Mathematical Problems in the Biological Sciences 10

REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and published by SIAM.

GARRETT BJRKHOFF, The Numerical Solution of Elliptic Equations D. V. LINDLEY, Bayesian Statistics—A Review R. S. VARGA, Functional Analysis and Approximation Theory in Numerical Analysis R. R. BAHADUR, Some Limit Theorems in Statistics PATRICK BILLINGSLEY, Weak Convergence of Measures: Applications in Probability i. L. LIONS, Some Aspects of the Optimal Control of Distributed Parameter Systems ROGER PENROSE, Techniques of Differential

Topology in Relativity

HERMAN CHERNOFF, Sequential Analysis and Optimal Design 3. DURBIN, Distribution Theory for Tests Based on the Sample Distribution Function SOL I. RUBINOW, Mathematical Problems in the Biological Sciences

MATHEMATICAL PROBLEMS

in the BIOLOGICAL SCIENCES

SOL I. RUBINOW Graduate School of Medical Sciences, Cornell University and Sloan-Kettering institute

SOCIETY for INDUSTRIAL and APPLIED MATHEMATICS P H I L A D E L P H I A , PENNSYLVANIA 1 9 1 0 3

Copyright 1973 by Society for Industrial and Applied Mathematics All rights reserved

Printed for the Society for Industrial and Applied Mathematics by J. W, Arrowsmith Ltd., Bristol 3, England

MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES

Contents Preface

v

Introduction

vii

Lecture 1 THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, I

1

Lecture 2 THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, II

13

Lecture 3 TRACER ANALYSIS OF PHYSIOLOGICAL SYSTEMS, 1

24

Lecture 4 TRACER ANALYSIS OF PHYSIOLOGICAL SYSTEMS, II

29

Lecture 5 ENZYME KINETICS, 1

38

Lecture 6 ENZYME KINETICS, II

46

Lecture 7 CELL POPULATIONS, 1

53

Lecture 8 CELL POPULATIONS, II

62

Lecture 9 DIFFUSION IN BIOLOGY, 1

73

Lecture 10 DIFFUSION IN BIOLOGY, II

83

Hi

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Preface This monograph is the record of ten lectures which I delivered at the Regional Conference on Mathematical Problems in the Biological Sciences of 5-9 June 1972 at Michigan State University under the Regional Conferences program sponsored by the Conference Board of the Mathematical Sciences with the support of the National Science Foundation. I would like to take this opportunity to express my appreciation to Dr. John R. Deitrick, former Dean of Cornell University Medical College, and the late Frank L. Horsfall, former Director of the Sloan-Kettering Institute for Cancer Research, whose original effort in the establishment of a biomathematics program in a biomedical milieu has made my experiences in biomathematics possible. I am also grateful to Dr. Robert Buchanan, the present Dean of Cornell University Medical College, for continuing this commitment, to Dr. Charles J. Martin of the Department of Mathematics of Michigan State University who conceived of and organized the conference, and to the Conference Board of the Mathematical Sciences and the National Science Foundation for their joint support of the conference of which these lectures were an integral part. Such sponsorship encourages and disseminates the cause of mathematics applied to the biological sciences. SOL I. RUBINOW

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Introduction I shall first make a few general remarks based on my experiences as a theoretical physicist and/or applied mathematician with problems in the biological sciences. Biology is a vast subject with many branches, so that what I shall talk about represents only a small and highly selective sample of the quantitative approach to biology. With regard to the lectures, it will be seen that they contain, inter alia, a record of some of the achievements of my colleagues and myself at the Biomathematics Division of the Graduate School of Medical Sciences of Cornell University and Sloan-Kettering Institute during the past eight years or so. Broadly speaking, biological information is semi-quantitative or "soft", in contrast to the "hard" or precise observations to be found, for example, in the sister science of physics. The reason for this difference is simply that the fundamental problems of biology such as cell differentiation or morphogenesis are several orders of magnitude more difficult than those of physics, at the present time. A cell, from the viewpoint of physics, is a large number of strongly coupled physico-chemical fields in interaction over such small distances that the distinction between microscopic and macroscopic is often lost. Furthermore, the system loses most of its properties in the non-living state, so that the fields cannot be isolated and studied separately, as is so easily done in physics. The ability to obtain quantitative data in the face of such difficulties is a tribute to the ingenuity and determination of biologists. Because of its soft character, biological data cannot support sophisticated mathematical models, and we do not often encounter them. One sometimes does see theoretical models in the literature which flourish in the face of a great poverty of biological facts. My own view is that such attempts must be considered at the present time to be exercises in mathematical speculation. Rather, it is often found to be the case that a phenomenological model, one integrated closely with experiment, and not based solely on first principles, is the best that one can formulate to represent a given set of observations. A spectacularly successful example of such a model is the Hodgkin-Huxley theory of nerve conduction. Finally, I would like to suggest to those mathematically-oriented scientists among you who are serious about making a contribution to the mathematical theory of biology, that you should first study biology. With regard to the latter task, I have found that it has not been difficult to do. Because biological knowledge is often descriptive or semi-quantitative, it is not long before one is at the frontiers of a branch of biology. The greatest difficulty encountered in learning this (as well as another) subject is the problem of learning its special vocabulary. vii

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LECTURE 1

The Circulatory System and the Flow of Blood, I The first two lectures of this series will be devoted to the subject of the flow of blood in the mammalian circulatory system. This subject is atypical of others in biology in that the underlying physical theory of the phenomena to be encountered is very well understood, being that of continuum mechanics, especially fluid dynamics and elasticity. Nevertheless, the theory of blood flow is still only in its infancy because the theoretical problems encountered in it are rather difficult as problems in mechanics go. The two principal reasons for this are the complex architecture of the circulatory system, and the anomalous mechanical properties of blood and its vessels. This lecture will be largely devoted to the properties of blood.

FIG. 1. Major pathways of circuit representing the dr. ulatory system, from Green [[]. 1

2

LECTURE 1

Figure 1 shows a block diagram of the human circulatory system [1], consisting of the heart plus a branching system of elastic tubes. Its objective is the distribution of blood to the tissue and back. The system is seen to consist of many parallel branches. The shortest passageway back is through the myocardium (heart tissue) itself. The heart is seen to consist of two power sources called ventricles. The auricles are merely collection chambers. Blood flow is pulsatile, so that the circulatory system from an electrical engineering viewpoing is an A.C. power system whose primary objective is D.C. power. The ventricles must maintain equal mean outflow, although they do not operate at the same mean pressure. The right ventricle, which supplies blood to the lungs only, is a low pressure pump (PRV ~ 20 mm Hg, read "mean pressure in right ventricle is approximately 20 millimeters of mercury"), while the left ventricle which supplies blood to the rest of the body, is a high pressure pump (pLV ~ 120mm Hg). Equal outflow is accomplished, not by a feedback control mechanism, but in a purely passive mechanical way, according to "Starling's law of the heart." Heart is mainly muscle which consists of muscle fibers, and according to Starling's law, "the energy of contraction is a function of the length of the muscle fiber." This relationship is shown schematically as follows:

If the ventricular flows are unequal, blood accumulates in the vessels and tissues between the ventricles, a serious condition known as congestive heart failure. Figure 2 represents a cast of a cat's aorta [2], which is the blood vessel emanating from the left ventricle through which blood leaves. Figure 3 shows a cast of the pulmonary arterial tree [3], which feeds the lungs. Harvey, who discovered the circulatory system, knew that blood left the heart through the arteries, and returned to the heart through the veins, but he did not know how blood got from the arterial side of the heart to the venous side. We now know that this is accomplished through the smallest blood vessels, the capillaries, shown in Fig. 4 [4], It is at this stage that oxygen, nutrients, and waste products are exchanged between the circulating blood and the tissue.

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, I

FIG. 2. Silicone cast of cat's aorta and major branches, from Frasher [2].

FIG. 3. Cast of the pulmonary arterial tree, from Attinger [3].

3

4

LECTURE I

FIG. 4. Capillary bed in margin of a rabbit ear. Large vessels on left and right are venules and small arteries, respectively. Taken from Burton [4, p. 69].

In sum, the geometry of the circulatory system may be viewed roughly as two trees, representing the arterial and venous sides of the circulation, whose leaves are pasted together. The heart joins the bases of the trunks of the trees. The capillaries and smaller blood vessels are contained in the leaves which represent the body tissues. For the remainder of the lecture, I shall discuss some important properties of blood. Blood is a suspension of blood cells in a liquid, the blood plasma. The cells are microscopically visible formed elements. The cells are of three types: (i) red cells (erythrocytes), actually a nonnucleated cell remnant; (ii) white cells (leukocytes), a nondividing cell; (iii) platelets (thrombocytes), a nonnucleated cell remnant. The overwhelming majority of blood cells are the red cells, which give blood its distinctive color. White cells make up 0.9 % of the cell volume. The volume concentration of cells in the blood is called the hematocrit: 0.41-0.44 is its range in normal man. Plasma consists of 91 % H 2 0,6.5 % protein (mostly albumin, globulins, fibrinogen), 1.4% substances transported by the blood stream (mostly glucose, urea), and 0.6 % inorganic salts (mostly NACL). Blood coagulation occurs in response to injury or exposure of blood to air through a complex sequence of biochemical reactions culminating in the for-

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, I

5

mation of fibrin. Clotting may be suppressed by anti-coagulant agents such as heparin, citrate, or oxalate. Serum is plasma minus fibrinogen, a protein needed for coagulation. The density of blood pBL = 1.06gm/cm3. However, p cells = 1.09gm/cm 3 , and ppiasma = 1.03 gm/cm3, and since p ce , Is > p p i asma , cells will sediment. After mixing blood with an anticoagulant, the erythrocyte sedimentation rate (ESR) may be measured. In disease, the ESR always increases. If cells were spheres, the ESR would be determined, in the first approximation of considering blood as a dilute suspensions of red cells, by Stokes' law: Here a is the cell radius, g is the gravitational constant, fj. is the viscosity of plasma, and v is the cell velocity or ESR. The reason for the increase in ESR during disease is the red cell clumping or rouleaux formation, in which the red cells stick together like a stack of coins. This has the effect of increasing a in (1) and hence the ESR. The red cell shape is that of a biconcave disk (see Fig. 5). The cell membrane is very distensible and can be pulled into the shape of a toothpick. In a hypotonic

FIG. 5. The shape of the normal red cell, with dimensions according to Ponder (Hemolysis and Related Phenomena. Grune and Stratton, New York, 1948). Left, illustrating a typical rouleaux formation, from Burton [4, p. 34].

6

LECTURE 1

solution (water, say), red cells will swell into spheres and burst. The intra-cellular contents (mostly hemoglobin) will spill out, leaving behind the cell membrane, which retains its original shape. These empty cell membranes are called "ghosts". An interesting unanswered question is, "Why does the red cell have the shape it does?" Purely elastic theories have been proposed [5], [6] as well as theories combining elasticity with either internal forces [7] or cell surface charge effects [8]. Support for the existence of surface charges comes" from some experiments [9] which show that red cells placed in electric fields assume a regular crenated appearance. Poiseuille was a French physician interested in the viscosity of blood. Because of the difficulties of preventing exposed blood from clotting, he worked with water and measured the efflux from long thin pipes or tubes of circular crosssection [10]. He found experimentally, when such a pipe connects two reservoirs maintained at pressures Pt and P2 respectively, that

where Q is the volume efflux rate from the tube, R is the tube radius, L is the tube length, n is the viscosity of water, and AP = Pl — P 2 , Pj > P2. Equation (2) was subsequently derived mathematically from the Navier-Stokes equations for a viscous fluid. The equation is called Poiseuille's law, and is now so well established that it may be used to determine the apparent viscosity of blood which we shall denote by //*, that is,

When fi* is measured utilizing tubes of different radii, the dependence of n* on R is that shown in Fig. 6. The result that n*(R)/n*(co) decreases as R decreases below a value of 100 n is called the Fahraeus-Lindqvist effect. It is further observed that there is a plasmatic zone or cell-free zone close to the wall of the tube when R < 100 fj.. This zone is greater if AP is increased. In rheology, where similar effects are observed for fluids containing particulate suspensions, the phenomenon is known as the sigma effect. We see that blood has an anomalous viscosity. This is caused entirely by the facts that blood is a suspension of red cells and that it contains fibrinogen, which affects the red cell-red cell interaction. Blood plasma displays no such anomalous property. The presence of the red cells is also the reason blood is characterized as a non-Newtonian fluid. For a Newtonian fluid the ratio of stress to rate of strain is a constant ju, the fluid viscosity. For blood, the measurements of stress as a function of strain rate [11] are shown in Fig. 7, taken from the work of Merrill [11]. The fact that the curve intersects the ordinate axis at a finite value indicates that blood has a yield stress. This is good for house paint because it should not sag under the influence of gravity. The property of yield stress is distinct from that of viscosity. Honey is very viscous but is a perfectly Newtonian fluid. It drips off a

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, I

7

FIG. 6. The viscosity of blood determined on the basis of equation (3) is shown as a function of tube radius R. The unit of viscosity is that of plasma. From [4], based on the work of Haynes (Amer. J.Physioi, 198 (1960), p. 1193).

FIG. 7. Shear stress r is shown as a function of strain rate y for normal blood. From Merrill [11].

