The number of neutral alleles maintained in a finite, geographically structured population

THEORETICAL

POPULATION

BIOLOGY

2, 437-453 (1971)

The Number of Neutral Alleles Maintained in a Finite, Geographically Structured Population* JAMESF. Genetics Laboratory,

CROW

of Wisconsin, Madison, Wisconsin 53706

University

AND

TAKEO MARUYAMA National

Institute

of Genetics, Mishima, Japan

Received March

In a geographically

structured

5, 1971

population 4N.u

1

n*T=fw

l-f0

where n,r is the effective number of neutral alleles maintained in the total population at equilibrium, f,, is the probability that two homologous genes from the same individual or from the same locality are identical, f is the corresponding probability for pairs of genes drawn at random from the entire population, I( is the mutation rate, and N. is the total population number (or effective number). This is true regardless of whether the population is divided into wholly or partially isolated subgroups or is geographically continuous with isolation by distance. This assumes an infinite number of potential mutant states. If the number of such states is K, with an equal mutation rate to each, the formula becomes

1 -f,,

+ 4N,,r+

Regardless of the number of potential alleles, the local effective number of alleles at equilibrium is n.,~ = n.,j/f,, , and the panmictic effective population number, defined as the size of a panmictic population that would maintain the same number of neutral alleles at equilibrium, is Nsp = N.(l -f)/( 1 - fJ. That is to say, NsP is equal to the effective number not taking geographical structure into account (NJ multiplied by the ratio of the global to the local heterozygosity. * Contribution No. 1430 from the Genetics Laboratory, University of Wisconsin, and No. 793 from the National Institute of Genetics, Mishima, Shizuoka-ken, 411 Japan.

437 653/2/4-6

438

CRO\V AND MARUYAMA 1. INTRODUCTION

The effective number of selectively neutral alleles at equilibrium, defined as the reciprocal of the probability that two randomly chosen homologous genes are identical, is given to a close approximation by

n, = 4N,u + I

(1)

where N, is the population number (more realistically, the effective population number) and u is the mutation rate. The potential number of alleles is assumed to be large enough that each new mutant is a type not already represented in the population; u is then the total mutation rate to all alleles (Kimura and Crow, 1964). If there is a fixed number of potential alleles, K, and the rate of mutation from one to any of the other K - 1 states is u/(K - l), the total rate being u as before, the effective number of alleles at equilibrium is

approximately (Kimura, 1968a; Crow and Kimura, 1970, pp. 324, 453). In this article we extend these results to a geographically structured population, either a population broken up into subunits with migration among them, or a population with limited dispersal so that there is some genetic differentiation by distance. It was already becoming evident at the time the 1964 paper was written that the number of possible mutant states at a locus is enormous and that many of these represent only the minutest changes in the function of the gene product. The possibility that many mutants are selectively neutral, or nearly so, seemed great enough to warrant an investigation of the consequences of this possibility. If not a realistic model, it could at least represent a limiting case with which comparisons of selective models might be made. Since that time the large amount of genetic variability in natural populations has been documented (Lewontin and Hubby, 1966; Harris, 1966, and many others) although the mechanisms by which this variability is maintained are still not understood. Kimura (1968, 1969) and King and Jukes (1969) have suggested that neutral mutations may account for a large fraction of the amino acid changes that are observed in homologous proteins in different species. The number of alleles maintained in a population of specified total size and the time required for a new mutant to reach fixation depend strongly on the structure of the population. Hence, it seems worthwhile to extend (1) and (2) to take geographical structure into account.

