Chapter 20 Adaptation and selection. In Genetics, Paleontology, and Evolution

· 20 · ADAPTATION AND SELECTION BY

SEWALL

WRIGHT1

THE process of adaptive evolution is one of central interest from both the philosophical and scientific points of view. As with other natural phenomena there is probably both an internal and an external aspect. To avoid possible duplication of the same factor under different aspects, an attempt at scientific analysis must be restricted to the latter. In the present discussion, no consideration will be given to hypotheses which attribute evolution to inscrutable creative forces whether these be supposed to bring about gradual orthogenetic advance (Nageli, Osborn), abrupt emergence (Lloyd Morgan) or are assigned to individual organisms (Lamarck). The discussion will also be restricted to factors for which there is a substantial basis in genetics. This consideration excludes the hypothesis that evolution is an extension of individual physiology (Lamarck, Eimer, Cope, etc.). Mutations of chromosomes and genes, and recombinations of these, are with minor qualifications the only known sources of hereditary change. The available evidence indicates that these occur independently of physiclogical adaptations of the individual. Weare left with the hypothesis that phylogenetic adaptation is ultimately preadaptive. The variations come first, the organisms do the best they can with them, and natural selection is the arbiter (Darwin). Under any hypothesis, the results of natural selection may be considered as the measure of adaptation, and some degree of guidance of the course of evolution is Virtually inevitable. It appears now that natural selection is the only verifiable factor making for cumulative adaptive change. This statement, however, by no means exhausts the possibilities of analysis. EVOLUTION FROM SINGLE MUTATIONS

The earliest hypothesis along Mendelian lines was that significant evolutionary change is due to the occasional occurrence of mutations of such a nature as to give rise to new species (or higher categories) at once (deVries, Goldschmidt). The abrupt origin of species by mutation requires that the most essential feature of the mutation be an isolating effect. This hypothesis has in its support the evidence that polyploidy has been important in the multi1 Ernest D. Burton Distinguished Chicago.

Service Professor of Zoology, The University

{ 365 }

of

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plication of species, especially in cases in which it has occurred in a sterile hybrid, converting the latter into a fertile amphidiploid. It is, however, impossible to base all evolution on the addition of chromosome sets. There is no other class of mutation which combines the properties of full fertility by itself, the production of sterile hybrids with the parent form or forms, a balanced organization favorable to adaptation, and (in amphidiploids) striking novelty. The other balanced types of chromosome mutation (translocation, inversion) result, when viable, in minor variants within the species of origin, often recognizable only by examination of the chromosomes themselves. Simple aneuploidy may bring complex character changes, suggestive of those distinguishing species, but not isolation. Most such changes are indeed incapable of existence except as heterozygous segregating varieties. Compound aneuploidy may, however, be a rare species-forming process (Darlington). Small duplications have no doubt been very important in evolution in increasing the number of genes, but hardly in the abrupt origin of new species. Such minor chromosomal changes can be treated statistically for the most part as gene mutations. EVOLUTION BY GRADUAL TRANSFORMATION

The most obvious alternative to abrupt origin of species by mutation is the conception of gradual transformation through the occurrence, orderly increase in frequency, and ultimate fixation of mutations that give a selective advantage to the individuals that carry them. Haldane has developed mathematically the nature of this process in a great variety of cases (1924 and later). Fisher (1937) has investigated the properties of the wave of diffusion from a portion of a population in which a mutation has become fixed. Acceptance of any hypothesis of evolution by statistical transformation of populations permits differences of opinion as to whether it is the fixation of rare major adaptive mutations or the accumulation of a large number of minor ones that is usually responsible for a new species, whether by transformation of the entire parent species or of an isolated portion of the latter. This issue is somewhat confused by the ambiguity of the term bigness as applied to a mutation. The most literal sort of bigness is in the degree of physical change in the chromosomes. There is, however, no consistent correlation between this sort of bigness and conspicuousnessor complexity of the effects on characters. There may be extensive rearrangements of the material of the chromosomes without conspicuous effect, and conversely single gene mutations may produce abnormalities of the most extreme sort. Again mere conspicuousness (as of many color mutations) may have little relation to morphological or physiological ( 366 )