8

LECTURE 1

spoon, unlike mayonnaise which has a yield stress but is not very viscous. The non-Newtonian behavior of blood seems to be important only for capillary flow and in venules (small veins) where the shear stress is small. The existence of a yield stress makes it easier to stop and harder to start blood flow. A theoretical expression for the viscosity of a suspension of spherical particles was first given by Einstein [12]. He found that

where /i* is the viscosity of the suspension, ^ is the viscosity of the suspending medium, and c is the volume concentration of suspended particles. Einstein's formula applies to dilute suspensions, and has been extended in several ways to include terms of order c2, as well as modified to apply to nonspherical particles, by altering the numerical coefficient 2.5. However, there is no theory of the viscosity of a high concentration suspension such as blood. Nevertheless, the Einstein formula has been made the basis for a phenomenological theory of blood flow in small vessels [13], [14], which I shall briefly describe. Assume that for steady viscous tube flow, the shear stress T = ^rdp/dz obeys the equation

as for a Newtonian fluid. Here r is radial distance from the z-axis which is the tube axis, v is the longitudinal velocity in the z direction, and p is the pressure in the tube. By analogy with Einstein's formula, set

where ^0 is viscosity of the suspending fluid and a is a constant, to be specified later. The concentration c is assumed to have a radial dependence,

where R is the tube radius, and d is also an unspecified parameter. 6 and a are determined by letting the mean blood concentration c equal the hematocrit value, or

and by fitting n to the observed anomalous behavior of/x blood . In this manner, it is found that a « 1.75, and d K 3 microns. The latter value agrees qualitatively with observations.

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, I

9

An alternative approach [15], [16] is to assume Casson's equation

which works well for printing ink and molten chocolate. Here k0 and /c t are constants. Then

where xf = k20. The region T < if leads to plug flow. The resultant velocity profile predicted is that of a truncated parabola (see Fig. 8) and in fact, observed blood flow profiles have this qualitative appearance.

FIG. 8. The predicted velocity profile of the flow through a pipe, according to equation (10). From Oka [16].

Approaches to a dynamical theory of the behavior of a red blood cell in a Poiseuille flow in a tube have been made, but I think it is fair to state that these are more of fluid dynamical than biological interest. Single red cells appear to move along the axis in small-bore tubes. In capillaries, the cells manage to get through by deforming themselves even though the diameter of a capillary is often smaller than the largest diameter of a red cell. Lighthill [17] has proposed a "hydrodynamic lubrication theory" to account for the red cell passage through capillaries. It is generally believed that the anomalous viscosity of blood is of physical or hydrodynamic origin. It has been observed [18] that rigid spheres placed in Poiseuille flow end up in an annular region downstream in the tube, regardless of their initial radial position. At the risk of alienating magnetohydrodynamicists and Englishmen, I call this the "bloody pinch effect". An adequate theoretical explanation of this effect is still needed. A lift force, perpendicular to the direction of motion of the tube flow, is required to help explain it. In the Stokes' approximation to the fluid dynamic equations, it is a theorem that an arbitrary shear flow produces no lift. Nevertheless, it is instructive to consider the Stokes' approximation, because the equations are linear. For a sphere in an incident

10

LECTURE 1

Poiseuille flow, the incident flow with respect to a coordinate system fixed at the center of the sphere may be broken up into its constituent parts and considered separately, as shown in the diagram.

The lag induced by the quadratic shear and the spin induced by the linear shear cause the sphere to behave like a curving baseball. This has led to the consideration of a spinning sphere (angular velocity Q) in a uniform slow flow ~v, taking into account the inertial or nonlinear terms in the Navier-Stokes equations. J. B. Keller and I [21] found, by considering these equations with the aid of a perturbation expansion in the Reynolds number, that there is in fact a lift force ^ which to first order in the Reynolds number is given by the expression where a is the radius of the sphere and p is the fluid density. If this force is applied to the problem of a sphere in a Poiseuille flow (neglecting boundary wall effects),

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, I

11

so that fl is set equal to the Einstein spin velocity, and IMS set equal to the Simha lag velocity, it is found that

FL is an inwardly directed force of the correct order of magnitude to explain the inward motion of particles initially near the tube wall in the bloody pinch effect. A former doctoral student of mine, S. Bjorklund [22] considered the NavierStokes equations for a sphere in a Poiseuille flow neglecting wall reflections, but including the inertial effect. He found, to first order in the Reynolds number, the above force (12), as well as several others. The net force was inward. There is as yet no theoretical explanation for the outward motion of spheres initially close to the axis. Presumably the origin of this motion is to be found by proper consideration of the tube wall, and the reflections it produces. Goldsmith and Mason [23] observed that a fluid droplet of one viscosity immersed in a Poiseuille flow of a fluid of a different viscosity moves rapidly to the axis of the tube while being carried downstream longitudinally. This occurs under conditions for which a rigid sphere of the same size as the drop displays no measurable transverse deviation from straight longitudinal motion. Inasmuch as the drop also deforms slightly away from its spherical shape, it seems obvious that the predominant cause of the transverse or lift force causing the axial motion is associated with the deformed shape of the particle. Furthermore, since this force is observed to be qualitatively greater than the inertial lift force previously discussed, it would appear to be sufficient to utilize the linearized or Stokes form of the Navier-Stokes equations. Another graduate student of mine, Philip Wohl, recently completed this calculation [24], based on a small perturbation expansion in the nondimensional parameter fiU/T. Here \i is the external fluid viscosity, U is a characteristic velocity, and T is the surface tension associated with the interface between the two viscous media. He found that there is a lift force in the drop, directed inward, whose physical origin is explicated by the following diagram:

12

LECTURE 1

Thus, to first order in the small parameter, there is a force Fl resulting from the uniform flow incident on the drop deformed by the linear shear, etc. The net force Fj + F2 4- F3 is always pointed inwards towards the axis, and there is reasonable agreement between the theoretically predicted drop trajectories and observations. This still leaves the problem of the dynamics of a single red cell in a tube flow for the future. Perhaps a fluid sphere surrounded by a thin elastic membrane represents a sufficiently good model of a red cell, for this purpose. REFERENCES [1] H. D. GREEN, Circulation: Physical Principles in Medical Physics, vol. 1, O. Glasser, ed., Year Book Publishers, Chicago, p. 210. [2] W. G. FRASHER, in Biomechanics, Proc. Symp. Appl. Mech. Div. ASME, New York, 1966, Y. C. Fung, ed., ASME, New York, p. 1. [3] E. O. ATTINGER, Circ. Res., 12 (1963), p. 623. [4] A. C. BURTON, Physiology and Biophysics of the Circulation, Year Book Medical Publishers, Chicago, 1965. The original source for Fig. 4 is R. L. de C. H. Saunders, J. Anat. Soc. India, 8 (1959), p. 1. [5] Y. C. B. FUNG AND P. TONG, Biophys. J., 8 (1968), p. 175. [6] P. B. CANHAM, J. Theor. Biol., 26 (1970), p. 61. [7] B. B. SHRIVASTAV AND A. C. BURTON, Can. J. Physiol. Pharmacol., 48 (1970), p. 359. [8] L. LOPEZ, I. M. DUCK AND W. A. HUNT, Biophys. J., 8 (1968), p. 1228. [9] R. P. RAND, A. C. BURTON AND P. CANHAM, Nature, 205 (1965), p. 977. [10] J. L. M. POISEUILLE, Ann. Chim. Physiol., 7 (1843), p. 50. [11] E. W. MERRILL, Physiol. Rev., 49 (1969), p. 863. [12] A. EINSTEIN, Ann. Phys.. 19 (1906), p. 289. [13] H. W. THOMAS, Biorheology, 1 (1962), p. 41. [14] T. WATANABE, S. OKA AND M. YAMAMOTO, Ibid., 1 (1963), p. 193. [15] E. W. MERRILL, A. M. BENIS, E. R. GILLILAND, T. K. SHERWOOD AND E. W. SALZMAN, J. Appl. Physiol., 20(1965), p. 954. [16] S. OKA, in Symposium on Biorheology, A. L. Copley, ed., Proc. 4th International Congress of Rheology, Interscience, New York, 1965, part 4, p. 89. [17] M. J. LIGHTHILL, J. Fluid Mech., 34 (1968), p. 113; in Circulatory and Respiratory Mass Transport, G. E. W. Wolstenholme and J. Knight, eds., J. and A. Churchill, London, 1969, p. 83. [18] G. SEGRE AND A. SILBERBERG, Nature, 189 (1961), p. 209. [19] G. G. STOKES, Camb. Phil. Trans., 9 (1851), p. 8. [20] R. SIMHA, Kolloid Z., 76 (1936), p. 16. [21] S.I.RUBINOWANDJ. B. KELLER, J. Fluid Mech., 11 (1961), p. 447. [22] S. B JORKLUND, On the force on a rigid sphere in unbounded Poiseuilleflow, Doctoral thesis, Stevens Institute of Technology, Hoboken, New Jersey, 1965. [23] H. L. GOLDSMITH AND S. G. MASON, J. Colloid Sci., 17 (1962), p. 448. [24] PHILIP R. WOHL, The transverse force on a drop in unbounded Poiseuille flow. Doctoral thesis, Cornell University, Ithaca, New York, 1971.

LECTURE 2

The Circulatory System and the Flow of Blood, II Here I shall consider blood flow problems associated with the larger arteries and veins. For such vessels, the anomalous viscosity of blood is not of significance, because of the large shear rates encountered in such flow. For the gross properties of blood flow with which we shall concern ourselves, it suffices to consider blood as a Newtonian fluid. The steady or "D.C." flow of blood through the entire circulatory system of a mammal such as a dog has been measured as a function of the difference between the mean pressure upstream, in the left ventricle or aorta, and the mean pressure downstream, in the vena cava. When the downstream pressure is varied, the pressure-flow curve shown in Fig. 1 results [1]. It makes no difference whether

FIG. 1. The steady flux of blood through the superior vena cava of a dog as a function of the pressure difference p t - 1 p2, where />, is the pressure in the jugular vein (upstream) held fixed, and p2 is the downstream pressure which was varied. Taken from Ruhinow and Keller [3], redrawn from the work of Brecher [1].

the intact heart governs the upstream pressure, or whether the upstream pressure is maintained by a pump. Physiologists have discovered that this remarkable nonlinear pressure-flow relation is also the property of the steady flow through a single elastic "collapsible" tube, as shown in Fig. 2 [1]. In fact, this similarity has 13

14

LECTURE 2

FIG. 2. The steady flux through a collapsible rubber tube as a function of the outlet pressure p2 for fixed inlet pressure pj and external pressure p 0 . Taken from Rubinow and Keller [3], redrawn from the work of Brecher [1].

led physiologists to name such a rubber tube a "Starling resistor" [2]. It has also led J. B. Keller and myself to propose a rather simple theory for the steady viscous flow through such an elastic tube [3], which I shall present here. Consider a long elastic tube of circular cross-section which varies with distance down the tube. A viscous fluid flows at a steady rate through the tube. Denote the upstream pressure by plt the downstream pressure by p2, and the pressure external to the tube by p0 (see diagram). We assume that Poiseuille's law holds

locally so that through any element of length dz of the tube,

Here Q is the volume flux of fluid, p = p(z) is the pressure at position z, and a is a proportionality constant called the conductance. It depends on the local cross-sectional area of the tube. Since the tube contracts or expands locally in response to the transmural pressure difference p(z) — p 0 , the cross-sectional area does also. Clearly, then,

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, II

15

Substituting (2) into (1) and integrating with respect to z, we easily find that

Equation (3) determines p as a function of z and the parameter Q. To determine Q, we set z = L in (3). Then

From this equation we can draw a number of significant conclusions. It is seen that Q depends on L,pl — p 0 ,andp 2 — p0. Q is an increasing function of pl - p0, and a decreasing function of p2 — Po (since a 5; 0). By the mean value theorem, QL is proportional to (PI — p 2 ) when (pl — p2) is small, as for a rigid tube. Finally, we see that Q -» limiting value as p2 — p0 -> — oo, assuming the integral exists. Thus, the principal qualitative features of Figs. 1 and 2, namely that Q is linear in (Pi ~ £2) f°r small pressure differences, and that it has an asymptotic value as p 2 decreases, are predicted by the theory. If the tube is circular, then according to Poiseuille's law,

where a is the tube radius. The radius of the tube is related to the transmural pressure difference and to the tension in the tube in an elementary physical way by the so-called "law of Laplace" (first suggested by Thomas Young):

where T(a) is the tension in the tube defined as force per unit length. Since the tension depends on the radius, the utilization of (5) and (6) requires the knowledge of the tension-length dependence T(a) for the elastic tube. As an application of the theory, we considered the flow through a more or less typical artery. Figure 3 shows a normal tension-length diagram for a section of the external iliac artery in man, as determined by static pressure-volume measurements by Roach and Burton [4]. From it, the dependence of p — p0 on a, as well as the inverse relationship, may be inferred. Let a0 be the resting value of the radius for which p - p0 = 0. Because the theory demands knowledge of p — p0 when a < a0, a functional dependence T(a) for a < a0 was assumed, which was symmetrically oriented with respect to the dependence of T(a) for a > a 0 . Other experimental evidence for such an assumption exists. On this basis, QL could be calculated, and some of the results of the calculation are shown in Fig. 4. The physical explanation of the theory is simply this: if a collapsible tube is used as a drinking straw, then increasing the suction beyond a given point does not appreciably increase the flux. Actually, increasing the external pressure relative to the internal pressure leads to buckling, which has not been taken into account in

16

LECTURE 2

FIG. 3. The solid line is a tension-length diagram for the external iliac artery, based on the experimental observations (circles) of Roach and Burton [4]. The dashed curve represents p - p0 as given by (6). From Rubinow and Keller [3].

the theory. When buckling occurs p2 — p0 is less than the critical buckling pressure which depends on the thickness of the tube. For arteries whose thickness/radius ratio is ~0.1, the critical buckling radius is a cril w 0.97a0. Other experimental observations, analogous to Figs. 1 and 2 and plotted in the form of Fig. 4, are shown in Fig. 5. The curves represent the steady flow through an isolated perfused cat lung [5] as a function of Pv (venous or downstream pressure) for fixed PA (arterial or upstream pressure). In these experiments, PT is just the constant tracheal pressure external to the lung. The quantitative agreement of the theory of the flow through a single tube with the observations of a complex network of branched elastic tubes may be understood as follows. A network may be approximated as a large inlet artery and a large outlet vein, connected together by a parallel arrangement of JV identical small elastic tubes, like the rungs of a ladder. Because the inlet and outlet tubes are of large bore, the pressure drop down their lengths is slight, so that they may,

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, II

17

FIG. 4. Theoretical curves oj QL as a function of [>2 - p0 for various fixed values of the pressure difference PI — p0, in nondimensiona! form. The dashed lines represent asymptotes when p2 — pa —> — x. From Rubinow and Keller [3].