NEUTRAL ALLELES IN ASTRUCTURED POPULATION

439

2. HISTORY The first studies of the genetic consequences of a structured population were made by Wright (1931, 1943, 1951), who used an island model. He assumed that a certain fraction, m, of the individuals in a subpopulation are replaced each generation by an equal number chosen at random from the entire population. Among other things, Wright showed that if 4Nm > 1, where N is the number in a subpopulation, there is very little local differentiation and the entire population behaves essentially as a single panmictic unit. On the other hand, if 4Nm < 1, there is considerable fluctuation in gene frequencies from one local population to another. A second model, more realistic for some populations, is the stepping-stone model (Kimura, 1953; Kimura and Weiss, 1964; Weiss and Kimura, 1965). As in Wright’s model there are discrete local populations, but in the steppingstone model migration is always between adjacent localities. As with the island model, the population may be linear or two dimensional (or of higher dimension). The island model may be included in the formulation by allowing also long-range migration from the entire population. Wright recognized, however, that the gene frequency of actual immigrants is usually closer to that of the recipient population than to the average of the population as a whole. In analyzing an actual population he used a modified migration rate, adjusted to take this into account (see, for example, Dobzhansky and Wright, 1941). Wright (1943, 1951) and Malecot (1948, 1967) have pioneered in the development of models for continuously distributed populations. Wright’s results were in terms of the “neighborhood size”, the size of an area from which the parents may be assumed to be drawn at random. Malecot considered a population with limited dispersal and derived formulae for the decrease in genetic identity with distance in terms of the distribution of the distances between the birth place of parents, or between that of parents and offspring. Ma&cot’s theory has been applied extensively to human population structure by Morton and his associates (for a recent review, see Morton 1969). Maruyama (1970, 1970a) has studied island, stepping-stone, and continuous models where the total population is finite. In particular he has obtained the steady state probability that two randomly chosen homologous genes are identical in terms of the mutation rate, the migration or dispersion rate, and the total population number. Although the solutions differ from caseto case, one relationship emerged for each of the models regardless of the migration law and of the number of dimensions of the habitat. This is an extension of (1) namely 4N,u ne = l--f0

(3)

440

CROW AND MARUYARIA

wheref, is the probability of identity of two alleles from the same locality. This equation has also been obtained for the island model by Maynard Smith (1970). We now show that this relationship is true for a population at a steady state under very general conditions and then discuss some of the implications. 3. DERIVATION OF EQUATION 3 We make no assumption about the geographical structure of the population except that the total size is constant and finite and that the structure is stable. Mutation occurs at a rate u and it is assumed that the number of potential allelic states is large enough that each mutant allele is a type not preexisting in the population. Under this assumption the probability of identity of homologous genes is the same as the probability of homozygosity. Finally, we assume that the whole population structure, regardless of the pattern of migration and dispersal, has reached a steady state. Equation 3 can be derived in several ways, but the simplest and most direct one, which we will use, follows a pattern suggested by Dr. Motoo Kimura. To fix ideas, we assume that the population is divided into discrete local populations within which mating is random; later we show that the formula can also be applied to populations with a continuous geographical distribution. We define the following quantities: Ni : the effective number of individuals in the i-th local population N, = x Ni : the total effective population number L: the number of local populations fii : the probability that two different homologous genes, chosen at random from the same colony, are identical fii : the probability that two homologous genes, one from colony i and one from j, are identical 24: mutation rate. With these definitions, the average probability of identity for two homologous genes chosen at random from the entire population is (4) Ignoring migration for the moment, the probabilities (designated by a prime) are - +,

fii] (1 - 24)’

fii] (1 - UY, fij = fij(l - U)“.

one generation later

(5) (6)

NEUTRAL ALLELES INASTRUCTURRD POPULATION

441

Equation 5 was first given by Malecot (1948), along with the slight modification needed to take separate sexes into account (and which we ignore). For a derivation, see Crow and Kimura (1970, pp. 101, 323). Substituting (5) and (6) into (4) gives

The migration between colonies may be following any rule and may be of a very complex sort. However, we don’t need to understand it. As long as the population is in a steady state, which we are assuming, the structure remains constant. In the population as a whole the input of new mutations equals the rate of loss through random drift. Migration and dispersal shift the genes around within the population so that the individual fii’s and fij’s change, but the averages are stable and the individual values can be ignored if we concentrate on mutational input and random extinction for the whole population. At equilibrium j’ =3. Using this, and substituting for C NiNjfij from (4), we obtain

or, after simplifying,

3 = (1 -fo)U - 4”

(8)

2N,(2u - u”) ’ where

Since u is ordinarily very small, (8) is satisfactorily approximated by

f=- 1 -fo 4N,u

which is the relationship we seek.