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-

-

-

-

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complexity. Finally bigness in any of these ways is far from being positively correlated with ecological potentiality. Analyses of the actual genetic differences between related species have demonstrated all of the viable types that have arisen in the laboratory (ef. Dobzhansky, 1941). Changes in chromosome number and rearrangements are common but appear to have only minor importance, where any, with respect to distinguishing characters other than of fertility of hybrids. Conspicuous unitary differences are sometimes found especially with respect to color. The mode of segregation for most characters, however, indicates a multiplicity of genetic changes with effects that are either individually slight, or (if the character itself is of an all-or-none sort) of low penetrance. The view that these species differences are largely due to mutations that were not only adaptive from the first but were carried directly to fixation by favorable selection is undoubtedly too simple, as indeed has probably been recognized in some degree by all who accept the general hypothesis of transformation. It makes the occurrence of adaptive mutations the limiting factor in evolution to an extent that severely restricts the possible rate of evolutionary change. There is evidence that evolution can proceed with great rapidity, given an adequate ecological opportunity. It is necessary to examine the genetic properties of populations to find whether these may not provide a basis for rapid exploitation of opportunities without waiting for the accidental occurrence of the right mutation. GENE

FREQUENCY

The basic concept of statistical genetics is gene frequency. Because of the symmetry of the Mendelian mechanism, gene frequencies tend to persist unchanged from generation to generation. The state of the species with respect to a particular locus can be described in such a form as ~~=1 (q;A)

where the A's designate k alleles and the q's are their frequencies. The random unions of eggs and sperms, each with the array of alleles characteristic of the species, give a stable genotypic array at each locus of the type [::$~=1(qiAi) P (Hardy, 1908; Weinberg, 1908). The frequencies of combinations among loci are in the long run those of random combination, in a random breeding population, irrespective of linkage, unless the latter is complete (Weinberg, 1909, 1910;Robbins, 1918; Geiringer, 1944, 1948). The array may be represented by

( 367 )

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where L is anyone of n different loci. Thus 4 alleles (10 combinations) at each of 100 loci provide the potentiality for 10100 different genotypes. A genetic description of a population can obviously be given much more economically in terms of gene frequencies than of genotypic frequencies. Moreover the tendency toward persistence of gene frequencies insures relative persistence of the statistical properties of the population, if the system of mating remains the same, even though no genotype is ever duplicated. A change in the system of mating changes the array of genotypic frequencies and consequently the statistical properties of the population, but does not in itself tend to change the arrays of gene frequencies. Thus a tendency toward mating of relatives, whether sporadic or from a breaking up of the population into more or less isolated groups, gives a genotypic array at each locus of the type (l-F)[~(qiAJF

+ F~(qiAiA)

where F is the inbreeding coefficient, defined as the correlation between uniting gametes with respect to additive gene effects (Wright, 1921,1922). The increased homozygosis, measured by F, gives rise to the well-known effects of inbreeding, but the q's are not changed, and a return to the original system of mating is followed by return to the original genetic situation. Similarly, pure assortative mating with respect to any character causes departures from randomness of combination among the loci and consequent changes in the statistical properties of the population, but again the process does not in itself affect the gene frequencies and random combination is gradually restored on resumption of random mating (Wright, 1921). THE

ELEMENTARY

FAcrORS

OF EVOLUTION

The elementary evolutionary process in a reasonably large homogeneous population may be considered to be change of gene frequency. It is convenient for the present purpose to distinguish three primary modes of change according to the degree of determinacy in the changes which they bring about. First are modes of systematic change, termed the evolutionary pressures (Wright, 1929, 1931). These are capable at least in principle of precise mathematical formulation which would make possible prediction of evolutionary trend if there were no processes of an indeterminate nature. Second are the random fluctuations in gene frequency, of which only the variance is determinate. Finally, it is convenient, although somewhat arbitrary, to distinguish from both of these, events that are unique or nearly so in the life of the species. { 368 }