FIG. 5. The steadv flow F through an isolated perfused cat lung as a function of venous pressure PY for fixed values of pulmonary arterial pressure PA and for fixed values of tracheal pressure Pr. From the work of Banister and Torrance [5].

in the first approximation, be assumed to be each at a constant pressure. Thus, the total flux through the system is N times the flux through one of the small identical tubes. The theoretical treatment of the buckling of a tube is the same as the classical problem of the buckling of a ring, and the problem of opposite wall contact cannot

18

LECTURE 2

be ignored. The nonlinear equations of elasticity are required and the problem has been considered in a subsequent work [6]. Figure 6 shows the cross-sectional appearance of actual buckled elastic tubes [7], and is identical in form to the results of the theoretical calculations. The vena cava has been seen to buckle in

FIG. 6. The cross-sections of Penrose rubber tubing in various stages of collapse during flow, as observed by Holt [7].

this fashion during breathing. Knowing the buckled cross-section, we can determine a by solving a suitable flow problem, and knowing a, we can determine Q. Since the circle is the optimum cross-section for maximizing the flux, we know that the asymptotic value of Q will be lower than that shown in Fig. 4, but the qualitative features of the pressure-flow curve will not be affected. I shall now turn to some "A.C." properties of the circulating system. Blood flow is pulsatile, and the blood pulse is a wave phenomenon, as was first recognized by Young (1808) [8]. He derived an expression for the velocity cof the pulse wave,

His model was a single long elastic tube of radius a, tube thickness h, tube Young's modulus E, and fluid density p0. Young's ingenious derivation was based on the recognition that an incompressible inviscid fluid contained in an elastic tube would have, by virtue of its container, an effective bulk modulus, and would therefore sustain a wave through it. He calculated this effective bulk modulus and then simply used Newton's formula for the wave velocity through a compressible medium. That the pulse wave was a new phenomenon unknown to the theories of fluid dynamics and elasticity can be recognized by observing that the pulse velocity depends on properties of the elastic tube and the fluid contained in it. I shall present here a subsequent version, due to W. Weber (1850) [9] and Resal (1876) [10], which displays the essence of the present-day theory.

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, II

19

Consider a long elastic tube of cross-sectional area A through which flows an incompressible inviscid fluid, as shown in the diagram:

The equation of motion and equation of continuity for the fluid assume the forms:

Here p0 is the constant fluid density, and p = p(z, t) is the fluid pressure at position z at time t. Equations (8) and (9) are supplemented by an equation of state for the tube wall which relates the cross-sectional area A to the fluid pressure p: In fact, equations (8)-{10) were all written down, not surprisingly, by L. Euler in 1776, although his investigation was first published posthumously in 1862 [11]. Euler, incidentally, complained about the difficulty of solving these nonlinear equations. In recent times, they have been solved numerically using the method of characteristics. The simplifying thought of Weber and Resal was to linearize them. Thus, assume

where A0 is the equilibrium value of A when the fluid is at rest, and p0 = p(A0). Substitute (11) and (12) into (8)-(10), neglect p 0 , and assume that a, w, and p and their derivatives are small quantities. We find that these quantities all satisfy the wave equation, for example,

20

LECTURE 2

where the velocity of propagation c is given by the expression

Now assume that the tube is of circular cross-section and obeys a linear elastic law (a linear approximation to the curve shown in Fig. 3). The tension T is then related to the radius r of the tube by the equation where E is Young's modulus, and r 0 is the equilibrium radius of the tube. If we combine this with the equilibrium equation, where h is the thickness of the tube, we find on eliminating T t h a t (dp/dA) 0 can be readily calculated (remember A = nr2) to be

Substituting (17) into (14) yields Young's formula. Korteweg [12] extended these considerations by permitting the fluid to be compressible, and by including the effect of tube inertia. Thus, he replaced (10) by the equation of motion of the tube. His result for c was

where c0 is the velocity of sound in the fluid. This result shows that c < c 0 . Equation (18) was subsequently rederived by Joukowsky (1900) [13] and Allievi (1902) [14], and is known in engineering practice as the water-hammer equation. When the tube inertia is taken into account, the fact that the velocity is frequency dependent becomes apparent. Korteweg showed that (18) is the long wave-length limit for the phase velocity of the wave. The first modern mathematical analysis of the problem was made by Lamb [15] in 1898. He showed that there were in fact two propagating modes for waves in a fluid-filled tube. Most of these theoretical investigations languished or were ignored, until Womersley's investigations in the 1950's [16]. He tried with some success to apply such theoretical considerations to actual problems of pulse propagation. Since his work, a great many similar investigations have been made, although there has been a certain monotony in them in that a single fluid-filled tube is always considered. Variety arises from trying to consider effects of viscosity, tube taper, tube viscoelasticity, modes with nonaxial symmetry, nonlinear inertia effects, etc. J. B. Keller and I have added our bit [17] by considering the effect of frequency on the phase velocity. Figure 7, taken from our work, shows the phase velocity as a function of frequency for all propagating modes. Each mode is associated with one curve, and the labels denote different values of

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, II

21

FIG. 7. The wave number k is shown as a function of the frequency to for propagating waves in a thin elastic tube filled with an inviscid compressible fluid. The labels denote various values of a mass parameter m. There are two "tube" modes, labeled + and -, and an infinite number of acoustical modes, shown at the right, with a common asymptote k = eo/c0. The diagram contains a change in scale at to = 2. From Rubinow and Keller [17].

a parameter m, where m is 1/2 the ratio of the tube mass to the fluid mass, per unit length of tube. It can be seen clearly that there are two propagating modes at low frequency labeled " + " and " — " in thefigure,and that one of the modes (having Young's velocity formula at zero frequency) displays a stop and pass band character, that is, it does not propagate beyond a certain "cut-off" frequency. The other mode, which does propagate at all frequencies, is only readily detectable in the tube wall, where the amplitude of oscillation is greatest. The remaining propagating modes are acoustical in nature and occur at rather high frequencies only. It must be stressed here, however, that pulse propagation is basically a long wave length or low frequency phenomenon. In man, the fundamental heart beat frequency v ~ 1 cycle/sec. Other representative values for the aorta are, in c.g.s. units, T ~ 107 dynes/cm2, r0 ~ 1 cm, h/r0 ~ 0.1, p0 ~ 1 gm/cm3. Thus, by Young's formula, c ~ 1 m/sec, which is a typically observed value for the pulse velocity. The associated wavelength 1 = c/v ~ 7 m, which is greater than the length or diameter of any blood vessel. Consideration of higher harmonics hardly alters the basic conclusion. The main goal of single tube analysis is to provide the foundation for the consideration of a network of such tubes and thus to provide a theory for at least the passive mechanical behavior of the arterio-venous system [18]. This goal is very far from being accomplished, for a number of reasons, both theoretical and experimental: 1. The parameters (Young's modulus, radius, etc.) of the system are not completely known. 2. The significant approximations to be made in the theoretical analysis are undetermined. 3. The problem of junctions where two or more vessels join is unsolved. I shall close with an illustrative example of the difficulties: Fig. 8 shows the variation in the pulse wave form that is observed as it travels down the aorta of a dog

22

LECTURE 2

FIG. 8. The change in wave form of the pressure P and longitudinal velocity V as it travels away from the heart and down the aorta in a dog. From the work of D. A. McDonald [19].

[19]. A pronounced "peaking" of the wave form is seen. There is as yet no satisfactory theoretical explanation for the variation, although the possible principal causes are many: (i) reflections from junction sites in the system; (ii) dispersio (pulse velocity changes with frequency of harmonic); (iii) variation in elasticity with distance; (iv) taper; (v) nonlinear effects (such as fluid inertia). REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

G. A. BRECHER, Amer. J. Physiol., 169 (1952), p. 423. F. P. KNOWLTON AND E. H. STARLING, J. Physiol. Lond., 44 (1912), p. 296. S. I. RUBINOW AND JOSEPH B. KELLER, (. Theor. Biol., 35 (1972), p. 299. M. R. ROACH AND A. C. BURTON, Can. J. Biochem. Physiol., 35 (1957), p. 681. J. BANISTER AND R. W. TORRANCE, Quart. J. Exp. Physiol., 45 (1960), p. 352. J. FLAHERTY, J. B. KELLER AND S. I. RUBINOW, SIAM J. Appl. Math., 23 (1972), p. 446. J. P. HOLT, Circulation Res., 7 (1959), p. 342. T. YOUNG, Phil. Trans. Roy. Soc., 98 (1808), p. 164. W. WEBER, Berichte der Sachs. Ges. der Wiss., Math.-Phys. Classe, 18 (1866), p. 353. H. RESAL, Comptes Rendus, Acad. des Sci., 82 (1876), p. 698. L. EULER, Principia pro motu sanguins per arterias determinando, Opera posthuma mathematica et physica anno 1884 detecta, P. H. Fuss and N. Fuss, eds., Petropoli: Apud Eggers et socios, 2 (1862), p. 814. [12] D. J. KORTEWEG, Annal. Phys. und Chem., Neue Folge, 5 (1878), p. 525. [13] N. W. JOUKOWSKY, Ueber den hydraulischen Stoss in Wasserleitungsrohen, Mem. de L'Acad. Imp. des Sci. de St. Petersburg, 8 serie, 9 (1900). [14] L. ALLIEVI, Theory of Water Hammer, E. E. Holmos, trans., R. Garroni, Rome, 1925.

THE CIRCULATORY SYSTEM AND THE FLOW OF BLOOD, II

23

[15] H. LAMB, Manchester Lit. and Phil. Soc. Mem. and Proc., 42, no. 9 (1898). [16] J. R. WOMERSLEY, Wright Air Development Center, Tech. Rep. WADC-TR56-614(1958) (compilation of all publications of the author devoted to blood flow). [17] S. I. RUBINOW AND JOSEPH B. KELLER, J. Acoust. Soc. Amer., 50 (1971), p. 198. [18] , in Hemorheology, Proc. First Int. Conference, Reykjavik, Iceland, 1966, Pergamon, Oxford/New York, 1968, p. 149. [19] DONALD A. MCDONALD, Blood Flow in Arteries, Edward Arnold, London, 1960, p. 271.

LECTURE 3

Tracer Analysis of Physiological Systems, I The subject of compartment analysis has arisen in the last twenty-five years or so, largely as the result of the introduction of radioactive tracers into biology [l]-[3]. We think of a living organism, an organ, a group of cells, or even a subcellular component, as made up of a set of "pools" or "black boxes". The system is usually an open system: it exchanges material with its environment. The exchange requires energy to drive it. The theory of such systems has a wide range of applicability, encompassing electrical engineering, economics, social sciences, etc. Here I shall present the theory [4] as it applies to physiological systems. Examples of such systems arise in the problem of determining the cardiac output, the problem of the metabolism of iodine or other metabolites, and the problem of the determination of the kinetics of potassium transport in heart muscle. The latter problem could be represented in a schematic way by the following diagram:

A compartment is defined by a characteristic material (chemical species, biological entity) occupying a given volume. The compartment is made unique either by its characteristic material, or by its unique spatial assignment. In addition, a compartment is characterized by its having a steady state flux of material into and/or out of it. A compartment system consists of n interconnected compartments. In general, there may be a flux between any two compartments ;' and j of the system. The flux is bidirectional, and is assumed to occur via separate channels, so that flux (' ~* ]) ^ flux (J ~* ')• An influx from the exterior at a steady rate is permitted into each compartment. We shall assume that no compartment is either a source or a sink. The system is studied experimentally in the following manner. A pulse of labeled material is introduced at time t = 0 into one or more of the compartments. The pulse initiates a transient into the system with the objective of obtaining information about the steady state properties. It is as if we had a black box containing a number of interconnected oscillators or bells and we tapped the box 24

TRACER ANALYSIS OF PHYSIOLOGICAL SYSTEMS, I

25

gently, sufficiently hard to ring the bells, but not so hard as to disturb the interconnections. It is assumed that: 1. The amount of labeled matter « the amount of unlabeled matter. 2. The steady state is unaltered. 3. The material in a compartment is homogeneous. This implies that: (A) labeled matter is representative of unlabeled matter; (B) material in a compartment is instantaneously mixed. 4. The transit time between any two given compartments is nil. The system may be conceptualized in the following manner: imagine a set of water reservoirs or bathtubs which are interconnected, two pipes (in and out) connecting any pair. Water flows through each pipe in the assigned direction. The level of water in each bathtub remains constant at all times. Aside from its interconnecting pipes, each bathtub has: (i) its own faucet for receiving water from the outside at a constant rate, turned on or off; (ii) its own drain to the outside, which may be open or closed; (iii) its own blender, continuously "homogenizing the water. At t = 0, a known amount of dye is injected into some of the bathtubs. Imagine that we can measure the concentration of dye in one or more bathtubs. From these measurements, made at more or less regular time intervals, we wish to determine the following: 1. The amount of water in every bathtub. 2. The rate of water flow between any pair of bathtubs, in both directions. The fundamental mathematical equations of the system are, in matrix notation,

