2

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 proportionsof complementary gametes, anda reduction of the genetic (and phenotypic) variance by the linkage disequilibrium. The model is compared withMather’s polygenic balance theory, withmodels thatinclude rnutation-selection balance, andothers that have been proposed to studythe role of linkage disequilibriumin quantitative ” inheritance.

HE population genetic description of quantita- tion on the phenotype, and how these changes affect T tive variation in natural populations has long the decomposition (1). been of major interest to natural historians (see, e.g., In oneof the earliest attempts to addressgenotypic FUTUYMA 1986, ch. 7). Besides the development of response to phenotypic selection, MATHER (1 942) de- an appropriate statistical framework to discuss such veloped a model of polygenic balance. According to variation, the questions of its maintenance and change thismodel “polygenes” are distributedalong chro- under natural selection have received much attention mosomes in such a way that the signs of their contri- in the theoretical literature (e.g., BARTONand TUR- butions to the phenotype alternate. Thus, polygenes ELL1 1989). with larger (+) and smaller (-) effects on the trait FISHER(1 9 18) proposed a variance decomposition follow each other in the sequence +-+-, etc. MATH- for a quantitative trait determined by a set of poly- ER’S model also supposes the presence of complemen- morphic loci in the absence of selection. There have tary pairs of chromosomes whose products produce been a number of extensions to the original theory the genotypic balance: +-+-a - ./-+-+e - . This po- that involve assumptions about the absence of the lygenic balance may be invoked to explain the greater linkage disequilibrium and interactionsbetween com- fitness under stabilizing selection of individuals with ponents of the variation. One of them has the form such complementary pairs of chromosomes relativeto (COCKERHAM1954; KEMPTHORNE1957) those in which configurations like ----. e. or v VA + VI + VE (1) ++++ - - predominate. Obviously, MATHER’Smodel allows production of less balanced polygene combi- where V is the overall phenotypic variance, VAis the nations by recombination, butin his view the balanced additive genetic variance, and VI is a component due complementary pairs are theprimary contributors to to interactions between alleles and amonggenes. VEis quantitative variability under stabilizing selection. a measureof (nontransmitted) environmental vari- Although MATHER’Sis an attractive hypothesis it ance. The genotypic componentsVA and VI are deter- has not been supported by experimental data. THO- mined by a set of constant parameters as well as the DAY (1961) and THOMPSON,HELLACK and TUCKER frequencies of alleles [see FALCONER(1989) for the (199 1) did not find a pattern of alternating contribu- details]. If, as if often assumed to be the case, there is tions, positive and negative to the phenotypes they stabilizing selection on the trait under study, then it studied. Further, THOMPSON,HELLACK andTUCKER becomes importantto understand how allelic and genotypic frequencies changein response to the selec- (1991) were unable to demonstrate the existence of complementary pairs of chromosomes in their study ’ To whom reprint requests should be addressed. population.Nevertheless, the kind of statistical ar-