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MODES OF IMMEDIATE

AND

SELECTION

CHANGE OF GENE FREQUENCY

1. Systematic change (L.q determinate in principle). A. Pressure of recurrent mutation. B. Pressure of immigration and crossbreeding. C. Pressure of intragroup selection. 2. Random fluctuations (Sq indeterminate in direction but determinate in variance). A. From accidents of sampling. B. From fluctuations in the systematic pressures. 3. Nonrecurrent change (indeterminate for each locus). A. Nonrecurrent mutation. B. Nonrecurrent hybridization. C. Nonrecurrent selective incidents. D. From nonrecurrent extreme reduction in numbers. Back of these processes are all factors that cause secular changes in mutation rates, conditions of selection, size and structure of the population and the possibilities of ingression from other populations. MUTATION

PRESSURE

Mutation pressure refers to the tendency of known genes to mutate to known alleles at definite rates. In the two-allele case, the net rate per generation obviously may be written L.q

= ~;= v(l-q)-uq

where u is the rate of mutation from the gene in question (frequency q) to its allele, v is the rate of reverse mutation, and time is measured in average generations. Haldane (1926) has shown that the effects of overlapping of generations are of only secondary importance in the transformation of populations. 6q may be treated as dq/dt in order to obtain the approximate amount of change in a given number of generations (t) by integration. The extents of systems of multiple alleles, known at many loci in many organisms, make it probable that each locus is capable of evolutionary change through a branching system of indefinitely great extent, and that many alleles normally coexist at many loci in any large population. Taking cognizance of this, the description of mutation pressure requires a system of simultaneous equations of the type 6qo

= ~~o = "(ucR)

- ("ui)qc

where Uci is the rate of mutation to allele Ac from one of its alleles Ai, Uio is the rate of reverse mutation and summation applies to all alleles of Ac• { 369 }

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It is obvious that the distinction between recurrent mutation and nonrecurrent mutation must be arbitrary. IMMIGRATION PRESSURE

Immigration pressure refers to the effects of recurring invasion of the territory under consideration by individuals, capable of producing crossbreds that enter into the population. Here

/sq, = -m( qe -

qe(l)

where qe(l) is the frequency of the gene among the immigrants and m is a coefficient measuring the amount of replacement of population by the immigrants per generation. The theory is closely similar to that of mutation pressure because of the linear nature of its effect. As with mutation pressure there is no sharp line between recurrent immigration and a type of hybridization so infrequent in the history of the species that each occurrence is best treated as a unique event. SELECTION PRESSURE

Immigration pressure may be considered as a process of intergroup selection, less drastic than expulsion or extermination. We shall, however, use the term selection pressure, where not qualified, as relating only to intragroup selection. Selection pressure in this sense may be defined so as to include exhaustively all systematic modes of change of gene frequency which do not involve physical transformation of the hereditary material (mutation) or introduction from without (immigration). It includes the effects of differences in mating rate, fecundity, mortality rate, rate of attainment of maturity, and emigration rate. It differs mathematically from the pressures of mutation and immigration in being nil for either q 0 or q = 1. There can be no selection in the absence of alternatives. The general formula for selection pressure is

=

/sq;

= oqc!'at = qe (We

- W)/W

where We is the momentary selective value of the gene in question, giving due weight to the frequencies and selective values of all types of zygote into which it enters, and w (= ~k i=l W Hi ) is the mean selective value for the population as a whole. The selective value of a given type of zygote (fertilized egg) is assumed to be measured by its average contribution under the prevailing conditions to the array of zygotes produced a generation later in such a { 370 }

