schemes to integrate CNFETs into larger structures using DNA self-assembly is one advantage carbon nanotubes have over traditional devices [4,13]. However, several challenges in carbon nanotube synthesis must be overcome before CNFETs can be used in the full scale fabrication of a computer system. The most important among these challenges is the lack of control during synthesis over the electrical properties (i.e., chirality) and lengths of the nanotubes. Several promising post-synthesis techniques are being developed to select nanotubes of a specific chirality [27,35] and length [18]. Such techniques may prove useful to self-assembled systems that are compatible with aqueous environments.

Self-Assembled Computer Architectures

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Fig. 8.1. An RG-FET. (a)–(b) Cross-section and (c) perspective views of the transistor. For colour version, see Colour Plate Section.

Interest in hybrid silicon-CNFET systems has identified new challenges in reliably nucleating nanotubes from patterned catalyst layers and the compatibility of catalyst materials with conventional silicon processes. Despite these difficulties hybrid systems are expected to be some of the first economically viable applications of carbon nanotubes in electronics. The potential of CNFETs is encouraging, but many fabrication challenges must be overcome to exploit their full potential. Therefore, investigations of alternative nanoscale devices are also being pursued. 2.1.2. Ring-Gated FET (RG-FET) Templated synthesis techniques have enabled the fabrication of free standing crystalline rod structures from common semiconducting materials including silicon [32]. The template provides control over the diameter of the rod and the growth time controls the rod length. This degree of control makes templated synthesis attractive for many applications including self-assembled computer systems. The RG-FET is a doped silicon rod fabricated using a templated synthesis method and is otherwise similar to conventional silicon FETs as illustrated in Fig. 8.1. The dopant profile along the rod axis is controlled during the growth of the rod (in the template) to create the drain, channel, and source regions. The diameter of the rod is defined by the template geometry [32]. The conductance channel is formed in the mid-section of the rod by an applied voltage on a surrounding metallic band, or ring. The ring of metal acts like a gate to modulate the conductance along the axis of the rod. The ring is isolated from the rod by a thin layer of silicon dioxide grown on the surface of the rod after synthesis similar to a conventional FET gate oxide. The ends of the rod are the drain and source and Ohmic contact must be made after assembly.

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Since the RG-FET is fundamentally a silicon FET device (after the drain and source contacts are made) it has electrical behavior similar to conventional FETs [3,14]. As with CNFETs, the similarity in behavior to conventional circuit components makes the circuit design task similar. The main body of the RG-FET is free from a substrate and can therefore diffuse in an aqueous solution (or a suspension) and be organized by DNA-guided self-assembly. Further, the substrate effects that dominate nanoscale FETs are reduced because the RG-FET is not surrounded by bulk silicon. Because of this the RG-FET has an electrical advantage over conventional FETs and consumes less dynamic power [9].

2.2.

Self-Assembled Organization

Both devices discussed in Section 2.1 are amenable to DNA-based self-assembly. In this section we briefly summarize DNA self-assembly and two approaches that use DNA to organize nanoscale components. DNA self-assembly is based on the thermodynamic behavior of complementary DNA. Complementary single stranded DNA can hybridize to form a double helix structure when the temperature of their environment crosses a critical point, or the melting temperature (Tm ). The Tm of two strands is sensitive to the nucleotide sequence and a host of other environmental factors including salt concentrations, pH, etc. The transition from single stranded (unordered) to double stranded (ordered) DNA is the fundamental property of hybridization that defines the action of self-assembly. The structural properties of double stranded DNA are an important aspect of DNA self-assembly. Single stranded DNA is free to rotate about its backbone and is therefore more flexible than double stranded DNA. The double helix constrains the geometry of the two strands and is more rigid than either strand alone. Like yarn that has been woven into a tapestry, the double stranded DNA can be woven into larger and more complex structures [16,33,34]. Figure 8.2 shows an atomic force microscope (AFM) image of the letter “A” fabricated in our lab from a DNA tile system [6]. The colorscale encodes height across the image and the specific binding of a protein on the grid creates the lettered pattern. The letter is approximately 60 nm in length on a side and we estimate that nearly 1014 formed during a single reaction. That is, the number of fabricated structures is comparable to the number of letters in all manuscripts and books held in the U.S. Library of Congress. Pure DNA self-assembly uses DNA hybridization to form structures that can, for example, be used to template other materials such as metals or semiconductors. That is, the DNA structure serves as a scaffold for the other materials and must provide a stable platform for further growth. This is distinct from DNAguided self-assembly wherein other materials are brought together by DNA hybridization but the DNA does not form a single monolithic structure.


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Fig. 8.2. An AFM scan of a DNA self-assembled letter “A”. The highlighted spots are proteins that have specifically bound to the DNA grid to form the pattern. The spacing between spots is ∼20 nm. For colour version, see Colour Plate Section.

DNA-guided self-assembly relies on chemical functionalization of other materials (e.g., a carbon nanotube, a silicon rod, colloidal particles, etc.) with DNA and subsequent DNA hybridization to organize the non-DNA materials [20,21, 23]. The DNA hybridization guides the materials to interact with specific geometric constraints. For example, a particle functionalized at one end with single stranded DNA can interact with another particle if they each carry complementary DNA as illustrated in Fig. 8.3.

3.

Challenges

The process of using DNA-based self-assembly to create nanoelectronic circuits presents several challenges that must be addressed when designing a system. The three primary aspects of the process are (i) small-scale control of placement and

Fig. 8.3. Three particles with unique but complementary DNA sequences can form a tethered system. For colour version, see Colour Plate Section.

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connectivity, (ii) large-scale randomness in node placement and interconnection, and (iii) high defect rates. These three aspects significantly impact architectural decisions, particularly since conventional architectures assume precise control at both the small and large scale.

3.1.

Technology Limitations

3.1.1. Small-Scale Control The ability of DNA-based self-assembly to achieve only small-scale control impacts architectural decisions in several ways. Three of the most significant are: limited space, limited coordination, and limited communication. 3.1.1.1. Limited Space Whether using DNA scaffolded or DNA-guided selfassembly, small-scale control of placement and connectivity will limit the size of any given circuit. This causes a commensurate limit on the functional capabilities of such a circuit. This limited node size presents a trade-off in node design. At one extreme, we could design just a single node type that contains both computation and storage capabilities. However, since the storage and computation circuits must share the node, each may be severely limited in capability. Alternatively, we could design a few specialized node types, some devoted to computation and others devoted to storage. Even when designing a specialized node, the limited space impacts architectural decisions. For example, large state machines are not an option since there is insufficient space for state storage. Similarly, the number of bits available in a storage node may be limited, thus affecting an architecture’s word size. 3.1.1.2. Limited Communication Without large-scale control, there is limited communication among nodes. In general, connections between nodes may be limited to either (i) a single shared broadcast media or (ii) a small number of point-to-point connections to a few neighboring nodes. In either case there is no long-haul communication between independent sets of nodes. Furthermore, the connections from a node to the broadcast media or to each of its neighbors is limited to a single wire. Although the degree of each node or the number of connections between neighbors could be increased, each connection occupies precious edge space. Conventional designs exploit multiple metal layers for long-haul communication and large-scale control to create multi-wire connections between components. Therefore, the architecture must avoid relying on sophisticated communication hardware. 3.1.1.3. Limited Coordination Conventional CMOS designs rely on precise control during fabrication to create sophisticated circuits (e.g., a 64-bit adder with


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carry lookahead). For our technology, if the most sophisticated node is a fulladder, then it is unlikely that 64 such nodes can be coordinated to implement a 64-bit adder. Coordination among nodes is very limited and it is difficult to a priori configure a group of nodes to operate in a coordinated manner. Each node can perform only limited coordination with its immediate neighbors. 3.1.2. Large-Scale Randomness DNA-based self-assembly provides excellent control at the small scale, however it cannot achieve such control at large scales. The resulting randomness introduces some additional issues that architectures must address. 3.1.2.1. Node Placement The self-assembly process does not guarantee where any particular node will lie in the final circuit. Each node simply attempts to connect to other nearby nodes. The architecture and machine organization must accommodate this arbitrary placement of functional blocks. 3.1.2.2. Node Orientation Similar to the random node placement, the assembly process we envision does not provide control over node orientation. Any system design must tolerate arbitrary node orientations, and cannot make a priori assumptions about the orientation. 3.1.2.3. Node Connectivity Connections between nodes are not guaranteed to succeed during self-assembly. Therefore, it is possible for any node to have between zero and four functioning connections to its neighbors. The architecture must not make any a priori assumptions about available connectivity. When combined with random orientation, it is possible for nodes to connect in a triangular shape rather than the rectangular grid one would assume with nodes that have degree four. 3.1.3. High Defect Rates An inherent aspect of any self-assembly process is defects. These fabrication defects can influence node functionality and connectivity. Some of the interconnect defects cause the above problems with connectivity. While some aspects of fabrication can reduce the likelihood of defects (e.g., purification steps or careful design of DNA tags), there will always be a significant number of defects and any architecture using these technologies must tolerate these defects.

3.2. Architectural Challenges The above discussion exposes several aspects of this fabrication technique for nanoscale circuits that must be addressed by any architecture and its corresponding implementations. In this subsection, we enumerate several important challenges to developing an appropriate architecture for this emerging technology.

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This list is not meant to be exhaustive, but rather to highlight some important challenges. 3.2.1. Node Design The architect must decide what functionality to place in each node. Should there be homogeneous nodes or heterogeneous nodes? If heterogeneous, then what types of nodes? How does node design affect connectivity/communication with other nodes, and what primitives should be provided? 3.2.2. Utilizing Multiple Nodes Since individual nodes do not contain sufficient computation and storage to perform much useful work in isolation, then an architect must determine how to exploit multiple nodes. This must be achieved given the above limitations on coordination, communication, placement, orientation, and connectivity. 3.2.3. Routing with Limited Connectivity Traditional routing techniques may not directly apply since there is limited space for the complexity of dynamic routing and there are insufficient guarantees on node placement and connectivity to use conventional static routing. The designer must develop a routing technique that overcomes these challenges. 3.2.4. Developing an Execution Model The execution model embodies the software visible aspects of the architecture and can be influenced by implementation constraints or instruction set requirements. For the envisioned fabrication technique, the execution model must overcome the severe implementation constraints outlined above while still enabling a reasonable instruction set. 3.2.5. Developing an Instruction Set Programmable systems require an interface that enables software to specify operations. Typically this is achieved by the instruction set architecture (ISA). The ISA may be influenced by the underlying capabilities of the technology. Given our fabrication technique, the architect must design an appropriate ISA that supports the above execution model. 3.2.6. Providing a Memory System Storage is a crucial component of most computing systems regardless of the execution model. The ability to retain values for future use and to name and find particular values is necessary for most computing paradigms. 3.2.7. Interfacing to the Microscale An important aspect of any nanoscale system is the interconnection to larger-scale components (e.g., microscale). This connection is necessary for at least providing an I/O interface for communication with the outside world. It may be possible for the architecture to exploit this interface in other ways.


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The challenge is to address each of these issues such that we arrive at a functioning system. There are likely many possible approaches to developing a functioning system. In the next section we present case studies of three proposed architectures.

4.

Architecture Case Studies

The previous section outlined many of the challenges that architectures for DNAbased self-assembly must address. This section summarizes three different approaches for creating functioning systems that exploit the potential capabilities of DNA-based self-assembled nanocomputers while overcoming the many challenges. Section 4.1 describes the Oracle, a novel computer system that performs much of its computation during the assembly process. The Decoupled Array Multi-Processor (Section 4.2) extends the oracle concept to enable slightly more general purpose computing with a SIMD style instruction set. Finally, Section 4.3 presents the Nanoscale Active Network Architecture that provides a generalpurpose instruction set.

4.1. Oracle An example system that can overcome some of the challenges mentioned in Section 3 is the proposed Oracle computer [7,8]. In this system, each self-assembled node is the solution to a predefined computational problem. For example, an Oracle for n-bit addition would self-assemble nodes that represent each of the possible n-bit additions the system can perform as well as their solutions. An Oracle computer performs virtually all of its computation at-fabrication [5]. An Oracle can be assembled using electrically active components (e.g., silicon rods) and DNA (metallized during post-processing) to form large arrays of simple circuitry. The Oracle is designed to solve a large portion of a particular problem during fabrication rather than relying on purely run-time computation. That is, the Oracles are a hybrid between general-purpose electrical designs and electrical circuitry inspired by DNA computing. Abstractly, an Oracle contains a large number of question and answer pairs. Questions are posed to the machine and it generates a response if the question is contained in any of the Oracle’s question/answer pairs. In this fashion, the Oracle is like a large content-addressable memory (CAM) that has been preloaded with the answers to a certain problem. An Oracle differs from a conventional CAM by the method in which the question and answer pairs are entered into the memory. To fully cover an input space of k-bits, a conventional CAM requires O(2k ) steps to load the answers (each of which must be computed). Each address represents a k-bit question and the associated data field the answer, like a lookup table. In comparison, the Oracle requires O(k) steps to assemble and no

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Fig. 8.4. The tiles for a 2D addition Oracle. Each entry in the full adder truth table is translated into one tile using the legend above. The carry-in (Ci) and carry-out (Co) bits must match between tiles for correct addition. Each tile encodes a carry bit in–out relationship by a single- or double-peak that fits into a single- or double-valley. Figure 8.5 illustrates a legal arrangement of tiles to form an addition string. For colour version, see Colour Plate Section.

run-time loading steps. The answers are determined by the manner in which the Oracle is assembled (e.g., through DNA hybridization that follows a rule set as in DNA computing). The self-assembly of each question and answer pair provides the Oracle with the answers (with a high probability but not a certainty that a given question and answer pair will exist within the Oracle). If a particular question and answer pair did not form during the Oracle’s assembly then the Oracle cannot solve that instance of the problem. For example, Fig. 8.4 shows the tile set and truth table for an addition Oracle. The tiles are directly mapped from the truth table following the key at the top of Fig. 8.4. A large number of the tiles (2k where k is the number of bits being added) are mixed and made to assemble into strings. Since the tiles can only bind according to the truth table, as represented by the shapes on the ends of each tile, each valid string is an instance of the addition problem. Figure 8.5 shows a particular (i.e., randomly chosen) string from the result. This particular string encodes the addition “3 + 5 = 8”. A more detailed treatment of the tile implementation and addition Oracle can be found in [7]. While the Oracle may be an interesting theoretical twist on DNA computing, the set of problems that can be solved with this technique are limited. Unfortunately, many of the problems (e.g., path problems, combinatorial graph problems, etc.) that are amenable to being solved by an Oracle have powerful conventional


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Fig. 8.5. An example string from the addition Oracle using the tiles defined in Fig. 8.4. For colour version, see Colour Plate Section.

approximations that are in many cases sufficiently accurate, though not optimal, for real applications.

4.2.

Decoupled Array Multi-Processor (DAMP)

The DAMP is an architecture that addresses some of the challenges discussed in Section 3 but can be used to solve a slightly wider field of problems than Oracles. This architecture is a parallel computer suitable for solving parallel problems that do not require large I/O bandwidth. The device architecture assumed for the DAMP is based on nanoscale rod/tube components assembled by DNA-guided self-assembly [8,9]. Each type of component can be individually assigned to a specific location in the circuit through DNA self-assembly. Further, the DNA junctions between components can be metallized to create Ohmic contacts between the rod ends. The DAMP is a single-instruction multiple-data (SIMD) system design that uses simple processing elements (PEs) in parallel to solve general optimization problems (e.g., the thermal intercept problem, or any problem with a succinct objective function). However, the constraints placed on the design by the selfassembly technology change the fundamental assumptions about how the system can work. For instance, the design relies on a single input and output channel that is shared between all PEs. This restrictive I/O design is indeed a performance

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Fig. 8.6. A single processing element (PE) from the DAMP. R0, . . . , R4 are general purpose registers and ACC is a serial accumulator. The status bits “BCDRSW” are set/cleared per the definition of each ALU operation. Registers with “∗” labels can be uniquely randomized at power-up. For colour version, see Colour Plate Section.

bottleneck but eliminates the challenge of interfacing a high density nanoscale device array with microscale circuitry. Feedback between the device and architecture levels identifies where such constraints are necessary to make the system feasible. The design approach involved modifying (severely) a typical assumption about I/O connectivity because the underlying technology did not readily support such connectivity. The scale of device integration supported by self-assembly is another example of a technological constraint that has dramatic implications on the computer architecture. Reduced levels of integration constrain the logical size of a PE, i.e., fewer devices per PE mean simpler functionality per PE. Figure 8.6 shows the schematic of a single PE. Through bit-serial operations the PE can manipulate an accumulator and six registers, each 16 bits long. The only way high performance can be achieved with the DAMP is if there are many PEs, even if they are simple. Self-assembly has the potential to organize molar-scale numbers of components which amount to ∼1012 to ∼1019 assembled devices (at 100 µM concentrations) depending on process yield and defect rates. This requires design feedback between the device layer and the architecture or the final design will be impractically large for a single PE. For example, each PE in the DAMP uses ∼1500 RG-FETs to implement a register file, an accumulator, and a set of simple bit-serial operations. The peak performance of the full DAMP (with ∼1012 PEs) on simple, local (i.e., no I/O involved) ALU operations is compared to some other machines in Table 8.1. It is clear at this point that the DAMP will do well on simple parallel problems but the severe I/O bandwidth penalty restricts the DAMP such that it will not


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Table 8.1. Peak performance comparison Machine

Peak ALU (16-bit) operations per second

DAMP IBM BlueGene/L NEC Earth Simulator Intel P4

∼1018 ∼1015 ∼1015 ∼1011

be suitable for a large array of general purpose applications. High performance general purpose computing requires a much larger degree of information sharing than can be accommodated by the technological assumptions underlying the DAMP.

4.3.

Nanoscale Active Network Architecture (NANA)

Our final approach for addressing the issues raised in Section 3, is an active network architecture that is compatible with our fabrication technology, specifically the DNA-scaffolded self-assembly [25]. The architecture is like an active network [29] in that execution packets that contain instructions and operands search through a loosely structured sea of processing and memory nodes for the functionality that they need at each step of execution. This architecture matches our technology characteristics since it (i) allows for differing node types with specialized functionality, (ii) tolerates a random interconnection of nodes, and (iii) tolerates node and interconnect fabrication defects. 4.3.1. System Model The system model is a random interconnection of various node types, in which all nodes contain circuitry for communication and each node has some specialized circuitry (e.g., processing, memory, etc.). Groups of nodes are organized into cells. A node communicates with a neighboring node via a single link that is asynchronous and bidirectional (time-multiplexed on a single physical wire). Each cell has a viaduct that is its connection to a microscale, and one of the many nodes connected to the viaduct acts as the anchor node for the cell. Inter-cell communication occurs through a microscale interconnection network. The memory nodes in each cell comprise a portion of the global memory space. Some fraction of memory nodes are configured as memory ports to provide an interface between execution packets and memory storage. Figures 8.7 and 8.8 illustrate our system model and physical model, respectively. To impose structure on the interconnection network and the memory system, there is a configuration phase that occurs before any execution [26]. Reconfigurable architectures [2,11,12,24] have demonstrated that this approach is important to achieve high performance in the context of highly focused (i.e., aggressive) or highly defective technologies, including nanotechnology.

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Fig. 8.7. One cell in the NANA system. Each node is labeled either “A” for anchor, “M” for memory, “M∗ ” for memory port, or “P” for processing. The block “V” is the microscale contact. Distinct cells can operate in parallel. For colour version, see Colour Plate Section.

Fig. 8.8. Synthetic rendering of the physical model for the NANA system. The waffle-shaped structures are nodes and the bulbous linear structures are links in the interconnection network. For colour version, see Colour Plate Section.

While node functionality is heterogeneous, all nodes have some common responsibilities. Each node generates its own local clock (10 GHz) and communicates asynchronously with its neighboring nodes using signaling techniques similar to push-style pipeline systems. High-level communication between two devices over a single wire can be managed using simple two- and four-phase single wire techniques [30]. Each node must also contain routing functionality for determining the outgoing link for an incoming packet (or the result of an opera-


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tion). This circuitry maintains node state (e.g., currently processing a packet) and handles link contention. 4.3.2. Execution Model The execution model relies on an accumulator-based ISA. Conceptually, the accumulator is initialized and then a sequence of operations are performed on the corresponding series of operands. The accumulatorbased ISA reduces the need for widespread a priori coordination and communication among many components (e.g., associative lookup in issue queues), since the only data dependence involves the accumulator and instructions are processed in order [15]. We support the accumulator-based execution by forming an execution packet that contains the operations, the accumulator, and all operands in appropriate order. The packet searches within a cell for the requisite functional units (or memory ports) in the order specified by the operation sequence. Logically, each functional unit performs its specified operation, removes the operand and forwards the new accumulator and the remaining operands to the subsequent functional units. Each subsequent functional unit performs a similar sequence until all operations in the packet are completed. Memory operations generate memory packets that are handled by the memory ports. To augment the defect tolerance of configuration and to protect against transient faults we could add a signature vector to each packet and verify the integrity of a computation performed by the packet. The signature vector is operated on like the operands field of a regular execution packet with the exception that the initial signature is not consumed by the operation. The order of instructions will be reflected by a characteristic signature vector and can be used to determine if the nodes performing those operations were functioning properly during the signature calculation. This approach can be further augmented with redundant execution packets and a voting mechanism. The system and execution model enable significant parallelism by instantiating multiple execution packets within a cell and in multiple cells. While this parallelism is an important aspect of the architecture that fully exploits the capabilities of the underlying technology, this work focuses primarily on the operation of a single cell and sequentially instantiate execution packets. 4.3.3. Evaluation We have implemented two different programs for execution on NANA: Fibonacci and string match. To demonstrate our system operation, we simulate its behavior at the bit serial link level executing the two programs. We model a single 32 × 32 cell and include a random distribution of 3% defective nodes, 25% ALU nodes, and we configure 64 16-bit memory nodes and 80 memory ports. The average time per loop iteration in Fibonacci is 2964 cycles, loop unrolling may improve this. While the absolute performance of this example does not surpass even current CMOS, it serves to demonstrate the operation of a single node. The greatest advantage of this technology arises from the scale of the system.

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The opportunity for massively parallel computation with self-assembled systems is tremendous. String searching is a common operation in many applications (e.g., searching for particular DNA sequences within a genome). Our string match program loads a 16-bit key and compares it to all data elements within the cell. A full self-assembled system could search a 32 GB database by using all 109 cells. The execution time within one cell is 35 ns per comparison, for a total of 28.5 × 106 comparisons/sec. The potential for massive parallelism would be exposed by having each of the 109 cells perform a unique comparison, yielding an overall rate of 2.85 × 1016 comparisons/sec. Developing these programs exposes the limitations of our current architecture: (1) underutilization of nodes and network connectivity, and (2) the memory system is a bottleneck. The peak performance of NANA, assuming 1/2 the nodes compute, is significantly larger than today’s supercomputers: NANA achieves 4.12 × 1021 bitops/sec, IBM BlueGene 4.6 × 1016 bitops/sec and NEC ES 5.2 × 1015 bitops/sec.

5.

Conclusions

The potential technological advantages of DNA self-assembly over conventional silicon-based photolithography have motivated intense study in this field. However, there are serious challenges in the device synthesis and materials science behind self-assembly that couple the design of higher-level computer architectures to the technology. The severe constraints placed on the architecture by the technology require a methodical re-evaluation of conventional design principles to resolve unanswered questions about the feasibility of such systems. The fundamental challenge to self-assembled architectures stems from a loss of control over the composition, placement, and orientation of the computing devices (e.g., logic gates, FETs, interconnect). The technological limitations of self-assembly require the architecture to cope in unconventional ways to overcome limited device capacity, communication bandwidth, high defect rates, and general randomness during fabrication. We have presented three architectures that overcome many of these challenges for both a DNA-guided and DNA scaffolded self-assembly technology. However, these systems are only suitable to a small set of application domains and further optimization is required to achieve high performance general-purpose computing.

References [1] A. Bachtold, P. Hadley, T. Nakanishi, C. Dekker, Logic circuits with carbon nanotube transistors, Science 294 (November 2001) 1317–1320. [2] P. Beckett, A. Jennings, Toward nanocomputer architecture, in: Proceedings of the Seventh Asia-Pacific Computer Systems Architecture Conference, 2002, pp. 141–150.


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[3] Y. Cui, C.M. Lieber, Functional nanoscale electronic devices assembled using silicon nanowire building blocks, Science 291 (February 2001) 851–853. [4] C. Dwyer, M. Guthold, M. Falvo, S. Washburn, R. Superfine, D. Erie, DNA functionalized single-walled carbon nanotubes, Nanotechnology 13 (2002) 601–604. [5] C. Dwyer, A.R. Lebeck, D.J. Sorin, Self-assembled architectures and the temporal aspects of computing, IEEE Computer 38 (1) (January 2005) 56–64. [6] C. Dwyer, S.H. Park, T. LaBean, A. Lebeck, The design and fabrication of a fully addressable 8-tile DNA lattice, in: Foundations of Nanoscience: Self-Assembled Architectures and Devices, April 2005, pp. 187–191. [7] C. Dwyer, J. Poulton, R. Taylor, L. Vicci, DNA self-assembled parallel computer architectures, Nanotechnology (2004) 1688–1694. [8] C. Dwyer, L. Vicci, J. Poulton, D. Erie, R. Superfine, S. Washburn, R.M. Taylor, The design of DNA self-assembled computing circuitry, IEEE Transactions on VLSI 12 (April 2004) 1214– 1220. [9] C. Dwyer, L. Vicci, R.M. Taylor, Performance simulation of nanoscale silicon rod field-effect transistor logic, IEEE Transactions on Nanotechnology 2 (2) (June 2003) 69–74. [10] M.S. Fuhrer, J. Nygard, L. Shih, M. Forero, Y.-G. Yoon, M.S.C. Mazzoni, H.J. Choi, J. Ihm, S.G. Louie, A. Zettle, P.L. McEuen, Crossed nanotube junctions, Science 288 (April 2001) 494–497. [11] S.C. Goldstein, M. Budiu, NanoFabrics: Spatial computing using molecular electronics, in: Proceedings of the 28th Annual International Symposium on Computer Architecture, July 2001, pp. 178–191. [12] J. Han, P. Jonker, A defect- and fault-tolerant architecture for nanocomputers, Nanotechnology 14 (January 2003) 224–230. [13] M. Hazani, F. Hennrich, M. Kappes, R.N. Naaman, D. Peled, V. Sidorov, D. Shvarts, DNAmediated self-assembly of carbon nanotube-based electronic devices, Chemical Physics Letters 391 (2004) 389–392. [14] Y. Huang, X. Duan, Y. Cui, L.J. Lauhon, K.-H. Kim, C.M. Lieber, Logic gates and computation from assembled nanowire building blocks, Science 294 (November 2001) 1313–1317. [15] H.-S. Kim, J.E. Smith, An instruction set and microarchitecture for instruction level distributed processing, in: Proceedings of the 29th Annual International Symposium on Computer Architecture, May 2002. [16] T.H. LaBean, H. Yan, J. Kopatsch, F. Liu, E. Winfree, J.H. Reif, N. Seeman, Construction, analysis, ligation, and self-assembly of dna triple crossover complexes, Journal of the American Chemistry Society 122 (2000) 1848–1860. [17] X. Liu, C. Lee, C. Zhoua, J. Han, Carbon nanotube field-effect inverters, Applied Physics Letters 79 (20) (November 2001) 3329–3331. [18] S.R. Lustig, E.D. Boyes, R.H. French, T.D. Gierke, M.A. Harmer, P.B. Hietpas, A. Jagota, R.S. McLean, G.P. Mitchell, G.B. Onoa, K.D. Sams, Lithographically cut single-walled carbon nanotubes: Controlling length distribution and introducing end-group functionality, Nano Letters 3 (8) (August 2003) 1007–1012. [19] R. Martel, V. Derycke, J. Appenzeller, S. Wind, P. Avouris, Carbon nanotube field-effect transistors and logic circuits, in: Proceedings of the 39th Design Automation Conference, June 2002, pp. 94–98. [20] B.R. Martin, D.J. Dermody, B.D. Reiss, M. Fang, L.A. Lyon, M.J. Natan, T.E. Mallouk, Orthogonal self-assembly on colloidal gold-platinum nanorods, Advanced Materials 11 (12) (August 1999) 1021–1025. [21] J.K.N. Mbindyo, B.D. Reiss, B.R. Martin, C.D. Keating, M.J. Natan, T.E. Mallouk, DNAdirected assembly of gold nanowires on complementary surfaces, Advanced Materials 13 (4) (February 2001) 249–254.

