On models of quantitative genetic variability: a stabilizing selection-balance model.

On Models of Quantitative Genetic Variability:

A Stabilizing Selection-Balance Model

Lev A. Zhivotovsky* and Marcus W. Feldman-f-”

*Institute of General Genetics, Russian Academy of Sciences, Moscow B-333, Russia, and ?Department of Biological Sciences, Stanford University, Stanford, Calgornia 94305

Manuscript received May 20, 1991 Accepted for publication January 2, 1992

ABSTRACT

A model of stabilizing selection on a multilocus character is proposed that allows the maintenance of stable allelic polymorphism and linkage disequilibrium. The model is a generalization of Lerner’s model of homeostasis in which heterozygotes are less susceptible to environmental variation and hence are superior to homozygotes under phenotypic stabilizing selection. The analysis is carried out for weak selection with a quadratic-deviation model for the stabilizing selection. The stationary state is characterized by unequal allele frequencies, unequal proportions of complementary gametes, and a reduction of the genetic (and phenotypic) variance by the linkage disequilibrium. The model is compared with Mather’s polygenic balance theory, with models that include rnutation-selection balance, and others that have been proposed to study ” the role of linkage disequilibrium in quantitative inheritance.

T

HE population genetic description of quantita- tive variation in natural populations has long been of major interest to natural historians (see, e.g., FUTUYMA 1986, ch.

7).

Besides the development of an appropriate statistical framework to discuss such variation, the questions of its maintenance and change under natural selection have received much attention in the theoretical literature (e.g., BARTON and TUR-

FISHER (1 9 18) proposed a variance decomposition for a quantitative trait determined by a set of

poly-

morphic loci in the absence of selection. There have been a number of extensions to the original theory that involve assumptions about the absence of the linkage disequilibrium and interactions between com- ponents of the variation. One of them has the form (COCKERHAM 1954; KEMPTHORNE 1957)

ELL1 1989).

v

VA

+

VI

+

VE (1) where V is the overall phenotypic variance, VA is the additive genetic variance, and VI is a component due t o interactions between alleles and among genes. V E is a measure of (nontransmitted) environmental vari- ance. T h e genotypic components VA and VI are deter-

mined by a set of constant parameters as well as the frequencies of alleles [see FALCONER (1989) for the details]. If, as if often assumed to be the case, there is stabilizing selection on the trait under study, then it becomes important to understand how allelic and genotypic frequencies change in response to the selec-

’

To whom reprint requests should be addressed. Genetics 130: 947-955 (April, 1992)

tion on the phenotype, and how these changes affect the decomposition (1).

In one of the earliest attempts to address genotypic response to phenotypic selection, MATHER (1 942) de- veloped a model of polygenic balance. According to this model “polygenes” are distributed along chro- mosomes in such a way that the signs of their contri- butions to the phenotype alternate. Thus, polygenes with larger (+) and smaller (-) effects on the trait follow each other in the sequence

+-+-,

etc. MATH-

ER’S model also supposes the presence of complemen-

tary pairs of chromosomes whose products produce the genotypic balance: +-+-a

-

./-+-+e

-

.

This po-

lygenic balance may be invoked to explain the greater fitness under stabilizing selection of individuals with such complementary pairs of chromosomes relative to those in which configurations like

----.

e . or

++++

-

-

predominate. Obviously, MATHER’S model allows production of less balanced polygene combi- nations by recombination, but in his view the balanced complementary pairs are the primary contributors to quantitative variability under stabilizing selection.

Although MATHER’S is an attractive hypothesis it has not been supported by experimental data. THO-

DAY (1961) and THOMPSON, HELLACK and TUCKER

L. A.

rangements among the genes contributing to a quan- titative trait remains a central evolutionary issue.