Here p = {pt} is a column vector with generic element pt representing the amount of labeled material in the rth compartment, i = 1, 2, • • • , n. L = {L0} is an n x n square matrix, with Lu (i ^ j) representing the material transport rate from compartment j to compartment i, per unit amount of material in the ;th compartment. Lfj is called the fractional turnover rate from j to ;'. Note the convention: the direction of transport is from the second subscript to the first. (- L n ) represents the fractional rate at which material in the zth compartment is replaced, or the total turnover rate of compartment /. Assume L is a constant matrix. To be more explicit, the first component of (1) has the form

On the right, the first term is negative and represents the rate at which labeled material leaves compartment 1, the second term is positive and represents the rate at which labeled material enters compartment 1 from compartment 2, the third term is positive and represents the rate at which labeled material enters compartment 1 from compartment 3, etc. In view of their physical significance,

26

LECTURE 3

the components of L satisfy the following conditions:

It follows from (2) and (3) that Lfi rg 0, and the equality sign holds only if all LIJ = 0, i fixed, for all j ^ i. The fractional excretion rate from compartment j, LQj, is defined as

From what was said before, it follows that L0i ^ 0 for all i. We say that a compartment is leaky if L 0i ^ 0, and leakproof if L0i = 0. If every compartment is leakproof, the system is said to be closed. If it is not closed (i.e., at least one L0| ^ 0), then the system is said to be open. The above linear system of n first order ordinary differential equations represents the transient response to tracer injection of many biological systems. Why should this be? In general, no justification has been given, in the sense of an appeal to the theory of more fundamental mechanisms, aside from plausibility. Ultimately, the test is whether the equations "work" or not, and the answer is they do. What is the mechanism of transport? I did not say, but it could be diffusion, convective flow, biochemical reactions, or something more complicated. In principle, each such mechanism could be examined individually so as to "derive" the compartment equations for it. For the examination of a simple case of diffusion, see [5]. The only general argument I can give is to imagine that the transport of unlabeled material p is governed by the nonlinear equation system

where The steady state of the system is denned by the system of equations which has the solution p = p0 = {pio}. To describe the transient case of a small perturbation away from the equilibrium state, set where p = {/?,} and pt « pi(l for all i. Then expand/in a Taylor series about the equilibrium point p 0 , retaining first order terms in p only. Thus,

Substituting (6)-{8) into (5) leads directly to (1).

TRACER ANALYSIS OF PHYSIOLOGICAL SYSTEMS, I

27

It is very useful to introduce the associated diagrams of a compartmental problem which will be recognized as the associated graphs of a matrix [6]. Thus, to each off-diagonal matrix element of L assign a directed line segment, and to each compartment a node or circle. In addition, each excretion rate L 0j is also assigned a directed line segment leaving the node i. Then the diagonal matrix elements of L may be inferred from the graph by utilizing (4). For example, a typical three compartment system might have an associated graph which appears as follows:

Here the absence of certain directed line segments implies that L 01 = L 23 = 0. The steady state behavior of the system prior to the introduction of tracer material is described by the equation

where I/ is the characteristic volume associated with compartment j, j = 1,2, • • • , n. It is also called the volume of distribution, or compartment size. I{ represents the constant material flux rate from the exterior into compartment / (the faucet flow rate, in the "bathtub" description). We shall now solve the direct problem of compartment analysis, which is a necessary preliminary to solving the central problem of compartment analysis. Let where x = {x;}, xi is the concentration of labeled material, or ratio of labeled to unlabeled material, in the rth compartment. It is sometimes measured by specific activity, or the amount of radioactivity per unit amount of material. Kis a diagonal matrix, Vti = V,. Assume V is constant. It is a theorem that all the components of x are nonnegative for t ^ 0 if they are nonnegative at f = 0 [7]. Substituting (10) into (1), there results

where M = V

LV. Note that V

-1

exists, because V is not singular. What is

28

LECTURE 3

the solution to (11)? The question posed is called the direct problem. To answer it, let where A is a constant n x n matrix, the solution matrix, and A, are the eigenvalues of M as well as of L, since these two matrices differ only by a similarity transformation. It has been shown [8] that A f have negative or zero real parts, and are never pure imaginary. In practice, I have never seen a compartmental system that had complex roots associated with it, although oscillations in biological systems certainly do exist. From (13) it follows that

where A is a diagonal matrix, A,-,- = A,-. Substituting (12) into (11) yields the result Thus, knowing M, we can determine A and A from (15) using standard methods. Equation (15) states that the columns of A are the eigenvectors of M. We have assumed that the eigenvectors are linearly independent and that the eigenvalues are distinct. There are some theoretical complexities which may result if the eigenvalues are not distinct, that is, if a multiple root occurs. Then powers of t appear in the solution. The case of a zero multiple root does not in fact induce the appearance of such powers [9], but a multiple nonzero root could. However, we do not have to concern ourselves with such singular matrices M, for a reason to be given later. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]

C. W. SHEPPARD AND A. S. HOUSEHOLDER, J. Appl. Phys., 22 (1951), p. 510. J. S. ROBERTSON, Physiol. Rev., 37 (1957), p. 133. C. W. SHEPPARD, Basic Principles of the Tracer Method, John Wiley, New York, 1962. S.I. RUBINOW AND ALICE WINZER, Math. Biosciences, 11 (1971), p. 203. H. COHEN AND S. I. RUBINOW, in Proc. Symp. System Theory (New York), Polytechnic Press, Brooklyn, New York, 1965, p.321. See, for example, R. B. MARIMONT, Bull. Math. Biophys., 31 (1968), p. 255. R. BELLMAN, Introduction to Matrix Analysis, McGraw-Hill, New York, 1960, p. 172. J. Z. HEARON, Bull. Math. Biophys., 15 (1953), p. 121. , Annals N. Y. Acad. Sci., 108 (1963), p. 36.

LECTURE 4

Tracer Analysis of Physiological Systems, II We continue our discussion of the previous lecture with the central problem of compartment analysis: given A and partial knowledge of A, what are the compatible matrices L and V. This problem which is inverse to the direct problem may be called the inverse problem. The first task posed by this problem is to convert the raw data, namely, the observations of x t ( t ) , x2(t), etc. at given time intervals, into continuous representations consisting of sums of exponentials, viz., This is not a trivial problem, and may be solved by the method of least squares in conjunction with Newton's method, or more usually by a simpler variant called the method of exponential peeling [l]-[3]. We shall assume that this conversion has been accomplished. It is at this point that the assumption is tacitly made that L and hence M is not singular. If it were, powers of t would appear in the solution. However, one attempts to fit the data only by an equation of the form of equation (12) in Lecture 3, without powers of t in it. Thus, the distinct A; that are found are assumed to be the only eigenvalues. The number of eigenvalues determines n. Clearly, if all compartments can be observed and fitted, then A and A are completely known, and so is M, from equation (15) in Lecture 3: Note that A~ l exists because its columns are eigenvectors of M and are linearly independent. To disentangle L and V from a knowledge of M, it is necessary to have independent knowledge of the values of /;. Suppose A is not completely known. What can be said about M? We may view (2) as n2 equations for the n2 elements of M, expressed in terms of (n2 + n) parameters, the n2 elements of A and the n elements of A. The initial condition corresponding to an experiment, can be considered as n equations of constraint on the elements of A, so that the right-hand side of (2) in fact contains only n2 unknown parameters. Suppose now that a particular component i is observed as a function of time. This observation determines (In — 1) of the parameters: (n — 1) amplitudes Ati and n eigenvalues &j. Observation of a second compartment 1 determines only (n — 1) additional parameters, the AIJ. If k parameters are left undetermined, (2) implies that a 29

30

LECTURE 4

/c-parameter manifold of compatible matrices exists. Berman and Schoenfeld [4] suggested that an initial guess be made of the k parameters, and the resulting matrix M\ be calculated and checked to see whether it satisfies the constraint equations (2) and (3) in Lecture 3. If not, the initial guess could be altered, and the procedure repeated. The troubles with this procedure of course are that (i) convergence is not assured, and (ii) only one compatible matrix is found. An alternative suggestion that the original data be fitted directly to the elements of M [5] suffers the same objections. It appears that the only sensible procedure is to determine the entire class of matrices. How can this be done? As an initial step, we suggest that all the relationships relating the elements of M to the observed data be set down. In other words, first eliminate the unknown parameters from consideration. To accomplish this, observe that (3) implies

Taking integral powers of (2) leads to p integral. If now we post-multiply (5) by x(0) and utilize (4), there results

We permit p to be no larger than (n — 1) because of the Cayley-Hamilton theorem that a matrix satisfies its own eigenvalue equation. Setting p = n would not lead to a new relation between M and the parameters because it would be a linear combination of the above relations (6). Further, we may set down the relations between the matrix M and its invariants:

On the left-hand side of each equation there appears the sum of all g-rowed principal minors; on the right, the sum of eigenvalues taken q at a time. Equations (6) and (7) constitute all the known independent relations between the elements of M and the determined parameters. For example, when one compartment i is observed, one row of A and all the elements of A are fixed. Therefore, the ith element of the vector on the right-h md side of (6) is determined, for any value of p. Thus, by varying p, (6) permits us to write (n — 1) equations for the elements of M. Together with the n equations appearing in (7), we can write a total of (2n — 1) equations relating the elements of M to known quantities. If

TRACER ANALYSIS OF PHYSIOLOGICAL SYSTEMS, II

31

another compartment is observed, an additional (n — 1) equations from (6) may be set down. However, these equations are nonlinear algebraic equations. They are an under-determined set so long as the number of observed compartments is less than n. Very often, other biological knowledge of the system under investigation permits one to say or to believe strongly that some matrix elements are null: Li} = 0 for given values of i and j. Therefore, it is important to pose the question: what are the minimal matrices compatible with observations. By a minimal matrix I mean one subjected to a maximum number of null constraints. We have tried to answer this question, and have reached a few interesting conclusions. 1. Setting a diagonal element of L equal to zero is forbidden. From Lecture 3, equations (2) and (3), Lti = 0 for a fixed i implies that Ljt - 0 for all j, which implies that / is a sink, contra ~y to assumption. 2. L cannot be quasidiagonal. If L is of the form

with A and B square

submatrices lying along the main diagonal, the system consists of two physically disjoint subsystems. Then less than n eigenvalues are needed to fit the observations of a given compartment, contrary to assumption. 3. L is reducible only if A. A "source-like" compartment was initially injected. B. A "sink-like" compartment was observed. L is reducible if it can be put (by rearrangement of rows and columns) into the form

, where A, B and C are submatrices. What this implies is

easily recognized with the aid of the associated graph :

The compartments represented by A ("source-like") are unidirectionally connected via C to the compartments represented by B ("sink-like"). Suppose a compartment in A were observed. Then no labeled material would be detected if a compartment in B had been initially '.-jected, and less than n eigenvalues would be required to describe the observations if a compartment in A had been initially injected. Both possibilities contradict the original assumption that n eigenvalues are needed to describe the observed compartment in A.

32

LECTURE 4

As a more specific example, we shall show the associated graphs representing the minimal matrices for a three-compartment system in which the same compartment is initially injected and subsequently observed (shown shaded in diagram). There are only four graphs possible:

The corresponding matrices constitute a two-parameter family. Further null restrictions can only be applied by setting some L0i = 0. Up to now, the two- and three-compartment systems have been more or less thoroughly examined, but little has been done for four or more compartment systems. It seems clear that graph theory has a useful role to play in the solution of such problems as the determination of minimal matrices. I believe that the present status of compartment analysis from a mathematical viewpoint is that it is a problem in computer science. It would be very desirable if a biological investigator could punch in all his data to a computer and have the computer punch out all compatible matrices, with a subsequent investigatorcomputer dialogue in which the investigator could examine the consequences of other constraints. As a postscript, I would like to indicate how the imposition of null constraints also aids in the disentangling of L and V from M. In particular, I shall show that there exist compartment systems for which the matrix elements of L and V can be uniquely determined (up to an (n — l)-fold multiplicity), even if only one compartment has been observed [6].

