For a Single Locus with Two Alleles Is Simply Heterozygote Superiority

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3Cavalli, L. L., and H. Heslot, Nature, 164, 1057 (1949). 4Lederberg, J., Science, 114, 68 (1951). 6 Cavalli, L. L., Boll. Ist. Sierotera, Milano, 29, 281 (1950). 6 Jacob, F., and E. L. Wollman, Symp. Soc. Exptl. Biol., 12, 75 (1958). 7 Richter, A., Genetics, 42, 391 (1957). 8 Hayes, W., Nature, 169, 118 (1952). 9 Cavalli, L. L., J. Lederberg, and E. M. Lederberg, J. Gen. Microbiol., 8, 89 (1953). 10 Lederberg, J., L. L. Cavalli, and E. M. Lederberg, Genetics, 37, 720 (1952). 11 Lederberg, J., J. Bacteriol., 71, 497 (1956). 12 Zinder, N. D., and J. Lederberg, J. Bacteriol., 64, 679 (1952). 13 Stocker, B. A. D., N. D. Zinder and J. Lederberg, J. Gen. Microbiol., 9, 410 (1953). 14 Lederberg, J., and P. R. Edwards, J. Immunol., 71, 232 (1953). 15 Zinder, N. D., Cold Spring Harbor Symposia Quant. Biol., 18, 261 (1953). 16 Baron, L. S., S. B. Formal, and W. Spilman, Proc. Soc. Exptl. Biol. Med., 83, 293 (1953). 17 Spilman, W., L. S. Baron, and S. B. Formal, Bacteriol. Proc., 51, 1954. 18 Sakai, T., and S. Iseki, Gunma J. Med. Sci., 3, 195 (1954). 19 Lennox, E. S., Virology, 1, 190 (1955). 20 Morse, M. L., E. M. Lederberg, and J. Lederberg, Genetics, 41, 142 (1956). 21 Luria, S. E., and J. W. Burrous, J. Bacteriol., 74, 461 (1957). 22 Baron, L. S., W. F. Carey, and W. M. Spilman, Proc. VIIth Intern. Congress for Microbiology, 50, 1958. 23 Lederberg, J., Meth. Med. JRes., 3, 5 (1950). 24 Baron, L. S., S. B. Formal, and W. Spilman, J. Bacteriol., 68, 117 (1954). 25 Englesberg, E., and L. S. Baron, submitted to J. Bacteriol. 26 Baron, L. S., W. M. Spilman, and W. F. Carey, Bacteriol. Proc., 29 (1959). 27 Lederberg, E. M., and J. Lederberg, Genetics, 38, 51 (1953). 28 Hartman, P. E., in The Chemical Basis of Heredity, ed. W. D. McElroy and B. Glass (Balti- more: The Johns Hopkins Press, 1957), 408. 29 Wollman, E. L., F. Jacob, and W. Hayes, Cold Spring Harbor Symposia Quant. Biol., 21, 141 (1956).

ROLE OF EPISTASIS AND OVERDOMINANCE IN STABILITY OF EQUILIBRIA WITH SELECTION* BY KEN-ICHI KOJIMA DEPARTMENTS OF GENETICS AND EXPERIMENTAL STATISTICS, NORTH CAROLINA STATE COLLEGE Communicated by Sewall Wright, May 11, 1959 The condition which leads to a stable equilibrium under constant selective values for a single locus with two alleles is simply heterozygote superiority. This condition is also necessary and sufficient for many loci as long as they recombine freely and do not exhibit epistasis. The purpose of this study is to investigate the conditions of stable equilibria when genes do exhibit epistasis. Two alleles per locus, constant genotypic values, free recombination, and random mating are assumed throughout. Genotypic Variance as Functions of Derivatives of the Mean.-The conditions for stable equilibria must be phrased in terms of derivatives of the mean of a population. These derivatives are essentially gene effects (mono-, di-, tri-genic, and so on) which lead to the same partitions of genotypic variance as those of CockerhamI and Kemp- thorne.4 This method of obtaining the partitions will be briefly illustrated to estab- lish notation and to tie the conditions for stable equilibria as closely as possible to the genotypic variances. It is worth noting also that the procedure is expeditious Downloaded by guest on September 25, 2021 VOL. 45, 1959 GENETICS: K. KOJIMA 985

in obtaining the partitions of genotypic variance for many genetic models where the mean can be written as a condensed function of gene frequencies and parameters of the model. The mean of the population, Y, may be written as