In the terminology of population genetics, MATH-

ER’S idea involves the presence of linkage disequili-

brium, since it involves complementary pairs of ga- metes. MATHER’S treatment did not lead to an explicit computation of linkage disequilibrium produced by

selection. It is well known, however, that selection can lead to stable linkage disequilibrium (LEWONTIN and KOJIMA 1960; BODMER and FELSENSTEIN 1967; KAR-

LIN and FELDMAN 1970; FRANKLIN and LEWONTIN 1970; FELDMAN, FRANKLIN and THOMPSON 1974). BULMER (1974, 1980) studied model (1) under stabi- lizing selection and concluded that it results in linkage disequilibrium. T h e arguments were not, however, structured as formal population genetic models. In particular he used a completely additive variant of model (1) that actually has no stable polymorphic state. LANDE (1 975) and TURELLI and BARTON (1990) recognized this instability and used a mutation-selec- tion balance to maintain polymorphic variation. Within this framework linkage disequilibrium is not an important contributor to the ultimate distribution of the quantitative character.

Stable polymorphic states may exist under epistatic selection and may exhibit linkage disequilibrium (see references above). It may therefore be the case that alternative models of stabilizing selection may be ap- propriate, and exhibit stable polymorphic equilibria with linkage disequilibrium. Some models of this kind have been discussed by ZHIVOTOVSKY and GAVRILETS (1992) and in this paper we return to these models. T h e nature of the stable polymorphic states that we are able to study, under stabilizing selection, suggests that MATHER’S polygenic balance is not a tenable hypothesis and that linkage disequilibrium may be important to the variation of quantitative characters at the stationary state under stabilizing selection.

T H E MODEL

Consider a quantitative character affected by n au- tosomal loci each with two alleles which we denote by A, and ai. T h e contribution of the ith locus to the character is measured by a weight a; (a;

>

0). T h e relative contributions of the alleles A; and ai at the ith locus to the character are 1 and 0 respectively. Let G = (II, 12,

. .

.

, In) denote a gamete where l i are

indicator variables such that 1, = 1 if the gamete contains Ai and li = 0 if the gamete contains a, at the ith position. If G = (1 {, 1 4 ,

.

. .

, 1:) is a second gamete the diploid genotype is then denoted by GG I . The

phenotypic value of the character in an individual whose genotype is GG is computed additively as

n

x = a.

+

a;(li

+

11)

+

e (2)

where e is an environmental deviation, namely a ran-

i= 1

dom variable with null expectation and variance V , that can depend on genotype. a0 is a constant that represents some baseline phenotypic value, and can be of any sign, although, by definition ai

>

0

(i = 1, 2,

.

. .

, n). Define

pi

= %(&I,

4;

= 1

-

pi

(3)

and

D . . = 8 _p[(li

_-

_Pi><lj

_-

_PJI,

where

5

is an expectation operator, so that

pi

is the frequency of A, and D , is the linkage disequilibrium between the ith and jth loci. Let denote the expec- tation with respect to the distribution of e. Under random union of gametes the mean of the phenotype defined by (2) is

n

2 = a0

+

2 a& (4)

I= 1

and its variance, V = gFe[(x

-

i)‘], satisfies

v

= V A

+

C L

+

V E (5)

where

v,

= 2 a:piq; (6)

i

is an additive variance.

CL = 2 CY;CY,D~ (7)

i#j

is an additional term due to linkage disequilibrium and VE = q ( V , ) . T h e expansion (5) can be found in WEIR, COCKERHAM and REYNOLDS (1980) and the notation C L in

(7)

is originally due to BULMER (1980). A proof of the decomposition (5) is given in APPEN-

Selection occurs on the phenotype according to a

DIX 1.

quadratic stabilizing regime

w(x) = 1

-

s(x

-

e)‘

(8)

where 0 is an optimal phenotype at which fitness is maximized, T h e parameter s 3 0 is a measure of the intensity of the selection, the greater is s the stronger is the selection towards

e.