TRACER ANALYSIS OF PHYSIOLOGICAL SYSTEMS, II

33

In the first instance, we see from the definitions of M (see Lecture 3, equation (11)) and V that

From this we see that diagonal elements of M and L are equal,

It also follows that for all off-diagonal elements,

If, in addition, a compartment system is closed, then L0j = 0 for all ;', so that from Lecture 3, equation (4), n

The closure property also implies that there is no net material influx into the system, so that /, = 0 for all i. It follows then from Lecture 3, equation (9) that

which may also be written, in view of (8), in the form

Combining (13) and (11) with (9) yields the relation

where the prime denotes exclusion of the term i = j in the summation. The n relations (14) are a property of all closed systems. Now let us consider a closed catenary compartment system. A catenary system is one for which Lti, L M ±1 ^ o, and all other off-diagonal matrix elements are null. It is commonly encountered in biological systems, for example, a sequence of biochemical reactions. The associated graph for a closed catenary system appears as follows:

34

LECTURE 4

For it, (10) and (14) take the form

with the convention Loi = L a+M = Mi0 = M j>n+1 = 0. Thus, we have here (2n — 1) equations relating the (2n — 2) off-diagonal elements of L with those of M. One of the equations is redundant. The consequence of these equations together with (9) is that Furthermore, in view of (8),

Suppose now that compartment 1 is injected at time t = 0 with a known amount of labeled material pj(0) and that this same compartment is observed for all subsequent times. From what was said previously in the discussion following (7), we can write (2n — 1) equations for the (2n — 1) unknown elements of M and determine M completely. However, because of the algebraic nature of the equations, there is an (n — l)-fold multiplicity of solutions. Thus, up to this multiplicity, L can be uniquely determined from (2). Furthermore, since YI = Pi(0)/X!(0) is known from the initial condition, (17) uniquely determines V. Because of the redundancy in equation (15), the above conclusions remain valid if one of the closure constraints is relaxed, that is, one L0j is permitted to remain nonzero. We call such a system an almost closed compartment system. Equations (16) and (17) are also true for compartment systems which are closed mamillary, or almost closed mamillary compartment systems. A mamillary compartment system is one which is "star-shaped", having one central compartment (call it 1) connected to (n - 1) other compartments which are otherwise disconnected. For it, LI,, Ltl, Lu ^ 0 for all i, while all other Ltj = 0. As a concrete example of the ubiquity of compartment analysis in biological investigations, I shall mention a recent example I encountered. Using tritium (3H) labeled thymine, the flux of thymine into E. coli DNA was studied [7] in an effort to determine the precursors of the DNA replication scheme. The essence of the investigation is that radioactive thymine is, inter alia, taken up by a number of molecules called TMP, TOP, TTP, as well as ultimately by DNA. The question to be answered is whether any of these molecules is a direct precursor of DNA. The radioactivity of the molecules as well as the DNA harvested from many cells was studied as a function of time following exposure of E. coli cells to a medium containing tritiated thymine. The entire system was presumably in a steady state with regard to thymine transport: a steady state flux of thymine is entering and leaving a precursor compartment p, and a steady flux of thymine is entering the DNA compartment. At t = 0, a steady flux of labeled thymine commences to enter the precursor compartment. Let p and n denote labeled thymine

TRACER ANALYSIS OF PHYSIOLOGICAL SYSTEMS, II

35

FIG. 1. The theoretical expression given by equation (21) for the amount of *H-labeled thymine in DNA is shown plotted as a function of the time, with parameter values chosen to fit the experimental data [7] (open circles). From Rubinow and Yen [8].

in the precursor and DNA compartments, respectively. Figure 1 shows the experimental results of tritium radioactivity in DNA measured in counts per minute (cpm) as a function of time, and Fig. 2 shows similar results for TMP, TDP and TTP. The solid curves are theoretically fitted curves based on the following model [8]:

where a, ft and j are assumed to be constants. (The astute reader will note that these equations are more general than equation (1) of Lecture 3 in that they contain a source term j.) The solution to the model equations is

and the solid curves shown in the figures utilize these expressions with a choice

36

LECTURE 4

FIG. 2. The theoretical curves for the amount of 3H-labeled thymine in a precursor, given by equation (20), versus the time. The parametric values of p^ and a have been chosen in each case to best fit the data points shown. From Rubinow and Yen [8].

of constants px, c and a for each molecule, based on the method of least squares. It is clear that the experimental results are fitted very well by these expressions. Moreover, according to the model, the precursor, whatever it is, must have a time constant« associated with it which is the same as the time constant a. appearing in the curve for «. When the a values for the precursor candidates TMP, TOP, TTP were compared to the a value for DNA, it was found that none of them was the same, although the value for TDP was closest to that for DNA. Thus we concluded that the experiments are consistent with the idea that there is a single but unidentified thymine pool which is a direct precursor of DNA. Allowing for experimental error, the direct precursor among the trio of precursor candidates observed is most likely TDP. These conclusions are not so very different from that reached by the experimental investigator [7]. The point is that compartment analysis permits the conclusions to be expressed in an unambiguous and more precise quantitative manner.

TRACER ANALYSIS OF PHYSIOLOGICAL SYSTEMS, II

37

REFERENCES [1] [2] [3] [4] [5]

R. E. SMITH AND M. F. MORALES, Bull. Math. Biophys., 6 (1944), p. 133. W. PERL, Int. J. Appl. Rad. and Isot., 8 (1960), p. 211. H. D. VAN LIEW, J. Theor. Biol., 16 (1967), p. 43. M. HERMAN AND R. SCHOENFELD, J. Appl. Phys., 27 (1956), p. 1361. M. HERMAN, in Proc. Fourth Berkeley Symp. Math. Statistics and Probability, Univ. of California Press, Berkeley, 1960, p. 87. [6] S.I. RUBINOW, unpublished. [7] R. WERNER, Nature New Biology, 233 (1971), p. 99. [8] S. 1. RUBINOW AND ANDREW YEN, Ibid., 239 (1972), p. 73.

LECTURE 5 Enzyme Kinetics, I Virtually all chemical reactions in the cell involve the direct participation of enzymes, which are proteins that act as catalysts. These biological catalysts differ from all other catalysts known to chemists in two essential ways: 1. They are exceptionally efficient under the mild conditions of the normal physiological state: aqueous medium, standard pressure, and low temperature. 2. They exhibit great specificity, acting rather selectively on definite compounds called substrates. Enzymes also act as regulators of biological processes. The principal way to understand enzymes and characterize their properties is to study enzyme kinetics. By this is meant the "motion" or temporal dependence of enzymes [1]. Here we wish to present the theory of the simplest and most basic enzymatic reaction: that of a single substrate interacting with an enzyme. It is called Michaelis-Menten theory. The fundamental assumption of the theory [2] is that the enzyme and the substrate react to form a complex initially. The complex subsequently breaks down to form the free enzyme plus one or more products. The reactions are represented schematically as follows:

Here E, S, C and P stand for enzyme, substrate, complex and product, respectively. The back reaction C . The tree value is the product of the branch values of a basic tree. A basic tree associated with a given base is the set of branches which touches all the nodes of a graph, does not form a loop, and is directed toward the basic node. (If n is the order of a graph, a tree has (n — 1) branches.) A particular node selected for consideration is called the basic node or base. As an example, we shall consider the problem posed by (1), or, equivalently, (2)-(4), previously considered. The associated graph (with c0 replacing e) is shown below.

Choosing in trun c0,c1 and c2 as bases,and using the following diagram,

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LECTURE 6

we readily calculate the basic determinants to be

By utilizing (13) and the definition of the reaction velocity, equation (6), the final result given by (7) readily follows. This example is perhaps too simple to indicate the full power of graph theory in solving such problems, but if we recall the labor required to solve n linear simultaneous equations by Kramer's rule when n ^ 4, the utility of graph theory may perhaps be appreciated. Even so, there still remains the problem of categorizing various possible reaction schemes so as to determine differences in predicted behavior of the reaction velocity with respect to a given parameter. In this connection, see especially the work of Alberty [6] and Cleland [7]. REFERENCES [1] [2] [3] [4]

M. V. VOLKENSHTEIN, Enzyme Physics, Plenum Press, 1969, Chap. 11. J. MONOD, J. WYMAN AND J. P. CHANOEUX, J. Mol. Biol., 12 (1965), p. 88. D. E. KOSHLAND, JR., G. NEMETHY AND D. FILMER, Biochem., 5 (1966), p. 365. M. V. VOLKENSTEIN AND B. N. GOLDSTEIN, Biochim. Biophys. Acta, 115 (1966), pp. 471, 478. See also [1], Appendix I. [5] S. J. MASON, Proc. I. R. E., 41 (1953), p. 1144; 44 (1956), p. 920. See also S. J. Mason and H. J. Zimmerman, Electronic Circuit Theory, Interscience, New York, 1960. [6] R. A. ALBERTY, Advan. Enzymol., 17 (1956), p. 1; Brookhaven Symp. Biol., 15 (1962), p. 18. [7] W. W. CLELAND, Biochim. Biophys. Acta, 67 (1963), pp. 104, 173, 188; Ann. Rev. Biochem., 36 (1967), p. 77.

LECTURE 7

Cell Populations, I The growth of a cell population generally follows the familiar logistic law [1], [2] described by the differential equation

Here N is the total population at time t, and a and ft are constants. The first term on the right, by the law of mass action, describes the fact that the net birth rate is proportional to the total number of cells present. The second term describes the limitations on cell growth imposed by "crowding". The crowding effect may be expected to be proportional to N2 because the number of cell-cell interactions is approximately A'2. The biochemical cause of this crowding is however not explained by (1). It may be due to lack of nutrients, lack of oxygen, changes in pH, a chemical "inhibitor" given off by the cells, etc. The exact solution of (1) is readily found to be

where N0 is the initial value of N. The solution has the form shown below, if JV0 < OL/P.

An alternate description of cell populations which contains more details of structure is offered by the age-time representation. Consider a population of living organisms which we shall assume to be cells, although it is equally possible to 53

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LECTURE 7

describe human populations, say, with it. Each member has a characteristic chronological age a, the time elapsed since its birth. The population of cells of any age is assumed to be large so that it can be represented as varying in a continuous manner. The continuous functional description of the population is provided by the cell density function n(a, t), where n(a, t) da is the number of cells at time t in the age interval a to a + da. Then the total cell population is obviously

Even though the age of a cell is not in general an observable quantity, many cell properties which are observable are age dependent: mitosis and cell division, for example. The cell density function is assumed to satisfy an equation first proposed by von Foerster [3],

Actually, the equation was proposed earlier by Scherbaum and Rasch [4] (with /I = 0), and the basic idea of n may be found in earlier work of demographers. Usually, / is a function of age only, and it may be divided into two parts, where xm represents the fractional probability per unit time for cell division to take place (the subscript m stands for mitosis, which culminates in division), and /.d represents the fractional probability per unit time for cell death or disappearance. The initial condition is where /(a) is a given age distribution function, and the boundary condition at a = 0 is

The factor two appears because of the assumption that cell division results in two new cells being formed. An alternative suggestion for describing cell populations is the following equation for the density function n,

Here fj, represents "physiological age", and v0 and D are given constants [5]. This equation states that cells on the average have a maturation rate f 0 . but that

CELL POPULATIONS, I

55

dispersion is present too, as shown by the diffusion term on the right. The latter has the conceptual disadvantage that a certain percentage of cells necessarily grows younger. A maturity-time representation has also been proposed [6] for the cell density function n = n(fi, t):

Here v is the velocity of maturation, a prescribed function, and X represents cell disappearance due to death or other means, but excluding cell division. The birth process is represented by the boundary condition There is also an initial condition, similar to (7), with gdu) a given maturation distribution. What do we mean by cell maturity? It is not necessary to say, but two possibilities are : (i) cell volume, and (ii) amount of DNA in a cell. In simple cells such as E. coli, DNA synthesis proceeds from the moment of birth at a linear rate, which makes the amount of DNA contained in a cell a good measure of its state of maturity. For the remainder of the lecture, I shall contrast the different predictions about the temporal evolution of cell populations made by the age-time and maturitytime formalisms. An example of this contrast is afforded by studying the longtime asymptotic behavior. In the age-time theory, assume for simplicity that Ad = 0, and let us look for a solution of the form where y is a constant. Nooney [7] showed that this asymptotic form is achieved if 1 is continuous. By substituting (13) into (4) and solving formally for the function «(a), we obtain the expression

where n n is a constant. Applying the boundary condition to (14) leads to the result

which may be viewed as a condition on the growth rate y. It is easy to show that the expression within the curly brackets represents the fraction of cells undergoing division per unit time. Thus, there is a characteristic steady state age distribution for a cell population in steady exponential growth. Similarly, in t' maturity-time formalism, seek a solution of the form

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LECTURE 7

Substituting (16) into (10), there results

From the boundary condition (11), the following condition on y is obtained,

For example, if v = /j/i, and if n1 = 2fi0 as is the case when n represents cell volume, then (17) reduces to

or, the steady state volume distribution for a cell population in steady exponential growth follows a characteristic inverse square law [8], [9]. A solution to the age-time equation will be presented here in which the biological identification of the population by generations is made [6]. First, we observe that a formal solution to (4) without regard to either the initial or the boundary conditions is given by

where a and ft are arbitrary functions [10]. We now seek a solution of the form

where j represents the y'th generation of cells. Thus, n^a, t) is chosen to satisfy the initial condition (7), and in disregard of the boundary condition. To accomplish this we set a = /and ft = a — t in (20). Then

Clearly, for a time t greater than a given age a, there are no longer any cells of age a because the youngest cells which were of age 0 at t = 0 are already older than a. The first generation gives birth to the second generation n 2 , and according to (8),

To obtain n2(a, t), observe that

CELL POPULATIONS, I

57

Substituting (23) into (24) yields the result

The density function nj(a, t) for the succeeding generations j = 3,4, • • • , may be obtained by iteration of the above procedure. The method of solution may also be applied to the maturity-time formalism. A fundamental problem in the kinetics of cell populations is the cause of the variability of cell generation times. The generation time of a cell is merely the time interval between birth and subsequent cell division. When a cell type is cultivated under the most ideal conditions of identical environment for each member of the population, it is found that the generation times of the cells are not the same. This variability appears to be an intrinsic component of cells viewed as an ensemble. Some experiments of Prescott [11] provide a good illustration of this and are shown in Fig. 1. The number of cells of Tetrahymena geleii with a given

FIG. 1. The data of Prescott [11] for the flux of cells of the HS strain of Tetrahymena geleii that divide, as a function of time. The cells were grown under uniform conditions, and all cells are of age zero at time zero. The solid line is an arbitrary functional form, a gamma distribution, with parameter v determined by the least square criterion. The distribution was otherwise constrained by the equations shown in the inset. From Rubinow [6].