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[22] P.L. McEuen, M.S. Fuhrer, H. Park, Single-walled carbon nanotube electronics, IEEE Transactions on Nanotechnology 1 (1) (March 2002) 78–85. [23] C.A. Mirkin, R.L. Letsinger, R.C. Mucic, J.J. Storhoff, A DNA-based method for rationally assembling nanoparticles into macroscopic materials, Nature (August 1996) 607–609. [24] K. Nikolic, A. Sadek, M. Forshaw, Fault-tolerant techniques for nanocomputers, Nanotechnology 13 (2002) 357–362. [25] J.P. Patwardhan, C. Dwyer, A.R. Lebeck, D.J. Sorin, Circuit and system architecture for DNAguided self-assembly of nanoelectronics, in: Foundations of Nanoscience: Self-Assembled Architectures and Devices, April 2004, pp. 344–358. [26] J.P. Patwardhan, C. Dwyer, A.R. Lebeck, D.J. Sorin, Evaluating the connectivity of selfassembled networks of nano-scale processing elements, in: IEEE International Workshop on Design and Test of Defect-Tolerant Nanoscale Architectures (NANOARCH’05), May 2005, pp. 2.1–2.8. [27] M.S. Strano, M. Zheng, A. Jagota, G.B. Onoa, D.A. Heller, P.W. Barone, M.L. Usrey, Understanding the nature of the DNA-assisted separation of single-walled carbon nanotubes using fluorescence and Raman spectroscopy, Nano Letters 4 (4) (2004) 543–550. [28] S. Tans, A. Verschueren, C. Dekker, Room-temperature transistor based on a single carbon nanotube, Nature 393 (1998) 49–52. [29] D.L. Tennenhouse, D.J. Wetherall, Towards an active network architecture, Computer Communication Review 26 (2) (1996). [30] K. van Berkel, A. Bink, Single-track handshake signaling with application to micropipelines and handshake circuits, in: Proceedings of the Seconds International Symposium on Advanced Research in Asynchronous Circuits and Systems, March 1996, pp. 122–133. [31] S.J. Wind, J. Appenzeller, R. Martel, V. Derycke, P. Avouris, Vertical scaling of carbon nanotube field-effect transistors using top gate electrodes, Applied Physics Letter 80 (May 2002) 3817–3819. [32] Y. Wu, Y. Cui, L. Huynh, C.J. Barrelet, D.C. Bell, C.M. Lieber, Controlled growth and structures of molecular-scale silicon nanowires, Nano Letters 4 (3) (2004) 433–436. [33] H. Yan, T.H. LaBean, L. Feng, J.H. Reif, Directed nucleation assembly of barcode patterned DNA lattices, Proceedings of the National Academy of Sciences 100 (14) (July 2003) 8103– 8108. [34] H. Yan, S.H. Park, G. Finkelstein, J.H. Reif, T.H. LaBean, DNA templated self-assembly of protein arrays and highly conductive nanowires, Science 301 (5641) (September 2003) 1882– 1884. [35] M. Zheng, A. Jagota, E.D. Semke, B.A. Diner, R.S. McLean, S.R. Lustig, R.E. Richardson, N.G. Tassi, DNA-assisted dispersion and separation of carbon nanotubes, Nature Materials 2 (May 2003) 338–342.

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Simulation of Self-Assembly Processes Using Abstract Reduction Systems Jean-Louis Giavitto* , Antoine Spicher** IBISC, FRE 2873 CNRS – Université d’Évry, Génopole, 523 place des Terrasses de l’Agora, 91000 Évry, France url: http://mgs.ibisc.univ-evry.fr Abstract. We present in this chapter the use of MGS for the abstract modeling and simulation of some self-assembly processes. MGS is a declarative and rule-based language dedicated to the modeling and the simulation of various morphogenetic and developmental processes. Such systems can be seen as dynamical systems but with a dynamical structure. This property is often exhibited by self-assembly processes. The MGS approach relies on the introduction of a topological point of view on various data structures called topological collections. This topological approach enables a uniform handling of these data structures by a new kind of rewriting rules called transformations. Using (local) rewriting rules to specify self-assembling processes is particularly adequate because it mimics closely the incremental building mechanism of the real phenomena. The MGS approach is illustrated on the fabrication of a fractal pattern, a Sierpinski triangle, using two approaches: by accretive growth and by carving. More generally, the notions of topological collections and transformations available in MGS enable the easy and concise modeling of cellular automata on various lattice geometries as well as more arbitrary constructions of multi-dimensional objects.

1. Introduction Self-assembly is a process that creates incrementally complex hierarchical spatial structures. Nature presents a lots of examples, ranging from crystallization in physics to morphogenesis in developmental biology. There is no unified general theory of self-assembling, nor a unique definition. However, understanding the principles underlying self-assembly processing will open entire new opportunities for our technological capabilities. In this chapter, self-assembled systems can be thought to be built of basic building elements (molecules, cells, etc.); together these basic elements exhibit a new, often highly, complex behavior. For a computer scientist, self-assembly processes are particularly inspiring because the dynamic organization of the involved entities emerges from many decentralized and local interactions that occur concurrently at several time and * E-mail: [email protected]. ** E-mail: [email protected].



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space scales. As a matter of fact, they have inspired several new computational models like amorphous computing [1] or autonomic computing [8]. The emergence of the global structure of self-assembled systems cannot be deduced from the individual composing elements. Simulation models are often the only available way to obtain a deeper insight of these complex systems. However, the modeling and the simulation of self-assembly can be very difficult to achieve, because of the representation of the underlying space and of the handling of complex spatial structures built in this space.

1.1.

Self-Assembly by Accretive Growth and by Carving

A central thema in the research in self-assembly processes is the organizational principles that can be used to structure a population of basic elements. The structure is incrementally built and often corresponds to a spatial structure. In this chapter we will focus on the modeling of two kinds of self-assembly that rely on the addition or the removal of materials. That is, we do not consider rearrangement of matter, nor its modification (e.g. by stretching). Our approach has a combinatorial nature and hide the burden of the physical implementation. 1.1.1. Self-Assembly by Accretive Growth One of the most fundamental kind of self-assembly is certainly processes where basic elements are aggregated into a shape during a growth process. An incremental growth process can be described as the iteration of a basic aggregation: in each growth stage, new basic entities (e.g., material) are added to the preceding growth stage [9]. The aggregation depends on the available material and also on the form of the object in the preceding growth stage. We use the term accretive growth to qualify a growing process that takes place on the boundaries of the system. This kind of growth is to oppose to “intercalary growth” where the growing process is from the inside of the assembly. 1.1.2. Self-Assembly by Carving Manca et al. have introduced a somewhat unusual type of computation strategy called computation by carving [11]. The idea is to generate a (large) set of candidate solutions of a problem, then remove the non-solutions such that what remains is the set of solutions. This idea to remove unwanted elements is also present in building shapes by space carving [10], an algorithm to compute a volume that is consistent with a set of photos of a 3D shape. Transposed in the domain of self-assembly, this leads to the idea to iteratively remove elements, starting from an initial shape. A better name is perhaps self-disassembling.

Simulation of Self-Assembly Processes

1.2.

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Domain Specific Languages for the Simulation of Self-Assembly

As noted above, the simulation of self-assembly can be very difficult to achieve. In this work, we advocate the use of a domain specific language (DSL) for the modeling and the simulation, in an abstract and uniform setting, of accretive growth and carving. DSLs are specially tailored programming languages designed for solving problems in a particular domain. To this end, a DSL provides abstractions and notations for the domain at hand. DSLs are usually small, and more declarative than imperative. Moreover, DSLs are more attractive for programming in the dedicated domain than general-purpose languages because of easier programming, systematic reuse, better productivity and flexibility. Our approach relies on two dedicated notions: – dedicated data-structures, called topological collections, are used to represent the space underlying a self-assembly process and/or the self-assembled system; and – rewriting rules on topological collection, called transformations, are used to implement the local evolution rules usually used to specify the self-assembly process. These two notions are studied in an experimental programming language called MGS. MGS is a vehicle used to investigate the notions of topological collections and transformations and to study their adequacy to the simulation of various biological and self-assembly processes [3,5]. Using (local) rewriting rules to specify the self-assembling process is particularly adequate because it mimics closely the incremental building mechanism of the real phenomenon. In addition, the declarative style enabled by rule-based programming corresponds to a mathematical specification of the self-assembly process, which opens the way to mathematical reasoning and proofs.

1.3.

Organization of the Chapter

The rest of this chapter is organized as follows. The next section provides a quick introduction to MGS. Group-based data fields are sketched: they are topological collections used to define various lattices used in the modeling of accretive growth. Section 3 presents three short and well-known examples of growth by aggregation processes in MGS. Section 4 details the self-assembly of Sierpinski triangles. Section 5 introduces the notion of abstract cellular complexes and their handling in MGS. Cellular complexes are used to model arbitrary shape for growth and carving. The use of abstract cellular complexes is illustrated to sketch the 3-dimensional assembly of proteins. The process sketched is rather abstract and does not refer to a physical example. However it gives a good flavor of the possibilities offered by abstract cellular complexes and the very concise style

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enabled by the MGS transformations. In Section 7 we build again Sierpinski triangles but in 3 dimensions and using a carving process. The conclusion reviews some previous, related and future work.

2.

A Short MGS Presentation

In this section, we present the notions needed to understand the MGS coding of the following computation processes. MGS is a declarative programming language aimed at the representation and manipulation of local transformations of entities structured by abstract topologies [3]. A set of entities organized by an abstract topology is called a topological collection. Topological means here that each collection type defines a neighborhood relation specifying both the notion of locality, path and sub-collection. A path is a finite sequence of elements ei where ei+1 is a neighbor of ei . A sub-collection B of a collection A is a subset of elements of A defined by some path and inheriting its organization from A. The global transformation of a topological collection C consists in the parallel application of a set of local transformations. A local transformation is specified by a rewriting rule r that specifies the change of a sub-collection. The application of a rewrite rule β ⇒ f (β, . . .) to a collection A: (1) selects a sub-collection B of A whose elements match the pattern β, (2) computes a new collection C as a function f of B and its neighbors, and (3) specifies the insertion of C in place of B into A. The collection types can range in MGS from totally unstructured with sets and multisets to more structured with sequences, “group-based data fields” and “abstract cellular complexes”. There are two kinds of patterns that can be used in a transformation for selecting a sub-collection: path patterns and patch patterns.

2.1.

Path Patterns

Path patterns match paths in a collection. A path pattern is a sequence of elements separated by a comma. Path pattern x,y defines a path of two elements, where y must be a neighbor of x. Arbitrary condition can be tested using guards inserted in a path pattern: (x/x>0),(y/y>x) matches two elements x and y such that the value of x is strictly positive and y is a neighbor of x and the value of y must be greater than the value of x.

2.2.

Patch Patterns

Patch patterns allow the matching of arbitrary sub-collection. A patch pattern is specified using a set of clauses. We will present the patch pattern features we need on Section 7.


2.3.

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Group-Based Data Field

Group-based data fields (GBF in short) are used to define topological collections with uniform neighborhood. A GBF is an extension of the notion of array, where the values are indexed by the elements of a group, called the shape of the GBF [4]. The elements of the group are called the positions of the GBF. For example: gbf Grid2 = < north, east >

defines a GBF collection type called Grid2, corresponding to the von Neumann neighborhood in a classical array (a cell above, below, left or right—not diagonal). The two names north and east refer to the directions that can be followed to reach the neighbors of an element. These directions are the generators of the underlying group structure. The right-hand side (r.h.s.) of the GBF definition gives a finite presentation of the group structure. The list of the generators can be completed by giving equations that constraint the displacements in the shape: gbf Hex2 = < east, north, northeast; east + north = northeast >

defines an hexagonal lattice that tiles the plane, see Fig. 9.2. Each cell has six neighbors (following the three generators and their inverses). We use an additive notation for the group operation. The equation east + north = northeast specifies that a move following northeast is the same as a move following the east direction followed by a move following the north direction. In this chapter we will use only commutative groups, that is, commutation equations between generators (like east + north = north + east) always hold implicitly. For convenience, we identify the type of a GBF with the presentation of the underlying group. A GBF g of type G can be formalized as a partial function g from the group specified by G to some set of values: g associates a value to some positions. In other words, the group elements act as indices of a generalized array. An empty GBF is the everywhere undefined function. A special constant can be used in path patterns to match a position in a GBF that has no associated value. The topology of the collections of type G is easily visualized as the Cayley graph G of G: each vertex in the Cayley graph is an element of the group G and vertices x and y are linked by an edge labeled by u if there is a generator u in the presentation of G such that x + u = y. The relationships between Cayley graphs and group theory are pictured in Fig. 9.1. A word (a sum of generators) is a path. Path composition corresponds to the group addition. A closed path (a cycle) is a word equal to e (the identity of the group, denoted also by 0). An equation v = w can be rewritten v − w = e and then corresponds to a cycle in the graph. There are two kinds of cycles in

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Fig. 9.1. Graphical representation of the relationships between Cayley graphs and group theory. The GBF pictured is Grid2 where, for brevity, a is used to denote north and b is used to denote east. For colour version, see Colour Plate Section.

the graph: the cycles that are present in all Cayley graphs and corresponding to group laws (intuitively: a backtracking path like east + north - north - east) and closed paths specific to the own group equations (e.g.: east north - east + north). The graph connectivity (there is always a path going from P to Q) is equivalent to say that there is always a solution x to equation P + x = Q.

3. 3.1.

Aggregation Processes in MGS Eden’s Process

We start with a simple model of growth sometimes called the Eden model [2]. The model has been used since the 1960s as a model for such things as tumor growth and growth of cities. In this model, a 2D space is partitioned in empty or occupied cells. We start with only one occupied cell. At each step, occupied cells x with an empty neighbor are selected, and the corresponding empty cell is made occupied (by a copy of the content of x). The Eden’s aggregation process is simply described as the following MGS global transformation: trans Eden = { x, => x, x }

The r.h.s. of the rule lists a set of values: in a path transformation, the first value in the r.h.s. replaces the first value matched in the left-hand side (l.h.s.), the second value in the r.h.s. replaces the second value matched in the l.h.s., etc.


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Fig. 9.2. Eden’s model on an hexagonal mesh (initial state, and states after 3 and 7 time steps). This shape corresponds to the Cayley graph of Hex2 with the following conventions: a vertex is represented as a face and two neighbors in the Cayley graphs share an edge in this representation. An empty cell has an undefined value. Only a part of the infinite domain is figured.

3.2. The Growth of a Snowflake A crystal forms when a liquid is cooled below its freezing point. Crystals start from a seed and then grows by progressively adding more molecules to their surface. As an idealization, the molecules of a snowflake lie on an hexagonal grid and when a piece of ice is added to the snowflake, the heat released by this process inhibits the addition of ice nearby. This phenomenon leads to the following cellular automata rule [18]: a black cell (value 1) represents a place of the crystal filled with ice and a white cell (value 0) is an empty place. A white cell becomes black if it has exactly one black neighbor, otherwise it remains white. The corresponding MGS transformation is: trans SnowFlake = { 0 as x / 1 == FoldNeighbor[+,0](x) => 1 }

The construction path as x enables the naming of a sub-collection matched by path: here the identifier x refers to some occurrence of 0 in the GBF. The construct FoldNeighbor is not a function but an operator available only within a rule: it enables to fold a function on the defined neighbors of an element matched in the l.h.s. Here, this operator is used to compute the number of neighbors with value of x (the accumulating function is the sum and the initial value is 0). This transformation acts on a value of type Hex2 and a possible run is illustrated in Fig. 9.3.

3.3. Diffusion Limited Aggregation In a diffusion limited aggregation process, or DLA [17], a set of particles diffuse randomly on a given spatial domain. Initially one particle, the seed, is fixed. When a mobile particle collides a fixed one, they stick together and stay fixed.

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Fig. 9.3. Formation of a snowflake. The pictured states are the step at time states 1, 4, 8, 12, 16, 18, 20 and 23.

This process leads to a simple lattice gas automata that could be easily done in MGS using topological collections and transformation: trans dla = { ‘mobile, ‘fixed => ‘fixed, ‘fixed ‘mobile, => , ‘mobile }

We use two symbols ‘mobile and ‘fixed to represent respectively a mobile and a fixed particle (MGS’s symbols are like Lisp’s atoms). The two rules of the transformation deal with: (1) the aggregation: the first rule specifies that if a diffusing particle is the neighbor of a fixed one, then it becomes fixed (at the current position); (2) the diffusion: if a mobile particle is the neighbor of an empty place (position), then it may leave its current position to occupy the empty neighbor (and its current position is made empty). Note that the order of the rules is important because, following the rule application semantics of MGS, the first one has priority over the second. Figure 9.4 presents the final state of the application of the transformation dla on two kinds of topological collections: On the left, the neighborhood relationship is homogeneous and a GBF is used. On the right, the dla transformation is applied on a meshed sphere modeled as an abstract complex, cf. Section 5 below. The elements are the facets, and two facets are neighbors if they share an edge. For more details, refer to [15].

4.

Accretive Growth of Sierpinski Triangles

The Sierpinski triangles (ST from now on) is a fractal described by Waclaw Sierpinski (Polish mathematician, 1882–1969) in 1915 but appearing at least back


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Fig. 9.4. Examples of DLA on two different topologies: an hexagonal mesh and a sphere. The plain hexagons and facets represent fixed particles. On the sphere, the empty positions are not drawn. Exactly the same transformation is used on the two collections.

from the 13th century (e.g. in Cosmati mosaics on the Anagni cathedral in Italy). It is also called the Sierpinski gasket or Sierpinski sieve [16]. The ST can be produced by iterating the 2-dimensional morphism defined on {0, 1} by 0 → 00 00 and 1 → 11 01 . Starting from 1, we obtain: 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 −→ −→ −→ −→ · · · . 1 1 1 0 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 Equivalently, it can be produced by taking the Pascal’s triangle modulo 2 (see Fig. 9.5). The formula for the binomial coefficient in Pascal’s triangle is: P (0, j ) = 1, P (i, j ) = 0 for i > j and P (i, j ) = P (i − 1, j − 1) + P (i − 1, j ) for the remaining cases. Considered modulo 2, this formula gives raise to the transformation below acting on a lattice Grid2:

Fig. 9.5. Taking the binomial coefficients modulo 2 produces the shape of the ST.

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Fig. 9.6. Growth of ST on GBF: the four figures present the growth of the ST on a Grid2 at different steps. Left: the initial state; right: the final state; on top middle: an intermediate step with the standard rule application strategy (maximal parallel); on bottom middle: an intermediate step using a stochastic application strategy (note the irregular boundary). On the grid, blank squares corresponds to the special value , light and dark gray squares are respectively the digits 0 and 1. trans ST1 = { |south> x |west> y => (x+y) mod 2, x, y }

In this rule, the comma is refined using a GBF generator: a |south> b means that b is a neighbor of a following the south direction. The transformation could be iterated on an initial lattice where the positions (0, j ) are filled with 1 and positions (i, 0) are filled with 0 for i > 0. The result is shown in Fig. 9.6. However, this transformation uses arithmetic operators (the + and mod). This is not acceptable if we want model more closely a physical growth process: then, the values 0 and 1 must be considered as arbitrary symbols and not as integers amenable to arithmetic operations. To avoid the arithmetic operations, the rule can be split into 4 rules where all the different cases are enumerated. Therefore, we obtain: trans ST2 }

= { |south> |south> |south> |south>

0 0 1 1

|west> |west> |west> |west>

0 1 0 1

=> => => =>

0, 1, 1, 0,

0, 0, 1, 1,

0 1 0 1

This more elementary computation is close to a specification of the ST growth using a tiling process where the tiles are DNA fragments, as in [14]. Following


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this work, we consider 4 tiles corresponding to the two boolean values a cell (i, j ) receives from the cells (i − 1, j − 1) and (i − 1, j ). This tiling is easily coded and then simulated in MGS. We use the four symbols ‘T00, ‘T10, ‘T01 and ‘T11 to represents the 4 types of tiles: tile ‘Txy at position (i, j ) means that x is the value of P (i − 1, j ) and y is the value of P (i − 1, j − 1). So the value 0 is represented by either ‘T00 or ‘T11 and the value 1 by ‘T10 or ‘T01. Finally, we use a transformation with 4 rules to specify the placement of the tiles: trans ST3 = { |south> (‘T00|‘T11) as x |west> (‘T01|‘T10) as y => ‘T01, x, y |south> (‘T00|‘T11) as x |west> (‘T00|‘T11) as y => ‘T11, x, y . . . two additional symmetric rules . . . }

The path pattern works as follow: the | operator in a pattern denotes an alternative: ‘T00|‘T11 matches the symbol ‘T00 or the symbol ‘T11; the as construct is used to bind the value of a pattern fragment to a variable: in (‘T00|‘T11) as x the pattern variable is bound to the actual value matched by the pattern. In [14], the tile corresponds to molecules and the self-assembly process to a crystallization. In this kind of process, at a given time, only a fraction of the available binding sites effectively accept a new molecule. This is not correctly specified using the standard rule application strategy in MGS. The standard rule application strategy is called maximal parallel: in this strategy, all distinct instances of the sub-collections matched by the l.h.s. pattern are “simultaneously replaced” by the r.h.s. This matching strategy ensures a maximal rule application over a collection. In other words, if a rule is not triggered, then there is no instance of a possible path that fulfills the pattern. Such strategy is for example used in the application of the production rules of a Lindenmayer system [13] and may give an account for the fact that the laws of physics apply everywhere. The non-deterministic character of the process described in [14] is due to errors and the kinetics of chemical reactions and can be taken into account using a stochastic application strategy. As a matter of fact, there are several kinds of application strategy in MGS. With the stochastic application strategy, only one rule is applied (asynchronous mode) and the rule is selected according to a given probability. A pseudoprobability p can be associated to a rule using a new kind of arrow: lhs ={ probability = p }=> rhs

(the pseudo-probabilities of the rules within a transformation are summed up and normalised to give a probability; an ordinary rule has implicitly a pseudoprobability of 1).

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The use of the stochastic application strategy is specified in the transformation application, not in the transformation definition. The expression: ST3[strategy = ’stochastic](c)

applies the transformation ST3 using the stochastic application strategy to collection c. Because the four rules in ST3 do not define any explicit probability, each rule as an equal chance to be chosen. In the building of a ST by the previous transformations, the final result is not affected by the choice of a stochastic application strategy instead of the maximal parallel one, because each rule application is exclusive from the other ones. However, intermediate results may differ, cf. Fig. 9.6.

5.

Handling Arbitrary Shapes

Using a group presentation to define the underlying lattice of a DLA process is a powerful mechanism. However, the sphere in the right of Fig. 9.4 cannot be specified as the Cayley graph of a group. Indeed any Cayley graph is a regular graph (i.e.: each vertex has the same number of neighbors) and this is not the case for the sphere. In addition, the self-assembly processes we have described so far take place in a predefined discretized space. This is a shortcoming in several situations, for example when there is a need to handle arbitrary rotation. In the rest of this chapter, we will use abstract cellular complexes to represent and build incrementally arbitrary shapes in arbitrary dimensions.

5.1.

Cellular Complexes

The notion of abstract cellular complex comes from the combinatorial algebraic topology theory. An abstract cellular complex represents a space built by gluing together cells of various dimensions [12]. This data structure generalizes the idea of a graph: cells of dimension 0 are vertices, cells of dimension 1 are edges, cells of dimension 2 are faces, etc. More formally, a cellular complex is a set K of abstract elements together with an antisymmetric, irreflexive and transitive binary relation B ⊂ K × K and a function dim : K → N. The relation B is called the boundary relationship, and the function dim, dimension. Moreover, B and dim verify: ∀(c1 , c2 ) ∈ B, dim(c1 ) < dim(c2 ). The elements of K are called topological cells. If c ∈ K and dim(c) = n, then we say that c is an n-cell. The structure of cellular complex can be easily handled using a lattice based on the boundary relationship B. The following notions will be used in the rest of the chapter:


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(1) incidence: let c1 and c2 be two cells, c1 and c2 are incident if (c1 , c2 ) ∈ B or (c2 , c1 ) ∈ B. Therefore, the lattice based on B is also called an incidence graph; (2) notion of face: let c be an n-cell, the faces of c are the set of the (n − 1)-cells cf such that (cf , c) ∈ B. As an example, if c is a triangle (a 2-cell), the faces of c are the three edges of c; (3) notion of coface: let c be an n-cell, the cofaces of c are the set of the (n + 1)cells ccf such that (c, ccf ) ∈ B. Of course, if c1 is a coface of c2 , c2 is a face of c1 . In this context, the corresponding MGS topological collection associates a value with each topological cell. This association corresponds to the notion of chain developed in algebraic topology.

5.2. Patch Transformations Path patterns, previously used to operate on GBF, are not well fitted to handle arbitrary cellular complex: an arbitrary sub-collection cannot be easily described by a path. Patch transformations have been created to handle any arbitrary cellular sub-complex. As any kind of transformations, patch transformations act on topological collections using rewriting techniques, and are defined by case where each case is a rewriting rule. The rest of this section describes the patch rules syntax. 5.2.1. Patch Patterns In patch patterns, a sub-collection is described through the elements that compose it (i.e., the n-cells) and their topological organization. We use two new operators < and > to represent the incidence relationship: a < b (or b > a) means a is a face of b (and obviously b is a coface of a). So we define a new syntax for pattern: a patch pattern PPat is a sequence of basic filters PBF separated by an incidence operator POp. Each basic filter matches one n-cell. The new grammar is defined by: PPat ::= PBF | PBF POp Pat POp ::= < | > | ’ ’ PBF ::= PVar | ~ PVar PVar ::= id | id:[dim=exp,...,clausei ,...]

where id ranges over the pattern variables and exp is an expression. A systematic interpretation for these patterns is given below. 5.2.2. Variable A pattern variable a:[dim=exp,...] matches exactly one element with a well-defined value. The pattern variable a denotes the matched element.

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For each variable, some constraints can be given between square brackets. The constraint dim=exp specifies the dimension of the filtered n-cell (the expression exp has to be evaluated into an integer). Other constraints can be given and some are detailed below. Contrary to what was required with path patterns, a pattern variable a may occur several times elsewhere in the whole rule. If the same variable is defined several times, both guards are grouped together in a conjunction. For example, a:[clause1 ] a:[clause2 ] is equivalent to a:[clause1 ,clause2 ]. 5.2.3. Neighborhood The expression e op p matches an element e that respect the incidence property denoted by op with the first element e matched by p. The incidence relation op is either: “” which means that e is a coface of e or “ ” (i.e. a white space) if e and e don’t have to respect any relationship. 5.2.4. Consuming Usually, if an element is matched by a pattern, it cannot be matched in another pattern matching. Indeed, two sub-collections matched by the l.h.s. of some rules of a transformation cannot overlap. We say that the elements matched by a pattern are “consumed”. Nevertheless, we need sometimes to refer to some particular elements (e.g. for the sake of the reconstruction in the r.h.s.), but without consuming those elements (that is: leaving this element free for another possible matching). This is the role of the tilde qualifier: ~ v is a basic pattern that leaves the element v free to be matched in a forthcoming rule application. To avoid any overspecification, any matched and unconsumed element can be matched but not consumed in another occurrence of a patch pattern. Note that using this syntax, the specification of a path of n-cells is easy: the path pattern a,b,c is equivalent to the patch pattern a < _ > b < _ > c

the _ is used to match a cell without giving it a name. As an example, the following pattern matches a triangle f with its boundary and without consuming edges e1, e2 and e3: f:[dim=2] v1 < ~ e1 > v2 < ~ e2 > v3 < ~ e3 > v1 ~ e1 < f ~ e2 < f ~ e3 < f

In order to have a more concise syntax, syntactic sugar is used to refer to a part of the neighborhood. The previous example can be rewritten as: f:[dim=2, (e1,e2,e3) in faces] v1 < ~ e1 > v2 < ~ e2 > v3 < ~ e3 > v1


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The clause “(e1,e2,e3) in faces” means that a matched cell must have at least 3 faces named e1, e2 and e3. These elements are not consumed. 5.2.5. Specifying the Right-Hand Side Basically, the left-hand side specifies a subpart of the incidence graph of the cellular complex. So, the reconstruction can be described by an incidence graph that will replace the one matched by the patch pattern. The syntax to specify this graph is very close to the patch pattern specification: it is a sequence of two kinds of elements. An element can be either id:[exp]

which refers to an existing matched and consumed element named id in the patch pattern and whose value has to be updated by the value of exp, or ‘id:[dim=expd , faces=(id1 ,id2 ,...,‘id1 , ‘id2 ,...), cofaces=(id1 ,id2 ,...,‘id1 ,‘id2 ,...), val=expv ]

which refers to a new cell of dimension expd , with the associated value expv . If the val clause is missing, there is no value associated to the cell. The symbol ‘id can be used elsewhere in another clause to refer to this newly created element. Two clauses faces and cofaces are given to specify the list of faces and cofaces of the new cell ‘id. In these lists, the variables refer to the matched elements (i.e. they are pattern variables) while the symbols refer to new elements created elsewhere during the reconstruction step. The default value for faces and cofaces is the empty list. Note that the list of faces and cofaces do not need to be complete: all the faces and cofaces specifications are collected and normalized (that is, if x appears in the face list of y, then y is added if necessary in the coface list of x and vice versa). Note that a pattern variable in an expression may refer to two different values: the topological cell corresponding to the matched element or the value associated with the topological cell. The denoted value depends on the context: for instance, in a face clause, a pattern variable refers to the matched cell. In an ordinary expression, a pattern variable refers to the value associated to the matched cell. The operator ^ applied on pattern variable can be used to force the denotation of the matched cell is necessary. 5.2.6. A Simple Example Here is a short and simple example of a patch rule that matches an edge and adds a new vertex on it: v1 < e:[dim=1] > v2 => ‘e1:[dim=1, faces=(v1,‘v)] ‘v:[dim=0, cofaces=(‘e1,‘e2)] ‘e2:[dim=1, faces=(v2,‘v)]

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Fig. 9.7. Insertion of a vertex on an edge.

In this example, three new elements are created ‘v, ‘e1 and ‘e2, and e is removed. Figure 9.7 gives an illustration of this rewriting step.

6.

Self-Assembled Polymers

In this section we will use abstract cellular complexes and patch transformation to model a polymerization process. The model does not refer to a real example. Its only purpose is to illustrate the ability of modeling arbitrary shapes using abstract complexes. In the next section we will use abstract complexes to build Sierpinski triangle and tetrahedra by carving. Polymers are long-chained molecules with repeating units. They are formed by reacting monomers (the repeating part of the chain) in a process known as polymerization. These reactions fall into 2 main groups—addition polymerization and condensation polymerization. Addition polymerization is adding monomers and polymers together to form the chain. Thus, it is an example of accretive growth. In condensation polymerization, a small molecule, often water, is removed as part of the process. Here we suppose that a monomer is represented by a rectangle, that is, a 2-cell. When a monomer is joined to another monomer or a chain, the junction may form an angle that depends of the monomer constituent. For the sake of the simplicity, we suppose that each monomer is associated with its corresponding angle, see Fig. 9.8. The self-assembly process consists in taking arbitrarily two monomers (or intermediate polymers) and gluing them together, as sketched in Fig. 9.9. We assume that a monomer or a polymer has only one active binding site which correspond to an edge. The functions mark and unmark are used to activate and inhibit a binding site. Polymerization is a patch transformation that takes two elements in the previous multiset and joins them. This operation is iterated until there is only one polymer left in the solution. The transformation is applied following a stochastic strategy. Here is the code of the reaction transformation: trans reaction = { a,b => let cc = mark(merge(a + b)) in translate(cc) } ;;

This transformation implements the chemical reaction in the chemical soup: two complexes are selected and merged using the patch merge. The expression a + b creates a complex made of the two disjoint complexes a and b. After


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Fig. 9.8. The initial population of monomers. They are represented with their corresponding joining angle. For colour version, see Colour Plate Section.