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way that W is the ratio of the effective size of the population in the following generation to that in the one under consideration. W = "Wf in terms of the selective values (W) and frequencies (f) of genotypes. It is important to distinguish two kinds of selection: that in which the selective values of genotypes are constant under standard conditions, and that in which they are functions of the relative frequencies of genotypes. Relative constancy is to be expected where selection depends directly on a constant environment external to the species. Secular changes in both absolute and relative values of the selective values may indeed be expected to occur with changes in the density of the population but these are independent of the relative frequencies of genotypes in this sort of selection. If, however, there is selection based directly on the relations of different kinds of individuals of the species to each other, the selective values are necessarily functions of the frequencies of these genotypes. In the following discussion, random mating is assumed unless otherwise specified. CONSTANT SELECTIVE VALVES. SINGLE LOCI

It is convenient to take up first the case in which the selection coefficients for genotypes of the locus in question are independent of frequencies at other loci and are constant and alike in the sexes. In this case the rate of change in frequency of gene. A, is as follows (Wright, 1937, 1942).

/s«;

= oqclot = qc( l-qc)

(oW /oqc) /2W

It is assumed that the frequencies of all alleles of the gene, Ac, are expressed in the form q, = ric(l- qc) in W before differentiating. Here ric is the frequency of A among the alleles of Ac• Thus Oqi/Oqc

= -ric = -qi/(I-

qc)'

The composition of the population relative to a series of alleles may be represented by a point in a (k - 1) dimensional space. Thus in the case of three alleles, the gene frequencies may be represented by the distances from the sides of an equilateral triangle of unit height. The mean selective values (W) for populations with given sets of gene frequencies, corresponding to points within this, may be represented by ordinates from the plane of this figure, defining a surface. Similarly in the case of four alleles the gene frequencies may be represented by distances from the four faces of a regular tetrahedron of unit height. The mean selective values { 371 }

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of populations require an additional dimension W which may be indicated by shells of equal selective value, the contours of a "surface" in four-dimensional space. This "surface" of selective values may be of various sorts. There may be a continuously upward gradient leading from each point toward a single peak value. If this is at one of the corners, it means that selection leads inevitably to fixation of the corresponding homozygote, except in so far as mutation or immigration maintains other alleles. If it is on an edge, or face, or in the interior of the figure, it is implied that selection by itself leads to a condition of stable equilibrium among two or more alleles. A depression from which gradients lead up in all directions implies a position of unstable equilibrium. There may also be saddle points, toward which selection drives the set of gene frequencies from certain directions but from which selection tends to cause ever-increasing departure, after a slight initial departure in certain other directions. All alleles are maintained in equilibrium, if all heterozygotes (A0j) are alike in selective value and are superior to all of the homozygotes (A~i, AjAj etc.). Let Wij=a,

Wu=a(l-si), Wjj=a(l-sj): W=a[l-~~ S.q2] >=1, i

= aqo(~~lsH/

!::.qo

go

- soq)/W = (l/S)/:S~l (l/sJ

go

where is the value of qo at equilibrium all !::.q'sequal O. If the heterozygotes differ in selective value, the conditions for maintenance of all alleles are somewhat restricted. Thus if all homozygotes (k alleles) have the same selective value, a, and all of the heterozygotes except one have the value a (1 s) with s positive, and that one (A1A2) has the value a (1 t)

+

+

W=a[l+s(l-~~lqn !::.ql = aqJs(~~lq2

=

+2(t-s)qRJ -

qJ -

(t - s)(2qR2

- q2)]/W

/sq, aq.[s(:S~lq2 - qa) - (t - S)2qlq)/W Ql=Q2=s/[ks(k-2)(t-s)] if 0 < t < 2s qa qk (2s - t) /[ks - (k - 2) (t - s)] if 0 < t

= =

< 2s

All alleles are maintained in equilibrium if t is between 0 and 2s. If t is negative, equilibrium is unstable; either Ai or A2 is eliminated according to the initial composition. If t is greater than 28, all alleles other than Ai and A2 tend to be eliminated. It is possible for a gene that would be eliminated if only certain alleles { 3713 }

..