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LECTURE 7

generation time are shown, with a smooth curve in the form of a gamma distribution fitted to the data points. The total number of cells observed by Prescott was 766. In a related experiment, Prescott very carefully followed in time the growth in population of 50 cells of the same strain, which were initially synchronized at t = 0 to be newly born. His results for N(t) provide a stringent test of any mathematical theory of the birth process. For example, in the age-time formalism, it is assumed that a probabilistic rule governs cell division for every cell and there is no memory from one generation to the next of the parental generation time. An alternative hypothesis which fits in naturally with the maturity-time formalism is that there is a heterogeneous distribution of velocities of maturation, with some memory from one generation to the next of generation time. The extreme form of this point of view is the assumption that each generation imparts to its daughters the same generation time, that is, perfect memory. I carried out the mathematical calculation of the population growth for Prescott's second experiment, based on each of these two opposing viewpoints. Thus, in the age-time formalism, set where N0 = 50 and d(a) is the Dirac delta function. With the fitted generation distribution of Fig. 1, call it u(T), the probability function A(a) could be determined uniquely from the equation

with n j determined from equations (22) and (25) (Xd is assumed to be zero). Then the total solution could be found by the generation expression (21). In fact, only a few generations are needed to compare with the experimental results. In the "perfect memory" model, let v = l/T, a constant. Then T is the generation time. Assume that the initial population consists of a distribution of subpopulations, each of generation T. Then

and nT(n, t) is assumed to satisfy (10) with k = 0. The initial condition and boundary condition for nr(/i, t) are, respectively,

It should be obvious that a different mathematical prediction for the total population is obtained in this manner. We could also have replaced the maturity level H by the age a above, and considered the model as a special example of the agetime formalism. Conceptually, I prefer to think of it as a maturation-time model. In this model we see that the generation time is preserved from mother to daughter.

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59

Figures 2 and 3 show the experimental results of Prescott superimposed on the predictions of the age-time formalism or probabilistic model, and on the maturity-time fownalism or perfect memory model, respectively. I was greatly surprised to find that the latter model fitted the experimental points much better

FIG. 2. The circles represent the data of Prescott [11] for the population growth of an initially synchronized group of 50 cells of Tetrahymena geleii HS. A state of asynchronous growth appears rapidly because of generation time variability among the constituents of the population. The solid curve is the theoretically predicted growth curve for such a population in the age-time representation. From Rubinow [6].

FIG. 3. The solid line shows the total population as a junction of time that is theoretically predicted by the maturity-time representation. The data of Prescott are repeated from Fig. 2. From Rubinow [6].

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LECTURE 7

than the probabilistic model. We can conclude from this that, at least over the order of several generations, such memory does exist, on average. However, such a model cannot be correct for large times because fast growing cells would ultimately swamp out the more slowly growing cells, and no generation distribution would be observed in the "wild" state. Therefore, the question as to how the distribution function u(T) arises in the first instance remains unresolved. I would like to mention some more recent thoughts about the resolution of this problem [12]. Assume, as before, that the total population consists of different sub-populations, each with its own characteristic generation time T. Here, we shall utilize an extension of the age-time formalism and introduce, as in (27), the density function associated with the sub-population having generation time T, n = n(a, £; T) . Then

The initial condition is a given function of a and T. The boundary condition is

where K(t, T') is the probability of cells of generation time T' giving birth to cells having a generation time in the interval T to T + dx. Clearly, the daughters of cells of generation time i' that divide end up with some generation time, so that

We have formally solved these equations for both long-time and short-time behavior. We have found that if a function of T alone, the model reduces to the age-time or probabilistic model previously considered. At the other extreme, if we set then we reduce to the maturity-time or perfect memory model. It appears that the kernel function K(i, T') must be determined experimentally. Only one theoretical constraint exists, which arises from consideration of the long-time behavior. Thus, when the population is growing experimentally, It then follows from (32) that

CELL POPULATIONS, I

61

Thus, the generation time distribution function u(i) may be viewed as an eigenfunction of the operator associated with the kernel K. REFERENCES [1] P. F. VERHULST, Notice sue la hi que la population suit dans son accroissemenl, Corr. Math, et phys. publ. par A. Quetelet, 10 (1839), p. 113. [2] R. PEARL AND L. J. REED, Proc. Nat. Acad. Sci., 6 (1920), p. 275. [3] H. VON FOERSTER, in The Kinetics of Cellular Proliferation, F. Stohlman, Jr., ed., Grune and Stratton, New York, 1959, p. 382. [4] O. SCHERBAUM AND G. RASCH, Acta Pathol. Microbiol. Scand., 41 (1957), p. 161. [5] R. N. STUART AND T. C. MERKLE, Calculation of Treatment Schedules for Cancer Chemotherapy, Part II, Univ. of California E. O. Lawrence Radiation Laboratory Rep. UCRL-14505, Univ. of California, Berkeley, Calif., 1965. [6] S. I. RUBINOW, Biophys. J., 8 (1968), p. 1055. [7] G. C. NOONEY, Ibid., 7 (1967), p. 69. [8] A. L. KOCH AND M. SCHAECHTER, J. Gen. Microbiol., 29 (1962), p. 435. [9] G. I. BELL AND E. ANDERSON, Biophys. J., 7 (1967), p. 329. [10] E. TRUCCO, Bull. Math. Biophys., 27 (1965), pp. 285, 449. [11] D. M. PRESCOTT, Exptl. Cell Res., 16 (1959), p. 279. [12] JOEL L. LEBOWITZ AND S. I. RUBINOW, unpublished.

LECTURE 8

Cell Populations, II The age-time and maturity-time formalisms can be adapted so as to provide a quantitative analysis of radioautographic studies of cell populations. To understand such studies, we must first define the phases of the cell cycle which is the interval between cell birth and cell division. It is known that cells undergo a phase called mitosis, or M-phase, which just precedes cell division. There is also a phase of the cell cycle during which its DNA content is duplicated. This phase is called the DNA synthesis phase or simply S-phase. The remaining phases of the cell cycle are naturally defined by the above two phases and called the G^-phase and G2-phase, as shown in the diagram.

In the procedure called radioautography, a radioactively labeled material such as tritiated thymidine is introduced to the medium surrounding some cells. Thymidine is a DNA precursor, so that it enters the cells, and is then incorporated into the DNA of those cells in S-phase only. If not incorporated, thymidine is degraded rapidly (within one half-hour) and lost. Subsequently, cells are removed, dried, and pressed against special photographic plates for a long period ranging up to two weeks. Some of the tritium in the incorporated thymidine decays by j8-decay, and each ft particle entering the photographic emulsion leaves as a trace a little black spot or grain. Such grains can be counted. The method is useful for studying the kinetics of normal and leukemic cell populations. With it, one can measure the following quantities: 62

CELL POPULATIONS, II

63

1. Initial grain count distribution: the number of cells in a given sample having a given grain count immediately following the labeling procedure. 2. Label index: the fraction of cells labeled at a given time, L(t). From the figure, it is seen that the initial label index is L(0) = J rs n(a, Q)da/$r n(a, 0)da. 3. Labeled mitotic index: the fraction of cells in mitosis which is labeled, L M (f). For example, suppose the population is in a steady state so that n(a, t) is constant in the above diagram, and all cells have the same generation time T. Then if at t = 0 all cells in S-phase are labeled, the theoretical form of L M (r) is shown below:

From such an experimentally determined curve, the phase durations of the cell cycle can be inferred. What is the descriptive temporal nature of leukemia? We shall attempt to answer this question for one particular type: acute myeloblastic leukemia, which occurs in children and mature adults, and is generally fatal. We first describe briefly the natural history of the most common type of leukocyte called the neutrophil or granulocyte. Such cells are produced in the bone marrow. The earliest recognizable form is called the myeloblast. An earlier form called the stem cell that contributes to myeloblasts as well as other blood cell lines has been postulated but not positively identified. Myeloblasts are proliferativc: they divide and produce more myeloblasts as well as promyelocytes, which in turn divide and produce myelocytes. The myelocytes are likewise proliferative. Some time after becoming myelocytes, cells undergo a period of maturation lasting 4-10 days. All granulocytes beyond the myelocyte stage are nonproliferative. Finally, they are expelled from the marrow by some unknown mechanism into the blood as young mature neutrophils. About one half of the mature neutrophils remains in the blood while the other half enters a "marginal blood pool" whose precise identification in the body.is not known. There appears to be a steady exchange of neutrophils between the blood and the marginal pool. From the blood, granulocytes exhibit random disappearance with a half-life TV2 ~ 7 hrs. This natural life history of the neutrophil is schematized by the diagram at the top of p. 64. Acute myeloblastic leukemia has the following characteristics by contrast: 1. Overproduction of myeloblasts. Leukemic myeloblasts are cytologically indistinguishable from normal myeloblasts.

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2. Appearance of myeloblasts in the blood. Normally this never occurs. Myeloblasts in the blood are nonfunctional as mature neutrophils, but are otherwise harmless. 3. Concomitant decrease in mature neutrophil number in the blood. It is this last characteristic that makes the disease fatal. Leukemic patients frequently succumb to pneumonia. Clearly, in leukemia, the control mechanisms that normally regulate leukocyte production have gone awry. An extensive investigation at Sloan-Kettering Institute and Memorial Hospital of two volunteer leukemic patients was made by Clarkson et al. [1]. The patients had to submit to repeated bone marrow extracts, which are necessary in order to make observations of the proliferating blood cells. In order to better understand the observations, Joel Lebowitz and I developed a theory of labeled cell populations [2] within the context of the following somewhat simplified model. Cells are assumed to exist in two states, an active state and a resting or "G0" state as shown in Fig. 1. In the active state, cells mature and divide. In the resting state, cells merely age. When cells divide, a certain

FIG. 1. Model of leukemic myeloblast proliferation, from Lebowitz and Rubinow [2]. The model originally permitted cells from the active state to disappear at a fractional rate per unit time /J,. The discussion in the text assumes /J, = 0.

CELL POPULATIONS, II

65

fraction 5 of the daughter cells enter the resting state, while the remainder enter the beginning of the active state. The fractional rate at which cells leave the resting state to return to the active state is a, and the fractional rate at which cells leave to enter the blood is /? 0 . These latter cells eventually die. We assume a steady state, which requires a = fi0. The theory introduced is as follows. The states are represented by cell density functions, H(^, t) and Q(a, t), where

The boundary conditions are

The population which is labeled is represented by n'(x, ^, r) and Q'(x, a, r), where x is the amount of radioactive label, treated as a continuous variable. A group of cells undergoing mitosis and having an amount of label between x and x + dx gives rise to daughters with label between x/2 and (x + dx)/2. This is because replication is semi-conservative: 1/2 the DNA content of each parent cell goes to each daughter cell. Thus, n' and Q' satisfy the same differential equations, (1) and (2), respectively, as the total population densities, but the boundary conditions are different,

These equations may be solved by the generation scheme described in the previous lecture. Assuming the total population is initially in a steady state, and if, at t = 0, a pulse of tritiated-thymidine is injected into the blood, then all the cells in S-phase of the active state are labeled with a given distribution of label (x), and the actual observations of the fractional number of cells with a given grain count x. The two curves denote the results for the two patients, R.R. and M.T. Figure 3 shows the fractional number of labeled cells in mitosis as a function of time, L M (f; 5). The argument 5 denotes the threshold for counting a cell as labeled: only cells with a grain count i> 5 are called labeled. The experiments indicate (see especially those points with a threshold of 10 grains) that there is no second wave

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FIG. 2. The solid line represents the assumed initial form of the normalized distribution function (j>(x). It is obtained by fitting an assumed functional form to the observed distribution of fractional cell number vs. grain count x which is displayed as a histogram [1], for each of two patients, (a) R.R., and (b) M.T. The dashed curve 2<j>(2x) in (a) represents the labeled distribution function expected in the second generation of cells. The dashed vertical line at x = 5 indicates that only cells with a grain count S 5 were considered to be labeled. From Rubinow, Lebowitz and Sapse [3].

CELL POPULATIONS, II

67

FIG. 3. The fractional number of labeled cells is shown as a /unction of the time t. The unit of time is 20 hr. The theoretical curve for o = 0.75 displays a second wave which is not a property of the observations [1], especially if 10 grains is taken as the threshold of observation for denoting a cell as labeled. From Rubinow, Lebowitz and Sapse [3].

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LECTURE 8

oflabeled cells in mitosis, as there would be for a proliferating population in which the daughter cells simply returned to the beginning of the active state. To agree with observations, it is necessary to assume that all newly born cells enter the resting state (5 = 1 in Fig. 1). Figure 4 shows the labeling index as a function of time. A characteristic feature of the leukemic state is the very small initial value

FIG. 4. The labeling index L(t; 5) is shown as a function of the time t. Four theoretical curves are shown based on different assumed values of the mean generation time of the active state TA. In (a) the value of L(0;5) was adjusted so as to make the theoretical maximum agree more closely with experiment, shown as solid circles [1], From Ruhinow, Lebowitz and Sapse [3].

of the labeling index, which is about 7.5 % in the cited data. An "average" value for normal man is about 30%. This difference is surprising, because the initial labeling index is a measure of those cells destined for cell division. Such observations contradict the intuitive notion of cancer being a state of rapid uncontrolled growth.

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It can be seen from the figure that a tolerable fit of theory with experiment can be made, and the parameters of the model can be determined [3]. For example, the low initial labeling index is achieved by assigning about 90 % of all cells to the resting state. We have also been led to formulate the following model of leukemic and normal granulocytic growth [4]. In it, we have tried to incorporate not only the previous results, but other known general features of the normal and leukemic states. For example, normal cell growth is subject to some unknown feedback control system which tries to maintain the number of neutrophils in the blood at the necessary level. The generic term for such biological control is homeostasis. The normal population is represented by the following model:

Here, A, G 0 , M and B represent the active state, resting state, maturation state, and blood plus marginal pool, respectively. The proliferative part of the model, A + G 0 , represents all proliferative granulocytic compartments: myeloblast, promyelocyte and myelocyte. We are forced to lump together proliferative compartments in this manner because not enough is known about the kinetic details of the individual cell types. Each pool or compartment is represented by a cell density function, thus.