It is well known that the allele frequencies (3) under the specification (2) and (8), with constant environmental variance V,, cannot achieve a stable polymorphic equilibrium (WRIGHT

1969; LEWONTIN 1964). Generalization of the above framework may allow stable multilocus polymorphism under the selection regime (8). Here we consider the case where the environmental deviation depends on the genotype so that

n

v,

= Eo

-

I:

pi(&

+

I (

-

2lil(), (9)

where

Pi

are constants. This is a generalization of

LERNER'S model of homeostasis in which the greater is the heterozygosity the smaller is the variance (LER- NER 1954). T h e scheme (9) differs from LERNER'S by

our inclusion of unequal contributions from different loci. It also extends the model of ZHIVOTOVSKY and GAVRILETS (1 992) to the case where allele effects are not all equal. T h e constant Eo is the variance corre- sponding to the state where all n loci are homozygous, and for V, to be positive for every genotype we require Eo

>

E:!,

Pi. For future reference write Vp =

Substitution of

(2)

into (8) yields the fitness of the 2CLlPipiqi.

genotype GG

w(GG') = p

+

[Zi(Zi

+

I()

+

2&ZiZ(]

i

(10)

+

C

kb{Z,

+

ll)(lj

+

,

)

!

Z

i#j j

where, as shown in APPENDIX 2,

T h e general dynamical system that determines the evolution of the gamete frequencies with fitnesses specified by (1 0 ) is extremely cumbersome to analyze. Using perturbation techniques, however, it is possible to determine the stationary states for small values of

s. These have the form

pi =

pp

+

sp:

+

s'p:

+

*

-

(1 2a)

D..

= Do

+

&'.

+

$2"'

+

...

.

(12b)

'I 'I

'I 'I

Of course we seek solutions with 0

<pp

<

1 (i = 1, 2,

.

. .

, n ) .

The properties of the perturbation in (1 2a, b) were demonstrated by ZHIVOTOVSKY and GAVRILETS

(1992, Equations 16c, 18, 19). They showed that

(pp),

{ p i ) ,

ID$)

and (06) satisfy

Ki

+

2Lipp

+

4 Me?? = 0 (1 3)

j#i

where rq # 0 is the recombination fraction between loci i and j . We shall restrict our attention to terms in (1 2) of first order in s. In APPENDIX 3 we show that on

substitution of (10) with (1 1) into (13), (14) and (15) the following versions of (1 2a) and ( 1 2b) are valid

Here x, = a.

+

ai is the mean value of the quanti- tative character when

pi

= 1/2 (all

i).

Thus f3

-

x, represents the deviation of the optimum value from the midpoint of the phenotypic range. T h e values ' P I ,

7,, +i are positive values which are cumbersome to

write down and are recorded in APPENDIX 3.

STABILITY OF ALLELE FREQUENCIES

It was shown by ZHIVOTOVSKY and GAVRILETS

(1992) that for small values of

k;,

i;

and in the fitness expression (1 0) the polymorphic equilibrium specified by (1 6) and (1 7) is stable under the assump- tion

i

q

= yaiaj provided that

ij

<

2ya? ( y

<

0).

Applying this result to our parameters (1 1) the con- dition for stability is

pi

> a

' (i = 1, 2,

. .

.

, n). (1 8)

At the stable polymorphism there is linkage disequi- librium and, returning to (5) and (7) its contribution to the variance V is CL with

DISCUSSION

T h e model defined by (2) and (8) with (9) is a possible alternative to existing models of quantitative variability. T h e stable polymorphic equilibrium spec- ified by (16) and (1 7) has a number of interesting qualitative properties which we document below.