Genetics 130: 947-955 (April, 1992) 948 L. A. Zhivotovsky and M.W. Feldman rangements among thegenes contributing to a quan- dom variable with null expectation and variance V, titative trait remains a central evolutionaryissue. that can depend on genotype. a0 is a constant that In the terminology of population genetics, MATH- represents some baseline phenotypic value, and ER’S idea involves the presence of linkage disequili- can be of any sign, although, by definition ai > 0 brium, since it involves complementary pairs of ga- (i = 1, 2, . . . , n). metes. MATHER’Streatment did not lead to an explicit Define computation of linkage disequilibrium produced by selection. It is well known, however,that selection can pi = %(&I,4; = 1 - pi lead to stable linkage disequilibrium (LEWONTINand (3) and KOJIMA1960; BODMERand FELSENSTEIN1967; KAR- = D.. 8p[(li Pi>mutation-selec- I= 1 tion balance to maintain polymorphicvariation. Within this framework linkage disequilibrium is not and its variance, V = gFe[(x - i)‘],satisfies an important contributor to theultimate distribution = VA CL VE (5) of the quantitative character. v + + Stable polymorphic states may exist under epistatic where selection and may exhibit linkage disequilibrium (see references above). It may therefore be the case that v, = 2 a:piq; (6) alternative models of stabilizing selection may be ap- i propriate, and exhibit stable polymorphic equilibria is an additive variance. with linkage disequilibrium. Some models of this kind have been discussed by ZHIVOTOVSKYand GAVRILETS CL = 2 CY;CY,D~ (7) (1992) and in this paper we return to these models. i#j The nature of the stable polymorphic states that we is an additional term due to linkage disequilibrium are able to study, under stabilizing selection, suggests and VE = q(V,). The expansion (5) can be found in that MATHER’S polygenic balance is nota tenable WEIR, COCKERHAMand REYNOLDS(1980) and the hypothesis andthat linkage disequilibrium may be notation CL in (7) is originally due to BULMER(1980). important to the variation of quantitative characters A proof of the decomposition (5) is given in APPEN- at the stationary state under stabilizing selection. DIX 1. THE MODEL Selection occurs on the phenotype according to a quadratic stabilizing regime Consider a quantitative character affected by n au- tosomal loci each with two alleles which we denote by w(x) = 1 - s(x - e)‘ (8) A, and ai. The contribution of the ith locus to the where 0 is an optimal phenotype at which fitness is character is measured by a weight a; (a;> 0). The maximized, The parameter s 0 is a measure of the relative contributions of the alleles A; and ai at the ith 3 intensity of the selection, the greater is s the stronger locus to the character are 1 and 0 respectively. Let is the selection towards e. It is well known that the G = (II, 12, . . , In) denote a gamete where li are . allele frequencies (3) under the specification (2) and indicator variables such that 1, = 1 if thegamete (8), with constant environmental variance V,, cannot contains Ai and li = 0 if the gamete containsa, at the achieve a stable polymorphic equilibrium (WRIGHT ith position. If G = (1 {, 14, . . , 1:) is a second gamete . 1969; LEWONTIN1964). Generalization of the above the diploid genotype is then denoted by GG I. The phenotypic valueof the character in an individual framework may allow stable multilocus polymorphism under the selection regime (8). Here we consider the whose genotype is GG is computed additively as case where the environmental deviation depends on n the genotype so that x = a. + a;(li + 11) + e (2) i= 1 n v, = Eo - I: pi(& + I( - 2lil(), (9) where e is an environmental deviation, namely a ran- i= 1 Stabilizing Selection-Balance Stabilizing 949 where Pi are constants. This is ageneralization of substitution of (10) with (1 1) into(13), (14) and (15) LERNER'Smodel of homeostasis in which the greater the following versions of (12a) and (1 2b) are valid is the heterozygosity the smaller is the variance (LER- NER 1954). The scheme (9) differs from LERNER'Sby our inclusion of unequal contributions from different loci. It also extends the model of ZHIVOTOVSKYand GAVRILETS(1 992) to thecase where allele effects are not all equal. The constant Eo is the variance corre- sponding to thestate whereall n loci are homozygous, and for V, to be positive for every genotype we require Eo > E:!, Pi. For futurereference write Vp = 2CLlPipiqi. Here x, = a. + ai is the mean value of the quanti- Substitution of (2) into (8) yields the fitness of the tative character when pi = 1/2 (all i). Thus f3 - x, genotype GG represents the deviation of the optimum value from the midpoint of the phenotypic range. The values 'PI, w(GG') = p + [Zi(Zi+ I() + 2&ZiZ(] 7,, +i are positive values which are cumbersometo i (10) write down and are recorded in APPENDIX 3. + C kb{Z, + ll)(lj + Z!), i#j j STABILITY OF ALLELE FREQUENCIES where, as shown in APPENDIX 2,

It was shown by ZHIVOTOVSKY and GAVRILETS (1992)that for small values of k;,i; and in the fitness expression (10) the polymorphic equilibrium specified by (16) and (1 7) is stable under the assump- tion iq = yaiaj provided that ij< 2ya? (y < 0). The general dynamical system thatdetermines the Applying this result to our parameters (1 1) the con- evolution of thegamete frequencies with fitnesses dition for stability is specified by (10) is extremely cumbersome to analyze. pi > a' (i = 1, 2, . . . , n). (18) Using perturbation techniques, however, it is possible to determine the stationary states for small values of At the stable polymorphism there is linkage disequi- s. These have the form librium and, returning to (5) and (7) its contribution to the variance V is CLwith pi = pp + sp: + s'p: + * - (12a)