Fig. 9.9. Gluing of two monomers: a triangular 2-dimensional cell is added between the two rectangles. Beware that the rectangles are in a general position in a 3-dimensional space (see Fig. 9.8) and consequently, the three cells are not in the same plane.

the merging, an edge of the boundary is selected to become the new active site using the function mark. The only rules in the merge patch is looking for two active edges e1 and e2. They have to belong to two disconnected complexes. The guards (the clause that starts with a “?”) check these properties. We only match and consume the vertices v11 and v21 because they are merged to create ‘v (see Fig. 9.9). The other

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Fig. 9.10. A final polymer, from two points of view, generated by the reaction and merge transformations. For colour version, see Colour Plate Section.

cells remain after the application of the rule. The patch merge must be applied in an asynchronous mode to link the molecules by a unique edge. patch merge = { v11 < ~ e1:[dim=1,?(. . .)] > ~ v12 v21 < ~ e2:[dim=1,?(. . .)] > ~ v22 => ‘v:[dim=0, cofaces=join(cofaces(v11), cofaces(v21)), . . .] ‘e:[dim=1, faces=(v12,v22), cofaces=(‘f), . . .] ‘f:[dim=2, faces=(‘e,e1,e2), . . .] } ;;

This patch searches for two edges e1 and e2. They have to belong to two distinguished complexes and correspond to a possible link between the molecules. The guards check these properties. We only match and consume the vertices v11 and v21 because they are merged to create ‘v. The other cells remain unchanged after the application of the rule. The patch merge must be applied in an asynchronous mode to link the molecules by a unique edge. Figure 9.10 illustrates a final result.

7.

Carving Sierpinski Triangles

In this last section, we will use abstract complexes to introduce a new kind of selfassembly: self-assembly by carving. We will use the shape already introduced in Section 4: the Sierpinski triangles and its 3-dimensional generalization. Building a ST by carving is illustrated in Fig. 9.11. This process is easily coded in MGS using patch patterns on abstract cellular complexes.


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Fig. 9.11. ST can also be produced by iterating the carving of a triangle inside another triangle.

7.1.

Carving Transformation

To represent the ST, we use an abstract cellular complex where the value of a vertex represents the coordinate of an embedding of the ST in the plane. The main advantage of using cellular complexes is that we can handle cells of various dimensions to represent all the elements that compose the ST. In fact, in the previous representation, the ST were patterns appearing on a matrix of digits, that is, on a predefined space. Here the concrete geometric structure of the ST is specified and the building of the ST also builds “its own embedding space”. Building the ST by carving means iterating the removing of the central part of a triangle. The limit of this process corresponds to the ST fractal (see Fig. 9.11). The initial step consists in a unique triangle. It corresponds to the digit 1 of the previous representation at the beginning of the series of transformations given in Section 4. It is composed of three 0-cells (vertices), three 1-cells (the edges that link the vertices pairwise) and a unique 2-cell (the unique coface of the edges): let rec v1 = new_cell(0,[],[e1,e3]) and v2 = new_cell(0,[],[e1,e2]) and v3 = new_cell(0,[],[e2,e3]) and e1 = new_cell(1,[v1,v2],[f]) and e2 = new_cell(1,[v2,v3],[f]) and e3 = new_cell(1,[v3,v1],[f]) and f = new_cell(2,[e1,e2,e3],[]);;

The primitive new_cell(d, , ) creates a new topological d-cell whose faces are given as the list and cofaces are . The carving, which is the unique step of the process, is the topological surgery described on Fig. 9.12. The MGS patch transformation follows:

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Fig. 9.12. Local topological transformation: the patch transformation CRV divides a triangle into 3 smaller ones, leaving a triangular hole in the middle. patch CRV = { ~ v1 < e1:[dim=1] > ~ v3 < e2:[dim=1] > ~ v5 < e3:[dim=1] > ~ v1 f:[dim=2, (e1,e2,e3) in faces] => ‘v2[dim=0, val=average(v1,v3)] ‘v4[dim=0, val=average(v3,v5)] ‘v6[dim=0, val=average(v5,v1)] ‘e1[dim=1, faces=(v1,‘v2), val=‘edge] ‘e2[dim=1, faces=(‘v2,v3), val=‘edge] ‘e3[dim=1, faces=(v3,‘v4), val=‘edge] ‘e4[dim=1, faces=(‘v4,v5), val=‘edge] ‘e5[dim=1, faces=(v5,‘v6), val=‘edge] ‘e6[dim=1, faces=(‘v6,v1), val=‘edge] ‘e24:[dim=1, faces=(‘v2,‘v4), val=‘face] ‘e46:[dim=1, faces=(‘v4,‘v6), val=‘face] ‘e62:[dim=1, faces=(‘v6,‘v2), val=‘face] ‘f1:[dim=2, faces=(‘e6,‘e1,‘e62), val=‘face] ‘f2:[dim=2, faces=(‘e2,‘e3,‘e24), val=‘face] ‘f3:[dim=2, faces=(‘e4,‘e5,‘e46), val=‘face] }

The function average, that is not detailed here, computes coordinates of the middle point between its 2 arguments. In this patch, all the elements of a triangle are matched. They are all consumed except the vertices that remain after the application of the rule. Moreover, they may be shared by other triangles, and therefore they cannot be destroyed and have to be matchable in a parallel application of the rule. Note that each edge has only one coface at any step of the building process. This is easy to prove considering the initial state and making an inductive reasoning at each step: in fact, the rule creates 9 new edges that have a unique coface. This kind of proofs on rewriting systems is usual and allows easy verifications of properties on self-assembly processes specified using rewriting techniques.


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Fig. 9.13. On the left, application of the CRV patch on a different initial state. On the right, the expected result.

The patch CRV is a little long and may be difficult to understand. Moreover, CRV depends on the property of having one coface per edge. Indeed, if we take another initial state that does not assume this property, some aberrations can appear as shown on Fig. 9.13. In this example, two triangles share an edge. If we apply CRV on one of the two triangles, the topology of the other one is modified. It is not a triangle anymore, but a square. To reach the expected result irrelevantly of the initial shape, the building process can be divided into two distinguished transformations. The first one, AV, adds a vertex in the middle of each edges (see Fig. 9.14): patch AV [gen] = { ~ v1 < e:[dim=1] > ~ v2 => ‘v:[dim=0, cofaces=(‘e1,‘e2), val=average(v1,v2,gen)] ‘e1:[dim=1, faces=(v1,‘v), val=‘edge] ‘e2:[dim=1, faces=(v2,‘v), val=‘edge] }

An additional argument, gen, is required to distinguish the new and the old vertices and is used in the next computation step. Conveniently gen can be an in-

Fig. 9.14. Carving a triangle. The first transformation AV adds vertices in the middle of every edge. The second transformation RV refines the central hexagonal face into three triangles.

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Fig. 9.15. Sierpinski sponge building process: initial state and steps 1, 2, 3 and 4.

teger representing the generation number of the vertex. The function average is modified to compute and store the position and the age of the vertex. The next step looks for all the hexagons and replaces them with three triangles (see Fig. 9.14): patch RF [gen] = { f:[dim=2, faces = (e1,e2,e3,e4,e5,e6)] ~ v1 < ~ e1 > ~ v2:[?ok(gen,v2)] < ~ e2 > ~ v3 < ~ e3 > ~ v4:[?ok(gen,v4)] < ~ e4 > ~ v5 < ~ e5 > ~ v6:[?ok(gen,v6)] < ~ e6 > ~ v1 => ‘e24:[dim=1, faces=(v2,v4)] ‘e46:[dim=1, faces=(v4,v6)] ‘e62:[dim=1, faces=(v6,v2)] ‘f1:[dim=2, faces=(e6,e1,‘e62)] ‘f2:[dim=2, faces=(e2,e3,‘e24)] ‘f3:[dim=2, faces=(e4,e5,‘e46)] }

Note the guards in the specification of the matched vertices: the predicate ok checks the vertices v2, v4 and v6 to have the right generation number gen. The ST is 2-dimensional. The corresponding 3-dimensional fractal is called the Sierpinski sponge, see Fig. 9.15. These pictures have been generated using the current implementation of the MGS language. The 3D program is the same as the 2-dimensional one. In fact, we use the patches AV and RV to perforate the faces of a tetrahedron. A last patch (not detailed in this chapter) is used to create the 4 smaller tetrahedrons associated with the 4 original vertices. It is interesting to


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note that the Sierpinski process is recursive within the dimension: we first applied an iteration of the 2D process (with AV and RV), and a last patch finalized the 3D construction.

8.

Conclusions

In this chapter we have presented the use of a DSL language for the modeling and the simulation of two kinds of self-assembly processes: by accretive growth and by space carving. Despite their specificities, we are convinced that they are paradigmatic of a full class of self-assembly processes. For instance, arbitrary Wang tiling [7] could be coded in MGS following the scheme presented in Section 4. The self-assembly of ST has been really implemented using DNA molecules [14]. Compared to the TAM and kTAM simulations used in this last work, the MGS modeling is more abstract: the purpose is not to study the implementation in DNA but to investigate the shape produced by some families of self-assembly processes. We insist on the expressiveness brought by the notions of topological collections and their transformation. For example, the patch language used in Section 7 is powerful enough to produce Sierpinski and Menger sponge (a generalization of carving a tetrahedron and a cube in 3D), see Fig. 9.16. The MGS programming style corresponds to the rule-based programming paradigm. Rule-based programming is currently experiencing a renewed period of

Fig. 9.16. Menger sponge building process: initial state and steps 1 and 2.

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growth with the emergence of new concepts and systems that allow a better understanding and better usability. However, the vast majority of rule-based languages (like expert-systems) are funded on a logical approach (computation is a logical deduction and a deduction step is specified by a rule) which is not adequate to describe various spatio-temporal processes. We hope that the previous section have demonstrated the ability of MGS to express easily and concisely the building of sophisticated spatial structures, like the ones needed to model selfassembly processes. Our topological approach is motivated by some considerations internal to computer science, and also by the needs expressed by some application domains. A target application domain for MGS is the modeling and simulation of dynamical systems and especially dynamical systems that exhibit a dynamic structure [3]. This kind of dynamical systems is very challenging to model and simulate. New programming concepts could be developed to ease their modeling and simulation. Another target application that should be investigated is robotics self-assembly. All the examples presented in this chapter have been developed and processed using the MGS interpreter. The results of the various programs have been stored in data files that are later used by separate external viewers for graphic representation. Others examples are presented on the MGS web page: http://mgs.ibisc. univ-evry.fr. The perspectives opened by this work are numerous. We want to develop several complementary approaches to define new topological collection types. One approach to extend the GBF applicability is to consider monoids instead of groups, especially automatic monoids which exhibit good algorithmic properties. Another direction is to handle other kind of combinatorial spatial structures. At the language level, the study of the topological collections concepts must continue with a finer study of transformation types. Several kinds of restrictions can be put on the transformations, leading to various kinds of pattern languages and rules. The complexity of matching such patterns has to be investigated. The efficient compilation of a MGS program is a long-term research plan. We have considered in this chapter only 1-dimensional paths, but a general n-dimensional notion of path exists and can be used to generalize the substitution mechanism of MGS. From the applications point of view, we are targeted by the simulation of more complex developmental processes [6].

Acknowledgements We are grateful to Natalio Krasnogor that give us the initial motivation to write this chapter. We are also grateful to the anonymous referees for comments and suggestions. A big thank is due to Olivier Michel at the University of Evry which is one of the designer of MGS. Further acknowledgments are also due


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to M. Gheorghe at the University of Sheffield, F. Delaplace at the University of Evry, C. Godin and P. Barbier de Reuille at CIRAD–Montpellier, the members of the Epigenomic group at GENOPOLE–Evry, P. Prusinkiewicz at the University of Calgary, and the participants of the CellCom FET proposal for helpful discussions, biological motivations, fruitful examples and challenging questions. This research is supported in part by the CNRS, GENOPOLE–Evry, the GDR ALP, IMPG, and the University of Evry.

References [1] H. Abelson, D. Allen, D. Coore, C. Hanson, G. Homsy, T. Knight, R. Nagpal, E. Rauch, G.J. Sussman, R. Weiss, Amorphous computing, in: CACM, vol. 43, ACM Press, 2000. [2] M. Eden, in: H.P. Yockey (Ed.), Symposium on Information Theory in Biology, New York, 1958, Pergamon Press, 1958, p. 359. [3] J.-L. Giavitto, Invited talk: Topological collections, transformations and their application to the modeling and the simulation of dynamical systems, in: Rewriting Technics and Applications (RTA’03), Valencia, June 2003, in: LNCS, vol. 2706, Springer, 2003, pp. 208–233. [4] J.-L. Giavitto, O. Michel, Declarative definition of group indexed data structures and approximation of their domains, in: Proc. of the 3rd Int. ACM SIGPLAN Conference on Principles and Practice of Declarative Programming, September 2001, ACM Press, 2001. [5] J.-L. Giavitto, O. Michel, Modeling the topological organization of cellular processes, Biosystems 70 (2) (2003) 149–163. [6] J.-L. Giavitto, O. Michel, Modeling developmental processes in MGS, in: Molecular Computational Models: Unconventional Approaches, Idea Group, 2004, pp. 1–46. [7] B. Grünbaum, G.C. Shephard, Tilings and Patterns, W.H. Freeman & Co., 1986. [8] P. Horn, Autonomic computing: IBM’s perspective on the state of information technology, Technical report, IBM Research, http://www.research.ibm.com/autonomic/manifesto/ autonomic_computing.pdf, October 2001. [9] J.A. Kaandorp, Modelling growth forms of biological objects using fractals, PhD thesis, University of Amsterdam, 1992. [10] K.N. Kutulakos, S.M. Seitz, A theory of shape by space carving, International Journal of Computer Vision 38 (3) (2000) 199–218. [11] V. Manca, C. Martin-Vide, G. Paun, New computing paradigms suggested by DNA computing: Computing by carving, Biosystems 52 (1–3) (1999) 47–54. [12] J. Munkres, Elements of Algebraic Topology, Addison–Wesley, 1984. [13] P. Prusinkiewicz, A. Lindenmayer, J.S. Hanan, et al., The Algorithmic Beauty of Plants, Springer-Verlag, 1990. [14] P.W.K. Rothemund, N. Papadakis, E. Winfree, Algorithmic self-assembly of DNA Sierpinski triangles, PLoS Biol 2 (12) (2004) e424, www.plosbiology.org. [15] A. Spicher, O. Michel, J.-L. Giavitto, A topological framework for the specification and the simulation of discrete dynamical systems, in: Sixth International Conference on Cellular Automata for Research and Industry (ACRI’04), Amsterdam, October 2004, in: LNCS, vol. 3305, Springer, 2004. [16] I. Stewart, Four encounters with Sierpinski’s gasket, Mathematical Intelligencer 17 (1995) 52–64. [17] T.A. Witten, L.M. Sander, Diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett. 47 (1981) 1400–1403. [18] S. Wolfram, A New Kind of Science, Wolfram Media, 2002.

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Computer Aided Search for Optimal Self-Assembly Systems Pablo Moisset de Espanés University of Southern California, Computer Science Department, USA Abstract. DNA self-assembly is a promising paradigm for nanotechnology. In this chapter we study the problem of finding tile systems of minimum size that assemble a given shape in the Tile Assembly Model, defined by Rothemund and Winfree [Paul W.K. Rothemund, Erik Winfree, The program-size complexity of self-assembled squares (extended abstract), in: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM Press, 2000, pp. 459–468]. We present a tile system that assembles an N × log2 N rectangle in asymptotically optimal Θ(N ) time. This tile system has only 7 tiles. Earlier constructions need at least 8 tiles [A. Goel, H. Chen, Q. Cheng, P. Moisset de Espanés, Invadable self-assembly, combining robustness with efficiency, in: Proceedings of Symposium on Discrete Algorithms, ACM Press, 2004]. We managed to reduce the number of tiles without increasing the assembly time. The new tile system works at temperature 3. The new construction was found by the combination of exhaustive computerized search of the design space and manual adjustment of the search output. The search algorithms are of independent interest. We generalize the search techniques to look for tile systems that assemble non-rectangular shapes as well. In addition to the precise definition of the new counter, this chapter describes the design process, including the fortunate accidents that lead to its discovery.

1. Introduction Many mathematically oriented research publications emphasize precision and rigor over intuition and insight. Although this tendency is understandable, it has the disadvantage of not providing the reader with useful information. The intuitions and heuristics that were very useful to obtain a result become unavailable to the reader. These underlying ideas, even when they are mathematically “wrong,” can still help the scientific community to make further progress. It is also true that, frequently, proofs are derived “by accident.” That is, mathematical theorems are proved while trying to solve another problem or trying to prove some conjecture that turned out to be false. Another trend in mathematics is the increased use of computerized exploration to derive theorems. Computers can help to analyze explicitly a huge number of cases in a proof. They are also useful to derive conjectures. Though critics of STUDIES IN MULTIDISCIPLINARITY VOLUME 5 ISSN 1571-0831/DOI: 10.1016/S1571-0831(07)00010-X


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computer aided proofs have raised concerns about the validity of this procedure, it is impossible to deny that computers are valuable tools to explore hypothesis. This chapter describes the process of designing self-assembled (SA) counters, which are special self assembly systems that produce rectangles. I have tried to combine the formal definitions and theorem statements with informal explanations describing why they are important and how they were designed. I also decided to leave most of the results without proof, in order to present the constructions and underlying intuitions in more detail. The constructions are interesting from a theoretical point of view, but I consider the process of finding them equally interesting. Section 3 describes the “manual” design of a counter which was an improvement over previously known constructions. In contrast, Section 4 describes the computer aided search for an even “better” counter. This result was interesting because the computerized search did not find the actual solution, but suggested a similar, yet interesting construction. The reasons why the counter works is so subtle that we believe finding the counter without the computer exploration would have been unlikely, as we have not yet found a simple underlying principle that explains it correctness.

2. 2.1.

Definitions The Tile Assembly Model (TAM)

The tile assembly model [1,11] extends the theoretical model of tiling by Wang [12] to include a mechanism for growth based on the physics of molecular selfassembly. We will present a succinct definition, with minor modifications with respect to those found in [1,11] for ease of explanation. A tile is an oriented unit square with the north, east, south and west edges labeled from some alphabet Σ of glues. For each tile t ∈ T , the labels of its four edges are denoted σN (t), σE (t), σS (t), and σW (t). Sometimes we will describe a tile t as the quadruple (σN (t), σE (t), σS (t), σW (t)). Consider the triple T , g, τ where T is a finite set of tiles, τ ∈ Z>0 is the temperature, and g is the glue strength function from Σ × Σ to Z0 , where Σ is the set of glues. We assume that (x = y) ⇒ g(x, y) = 0 for all x, y ∈ Σ. In [3,4] the case when the assumption is removed was explored. Given p = (x, y), p = (x , y ) ∈ Z2 , we say p and p are position adjacent iff |x − x | + |y − y | = 1. A shape S is a finite subset of Z2 , such that for all pairs of positions (p, p ) in S 2 , either p = p or there exists a sequence of positions p1 , p2 , . . . , pn such that p1 = p, pn = p and for all 1 k < n, pk and pk+1 are position adjacent. A configuration is a partial function from Z2 to T . Let Dom(f ) denote the domain of a function f . Let C and D be two configurations. Suppose there exist some t ∈ T and some (x, y) ∈ Z2 such that (x, y) ∈ / Dom(C), D(x, y) = t and D = C except at (x, y). Let fN,C,t (x, y) =

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g(σN (t), σS (C(x, y + 1))) if (x, y + 1) ∈ Dom(C) and fN,C,t (x, y) = 0 otherwise. Informally fN,C,t (x, y) is the strength of the bond between C and the north side of t. Define fS,C,t (x, y), fE,C,t (x, y) and fW,C,t (x, y) similarly. Then we say that tile t is attachable to C at position (x, y) iff fN,C,t (x, y)+fS,C,t (x, y)+ fE,C,t (x, y) + fW,C,t (x, y) τ , and we write C →T D to denote the transition from C to D in attaching a tile to C at position (x, y). Informally, C →T D iff D can be obtained from C by adding a tile t such that the total strength of interaction between t and C is at least τ . A tile system is a quadruple T = T , s, g, τ , where T , g, τ are as above and s is a special configuration called the “seed,” whose domain is a shape. If Dom(s) is a singleton, we will say T is a unit seed tile system. We define the notion of a derived supertile of a tile system T = T , s, g, τ recursively as follows: (1) the seed configuration s is a derived supertile of T, and (2) if C →T D and C is a derived supertile of T, then D is also a derived supertile of T. Informally, a derived supertile is either just the seed (condition (1) above), or obtained by legal addition of a single tile to another derived supertile (condition (2)). We will often omit the word “derived” in the rest of the chapter, and use the terms “seed supertile” or just “seed” or s to denote the special supertile in condition (1). A terminal supertile of the tile system T is a derived supertile A such that there is no supertile B for which A →T B. If there is a terminal supertile A such that for any derived supertile B, B →∗T A, we say that the tile system uniquely produces A. Given a tile system T which uniquely produces a supertile, we say that the program size complexity of the system is |T |, i.e. the number of tiles. The shape of a supertile Γ is Dom(Γ ). We will use the notation [Γ ] to denote the shape of Γ . A tile system T is said to uniquely produce a shape W iff it uniquely produces some supertile Γ and [Γ ] is identical (up to translation) to W . We will now add the notion of running time to this model. We associate with each tile t ∈ T a nonnegative probability P (t), such that t∈T P (t) = 1. We assume that the tile system has an infinite supply of each tile, and P (t) models the concentration of tile t in the system. Now self-assembly of the tile system corresponds to a continuous time Markov process where the states are in a one– one correspondence with derived supertiles, and the initial state corresponds to the seed s. Suppose a single tile t can be added to supertile B to produce supertile C. Then there is a transition from state B to C in the Markov chain, and the rate of the transition is P (t). Suppose the tile system produces a unique terminal supertile AT . In the Markov chain, the time for reaching AT from s is a random variable. The “running time” of the self-assembly process is defined as the expected value of this random variable. Note that the Markov process modeling the self assembly process is inherently parallel. For details see [1].

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All definitions up to this point are to describe the “rules of the game.” In this setting, “square molecules” can be attached to a growing assembly. The following definitions describe the problem we are going to address in this chapter. A tile system T = T , s, g, τ is said to be line-based iff: 1. T uniquely produces some coordinated supertile. We will denote this coordinated supertile as ΓT . 2. There exist integers l, r and h such that Dom(s) = {(x, y) | l x r ∧ y = h}, i.e. the coordinated shape of the seed is a horizontal line. 3. ∀(x, y) ∈ Dom(ΓT ) − Dom(s), y > h, where h is defined as above. The reduced-size of a line-based tile system T = T , s, g, τ is the number of elements in the range of the restriction of ΓT to Dom(ΓT ) − Dom(s). Informally, the reduced-size is the number of tiles that are used to construct ΓT , not considering the tiles used in the seed. Minimum line-based tile system problem. Given a shape S, find a line-based tile system of minimal reduced size that uniquely produces S. The General Purpose Counter (GPC) problem. Find a tile system such that for all positive integers h, for all K log h, there exists a seed row such that the tile system uniquely produces a supertile whose shape is an h × K rectangle. We would like to find a small set of tiles that assembles a rectangle whose size is determined solely by the initial configuration, i.e. the seed, of the SA process. We will also assume the shape of the seed has to be a horizontal line. We would like the size of this set of tiles to be independent of the size of the desired rectangle. Constructions with these properties were used by Rothemund and Winfree [11] and by Adleman et al. [1] as a “subroutine” to assemble squares. Since the techniques to assemble rectangles in [11] and [1] are based on repeated addition of binary numbers we refer to these constructions as counters. Why are counters important? They are useful subroutines to solve other problems. For example, we know how to use tiles to sort numbers [5]. The sorting construction uses a counter. Conceptually, counters are structures that grow to a specific size and they resemble the “for” loop of a programming language. If counters ever work in the lab (for example using DNA tiles), they could help to produce relatively big assemblies with sizes accurate up to one molecule. It has been noted that the pattern of tiles in some counters could be used to build interesting electronic circuits [7]. Why is it important to minimize the reduced size of a line-based system? Simply to reduce the cost of implementing this idea in the real world. If a certain construction can be achieved with only a few tiles, it is likely to require only a few “species”, such as DNA strands, to complete the experiment. Also note that in our definition of counter we want to minimize the number of tiles that appear


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above the seed line and are independent of the size of the assembled rectangle. To obtain a rectangle of a given size, we would have to synthesize the base-line only. Since counters can be as thin as n × log n, with a “short” and presumably cheap base-line we can obtain a long structure.

3. Counting to n at τ = 2 Rothemund and Winfree’s counter uses 12 tiles (not counting those that appear in the seed row), works at temperature 2 and has a O(n log n) time complexity, where n is the height of the assembled rectangle. The counter presented in [1] is faster, as it has a O(n) time complexity, uses 15 tiles and works at temperature 3. This counter is an involved construction that uses a parallel binary addition algorithm. We will present now a simpler counter which has O(n) time complexity, uses only 8 tiles, requires temperature 2. Proving correctness of this counter as well as determining the running time is also easier than analyzing the counter in [1].

3.1.

Designing a Counter Manually

Adleman et al. [1] posted the problem of finding a counter that works at temperature 2 and assembles a rectangle in asymptotically optimal time. The construction in this section solves that problem. My first attempts to solve the problem were based on tweaking the counter in [1], trying to reduce the temperature from 3 to 2. One of the paradigms I used was to convert each tile in the original system into a 3×3 block of tiles in the new system. The approach was not successful. I don’t know if the idea was doomed but I could not find a working construction. Nonetheless, the idea of converting single tiles into blocks was useful to solve other problems, see for example [6]. I also tried to design a counter based on a heavily pipelined hardware adder I had designed in the past. That approach was also unsuccessful. Most of my designs involved a few dozen tiles and were rather complicated. I decided to change the design strategy. I tried to find a very simple counter that worked at temperature 2. After that, I planned to determine why it did not assemble in optimal time and I was going to optimize the design to speed it up. Of course, I did not expect a naive design to be fast, because a fast design requires a level of parallelism in the assembly process that did not look easy to achieve. Since all the counters we knew were based on simulating repeated additions of 1 in binary, I decided to continue with that line of thinking and I asked myself: “If I had to explain how to add 1 in binary to a 5 year old kid, how would I do it?” The answer I came up with is depicted in Fig. 10.1. The resulting algorithm creates a table of consecutive integers written in binary. The simple recipe can be

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Fig. 10.1. Algorithm to add 1 in binary.

summarized as follows: (a) Locate the rightmost zero in the binary number. (b) Write a ‘1’ on top of it. (c) To the right of this ‘1’, pad with zeros. To the left, copy the original digits. (d)(e)(f) Repeat the operations. I mimicked this algorithm using tiles, and the resulting design was a very efficient counter that works at temperature 2.

3.2.

The Tile System

We begin by showing a pictorial representation of the tile system (Fig. 10.2) and of the supertile it produces for n = 8 (Fig. 10.3).

Fig. 10.2. The tiles for the SA-counter. The number of lines jutting from each edge of the tile represent the strength of the bond (1 or 2) and the labels on the edges represents the glues. Some edges do not show any glue, meaning they have a zero strength glue.


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Fig. 10.3. Representation of an SA-counter supertile. The dashed line encloses the seed row.