Property 1: At the stable state, irrespective of the values of the contributions of the different loci to the phenotypic value, the frequencies of alleles with a positive effect on a character under weak stabilizing selection are all on the same side of 1/2. That is, if

ai

>

0 for all i, then all pi are greater than 1/2 or all

are less than 1/2. In fact, for sufficiently small selec- tion

if f3

>

xm then every

pi

>

0.5

if f3

<

x,,, then every

p i

<

0.5.

and M.W.

in a population subject to the constraints of the model there should be a preponderance of gametes (or chro- mosomes) with mainly A-alleles, or a-alleles, according to whether B

>

x, or B

<

xm, respectively. Also from

(1 6) we have

Property

2: At the polymorphic equilibrium the allele frequencies

pi

are, in general, different from one another. Indeed from (1 6) we see that

pi

#

p,

for each s if a,/(&

-

a?) # a,/(@,

-

a;). Thus, differences

in the contributions from the different loci to the character under selection result in different allele frequencies. Even if all such contributions are equal, however, allele frequencies may differ one from the other if the loci are linked. Indeed, set ai = a, pi =

0

and take n very large. Then in APPENDIX 4 we show that

where Bo =

[(e

-

ao)/an]

-

1 is a relative measure of the deviation of the optimal value B from the midpoint x,, with BO = 0 if 8 = x,; Bo = -1 when B = ao, the minimum value of x ; Bo = +1 when B = a.

+

2na, the maximum value of x. Here

are harmonic averages of the recombination fractions. Thus, in our model the Pi's are all equal if and only if the loci contribute equally to the phenotype and all the loci recombine freely.

Property 1 has an obvious but interesting corollary. Suppose for example that B

>

x, so that every

p i

is larger than 1/2. Then the majority of the gametes carry more Ai-alleles (+ alleles in MATHER'S terminol- ogy, i.e. alleles that increase the value of the trait) than ai-alleles. Thus the proportions of complemen- tary gametes are expected to be unequal.

Property 3: T h e greater the deviation of the opti- mal value, 0, from the midpoint of the character range

x,, the smaller is the frequency of the complementary gametes. In order to demonstrate this phenomenon we make the simplifying assumptions that all loci contribute equally to the trait, that allele frequencies are equal and that there is linkage equilibrium. That is terms order s are neglected from the allele frequen- cies. These assumptions do not qualitatively alter the validity of Property 3. Let pi =

p * ,

for all i with

1/2

<

p .

<

1. Denote gametes carrying k alleles of type A, by G ( k ) . Then the frequency Pk of G ( k ) is, neglecting terms O@),

where q. = 1

-

p..

Pk achieves its maximum at

k

= np* =

K.,

say. Thus the relative frequencies of those gametes carrying close to

k*

alleles of type Ai may be substantially greater than those of their complemen- tary gametes G(n

-

h). T o see this define X k = Pn-k/ P k as the ratio of the frequency of complementary gametes G(n

-

k ) to G(k). Then = X ~ k - " where XI =

p / p *

<

1, and

Pk

is the probability that the ratio takes the value X k ( k = 1, 2,

.

.

.

, n). Fix a small number 7. Then X k

<

7 if ( 2 k

-

n)ln X* G In 7, i.e., if

k 2

(

n + - / Z = n , ,

l

?

)

say. Then the probability that Xk

<

7 is the probability that k 2 no, say H,:

n

n,

=

2

Pk. k=n,

For example in the case

p t

= 0.7, n = 12, In X* =

-0.85 so that if 7 = 0.1 then n, = 7.3 and

no,,

>

0.72.

If 17 = 0.2, then = 0.88. Hence the ratio of the

frequencies of complementary gametes is less than 0.1

(0.2) with probability 0.72 (0.88).

An alternative way to view the skewness of this distribution is to observe that for k = np., where Pk achieves its maximum, the value of Xk is X:(2f'*") =

X , , ,

, say. For the same example

pt

= 0.7, n = 12, ,,X,

= 0.017, so that the modal expected value for the

ratio of complementary gamete frequencies G(n

-

K)

to G(K) is less than 2%.