1' = q2Y2 t + 2q%(1 - qA) Pit + (1 qk)2Yok (1) where qk is the frequency of allele, A, at the kth locus, and Y2k, rt and Pok are the mean of three groups of individuals according to the three genotypes'-A Ak Akak and akak, respectively. The partial derivative of the' mean (1) with respect to qk is

a = 2(alO)k = 2{qk(Y2 - Ylk) + (1 qk) (lk - YOk)O} X (2) aqk Expression (2) was used by Wright8 to formulate-the change in gene frequencies due to differential selective values of genotypes. It is twice the average excess or average effect (they are the same in random mating populations) of the kth locus defined by Fisher.2' The contribution of the kth locus to the additive genetic variance, o0i2, is

(olo2)k = 2q(l- qk)(alO)2 and lo = (ui12)*, (3) k=l where s is the total number of loci. The second derivative of the mean with respect to qk is

- = 2(aOl)k = 2{YPk - 2rlk + YOk1, (4)

which is twice the dominance comparison for the kth locus. The contribution of the kth locus to the dominance variance, 0o102, is

s (-012)k = qk2(1 - qk)2 (aOl)k2 and. Cor2= (012)kl (5) k=l The mixed partial derivative of the mean with respect to qk and qj is 52y tqq = 4(a2O)jk = 41qkqj12k2j + qk(l -qj)l2klj + (1 -qk) qjllk2j + (1 - qk)(1 - qj) llkj}, (6) where

= + = + 12k2J Y2k2j - Y2k1j - Yk2 Ylkly 12k1 -2kl Y2kOj - Yklj Y1k0o '112k = Ylk2j - - Yok2j + YOklj, 1lklj = -lkf Y o1j - Pk-+ Yok and Y2k2j, Y2klJ. . . Yokoj are the means of individuals with genotypes, AkAkA1Aj, AkAkAjaj, ... ., ajtaktaja, respectively. This expression (6) is the additive X additive comparison for the kth andjth loci. The contribution of the kth and jth loci to-the additive X additive variance, a2C2 is

= - t0 (1 k and or2o2 = E ( (7) 4qk(l -q)qj qj)(a2O)2 k

The expressions (3), (5), and (7), and other partitions of variance are the same as those given by Cockerham,1 when there is no inbreeding. Generally, if a quantity, (aLQ) = 2(L+Q) L Q forj $ k, (8) 2L+)L Q fl5qjlbqk2 j k then a contribution of the comparison (L-additive) X (Q-dominance) from a set of (L + Q) loci to the (L-additive) X (Q-dominance) variance is L Q (aLQ2) = 2L I qj(l - qj) f qk2 (1 - qk) (aLQ)2, (9) j k Conditions for Stable Equilibria.-Consider a surface of the means of a given character with respect to s coordinates, each corresponding to the gene frequency of one of s loci. When the character is a measure of the adaptive value of genotypes, such a surface is called the adaptive surface.7 The method of finding the conditions for an equilibrium to be stable under mass selection is to investigate the conditions for a point on the surface to be a relative maximum, i.e., a slight shift of gene frequencies in any direction results in a lower mean. For a given point (Y, qj, q2....,.q,) on the surface to be a relative maximum, the following conditions are necessary and sufficient:

- 0 forj = 1, 2,..., s. (10) bqj Let the nontrivial solutions (i.e., qj $ 0 or 1) of the simultaneous equations in (10) be (qj, q2, ** *, ' ) ii. The quadratic form, of which the element djk is 62 ?/6qj6qk for all j and k (j =

1, 2, ...,I ; k = 1, 2, .. s), must be negative definite at the point Q 2, .. ). This means that the principal minors, J1, of the matrix of the quadratic form must havesign (- 1)'forl = 1, 2, ... ., s. The genetic implication of these two conditions are now considered. From (2) the solutions of the equations in (10) have the form

A ~~~~1 qj 1 + Y - - or j = ' ' (11) Ylj - YOj This indicates that two differences (Yij - 12j) and (PFjj- ?oj) must have the same sign for eachj in order that the q's be nontrivial. In other words the heterozygote must be either superior or inferior to both homozygotes in terms of marginal means of three genotypes at each locus. This is a necessary condition for a point to be an equilibrium. From (4) and (6), the elements djk in the quadratic form are identified as, djj = 2(aOl)j (12) dik = 4(a2O)jk Downloaded by guest on September 25, 2021 VOL. 45, 1959 GENETICS: K. KOJIMA 987 where djj's are diagonal elements and dl,'s are off-diagonals. Then the sufficient conditions for a point of equilibrium ('q, q2, . . ., q() to be stable is (aOl)j < O for all jcl (aO1)j 2(a20),k > 0 for all j and k C2 2(a20)Jk (aOl)k (aOl)j 2(a20)ik 2(a20)jm < 0 for all j, k and m C3 2(a20)Jk (aOl)k 2(a20)km (13) 2(a20)jm 2(a20)km (a )m