4. SOME SPECIAL CASES,INCLUDING A POPULATION WITH A CONTINUOUS DISTRIBUTION

If the population is completely panmictic, then f= in agreement with (l), since 1z,= l/f.

1 4N,u+l

f.

= 3=

f

and (10) becomes (11)

442

CROW

If there are I, local populations

AND

RIARUYARIA

of equal effective size, Nr then

and fii =fjj =fo . The extreme case, opposite to panmixia, is a population in which there is no migration between colonies. In this case fij = 0 and, for L local populations of equal size N,

f. =

Lf==

I 4N,u + 1 ’

as expected by analogy with (11). The most important application of Eq. 10 is likely to be in a population of continuous geographical distribution where it is not possible to identify separate subpopulations. In this case f. can be measured as the probability of identity of the two homologous genes in an individual, or equivalently, of two homologous genes chosen at random from mates. Alternatively, f. can be measured from individuals chosen at random from an area small enough to be sure that mating within the area is random. Then,

f.

= average inbreeding coefficient of individuals or the coefficient of kinship of mates or of individuals within an area small enough that random mating can be assumed;

f = average coefficient of kinship of two different at random from the entire population.

individuals

chosen

In an actual population these two quantities could be readily determined if there were loci with a large number of distinguishable neutral alleles. The quantity f. is simply the probability of homozygosity if it is measured from single individuals. Otherwise it can be estimated from local gene frequencies. The quantity j is estimated from the average gene frequencies for the entire population. If pi is the frequency of a particular allele, L4, , then f,, or f (as the case may be) is given by

f=CPA

(14)

where the frequencies are measured locally or globally depending on whether or j is being measured. In some human populations it may be possible to estimate these quantities from pedigree studies or by isonymy (Crow and Mange, 1965; Morton, 1969). The quantities f. , 3, and the related estimate of current inbreeding, (fo -3)/U - 319were determined for shorthorn cattle by pedigree sampling by McPhee and Wright (1925).

f.

NEUTRAL ALLELES IN A STRUCTUREDPOPULATION

443

If there is local nonrandom mating, then the theory would have to be extended, since we have assumed random mating in each locality. This could be important in species, such as some plants, where there is some self-fertilization. In human populations there is usually avoidance of marriage between very close relatives, but unless the local population is extremely small this is not likely to alter the estimates of f0 appreciably. Also, in very small local populations a correction of the formulae to take account of separate sexes would slightly improve their accuracy. 5. A TEST OF THE INFINITE NEUTRAL ALLELE HYPOTHESIS

It is pleasing that the two quantities,f, andJ, that appear in (10) are the quantities that are likely to be measured in any field survey. They can be estimated from isozymes or other proteins (and ultimately, nucleotide sequences) where there is a good chance that physiologically unimportant differences can be detected by chemical means. If alleles at several loci can be studied in the same population and if the mutation rates are known, or can be assumed, then the quantity (1 - fa)/‘~ should be constant for all such loci. A major difficulty is that the equilibrium is slow to be attained and the population structure may not be stable enough to make the equilibrium theory applicable. 6. THE EFFECTIVE NUMBER OF ALLELES It might at first seem that the number of different alleles, tt, , maintained at a locus would be a good index of the amount of variability in the population, but it has several drawbacks. The distribution of allele number is highly skewed with many of the alleles being represented only a few times or even only once in the population. These very rare alleles contribute very little to the genetic variance of the population. Furthermore the extreme skewness of the distribution means that ordinary sampling procedures are inadequate to measure the number of alleles with any precision. Likewise, it is unsatisfactory to classify loci simply as monomorphic or polymorphic. For one thing, it is necessary to define an arbitrary frequency to provide a cutoff point between alleles that are rare polymorphs and the more frequent idiomorphs. A second problem arises from sampling; the probability of finding the rarer alleles, and therefore the likelihood that a population will be classified as polymorphic, increases with the size of the sample. For these reasons, the concept of e&tiwe number of alleles is useful (Kimura and Crow, 1964). It is defined as