Finally, observe that at the stationary state (1 6) and (1 7) entail that the disequilibrium is negative to first order in s. In fact, the form of C t in (19) demonstrates

Property 4: At the stable polymorphic equilibrium both genotypic and phenotypic variance are less than those expected in the absence of linkage equilibrium. This follows because Vc

+

V,

+

CL and V = VC

+

VE.

In (1 9) CL is clearly negative.

T h e properties described above distinguish our treatment from earlier studies. Consider first the polygenic balance model of MATHER (1942, 1943,

1973), MATHER and JINKS (1982). Although this was

not expressed in terms of an evolutionary dynamical system, its main qualitative properties can be deduced. Implicit in MATHER'S discussion is the supposition of equal frequencies for complementary gametes, which differs from Property 3 above. Further, MATHER sup- poses a genotypic structure of the type

+-+-.

.

./

-+-+.

.

.

which entails negative linkage disequili- brium between loci 1 and 2 and between loci 2 and 3, but positive linkage disequilibrium between loci 1 and 3 and

2

and 4, etc. Clearly this differs from Property 4 of our model which affirms the negative sign of the linkage disequilibrium.

on normal distribution theory and supposes linear regression equations that connect the genotypic values of relatives. T h e main qualitative property of BUL-

MER’S treatment is that it results in negative linkage

disequilibrium which reduces genotypic and pheno- typic variance. This coincides with our Property 4. T h e other properties 1, 2, and 3, however, do not appear to be shared with BULMER’S model which was designed primarily to describe transient effects of selection on genetic variation.

LATTER (1960), KIMURA (1965), BULMER (1972), LANDE (1 975), CAVALLI-SFORZA and FELDMAN (1 976), FEUENSTEIN (1 977), FLEMING (1 979), TURELLI

(1 984), BARTON (1 986), TURELLI and BARTON (1 990), KEIGHTLEY and HILL (1 990), ZHIVOTOVSKY and GAV-

RILETS (1 990) all examine versions of a model in which

a stable equilibrium of allele frequencies is maintained

by a balance between mutations that produce varia- tion and stabilizing selection against it. This mutation- selection balance model allows a quantitative trait to maintain variability. These models generally result in negative linkage disequilibrium and thus share our Property 4. Property 2, that allele frequencies differ across loci (even when all loci have the same additive- effect parameters) is shared in the results of LANDE (1 975) and TURELLI and BARTON (1 990). Our prop- erties 1 and 3, however, do not appear to emerge from these models.

T h e class of models that produce stable polymor- phism under stabilizing selection might be called sta- bilizing selection-balance models to emphasize that mu- tation is not required for the maintenance of the polymorphism. T h e multilocus “optimum models” in- vestigated numerically by LEWONTIN (1964) had ad- ditive contributions across the loci and stabilizing se- lection on the resulting phenotype, and belonged to our class when they were homeostatic. As in our models, this occurred when the environmental vari- ance associated with genotypes decreased as the num- ber of heterozygous loci increased. GILLESPIE (1 984) studied the stabilizing selection balance produced by the quadratic selection function (8) with the added assumption that if a locus is homozygous, the fitness decreases by a fixed amount. This “pleiotropic over- dominance” allowed quantitative genetic variability to be maintained at equilibrium, but the analysis was made under the assumption of linkage equilibrium. T h e model of GILLESPIE and TURELLI ( I 989) included a genotype-dependent component of environmental effects on the phenotype. Whereas we write in Equa- tion 2 that V, is specified by (9), GILLESPIE and TUR-

ELLI take the genotype-dependent part of e to be due to random effects contributed by each allele and summed over the loci. T h e relationship among these random effects is assumed to be extremely symmetric, specified by a within-locus correlation and a between-

locus correlation. T o this extent our model can be viewed as an extension of GILLESPIE and TURELLI’S, and our results are qualitatively in agreement. We have kept track here of the contribution of linkage equilibrium to the phenotypic variance, and this is not made explicit by GILLESPIE and TURELLI.