In this model, feedback control is exercised by making a and /? depend on N, the total population of all pools. There is some support for such a mechanism in the recent discovery of chalone as the hormone regulating white cell production. Such a mechanism is also in accord with the mechanism by which erythropoietin

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regulates red cell production. The schematic dependence of a and /? is shown in the next diagram.

It should be intuitively clear that the system attempts to maintain a homeostatic level of N cells by overproducing if, as a result of some trauma, N drops below N, and conversely by underproducing if N becomes greater than N. The leukemic cell population has apparently in some manner lost the capacity to undergo the normal maturation process. Our fundamental point of view is that the leukemic state is not "uncontrolled growth", but rather, controlled growth

FIG. 5. A "normalized" Gompertz plot in which the growth data for 19 examples of 12 different animal tumors have been superimposed by adjusting the units of the axes for each example. The ordinate is tumor size and the abscissa is time. From the work of A. K. Laird [5].

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71

with an abnormal set of controls. The evidence indicates that all observed animal tumors obey the Gompertz growth law [5], as shown in Fig. 5. According to this law, which is completely empirical, population growth obeys the equation

rather than the logistic law. It has also been observed that the mean generation time of leukemic cells increases in later stages of growth. Most leukemic cells end up in a "resting" state in late stages of the disease, as we have previously indicated. Our model of leukemic cell growth is then similar to the normal state, with the maturation compartment missing, and a different set of parametric functions a', /?', A' (see diagram).

The functions a' and /?' are similar in form to a and /?, respectively. However, the equilibrium value N' is assumed to be substantially greater than N, for example, N' ~ 3N. We further assume that when the two populations coexist, the feedback control depends on the total population N + N' of normal cells N and leukemic cells N'. The dependence of the control functions a, /?, a' and /?' on (N + N') is shown schematically below.

It may be shown that the total system, given that some leukemic cells are present, approaches the stable value (N + N') = N'. If no leukemic cells are present, the stable value is of course N. Speaking anthropomorphically, the normal cells are fooled by the presence of the leukemic cells and recognize them as normal cells.

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When N + N' is > N, normal cells believe there are too many cells present, which leads them to turn off production and gradually eliminate themselves. We are currently solving numerically the equations representing the system to see whether we can simulate the natural history of leukemia. Some chemotherapeutic agents are known to be cell-cycle specific, that is, they kill cells which are in a certain phase of the growth cycle only, in S-phase, for example. Unfortunately no drug (with the exception of L-asparaginase) in its lethal effect distinguishes between cancerous and normal tissue. It is the lethal effect on normal cells that is called drug toxicity. Therefore, we ultimately want to impose on our model a drug-treatment regimen which will optimize treatment, that is to say, maximize the killing of leukemic cells and minimize the killing of normal cells. Hopefully, this can be accomplished by taking advantage of the different kinetic characteristics of normal and leukemic cells. It should be observed that treatment success does not require the complete elimination of the leukemic population. Merely keeping it below a certain level is good enough. The model presented is, I believe, a good example of the difficulties a theoretician faces in trying to make a useful contribution to a medical area in which the underlying biological mechanisms are poorly understood. REFERENCES REFERENCES [1] [2] [3] [4] [5]

B. CLARKSON, T. OHKITA, K. OTA AND J. FRIED, J. Clin. Invest., 46 (1967), p. 506. J. L. LEBOWITZ AND S. I. RUBINOW, J. Theoret. Biol., 23 (1969), p. 99. S. I. RUBINOW, J. L. LEBOWITZ AND ANNE-MARIE SAPSE, Biophys. J., 11 (1971), p. 175. S. I. RUBINOW AND J. L. LEBOWITZ, work in progress. A. K. LAIRD, Brit. J. Cancer, 19 (1965), p. 278.

LECTURE 9

Diffusion in Biology, I As it is natural to expect, a significant mechanism of intracellular and extracellular transport is diffusion. Here I shall present some simple examples of problems in biology which are related to diffusion. 1. OLFACTORY COMMUNICATION IN ANIMALS. One method of communication among animals is accomplished chemically by the release of an "odor". The chemical released is called a pheromone. For example, the harvester ant Pogonomyrmex badius gives off a very volatile alarm substance under suitable provocation [1]. The substance originates in the mandibular glands of the ant. A sister ant that detects the alarm substance displays mild excitement and is attracted toward the source. At a greater concentration of the alarm, occurring when the ant is closer to the source or at a later time at the original position of detection, the ant abruptly begins to run in circles. In this second state the ant sometimes appears to emit the alarm substance itself. Such is the presumed manner of alarm communication. In order to make quantitative measurements of this system, experiments with harvester ants were performed [2] in which the puff of the alarm substance was simulated by crushing the head of a worker ant at one end inside a long closed tube, and removing it a few seconds later. To detect the alarm, ants were placed at various positions in the tube. To prevent stimulation to highest excitement, a screen barred the ants from approaching too closely to the source. The location and time of detection of the ants were observed. The theory of this experiment requires a simple and straightforward application of the mathematics of a diffusing source [2]. Thus the pheromone chemical obeys the diffusion equation

where c is the pheromone concentration, A is the Laplacian, and D is the pheromone diffusion constant in air. A source consisting of N molecules located at the origin at r = 0 is represented by the solution

Here r is the radial distance from the origin. Since pheromones are released close to the ground, the space under consideration is the infinite medium z > 0. The factor 2 appears because of reflection at the surface of the earth, z = 0. This 73

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solution is the limit as z0 -» 0 of a source located at height z 0 and an image source located at a depth (— z 0 ), subject to the condition that no particles diffuse into the ground, or

Let the threshold for detection be K, so that the detection condition is

The equality sign determines a sphere of influence of the pheromone. The radius of influence is given by (2) and (4) as

where

The quantity I represents the duration of influence or fadeout-time. The diagram below shows the solution c schematically at three successively later times tl,t2,t3.

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To apply the theory to the experiments in the tube, the solution in one dimension was selected as a reasonably approximate solution,

and A is the cross-sectional area of the tube. From the measurements of x vs. f, tl and D can be determined, and from these N/K. may be inferred. These inferences tell us something about the pheromone, which is difficult to analyze chemically because of its great volatility. The fact that ants run towards the source indicates that they are able to detect the direction as well as the presence of the alarm chemical. The theory has also been applied to chemical trails (source point emitting continuously and moving with constant velocity), and to the presence of a steady wind. 2. SEDIMENTATION IN A CENTRIFUGE. A powerful instrument in the hands of the modern biologist is the ultracentrifuge. This is a device for spinning biochemical solutions at very large angular velocities, thereby subjecting any solute or suspension in the solution to a strong centrifugal force. With it, macromolecular elements of a cell are separated, and molecular weights are routinely determined. The following discussion is taken from van Holde [3]. Consider a solution in a sector-shaped container in which there is a single solute molecule of mass m as shown in Fig. 1. For a single particle, the balance

FIG. 1. Diagrammatic representation of a sector-shaped container spinning about the axis A in an ultracentrifuge. The molecule is acted on by buoyant, centrifugal, and drag forces. From Van Holde [3].

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of forces requires Here a> is the angular velocity of the container, r is the radial position of w, m0 is the mass of an element of solute having a volume equal to that of m, v is the particle velocity in the radial direction, and /is the frictional coefficient. The first term on the right is the centrifugal force on the particle due to rotation (positive outwards); the second term is the buoyant force on the particle (positive inwards); and the last term is the drag force (positive inwards). From (9), v can readily be determined. The generalization of the above considerations to a binary solution containing a solvent and a solute is expressed as follows. The total flux density j of solute particles is assumed to consist of two parts, a contribution due to sedimentation as above, and a contribution due to diffusion. Thus where the first term on the right represents the sedimentation flux density, and the second term represents the diffusional flux density. The expression for v resulting from (9) is generalized for a solute to be where, as for (9),

But now m = M/A, where M is the molecular weight of the solute species and A is Avogadro's number; m0 = (M/\)vp, where v is the partial specific volume of the solute which is assumed constant, and p is the density of the solution (solute plus solvent); and/is the drag coefficient for the solution, which is concentration dependent as a consequence of its dependence on the viscosity of the solution. The quantity s is called the sedimentation coefficient, and in general s = s(c). It is usually determined experimentally by measuring v and to2r. The customary unit of s is a Svedberg = 10"13 sec. By Pick's law,

Substituting (10) into (13) and recognizing that y" has only a radial component, it follows that

This is called the Lamm equation for the ultracentrifuge [4]. Equation (14) is its simplest manifestation. More complications are introduced by the consideration of a multicomponent system, when c is a vector and D is a matrix. Also, many

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protein systems undergo association-dissociation reactions while being centrifuged. The authoritative treatise on the mathematics of centrifugation is the book by Fujita [5]. As a simple but important application of the theory, we shall consider the equilibrium distribution of the concentration when the solution contains a density gradient. One way to produce such a gradient is to fill the container in a continuous fashion by a sucrose solution of gradually increasing density. Such a solution is not stable and ultimately disperses due to diffusion. However, over the time scale of ultracentrifugation experiments of the order of several hours or less, little change in the gradient occurs and it may be assumed to be constant. From Lamm's equation, in a steady state, c = c(r) and is determined by the equation

Because

another result due to Einstein, it follows from (12) and (16) that Now assume that the density gradient is chosen with particular regard for the solute under investigation so that

where r0 is the equilibrium radius value for which p(r0) = l/v, and higher order terms in the density dependence on radius are neglected. From (15), (17) and (18),

whence

Thus, at equilibrium, the solute molecule exhibits a Gaussian concentration distribution about its equilibrium value at r 0 . The breadth of the observed concentration band is proportional to M~ 1 / 2 , so that observations of it may be used to determine M. For example, such a procedure has been used to determine the molecular weight of DNA centrifuged in a cesium chloride density gradient [6]. 3. CAPILLARY-WALL EXCHANGE. When blood flows through a capillary, a particular solute component of the blood with concentration c is being transported across the capillary wall. This transport arises usually as the result of either of two

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driving forces. One driving force is a transmural pressure difference, which may be either hydrostatic or osmotic in origin. The second driving force is a concentration difference of the solute inside and outside the capillary wall. The first mechanism is primarily responsible for maintenance and control of plasma and interstitial fluid volumes. The second mechanism is primarily responsible for tissue metabolism by transcapillary transport of specific solutes. We shall consider here this latter mechanism, a purely passive process. Denote the concentration in the tissue by CT . We neglect diffusion inside the capillary and assume that diffusion in the tissue is so rapid that it behaves as a homogeneous compartment. Diffusion is assumed to occur across the capillary wall only. Consider a cross-section of the capillary of area A. Let the blood be

treated as inviscid, and assume its steady velocity is u. Conservation of flux (see diagram) requires that

Here P is the permeability constant of the solute in the membrane, S is the total surface area of the capillary, L is its length, and S/L is the perimetric length of a cross-section of the capillary, assumed to be constant. In the limit as Ax -> 0, (21) becomes the convective diffusion equation

where c = c(x,t). The rate of change of solute matter in the tissue volume associated with a given capillary is

where VT is the associated tissue volume, CT = c r (t), the last term in (23) is the consumption rate of solute in the tissue, and / 0 is the net mass efflux from the capillary given as

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The boundary condition is where CA is a constant, and the initial conditions are

This system of equations can be solved with the aid of the Laplace transform [7]. However, the initial condition is somewhat arbitrary and the initial transients in any case cannot be observed. Therefore, it suffices to consider the steady state solution c = c(x) [8] when CT = CTO and (22) becomes

where Q = Au is the volume flux of blood through the capillary, per unit time. Then and

Note that PS is a measure of the capacity of a capillary to transport matter across its wall by means of diffusion, while Q is a measure of the capacity of a capillary to transport blood through it. Thus, if Q » PS, which means that the rate-limiting process governing transport is diffusion. In other words, rapid flow maintains c a CA in the capillary, which is the best that can be done to expedite transport. On the other hand, if Q « PS, which means that the rate-limiting process when diffusion is "fast" is blood flow. In general, I0 depends on the rate-limiting process, or "bottle-neck". / 0 can be measured for different solutes as a function of Q and fits (29) tolerably well. The approach to equilibrium can also be considered in a simple manner. Then

and it may be shown [6] that

provided SL « VT, and ft « PS/VT. I have not been able to find any experimental

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evidence of this law, which is a commentary on both the paucity and the difficulty of quantitative measurements in this and so many other areas of biology. 4. STANDING-GRADIENT OSMOTIC FLOW. Most epithelia absorb or secrete specific fluids such as bile, gastric juice, cerebrospinal fluid, and sweat. The primary transported fluid may be either isotonic or hypertonic to plasma. A solution separated from plasma by a membrane is said to be hypertonic (isotonic, hypotonic) with respect to the plasma if the osmotic pressure in the solution is greater than (equal to, less than) the osmotic pressure in the plasma. Physiological experiments indicate that water transport is a passive, secondary consequence of local osmotic gradients set up within the epithelia by active transport of some solute. Examples of the latter process are gall bladder—NACL, stomach—HCL, liver—bile salt. A common feature of epithelial structures is the existence of long, narrow fluidfilled spaces bounded by cell membrane, and closed at one end. Transported fluid leaves the open end. Suppose an active transport mechanism deposits solute within the channel. This deposition makes the channel fluid hypertonic with respect to the surrounding tissue and attracts water osmotically. Solute moves toward the open end by diffusion and by being convected by water leaving the open end. In a steady state, a fluid of fixed osmolarity (concentration) would continually emerge. A steady osmotic gradient would be maintained over the length of the channel, with the osmolarity decreasing towards the open end. These considerations, taken from Diamond and Bossert [9], are the physical basis of the following mathematical model of a "standing-gradient osmotic flow" system, which they proposed. Consider a channel closed at one end, of fixed crosssectional area A, length L, and total surface area S (excluding the closed end), as shown in Fig. 2. Assume that an active transport mechanism exists at the closed end between x = 0 and x = d, and that the rate of transport of solute per unit area is N 0 . Fluid flows at a rate v(x) through the cross-section of the channel

FIG. 2. Model of the standing-gradient flow system, from Segel [10].