We initially assume that n = 2K for some positive integer K—we will later show how this assumption can be removed. The fully assembled counter is going to be a rectangle, with K tiles in each row. Each row represents a binary number between 0 and n − 1, with the row representing i being immediately above the row representing i − 1. While each tile is completely specified by its four glues, it is convenient for the purpose of exposition to allow tiles to have labels. Each tile has one main label (either 0 or 1). We will refer to tiles representing the digits 0 and 1 as “0-tiles” and “1-tiles” respectively. The number represented by each row can be obtained by reading the main labels on the tiles in that row, with the most significant bit being the west-most in the row. We will refer to the row representing the number i as the ith row of the counter. We define TSA (K) as the tile system in Fig. 10.2. The glue strength function is as indicated in the same figure. The temperature is 2, and there is a single seed supertile called SK . SK is a row of K special tiles. The glues on the north sides of these K tiles represent the initial value of the counter in binary. To represent

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the number 0, the glue on the north side of the east-most tile in SK is b, and all the other glues on the north side of SK are n. We will assume that all tiles which are not in SK have the same constant concentration pSA which is a constant independent of n. The increment operation going from row to row uses 8 different tiles as well as K special tiles that appear in the seed row only (see Fig. 10.2). For all positive i less than n, the assembly of the ith row always starts as the attachment of a 1-tile on top of the east-most 0-tile in the (i − 1)st row. Let pi denote the position of the first tile being attached in the ith row. All the remaining tiles in the ith row to the east of pi will be 0-tiles. All the remaining tiles in the ith row to the west of pi will “copy” the labels of the tiles immediately below, i.e. if p is to the west of pi , then a 1-tile will attach to the growing supertile at p iff there is a 1-tile at p − (0, 1). We will assume that the temperature for this counter construction is 2. We now describe the tiles in each of the rows and their glues. In addition to the main label, a 0-tile may have a star symbol. 0∗ -tiles denote the east-most 0-tile in each row. 0-tiles: There are the following 0-tiles: 0∗E , 0∗M , 0Z , 0C . 1-tiles: 1E , 1M , 1S , 1C . The glues and their strengths are depicted pictorially in Fig. 10.2. The label suffixes indicate the position in the row each row a tile may occupy and/or the function in the assembly. E: The tile can be only in the east-most column of the counter. Only tiles with the E suffix can be in the east-most column of the counter. M: The tile is either a 0∗ -tile or a 1-tile that is directly on top of a 0∗ -tile. M tiles do not appear in the rightmost column. There are special E tiles to describe 0∗ -tiles in that column. Z: Attach a 0-tile to the east. Tiles with the Z suffix can appear only to the east of the 0∗ -tile in the previous row and to the west of the 0∗ -tile in the same row. C: Copy to the west the digit in the lower row. All tiles to the west of an M tile in the same row are C tiles. Every C tile has the same main label as the tile immediately to its south. S: Search in the east to west direction for the position of the first 0-tile in the row immediately below. Tiles with the S suffix can appear only to the east of 0∗ -tiles. Theorem 1. The tile system TSA (K) uniquely produces a supertile which is a 2K × K rectangle. Initially, I wrote a long inductive proof from first principles, but Erik Winfree suggested a more elegant approach, which was a straightforward application of a


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theorem proved by Rothemund [10]. The proof is omitted, details can be found in [5]. Removing the assumption h = 2K is straightforward. For all integers h > 1, for all K log h, there exists a seed row such that the tile system uniquely produces a supertile which is an h × K rectangle. To find that seed row given h and K, define d = 2K − h. Note that d 0. Let dK−1 , dK−2 , . . . , d0 be the binary representation of d, with d0 being the least significant digit. Let p be the least index such that dp = 0. Call gj the glue on top of the j th tile of the seed row, counting from east to west. If p = 0 then make g0 = b. If p > 0 then make gp = a and g0 = l. For all j in {1, 2, . . . , K − 1} − {p} make gj = u iff dj = 1 and make gj = n otherwise. Intuitively, d is the initial value of the counter and the counter always stops when it reaches the value 2K − 1. Note that it is possible to define the seed supertile without adding additional tiles to the system. This means that the total number of tiles is 8, regardless of h and K. If we wanted to construct the rectangle using a unit seed tile system, the number of tiles would have to depend on h. It is possible to use O(log h/ log log h) tiles to solve the problem, as shown in [1]. At this point I analyzed the assembly time expecting it to be sub-optimal. The detailed analysis was published in [9]. To my surprise, the assembly time was asymptotically optimal! This simple, small and “dumb” counter was smaller that the smallest previously know counter (by Rothemund and Winfree [11]) without sacrificing assembly time. Also, I believe it is slightly faster than the counter described by Adleman et al. [1]. That counter also assembles in asymptotically optimal time, but I believe that the design in Fig. 10.2 is faster than the one in [1] by a constant factor. Since my intuition indicates that the improvement in the constant is not dramatic and the analysis to determine it looks complicated, I decided not to study the running time in more detail.

4.

Finding Optimal Reduced-Size Tile Systems

I started considering the problem of finding a counter with a smaller reduced size. This seemed to be a hard optimization problem. My attempts to slightly modify the counter in Fig. 10.2 failed. Fortunately, the optimization process looks simple enough to be solved with a computer. Roughly speaking, we can try to brute force the problem by enumerating all possible sets of tiles of size 7 or less. Each one of these sets can be tested to determine whether they produce rectangles of predefined size or not. Figure 10.4 represents the process. 1. Generation of all sets of at most 7 tiles, using glues of strength not greater that 2. 2. Verification of each one of the generated tile systems. The verification process involves determining if the tested set of tiles can uniquely produce rectangles

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Fig. 10.4. Outline of the search procedure.

of certain predefined sizes. Since we are looking for tile systems that can uniquely assemble rectangles of all sizes h × w, as long as w log h, the choice of sizes is not really very important. The testing algorithm should know what the seed row is like. The seed row is not known, but this problem can be solved through exhaustive search. If the tested set fails to uniquely produce at least one of the rectangles tried, then, that set cannot be used for a counter and is discarded. 3. If the tested set succeeds to uniquely produce all of the rectangles tried, then, the set is probably a good counter. However, it may be the case that a set that passed all the tests during stage 2 fails to assemble a rectangle of a size not included in the test cases. Nonetheless, if we try many sizes in stage 2 and some set of tiles passes all the test cases successfully, that would be strong evidence of that set being a valid counter. We still have to prove correctness of the construction manually, but the probability of facing a “false positive” seems to be very small. Also, verifying a given design should be, in principle, easier than finding it. Note that if we did not know that there exists a counter with reduced size 8, the computerized approach would not look very promising. Rough estimates suggested that generating all sets of tiles of size up to 11 (1 less than the counter by Rothemund and Winfree) would be unfeasible. Even for sizes up to 7 the problem is non-trivial. We can derive a trivial upper bound. There are 4 × 7 = 28 sides in a set of 7 tiles. Since we are interested in tile systems that work at temperature 2, we need at most 57 glues, 28 glues of strength 1, 28 of strength 2 and a glue with strength 0. The total number of assignments of these glues to the 28 sides is at most 5728 < 1.5 × 1049 . Even if this upper bound is not tight at all, it shows that the naive approach of generating the sets of tiles using a handful of nested loops will not work in practice.


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Fig. 10.5. Idea to reduce the search space.

4.1.

A More Formal Description of the Problem

To simplify the discussion, we will modify slightly the definition of tile system. Because of programming issues, dealing with sequences was easier than manipulating sets in this problem. Therefore: A tile system is a quadruple T = T , s, g, τ , where s, g and τ are as defined in Section 2, and T is a finite sequence of tiles (T1 , T2 , . . . , T|T | ) with no duplicates. To find tile systems with reduced size n and a given temperature τ , there are at most 4n∗τ +1 glues to be assigned to 4n tiles. By enumerating all the tile systems and testing them by simulating the assembly it is possible to solve the problem. Unfortunately, the naive enumeration approach requires testing Ω((4n∗τ +1)4n ) systems. Besides, testing each system requires running the simulation with all possible seeds for that system. The problem becomes intractable for small choices of n and τ . We need a dramatically better generation method. I designed such a technique using the following idea: Call S the class of all sets of tiles of size up to 7. Call C1 , C2 , . . . , C7 the sets of counters with sizes 1, 2, . . . , 7 respectively. Note that some or all of the Ck sets may be empty. In fact, it is easy to prove that C1 is empty. We wish to define some subset E of S such that for all 1 k 7, if Ck is non-empty, then E ∩ CK is non-empty. Testing if elements in E are counters or not is enough to solve the optimization problem. Trivially, we can make E = S, but we wish to find an E that is reasonably small and yet “easy” to enumerate using a computer. The approach is outlines in Fig. 10.5 I chose the set E by requiring the sequences of tiles in E to have some simple properties. Theorem 2 shows that only a relatively small subset of all the tile systems have to be tested. Before presenting Theorem 2, we will need some definitions: Without loss of generality, assume that the alphabet of glues used is Σ = {0, 1, 2, . . . , 4|T |}, and that ∀σ ∈ Σ, g(σ ) = 0 ⇔ σ = 0. Consider two tile systems T = T , s, g, τ and T = T , s , g , τ . T is said to be isomorphic to T iff: 1. τ = τ ∧ |T | = |T |.

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2. There exists a one to one mapping f : Σ → Σ such that: (a) ∀t ∈ T , ∃t ∈ T such that f (σN (t)) = σN (t ) ∧ f (σS (t)) = σS (t ) ∧ f (σE (t)) = σE (t ) ∧ f (σW (t)) = σW (t ), (b) ∀σ ∈ Σ, g(σ ) = g (f (σ )). 3. Dom(s) = Dom(s ). 4. ∀p ∈ Dom(s), f (σN (s(p))) = σN (s (p)) ∧ f (σS (s(p))) = σS (s (p)) ∧ f (σE (s(p))) = σE (s (p)) ∧ f (σW (s(p))) = σW (s (p)). Intuitively, two tile systems are isomorphic if they are the same up to glue renaming. Note that tile system isomorphism is an equivalence relation. Also observe that if T uniquely produces a coordinated supertile ΓT , then T uniquely produces a coordinated supertile ΓT , which can be thought of ΓT after renaming the glues as f indicates. Consider a tile system T = T , s, g, τ . For all tiles t and t in T , we say t glueprecedes t , and we will denote it as t ≺g t , iff t lexicographically precedes t . For all tiles t in T , we define the glue-strength signature of t as the ordered quadruple g(σN (t)), g(σE (t)), g(σS (t)), g(σW (t)). For all tiles t and t in T , we say t strength-precedes t , and we will denote it as t ≺s t , iff the glue-strength signature of t lexicographically precedes the glue-strength signature of t. Note that T is well ordered with respect to both ≺g and ≺s . T is said to be strengthordered iff for all i, j in {2, 3, . . . , |T |}, Ti ≺s Tj ⇔ i < j or (Ti and Tj have the same glue-strength signature and Ti ≺g Tj ). Consider a sequence of tiles T = (T1 , T2 , . . . , T|T | ). For all k ∈ {1, 2, . . . , |T |}, let NT ,k = max1ik σN (Ti ). Define ET ,k , ST ,k and WT ,k similarly. A tile system T = T , s, g, τ is said to be glue-compact iff for all i ∈ {2, 3, . . . , |T |}, σN (Ti ) NT ,i−1 + 1 and σE (Ti ) ET ,i−1 + 1 and σS (Ti ) ST ,i−1 + 1 and σW (Ti ) WT ,i−1 + 1. We will define a set of properties a tile system T = T , s, g, τ may have. 1. T has the P0 property iff for all t ∈ T g(σE (t)) + g(σE (t)) +g(σS (t)) + g(σW (t)) τ . 2. T has the P1 property iff T is strength-ordered. 3. T has the P2 property iff T is glue-compact. 4. T has the P3 property iff σN (T1 ) = 1 ∧ g(1) = τ ∧ σS (T1 ) = 1 ∧ g(σE (T1 )) + g(σS (T1 )) + g(σW (T1 )) τ ∧ g(σE (T1 )) g(σW (T1 )) ∨ g(σE (T1 )) = g(σW (T1 )) ∧ σE (T1 ) σW (T1 ). 5. T has the P4 property iff there exists exactly one tile t ∈ T such that σS (t) = 1. 6. T has the P5 property iff for the tile t ∈ T such that σS (t) = 1 ∧ (σE (t) = 0 ∨ σW (t) = 0 ∨ g(σS (t)) τ ) [σE (t) = σW (t) ⇒ g(σE (t)) < τ ]. |T | 7. Let NT = i=1 {σN (Ti )} − {0}. Define ET , ST and WT similarly. Let GN,T ,σ = {t | t ∈ T ∧ σN (t) = σ }. Define GE,T ,σ , GS,T ,σ and GW,T ,σ similarly. T has the P6 property iff ET = WT and ∀σ ∈ NT [(GN,T ,σ ⊆ GS,T ,σ ) ∧ g(σ ) = τ ⇒ (|GN,T ,σ | 1 ∨ |GS,T ,σ | 1)].


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Theorem 2. For all shapes S, if there exists a line-based tile system TS that uniquely produces S, then there exists a line-based system T S with properties P0 , P1 , P2 , P3 , P4 , P5 and P6 such that T S uniquely produces S and the reduced size of T S is not greater than the reduced size of TS . Proof outline. Trivially, P0 is a necessary condition, otherwise there would exist a tile that would not be attachable ever. P1 can be always achieved by permuting tiles in T . P2 can be always achieved by renaming glues. P3 follows from the necessary condition of having at least one glue with strength equal to the temperature on the south side of some tile, and the possibility of renaming glues. P4 is a necessary condition for termination. P5 follows from the necessary condition of having at least one glue with strength equal to the temperature on the south side of some tile, and the possibility of renaming glues, also from the need of continuing the assembly after the attachment of that tile. P6 means that there is no point in using a glue with positive strength on a west side if it is never used on an east side, and also because of the analogous argument for the north, south and east directions. Note that this result is not limited to the problem of finding counters at temperature 2. Even though the derivation of the properties described here was motivated by that problem, subsequent analysis showed that the theorem was applicable to a much wider set of shapes. The extension to the more general setting of line-based systems was so natural that I could not resist generalizing the results. 4.1.1. Searching for the Smallest Counter Given n and τ , it is relatively easy to write a program that outputs exactly the set of all tile systems with n tiles, temperature τ and with properties P0 , P1 , . . . , P6 . To address the problem of testing if a tile system uniquely assemble a rectangle of a given size, it is possible to use an algorithm derived by Kempe and Rothemund [2]. This algorithm receives as inputs a tile system T and a shape A. In polynomial time (in the size of A) the algorithm determines whether T uniquely produces A or not. The algorithm works essentially by simulation of the assembly process. Since in our exploration of tile systems we do not know what the seed-line looks like the tester had to try out every possible seed-line and test if the resulted tile system uniquely assembled the desired rectangle. The implementation of the Kempe–Rothemund algorithm I used was not complete. It was, in fact, a weaker test. My version of the test could accept a tile system that did not uniquely assembled a given rectangle, as long as it assembled the rectangle. There was no real reason for me to use this incomplete implementation, other than not having to write an extra few dozen lines of code. Some heuristic reasoning convinced me that it was almost impossible for the program to give a wrong result. Fortunately, as we will see later, I was wrong.

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Although the size of the search space was dramatically reduced by using Theorem 2, the improvement was not good enough to solve the problem completely. It took two weeks of computer time to generate and test all sequences of size 5 or less in E. Simple extrapolation suggested that it is possible to explore all sequences of size up to 6 using a small computer cluster (16 machines) in about one week. The case of 7 tiles could be cracked using a fast supercomputer in about one year. Using that much computer power to solve this problem would wasteful, but it is clear that the simple search optimization techniques I used were effective. A handful of simple properties transformed a completely intractable problem into a tractable one. Computer based exhaustive search based on Theorem 2 yields the following result: Theorem 3. There is no tile system with reduced size less or equal to 5 that solves the GPC problem at temperature 2. Since it did not look easy to obtain substantial improvements in the running time, I decided to restrict the GPC problem. The restriction did not look as interesting as the original problem, but at least it was possible to solve it. The general idea was to “lower the standards” and look only for counters of a very special kind. For all tile sets T , a binary labeling map is a function m : T → {0, 1}. Intuitively, it labels each tile with either a 0 or a 1. A tile system T = T , s, g, τ is a full-binary system iff: 1. T solves the GPC problem. 2. All coordinated supertiles of T are full. 3. There exist a labeling map for T such that the terminal coordinated supertiles with rectangular shapes have the following property. Each row after the labeling has to represent a binary number one unit bigger than the number represented in the row immediately below it. In addition, the top-most row has to represent 2w − 1, where w is the width of the rectangle. The Full-Binary Counter (FBC) problem: Find a full-binary tile system. We already know a full-binary tile system, namely, the one described in Section 3. We may wonder if there exists an FBC with smaller reduced size. The gist of the idea to reduce the search space is easily explained with help of Fig. 10.6. Assume there exists a FBC with reduced size 7. Also assume we know the labeling of the tiles. In particular, the counter should assemble a 6 × 3 rectangle. Now assume we know the tile at each position in that rectangle. Now consider the particular example depicted in Fig. 10.6. Since the supertile must be full, it is clear that the north glue of tile T4 should be the same as the south glue of


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Fig. 10.6. Derivation of additional constraints to speed up the search for full binary counters.

tile T3 . We can derive a set of similar constraints if we analyze all the junctions in the rectangle. All these constraints reduce the total number of sequences of tiles to enumerate. Of course, we do not know in advance a correct labeling and the mapping of tiles into positions. Therefore, we are forced to try out every possible labeling and every possible mapping. The only mappings we are interested in are those that generate rectangles that “look like” a sequence of consecutive binary numbers. However, the constraints derived from analyzing the junctions allow to reduce the size of the search space so much that the whole approach achieves a significative speedup. It was fairly easy to add these ideas to the previous generator. The obvious drawback of the approach is that we are looking for very special counters. Perhaps there are counters with reduced size at most 7 that assemble rectangles with glue mismatches in some junctions. The generation method we just described is not guaranteed to find them. It was only possible to prove that if there is a FBC with reduced s 7 the generator will generate a FBC with reduced size s. The testing program was not changed at all. After two weeks of computation, the generator/tester programs finished exploring the search space. One conclusion extracted from the exploration was: Theorem 4. There is no tile system with reduced size less or equal to 6 that solves the FBC problem at temperature 2. The output of the complete run was a list of about 80 candidate counters. The first candidate in the list was the one depicted in Fig. 10.7. Figure 10.8 shows how the tiles can assemble a 7 × 3 rectangle. Interestingly, it is easy to see just

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Fig. 10.7. Set of tiles generated by the computer search.

Fig. 10.8. Supertile assembled by the counter with 7 tiles.

by analyzing Fig. 10.8 that the counter is not binary! In any case, it just seemed to generate rectangles of all sizes when the appropriate seed row was used. The tester was run repeatedly with many rectangle sizes, and in every case the tester found a sequence of attachments that produced the desired result. The biggest example I tried was a 2048 × 12 rectangle.


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Fig. 10.9. Fixed set of tiles that actually solves the GPC problem at temperature 3.

In spite of the overwhelming computational evidence, we still had to rule out the possibility of the construction being a false positive. All my attempts to prove the construction was correct failed. The main difficulty in analyzing the construction was that I could not find a simple underlying principle that makes the process work, such as the “add one in binary” process. Eventually, Ashish Goel found a trivial counterexample, proving the construction was flawed because it did not uniquely produced rectangles. The counterexample would have been detected by a correct implementation of the Kempe–Rothemund algorithm and the set of tiles would have been discarded. However, the “impossible” happened and we obtained a false positive. The striking surprise was that Ashish Goel also found a simple way to modify the construction, shown in Fig. 10.9, that results in a correct counter. The fix consisted in adding one to the strength of all glues that appear on horizontal sides and increasing the temperature to 3. Proving that the fixed construction works is not trivial. Even though we understand the problem well enough to produce an inductive proof, we have not yet been able to find an intuitive principle that describe the underlying algorithm that makes the counter work. Another pleasant surprise came from the analysis of the time complexity. Even when this counter was found solely by trying to minimize the reduced size, it still assembles rectangles in asymptotically optimal time. It is in fact slightly faster (constant improvement) over the construction described in Section 3. Both the proof of correctness and the time complexity analysis have not been published yet [8].

5.

Conclusions

The following table summarizes all the results obtained during the research described in previous sections.

242

τ =3 τ =2

– – – – – – – –

Chapter 10 n=2

n=3

n=4

n=5

n=6

n=7

n=8

B ? ×

B ? ×

B ? ×

B ? ×

B ? ×

B ? †

B √ √

G ? ×

G ? ×

G ? ×

G ? ×

G ? †

G √ ?

G √ √

τ : The temperature. n: The reduced size of the tile system. G: Solves the GPC problem. B: Solves the FBC problem. √ : There is solution and it is known. ×: No solution is possible. †: Open problem, but tractable. ?: Open problem.

The remarkable aspects of the derivation of the theorems described here can be summarized as follows. – The counter with reduced size 8 was designed by trying to find a very easy to understand tile system that worked at temperature 2. There was no initial attempt to make the reduced size small or the running time optimal. The simple design just had the two characteristics. – The counter with reduced size 7 is not an FBC and finally worked at temperature 3, while the computerized search was designed to look for an FBC at temperature 2. In short, it was outside the search space. – The counter with reduced size 7 assembles in asymptotically optimal time, even when the computerized search was just trying to optimize the reduced size. – The set of 7 tiles suggested by the computer did not work, and yet it passed several tests based on simulations. The Kempe–Rothemund algorithm is able to detect the particular problem the set has, but the incomplete implementation I used did not reject the set. There was not a good reason to avoid programming the full Kempe–Rothemund test, but this “mistake” was pivotal in discovering the counter. If I had done the “right thing” we would have never considered this flawed, but clever construction that was easy to fix. From a higher level point of view, we can see in this chapter what it is seldom shown in many research publications. Even in theoretical research, serendipity can play a vital role. Also, strictly incorrect constructions may have some merit because they can serve as inspiration to derive correct results.


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Acknowledgements I would like to thank Ashish Goel, Len Adleman, Ming-Deh Huang, Qi Cheng, Dustin Reishus, Erik Winfree, Paul Rothemund and David Kempe for their collaboration and useful discussions.

References [1] L. Adleman, Q. Cheng, A. Goel, M. Huang, Running time and program size for self-assembled squares, in: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, ACM Press, 2001, pp. 740–748. [2] L. Adleman, Q. Cheng, A. Goel, M. Huang, D. Kempe, P. Moisset de Espanés, P. Rothemund, Combinatorial optimization problems in self-assembly, in: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM Press, 2002, pp. 23–32. [3] G. Aggarwal, M. Goldwasser, M. Kao, R.T. Schweller, Complexities for generalized models of self-assembly, in: Proceedings of Symposium on Discrete Algorithms, ACM Press, 2004. [4] Q. Cheng, P. Moisset de Espanés, Resolving two open problems in the self-assembly of squares, Technical Report 03-793, University of Southern California, 2003. [5] P. Moisset de Espanés, Fast, small and robust self-assembly, PhD thesis, University of Southern California, Los Angeles, 2005. [6] A. Goel, H. Chen, Q. Cheng, P. Moisset de Espanés, Invadable self-assembly, combining robustness with efficiency, in: Proceedings of Symposium on Discrete Algorithms, ACM Press, 2004. [7] P. Rothemund, M. Cook, E. Winfree, Self assembled circuit patterns, in: Proceedings of DNA Computing, Springer-Verlag, 2003. [8] P. Moisset de Espanés, A. Goel, Toward minimum self assembled counters, in: Proceedings of DNA Computing, Springer-Verlag, 2007. [9] A. Goel, Q. Cheng, P. Moisset de Espanés, Optimal self-assembly of counters at temperature two, in: Foundations of Nanoscience, 2004. [10] P. Rothemund, Theory and experiments in algorithmic self-assembly, PhD thesis, University of Southern California, 2001. [11] Paul W.K. Rothemund, Erik Winfree, The program-size complexity of self-assembled squares (extended abstract), in: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM Press, 2000, pp. 459–468. [12] H. Wang, Proving theorems by pattern recognition ii, Bell Systems Technical Journal 40 (1961) 1–42.

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Programmable Self-Assembly—Theoretical Aspects and DNA-Linked Nanoparticles B. Högberg, J. Boo, J. Liu-Helmersson, L. Glans, H. Olin* Mid Sweden University, Department of Engineering, Physics and Mathematics, Sundsvall, Sweden Abstract. We briefly present a method for the parameterization of assembly systems derived from their ability to form unique structures. The concept of bond uniqueness is introduced and we show how it influences the number of unique structures that a programmable, or algorithmic, selfassembly system can create. Further, we argue that programmable self-assembly systems create embedded, additional computation that is reflected in the complexity of the generated structures and show how this complexity is related to the bond uniqueness of the building blocks. A brief introduction to sticky graphs, a mathematical tool for modeling self-assembly systems, is given. From the theoretical discussions it becomes clear that building blocks for programmable self-assembly need to have at least four distinct, geometrically separated bonds. A scheme for the production of building blocks with well-directed bonds for programmable self-assembly using DNA-nanoparticles is presented. The introduced procedure is a completely bottom–up approach and can be used to produce quite advanced PSA building blocks like nanoparticle eight-mers with eight bonds. Initial experiments are presented.

1. Introduction The motivation for the increased interest in self-assembling systems is both technological and fundamental. The miniaturization of electronic components results in increasing fabrication costs [1]. A bottom–up building approach based on selfassembly has been discussed as an alternative method for nanofabrication. However, it is not clear how to build complex systems using simple self-assembly. One possible path might be to first self-assemble a scaffold of technologically simple building blocks, then, self-assemble the interesting devices on the addressable sites created by the scaffold assembly. Potential scaffold materials are for example DNA [2] or DNA-linked nanoparticles [3,4] as well as more advanced biological systems like proteins [5] or viruses [6]. Using metallization of the biomolecules, the scaffold can also act as an electrical contact between the devices assembled upon the scaffold [7]. * E-mail: [email protected].



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From a fundamental point of view, self-assembly has been studied as a naturally occurring computation process. One formal model for studies of programmable, or algorithmic, self-assembly is the Tile Assembly Model [8] where the building blocks, called tiles, are self-assembled into a square lattice. The model is an extension of Wang’s theory of tiling [9]. Programmable self-assembly is powerful, for example, it has been shown to be capable of universal computation [10]. The Tile Assembly Model has been used for investigation of, for example, the minimum number of tiles needed to self-assemble a square of a certain size (program size complexity) [8], the time complexity [11], optimal size and time complexity [12], or whether a given tile system uniquely produces a given shape [13]. Experimentally, DNA double-crossover molecules with four sticky ends, analogues to the four sides of the Wang tiles, have been demonstrated to assemble into two-dimensional lattices [2]. Another experimental demonstration, of a much more complex pattern, is the algorithmic assembly of DNA Sierpinski triangles [14]. The use of DNA as a scaffold material has been investigated experimentally [15]. Macroscopic systems based on millimeter-scale plastic tiles that float on a liquid [16,17] can also be used for experimental realization of algorithmic self-assembly [18]. For reviews on some of these self-assembly systems see [19,20]. In this chapter we review the different fundamental classes of self-assembly systems and the concepts of unique structures and bond uniqueness. We present our formal model for modeling of self-assembly systems: sticky graphs, and we briefly review some experiments using DNA modified nanoparticles to build nanostructures.

2.

Fundamental Aspects of Programmable Self-Assembly

In future applications where nanostructures are formed using a self-assembled scaffold, one would optimally like to assemble specific nanodevices on specific locations on the scaffold. These locations must thus be uniquely addressable. An addressable location should have a certain unique pattern that the nanodevice will recognize and stick to. The pattern could for example be a number of single stranded DNA sticking out form the scaffold and the nanodevice would have the complementary strands attached to it. Thus, a good self-assembly technology should produce patterns containing irregular, unique, structures. In the following we use the word tile to denote a general assembly building block. A tile have one or more functional edges with specific types of glues, or bonds. Two open bonds can form a bond if they are bitwise complements of each other. Depending on the types of open bonds on the self-assembly tiles and on the physical process that makes the tiles assemble, we classify the process according to the type of assembly it can produce.

Programmable Self-Assembly

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Fig. 11.1. Classes of self-assembly systems. (a) Crystal type assembly (CA). (b) Unique addressing type assembly (UA). In programmable, or algorithmic, self-assembly (PSA) (c) the bond type does not uniquely determine the tile type to be bonded (bu > 1). An incoming tile must interact with at least two other tiles in order to assemble correctly. The unique structures generated in each case are encircled in gray. For colour version, see Colour Plate Section.

In crystal self-assembly (Fig. 11.1(a)) patterns of tiles are reproduced throughout the assembly. Since a crystal self-assembly (CA) repeats itself, the neighborhood of each tile must be identical everywhere in the assembly. This leads to the conclusion that each open bond of a crystal tile can only make a bond with one specific type of tile. This constraint on the assembly process assures that each tile of a certain type has the same type of tile neighbors everywhere. In a unique addressing assembly (UA) (Fig. 11.1(b)), each tile type only occurs once and the tile position is completely defined [8]. Each bond must be specific as to what type of tile it can accept, like in the CA case above. There must also exist a unique way to arrange the tiles so that there are no more open bonds or so that the remaining open bonds are blocked by the assembly itself. A programmable self-assembly (Fig. 11.1(c)) (PSA) system produces assemblies where one can find both repeating and unique patterns of tiles. For this to happen, some, or all, of the open bonds of a PSA tile must be able to bond to more than one type of tiles. Since an assembled PSA tile must, according to the above definition, accept at least two different types of new neighbors, the process is not deterministic and thus not programmable; a single PSA tile alone cannot uniquely define the next tile in the assembly. One thus introduces the added criterion that a new tile must bind to at least two, already assembled, tiles; i.e. two tiles is the smallest structure that define the type of the third tile to be assem-

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bled. This feature seems to be the single most important factor for the creation of complex assemblies. In nature, this type of dependent binding has been shown to be an important feature. For example in the self-assembly of ribosomes certain proteins do only assemble if the growing ribosome assembly contains a certain pair of previously assembled proteins [21]. As noted above, one issue that makes self-assembly interesting for nanotechnology is the ability to construct scaffolds with uniquely addressable structures. In the following sections, we will show how it is possible to estimate the number of unique structures that a certain assembly system can produce. The discussion follows paper [22]. In each finite assembly, one can identify a certain number, S, of unique structures. A unique structure is a number of connected tiles that is not reproduced anywhere else in the assembly. As an example, the string ac is a unique structure in aaabacabca while ab is not. The minimum number of tiles needed to define a unique structure is denoted η or, number of tiles per unique structure. The symbol N will be used to specify the total number of tiles that constitute the assembly in the discussion that follows. The number of unique tile types will be denoted by w. The patterns in Fig. 11.1 are examples of a few simple cases where finding S and η is straightforward. In general it is always possible to divide an assembly into S unique structures of η tiles in each structure. A unique structure parser algorithm for finding the values of the pair η and S in any finite assembly, can be constructed. The pair (S, η) is defined to be: the maximum possible S, and the η that gives this maximum. The bond uniqueness, bu , for an assembly system is defined as the average number of tile types that each open bond can make a bond to. For example in the assembly system in Fig. 11.1(c) the bond uniqueness is equal to two because each specific open bond can only harbor two different tile types on average. The first tile in a unique structure can be chosen at will from the w available tile types. Once this tile is specified however, the following tile types can only be chosen from a subset of the tile types, containing bu tiles. If we consider a twodimensional assembly, the interior of a square of η tiles is completely determined by two of its edges. (Because each incoming tile is completely determined by its two neighbors, see page 247.) The two edges can be chosen in the following number of ways, which is the maximum number of possible ways to create an η-structure: √ 2 η−2

(1) √ Consider a two-dimensional assembly with N tiles (assume that it is N × N √ √ √ √ tiles big). There are ( N − η + 1)2 locations where a η × η-tiles structure may be located (result from 2D-word pattern matching, see for example [23]). Assume that each one of these structures must be unique and that we use the wbu

.