It is worth reiterating the point made by GILLESPIE and TURELLI that the conditions under which geno- type-dependent environmental interactions may allow genetic variation are not as restrictive as claimed by VIA and LANDE (1987). T h e extreme symmetry as- sumed by GILLESPIE and TURELLI in their analytic treatment is not necessary; as we have seen here their finding holds in considerable generality.

T h e structure of our analysis permits us to obtain an upper bound for the heritability of trait described by (6) and (9). We are grateful to a reviewer for pointing out that since Eo >

Pi

(following Equation 9 above), and piqi

<

1/4. Also, by (18),

Pi

>

a: and we deduce that the heritability h2 satisfies

Since the contribution of linkage disequilibrium is negative by (19), the heritability will be less than 1/2. Our model might be applicable for higher levels of heritability, because extremely heterozygous individ- uals are likely to be very rare, and the maximal her- itability may approach unity as n becomes large.

The authors are grateful to MICHAEL TURELLI and two anony- mous reviewers for their substantial suggestions that significantly improved the manuscript. This research was supported in part by National Institutes of Health grants GM 28016 and 10452 and a grant from the MacArthur Foundation.

L I T E R A T U R E C I T E D

BARTON, N. H., 1986 The effect of linkage and density-dependent regulation on gene flow. Hereditary 57: 415-426.

BARTON, N. H., and M. TURELLI, 1989 Evolutionary quantitative genetics: how little do we know? Annu. Rev. Genet. 23: 337- 370.

BODMER, W. F., and J. FELSENSTEIN, 1967 Linkage and selection: theoretical analysis of the deterministic two locus random mat- ing model. Genetics 57: 237-265.

BULMER, M. G., 1974 Linkage disequilibrium and genetic varia- bility. Genet. Res. 23: 281-289.

BULMER, M. G . , 1980 The Mathematical Theory of Quantitative Genetics. Claredon Press, Oxford.

CAVALLI-SFORZA, L. L., and M. W. FELDMAN, 1976 Evolution of continuous variation: direct approach through joint distribu- tion of genotypes and phenotypes. Proc. Natl. Acad. Sci. USA

73: 1689-1692.

COCKERHAM, C. C., 1954 An extension of the concept of parti- tioning hereditary variance for analysis of covariances among relatives when epistasis is present. Genetics 3 9 859-882.

FALCONER, D. S., 1989 Introduction to Quantitative Genetics, Ed. 3.

Longman, London.

FELSENSTEIN, J., 1977 Multivariate normal genetic models with a finite number of loci, pp. 227-245 in Proceedings of the Inter- national Conference on Quantitative Genetics, edited by E. POL- LACK, O . KEMPTHORNE and T. B. BAILEY JR. Iowa State Uni- versity Press, Ames.

FISHER, R. A , , 1918 The correlation between relatives on the supposition of Mendelian inheritance. Trans. R. SOC. Edinb.

FLEMING, W. H., 1970 Equilibrium distributions of continuous polygenic traits. SIAM J. Appl. Math. 3 6 148-168.

FRANKLIN, I., and R. C. LEWONTIN, 1970 Is the gene the unit of selection? Genetics 65: 701-734.

FUTUYMA, D. A., 1986 Evolutionary Biology. Sinauer Associates, Sunderland, Mass.

GILLESPIE, J. H . , 1984 Pleiotropic overdominance and the main- tenance of genetic variation in polygenic characters. Genetics 107: 321-330.

GILLESPIE, J. H., and M. TURELLI, 1989 Genotype-environment interactions and the maintenance of polygenic variation. Ge- netics 121: 129-138.

KARLIN, S., and M. W. FELDMAN, 1970 Linkage and selection: two-locus symmetric viability model. Theor. Popul. Biol. 1: 39- 71.