= Do + + $2"' (12b) 'I 'I 'I 'I 'I 'I D..'I &'. + ... . Of course we seek solutions with 0 0 for all i, then all piare 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 pi < 0.5. where rq # 0 is the recombination fraction between loci i and j. We shall restrict our attention to termsin This conclusion concerningthe allele frequencies (1 2) of first order in s. In APPENDIX 3 we show that on (which is a direct consequence of (16)) suggests that 950 L. A. Zhivotovsky and M.W. Feldman in a population subject to theconstraints of the model where q. = 1 - p.. Pk achieves its maximum at k = there should be a preponderanceof gametes (or chro- np* = K., say. Thus the relative frequencies of those mosomes) with mainly A-alleles,or a-alleles, according gametes carrying close to k* alleles of type Ai may be to whether B > x, or B < xm, respectively. Also from substantially greater than those of their complemen- (1 6) we have tary gametes G(n - h). To see this define Xk = Pn-k/ Property 2: At the polymorphic equilibrium the Pk as the ratio of the frequency of complementary allele frequencies pi are, in general, different from gametes G(n - k) to G(k). Then = X~k-" where XI = one another. Indeed from(1 6) we see that pi # p, for p/p* < 1, and Pkis the probability that the ratiotakes each s if a,/(&- a?) # a,/(@, - a;). Thus, differences the value Xk (k = 1, 2, . . . , n). Fix a small number 7. in the contributionsfrom thedifferent loci to the Then Xk < 7 if (2k - n)ln X* G In 7,i.e., if characterunder selection result in different allele frequencies. Even if all such contributions are equal, k2 n+-/Z=n,, however, allele frequencies may differ one from the ( l?) other if the loci are linked. Indeed, set ai = a,pi = 0 and take n very large. Then in APPENDIX 4 we show say. Then theprobability that Xk < 7 is the probability that that k 2 no, say H,:

n n, = 2 Pk. k=n, where Bo = [(e - ao)/an] - 1 is a relative measure of For example in the case pt = 0.7, n = 12, In X* = the deviation of the optimal value B from the midpoint -0.85 so that if 7 = 0.1 then n, = 7.3 and no,,> 0.72. x,, with BO = 0 if 8 = x,; Bo = -1 when B = ao, the If 17 = 0.2, then = 0.88. Hence theratio of the minimum value of x; Bo = +1 when B = a. + 2na, the frequencies of complementary gametesis less than 0.1 maximum value of x. Here (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,,,, X,,,, say. For the same example pt = 0.7, n = 12, X,,, = 0.017, so that the modal expected value for the are harmonic averages of the recombination fractions. ratio of complementary gamete frequencies G(n - K) Thus, in our model the Pi's are all equal if and only if to G(K)is less than 2%. the loci contribute equally to the phenotype and all Finally, observe that at thestationary state (16) and the loci recombine freely. (1 7) entail that the disequilibrium is negative to first Property 1 has an obvious but interesting corollary. order in s. In fact, the form of Ct in (19) demonstrates Suppose for example that B > x, so that every pi is Property 4: At the stable polymorphic equilibrium larger than 1/2. Then the majority of the gametes both genotypic and phenotypic variance are less than those expected in the absence of linkage equilibrium. carry more Ai-alleles (+ alleles in MATHER'S terminol- ogy, i.e. alleles that increase the value of the trait) This follows because Vc + V, + CL and V = VC + VE. than ai-alleles. Thus the proportions of complemen- In (19) CLis clearly negative. tary gametes are expected to be unequal. The propertiesdescribed above distinguish our treatmentfrom earlier studies. Consider first the Property 3: The greater the deviation of the opti- mal value, 0, from the midpointof the character range polygenic balance model of MATHER (1942, 1943, x,, the smaller is the frequency of the complementary 1973), MATHERand JINKS (1982). Although this was gametes. In order to demonstrate this phenomenon not expressed in terms of an evolutionary dynamical we make the simplifying assumptions that allloci system, its main qualitative propertiescan be deduced. contribute equally to the trait, that allele frequencies Implicit in MATHER'Sdiscussion is the supposition of are equal and that there is linkage equilibrium. That equal frequencies for complementary gametes, which is terms order s are neglected from the allele frequen- differs from Property 3 above. Further, MATHERsup- cies. These assumptions do not qualitatively alter the poses a genotypic structure of the type +-+-. . ./ -+-+. . . which entails negative linkage disequili- validity of Property 3. Let pi = p*, for all i with 1/2 < p. < 1. Denote gametes carrying k alleles of brium between loci 1 and 2 and between loci 2 and 3, type A,by G(k). Then the frequency Pk of G(k) is, but positive linkage disequilibrium between loci 1 and neglecting terms O@), 3 and 2 and 4, etc. Clearly this differs from Property 4 of our model which affirms the negative sign of the linkage disequilibrium. The model of BWLMER (1974, 1976, 1980) is based Stabilizing Selection-Balance 95 1 on normaldistribution theory and supposes linear locus correlation. To this extent our model can be regression equationsthat connect the genotypic values viewed as an extension of GILLESPIEand TURELLI’S, of relatives. The main qualitative property of BUL- and our results are qualitatively in agreement. We MER’S treatment is that it results in negative linkage have kept track here of the contribution of linkage disequilibrium which reduces genotypic and pheno- equilibrium to thephenotypic variance, and this is not typic variance. This coincides with our Property 4. made explicit by GILLESPIEand TURELLI. The other properties 1, 2, and 3, however, do not It is worth reiterating the point madeby GILLESPIE appear to be shared with BULMER’Smodel which was and TURELLIthat the conditions under which geno- designed primarily to describetransient effects of type-dependent environmental interactionsmay allow selection on genetic variation. genetic variation are not as restrictive as claimed by LATTER(1960), KIMURA (1965), BULMER(1972), VIA and LANDE(1987). The extreme symmetry as- LANDE (1 975),CAVALLI-SFORZA and FELDMAN (1 976), sumed byGILLESPIE and TURELLIin their analytic FEUENSTEIN (1977), FLEMING (1979), TURELLI treatment is not necessary; as we have seen here their (1984), BARTON(1 986), TURELLIand BARTON(1 990), finding holds in considerable generality. KEIGHTLEYand HILL (1 990),ZHIVOTOVSKY and GAV- The structure of our analysis permits us to obtain RILETS (1 990)all examine versions of a model in which an upper bound for theheritability of trait described a stable equilibrium of allele frequencies is maintained by (6) and (9). We are grateful to a reviewer for by a balance between mutations that produce varia- pointing out that since Eo > Pi(following Equation tion and stabilizing selection against it. This mutation- 9 above), and piqi < 1/4. Also, by (18), Pi > a: and we selectionbalance model allows a quantitative trait to deduce that the heritability h2 satisfies 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- Since thecontribution of linkage disequilibrium is effect parameters) is shared in the results of LANDE negative by (19), the heritability will be less than 1/2. (1 975) andTURELLI and BARTON(1 990). Our prop- Our model might be applicable for higher levels of erties 1 and 3, however, do notappear to emerge heritability, because extremely heterozygous individ- from these models. uals are likely to be very rare, and the maximal her- The class of models that produce stable polymor- itability may approach unity as n becomes large. phism under stabilizing selection might be called sta- bilizing selection-balance models to emphasize that mu- The authors are grateful to MICHAELTURELLI and two anony- mous reviewers for their substantial suggestions that significantly tation is notrequired for the maintenance of the improved the manuscript. This research was supported in part by polymorphism. The multilocus “optimum models” in- National Institutes of Health grants GM 28016 and 10452 and a vestigated numerically by LEWONTIN(1964) had ad- grant from the MacArthur Foundation. ditive contributions across the loci and stabilizing se- lection on the resulting phenotype, and belonged to LITERATURECITED our class when they were homeostatic. Asin our BARTON,N. H., 1986 The effect of linkage and density-dependent models, this occurred when the environmental vari- regulation on gene flow. Hereditary 57: 415-426. ance associated with genotypes decreased as the num- BARTON,N. H., and M. TURELLI,1989 Evolutionary quantitative ber of heterozygous loci increased. GILLESPIE(1 984) genetics: how little do we know? Annu. Rev. Genet. 23: 337- 370. studied the stabilizing selection balance produced by BODMER,W. F., and J. FELSENSTEIN,1967 Linkage and selection: the quadratic selection function (8) with the added theoretical analysis of the deterministic two locus random mat- assumption that if a locus is homozygous, the fitness ing model. Genetics 57: 237-265. decreases by a fixed amount. This “pleiotropic over- BULMER,M. G., 1974 Linkage disequilibrium and genetic varia- bility. Genet. Res. 281-289. dominance” allowed quantitative genetic variability to 23: BULMER,M. G., 1980 TheMathematical Theory of Quantitative be maintained at equilibrium, but the analysis was Genetics. Claredon Press, Oxford. made under the assumption of linkage equilibrium. CAVALLI-SFORZA,L. L., and M. W. FELDMAN,1976 Evolution of The model of GILLESPIEand TURELLI(I 989) included continuous variation: direct approach through joint distribu- a genotype-dependent component of environmental tion of genotypes and phenotypes. Proc. Natl. Acad. Sci. USA 73: 1689-1692. effects on the phenotype. Whereas we write in Equa- COCKERHAM,C. C., 1954 An extension of the concept of parti- tion 2 that V, is specified by (9), GILLESPIEand TUR- tioning hereditary variance for analysis of covariances among ELLI take the genotype-dependent part of e to be due relatives when epistasis is present. Genetics 39 859-882. to random effects contributed by each allele and FALCONER,D. S., 1989 Introduction to Quantitative Genetics, Ed. 3. summed over the loci. The relationship among these Longman, London. FELDMAN,M. W., I. FRANKLINand G.J. THOMSON,1974 Selection random effects is assumed to be extremely symmetric, in complex genetic systems. I. The symmetric equilibria of the specified by a within-locus correlation and a between- three-locus symmetric viability model. Genetics 76 135-1 62. 952 L. A. Zhivotovsky and M.W. Feldman