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were associated with it in a population to be maintained if others also are present. The conditions are decidedly restricted, however. In many cases, some of which have been indicated above, there may be two or more peak values. Obviously any homozygote that is superior to all of the heterozygotes in which its allele enters is at a distinct peak. Which allele becomes fixed depends on the initial composition of the population. Any two or more alleles may form a group capable of stable equilibrium within itself but in unstable association with other such groups. The rate of change in the mean selective value of the whole population is given approximately by the following formula, if the selective differentials are small.

It is assumed, as before, that the frequencies of all alleles are expressed in the form qj = rji( 1 - qi) in W, before differentiating with respect to any frequency, qi' CONSTANT GENOTYPIC

SELECTIVE VALVES

The treatment of selection as acting on single loci independently of all others is however highly artificial. Selective value, in a given setting, is a property of the organism as a whole and hence of its whole genotype. A gene is likely to be more favorable than an allele in some combinations, less favorable in others. The composition of the species may be thought of as located in a space of ~n (k - 1) dimensions, assuming n loci and a variable number (k) of alleles at each locus. If the selective value, W, for each genotype as a whole is constant, and if the frequencies at the loci are combined at random (as is approximately true under long continued random mating and small net selection coefficients of the genes), the formula for rate of change of gene frequency per generation is still f'::,qa= qa(l-

qa) (oW /oqa)/2W

under the same convention as before but W( =~Wf) is here a function not only of the gene frequencies at the locus in question but of the genes at all other pertinent loci. It is again convenient to add another dimension to the geometric' model: mean selective value (W) of populations. The "surface" defined by W may be expected to be a very rugged one with innumerable peaks and subpeaks, corresponding to different harmonious combinations of genes. These peaks may be of three sorts phenotypically: (a) peaks centering in different genotypes that give the same phenotype (as under the conventional multiple-factor hypothesis for quantitative variability and an { 373 }

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intermediate optimum), (b) peaks that center in phenotypes that differ but give adaptation to the same conditions, and (c) peaks that center in adaptations to different conditions within the array of conditions to which the population has access, corresponding to the different adaptive zones and subzones of Simpson (1944). The formula for 6.W is the same as that above except that summation applies to all genes at all loci. The control by the gradient is obviously qualified by the term qe(lqc)/2W in the formula for each 6.qc. With constant genotypic selective values, the species tends to move toward one of the peaks, though not, as carelessly stated previously, up the steepest gradient in the surface W from the point at which it is located. The peak toward which it moves is not likely to be the highest one. EFFECTS OF DEVIATIONS FROM ASSUMED CONDITIONS

The situation is altered more or less by various types of deviation from the idealized case to which the preceding formulae apply. If there are very large differences between the sexes in selection, special formulae must be used; but even with moderately large differences the results are almost as if the average selective coefficientsfor each genotype applied to both sexes as shown (with a different mathematical formulation) by Haldane (1926) (ef. Wright and Dobzhansky, 1946). If there are moderate deviations from random mating due to inbreeding, and the net selection coefficients of genes are small, the expression oW loqe of the preceding formulae may be replaced by o(W FWI )Ioqe where WI is the mean selective values of genotypes that are homozygous at the locusin question. The expression WI is identical with W at loci at which there is no dominance but otherwise is different (Dobzhansky and Wright, 1941). Selective mortality at different ages, and selective fecundity have diverse effects on observable frequencies. These must be carefully considered in analysis of concrete cases and are likely to make it necessaryto go back to the basic formula. Selection itself always produces departures from complete randomness of combination among loci if there are nonadditive interactions, but these are negligible for most purposes unless selection or linkage is strong (d. Wright, 1942). Assortative mating, however, may produce very great departure from randomness (Wright, 1921) which requires considerable increase in complexity in its treatment. Subdivision of the population into nearly isolated strains also has effects which alter the situation drastically and are best treated under the head of intergroup selection by means of differential migration pressure. In view of a criticism by Fisher (1941) it should be emphasized that the formula for 6.q above, to which he re-