KEIGHTLEY, P. D., and W . G. HILL, 1990 Variation maintained in quantitative traits with mutation-selection balance: pleio- tropic side-effect on fitness trait. Proc. R. SOC. Lond. Ser. B.

KEMPTHORNE, O . , 1957 A n Introduction to Genetic Statistics. John Wiley, New York.

KIMURA, M., 1965 A stochastic model concerning the mainte- nance of genetic variability in quantitative characters. Proc. Natl. Acad. Sci. USA 54: 731-736.

LANDE, R., 1975 The maintenance of genetic variability by mu- tation in a polygenic character with linked loci. Genet. Res. 2 6

221-235.

LATTER, B. D. H., 1960 Natural selection for an intermediate

LERNER, I. M., 1954 Genetic Homeostasis. John Wiley, New York. LEWONTIN, R. C., 1964 The interaction of selection and linkage.

LEWONTIN, R. C., and K. KOJIMA, 1960 The evolutionary dynam-

MATHER, K., 1942 The balance of polygenic combinations. J.

MATHER, K., 1943 Polygenic inheritance and natural selection.

MATHER, K., 1973 The Genetical Structure of Populations. Chapman MATHER, K . , and J. L. JINKS, 1982 Biometrical Genetics, Ed. 3.

THODAY, J. M., 1961 Location of polygenes. Nature 191: 368-

THOMPSON, J. N., J R . , J. J. HELLACK and R. R. TUCKER

199 1 Evidence for balanced linkage of X chromosome poly- genes in a natural population of Drosophila. Genetics 127:

TURELLI, M., 1984 Heritable genetic variation via mutation-selec- tion balance: Lersh's zeta meets the abdominal bristle. Theor. Popul. Biol. 25: 138-193.

TURELLI, M., and N. H. BARTON. 1990 Dynamics of polygenic characters under selection. Theor. Popul. Biol. 38: 1-57.

VIA, S., and R. LANDE, 1987 Evolution of genetic variability in a spacially heterogeneous environment: effects of genotype-en- vironment interactions. Genet. Res. 4 9 147-156.

WEIR, B. S., C. C. COCKERHAM and J. REYNOLDS, 1980 The effects of linkage and linkage disequilibrium on the covariances of noninbred relatives. Heredity 45: 35 1-359.

52: 399-433.

242: 95-100.

optimum. Aust. J. Biol. Sci. 13: 30-35.

11. Optimum models. Genetics 50: 757-782.

ics of complex polymorphisms. Evolution 1 4 458-472.

Genet. 43: 309-336.

Biol. Rev. 18: 32-64.

1L Hall, London.

Cambridge University Press, Cambridge.

370.

117-123.

WRIGHT, S., 1969 Evolution and Genetics of Populations, Vol. 2.

University of Chicago Press, Chicago.

ZHIVOTOVSKY, L. A., and S. Yu. GAVRILETS, 1990 Stabilizing selection and linkage disequilibrium (English translation). Sov. Genet. 2 6 N2 222-23 1 .

ZHIVOTOVSKY, L. A., and S. GAVRILETS, 1992 Quantitative vari- ability and multilocus polymorphism under epistatic selection. Theor. Popul. Biol. (in press).

Communicating editor: B. S. WEIR

APPENDIX 1

Define the expectations

8

and

&

with respect to the distribution of e and (1;) respectively. Then the variance

v

= gpe(x

-

Incorporating (A3) into (A2) we obtain frequency distribution, namely

/ \ 2

=

2

' Y ~ c x ~ D ~

+

2

x

a:f+qi

+

V,

iZj i

= V A

+

C L

+

q)'PlVJ

with V A and CL defined in (6) and

(7)

APPENDIX 2

Fitness with LERNER'S homeostasis: Combining

(2),

(8) and (9) we have

w(x) = 1

-

s[ao

-

8

+

ai(li

+

I:)

+

e]' (A5)

I

where e has zero expectation and variance

V, = Eo

-

2

&(Zi

+

I:

-

2 1 i Z ( ) .

i

Expanding,

~ ( x ) = I

-

s

i

(ao

-

e)'

+

e*

+

Pe(a0

-

e)

+

2(ao

-

e)

(Yi(l1 + I ( )

+

2e

2

ai(&

+

II)

1

i

Taking expectations with respect to the distribution of e we have

which proves (10) with (1 la,

b,

c).