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located at x. Denote the solute concentration in the channel by c(x). The concentration of solute in the tissue surrounding the channel is assumed to have a fixed value c 0 . The solute is permitted to diffuse in the longitudinal direction only. Conservation of fluid through a cross-sectional element of the tube of thickness Ax requires that

where P is the water permeability coefficient. The first term on the left represents water influx through the sides of the channel element. The perimetric length of the cross-sectional element is S/L. Conservation of solute mass implies

where

and

In the limit as Ax -> 0, (34) and (35) become

The boundary conditions are that

By integrating (39) and substituting (36), there results

Diamond and Bossert resorted to a numerical solution of this nonlinear equation system. This equation system was studied analytically by Segel [10] who showed that, in nondimensional form, equations (38) and (42) contain only 3 nondimensional parameters, v = A/ 0 /Pco, /I = L/d and v\ = DAc0/SN0d2. An examination of representative physiological values shows that v is typically small, while r] is typically large. Consequently, a small perturbation expansion in v was

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attempted. As in the mathematical problem discussed in Lecture 5 of the substrateinhibitor-enzyme system, it is necessary to prescribe the dependence of r\ relative to v. Thus, when rj = 0(1) was assumed, it was found that no perturbation solution was possible. However, a perturbation solution is possible if t] is proportional to v ~ 1 . From the approximate solution so obtained, the emergent osmolarity, an observable quantity, could be expressed analytically in terms of a single nondimensional parameter ( = l/(riv)1/2). Such mathematical considerations both simplify and deepen the understanding of the underlying physiological processes. Similar considerations may be expected to be useful in problems involving coupled solute and water flows in channels, such as occur, for example, in the kidney. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

E. O. WILSON, Psyche, 65 (1958), p. 41. W. H. BOSSERT AND E. O. WILSON, J. Theor. Biol., 5 (1963), p. 443. K. E. VAN HOLDE, Physical Biochemistry, Prentice-Hall, Englewood Cliffs, N.J., 1971, Chap. 5. O. LAMM, Z. Phys. Chem. (Leipzig), A143 (1929), p. 177; Arkiv Mat. Astron. Fysik, 213 (1929) (2). H. FUJITA, Mathematical Theory of Sedimentation Analysis, Academic Press, New York, 1962. M. MESELSON, F. W. STAHL AND J. VINOGRAD, Proc. Nat. Acad. Sci., 43 (1957), p. 581. J. A. JOHNSON AND T. A. WILSON, Amer. J. Physiol., 210 (1966), p. 1299. E. M. RENKIN, Ibid., 197 (1959), p. 1205. J. M. DIAMOND AND W. H. BOSSERT, J. Gen. Physiol., 50 (1967), p. 2061. L. A. SEGEL, J. Theor. Biol., 29 (1970), p. 233.

LECTURE 10

Diffusion in Biology, II Cellular slime mold amoebae (Acrasiales) are normally found in the soil. Because their life cycle is so short (~ 4 days), and because the conditions for their culture are well understood, they have been extensively studied in the laboratory. This is especially true of the species Dictyostelium discoideum. The following information about them is taken from the work of Bonner [l]-[4]. Let us arbitrarily begin the description of their life cycle with the spore stage. The spore is an encapsulated dormant organism, which, when placed on moist agar, splits and hatches one unicellular amoeba. The amoeba moves about by means of pseudopod formation, feeds on bacteria by engulfing them, and divides repeatedly by binary fission. Division and propagation continue so long as a sufficient food supply is available. When the food supply is exhausted, the amoebae appear to spread themselves uniformly in a monolayer. After an interphase period lasting several hours, they begin to aggregate about a number of centers. They do not individually move radially towards a center, but rather come together to form streams. The aggregate cell mass or slug contains anywhere from 102 to 105 cells and varies in length from 0.1 to 2 mm. This is illustrated in Fig. 1.

FlG. 1. Four stages in the aggregation of Dictyostelium taking place under water on the bottom of a glass dish: A. Beginning of aggregation. B. Formation of streams. C. Coalescence. D. The final cell mass. From the work of Bonner [4]. 83

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The slug then begins a second movement, that of migration, as shown in the middle of Fig. 2. It moves at a speed comparable to the speed of an amoeba: 0.5-2 mm/hr. The amoebae themselves within the slug are in active pseudopodial motion. They exude a sheath and then walk on it. The sheath remains motionless. The length of the migration period is somewhat variable, extending up to two weeks. In the later stages of slug formation, the cells become sticky.

FIG. 2. Developmental stages of the life cycle of Dictyostelium are shown, from aggregation at the left, to slug migration in the center, to spore formation at the tip of a stalk, on the right. From Banner [3].

FIG. 3. Upper photograph shows hundreds of amoebae placed near cyclic AMP contained in small block of agar (bottom). Lower photograph shows amoebae streaming towards the block. From Banner [3].

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The third stage is somewhat more involved. The cell mass rights itself, and shoots up into the air to form a delicate stalk at the top of which there is a round smooth spore mass (see the right side of Fig. 2). The process is the reverse of a fountain: the cells appear to pour up the outside and move down the center, becoming the stalk. Here differentiation has taken place. Some cells are destined to become stalk cells and die. Others are destined to become encapsulated spore cells. The entire system is in many ways one of the simplest models of morphogenesis— the development of structure during early growth—displaying in particular the phenomena of morphogenetic movements, differentiationandchemotaxis.The origin of the aggregating motion is chemical, hence the term chemotaxis. The cells secrete a chemical called acrasin and they move preferentially to regions of high concentration of this chemical. This is illustrated in Fig. 3, which shows amoebae moving towards a highly concentrated source of acrasin. Acrasin has since been identified as cyclic 3', 5'-adenosine monophosphate, or simply cyclic AMP. The attractive activity of cyclic AMP decays rapidly with time after a few minutes. The amoebae are also known to produce an enzyme acrasinase which converts cyclic AMP into the chemotactically inactive form 5'-AMP. Cyclic AMP is not produced in constant amounts but undergoes a hundredfold increase in

FIG. 4. The hundredfold increase in the secretion of cyclic AMP by Dictyostelium is shown as a function of the time, and is correlated with the aggregating motion of the amoebae, shown at the top of the figure. The maximum is attained at the completion of aggregation. From Banner [3].

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amount secreted at the onset of aggregation (see Fig. 4). The sensitivity to acrasin also increases a hundredfold at about the same time. I shall present here the simplest of the models put forth to explain the onset of aggregation, that due to my erstwhile colleagues Evelyn Keller and Lee Segel [5]. The theory attributes aggregation to an instability of the uniform distribution of both amoebae and acrasin, which characterizes the interphase period. The instability is regarded as being brought about as a result of changes in amoeba properties such as acrasin secretion rate. Let a denote the number density of amoebae per unit area in the x, y-plane. In the absence of proliferation, a satisfies

where j is the amoeba flux density vector. The amoeba flux is of two types, a diffusive random motion of cells, and a directed chemotactic motion due to the presence of a surface concentration of acrasin c, for example,

The minus sign is present because diffusive motion is away from regions of high amoeba concentration. The coefficient (i is called the motility. The sign in front of the last term on the right is positive because the chemotactic motion is toward the regions of high concentration of acrasin. The coefficient x is called the chemotactic coefficient. T. The concentration c is assumed to satisfy the equation

On the right, the first term represents the decay in time of acrasin, the second term represents the production of acrasin by amoebae, and the last term represents the ordinary diffusion of acrasin in the medium, with diffusion constant D. The coefficients //, #, k and/are in general expected to depend on c, and there is some evidence to support this expectation. Equations (2) and (3) possess the uniform solution

provided that the constants a0 and c0 satisfy the relation

Let us examine the stability of this uniform solution by making a perturbation analysis of it. Thus, set

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and treat £, rj, and their derivatives as small quantities. Then

where fi0 = n(c0), p'0 = ft'(c0), etc., and /i0, # 0 , k0 and/ 0 are positive constants. From (2), (3) and (6), the linearized equations for £, and r\ are

Equations (8) admit a solution of the form

where This solution represents a wave traveling in the direction K which either grows or decays in time, depending on whether a is positive or negative, respectively. Substituting (10) into (8) and letting K2 = K\ + K\, it follows that

A nontrivial solution requires that the coefficient determinant vanishes, or This is the dispersion equation, which is viewed as a quadratic equation in 0. Stability requires both roots to be negative which will be true if and only if b > 0 and c > 0. After some algebraic rearrangement, it is found that stability requires

The most stringent form of this condition occurs when K = 0 (or the wave length X = 2-K/K is infinite), because then the right-hand side is a minimum. Therefore, instability first occurs when K — 0 and the inequality is reversed, or The factors on the right represent stabilizing forces which cause the amoebae to disperse: by increasing the motility fi0, or increasing the effective decay constant of acrasin (fc0 + k'0c0). The latter factor may also be thought of as representing

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an increase in the amount of the enzyme acrasinase. The factors on the left represent destabilizing forces because they tend to increase the amoeba concentration : an increase of #0 intensifies the chemotactic response, and an increase in /0 represents an increase in acrasin secretion. The fact that x0, /0 and f'0 all seem to increase rapidly at about the time of aggregation lends qualitative support to the theory. Quantitative support must await the discovery of the values of the parameters and how they evolve. It has been observed that a "reticular" aggregation pattern develops in some circumstances [6], which is the inverse of the usual aggregation. Cells appear to move outward and form a mosaic pattern. The theory does not actually distinguish between such a pattern and the previous aggregation pattern because the sign of the perturbation (±£ 0 , ±f/ 0 ) is not known. The distinction between the two possibilities requires a nonlinear stability analysis. In some species, namely Polysphondylium violaceum and Dictyostelium minutum, aggregation appears to be initiated by founder cells which act as attractive centers [7], [8]. Also, certain species display rhythmic pulsations during aggregation, at least some of the time [9]. These and other observations form the basis of an alternate theory of cell aggregation due to Cohen and Robertson [10], which I shall not discuss here. Aside from the observations mentioned above, there are other aspects of the behavior of the amoebae which are not explained by the theory of Keller and Segel. For example, alternating straight bands of amoebae do not occur in aggregation, as assumed in (10). The significance of cell stickiness has been ignored. A threshold gradient of acrasin production perhaps exists. The theory does not predict "territory size". Under certain environmental conditions, it is found that the territory size is independent of cell density. It is natural to associate territory size with a critical wave length. So far, the critical wave length is infinite. Keller and Segel found it possible to identify a territory size by including in the theory a threshold value for the acrasin gradient necessary to initiate a chemotactic response. They have also included a theory of acrasinase production. I think it is fair to state, however, regardless of which of the competing theories of slime mold aggregation is correct, that the theory of Keller and Segel is important in the more general context of providing a specific physical method of morphogenesis. Such models were first proposed in the pioneering albeit speculative work of Turing [11]. I would like to present a second novel application of the foregoing ideas, this time to the observation of traveling wave bands of bacteria. Certain strains of E. co/i are motile by virtue of possession of flagellae. When cells of such a strain are placed at one end of a sealed long glass tube containing an energy source such as glucose, a number of them in the form of a band are seen to move steadily down the tube. After a while a second band emanates from the original source and also travels down the tube. The bacteria remaining at the end of the tube are viable but no longer motile. If they are placed in fresh tubes, they are capable of forming new bands [12].

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Chemical analysis shows that bacteria consume all the oxygen in their vicinity plus some glucose. The first band of bacteria forms to avoid the low oxygen concentration, and consumes most of the remaining oxygen in the tube. The second band appears if the remaining energy source can be utilized anaerobically by the cells. Thus, the experiments can be understood qualitatively in terms of a chemotactic response of the cells of the first band to oxygen, and of cells in the second band to the energy source. The quantitative phenomenological theory of Keller and Segel [13] follows. Let b represent the bacterial cell population density, and c be the concentration of a critical substrate chemical. It is assumed that cells can diffuse, or respond chemotactically to the presence of the chemical. In one spatial dimension'x, the density function b is assumed to satisfy the equation

Here /i is assumed to be constant, but % is assumed to be of the form

Thus x is singular. This is a necessary mathematical assumption, and in fact, (17) is the least singular form that will permit a traveling wave band solution, which is the type sought. Biologically, (17) receives some support from other areas as a kind of "Weber-Fechner law", according to which the response of a sensory mechanism is proportional to the relative stimulus which in this case is Vc/c. Assuming diffusion of the substrate as negligible, the substrate concentration c satisfies the equation

where k is a constant. A traveling wave solution of equations (17) and (18) is sought of the form:

where v is the wave velocity, a constant. The boundary conditions are that Here x -> oc implies c -» oo. Substituting (19) into (17) and (18) leads to a nonlinear system of ordinary differential equations, which are easily solved. The solution is:

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The quantities c and b remain finite if and only if $ > 1, or This relation implies that the disordering effect of random motion (/i) must be overcome by the organizing effect of chemotaxis (