+

{ 374 }

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ferred, was explicitly based on the assumption of random mating of diploids and of constancy of genotypic selection coefficients. Other formulae had been used for other conditions. Fisher (1930, 1941) himself has developed a different mode of approach for dealing with selection apart from other factors. According to his "fundamental theorem of Natural Selection": "The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time." Genetic variance here refers only to the additive effects of genes. There is approximate agreement between our formulae for small selective differences and constant selective values, the difference being that Fisher's method does not introduce the term W in the denominator. The application of this theorem to cases in which there are variable genotypic selection coefficients is not, however, clear. VARIABLE GENOTYPIC

SELECTIVE V ALVES

With variable genotypic selection coefficients, but otherwise the same assumptions as above and the same convention with respect to differentiation, L,qo = qo( 1- qo)~[W( olloqo)

]j2W

where the f's are frequencies of genotypes, in general involving many loci, and the summation applies to all genotypes. The term ~[W (of /oqo)] may be written [(oW/oqo)

-

(oW/oqo)]

(Wright, 1942). In this, oW/oqo measures the effect of change in the frequency of the gene in question on the average selective value of the population as a whole, while (oW/oqo) measures the average effect on the selective values of all genotypes and is of course absent if these selective values are constant. An important case is that of competition between members of the same species such that the relative degree of competitive success of each of the genotypes is constant but the net rate of increase of the population is unaffected by changes in the relative frequencies. This situation can be represented by writing Wo aRo/If when a is a general constant, B; is a constant pertaining to the particular genotype, and R ( = ~Rd) is the variable average value of R. The surface of selective values for the population as a whole is constant (W = a) in accordance with the hypothesis, but gene frequencies change according to the rule:

=

L,qo

= qo(1-

qc) (0R;oqc)/2R

The surface Ii is not level and selection pressure tends to drive the species toward a peak value of R. { 375 }

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If there is a combination of both kinds of selection, we can write W = a[(R/R) W=a(l+s)

+ s]

/sq, = aqc(l-

qc)

o(logR+ oqc

s)

-

/2W

The surfaces Wand log Ii may both be rugged and the peaks and valleys need not agree. The species moves toward a peak in the resultant surface [ (log Ii) If log if and are systematically opposite in sign and log R is numerically greater, selection leads to a deterioration of the species in relation to its environment. This puts in symbolic form an interpretation of apparently orthogenetic trends toward extinction given by Julian Huxley. The special case considered by Huxley was that of selection for characters useful to males in competition but somewhat deleterious in relation to the environment. A case that is similar in its consequences although somewhat more complicated from the mathematical standpoint is that of competition among members of the same brood. If there is no relation to success of the population and Wll, W12 and W22 are the selective values of AlAu A1A2 and A2A2 when present in the same brood, W a

+s].

s

=

6.

q1 = aq1

(1

- q1

2ql(1-

)[(W22 - W12) 12 +(;22 w.. W 22 W 12 q W 22 2W 12

+

q1)

+

+ (W12 - W )(I_ W12 + w.. ll

+W ) 11

q1)2]

If there is a negative correlation between success in intrabrood competition and success in meeting the external environment, the process may lead to deterioration of the species. This case has been treated from a different mathematical viewpoint by Haldane (1924a). Where advantages with respect to social relations and external environment are negatively correlated but closely balanced, it is possible for the opposing selection pressures to maintain alleles in equilibrium. A genotype may have qualities that are of advantage to the species in proportion to some function of the frequency of such individuals but of no additional advantage to these individuals themselves. If this is the case, all selective values (W) are to be multiplied by the same term [1 'I'( qc)] in which 'I'( qc) is the function of the frequency of the genotype in question. If the W's are otherwise the same, W = a[l '1'( qc)] with a peak or peaks corresponding to one or more values of qc' There is, nevertheless, no tendency for qc to change at all (6.qc = 0). If the gene has a deleterious effect on its possessors it will tend to be eliminated in