444

CROW AND

MARUYAMA

where f, as before, is the probability of gene identity and pi is the frequency of the i-th allele. If there are 7t equally frequent alleles, then pi = l/n and n, = n. If the frequencies are not equal, rz, is less than the actual number, n, . The effective number of alleles is readily estimated from gene frequency data and its value is not unduly sensitive to rare alleles. It is closely related to the genetic variance of the population, since the genie variance is proportional to the heterozygosity, which in turn is 1 - for (n, - 1)/n, . Another advantage of measuring variability in terms of the effective number of alleles rather than the actual number is that for many purposes the mathematics are simpler; often it is not necessary to know the distribution of allele frequencies. If the population is divided into L localities, each of size N, , then the effective number of alleles for the whole population is n

(16)

from (10). Note that (16) can be rewritten 11eT =

4N,u + fo 3

(17)

and n,L , the effective number of alleles in a locality, is l/J, , or eL = 4Np 73 + 1 = nsr f

.

11

0

0

(18)

When the L local populations are equally sized

3 = +r,

+ (1 - +) fig.

(19)

If, in addition, they are completely isolated then fij = 0 and

Since (20) can be written neT = LneL the number of alleles in the total population is the sum for all the subpopulations, as expected when there is total isolation of the subpopulations. Equation 20 suggests that the ratio fo/j is a measure of the extent of geographical subdivision and isolation. If the ratio is one, the whole population is a single unit and there is no local differentiation. At the opposite extreme, if there are L completely isolated colonies, fo/3is equal to that number. It can therefore be regarded as an “effective number of colonies”.

NEUTRAL ALLELES IN A STRUCTUREDPOPULATION

445

Table I gives some numerical examples of the effect of geographical structure on the number of alleles maintained in the population. The columns for falJ = 1 give the effective number of alleles in a population with no geographical subdivision or isolation by distance. The other columns show the greater total number of alleles that are maintained in the total population and the smaller number per locality if the population has some sort of structure. TABLE

I

The Effective Number of Neutral Alleles Maintained in a Population of Total Effective Size N. and Mutation Rate u for Various Values of fa/f The Ratio fa/f Is a Measure of the Degree of Geographical Separation Or Restriction of Migration. The Quantities n,r and n,r Are the Local and Global Effective Number of Alleles 4N.u

10

100 1

10

100

1

10

%L

101

11

2

11

2

%T

101

110

200

11

20

f0l.f

1 loo

1

1.1

2

110

2

10 1.1 11

loo 1.01 101

7. EFFECTIVE POPULATION NUMBERS We can ask two related questions: (1) What is the size of an isolated population that would have the same equilibrium probability of gene identity as observed in the local population, i.e., f. ? (2) What is the size of a panmictic population that would have the same equilibrium probability of gene identity as two random alleles in the structured population being studied, i.e., f ? We shall call these the isolated, N,, , and pannictic, numbers. Equation 10 can be written

From

analogy

with

(1) we can write

the

isolated

N,? , effective population

effective number from the

relation l/f, = 4N,, u + 1, giving

N=l'-fO-N-f-_ et 4l4 fo

efo *

(22)

446 Likewise

CRO\V AND MARuYAhI.4

the panmictic

effective population

number is (23)