√


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maximum number of variations of η-structures that the tile system may produce. Then from Eq. (1) we get that √ √ 2 √ (2 η−2) wbu (2) = N − η+1 . Using (2) with the condition that Sη = N we get: 2 2 log bu S=N b2 √ 2 . log wu N − NS + 1

(3)

Equation (3) gives the maximum number of unique structures of a twodimensional PSA process of N total tiles with w unique tile types and bond uniqueness bu . For details of the derivation see Ref. [22]. The bond uniqueness parameter, bu , is strongly influencing the structural complexity of the generated assembly patterns. The reduced bond uniqueness, b˜u is defined b˜u = (bu − 1)/(w − 1). b˜u is 0 when the assembly is crystal or uniquely addressed and it is 1 when the assembly is random, i.e. any tile fits any bond. For PSA assemblies 0 < b˜u < 1. Deterministic complexity (introduced by Solomonoff–Kolmogorov–Chaitin [24]) measures the amount of randomness, or information, in a structure whereas statistical complexity [25,26] is based on the notion that both randomness and periodicity are equally simple to describe statistically. Objects are considered complex (having a high statistical complexity) if the amount of computation required to produce them is large. Additional computation is defined as the amount of computation needed to correctly place a tile at a growth site, in excess of bond matching computation. To see if an open bond matches another open bond is fundamentally equivalent to performing cb AND operations, where cb is the bond complexity (number of bits necessary to describe the bond). This computation must be performed in any assembly processes. The additional computation embedded in PSA processes is the ability to select the correct tile given more than one open bond (in principle, this is equivalent to performing nested AND operations). Here, we propose to use the additional computation as a statistical complexity measure. The amount of additional computation needed to select the appropriate tile type to attach (bind) to an assembly is proportional to the number of incorrect tiles that needs to be sorted out times their bond complexity. The number of bonds an incoming tile must make in a PSA process is denoted τ (called the temperature parameter in [8]). The number of tile types that the assembly process need to take “into consideration” at each assembly step is the bond uniqueness times the τ -parameter, bu τ . At least τ out of these bu τ tile types will be the same, namely the correct tile type(s) that fits all bonds that need to be filled. The number of incorrect tile types the process must consider is thus (bu − 1)τ or, using the reduced bond uniqueness, b˜u (w − 1)τ . This value needs

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Fig. 11.2. A plot of S (total number of unique structures) versus η (building blocks per unique structure) for different kinds of assemblies at constant number of total building blocks, N . The η-axis is logarithmic. At the top left corner unique addressing (UA) is found, where each building block constitutes a unique structure. At the opposite corner crystal type assembly (CA) is found, where only one unique structure is possible (the whole assembly) because of the periodicity of the pattern. In the middle different kinds of programmable, or algorithmic, self-assembly (PSA), is situated. The solid line represents the upper bound on S N/η. The inset shows the qualitative dependence (Eq. (4)) of the amount of embedded additional computation in an assembly with respect to the reduced bond uniqueness b˜u of the building blocks.

to be corrected for the fact that, as bu increases, the fraction of incorrect tile types is decreasing. (When bu reaches w all tile types will fit and there are no incorrect tile types to consider.) The fraction of “available” incorrect tile types is 1 − b˜u so the maximum additional computation, C, is thus given by C = cb τ (w − 1)b˜u (1 − b˜u )

(4)

where w, τ > 1, cb 1 and 0 b˜u 1. The maximum additional computation, in other words the maximum structural complexity, is maximal around b˜u = 1/2 (PSA) and vanishing for b˜u = 0 and b˜u = 1 (CA-, UA- and random-type assemblies), see the inset of Fig. 11.2.

3.

Sticky Graphs

Here, we will present a system for modeling self-assembly, that we call sticky graphs. We expect our system to develop over time, and the version presented here is just our first approximation. The sticky graphs we present are described in


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Fig. 11.3. An example of a sticky graph. “Atom” A is connected to atoms B and C, and has no more possible connections left, while each of the atoms B and C has one more unused potential connection, that other atoms might stick to. We think of the system showed as a molecule, that yet can grow larger.

the preprint [28] by one of the authors. Mathematical details can be found there; in the present text we will, as far as possible, avoid mathematical technicalities. Like many other modeling systems for self-assembly [27], the sticky graph system is based upon (mathematical) graphs and grammars. Most such systems have a lot of mathematical properties in common, and the mathematical theories applying to them will be very similar. The sticky graphs are no exception, and we believe that from a strictly mathematical point of view, the system does not contribute with anything new. However, we think that the modeling properties of the sticky graphs are nice, and therefore we have chosen to develop them as our modeling tool. Sticky graphs are a priori highly independent of geometry, and can be used for modeling in one, two or three dimensions, or whatever. By specifying the rules describing a certain set of sticky graphs in a suitable way, a desired geometry may be implemented when needed. The idea behind sticky graphs is a kind of molecular analogue. We think of particles as atoms, with certain adhesive properties. Each particle is of a certain type (type of atom), and the type of atom determines how the atom can bind to other atoms. This is done in terms of connections, where each (instance of a) type of atom has a certain set of possible/potential connections. A connection can be unused, or it can be used. An unused connection describes a way for the atom to—potentially—connect to another atom, and a used connection describes that the atom really is attached to another atom. This is exemplified in Fig. 11.3, where a simple graphic representation for sticky graphs is used. In Fig. 11.3 we see that sticky graphs are not graphs in the usual mathematical sense; there are no “unused connections” in a graph. Sticky graphs constitute a generalization of graphs, where we have both used and unused connections. A sticky graph can be used to describe the statics of a self-assembly situation. In the sticky graph we can picture every atom in our universe under consideration, of what type each atom is, what atoms are connected to what atoms, and what possible connections are left. Hence we can see (continuing to use the molecular analogue) all the molecules and what unused connections we have. Thus, a sticky graph is a snapshot of the state of our universe.

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Since self-assembly is a dynamic process, we also need to take the dynamics in account, and this is done by means of a grammar. A grammar is, loosely speaking, a way of describing how changes may take place in the universe. It consists of a starting state, i.e., a sticky graph describing the situation when the process starts, and a set of rules for how local changes can occur. These rules, that we call productions, give precise information on how unused connections can combine to give a new binding between atoms, or, how an actual binding can be broken up, leaving two unused connections. To take care of this, one needs to have a way of identifying substrates in a state (sticky graph). In [28], we define morphisms between sticky graphs. As in normal mathematical language, a morphism is a mapping that respects structures, and a morphism between sticky graphs is a mapping from one sticky graph to another in such a way that fundamental structural properties are preserved. By morphisms, we can describe what it means for one sticky graph to be embedded into another, and the concept of embedded sticky graphs is fundamental for describing productions, and hence the dynamics of sticky graphs. A production is a rule that says that if a certain sticky graph is embedded into the state we are considering, then we can cut this substructure out, replacing it by another substructure containing the same types of atoms, but with some alterations in the connections. Figure 11.4 gives a graphic description of a production, and in Fig. 11.5, we see how this production is applied to a subsystem of a slightly larger sticky graph. Here, a in the sticky graph corresponds to y in the production, b to z, and c to x. Some care has to be taken when describing productions. The stars (∗) appearing in the left-hand side of the production in Fig. 11.4 indicates a demand that the corresponding connection be unused. Unused connections without a star in the production means that it is insignificant whether or not the connection is used. Naturally, a used connection in the left-hand side production indicates a connection that must be used for the production to apply. For example, the production in Fig. 11.4 could not have been applied to the sticky graph in Fig. 11.6. All this, and some more, have to be carefully thought of when designing productions. With this care, sticky graphs is a powerful tool to in a suggestive way picture self-assembly situations. The mathematical rigor they are based upon (as

Fig. 11.4. A production.


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Fig. 11.5. The production in Fig. 11.4 applied to a sticky graph. Note that the production also could have been applied to connect b to d.

Fig. 11.6. A sticky graph for which the production in Fig. 11.4 is not applicable.

presented in [28]) gives a theoretical framework and is also done so that it allows construction of computer programs to handle e.g. simulation.

4.

Programmable Self-Assembly Using DNA-Linked Nanoparticles

It is clear from the previous sections that to implement PSA one needs building blocks of a certain complexity and diversity. It is not, for example, possible to implement PSA using parts that have the same kind of connections, or bonds, on all faces. Using these kinds of building blocks will inevitably lead to periodic, non-complex structures. A PSA building block must have at least two bonds, in order to be able to interact with at least two, already assembled blocks as discussed on page 247. Furthermore, if the assembly process is to continue and not be blocked, it must also have two bonds left for new, incoming, building blocks. Having only one bond left is not enough, because if an incoming block has just one connection left after it has made bonds to two already assembled blocks, it has reduced the number of open bonds in the assembly by one. This leads to the conclusion that to PSA assemble a structure of n tiles one need to start out with a seed with n+1 open bonds. So in practice a general PSA building block must have at least four bonds. It is possible to build three-dimensional structures

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using building blocks with only four bonds. However, one would then be required to create a kind of diamond like structure (using tetrahedron building blocks with functionalized corners). A more practical approach is to use space-filling building blocks like triangular prisms or cubes having at least 5, respectively 6 bonds (6, respectively 8 bonds, if the bonds are situated at the corners). Building blocks that have a size of tens of micrometers can be manufactured and functionalized using conventional microtechnology. Standard techniques has also been tried on colloidal particles. One example of a top–down approach for the functionalization of two distinct areas of 1 µm particles can be found in [29]. However, as the dimensions of the building blocks are reduced, it is increasingly difficult to functionalize different parts of the building blocks with different functions. When dealing with nanoscale objects, a top–down procedure for selective functionalization is no longer feasible. A new approach is needed. We here propose such a new method for the fabrication of nanoscale PSA building blocks using only bottom–up methods, taken from [30] and [31]. Note that the idea to use DNA-linked nanoparticles to build nanostructures is not new [3,4]. What is new here is that we introduce a method to break the symmetry and produce anisotropic building blocks, thus opening up a path towards self-assembly of complex, non-periodic, nanoparticle structures. Firstly, nanospheres of a suitable material are functionalized with two different types of single stranded DNA (ssDNA) (called particles of type I below). For example, for gold [32,33] and SiO2 [34] nanoparticles, there exist standard protocols for this. Secondly, nanospheres of a second kind are prepared in the same way but using yet another set of different DNA sequences (these are called type II particles). These spheres are then mixed in solution, see Fig. 11.7(a). Consider the case when nanospheres of type I have ssDNA sequences A and X and the nanospheres of type II have sequences B and Y. Linker molecules, consisting of a chain of complementary sequences to Y (cY) and X (cX) coupled by some sequence of bases that are irrelevant to the assembly process (cL), are added to the solution, see Fig. 11.7(b). As the linker molecules are added the spheres of types I and II will adhere to one another and form aggregates of different sizes. By controlling the temperature of the solution we can promote the formation of dimers. After a while the linker DNA is rinsed away, stopping any further aggregation of the nanoparticles. The dimers could then be separated from the rest of the aggregates by centrifugation, gel electrophoresis or some other mass separation technique. If necessary, the sticky ends of the linker molecules could be passivated by adding a solution of LY and LX DNA (Figs. 11.7(d)–(e)). The dimers produced in this way constitute simple building blocks with ssDNA of types A and B sticking out at different ends of the dimers. By starting out with particles coated with four specific DNA sequences and using several iterations of this technique one could make four-mers and eight-mers to be used as more advanced PSA building blocks, Fig. 11.7(g).


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Fig. 11.7. Scheme for nanoparticle PSA building block assembly. (a) Two different sets of nanospheres are each coated with two (or more) single stranded DNA sequences A (dotted), X (gray) and B (light gray), Y (dash-dotted) respectively. These nanospheres are mixed in solution and DNA-linker molecules cX-cL-cY are added (b). The linker molecules hybridize with the complementary parts on the nanospheres, making the nanospheres bind together (c). Dimers can be purified out of the solution by chromatographic methods. Adding the complementary molecules (X-L and L-Y) (d) passivates the remaining sticky ends of the linker molecules (e). The only single stranded DNA left is of type A on one sphere and of type B on the other so the dimer in (e) is functionally equivalent to the one in (f). Using several iterations of these processes one can, for example, produce eight-mers with eight separate and specific binding sites to be used for PSA. For colour version, see Colour Plate Section.

Following the above method one can create a large number of different basic building blocks with specific sticky ssDNA on different faces of the blocks. As the basic building blocks are mixed together with blocks having complementary DNA, self-assembly will occur. Gold colloids with mean diameters ranging from 10 to 50 nm were purchased from G. Kisker GbR. Typical size distributions were about 15%, concentrations are approximately 0.07 nM for the 50 nm up to about 7 nM for the 10 nm particles. Thiol modified (5’ and 3’) and unmodified ssDNA were purchased from Cybergene AB. All oligos were HPLC-purified by the manufacturer. The lengths of the thiolated oligonucleotides were 16 code bases and 10 bases of consecutive T’s between the thiol group and the coding sequence. The ten T’s act as a separator between the particle and the hybridizing sequence. The coding sequences

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Fig. 11.8. (a) Photograph of the discrete red lines formed by agarose gel electrophoresis of 50 nm DNA coated nanoparticles with linker molecule added. (b) Electron micrograph of electrophoresis-purified nanoparticle dimers. For colour version, see Colour Plate Section.

were optimized to have cross-hybridization probability as small as possible. The sequences were also checked for unwanted hairpin and dimer formation. To this end we used software developed by ourselves and some online tools [35]. We have primarily used a subset of the DNA library for DNA-computing by Penchovsky and Ackermann [36]. This set of oligos is well optimized for uniqueness and the sequences are also optimized to have similar melting temperatures. The functionalization of the gold particles followed a modified version of the Storhoff protocol [37]. Equal amounts of 4 different thiolated oligos to a total of 1 nmole DNA are mixed separately and then mixed with 1.3 ml of the gold colloid. These samples are stored at 30 ◦ C overnight followed by the addition of a sodium phosphate buffer to 0.01 M (pH 7) and a NaCl solution up to 0.1 M. The addition of salt is done drop wise and in two steps separated by roughly two hours. After another 48 h at 30 ◦ C the samples are brought to 0.3 M NaCl. The colloidal solutions are then centrifuged twice at 14,000 rpm for 10–60 min (depending on particle size) with an intermediate rinsing. Finally the pellet containing the DNA-modified particles is redispersed in a 0.01 M phosphate, 0.3 M NaCl, 0.01% Azide solution. Particles of different DNA types were mixed together and the appropriate linker molecules added. After incubation for about 60 min at 20–60 ◦ C gel-loading buffer (dextrose) was added and the samples were allowed to cool down during gel loading. Agarose gel (0.8% w/v) electrophoresis was performed to separate the dimers [38] from single particles and from larger aggregates, see Fig. 11.8(a). The dimer bands were cut out from the gel using a scalpel and diced into smaller pieces. The dimers were retrieved from the gel slices using Nanosep MF centrifugal filters (Pall Corporation). SEM studies were performed on the 50 nm gold particles using a LEO-1450 EP electron microscope. The ssDNA successfully attached to the particles. This is clear from the fact that the gold colloids are stable in high salt concentration. The formation of


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dimers, trimers and larger aggregates has been verified by gel electrophoresis, see Fig. 11.8(a) and SEM photos, Fig. 11.8(b). The method seems to be a potential path towards programmable self-assembly of nanoparticle structures.1

5.

Conclusions

We have presented a method for the parameterization of assembly systems derived from their ability to form unique structures. We have also introduced the concept of bond uniqueness and showed how it influences the number of unique structures that a programmable self-assembly system can create. Furthermore, we have argued that programmable self-assembly systems creates embedded, additional computation that is reflected in the complexity of the generated structures. This structural complexity is directly dependent on the reduced bond uniqueness of the system. Sticky graphs, a mathematical tool for modeling self-assembly systems, is introduced to the reader. From the theoretical discussions it is clear that building blocks for programmable self-assembly need to have at least four distinct, geometrically separated bonds. A principal scheme for the production of building blocks, with well-directed bonds for programmable self-assembly using DNA-nanoparticles, has been presented. The introduced procedure is a completely bottom–up approach and can be used to produce quite advanced PSA building blocks like nanoparticle eight-mers with eight bonds. Initial experiments are presented.

References [1] R. Compañó (Ed.), Technology Roadmap for Nanoelectronics, European Commission, Office for Official Publications of the European Communities, Luxembourg, 2001. [2] E. Winfree, F.R. Liu, L.A. Wenzler, N.C. Seeman, Nature 394 (6693) (1998) 539–544. [3] C.A. Mirkin, R.L. Letsinger, R.C. Mucic, J.J. Storhoff, Nature 382 (6592) (1996) 607–609. [4] A.P. Alivisatos, K.P. Johnsson, X.G. Peng, T.E. Wilson, C.J. Loweth, M.P. Bruchez, P.G. Schultz, Nature 382 (6592) (1996) 609–611. [5] Shuguang Zhang, Biotechnology Advances 20 (5–6) (2002) 321–339. [6] A.S. Blum, C.M. Soto, C.D. Wilson, J.D. Cole, M. Kim, B. Gnade, A. Chatterji, W.F. Ochoa, T.W. Lin, J.E. Johnson, B.R. Ratna, Nano Letters 4 (5) (2004) 867–870. [7] Hao Yan, Sung Ha Park, Gleb Finkelstein, John H. Reif, Thomas H. LaBean, Science 301 (5641) (2003) 1882–1884. [8] P.W.K. Rothemund, E. Winfree, in: Proc. 32nd Ann. ACM Symp. Theor. Comp., New York, NY and Portland, OR, USA, 2000, ACM, 2000, pp. 459–468. [9] H. Wang, Bell System Tech. Journal 40 (1961) 1–42. 1 Recently this approach has been used to make dimer building blocks that assembles into curved structures [39]. This effect proves that the building blocks actually are anisotropically functionalized.

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[10] E. Winfree, in: R.J. Lipton, E.B. Baum (Eds.), DNA Based Computers, in: DIMACS Series in Discrete Math. and Theor. Comp. Sci., vol. 27, Amer. Math. Soc., Princeton, 1996, pp. 199– 221. [11] L.M. Adleman, Q. Cheng, A. Goel, M.D. Huang, in: ACM Symposium on Theory of Computing (STOC’01), 2001, pp. 740–748. [12] Q. Cheng, A. Goel, P. Moisset de Espanés, in: Proceedings of Foundations of Nanoscience: Self-Assembled Architectures and Devices, Snowbird, Utah, 2004, Sciencetechnica, 2004, pp. 219–226. [13] L.M. Adleman, Q. Cheng, A. Goel, M.D. Huang, D. Kempe, P. Moisset de Espanés, P.W.K. Rothemund, in: ACM Symposium on Theory of Computing (STOC’02), 2002, pp. 23– 32. [14] P.W.K. Rothemund, N. Papadakis, E. Winfree, Plos Biology 2 (2004) 2041–2053. [15] S.H. Park, H. Yan, J.H. Reif, T.H. LaBean, G. Finkelstein, Nanotechnology 15 (10) (2004) 525–527. [16] N. Bowden, A. Terfort, J. Carbeck, G.M. Whitesides, Science 276 (1997) 233–235. [17] K. Hosokawa, I. Shimoyama, H. Miura, Sensors and Actuators A: Physical 57 (1996) 117– 125. [18] P.W.K. Rothemund, Proc. Natl. Acad. Sci. USA 97 (3) (2000) 984–989. [19] N.C. Seeman, Nature 421 (2003) 427–431. [20] G.M. Whitesides, B. Grzybowski, Science 295 (5564) (2002) 2418–2421. [21] G.M. Culver, Biopolymers 68 (2) (2003) 234–249. [22] B. Högberg, H. Olin, J. Comput. Theor. Nanosci. 3 (3) (2006) 391–397. [23] A. Capri, A. de Luca, Int. J. Found. Comp. Sci. 15 (2004) 355. [24] R.J. Solomonoff, Inform. Contr. 7 (1–22) (1964) 224–254. [25] C.H. Bennett, in: Rolf Herken (Ed.), The Universal Turing Machine—A Half-Century Survey, Oxford University Press, Oxford, 1988, pp. 227–257. [26] J.P. Crutchfield, Physica D 75 (1–3) (1994) 11–54. [27] E. Klavins, IEEE Proc. Int. Conf. Robotics and Automation 5 (2004) 5293–5300. [28] J. Boo, Basic properties of sticky graphs and their morphisms, Report no 2-2005, Mathematics Mid Sweden University, 2005. [29] Z. Bao, L. Chen, M. Weldon, E. Chandross, O. Cherniavskaya, Y. Dai, J.B.-H. Tok, Chemistry of Materials 14 (1) (2002) 24. [30] B. Högberg, J. Helmersson, S. Holm, H. Olin, in: Proceedings of Foundations of Nanoscience: Self-Assembled Architectures and Devices, Snowbird, Utah, 2005, Sciencetechnica, 2005, pp. 219–226. [31] B. Högberg, J. Helmersson, S. Holm, H. Olin, Appl. Surf. Sci. 252 (15) (2006) 5538–5541. [32] L.M. Demers, M. Ostblom, H. Zhang, N.-H. Jang, B. Liedberg, C.A. Mirkin, J. Am. Chem. Soc. 124 (38) (2002) 11248–11249. [33] R.C. Jin, G.S. Wu, Z. Li, C.A. Mirkin, G.C. Schatz, J. Am. Chem. Soc. 125 (6) (2003) 1643– 1654. [34] Csogor, M. Nacken, M. Sameti, C.M. Lehr, H. Schmidt, Materials Science and Engineering: C 23 (1–2) (2003) 93–97. [35] Online tools for oligo-design, rna.tbi.univie.ac.at/cgi-bin/alifold.cgi, scitools.idtdna.com/ analyzer/. [36] R. Penchovsky, J. Ackermann, Journal of Computational Biology 10 (2) (2003) 215–229. [37] J.J. Storhoff, R. Elghanian, R.C. Mucic, C.A. Mirkin, R.L. Letsinger, J. Am. Chem. Soc. 120 (9) (1998) 1959–1964. [38] D. Zanchet, C.M. Micheel, W.J. Parak, D. Gerion, S.C. Williams, A.P. Alivisatos, J. Phys. Chem. B 106 (45) (2002) 11758–11763. [39] B. Högberg, H. Olin, Eur. Phys. J. D 43 (2007) 299–302.

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From Microscopic Rules to Emergent Cooperativity in Large-Scale Patterns Bosiljka Tadić* Department for Theoretical Physics, Jožef Stefan Institute, P.O. Box 3000, SI-1001 Ljubljana, Slovenia Abstract. Assembled large-scale systems of nano-particles or molecules can be represented by graphs (network structures), in which nodes and links have specified physical attributes. We consider two classes of complex spin-networks, in which spins with two degrees of freedom are attached to nodes of the network. We first demonstrate how different network geometries emerge when the microassembly rules and constraints are varied. We then study the dynamic response when these large-scale structures are driven by the magnetic field and show how the hysteresis curves and memory properties are affected by structural complexity of the underlying network.

1. Introduction Self-assembly processes, in which structurally complex large-scale systems of nano-particles [1,2] or molecules emerge, offer challenging prospectives both for applications and condensed matter physics. The key feature of the self-assembled systems is their structural and functional complexity. Even when the physical, chemical or biological properties of the integrating elements (i.e., molecules, nano-particles) are known in detail, the assembled large-scale structure often has new emergent properties that are not present at the level of individual elements. In this way, the self-assembled systems represent true complex systems, owing to the nonlinearity of the self-assembly processes. Considering several studied examples, one can find that the self-assembly often utilizes a distinct physical principle which makes the process possible. So far known processing procedures can be roughly divided into following categories: – Synthesis (e.g., evaporation methods), in which nano-crystals [3] or nanocrystal clusters [4] and nano-tubes [5,6] are obtained. In the classical case of the carbon nano-tubes, for instance, the assembly process can be viewed as adding two carbon atoms per evolution step and utilizing the underlying line symmetry of the interaction [7]. * E-mail: [email protected], url: http://www-f1.ijs.si/~tadic/.



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– Template-Patterning. The organization of nano-particles takes place by deposition and diffusion on a solid substrate with a pre-defined structure [8–11]. Examples include the organizing of the metallic nano-particles on the disordered alloy substrate with a pattern of domain walls [8,9], dislocations that form to relieve strains [10], and on the bacterial S-layers [11]. – Process-Aided Patterning. Far from equilibrium dynamic processes, usually fluid dynamics as, for instance, phase separation mixtures in polymers on a substrate [12], fluid-coupled motion with a nano-scale object immersed in viscous fluid [13], or irreversible evaporation of a solvent [14] have been employed to help pattern formation in thin films and restricted geometries. The instability in the semiconductor-heterostructure growth was shown to be a good environment for the self-organized formation of various nano-structures [15]. The assembly of linear and cross-linked patterns of nano-scale colloidal particles immersed into a nematic liquid crystal was aided by the ordering dynamics of the nematic near the phase transition point [16,17]. – Molecular Cross-Linking. The pre-fabricated spin-coated [18] nano-particles or C60 molecules can be cross-linked by chemically binding polymer chains [19]. The “programmed” cross-linking of the nano-objects attached to the DNA molecules into large-scale structure [20] is based on the evolution-refined recognition between compatible DNA strands. – Mechanical Cross-Linking. Virtually more freedom is experienced in design of the integrated electronic circuits of components or gates linked by wires [21], and in networks of Josephson junctions on superconducting films [22]. – Evolutionary Tunable Bio-Patterns. The other end of self-assembly processes are the evolutionary processes in which various network structures emerged in the living cells. These processes can be understood as self-correcting with the aim to optimize the cell function through the evolution-mutation steps. The laboratory experiments can only imitate these types of self-organized dynamics through repeated experiments on a given initial structure and by imposing an ad hoc optimization principle [23].

1.1.

Research Aims

The above short list of examples illustrates a variety of elements and processes that might be useful in the self-assembly of the integrated nano-scale devices. Having the elements to be integrated, a number of questions arise: What patterns are better with respect to the practical purpose of the large-scale device? What assembly processes are suitable for a given type of integration elements? What parameters in the assembly processes are relevant, i.e., whose manipulation may result in different structure and properties of the large-scale devices? Unfortunately, there are no simple answers to these questions. Almost in all large-scale “patterns” amenable to practical use a long history of experimenting

From Microscopic Rules to Emergent Cooperativity

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with the elements, parameters, and processing precedes the current knowledge of their properties and reproducibility. Future of the nano-technology depends on the processes of integration, controllable fabrication, and stability of the integrated devices. The current research of the self-assembly processes aims on finding some unifying principles behind certain classes of integrated complex patterns. In this sense, the science of self-assembly is only at its beginning. Abundance of the potential integrating elements and the appropriate assembly processes requires particular study in almost each case separately. This bottom–up approach is cutting various ways towards the potentially stable and reproducible assembly processes. On the other side are two conceptual problems that the theory is concerned with: (i) to identify classes of emergent structures with respect to varying microassembly rules and parameters; and (ii) to identify the structural elements of the emergent geometries which are essential for (expected) functional properties of the large-scale systems. The ultimate idea is to understand and control the procedure from the elements and processes to structure and function.

1.2.

Theoretical Background

In general, the self-assembly processes are nonlinear dynamic processes which involve driving, constraints and self-organization. Therefore, a controlled selforganized dynamics in large systems (as opposed to controlled chaos [24] in the case of few degrees of freedom) is needed in order to direct the self-assembly of materials. In this case one can expect that, by varying a relevant control parameter in an appropriate way, the system will converge to one or the other attractor of the dynamics. Near the attractors the system is in a metastable state, which is characterized with a large-scale ordered structure [25]. In fact, the occurrence of attractors of the dynamics is the only guarantee that different classes of structures can be identified. These classes can be quantitatively characterized by defining the appropriate measures. Such complex geometries are best represented by mathematical objects known as graphs (or networks). The nodes of the networks carry the objects with certain attributes, such as size and shape, chemical composition and valence, magnetic moment, anisotropy, surface roughness, etc., all of which can in a way contribute to the interactions between the integrating elements and to the parameters in the assembly process. Recognizing the relevant parameters and defining the micro-assembly rules is an important part of the research in the nano-patterned systems. Many sciences and methods are involved on this level of research [1–6,8–22]. On the other hand, the theoretical concepts are necessary in order to understand the emergence of structures from given microscopic rules and the functional characteristics of the large-scale geometries. This part of the research is suitably based on numerical experiments and modeling. The schematic flow diagram in Fig. 12.1 is showing

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Fig. 12.1. Schematic flow diagram of steps in self-assembly: From elements to rules (A), from rules to structure (B) and from structure to function (C) and back to the elements and/or rules (D).

the distinguishable steps in the research of self-assembly—from the elements to the integration rules, the emergent structures, and the physical properties of large-scale systems. In this chapter we discuss the theoretical approach by using numerical methods: Starting from defined assembly rules we show how the complex network structures emerge. In particular, we show how a small variation in the rules and constraints results in statistically different patterns. Then we study the collective dynamic response of these structures decorated with spins and driven by the magnetic field [26]. We relate the quantitative characteristics of the response to the relevant structural properties of the underlying pattern.

2.

Emergence of Complex Geometries

In this section we demonstrate the micro-assembly rules at work. First we introduce the rules of linking the elements (nodes) of a graph that evolves in time by addition of the nodes one after the other from i = 1, . . . , N the full size of the network. In the other example we use the same linking rules but applied within the whole network of N nodes, where we add links one after the other until = 1, . . . , L links are established. Subsequently, we explore the emergent structures in several quantitative details as described below.

2.1.