FELSENSTEIN,J., 1977 Multivariate normal genetic models with a WRIGHT,S., 1969 Evolutionand Genetics of Populations, Vol. 2. finite number of loci, pp. 227-245 in Proceedings of the Inter- University of Chicago Press, Chicago. national Conference on Quantitative Genetics, edited by E. POL- ZHIVOTOVSKY,L. A., and S. Yu. GAVRILETS,1990 Stabilizing LACK,O. KEMPTHORNE and T. B. BAILEYJR. Iowa State Uni- selection and linkage disequilibrium (English translation). Sov. versity Press, Ames. Genet. 26 N2 222-23 1. FISHER,R. A,, 1918 The correlation between relatives onthe ZHIVOTOVSKY,L.A., and S. GAVRILETS,1992 Quantitative vari- supposition of Mendelian inheritance. Trans. R. SOC.Edinb. ability and multilocus polymorphism under epistatic selection. 52: 399-433. Theor. Popul. Biol. (in press). FLEMING,W. H., 1970 Equilibrium distributions of continuous Communicating editor: B. S. WEIR polygenic traits. SIAM J. Appl. Math. 36 148-168. 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. APPENDIX 1 GILLESPIE,J. H., and M. TURELLI,1989 Genotype-environment interactions and the maintenance of polygenic variation. Ge- Define the expectations 8 and & with respect to netics 121: 129-138. the distribution of e and (1;) respectively. Then the KARLIN,S., and M. W. FELDMAN,1970 Linkage and selection: variance v = gpe(x - 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. 242: 95-100. KEMPTHORNE,O., 1957 An Introduction to GeneticStatistics. 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. 26 221-235. LATTER,B. D. H., 1960 Natural selection for anintermediate since e has the null expectation. Hence, expanding, optimum. Aust. J. Biol. Sci. 13: 30-35. LERNER,I. M., 1954 Genetic Homeostasis. John Wiley, New York. LEWONTIN,R. C., 1964 The interaction of selection and linkage. 11. Optimum models. Genetics 50: 757-782. LEWONTIN,R. C., and K. KOJIMA, 1960 The evolutionary dynam- ics of complex polymorphisms. Evolution 14 458-472. MATHER,K., 1942 The balance of polygenic combinations. J. Genet. 43: 309-336. MATHER,K., 1943 Polygenic inheritance and natural selection. Biol. Rev. 18: 32-64. MATHER,K., 1973 The Genetical Structure of Populations. Chapman 1L Hall, London. MATHER,K., and J. L. JINKS,1982 BiometricalGenetics, Ed. 3. Cambridge University Press, Cambridge. THODAY,J. M., 1961 Location ofpolygenes. Nature 191: 368- 370. THOMPSON,J. N., JR., J. J. HELLACKand R. R. TUCKER 199 1 Evidence for balanced linkage of X chromosome poly- genes in a natural population of Drosophila. Genetics 127: 117-123. 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. 49 147-156. WEIR,B. S.,C. C. COCKERHAMand J. REYNOLDS,1980 The effects of linkage and linkage disequilibrium on the covariances of noninbred relatives. Heredity 45: 35 1-359. Stabilizing Selection-Balance 953