N,, and NeP are useful for different purposes. Ner is a measure of local gene fixation and random drift caused by restricted migration or limited dispersal. Nep measures the effect of geographical isolation in increasing the amount of genetic variability in the whole population. It can be seen from (23) that NrP is ordinarily larger than N, . The concept of effective population number was first introduced by Wright (1931). Actual populations depart from the ideal model of binomial distribution of progeny numbers and discrete generations. Actual populations may contain many individuals past reproducing ages, may have unequal numbers of the two sexes, and the progeny number variance may be greater than binomial (although this may not be true for some human populations and for livestock breeding systems). Especially important are gross differences in population size. If the population goes through small bottlenecks, the effective number is much closer to the smaller size than the larger ones. The original N, , and the derived quantities N,, and Nep , assume that these factors have been taken into account. Ordinarily, N, is smaller than the actual total number of individuals in the population, N, . It is of interest to ask whether NEp is usually larger or smaller than the actual number N, . There is no necessary reason why either should be true. For the reasons given above, N, is usually less than N, . But Nep is larger than N, . Wright (e.g., 1951) and others (e.g., Maruyama, 1970, 1970a) have shown that a very small amount of migration or dispersal is sufficient to convert an entire population into nearly a panmictic unit. For example, in the pure island model an exchange of one gamete every generation is sufficient to nullify the effects of population subdivision. So, unless the population is very widely dispersed or very strongly isolated, Nep is probably less than Nr . For a more detailed study there must be more specific knowledge of the geographical structure (for a discussion, see Morton, 1969). An analysis of the island, stepping-stone, and geographically continuous models has been made by Maruyama (1970, 1970a), extending the results of Wright, Kimura and Malecot to include a finite total population. The effective number of alleles is expressed in terms of mutation rate, total population size, the size of the local population (or density in the continuous case) and the amount of migration or dispersion.

8. FINITE

NUMBER

OF ALLELES

The model of an infinite number of potential alleles (more specifically, a number large enough that each new mutant is not already represented in the

NEUTRAL

ALLELES

IN A STRUCTURED

POPULATION

447

population) may be unrealistic, at least for some loci. Our various formulae are readily modified for a model with a limited number of possible alleles. Assume that there are K possible alleles and the rate of mutation from one specified allele to another is u&K - 1). The total mutation rate from one allele to all K - 1 others is then u, as before. With this assumption, the equation corresponding to (7) is

- 2) +[2TI;’ + U2(K (K-1)*]&x

[; Ni2 (1 -

&

z ) (1 -hi)

+ i$i NdNiU -hA].

(24)

The term 2u(l - u)/(K - 1) is the probability that two different alleles become identical by one mutating to the other and u2(K - 2)/(K - 1)” is the probability that they become identical by both mutating to the same allele. Again we substitute from (4) for x NJVjf,j and set j’ = f to give the equilibrium values. After some algebraic rearrangement, this leads to

(1 -fs) [ 1 - (2U - U”) &

- (gf

1)2] + 4Arg”

; U, + ‘;F-;;;

U2

J= 2Ne [&-u2)

A-

4$] (25)

or, to a satisfactory approximation

(26)

These become equivalent to (8) and (10) when K becomes indefinitely large, as expected. We now give the key formulae from the earlier section modified for a finite number of alleles, rewriting the earlier ones for comparison.

CROW AND MARUYAiV.4

448 Equation no.

+ fo , 3 _

17

neT = 4N,u

18

neL

21

N e =_L’-fo 4u J

22

NeI=L'-fo

=

%T

:,

4u

22a 23 23a

Finite allele no.

Infinite allele no.

= N,f

lz,L

,

’ fo

fo ' NeP =_L'-3 4u 3 =N '-3 Tq'

%T

4NeuA + h f = 4NpB + 1 ’ =

%T

-

3

fo

,

Nd =“-fo

4uAf’

N el =L

‘-fo 4uAJ-B

=N

A3-B

e Afo - B ’

NeP ='l-3

4uAj-B

=N

‘-3 Tq

(27) W-9 (29 (30) (3W (3’) (314

where K A=K-‘j

1 B’K-1.

Note that Eqs. 18 and 28 are identical. This is as it should be; iff,/j is interpreted as the effective number of local populations, then the total effective number of alleles should be the effective number per local population multiplied bY fo/J Another relation of special interest is (23a), which is the same as (31a). In both cases the total effective number, taking structure and geographical isolation into account, is equal to the effective number in which this is not considered N, , multiplied by (1 - 3)/(1 - fo), that is to say, multiplied by the ratio of the global to the local heterozygosity. This again is expected for, regardless of the number of alleles, the heterozygosity decreases by random drift at a uniform rate and it would be expected that this ratio at equilibrium would be independent of the number of alleles. 9. COMPARISON WITH THE RATE OF DECREASE OF HETEROZYGOSITY IN BREEDING SYSTEMS

We have been discussing this question entirely from the standpoint of a natural population with some form of geographical isolation. In this casefo is greater than f. The direction of the inequality may be reversed in animal or plant breeding