From Rules to Structures: The Web Graph

We first consider the evolving graph structure, which is grown by a systematic addition of nodes and linking. Starting from few initial nodes M0 , a node i is added at each growth step i = 1, . . . , N. The node is first attempted to link with a probability α˜ to a target node selected among i − 1 already present nodes. Else, with probability 1− α˜ one of the already existing nodes rewires or adds a new link to the selected target. The target node k and the source node n are selected with some probabilities pin and pout , which can depend on the properties of nodes. More precisely, here we select the pair of nodes to be connected according to the number of links of a given kind (out-going and in-coming links) that they


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have at the moment of linking [27]: pin (k, i) =

α + qin (k, i)/M , (1 + α)i

(1)

and α + qout (n, i)/M (2) , (1 + α)i where qin (k, i) and qout (n, i) denote the actual number of in-coming links at the target node k and the number of out-going links at the origin node n, respectively. The preferential attachment rules, originally introduced for in-linking [28–30] are known to lead to the scale-free network structures of incoming links. The implementation of the rules (1)–(2) as explained above is shown to result in a directed cyclic scale-free graph with both in-coming and out-going links hierarchy and link correlations [27]. The emergent structure is statistically similar to the structure of the real world-wide Web [27]. For this reason we term it the Web graph, to stress its cyclic and scale-free character, as opposed to often used uncorrelated scale-free graphs [28–30] and to random-graphs [31]. Here we apply the original model [27] with α˜ = α = 1/4 (the Web structure) and vary the parameter M (links-to-nodes ratio), which controls the level of clustering. An example of the emergent structure is shown in Fig. 12.2 (left). The striking feature of this topology is the occurrence of the high-connectivity nodes—hubs. When directions of links are observed, the two types of hubs are found—large in-coming connectivity and large out-going connectivity hubs [27]. pout (n, i) =

2.2.

The Preferential Static Network

In the other example we use the same rules as given in Eqs. (1)–(2), now applied to the fixed number of nodes i = N , representing the full size of the network. The links between selected pairs of nodes (n, k) are added one after the other until a given number L = M × N of links is placed. In addition, we observe the constraint that a given pair of nodes cannot have multiple directed links. If the selected pair (n, k) appear to already have a link n → k, then another target node is searched. A part of the network with N = 1000 nodes is shown in Fig. 12.2 (right) after L = 2000 links have been added. The visual difference between the two structures—which belong to two distinct classes also known as causal and homogeneous graphs—is quantified in more details below.

2.3.

Quantitative Characteristics of the Emergent Structures

Compared to the regular arrays, the complex networks may exhibit diversity of structural characteristics, which require specific algorithms for their identification and classification. Moreover, the network structure is of a crucial importance for

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Fig. 12.2. Part of the scale-free Web graph near the main hubs (left) and of the preferential static network (right) for N = 1000 nodes and L = 2000 links. (Plots are made by using Pajek [32].) For colour version, see Colour Plate Section.

its function, or the dynamic processes taking part on the network. Therefore, two primary tasks in the research of functional complex structures are: (1) to determine the adequate quantitative measures of the complexity of the structure, and (2) to find the structural characteristics which are relevant for the particular function of the network. In the graphs theory [31] mathematically rigorous characterizations of the random graphs have been developed. More recently similar approaches are being extended to the new class of structured scale-free graphs. On the other hand, the characterization of networks with respect to their function remains elusive for theoretical methods. The problem is more amenable to the numerical modeling. In particular, within the numerical models where both the structure and the parameters of the dynamic processes can be varied, attempts were made to classify network structures with respect to efficiency of the information transport [33,34]. A more systematic study of the structure–dynamics interdependences on complex networks and other types of dynamics still remains to be done. Here we show some basic structural characteristics of the two types of graphs that we introduced above. In general, the complex geometry of networks can be described by sets of parameters and functions which reveal local and global structural property, and elementary and hidden structure. All of these characteristics can be relevant for the ultimate efficiency of the network’s function, and, in principle, can be manipulated by changing the rules of linking. A short summary of the properties of the Web graph and the preferential static network is given in Table 12.1 and in Figs. 12.3 and 12.4. More detailed analysis of these network structures can be found in Refs. [27,33].


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Table 12.1. Structural properties of the emergent Web graph and the preferential static network Property / Graph

Web graph

Preferential static net

Connectivity profile

qin (n) ∼ n−γin qout (n) ∼ n−γout

qin = qout = An−τ exp[−(n/ξ )σ ]

Emergent distributions

P (qin ) ∼ qin in −τ P (qout ) ∼ qoutout

P (qin ) ∼ P (qout ) = Bq −τ exp[−(q/ξ )σ ]

Clustering profile Average clustering M = 1 In-coming–out-going link correlations Scaling relations Hidden structures

−τ

Δ(n) ∼ n−φΔ 0.3601 qout (j )i ∼ qin (i)−κ (disassortative) τin = 1/γin + 1 τout = 1/γout + 1 Superstructure

Δ(n) ∼ Cn−τ exp[−(n/ξ )σ ] 0.0075 ∼ const (uncorrelated) τ =1−σ – Communities

Graph inhomogeneity is the property that the number of links attached to a node vary throughout the graph. It is measured by the connectivity profile and emergent connectivity distributions. In causal graphs, as a rule, the nodes that are added to the network in earliest stages of growth collect large number of links, leading to the functional dependence of the connectivity on the age of the node [30]. In static networks the profile is made by using the ranking statistics, where rank one is the node with largest number of links, etc. For comparison, we show the ranking statistics of local connectivity in Table 12.1 and in Fig. 12.3. For the Web graph the connectivity varies with a power law both for in-coming and out-going links with different exponents. The occurrence of the power laws are proved analytically for the evolving networks with the linking rules given in Eqs. (1)–(2) [30]. On the other hand, the profile in the preferential static network can be fitted by a stretched-exponential function. This quantifies the visual differences between the two networks shown in Fig. 12.2. In addition, the global characteristics of the inhomogeneity is given by the probability distribution of the connectivity, shown in Fig. 12.3 (lower panels). Again, in the Web graph the asymptotic behavior is a power law with two exponents for in- and out-links, whereas it is a stretched-exponential in the preferential static network. Moreover, the exact scaling relations between the local and global connectivity hold for the causal scale-free networks [27,30]. Clustering is a measure of connectivity between neighbours of a node [30,31, 35]. It can be quantified by the number of elementary triangles attached to a node. The ranked profile of the number of triangles regardless of the direction of links are shown for the two networks in Table 12.1 and in Fig. 12.4 (bottom panels). The wide variation of the clustering in the Web graph is compatible with the occurrence of hubs, which are absent in the static network. The average clustering

266 Chapter 12 Fig. 12.3. Structural characteristics of the Web graph (left column) and the preferential static network (right column). In the top row: in- and out-connectivity qin and qout of node n averaged over several network realizations in the Web graph, left, and ranked connectivity against node rank n for one network sample in the preferential static network, right. Lower panels: Corresponding emergent probability distributions of the in- and out-connectivity, P (qin ) and P (qout ), for the two networks types and for the random-graph. Exponents γin , γout and τin , τout are defined in Table 12.1. For colour version, see Colour Plate Section.


Fig. 12.4. Bottom: Clustering profiles for the Web graph for link-to-node ration M = 1 (left) and for the preferential static network for M = 24 (right). Fit lines are showing the power-law and stretched-exponential dependences, respectively. Top: Link correlations in these graph structures. For colour version, see Colour Plate Section.

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coefficients, which are listed in Table 12.1 for the lowest link-to-node ratio, suggest different nature of the two networks: the highly clustered scale-free graph against low-clustering, almost random-graph structure of the preferential static network. Link correlations are usually considered as higher structural characteristics, which can be found by detailed inspection of the properties of the neighboring nodes, i.e., the average out-linking degree qout nn of the nodes attached to a node with a given in-linking degree qin . The results for the two networks are shown in Fig. 12.4 (top panels). According to the link correlations, the networks assortativity can be determined [35]. In particular, the Web graph appears to be disassortative, suggesting that the low-connectivity nodes are directly linked to the hubs. Whereas, the absence of correlations is characteristic for the main part of the preferential static network. The poorly connected parts in both graphs show assortative features. Higher-order structures. Apart from the elementary triangles, other types of distinguishable sub-graphs, motifs, modules, or communities—sets of nodes better connected among themselves than with the rest of the graph—can be present in structured networks. Some of these structures can be defined only dynamically as units performing a specific function, for instance feedback regulation in the genetic networks [36]. In the Web graph a super-structure consisting of nodes interlinked between the two hubs occurs, which acts as a powerful unit in the information traffic with local navigation rules [37]. The geometry of local communities on the graph can be detected by specifically designed algorithms [38]. Such communities are expected to be present on the preferential static network. Number of specific algorithms are developed for search of other hidden structural and functional properties of graphs and networks [39]. In the next section we consider spin dynamics on these two network topologies, the scale-free Web graph and the preferential static network, on which spins are attached to nodes and interact along the network links. In particular, we demonstrate how the field-driven spin reversal dynamics on these two network types depends on the specific structural elements shown in Figs. 12.3 and 12.4. It should be stressed that the same structural motif, say a triangle of nodes, may have different effects when it is a part of a graph with directed links, as compared to symmetric undirected graphs. In the present study, due to the symmetry of spin–spin interactions, it is appropriate to considered the underlying graphs are as undirected.

3.

Dynamic Response of Complex Spin Networks

In various technological applications, e.g., information transport and memory devices, the hysteresis loops with specific characteristics and stable magnetization reversal processes are required. In the classical memory materials with domain


269

structure, the reversal processes are related to the motion and pinning of domain walls. These are usually accompanied by the magnetization bursts, which are known as Barkhausen avalanches [40]. In recent years much effort was devoted to understanding the apparent fractal nature of the Barkhausen noise (trains of the magnetization bursts), whose universal scaling features appear as a finger-prints of the underlying domain structure. As a rule, in the field-driven disordered ferromagnets [40], ferroelectrics [41], and in systems with structural transformations [42], the strong pinning at defects induces small domains and thus small magnetization avalanches and reduced coercive fields. On the other hand, novel “artificial solids” of magnetic nano-particles assembled into complex network-like patterns offer a challenging possibility of design of hysteresis curves with different characteristics [26]. In these materials, due to complex topology and small depth of networks, the concepts of the domain walls and domain-wall motion are entirely changed. The structural inhomogeneity, quantified through the variation in the local connectivity and clustering of the network discussed in Section 2, are responsible for a range of new features in the reversal processes. Recently we have shown how the reversal occurs in the field-driven antiferromagnetic spin-networks with scale-free and random-graph structure [26]. The study of the spin reversal on these extreme types of network geometries suggested the mechanisms of genesis of the magnetization avalanches and their scaling features [43]. Furthermore, with increased clustering parameter M and the presence of closed cycles (triangles) on networks, it was shown [26] that the reversal can be affected by “spin frustration” [44], the incompatibility of the antiferromagnetic interactions along the cycle. Furthermore, the density of the cycles attached to a node, Δ(n) in Fig. 12.4, varies through the network, which can strongly affect the propagation of avalanches.

3.1.

Collective Dynamics of Spins on Structured Graphs

To investigate the spin dynamics in detail, in this section we implement a numerical algorithm of driven spin dynamics on networks. We consider the graphs shown in Fig. 12.2, the structure of which is specified by the adjacency matrix C, and by attaching a spin with two degrees of freedom S = ±1 to each node we make spin networks. On these spin network the pairs of spins interact along the links connecting neighboring nodes. Similar spin networks were studied recently in the literature [26,30,43,45,46]. In the present study we assume that the interaction is uniform and anti-ferromagnetic J = −1 among pairs of neighbor spins and the external field hext is applied to all spins. The Hamiltonian is given by [26] hi Si = −J Cij Si Sj − hext Si . H ≡− (3) i

i,j >i

i

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The sum over pairs of nodes is restricted to the locally available links, which are given by the positive elements Cij of the adjacency matrix. As mentioned above, the interactions along these links are symmetrical. The local fields hi , which are defined in Eq. (3) at all nodes on the network i = 1, 2, . . . , N, vary with the node connectivity profile q(i, N) of the graph (see Table 12.1). The system is slowly driven by increasing the external field in small steps δh starting from the negative saturation field hext = hmax = qmax +η, which is strong enough to make the homogeneous initial state (all spins “down”). The value of the saturation field depends on the largest connectivity qmax at the hub node in the network. In the algorithm, after each increase of the external field system is updated so that first the set of local fields {hi } for each node on the network i = 1, 2, . . . , N is computed according to Eq. (3); when the external field hext exceeds the local field at a node k, the spin at that node is flipped to align with the external field; then the local fields at the affected nodes, i.e., nodes which are directly connected to the node k, are examined for the instability and flipped if the energy condition for flipping is satisfied; the process is then repeated for the neighbors of these spins and so on, until no more unstable spins are found. Subsequently, the external field is increased again. The events between two successive updates of the external field comprise a magnetization avalanche. Note that due to frustration effects some spins can flip back during the same avalanche or in the subsequent avalanches. As a result, the net magnetization increases by certain fraction which is in general different from the number of flips at each field advance. In Fig. 12.5 we show the time dependence of three quantities which characterize the reversal process. It appears that the large number of flips (large avalanches) occur in the middle of the hysteresis loop, where the magnetization changes sign. This is compatible with the large magnetization jump near the coercive field hc , as opposed to the small steps in the tails of the hysteresis loop. As Fig. 12.5 shows, qualitatively similar behavior is found in both classes of networks. However, the structure of noise signal and consequently the increase of the magnetization in time is more steady in the case of less structured static network (bottom panel). Further difference between the two networks is found in the evolution of the density of domain walls, defined as the number of links with opposite spin sign at each end. In the less structured network the profile of the domain-wall density builds up gradually with the magnetization increase and reaches a maximum at the coercive field. On the contrary, in the disassortative scale-free network, the domain-wall density profile jumps from zero at the beginning of the hysteresis loop and remains finite during the entire reversal, eventually jumping back to zero when the reversal is completed. It suggests that the periphery nodes of the graph, that are attached with a single link to the hub (cf. Fig. 12.1), flip at early stage of the process and are held fixed until the field is large enough to reverse the hub itself. Then all the spins attached to the hub flip at once causing a huge avalanche. The network structure is also reflected on the


271

Fig. 12.5. Time dependence of the magnetization, number of flips (Barkhausen noise) and the domain-wall density in the Web graph (top panel) and in the preferential static network (bottom panel). For colour version, see Colour Plate Section.

distribution of size of the magnetization reversal avalanches. The distributions of avalanche sizes for two networks are shown in Fig. 12.6. The avalanche size distributions have a typical form with small avalanches following a power-law decay and large avalanches occurring in an oscillatory pattern [26], which is characteristic for each network topology. The largest peak corresponds to the flips of spins at nodes with least number of links at the graph boundary. In the disassortative scale-free graph a large number of such nodes is connected directly to the main hub (Fig. 12.2). The series of the smaller peaks correspond to flips of increasingly better connected nodes until eventually the

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Fig. 12.6. Distribution of sizes of the Barkhausen avalanches in two spin-networks: the cyclic scale-free Web graph topology (top) and the cyclic stretched-exponential network (bottom) for N = 103 nodes and for two values of the clustering parameter M. For colour version, see Colour Plate Section.

highly interconnected super-structure in the dense part of the graph is reached, where spin flips are strongly correlated to each other. The avalanches in this region of the graph are found to have a power-law distribution of size s according to the formula P (s) ∼ s −τs , with the exponent τs which can be related to the graph structure, cf. Fig. 12.6. We find τs = 2 for the spin avalanches on the Web graph, and τs = 1.5, for the case of the stretched-exponential static network structure. Furthermore, the even structure of the static network leads to predominantly power-law distribution of avalanche sizes, as compared to the small-depth Web


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graph, where the collective response with the oscillatory part is more pronounced. The slopes of the power-law distributions remain unchanged when the clustering parameter M is increased in both network types. This indicates that the clustering does not affect the scaling behavior, hence the inhomogeneous connectivity of the graph is mainly responsible for the observed universality of the avalanche dynamics. The numerical values of the scaling exponents are in the range from τs = 1 in the case of least organized random tree graph [26] up to τs = 2 in the case of highly structured scale-free Web graph. However, a precise analytical relation between the local connectivity profile and the scaling of the avalanche size distribution remains unknown.

3.2.

Hysteresis Loop and Memory of Complex Spin Patterns

The intermittent avalanche-type relaxation of spins on networks is an example of cooperative spin dynamics induced by the driving field. The avalanches are manifested as irregular magnetization jumps on the field–magnetization curve, Fig. 12.7. Formally, the occurrence of avalanches can be attributed to a nonlinear character of the dynamics, in which the energy is stored and then released when the local stability criteria are exceeded. This kind of dynamics is essential in the case of self-organized criticality, where the dynamic steady states with characteristic long-range correlations are built [25]. In these states characteristic system-wide fractal patterns are usually formed which can be quantified, among other measures, by the power-law distributions of the avalanche sizes bounded by the system size only. Moreover, in some cases, e.g., in the critical sandpiles, a spatial profile of the density can be related to the nature of the microscopic dynamics [47]. The role and a potential use of the self-organized criticality in other driven dynamical systems is still not clear. Occurrence of the dynamic critical states on the hysteresis loop in driven disordered solids has been discussed extensively [48]. In these systems, the avalanches appear in the result of motion and pinning of the domain walls at pinning centers due to disorder, two types of critical states can be identified. Based on the numerical evidence [48], the dynamic critical points (or lines) may describe the depinning of long domain walls at low disorder. Alternatively, by increased disorder a dynamic critical point may control the transition from the states with the depinning and an infinite magnetization jump, on one side, from the states with finite avalanches, on the other. In both cases the universal critical exponents of the collective critical behavior can be used to identify the domain structure of the solid. However, the true criticality—occurrence of an infinite magnetization jump when the system size tends to infinity—has been proved analytically only in the case of low disordered Bethe lattice [49]. The spin-networks that we study here represent entirely different type of systems, in which no disorder in the classical sense is present. These disorder-free

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Fig. 12.7. Hysteresis loops in the cyclic scale-free Web graph (top) and in the preferential static network (bottom) for different values of the clustering parameter M. For colour version, see Colour Plate Section.

geometries exhibit the hysteresis due to the topological inhomogeneity in the connectivity and clustering of nodes [26]. Consequently, the properties of the hysteresis curves are changed when the specific details in the structural characteristics of the networks are varied, as shown on Fig. 12.7. However, the structure effects are different in different network classes. In particular, in the cyclic scale-free network the hysteresis curves have a characteristic step-like structure with a finite magnetization jump at the coercive field hc . Furthermore, the coercive field increases with increased average number of links per node M, i.e., with increased clustering and “frustration”, whereas the


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finite magnetization jump is preserved. On the other hand, in the less structured preferential static network, the hysteresis loop is narrow and vertical for M = 1, corresponding to low average connectivity and a mild connectivity profile on this network, Fig. 12.2. When the number of links per node M increases, more clustering occurs and as a consequence, the spin “frustration” rises. The hysteresis loop becomes more horizontal and with decreased coercive field, similar to the classical disordered solids [50]. Therefore, in the case of the static network the mild stretched-exponential inhomogeneity profile can be considered as a sort of (topological) disorder and the clustering as a measure of spin “frustration”, in a full analogy to the spin-glasses and other strongly disordered solids with hysteresis behavior [40,50]. However, the above analogy with the classical materials does not apply to the spin networks with scale-free structures, having a power-law profile of the connectivity and clustering. The reasons can be directly attributed to the occurrence of the hubs with large connectivity, and the super-structure attached to them in the cyclic scale-free graph (Fig. 12.2). With increased network size N, the number of nodes which appear attached to the main hub increases as a power of N. Together with the disassortativity of the network, this preserves the large number of periphery nodes being linked to the hub, which thus contributes to the proportionally large magnetization jump at the coercive field. The clustering among the nodes in the super-structure also increases with number of links per node M, which is accompanied by the increased coercive field, Fig. 12.7 (top). Further comparisons can be made between different classes of scale-free networks. The observed increase of the coercive field on highly clustered Web graph is strongly related with the graph’s correlation property (disassortativity), as compared with weaker increase found on the uncorrelated scale-free graphs in Ref. [26]. In addition, the remanent magnetization—value of the magnetization at zero field—exhibits a small drop with increased clustering, Fig. 12.7 (top), compared to a large decrease of the remanent magnetization in uncorrelated scale-free graphs [26] and in the static network, Fig. 12.7 (bottom).

4.

Conclusions

Research of new functional materials with the self-assembly of complex patterns of nano-particles and analysis of their physical properties represents an appropriate basis for new technologies and, at the same time, a challenging interdisciplinary research field. We attempted to divide the current research into three interconnected strands, shown schematically on the diagram in Fig. 12.1. Obviously, the material research (strand A), where new materials are being synthesized, their stability and functionality tested using a variety of methods developed in chemistry, physics and biology, and adjusted to the nano-scale world, is of a key importance. The strands B and C, on the other hand, are concerned with a

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systematic search for the underlying principles, which will ultimately reduce the number of unnecessary trials, but more importantly, will make a reliable classification of structures, bridging the gap between the microscopic assembly rules and practical applicability of the large-scale devices. In this respect the numerical modeling approach is indispensable. We have demonstrated the breath of this approach using two model spin-networks and their hysteresis properties. The main features can be summarized as follows: – Emergence of Patterns. The essential feature of the complex large-scale systems is in their emergent structural properties, which are not present at the small-scale level. Furthermore, these properties depend crucially on certain details in the assembly process, though not all details are equally important. In the studied examples we have shown that highly complex scale-free structures can be grown using a minimal set of rules with preferential attachments. Moreover, we have demonstrated the importance of constraints in the distinction between evolving and static patterns: The same set of rules leads to different emergent structures due to different constraints that apply for the grown and the static situation. – Emergence of Function. The function of a complex pattern, measured by the properties of the dynamic processes on it, crucially depends on the topology of the underlying graph. However, some processes can utilize the topology in a better way than the others. Generally, a more complex structure induces more possibilities for the process adjustment and improvement. We have shown how the differences in the reversal processes arise on the two network topologies. These differences can be monitored in the process details in the numerical experiment, but also on the level of the hysteresis curves, which can be measured in the laboratory. – Tunability. The essential properties of the hysteresis curves can be related to the topology of the spin-network. In this way, the properties of the magnetization curves can be tuned by changing the structural characteristics of the network, which, on the other hand, can be controlled to certain extent by changing the assembly rules. On the scale-free spin networks new classes of behavior are distinguishable, in which an increasingly stable reversal process is obtained when the clustering of the network (and spin frustrations) is increased, in a marked contrast to the classical memory materials. – Cooperativity and Universality. The collective dynamic response of the fielddriven spin networks occurs in avalanches of spin flips due to both spin interactions and network topology. The distribution of avalanche sizes obeys a power-law behavior with a universal scaling exponent and a characteristic oscillatory behavior at large avalanches. Both of these features are unique for the network topology and can be used as noninvasive methods for the structure characterization. The cyclic scale-free spin networks have additional features


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which are not present in less structured random or stretch-exponential networks. This is the occurrence of a finite magnetization jump when the network size increases, which indicates a self-organized critical behavior on the hysteresis. The concept of the critical behavior is useful in the sense that classes of universality can be identified and marked by the sets of scaling exponents. Then measurements of the scaling exponents of the Barkhausen avalanches will suffice to determine the relevance of the parameters of the system at the assembly level. – Other Types of Dynamics. In our examples the cyclic scale-free network provides a better control of the reversal processes through well-defined coercive field, which increases with the clustering property of the network. These qualities are required in the memory devices. However, when other dynamic processes are considered, the two topologies may have both advantages and disadvantages. For instance, in the transport of the information packets the cyclic Web graph appears to be very efficient when a two-level local search is applied [33]. The efficiency in the packet transport is based on the central role of the hubs. However, this structure appeared to be vulnerable for jamming when the packet density increases over a critical threshold [34,37]. Whereas, the absence of hubs in the preferential static network, makes the majority of nodes almost equally involved in the traffic process. In both cases, the cooperativity of the network activity is remarkable, and can be measured for instance by the correlations in packet streams. In the constant packet density regime, the power spectrum of the network activity signal exhibits a long-range correlated (antipersistent) behavior, with a scaling exponent which is unique for each network structure [51]. To make the theory practical, the ultimate goal would be to go back to the synthesis of the selected network structure with best functional properties. In this respect a range of practical problems needs to be solved, depending on the specific example. For instance, growing an electronic circuit as the scale-free structure can be easier than synthesis of such a structure of nano-particles within a reaction–diffusion process. In particular, a careful pre-processing of nanoparticles might be needed to make them suitable for nonrandom (preferential) linking. In conclusion, we have demonstrated that the practical assembly of functional materials, which involves nonlinear dynamic processes, may benefit from the numerical modeling in two ways. First, numerical experiments reveal the nature of the structure–function interdependence and help selecting the classes of structures with desired characteristics. Second, for a given structure type the relevant control parameter can be determined and the range of its values suggested in which the parameter tuning can be beneficial for the process.

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Acknowledgements Author thanks support by the Program P1-0044 of the Ministry of High Education, Science and Technology (Slovenia), and partial support and motivation for this work through EU Project MRTN-CT-2004-005728.

References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19]

[20] [21] [22] [23]

[24] [25]

M. Li, H. Schnablegger, S. Mann, Nature 402 (1999) 393. P. Moriarty, Rep. Prog. Phys. 64 (2001) 297. C.B. Murray, C.R. Kagan, M.G. Bawendi, Science 270 (1995) 1335. Z.L. Wang, Adv. Mater. 10 (1998) 13. M. Remškar, A. Mrzel, Z. Škraba, A. Jesih, M. Cheh, et al., Science 292 (2001) 479. L. Krusin-Elbaum, D.M. Newns, H. Zeng, V. Derycke, J.Z. Sun, R. Sandstrom, Nature 431 (2004) 672. M. Damnjanović, I. Milošević, T. Vuković, N. Sredanović, Phys. Rev. B 60 (1999) 2728; M. Damnjanović, I. Milošević, E. Dobradžić, T. Vuković, B. Nikolić, Phys. Rev. B 69 (2004) 153401. R. Plass, J.A. Last, N.C. Bartelt, G.L. Kellog, Nature 412 (2001) 875. R. van Gastel, R. Plass, J.A. Last, N.C. Bartelt, G.L. Kellog, Phys. Rev. Lett. 91 (2003) 055503. H. Brune, M. Giovannini, K. Bromann, K. Kern, Nature 394 (1998) 451. W. Shenton, D. Pum, U.B. Sleytr, S. Mann, Nature 389 (1997) 585. M. Böltau, S. Walheim, J. Mlynek, G. Krausch, U. Steiner, Nature 391 (1998) 877. M.R. Paul, M.C. Cross, Phys. Rev. Lett. 92 (2004) 235501. E. Rabani, D.R. Reichman, Ph.L. Geissler, L.E. Brus, Nature 426 (2003) 271. J. Stangl, V. Holý, G. Bauer, Rev. Mod. Phys. 76 (2004) 725. I.I. Smalyukh, S. Chernishuk, B.I. Lev, A.B. Nych, et al., Phys. Rev. Lett. 93 (2004) 117801. I. Muševiˇc, M. Škrabot, D. Babiˇc, N. Osterman, et al., Phys. Rev. Lett. 93 (2004) 187801. P. Moriarty, M.D.R. Taylor, M. Brust, Phys. Rev. Lett. 89 (2002) 248303; J.N. O’Shea, M.A. Philips, M.D.R. Taylor, P. Moriarty, et al., Appl. Phys. Lett. 81 (2002) 5039. B. Narymbetov, A. Omerzu, V.V. Kabanov, M. Tokumoto, H. Kobayashi, D. Mihailovic, Nature 407 (2000) 883; A. Omerzu, M. Tokumoto, B. Tadić, D. Mihailovic, Phys. Rev. Lett. 87 (2001) 177205. E. Winfree, F. Liu, L.A. Wenzler, N.C. Seeman, Nature 394 (1998) 539; A.P. Alivisatos, K.P. Johnson, X. Peng, T.E. Wilson, et al., Nature 382 (1996) 609. R. Ferrer i Cancho, Ch. Janssen, R.V. Solé, Phys. Rev. E 64 (2001) 046119. G. Ghigo, A. Chiodoni, R. Gerbaldo, L. Gozzelino, et al., Supercond. Sci. Technol. 12 (1999) 1059. K. Kaneko, On recursive production and evolvability of cells: Catalytic reaction network approach (Chapter 27), in: M. Toda, T. Komatsuzaki, T. Konishi, T. Berry, S.A. Rice (Eds.), Geometric Structures of Phase Space in Multidimensional Chaos: A Special Volume of Advances in Chemical Physics, Part B, vol. 130, John Wiley & Sons, Inc., 2005. A.S. Mikhailov, A.Yu. Loskutov, Foundations of Synergetics II: Chaos and Noise (Section 6.5), Springer, Berlin, 1996. H.J. Jensen, Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems, Cambridge Lecture Notes in Physics, Cambridge University Press, Cambridge, 2003.