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

/ \2

= 2 'Y~cx~D~+ 2 x a:f+qi + V, iZj i

= VA + CL + q)'PlVJ with VAand 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: - 21iZ(). i Expanding, ~(x)=I - s i(ao - e)' + e* + Pe(a0 - e) + 2(ao - e) (Yi(l1 + I() 1

+ 2e 2 ai(&+ II) i

Taking expectations with respect to the distribution of e we have 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

which proves (10) with (1 la, b, c). APPENDIX 3 Under the assumption of random union of gametes Derivation of polymorphic equilibria (16), (17): now take expectations with respect to the population We use the relations (1 3), (1 4) and (1 5). (1 3) together 954 L. A. Zhivotovsky and M.W. Feldman with (11 b) can be rewritten as which can be rewritten, using (A16)and T' = &ajp,!, as Ki =pi - CY? + 2(0 - (~o)(~i .?p oq 0 = 2(pi + a?)pP + 4 aiajpj [(pi - af)p! + 2aiT'I = (9: - P;)af2 I , j#i j#i rq = 2(pi - a?)pP + 4ai a,pj since PPqP # 0. Hence, since 9;- pp= -2Bi/(pi - a3,

= 2(p - af)pP + 4ap0

where Ti = 2 I;#iajpjqj/rq. Now multiply (A18)by ai and sum to produce

T' = N/(1 + (01) (A19) where N = -2 C,B,aST,/(& - a:)'. Thus the complete solution (A18) may be expanded as and - 2afBiTi (pi - ay i vo = Define 7,= afTi/(Pi- a'). Then the value 1 + s af(pi - a?) i = A/2(1 + VI), where is an average of these 7's with weights 2aj/QI(pj- aj) whose sum is unity. Upon substitution of 7, and Ti into (A18), using (A19) we find = x a: + (e - ao)Q1 i with

Q1 = 2 a'/(p; - a?). i Since A = cia, + (e - a0)Ql and Bo = 6 - (YO - A/(1 + Returning to (A11) we have therefore 'PI) we have

1 (e - a0)ai ai A p+-+ " 2 pi - a' pi - a; (1 + 91) =-+-1 B; 2' 2 pi - ai where Bi = ai[O - a. - A/(1 + V1)]= a,Bo,say. Using (11 c) write (14) as

Now combine (A15) and (A23) with (12a) to yield (16). Then, returning to (15)with (1IC) we have

APPENDIX 4

Derivation of (20) from (16): Suppose that ai = a, pj = 0. Then 'P1 reduces to 2na2/(8 - a') so that as n + m, Q1/(l+ 'PI) + 1. For n large we therefore Stabilizing Selection-Balance 955 take 1 + (01 = PI. Now with these assumptions the Finally weights of ri used to compute ? in (A21)reduce to

Define 80 according to the relation so that ? = CTi/n. Write DEF 6' - x, = 8 - a. - na = ando. Then

(see Equation 16)

= (ri - ;)/2an DEF where Vi = 2na2poq0and Vj = 2n/3p0q0. Hence under the assumption made above. But ri = a'/T,/(p - a'), where, since pj' = Po

T; = 2 E, a 200pjqj/rc = 2(n - 1)a2p0q0?;'. J+l

That is Finally a2 Ti = -2(n - 1)a2p0qoi;' p - a'

(2na2 p 0 q 0) a' "I = -p - *2 so that and

as claimed in (20).