+

+

{ 376

j

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AND

SELECTION

spite of its advantage to the species, still assuming random mating (ef. Haldane, 1932; Wright, 1945a). The relative as well as the absolute selective values of genotypes may change with changes in their frequencies. It must suffice here to consider a few simple cases involving only one series of alleles. It will be assumed that the selective values of heterozygotes are always exactly intermediate between those of the corresponding homozygotes to avoid complications due to dominance or superdominance of the sorts that occur also where the selection coefficients are constant. First consider the case in which the selective value of a homozygote rises in proportion to the frequency of its allele in the population. This may hold where the individuals of each type receive a social advantage from the presence of others of their own kind or are injured by the presence of other types. Let Wll = a(l 2S1ql)' W12 a[l 81ql S2q2]' W22 = a[l 2s2Q2]' etc. W a[l 2~k 8.Qi 2]

+

+

=

=

+

+

,£=1

/sq, = aqJ8cQc There is unstable equilibrium

(8'S

q = (1/8) c

-

+

~

~~=18H/l!W

all positive) at the set of values

/~~=l(1/8()

One allele or another tends to become fixed according to the initial composition of the population. The selective advantages of genotypes may, however, decrease as the corresponding allele increases in frequency. This occurs if there is a division of labor among types to their mutual benefit, or if the population occupies a heterogeneous territory in which rare genotypes can always find a favorable niche but abundant ones are forced to live in part under unfavorable conditions. In this case the 8'S in the above formula are negative. There is stable equilibrium at the values = (1/8c)l~(1/8i), exactly as if all heterozygotes had the same constant selective value and each of the homozygous types had a certain disadvantage [Wc=a(1-2sc)] etc. In the case with variable selection coefficients considered here, gene frequencies move in exactly the same type of surface W in the same way except that the rate is only half as great in relation to the slope as with constant coefficients (Wright and Dobzhansky, 1946). A somewhat similar case which has been analyzed in detail in a previous paper (Wright, 1938) is that of self-sterility alleles in plants. Alleles are favored when rare and opposed when common, but as the selection depends on a relation between male gametophyte (haploid) and female sporophyte (diploid), it is more complicated mathematically than cases in which selection depends on relations among diploids, and is symmetricalor nearly so with respect to sex.

qc

( 377 )

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There may be cyclic movement of gene frequencies under certain limiting conditions which are probably of more mathematical than evolution. ary interest. Thus if the selective values are as follows, and s is very small, there is an approach to perpetual cyclic motion although the surface of values is completely

flat (W

+ + +

= a)

Wu = a[l 2S(q2 - q3)] W 22 = a [1 2s ( q 3 - q 1) ] W33 = a[l 2S(ql - q2)] /sq, = Sql(q2 - q3) Dq2 = Sq2( q3 - ql) Dq3 = Sq3( ql - q2)

W12 = a[l W23 a[l W31 = a[l

=

+ S(q2 + s( q3 + si q; -

ql)] q2)] q3)]

With the slightest departure from the limiting situation the species tends to move in a spiral in the field of gene frequencies either toward or away from a peak, if there is one, or toward or away from a depression to a point at which all Dq's are zero. Somewhat similar cases may involve more than one locus. The formula for D W is the same as with constant selection coefficients. NET SYSTEMATIC PRESSURE

The net effect of all of the systematic pressures on the frequency of a gene is the sum of the various components. Thus in a random breeding population of diploids, the rate of change in the frequency of a gene Ac as one of a system of simultaneous equations is approximately as follows:

Dqc=

~(Uciqi) - (~UiC)qc-m(qc

- qC(l))

+qc(l-qc)[ow 2W

oqc -

(OW)] oqc

This type of pressure on all of the genes tends to carry the species to some position at which /sq 0 for each of them (except in certain limiting cases referred to above). This position tends to be in the neighborhood of a peak in the surface of mean selective values in so far as due to selection related to the environment, but this is not likely to be the highest peak. Mutation and immigration pressure, if present, keep the population from attaining the exact peak. The position of stability may indeed not even be near a peak in cases in which selection depends on certain types of social relations. We have here a theory of stability of species type in spite of constant mutation, extensive variability at all times, constant action of selection, and substantial possibility of improvement if the species could arrive at certain combinations of genes already present, other than the combinations that prevail.