From Microscopic Rules to Emergent Cooperativity [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

[41]

[42]

[43] [44] [45] [46] [47] [48]

[49] [50] [51]

279

B. Tadić, K. Malarz, K. Kulakowski, Phys. Rev. Lett. 94 (2005) 137204. B. Tadić, Physica A 293 (2001) 273; Physica A 314 (2002) 278. R. Albert, A.-L. Barabasi, Rev. Mod. Phys. 74 (2002) 47. S. Dorogovtsev, J.F.F. Mendes, A.N. Samukin, Phys. Rev. Lett. 85 (2000) 4633. S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks: From Biology to the Internet and the WWW, Oxford University Press, Oxford, 2003. B. Bollobás, Modern Graph Theory, Springer, New York, 1998. V. Batagelj, A. Mrvar, Pajek program for large network analysis, http://vlado.fmf.uni-lj.si/ pub/networks/pajek/doc/pajekman.htm. B. Tadić, S. Thurner, Physica A 332 (2004) 566; Physica A 346 (2005) 183. R. Guimera, A. Diaz-Guilera, F. Vega-Redondo, A. Cabrales, A. Arenas, Phys. Rev. Lett. 89 (2002) 248701. M.E.J. Newman, The structure and function of complex networks, SIAM Rev. 46 (2003) 167. J.M. Bower, H. Bolouri (Eds.), Computational Modeling of Genetic and Biochemical Networks, MIT Press, Cambridge, 2001. B. Tadić, S. Thurner, G.J. Rodgers, Phys. Rev. E 69 (2004) 036102. H. Zhou, Phys. Rev. E 67 (2003) 041908; Phys. Rev. E 67 (2003) 061901. R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows—Theory, Algorithms, and Applications, Prentice–Hall, New Jersey, 1993. K.L. Babcock, R.M. Westerfield, Phys. Rev. Lett. 64 (1990) 2168; P.J. Cotte, L.V. Meisel, Phys. Rev. Lett. 67 (1991) 1334; D. Spasojević, S. Bukvić, S. Milošević, H.E. Stanley, Phys. Rev. E 54 (1996) 2531; E. Pupin, Phys. Rev. Lett. 84 (2000) 5415. V.Ya. Shur, Phase Transitions 65 (1998) 49; E.V. Colla, L.K. Chao, M.B. Weissman, Phys. Rev. Lett. 88 (2002) 017601; B. Tadić, Eur. Phys. J. B 28 (2002) 81. E. Vives, J. Ortín, L. Mañosa, I. Ráfols, et al., Phys. Rev. Lett. 72 (1994) 1694; F.J. Pérez-Reche, E. Vives, L. Mañosa, A. Planes, Phys. Rev. Lett. 87 (2001) 195701; F.J. Pérez-Reche, B. Tadić, Ll. Mañosa, A. Planes, E. Vives, Phys. Rev. Lett. 93 (2004) 15701. K. Malarz, J. Krapińska, K. Kulakowski, B. Tadić, Physica A 373 (2007) 785. K.H. Fisher, J.A. Hertz, Spin Glasses, Cambridge University Press, New York, 1991. D.-H. Kim, G.J. Rodgers, B. Kahng, D. Kim, Spin glass phase transition on scale-free networks, Phys. Rev. E 71 (2005) 056115. M. Bauer, S. Coulomb, S.N. Dorogovtsev, Phys. Rev. Lett. 96 (2006) 109901. B. Tadić, D. Dhar, Phys. Rev. Lett. 79 (1997) 1519. J.P. Sethna, K. Dahmen, S. Kartha, J.A. Krumhansl, et al., Phys. Rev. Lett. 70 (1993) 3347; E. Vives, A. Planes, Phys. Rev. E 52 (1995) R5; B. Tadić, Phys. Rev. Lett. 77 (1996) 3843; B. Tadić, U. Nowak, Phys. Rev. E 61 (2000) 4610. S. Sabhapandit, D. Dhar, P. Shukla, Phys. Rev. Lett. 88 (2002) 197202. J.R. Petta, M.B. Weissman, K.P. O’Brien, Phys. Rev. E 54 (1996) R1029; J. Toulouse, B.E. Vugmeister, R. Pattniak, Phys. Rev. Lett. 73 (1994) 3467. B. Tadić, G.J. Rodgers, S. Thurner, Transport on complex networks: Flow, jamming and optimization, Int. J. Bifurcation and Chaos 17 (2007) 2363–2385.

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Automated Self-Assembling Programming Lin Li1 , Peter Siepmann1 , James Smaldon1 , German Terrazas1 , Natalio Krasnogor* Automated Scheduling, Optimisation and Planning Research Group, School of Computer Science and IT, University of Nottingham, NG8 1BB, Nottingham, United Kingdom Abstract. In this chapter we explore various facets of the interplay between natural computing and self-assembly as they pertain to automated self-assembling programming. In particular we focus on two complementary research issues, namely, the automated control and programming of model systems that self-assemble into specific configurations and, on the other hand, the use of self-assembling metaphors and model systems to implement new ways of performing computation. These “two sides of the self-assembly/computation coin” are tightly linked together, and advances in one of them could help to pave the way for advances in the other. Thus, we hope that this chapter will serve as an inspiration to other computer scientists to immerse themselves in the wealth of opportunities and problems begging for solutions [L. Adleman, Q. Cheng, A. Goel, M. Huang, D. Kempe, P. Moisset de Espanes, P.W.K. Rothemund, Combinatorial optimization problems in self-assembly, in: Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), ACM Press, 2002], that self-assembly related research presents.

1. Self-Assembly, Self-Organisation and Natural Computation Nature inspired computation is older than most people think. In the early fifties, A. Turing, one of the founding fathers of the computer sciences, proposed computational models of pattern formation in living systems [2] based on systems of coupled reaction–diffusion equations giving rise to spatial patterns, i.e. selforganisation, in chemical concentrations. J. von Newman demonstrated that it was possible to build self-replicating abstract machines [3] while A. Lindenmayer, soon after, introduced L-systems for modelling artificial plants [4]. Perhaps the best known example of a paradigm shift affecting natural computation research was the publication of John Holland’s book Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence [5]. Although some previous papers did use natural selection as an inspiration for new ways of solving problems, Holland’s book * E-mail: [email protected], url: http://www.cs.nott.ac.uk/~nxk. 1 These authors contributed equally.



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effectively launched the field of evolutionary computation and all its derivatives, thus transforming many fields far beyond the narrow remit of natural computation. It is our belief that a revolution in nature inspired computation is taking place today. Self-assembly is one of nature’s most powerful and ubiquitous engineering mechanisms. It is not surprising then that computer scientists are starting to investigate both the computational roots of self-assembly as well as its potential for new ways of carrying out computation. That is, computer scientists are starting to notice the fundamental appeals of self-assembly (as they did before with natural selection, self-reproducing automata, etc.) for solving computational problems.2 The remarkable observation by Rothemund [6] “Self-assembly and computation are linked by the study of mathematical tiling” lies at the core of our research programme. More specifically, Rothemund, Winfree, Reif, LaBean and others (e.g. [7] and references therein) have shown that universal computation may be carried out by self-assembling discrete DNA tiles in a 2D plane and, more generally, Benenson showed [8] that by using nanostructures and nanomachines, complex computation can be implemented at the nanometer scale. Indeed, the linking of self-assembly, self-organisation and computation provides a powerful new approach to addressing profound questions about the controllability of complex physico-chemical nanosystems [71]. In turn, an improved understanding of self-assembly and self-organisation could help us develop better, cheaper and smaller computational systems. Thus, advances in nanosciences and computational sciences are tightly interdependent. The rich interplay between these two complementary research areas presents both enormous challenges and promises. In this chapter we briefly showcase what we have termed the Automated Self-Assembly Programming Paradigm (ASAP2 ). ASAP2 attempts to systematically investigate the links between self-assembly and computation by recognising that (1) self-assembly and self-organisation can be understood from an information theoretic and computational viewpoint, (2) that advances in self-assembly and self-organisation will enable the emergence of new computational substrates and (3) that the previous two points, taken together, require a reconsideration of the way we design self-assembly and the way in which we compute with self-assembling systems; hence the need for both automation and a new paradigm of thought. Self-assembly is essentially a distributed process that arranges a collection of autonomous components into aggregated structures. There is no “master plan” followed by the individual components, nor any top–down control mechanisms governing the assembly process. The self-assembling entities interact only with 2 Here we take a very general perspective: although it is true that self-assembly has been used quite systematically in DNA-based computing, it has not yet captured the imagination to the extent of other aspects of the computer sciences.

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one another and with their local environment [70]. As the individual components that take part in the self-assembly process become simpler, the more distributed the “control” or “intelligence” that is required to achieve a specific self-assembled pattern of behaviour should be. Likewise, as the system’s intelligence gets diluted into miriads of smaller and simpler components, its programmability (or designability) becomes much more challenging. Indeed, virtually all computer science research has been mainly concerned with programming single systems with considerable computational power rather than with programming a large number of simple-minded computational components, as self-assembly requires. Thus a considerable challenge remains in regard to programming this new type of systems. This is true even when considering the advances in software engineering for large distributed and parallel systems precisely because the latter are designed and built with enormous built-in computational capability. In programming self-assembly and self-organising systems, one must rely on the existence of extremely rudimentary computational operations that are often unreliable, stochastic in nature, with a changing interconnection topology, without the possibility of individually addressing a component, etc. As several chapters in this book suggest, we could turn to nature for inspiration on how to program these systems. In this chapter we review the application of evolutionary algorithms and analysis techniques to the design and optimisation of self-organising cellular automata and Wang tile models. These two examples demonstrate that artificial evolution is capable of both parameter and structural design (automated programming) of self-assembling systems. The other two examples in this chapter cover the “other side of the coin”, that is, the use of self-assembly for the purpose of implementing unconventional computation. One of the examples demonstrates how a virtual network of computational compartments that is kept far from equilibrium can produce an increasingly rich “soup” of programs. The other example proposes, for the first time, a possible route for the implementation of vesicle-based computing by simulating—using dissipative particle dynamics—the self-assembly of vesicles that act as the virtual machine for a P-system. In this chapter we have opted to give a general overview of the themes mentioned above, rather than a detailed exposition of a particular theme, as it is our intention to highlight some of the many opportunities that lie at the interface of computer science and the self-assembly related disciplines.

2.

Evolutionary Algorithms for Parameter and Structural Learning in Self-Assembly Model Systems

This section presents a brief overview of recent advances in the ASAP2 , covering parameter and structural learning for self-assembling models. As recognised in [1,9] and elsewhere, self-assembly presents many challenging design and optimisation problems. Often, these problems can be cast as either

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parameter learning, structural learning or as mixed problems. Structural learning problems arise when one must decide the (close to) optimal choice of features a system must have among a set of discrete structural alternatives. These discrete entities could be molecular species, morphological varieties, etc. When the problem at hand is more closely related to specifying the values for continuous or discrete variables, then it can be classed as parameter estimation. Often, selfassembly research requires the solution of mixed problems where both parameter and structural optimisation are needed. Evolutionary computation (EC) is a powerful toolbox for the solution of hard problems whether of a combinatorial or continuous nature. EC is based on the core premises of evolution by natural selection: in the presence of (1) limited resources, (2) feature inheritance under (3) imperfect (mutation susceptible) replication, evolution occurs as a secondary emergent process of the selection, itself an emergent process that is due to the lack of infinite resources, for the best replicators. Evolutionary computation translates these three premises as follows: Limited resources are implemented as finite size populations of individuals, representing solutions to problems. The characteristics of “individuals” are encoded by their “genes” that represent specific building blocks for solving a given problem. When individuals reproduce, some of these building blocks are passed onto the offspring with a possible degree of mutation. The selection process operates by measuring the quality of a given individual in relation to how well it solves the problem at hand and its relative fitness in relation to the rest of the population. Hence, over time, new and better solutions are obtained as these principles are iterated. Evolutionary computing is not only good at providing exceedingly robust and near optimal solutions to hard problems; it often provides original solutions that sometimes are counter-intuitive from a human viewpoint. Hence using evolutionary algorithms for optimising the design of self-assembly through parameter and structural learning can also shed light on new types of solutions for the problems of interest. Evolutionary algorithms have been applied to many challenging domains [10–12] and they enjoy substantial theoretical foundations [13–15]. A canonical evolutionary algorithm is shown in Fig. 13.1. Every application of an evolutionary algorithm to a specific problem requires the precise specification of an individual’s fitnesses, encoding, mutation, reproduction and selection strategies as well as several algorithmic parameters. In what follows we present the salient features of an application of EAs to parameter learning for a self-organised system and structural learning for a self-assembly model. We have opted to detail some methodological tools, including fitness distance correlation, cluster analysis and evolutionary activity waves, available to EC rather than giving a full description of the algorithms themselves. The rationale behind this is that we have provided full detail of these algorithms elsewhere (see text below), and moreover, we prefer to emphasise the potential for insight that can be derived from an evolutionary methodology.


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initialise population while (stopping criterion == false) { parents = select parents from population for reproduction; with a defined probability ‘mate’ the parents to form (usually) two children; else children = parents; with a defined probability mutate children; insert children into population; evaluate and cull population; update stopping criterion; } Fig. 13.1. Pseudocode for a canonical evolutionary algorithm.

2.1.

Parameter Learning for Self-Organising Cellular Automata

Self-organisation, also called “dynamic self-assembly” (see Chapter 2 of this book), is a specific type of self-assembly where not only thermodynamics but also kinetic control, or other far-from equilibrium mechanisms, are at play [16]. Self-organising systems have often been studied by means of cellular automata (CA) [17–19]. A cellular automaton is defined as an infinite, regular grid of cells, each of which can be in one of a finite number of states. At a given time step, t, the state of a cell is a function (defined by a set of rules) of that cell’s neighbourhood at t − 1. There are a number of possible definitions of a neighbourhood in CA systems, from simply using the states of the eight surrounding cells (the Moore neighbourhood) to more complex definitions [20]. CA are an important modelling technique in many diverse domains of science [21,22], and are very useful in studying the dynamics of complex, self-organising systems. 2.1.1. Parameter Identification For those involved in the modelling of processes with CA, building a system (or directing existing systems, natural or man-made) such that the resultant behaviour matches some pre-defined target [23] is key. References [24] and [25] report on a methodology that uses a genetic algorithm [5] to derive the rules that define a CA system. We extend this work, concerning ourselves not with how to derive the functions themselves, but rather the parameters to these functions. In most CA models, the behaviour is carefully governed by a set of input parameters, and so a widely studied problem related to the study of complex systems by means of CA is that of parameter identification, that is to say, the parameter values a CA-based model of a complex system requires in order to attain a specific behaviour. Identifying these parameters enables the user to effectively program the CA system to perform some desired function. Analysis of the parameters in the model can give valuable insight into the problem domain and thus take us closer to the goal of “programming” real-life

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self-organising systems. In some cases, hand engineered methods are available for this parameter identification process but in many others, due to the extreme complexity and often counter-intuitive nature of the underlying processes, coupled with an extremely large search space (given four integer parameters in the range [0, 100], the search space comprises one hundred million points), often the most effective method for CA parameter identification is through the use of a metaheuristic search process, such as an evolutionary algorithm. If we are to achieve successful parameter identification, we must have some metric that compares a candidate behaviour to the target behaviour (where, in this case, “behaviour” is most often represented by a spatio-temporal pattern generated by the CA model) and outputs a numerical value that represents the similarity between the two. The quality of this metric is crucial to the success of the search, and a number of possible metrics are discussed below. Not only, however, is it important to use a metric that directs a successful search, but also, crucially, that is robust, that is to say, effective across all areas of the search space. Finding the most robust way for predicting when an evolutionary metaheuristic such as a genetic algorithm (GA) will be an effective method of optimisation is still an open topic of research in evolutionary computation theory. The issue is only further complicated by the nature of the mapping from, to use evolutionary computation vocabulary, genotype (the input parameters to the CA model) to phenotype (the spatio-temporal pattern generated by the model) and then from this phenotype to fitness (the numerical measure of similarity to the target pattern), being highly complex and non-linear. In order to perform any sort of analysis of a candidate metric, a suitable dataset must be defined. This set must not be unwieldy in size, but yet must effectively represent the entire search space. Such a set is most efficiently constructed by dividing each model parameter into a number of bins and then systematically exploring each and every combination of parameter bins, for example, given three parameters each taking integer values in the range [0, 100], each parameter could be split into three bins, 0, 50 and 100, and a dataset generated as {{0, 0, 0}, {50, 0, 0}, {100, 0, 0}, {0, 50, 0}, {0, 100, 0}, {50, 50, 0}, {50, 100, 0}, . . .}. Fitness Distance Correlation (FDC) [26] is a measure of how closely the fitness of an individual correlates to its genotypic distance to a known target. That is to say, given two different parameter sets (genotypes), FDC measures the correlation of the Euclidean distance between these genes against the value assigned by the similarity metric (fitness function). Numerically speaking, assuming a maximisation problem, a large positive correlation gives good support for evolutionary optimisation. If there is only a small correlation, a parameter optimisation GA, or any metaheuristic using the same similarity metric, will have little effect. The FDC of a candidate metric can be easily analysed by the generation of a number of scatter plots of fitness against distance, usually one plot for each variable


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Fig. 13.2. Fitness Distance Correlation scatter plot. For colour version, see Colour Plate Section.

parameter in the dataset, such as in Fig. 13.2. The formal correlation coefficient is calculated as in Eq. (1) where r is the correlation coefficient, n is the number of individuals under consideration, fi is the fitness (similarity) of individual i, and di is its genotypic distance to the target, f and SF are the mean and standard deviation of the set of fitnesses, and d and SD are the mean and standard deviation of the set of distances, (1/n) ni=1 (fi − f )(di − d) . r= (1) SF SD Fitness Distance Correlation analyses the genotype → fitness relationship, but in complex systems, the phenotype → fitness relationship must also be verified, i.e., can the metric (fitness function) differentiate between dissimilar phenotypes and effectively classify similar phenotypes for the purpose of selection. If the fitness function cannot achieve this, a parameter optimisation GA will have little effect. Data clustering is a simple and convenient method for verifying this phenotype → fitness relationship. Each pattern in the dataset is compared, using the similarity metric under consideration, to every other, and thus a square distance matrix is formed. Any clustering algorithm could be used (see for example [27] and references therein). The clusters can most conveniently be inspected using a logarithmic tree representation, such as that shown in Fig. 13.3. 2.1.2. Similarity Metrics The choice of metric is, of course, crucial for the success of the search. In the case of CA-based models, the outputs will usually be spatio-temporal images representing the behaviour of the model, and so the metric used must output a numerical value representing the similarity between the two graphical input patterns. There are a great number of possible techniques for image comparison, from simple Hamming distance [28] methods right through

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Fig. 13.3. Logarithmic cluster tree. For colour version, see Colour Plate Section.

to highly complex image processing techniques. If it is to be used within an evolutionary search process, the metric should not only be accurate, but fast. Moreover, if a number of CA systems are to be considered, the metric must not be specific to any one type of pattern. In [20], parameter identification for a number of CA-based systems is performed using a metric called the Universal Similarity Metric [29]. This is a measure of similarity between two generic objects (in the case of CA models, this is most often a graphical pattern, but the USM, by its very nature is universal, so can be used with any data) based on Kolmogorov complexity [30]. The USM has been used in many different topics of research ([31,32] and associated references). The GA system used to obtain optimised parameters for the cellular automaton systems in our work, is a generic optimiser developed in conjunction with the CHELLware project [33]. As it is used to solve a broad range of optimisation problems, the number and data type of genes in the chromosome, along with the parameters for the GA are all user-specifiable in a XML input script. This script, along with a plug-in-style problem specification class (which, most importantly, defines the fitness function), configures the GA to the specific problem to be optimised. Fitness evaluations can be parallelised over a large high performance computing cluster. In [20], we use the USM to evolve parameters so as to obtain a target behaviour in a number of different cellular automaton-based models. In many of them, we find that the USM is capable of producing highly similar patterns, especially when the target is particularly visually distinctive. However, some shortcomings


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Fig. 13.4. An evolved nanostructured pattern. For colour version, see Colour Plate Section.

of the USM are demonstrated, such as its blindness to negative (reverse colour) and mirror images. In [34], parameter identification is considered for specific type of CA system called a Gas Lattice model [35]. The system is that introduced in Chapter 1 of this book, and models the self-organisation of gold nanoparticles when adsorbed onto a substrate [36]. In this context, the USM was found to be largely ineffective. Hence, a more geometrically/morphologically aware metric was introduced, the Minkowski functionals [37]. These characterise a binary pattern in terms of area, perimeter and Euler characteristic (a measure of connectivity). The fitness function was derived by taking the error between the target Minkowski values and those derived from the evolved pattern. This was seen to be much more successful in this particular domain, and was able to produce strikingly similar patterns for four very morphologically different targets. Figure 13.4 shows an example result where the target image is an atomic force microscope image of an experimentally produced nanostructure pattern; the evolved pattern is visually very similar indeed. 2.1.3. Further Issues This work raises a number of important further considerations relevant to parameter identification for models of self-organisation processes. In the case of models that are stochastic in nature, it is important that each individual (parameter set) must be evaluated a number of times, that is, noisy fitness functions must be dealt with appropriately (see the proposals introduced in [38–40]). Furthermore, if the fitness function comprises a number of components (such as, in the latter case, the three Minkowski functionals), it is possible that each component might take values over widely different intervals, and so some sort of scaling process is vital in order to give each component adequate weighting within the fitness function. Alternatively, a multi-objective

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evolutionary algorithm could be used and the Pareto front approximated. These and other practical and research aspects related to the design of competent EAs are explored in detail in [41]. It is interesting to note that when searching for a matching pattern, it is not unusual for a visually similar pair to be produced using parameters with a relatively high discrepancy. This is just further illustration of the highly complex, non-linear and stochastic nature of the genotype → phenotype mapping discussed earlier. In this case, analysis of the FDC plots for the offending parameter would show a lower than usual correlation. It is to be expected that some parameters will have greater impact (that is to say, when combined with certain values of the other parameters) on the resulting phenotype in some areas of the search space than in others. Likewise, it is not unusual for a pattern that is visually similar to the target to be given a low fitness value, which illustrates the complex nature of the phenotype → fitness mapping. That is, analysis of the cluster tree in these cases would show that patterns belonging to different clusters look visually very similar, yet are quite far apart in the tree. This is often down to the fact that the similarity metric is more specific than human vision, so two images that look similar might actually have unexpectedly low similarity values. Furthermore, analysis of the cluster tree will often reveal that the search space can be partitioned into a number of ‘families’ of graphical similarity. In general, target behaviour patterns belonging to larger ‘families’ are quicker and easier to evolve through a metaheuristic search process. Thus the utilisation of a correct similarity measure is paramount for the success of the approach. Parameter identification and verification of the robustness of the identification process are techniques of great importance when studying self-organising systems and comprise a key step on the way to being able to program self-organising systems in within a physical, chemical or biological setting.

2.2.

Structural Learning for Self-Assembling Tile Systems

In the previous section we made an overview of some of the evolutionary computing tools that are available for the parameter estimation and analysis methods involved in designing self-organising models. In this section we also apply evolutionary algorithms to the structural learning of discrete self-assembling tile systems and we show some additional tools that allow us to better understand how an evolutionary process “programs” the tile system. Tiling theory has been widely used as a base model for many applications ranging from biology to computer science [7,42–44] and also serves as a link between computation and selfassembly [45]. One of the most widely known formal systems is the Wang tiles model, defined as a set of square coloured-edge tiles moving with Brownian motion across the Euclidean plane. In the case of a collision, tiles either attach to


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one another or bounce off. As they stick to each other, they aggregate in complex structures and the emerging shapes are thus self-assembled. Many problems associated to the design of optimal tile systems have been investigated [1] (see Chapter 10 in this book). A formal definition of a Wang tile set T is given in Eq. (2) where Σ is a collection of colour codes capturing the “glue” types associated to each edge. Thus, two tiles ti , tj will self-assemble if the glue types and strengths at their touching edges are compatible. This compatibility is given through the use of an interaction matrix M, which specifies for each pair of colours, the strength of their binding, and a temperature parameter τ as defined in [1]. If the bond strength exceeds the value given by τ , then the two colliding sides will bind. T = t | t = (x0 , x1 , x2 , x3 ) where x0 , x1 , x2 , x3 ∈ Σ. (2) 2.2.1. Evolvability of Wang Tiles As discussed above, evolutionary algorithms (EA) are very well suited to solving, among others, combinatorial problems, and one such example is the problem related to learning the structure of a tile family. Given an EA and a Wang tile system with a fixed τ and M, our aim is to evolve the design of a set of tiles capable of self-assembling into a user-defined shape. A GA system (as detailed in [46]) was employed where the genotype specified a set of tiles and conventional crossover and mutation operators were used. The evaluation phase consisted of two stages: tile simulation and structure comparison. 2.2.1.1. Tile Simulation For each tile family T i encoded by an individual, an equal number of instances are placed into empty positions on a two-dimensional square lattice. Thereafter, tiles move around the lattice sticking or bouncing off each other until the simulation ends. The way tiles interact with one another is defined by the following increasingly complex simulation models. Model 1. When two tiles reach adjacent locations, the interaction between the colours at the touching edges is looked up in M. If the resultant value is greater than τ , both tiles self-assemble and remain in their current location; otherwise they continue their random walk through the lattice. Model 2. This model extends the above by the introduction of a probabilistic moving criterion, ρ, for each tile. Its calculation is performed in terms of the interaction value given by each edge of the tile, its neighbours and τ . Initially, ρ is set to 1.0 and the tile is thus free to move on the lattice. As a tile interacts with others, ρ either decreases when the tile becomes part of an aggregation, or rises when the tile leaves an aggregation. To make the model richer we also introduce the possibility of tile rotation. Thus, we define Model 3 and Model 4 as the combination of the two previous models but extended with rotation, i.e. tiles without probabilistic motion plus rotation (Model 3) and tiles with probabilistic motion and rotation (Model 4).

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Fig. 13.5. Four types of self-assembly conformations: (a) two-tiles self-assembly, (b) three-tiles self-assembly, (c) four-tiles self-assembly, and (d) five-tiles self-assembly. Stable versions of these act as “generalised secondary structures”, i.e. milestones, for the self-assembly process. For colour version, see Colour Plate Section.

2.2.1.2. Structure Comparison The resulting self-assembled tile structures spread over the lattice are, in turn, compared to the target shape (rectangular aggregates). Each comparison measures a morphological similarity value using the Minkowski functionals introduced in 2.1. Hence, the more similar the shapes are to the goal, the more likely an individual is to reproduce and pass onto the next generation. The results of the experiments mentioned above, presented in [46,47], demonstrate the suitability of EAs for the design of self-assembling tiles. Some of the most interesting questions, however, are related to how the artificial evolution process programmes the tiles in order to achieve a specific target shape. It is to this issue that we now turn our attention. 2.2.2. Emergence of Generalised Secondary Structures Understanding how information is processed on a distributed system such as with the evolved tile lattice is non-trivial. Each tile has a very limited computational capability, using only very rudimentary if-then-else rules for deciding whether to stick to another tile or bounce off depending on the glue strength and temperature of the system. Thus understanding how the larger structures emerge out of these local interactions requires a careful analysis. We investigated the set of all possible self-assembling conformations that persisted through the evolution of the best tile set. That is, we studied combinations of two, three, four and five tiles as depicted in Fig. 13.5 and assessed which intermediate structures were stable and hence could be used as milestones towards achieving the final target structure. We call these intermediate structures “generalised secondary structures”, in homage to protein secondary structures, which are deemed to be building blocks for their tertiary or native state. Two-tile assembly conformations were found in some of the experiments using Model 1 and Model 3. Therein, some of the best solutions involved a particular single tile that stuck to all the others. This seed tile increases the chances of the formation of small self-assembled aggregates, but also makes exploration difficult as the resulting structures are morphologically dissimilar to the target shape, resulting in a low fitness value due to this premature aggregation.


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Table 13.1. Generalised secondary structures for three-tiles assembly Model 1

Model 2

Model 4

1

15

1

1

16

1

1

16

1

1

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1

1

14

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1

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1

For colour version, see Colour Plate Section.

We also consider the notion of cooperation [48] where a combination of three tiles or more is required to initiate self-assembly. Using Model 1, a small number of three-tile self-assembly conformations were employed. Conversely, for Model 2 and Model 4, a significant number of instances were identified. Table 13.1 shows the resulting outcome of this analysis with the type and amount of conformations per model. Further analysis shows the additional presence of four-tile and five-tile cooperative behaviour, both contributing further to the self-assembly process. These types of primary aggregates were found in the resulting individuals belonging to Model 1 and Model 2, with 26 examples of the four-tile self-assembly conformations and just a few occurrences for the five-tile patterns. Table 13.2 shows the resulting analysis. Figure 13.6 shows a concrete example of how generalised secondary structures evolved for Model 2 are used as milestone for the self-assembly of a 3×3 square. Each potential binding site, denoted with the symbol “?”, has a glue strength at least equal to the temperature. Starting from a three-tile assembly structure in the first step, an incoming tile attaches to the upper right binding site. Later on, Table 13.2. Generalised secondary structures for four-tiles and five-tiles assembly

Model 1 Model 2

– 7

1 5

For colour version, see Colour Plate Section.

1 9

– 5

1 –

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Fig. 13.6. Simulation of an individual using generalised secondary structures and probabilistic motion. For colour version, see Colour Plate Section.

step 4, it moves up due to the probabilistic motion leaving two bonding sites ready to receive tiles at step 5. In the following step, it moves back and, at step 7, the tile located below moves aside leaving two new binding sites ready to receive tiles at steps 8 and 9. Due to the probabilistic motion, this latterly attached tile moves up at step 11 allowing for the completion of the figure in step 12. 2.2.3. Measuring the Strength Activity In the previous section we explained how the evolved tiles self-assemble into a specific shape. We analyse next how the generalised secondary structures, which act as milestones during the selfassembly process, emerge. Selection of the appropriate glue strengths for the generalised secondary structures is a key to the discovery of the tile families capable of self-assembling into the target shape. The adaptation of the glue strengths involved in the evolved tiles is performed through an analysis method that considers the genotypic evolutionary activity as defined in [49]. Unlike [49] and the example in Chapter 4 of this book, however, we define the evolutionary activity for the generalised secondary structures mentioned above. That is, rather than measuring the activity for genes we measure it at the phenotypic level as this is a more combinient method when genes are highly epistatic. Hence, the evolutionary activity ai (t) represents the cumulative usage (or utility) for a glue of strength i, as instantiated by some generalised secondary structure, up to generation t. This is shown in Eq. (3) where Ci (t) is the concentration in the population of glues with strength i at generation t. A cut-off of 0.1 was used for the concentration as to capture only significant ones, hence when it falls below 0.1 the counting is restarted. j =t j =0 Ci (j ) if Ci (j ) > 0, ai (t) = (3) 0 otherwise. The chart in Fig. 13.7 shows the resulting activity waves of the possible glue strengths (0, . . . , 9) for a representative run of the evolutionary algorithm under Model 4. In this case, the measurement is applied to two-tiles interactions (shown


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Fig. 13.7. Evolutionary activity of generalised secondary structures glue strengths. For clarity we only show the activity of glue strengths related to intermediate assemblies of the form shown in Fig. 13.5(a). For colour version, see Colour Plate Section.

in Fig. 13.5(a)). Since this model is invariant under rotation glue strengths arising from interactions in the X or Y axis are equivalent. The evolutionary activity plots show how at the beginning of the run the algorithm explores several strategies for building two-tiles interactions, that is, all the glue strengths are (to a degree) represented. As the evolutionary process progresses, several glue strengths are discarded and they become extinct, i.e. their evolutionary waves vanish, only to be replaced with a glue strength of 2, 1 and 3 until generation 80. From generation 115, glue strengths 3 drives all the rest to extinction. The reader should note that, as the temperature τ is 4, this glue strength is the smallest one that is closer to τ . Hence the two-tile (partial) assemblies, although do not stick permanently they do have a high chance of staying together (as ρ is smaller than 1). Thus evolution “tunes” the stickiness of two-tiles generalised secondary structures as to avoid premature aggregations but still having a high chance of building up higher order GSSs. In general, the less sophisticated the tile model, the smaller the resulting aggregations, as two-tile interactions become not just common, but frequent building blocks that lead the system to premature uncontrolled aggregation. Conversely, experiments using tiles assisted with probabilistic motion result in larger

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aggregations due to the use of three-tile, four-tile and five-tile cooperation. In other words, probabilistic motion requires the use of intermediate building blocks—generalised secondary structures [50]—for adequate self-assembly to occur. Moreover, it seems that a conservation of complexity principle is operating whereby the richer the environment, the simpler the individuals and, conversely, the simpler the environment the more complex the individuals.