NEUTRAL ALLELES INASTRUCTURRD POPULATION

449

systems where the least related individuals are mated. This question was greatly illuminated by Robertson (1964), who showed that the decrease of heterozygosity of a population at time t is given by 1 - h, , where

1

-A-

t

1 1--L 4N

I-

c, *

(32)

In our notation, C,* isfa at time t, Ct isjt , and N is the population number. (If the 4 in the denominator instead of 2 seems surprising, this is because Robertson was considering a mating system in which each individual is constrained to have the same number of progeny rather than allowing this to vary binomially, the consequence of which is to approximately double the effective population size.) Systems of mating such as Wright’s “maximum avoidance of inbreeding” and “circular mating” (Kimura and Crow, 1963a) represent contrasting types. With maximum avoidance,f, f, early decrease in heterozygosity is greater because the matings are between half sibs, but the ultimate rate is less.

10. TIME FOR FIXATION OF A NEUTRAL MUTATION Kimura (1962) has shown that the probability of ultimate fixation of a neutral gene is simply its initial frequency. From this he showed (1968) that the rate of evolution of neutral mutants, that is to say the reciprocal of the average interval between the occurrence of successful mutants, is therefore u. This rate is correct regardless of the population structure or effective population number. It is only necessary that the period of observation be long compared to the time required for the mutant to spread through the population, and this means that the local populations cannot be completely isolated from each other; but within this very weak limitation the amount and pattern of migration and dispersion are irrelevant (Maruyama, 1970e). On the other hand, the time required for an individual mutant to increase from its initial rarity to a frequency of 1 depends on the effective population number and the population structure. As shown by Kimura and Ohta (1969), the average time from the occurrence of a mutant until its fiuation, given that it is destined to be fixed, is approximately 4N,. The question is, what is the appropriate effective number for this relationship ? If we rewrite Eq. 23a, or its equivalent (31a), as

-= 1 2Nep

1 1-h 2N,Tq’

(33)

450

CROU’

AND

MARUYAMA

the similarity to Robertson’s formula (32) is apparent, except for the previously mentioned 4 in the denominator. Because of this similarity we might be tempted to use l/ZN,,, as a measure of 1 - h and therefore regard ArePas the effective number appropriate to the rate of decrease in heterozygosity in a population at a state of steady decay, in which case it would be appropriate for the rate of increase of a neutral mutant. But our Eq. 33 applies only to a population at equilibrium between mutation and random loss of genes without regard to the rate at which the limit is attained, which is a different situation. One example will show the difference. Consider a population that is divided into completely isolated subgroups. Our equations give the appropriate answers for a population at equilibrium. On the other hand, if we write

or, when a state of steady decay is reached, l--x=

&-(3) I

(0

,

(35)

the relation cannot hold for completely isolated populations. There is no constant ratio of (1 - fa)/( 1 - J) in this situation because, as 1 - f,, approaches zero, 1 - j approaches 1 - l/L. As Robertson (1964) pointed out, equations like (35) apply only when the migration or dispersion is sufficient for the ratio to approach a constant value. Therefore, the effective number to use to get the average number of generations until an ultimately successful new mutant is fixed in a species is not simply NeP as given by Eq. 23a and 3 la. However, it is clearly related to the global population size rather than the local size. One way of arriving at an effective number for this purpose would be to equate it to l/2(1 - h) in those populations where a steady rate of decay is reached. The value of h has been worked out for specific geographical structures in a finite population by Maruyama (197Oc, 1970d). For the circular steppingstone model, when m is the rate of exchange of genes with the two neighboring local populations (half with each),

when Nm
and l--h%&T

(37)