=

ACCIDENTS OF' SAMPLING

Accidents

of sampling

may cause fluctuation { 378 }

from the position of

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SELECTION

stability, but these are negligible in a large random breeding population. They become important in small populations where they are responsible for the well-known effects of inbreeding. They may be responsible for nonadaptive differentiation of small island populations but are more likely to lead to ultimate extinction in a small population, completely isolated from its kind, than to evolutionary advance (Wright, 1931 and later). These conclusions follow from the chance of fixation of a completely indifferent gene which is 1/ (2N) where N is the effective size of population, and from the chance of fixationof a deleterious gene (selective disadvantage s in heterozygote, 2s in homozygote) which is 2s/ (e4NS - 1). NONRECURRENT MUTATIONS

Stability in a species living under constant conditions may be upset by the occurrence and establishment of a favorable mutation that has never occurred before or has previously been lost by accidents of sampling if it ever has occurred. Even a mutation with a considerable selective advantage is likely to be lost by accidents of sampling when present in only a few individuals, although it is practically certain to reach its equilibrium frequency if it ever reaches moderate frequencies in a large population. The chance of fixation of a semidominant mutation with selective advantage s in the heterozygote is 2s/ (1 _e-4Ns), or approximately 2s as given by Haldane (1927b). A recessive mutation with selective advantage s has a chance of fixation given by Haldane as of the order of ys/N and by Wright (1942) as Ys/2N. With v the rate of recurrence of a mutation, the chance that all recurrences within a period of T generations are lost by accidents of sampling is P = (1 - 2s) 2NvTin the former case, P = [1 - ys/2NFNvT, in the latter. The line between recurrent and nonrecurrent is necessarily arbitrary. A type of mutation with an even chance of becoming established within 1000 generations must be considered recurrent from the standpoint of a species with a life (T) of hundreds of thousands of generations; but if there is an even chance only in 100,000generations, the mutation may be considered nonrecurrent in this case. Putting P = ~f in the period T generations, rate v < 0.69/4NsT may be used to define nonrecurrent mutations with dominant favorable effect and v < 0.69/Y2NsT for recessives with favorable effect. Thus in' a population of 1010 and T = 10i:he"achie~~ment of a major adaptation, improvement should be as rapid as the store of variability permits. When decisive superiority is attained or unoccupied territory is reached, a very extensive and rapid adaptive radiation should \' follow under the divergent selection pressures toward exploitation of the various special ecological niches, opened up by the general adaptation (in the first case) or the mere absence of competition (in the second). The most favorable conditions are those in which the population is sufficiently sparse and the territory sufficiently extensive and interrupted to permit a fine-scaled subdivision grading into complete isolation in many places. Differentiation of two forms must ordinarily start in geographically isolated regions but may be accelerated by competition if one form reinvades the territory of the other after genetic isolation has been reached. This rapid evolution in the origin and adaptive radiation of a higher category constitutes the tachytely of Simpson. The primary adaptive radiation may be expected to be followed by secondary and tertiary radiations of progressively lower scope, and by extensive selective elimination of other higher categories especially in the { 387 }

ADAPTATION

case in which an original general adaptation has outmoded older forms in a great variety of ecological niches. As this process continues i..rne ~ornes when. each of the succf)s~f!ll_.. •. .... on X .on~. in«:;"2Llli!Yi!!Jge openfcdt, .~...single ridge in our model, There slow· more or less ..orthogenetic advance,· Simpson. There may be extensive diversification of genera and species, but these are rarely of significance in starting new trends. Ultimately forms may become bound each to a

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