3.

Self-Assembly Models for Unconventional Computing

In this section we describe two self-assembly models for unconventional computing. In the first model we take a step back from Genetic Programming and we ask what contributions (if any) a detailed analysis of self-assembling programs could make to the automated synthesis of computer programs. In the second example we propose that dissipative particle dynamic simulations of self-assembling vesicles could help bridge the gap between recent theoretical developments in P-systems and their chemical implementation.

3.1.

Before Genetic Programming: The Self-Assembly of Program Trees

Although it is generally accepted that Genetic Programming (GP) is the conventional means for automatic program synthesis [51–53], we seek to answer whether a more systematic analysis of self-assembly could shed light on program synthesis. A second motivation for our work is the observation that the “compartmentalistic school of thought” of the origin of life argues that compartment structures formed spontaneously through self-assembly processes, and perhaps provided the original membrane-bounded structures required for maintaining a sufficiently diverse and far-from equilibrium environment for cellular life to begin [54,55]. In this section, we discuss some insights on the use of compartments and self-assembly for the automated synthesis of computer programs. 3.1.1. Automated Software Synthesis Close and Far from Equilibrium In our previous work [56] we introduced the automated self-assembly programming paradigm (ASAP2 ). In ASAP2 , portions of software programs are embodied within a virtual compartment, i.e. a confined space, and allowed to perform a random walk. When two software components collide, a check is made on their input and output data types (signatures), and if they are compatible, the two components self-assemble into a new, larger component. In [56,57] we explored the dynamics of ASAP2 using a single compartment and in equilibrium (no net flux of components or energy into the system was allowed). In [58], we extended our work by analysing the effects of a network of compartments on the complexity and diversity of the self-assembly programs when the network as a whole was kept at equilibrium. In that setting, components could move freely from one compartment to any other neighbouring one and we explored the impact of network


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Fig. 13.8. Example β-graphs: (a) β = 0, (b) β = 0.2, (c) β = 0.6, (d) β = 1.0. For colour version, see Colour Plate Section.

topology on the self-assembled programs. In this section we extend our previous work by introducing a net flux of components into the network thus resulting in a system that is kept far from equilibrium. We overview the impact that this addition to the model has on the complexity of self-assembled programs. 3.1.2. β-Graph Model We use β-graphs [59] as a rich source of model network topologies. The graphs cover a range of network topologies, from the completely random to the highly ordered. In order to study meaningful configurations, the generated graphs are assumed to be connected, undirected and unweighted. The β parameter is a probability value that determines the random rewiring rate of edges from an initially completely ordered ring structure. Figure 13.8 shows examples of β-graphs constructed with β ∈ {0.0, 0.2, 0.6, 1.0}. As can be seen from Fig. 13.8, β-graphs can be gradually transformed from ordered graphs to random graphs with increasing β values. A highly ordered structure is shown in Fig. 13.8(a) where β is set to 0. At the other extreme, Fig. 13.8(d) illustrates a completely stochastic graph with all edges rewired when β is set to 1.0. 3.1.3. ASAP 2 : An Investigation of the Impact of Compartments Once a β-graphs is instantiated, each vertex, i.e. compartment, receives an equal num-

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ber of randomly chosen software components. Software components are sampled from man-made program libraries as we are interested in investigating the self-assembly properties of software as it is rather than as it could be. An edge connecting vertices vi , vj indicates that software components can move from compartment vi to vj and vice versa. When a software component that is moving within a compartment touches its boundary, it will be transferred to one of its neighbouring compartments. The choice of which compartment will receive the component is made in a uniformly random fashion across all the neighbours of a given vertex. When a software component is close to another component within a threshold distance (dε ) within the same compartment, they will either selfassemble into a more complex aggregate structure or repel, depending, as mentioned before, on the type of their binding sites [56]—binding sites are deemed compatible if their types match. This self-assembly process will achieve an equilibrium state when there are no more possible binding actions between any of the remaining self-assembled trees across the whole network. This is so because once the simulation starts no more components are allowed into the system, hence the simulation proceeds under thermodynamic control. In order to further investigate ASAP2 far from equilibrium, we introduced a net flow of software components. When a β-graph is constructed, a random node in the graph is selected as an influx compartment. The outflux compartment is selected in such a way that it has the greatest distance to the influx compartment in the network. In Fig. 13.8, an influx compartment is represented by a node with an inwardly pointing arrow, and an outflux compartment is indicated by a node with an outgoing arrow. At each time step, a random software component is added to the system at the influx compartment with an externally defined probability value pf . If a new software component is introduced into the system, a component of the same size (i.e. with the same number of nodes) will be extracted from the outflux pool. However, if there are no components of the same size in the outflux compartment, no software components will be removed (i.e. influx and outflux of components are not always balanced). Hence, the system is kept far from equilibrium and following a dynamic self-assembly process. Figure 13.9 shows how the diversity of generated programs changes for regimes at and far from equilibrium. Diversity of self-assembled programs (Dε ) is assessed by the number of different tree classes. Two programs are considered to belong to the same parse tree class if both their structure and content are identical [60]. We investigated how the distance from each compartment to the influx compartment (dα ), as well as time affects Dε . The terminating condition for dynamic ASAP2 is calculated as in [57], i.e., it reflects the time to equilibrium under static ASAP2 . Figure 13.9 also shows that dα has a clear and significant impact on Dε . It is expected that a compartment with a shorter distance to the source has a greater density of components. Figure 13.9 shows that in general, Dε increases as dα


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Fig. 13.9. Comparison of Dε between static and dynamic ASAP2 . The plane indicates total number of program trees assembled at equilibrium in static ASAP2 , whereas the lines indicate Dε at various time step in dynamic ASAP2 as a function of dα . For colour version, see Colour Plate Section.

decreases. The Characteristic Path Length (CPL) measures the mean shortest distance between an arbitrary pair of vertices (i, j ) in a graph. In Fig. 13.9, it is possible to observe a local peak at Dα = 4, which is close to the calculated CPL for the graph (CPL = 4.18). The same similarity was also observed in other experiments, showing it to be a general phenomenon. Moreover, it can also be seen that Dε increases over time under the same dα . Since there are no influx and outflux compartments in static ASAP2 , dα is not a factor that influences Dε . Hence, Dε is shown as a plane in Fig. 13.9 at its equilibrium. Comparing Dε at the termination time in dynamic ASAP2 with Dε in static ASAP2 , it can be seen that dynamic ASAP2 exhibits greater Dε . Our results confirm the results recently presented by [61] albeit in a more general (and perhaps practical) system as their work focuses on the hierarchical selforganisation of finite state machines. Gornerup and Crutchfield’s work, as well as ours, clearly demonstrates the need to maintain the system far from equilibrium as a means of maintaining, and in some cases sustainably increasing, the diversity and complexity of self-assembled programs.

3.2.

Towards Chemical Implementations of Membrane Computing Through the Controlled Self-Assembly of Virtual Vesicles

Membrane computing [62] is a promising computational paradigm based on symbol manipulation and compartmentalisation, analogous to the compartmentalisa-

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tion of chemical processes in biological cells. In this section, we discuss a method for modelling membrane computation with the intention of investigating a possible implementation of a membrane computer. 3.2.1. Bridging the Gap between Nano- and Macrocomputation In biological cells, processes such as photosynthesis and protein modification are encapsulated in membrane bound organelles. In the membrane computing paradigm and its P-systems formalism (discussed below), the idea of encapsulation of functionality is applied to symbol manipulation systems to create a computing paradigm that is inherently parallel and distributed. We propose the use of a coarse grained particle dynamics model as a “virtual machine” for P-systems, allowing investigation into the feasibility of implementing P-systems as real chemical membrane systems in vitro. In this simulation, artificial chemical vesicles will represent the abstract P-system membranes. A vesicle is a fluid compartment encapsulated by a bilayer membrane in a solution, typically less than a few microns in diameter. Vesicles correspond well with the simple membrane bound regions described by the P-systems formalism. 3.2.2. P-Systems P-systems [63] are a mathematical formalism of the Membrane Computing paradigm. In P-systems, computations are performed when symbols (analogous to chemicals in a real cell) are transformed by rules (analogous to chemical reactions in a real cell) within regions delimited by membranes. It has been shown [63] that P-systems that contain only a few membranes are computationally universal, but what is not clear is whether such a Turingcomplete P-system is a feasible membrane computer when implemented in vitro. Membranes in P-systems are organised hierarchically, with each membrane delimiting a region which may contain other membrane bound regions. Each region contains a multiset of symbol objects and a set of rules which act on those objects. Transition between states in a P-system involves applying rules to objects in each region “in a non-deterministic maximally parallel manner” [63] meaning that the rules to be applied to the objects in the region are chosen non-deterministically until no more rules can be applied, and that all regions evolve concurrently (in a synchronised manner). The structure of the membranes in a P-system is highly abstract when compared with a biological membrane, as spatial and compositional properties are not considered at all. Figure 13.10 shows a graphical representation of a simple P-system, computing multiples of four. In this P-system, the a → affff rule will be applied to the starting symbol a to produce one a symbol (replacing the one that was consumed when applying the rule) and four f symbols. This process repeats producing four more f symbols at each iteration until the a → δ rule is applied which dissolves membrane 2. The rule applied at each step of the computation is chosen stochastically. Once membrane 2 is dissolved, the f symbols move to membrane


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Fig. 13.10. A simple P-system.

1 and the computation halts as there are no further rules which can be applied. The output of the system is the number of symbols contained in the output membrane (membrane 1). 3.2.3. Dissipative Particle Dynamics as a Virtual Machine for P-Systems Although it is possible to perform P-systems computations by implementing the P-systems model on a standard desktop computer, the parallel and distributed benefits of the computational paradigm can only be utilised in vitro, with a real chemical implementation of P-systems. Moving from the P-systems representation of a membrane computer to a real implementation poses a number of difficult questions. It is not yet known whether it is possible to create the structure of a given P-system with vesicles, or if such a representation would remain stable, and perform as specified by the P-system. If application of rules to symbols is represented in solution as chemical reactions, then the chemicals may affect the vesicle structure in unexpected ways, and the abstract P-system formalism does not consider processes such as diffusion and membrane degradation that occur in real vesicles. As it is currently not possible to create chemical membrane computers in the lab, we propose the use of a modelling technique, Dissipative Particle Dynamics, to investigate the plausibility of P-systems in vitro. Dissipative Particle Dynamics (DPD) [64] is a hydrodynamic model that coarse grains atomic interactions by representing a number of molecules of a given type as a single “bead”. Conservative, dissipative and random forces between beads are calculated in a pairwise fashion, subject to a hard distance cut-off. The coarse graining of molecules and the hard force cut-off make DPD much less computationally intensive than other modelling techniques such as molecular dynamics, and the model strikes a good balance between the level of detail at the hydrodynamic length scales and computational complexity. The addition of spring based forces between particles to DPD has allowed the modelling of polymers [65] and with this extension, DPD has been used to simulate various different phases of amphiphillic molecules (molecules that have both hydrophobic and hydrophillic properties), such as micelles, bilayers [66] and vesicles [67–69]. Figure 13.11 shows images from a visualisation of a DPD vesicle simulation.

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Fig. 13.11. Vesicle simulation in DPD: (a) 3D DPD simulation of a vesicle and (b) cross section of (a) with solvent beads displayed. For colour version, see Colour Plate Section.

Fig. 13.12. Vesicle formation. For colour version, see Colour Plate Section.

Figure 13.12 illustrates the process of vesicle formation from a random initial configuration (a), as described in [67]. The amphiphiles first organise into micelles (b), then larger oblate micelles as the individual micelles fuse (c), and finally the large flattened micelles “curl” into bowl shapes which eventually close over to form vesicles (d). By representing the membrane structure of a P-system as simulated vesicles in DPD, and representing the rules and symbols of the P-systems formalism using a chemistry extension to the DPD model (see Chapter 4), DPD will act as a virtual machine for membrane computation. The virtual machine would take as its input a P-system specification, create an initial configuration for a DPD simulation based on that specification, and then allow the simulation to run for a specified simulation time or until some stopping criteria are fulfilled (e.g. only one vesicle remains in simulation space). Analysis of the resulting amphiphile configurations can then be applied to determine if the P-system computation occurred as intended. In this way, P-systems could be easily constructed and tested


303

in simulation, allowing insight into the feasibility of a real implementation of that system. Although the P-system virtual machine may not perform exactly as a real membrane computer might, we hope that the model is sufficiently realistic to allow investigation into the feasibility of different P-system structures as in vitro membrane computers and that such a tool will help to guide the creation of a vesicle based membrane computer in the near future.

4. Conclusions In this chapter we have discussed the link between natural computation, in particular evolutionary computing and P-systems, with self-assembly. We have also briefly mentioned ongoing research on the self-assembly of computer programs and a possible route for the implementation of vesicle computers. As our understanding of nature’s preferred engineering methods, self-assembly, selforganisation and natural selection, increases, computer scientists will be exposed to new challenges and innovative ideas. We believe that the interplay between computation and self-assembly goes well beyond what has already been achieved with DNA-based computing and we hope that this chapter will serve as inspiration to other computer scientists to delve into the computational nature of selfassembly.

Acknowledgements NK would like to acknowledge the EPSRC for funding projects EP/E017215/1, EP/D021847/1 and GR/T07534/01. He would also like to thank P. Moriarty, M. Gheorghe, G. P˘aun, M. Bedau, S. Gustafson and D.A. Pelta for very many useful discussions.

References [1] L. Adleman, Q. Cheng, A. Goel, M. Huang, D. Kempe, P. Moisset de Espanes, P.W.K. Rothemund, Combinatorial optimization problems in self-assembly, in: Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), ACM Press, 2002. [2] A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B 237 (1952) 37–72. [3] J. von Newman, Theory of Self-Reproducing Automata, University of Illinois Press, 1966. [4] A. Lindenmayer, Mathematical models for cellular interactions in development, parts I and II, Journal of Theoretical Biology 18 (1968) 280–315. [5] J.H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, 1975.

304

Chapter 13

[6] P.W.K. Rothemund, Using lateral capillary forces to compute by self-assembly, Proceedings of the National Academy of Science USA 97 (2000) 984–989. [7] T.H. LaBean, E. Winfree, J.H. Reif, Experimental progress in computation by self-assembly of DNA tilings, in: DNA Based Computers V: DIMACS Workshop—American Mathematical Society, 1999. [8] Y. Benenson, T. Paz-Elizur, R. Adar, E. Keinan, Z. Livneh, E. Shapiro, Programmable and autonomous computing machine made of biomolecules, Nature 414 (6862) (2001) 430–434. [9] N. Krasnogor, S. Gustafson, A family of conceptual problems in the automated design of systems self-assembly, in: 2nd Annual Conference on Foundations of Nanoscience: SelfAssembled Architectures and Devices, Sciencetechnica Publishers, 2005. [10] S. Harding, J. Miller, E. Rietman, Evolution in Materio: Evolving logic gates in liquid crystal, in: 2005 European Conference on Artificial Life: Workshop on Unconventional Computing, 2005. [11] A. Thompson, P. Layzell, R.S. Zebulum, Explorations in design space: Unconventional electronics design through artificial evolution, IEEE Trans. Evol. Comp. 3 (1999) 167–196. [12] J. Rieffel, J. Pollack, Evolutionary fabrication: The emergence of novel assembly methods in artificial ontogenies, in: Proceedings of the Genetic and Evolutionary Computation Conference: SEEDS Workshop, 2005. [13] J.L. Shapiro, Theoretical aspects of evolutionary computing, in: L. Kallel, B. Naudts, A. Rogers (Eds.), Springer, 2001, pp. 87–108. [14] N. Krasnogor, J.E. Smith, Memetic algorithms: The polynomial local search complexity theory perspective, Journal of Mathematical Modelling and Algorithms (2007), Springer, doi: 10.1007/S10852-007-9070-9. [15] R. Poli, N. Freitag McPhee, J.E. Rowe, Exact schema theory and Markov chain models for genetic programming and variable-length genetic algorithms with homologous crossover, Genetic Programming and Evolvable Machines 5 (1) (2004) 31–70, Kluwer Academic Publishers. [16] P.L. Luisi, The Emergence of Life, Cambridge University Press, ISBN 0-521-82117-7, 2006. [17] T. Toffoli, N. Margolus, Cellular Automata Machines—A New Environment for Modelling, MIT Press, 1987. [18] B. Chopard, M. Droz, Cellular Automata Modeling of Physical Systems, Cambridge University Press, 1998. [19] L. Kier, P. Seybold, C. Cheng, Cellular Automata Modeling of Chemical Systems, Springer, 2005. [20] G. Terrazas, P. Siepmann, G. Kendall, N. Krasnogor, An evolutionary methodology for the automated design of cellular automaton-based complex systems, Journal of Cellular Automata 2 (1) (2007) 77–102 (special issue). [21] E. Berlekamp, J. Conway, R. Gut, The game of life, in: Winning Ways for Your Mathematical Plays, vol. 2, Academic Press, 1982, pp. 817–849. [22] R. Subrata, A.Y. Zomaya, Evolving cellular automata for location management in mobile computing networks, IEEE Transactions on Parallel and Distributed Systems 14 (1) (2003). [23] G. Green, D. Newat, Towards a theory of everything?—Grand challenges in complexity and informatics, Complexity International 8 (2001). [24] M. Sipper, Evolving uniform and non-uniform cellular automata networks, in: D. Stauffer (Ed.), Annual Reviews of Computational Physics, vol. V, World Scientific, 1997. [25] M. Sipper, M. Tomassini, M.S. Capcarrere, Designing cellular automata using a parallel evolutionary algorithm, Nuclear Instruments and Methods in Physics Research A 389 (1997) 278– 283.


305

[26] T. Jones, S. Forrest, Fitness distance correlation as a measure of problem difficulty for genetic algorithms, in: Proceedings of the Sixth International Conference on Genetic Algorithms, San Francisco, CA, USA, pp. 184–192. [27] M. Halkidi, Y. Batistakis, M. Vazirgiannis, On clustering validation techniques, Journal of Intelligent Information Systems 17 (2) (2001) 107–145. [28] R.W. Hamming, Error detecting and error correcting codes, Bell System Technical Journal 26 (2) (1950) 147–160. [29] M. Li, X. Chen, X. Li, B. Ma, P. Vitnyi, The similarity metric, in: Proceedings of the Fourteenth Annual ACM–SIAM Symposium on Discrete Algorithms, 2003, pp. 863–872. [30] P.M.B. Vitani, M. Li, Kolmogorov Complexity and Its Applications, MIT Press, 1991. [31] N. Krasnogor, D. Pelta, Measuring the similarity of protein structures by means of the universal similarity metric, Bioinformatics 20 (7) (2004) 1015–1021. [32] M. Ling, R. Sharp, Melody classification using a similarity metric based on Kolmogorov complexity, in: Proceedings of the Conference on Sound and Music Computing, Paris, 2004. [33] L. Cronin, N. Krasnogor, B.G. Davis, C. Alexander, N. Robertson, J.H.G. Steinke, S.L.M. Schroeder, A.N. Khlobystov, G. Cooper, P. Gardner, P. Siepmann, Is it alive? Recognising cellular systems: A computational-chemical perspective, Nature: Biotechnology 24 (2006) 1203–1206. [34] P. Siepmann, C.P. Martin, I. Vancea, P.J. Moriarty, N. Krasnogor, A genetic algorithm approach to probing the evolution of self-organised nanostructured systems, Nano Letters 7 (7) (2007) 1985–1990. [35] G. Doolerd (Ed.), Lattice-Gas Methods for Partial Differential Equations, Santa Fe Studies in the Science of Complexity, Addison Wesley Longman Publishers, 1990. [36] C.P. Martin, M.O. Blunt, P. Moriarty, Nanoparticle networks on silicon: Self-organized or disorganized?, Nano Letters 4 (2004) 271–274. [37] K. Michielsen, H. de Raedt, Integral-geometry morphological image analysis, Physics Reports 347 (2001) 461–538. [38] A. Di Pietro, L. While, L. Barone, Applying evolutionary algorithms to problems with noisy, time-consuming fitness functions, in: Proceedings of the Congress on Evolutionary Computation, vol. 2, 2004, pp. 1254–1261. [39] H. Kita, Y. Sano, Genetic algorithms for optimization of noisy fitness functions and adaptation to changing environments, in: Proceedings of the Workshop on Statistical Mechanical Approach to Probabilistic Information Processing, Okayama, Japan, 2003. [40] G. Rudolph, A partial order approach to noisy fitness functions, in: Proceedings of the Congress on Evolutionary Computation, vol. 1, 2001, pp. 318–325. [41] D.E. Goldberg, The Design of Innovation: Lessons from and for Competent Genetic Algorithms, Kluwer Academic Publishers, 2002. [42] V.R. Syrotiuk, C.J. Colbourn, J. Pachl, Wang tiling and distributed verification on anonymous torus networks, Theory Computers Systems 30 (1997) 145–163. [43] M. Alsweis, O. Deussen, Wang-tiles for the simulation and visualization of plant competition, in: 24th Computer Graphics International Conference 2006, in: LNCS, vol. 4035, SpringerVerlag, 2006, pp. 1–11. [44] M.F. Cohen, J. Shade, S. Hiller, O. Oliver Deussen, Wang tiles for image and texture generation, in: SIGGRAPH ’03: ACM SIGGRAPH 2003 Papers, ACM Press, 2003, pp. 287–294. [45] H. Wang, Proving theorems by pattern recognition, Bell Systems Technical Journal 40 (1961) 1–42. [46] G. Terrazas, N. Krasnogor, G. Kendall, M. Gheorghe, Automated tile design for self-assembly conformations, in: Proceedings of the 2005 IEEE Congress on Evolutionary Computation, Edinburgh, UK, September 2005, vol. 2, pp. 1808–1814.

306

Chapter 13

[47] G. Terrazas, M. Gheorghe, G. Kendall, N. Krasnogor, Evolving tiles for automated selfassembling design, in: Proceedings of the 2007 IEEE Congress on Evolutionary Computation, 2007, pp. 2001–2008. [48] P.W.K. Rothemund, E. Winfree, The program-size complexity of self-assembled squares, in: Proceedings of STOC, 2000. [49] M.A. Bedau, C.T. Brown, Visualizing evolutionary activity of genotypes, Artificial Life 5 (1999) 17–35. [50] N. Krasnogor, G. Terrazas, D.A. Pelta, G. Ochoa, A critical view of the evolutionary design of self-assembling systems, in: Proceedings of the 7th International Conference on Artificial Evolution, Special Track on Self-Assembly, Lille, France, October 2005. [51] J.R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press, Cambridge, MA, USA, ISBN 0-262-11170-5, 1992. [52] C. Ryan, Genetic programming 3: Darwinian invention and problem solving (book review), Genetic Programming and Evolvable Machines 1 (4) (2000) 1389–2576. [53] M. O’Neill, A. Brabazon, C. Adley, The automatic generation of programs for classification problems with grammatical swarm, in: Proceedings of the 2004 IEEE Congress on Evolutionary Computation, Portland, Oregon, 2004, pp. 104–110. [54] H.J. Morowitz, Beginning of Cellular Life, Yale University Press, 1992. [55] F.J. Dyson, Origins of Life, Cambridge University Press, 1985. [56] L. Li, N. Krasnogor, J. Garibaldi, Automated self-assembly programming paradigm: Initial investigations, in: Proceedings of the Third IEEE International Workshop on Engineering of Autonomic & Autonomous Systems, Potsdam, Germany, 2006, pp. 25–34. [57] L. Li, N. Krasnogor, J.M. Garibaldi, Automated self-assembly programming paradigm: A particle swarm realization, in: Proceedings of the Workshop on Nature Inspired Cooperative Strategies for Optimization, Granada, Spain, 2006, pp. 123–134. [58] L. Li, N. Krasnogor, J.M. Garibaldi, Automated self-assembly programming paradigm: The impact of network topology, in: Special Issue on Nature Inspired Cooperative Strategies for Optimization in the International Journal of Intelligent Systems (2007), in press. [59] D.J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness, Princeton Studies in Complexity, Princeton University Press, ISBN 0691117047, 2003. [60] E. Burke, S. Gustafson, G. Kendall, N. Krasnogor, Advance population diversity measures in genetic programming, in: Proceedings of Parallel Problem Solving from Nature, 2002. [61] O. Gornerup, J.P. Crutchfield, Hierarchical self-organization in the finitary process soup, in: ALIFE X, Tenth International Conference on the Simulation and Synthesis of Living Systems, Workshop Proceedings, 2006, pp. 75–80. [62] G. P˘aun, Computing with Membranes, Turku Centre for Computer Science, 1998. [63] G. P˘aun, Membrane Computing: An Introduction, Springer-Verlag, New York, 2002. [64] P.J. Hoogerbrugge, J.M.V.A. Koelman, Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, Europhys. Lett. 19 (1992) 155–160. [65] Y. Kong, C.W. Manke, W.G. Madden, A.G. Schlijper, Simulation of a confined polymer in solution using the dissipative particle dynamics method, International Journal of Thermophysics 15 (1994) 1093–1101. [66] J.C. Shillcock, R. Lipowsky, Equilibrium structure and lateral stress distribution of amphiphilic bilayers from dissipative particle dynamics simulations, The Journal of Chemical Physics 117 (2002) 5048–5061. [67] S. Yamamoto, Y. Maruyama, S.A. Hyodo, Dissipative particle dynamics study of spontaneous vesicle formation of amphiphilic molecules, Journal of Chemical Physics 116 (2002) 5842– 5849.


307

[68] V. Ortiz, S.O. Nielsen, D.E. Discher, M.L. Klein, R. Lipowsky, J. Shillcock, Dissipative particle dynamics simulations of polymersomes, Journal of Physical Chemistry B 109 (2005) 17708–17714. [69] M. Laradji, P.B. S Kumar, Dynamics of domain growth in self-assembled fluid vesicles, Physical Review Letters 93 (2004) 198105. [70] G.M. Whitesides, B. Grzybowski, Self-assembly at all scales, Science 295 (2002) 2418–2421. [71] G.M. Whitesides, M. Boncheva, Beyond molecules: Self-assembly of mesoscopic and macroscopic components, Proceedings of the National Academy of Science (PNAS) 99 (2002) 4769– 4774.

309

Index

A accretive growth, 200 adaptability, 25, 291 automated software synthesis,

high defect rates, hysteresis, 273 L

296

language theory, 51 large scale randomness, 187 length scale, 32 lithographic fabrication, 181

C carving, 200 cellular complexes, 210 CMOS, 181 competitive interactions, 32 complex spin networks, 268 computing with shapes, 56 cross-carbon nanotube FET, 182 cytoskeleton, 29

M

D decoupled array multiprocessor, 191 defect-tolerance mechanism, 125 diffusion limited aggregation, 205 Dissipative Particle Dynamics (DPD), 81, 301 DNA, 27, 53, 181, 253 E energy dissipation,

187

32

melanophore programs, 165 MGS, 202 micelle, 84, 301 microfabrication, 101, 106, 114 minimum line-based tile system problem, 228 medium, 32 Monte Carlo algorithm, 13 morphology, 8 N nanoparticle, 3, 253 nanoscale active network architecture, natural computation, 281 O optimisation, 83 Oracle, 189

F fault-tolerance, 126 flagella, 30 fluidic self-assembly, 106 Full-Binary Counter problem (FBC),

P 238

G General Purpose Counter problem (GPC), 228 generalised secondary structures, 292 genetic algorithm, 83, 283

P-systems, 59, 299, 300 parameter learning/identification, 285 pattern, 3, 36, 202, 211, 273 photosynthesis, 29 polyominoes, 56 populations, 31 probabilistic defect map, 124 probabilistic design methodology, 133 programmed dynamic self-assembly, 171

H Hamiltonian, 12, 269 hierarchical RAM computing,

193

R 163

RAM programs, 165

310

Index

Random Access Machine (RAM), 154, 156 ring-gated FET, 183 RNA, 27

similarity metric, 287 sticker system, 53 sticky graphs, 250 structured graphs, 269, 297

S

T

self-assembly, 2, 22, 49, 79, 106, 154, 184, 199, 226, 245, 259, 281 self-healing, 26 self-organisation, 2, 22, 281 self-replication, 26 Sierpinsky triangle, 206 silicon networks, 112

template directed ligation, 90 Tile Assembly Model (TAM), 226, 291 topology, 8, 262 U universality,

64, 74

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