NEUTRAL

ALLELES

IN A STRUCTURED

POPULATION

451

when the inequality is reversed. Thus the ultimate rate of decrease in heterozygosity is proportional to m (but not to N) when m is small and becomes equivalent to a panmictic population when m is greater than about L/2ON. This is a very small amount of migration; for example if the number of local populations, L, is 20, an exchange of one gamete each generation with each of the two adjacent localities is sufficient to maintain a nearly panmictic state. For a circular continuous model the relation is

when pa2 < L/2m2, and l--x& T

when the inequality is reversed, following the pattern of (36) and (37). In this case p is the density of the population, u2 is the variance of the distance that a gene migrates in one generation, and L stands for the length of habitat (such that Lp = NT). These are for specific one-dimensional models. More generally, it is possible to set a limit that may be useful (Maruyama, unpublished). The upper limit on the rate of decrease of heterozygosity is given by l--h
’

L ’

where m is the migration rate (replaced by u2 if the population is geographically continuous). Although this may be too high an upper bound to be very useful for linear habitats, it is likely to be helpful for the more usual problem of a 2-dimensional habitat where the effect of local differentiation is less, since in this case 1 - X is proportional to l/L rather than 1/L2.

11.

DISCUSSION

We should emphasize that Ner and NeP are not proposed as standard allpurpose definitions of effective population number for geographically structured or isolated populations. The same demographic and geographical pattern may have different consequences for different genetic processes. The definitions that we have given are appropriate to define the effective number with regard to equilibrium homozygosity for multiple neutral alleles. The effective number may

452

CROW AND

MARUYAMA

be different for other properties, such as rate of fkation as we have discussed, as the inbreeding and variance effective numbers of a population are not always the same.

just

ACKNOWLEDGMENT We should like to thank Maynard Smith, and Tomoko script.

Motoo Kimura, Sewall Wright, Newton Morton, John Ohta for useful suggestions and comments on the manu-

REFERENCES

CROW, J. F. AND KIMURA, M. 1970. “An Introduction to Population Genetics Theory,” Harper and Row, New York. CROW, J. F. AND MANGE, A. P. 1965. Measurement of inbreeding from the frequency of marriages between persons of the same surname, Eugen. Quart. 12, 199-203. DOBZHANSKY, TH. AND WRIGIIT, S. 1941. Genetics of natural populations: V. Relations between mutation rate and accumulation of lethals in a population of LhosophiZa pseudoobscura, Genetics 26, 23-51. HARRIS, H. 1966. Enzyme polymorphism in man, Proc. Roy. Sot. &ties B 164, 298-310. KIMURA, M. 1953. “Stepping stone” model of population, Ann. Rep. Nat. Inst. Genetics Japan 3, 62-63. KIMURA, M. 1955. Random drift in a multi-allelic locus, Evolution 9, 419-435. KIMURA, M. 1962. On the probability of fixation of mutant genes in a population, Genetics 47, 713-719. KIMURA, M. 1968. Evolutionary rate at the molecular level, Nature (London) 217, 624-626. KIMURA, M. 1968a. Genetic variability maintained in a finite population due to mutational production of neutral and nearly neutral isoalleles, Genet. lies. 11, 247-269. KIMURA, M. 1969. The rate of molecular evolution considered from the standpoint of population genetics, PYOC.Nat. Acad. Sci. 63, 1181-1188. KIMIJRA, M. AND CROW, J. F. 1963. The measurement of effective population number, Evolution 17, 279-288. KIMURA, M. AND CROW, J. F. 1963a. On the maximum avoidance of inbreeding, Genet. Res. 4, 399-415. KIMURA, M. AND WEISS, G. H. 1964. The stepping stone model of population structure and the decrease of genetic correlation with distance, Genetics 49, 561-576. KIMU~A, M. AND CROW, J. F. 1964. The number of alleles that can be maintained in a finite population, Genetics 49, 725-738. KIMURA, M. AND OHTA, T. 1969. The average number of generations until fixation of a mutant gene in a finite population, Genetics 61, 763-771. KING, J. L. AND JUKES, T. H. 1969. Non-Darwinian evolution, Science 164, 788-798. LEWONTIN, R. C. AND HUBBY, J. L. 1966. A molecular approach to the study of genetic heterozygosity in natural populations:II. Amountof variation and degree of heterozygosity in natural populations of Drosophila pseudoobscura, Genetks 54, 595-609. MAtiCOT, G. 1948. “Les Mathematiques de l’HCr&lite,” Masson, Paris.

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ALLELES

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