Under the assumption of random union of gametes now take expectations with respect to the population

On the other hand, WRIGHT (1 969,

p.

106) expressed the mean fitness as

zs

= 1

-

S[(?

-

0)2

+

VJ,

which entails, in our terminology,

5 = 1

-

S[(z

-

-k VA

+

c~

+

VE] (A10)

where

X,

VA and CL are defined by (4), (6) and

(7)

and VE = EO

-

2

APPENDIX 3

with ( 1 1 b) can be rewritten as

Ki =pi

-

CY?

+

2(0

-

( ~ o ) ( ~ i = 2(pi

+

a?)pP

+

4 aiajpj

= 2(pi

-

a?)pP

+

4ai a,pj

= 2(p

-

af)pP

+

4 ap0

j#i

and

where

with

vo = i

1

+

s a f ( p i

-

a?)

i

= A/2(1

+

VI),

=

x

a:

+

(e

-

ao)Q1

i

Q1 =

2

a'/(p;

-

a?).

i

Returning to (A1 1 ) we have therefore

1

(e

-

a0)ai ai A

p+-+

"

2

pi

-

'a

pi

-

a; ( 1

+

91)

1 B;

= - + -

2 pi

-

ai 2 '

where Bi = ai[O

-

a.

-

A/(1

+

V 1 ) ] = a,Bo, say. Using ( 1 1 c) write ( 1 4) as

Then, returning to (15) with ( 1

IC)

we have

which can be rewritten, using (A 1 6) and T' = &ajp,!, as

.?p

oq 0 [ ( p i

-

af)p! + 2aiT'I = (9:

-

P;)af2 I ,

j # i rq

since PPqP # 0. Hence, since 9;-

pp=

-2Bi/(pi

-

a3,

where Ti = 2 I;#iajpjqj/rq. Now multiply (A 18) by ai and sum to produce

T' = N/(1

+

( 0 1 ) (A1 9)

where N = -2 C,B,aST,/(&

-

a:)'. Thus the complete solution (A1 8) may be expanded as

2afBiTi

-

( p i

-

a y

Define 7, = afTi/(Pi

-

a'). Then the value

is an average of these 7's with weights 2aj/QI(pj

-

aj) whose sum is unity. Upon substitution of 7 , and Ti into ( A 18), using ( A 19) we find

Since A = cia,

+

(e

-

a0)Ql and Bo = 6 - (YO

-

A/(1

+

' P I ) we have

Now combine (A1 5 ) and (A 23 ) with ( 1 2a) to yield (16).

APPENDIX 4

Derivation of (20) from (16): Suppose that ai = a,

take 1

+

( 0 1

=

P I . Now with these assumptions the

weights of ri used to compute ? in (A21) reduce to

so that ? = CTi/n. Write

(see Equation 16)

= (ri

-

; ) / 2 a n

under the assumption made above. But ri =

a'/T,/(p

-

a'), where, since

pj'

=

P o

T; = 2

E,

a 2p j q j / r c 0 0 = 2(n

-

1)a2p0q0?;'.

J+l

That is

a2

p

-

a'

T i =

-

2(n

-

1)a2p0qoi;'

=

( 2 n a 2

p

0 0 q )

-

'a "I

p

-

*2

and

Finally

Define 8 0 according to the relation

DEF

6'

-

x, = 8

-

a.

-

n a

=

ando.

Then

DEF

where V i = 2na2poq0 and V j = 2n/3p0q0. Hence

Finally

so that