Since the dominance comparisons are of the form (F2j - 2 YRi + Y0o) and must be negative, the equilibrium is unstable when the heterozygote is inferior to the both homozygotes just as it is when the loci do not interact. These conditions, C1, C2, . . . C3 may be viewed stepwise. For a single locus or noninteracting loci only heterozygote superiority of each is sufficient. For two loci which interact an additional condition, c2, is necessary for them to be jointly held in a stable equilibrium. For three interacting loci, not only do cl and c2 have to hold but also C3 must be satisfied for them to be simultaneously kept in a stable equilibrium and so on for more loci. These conditions are not easy to visualize without making some simplification. The general case for two loci will be considered, and for more loci, they will be considered to be identical in their effects. For s = 2 the inequality c2 in (13) limit the magnitude of the additive X additive comparison in such a way that (aOl)l. (aOl)2 > 4(a20)122, (14) From (9) the inequality (14) is equivalent to saying that the geometric mean of the two dominance variances must be larger than the additive X additive variance for the two loci, i.e., (0o1)1. (O01)2 > (o202)12 which is ¢12 > 2o202 if (aol), = (a1)2. For more than two loci the simplification that all loci have equal effects and that they have the same gene frequencies when at equilibrium, will be imposed. Then the dominance comparisons are the same, and the additive X additive comparisons are the same for all loci. The conditions now simplify to (aOl) <0 cl 0 < (a20) < - (aOl)/2(s - 1), if (a20) > 0 C21 or (aOl) < 2(a20) < 0, if (a20) < 0 C2/t The first condition, cl, is the same one insuring heterozygote superiority, and in addition, one of two conditions, C2I or c2", which limits the additive X additive comparison relative to the dominance comparison, is necessary for the s loci to be jointly held in a stable equilibrium. Downloaded by guest on September 25, 2021 988 GENETICS: K. KOJIMA PROC. N. A. S.

Translating c2' and c2" into variances, these conditions are 0.202 < 0oo2/2(s - 1), if (a20) > 0 C21 0202 < (s - 1)o002, if (a20) < 0 C2// 2 Thus, if the additive X additive comparison is positive, the amount of additive X additive variance relative to the dominance variance is severely limited for an equilibrium to be maintained, and when there are many loci. On the other hand if (a20) is negative, 0202 can be as large as (s - 1)/2 times 0102 for an equilibrium to be stable. The two conditions are the same when only two loci are considered. Discussion. The results of this study are specifically for two alleles at each locus. The conditions for stable equilibria of interacting loci with multiple alleles will probably be very different from those for interacting loci with only two alleles. Kimura5 gave algebraic rules for testing stability of equilibria for multiple alleles at a single locus and indicated that simple heterozygote superiority was no longer a necessary condition for stable equilibrium. The assumption of constancy of genotypic values may be valid in certain situa- tions but not in others. It is not clear how large the fluctuations in genotypic values can be, random or otherwise, and the conditions derived in this paper still be valid. In some cases the genotypic values may vary as functions of gene frequencies as was considered by Wright9 and Lewontin6 for a single locus with two alleles. Lewontin found that even heterozygote superiority was not necessarily required for a stable equilibrium. Although only unlinked genes are considered, linkage disequilibrium can arise from selection even when random mating is maintained. This linkage disequilibrium is ignored in this paper. However, the amount of linkage disequilibrium depends largely on the amount of shift in genotypic frequencies each generation. Since changes in genotypic frequencies are small near points of equilibria, it is believed that the effect of linkage disequilibrium is trivial and that the conditions found for stable equilibria are valid. The effect of linkage and linkage disequilibrium needs further study. While general conditions for stable equilibria of two loci were given in terms of components of variance, this could not be done for more than two loci without making restrictions. The restrictions of equal effects and equal frequencies of genes are unrealistic. However, the results indicate that generally only so much additive by additive epistatic variance can be afforded relative to the dominance variance and an equilibrium be maintained. Summary. The conditions for maintaining stable equilibria in a population undergoing mass selection and for interacting genes are studied. For two loci to be jointly held in a stable equilibrium, overdominance is required at each locus and the geometric mean of the dominance variances from each locus must be larger than the additive X additive variance for two loci. For more than two loci explicit expressions in terms of partitions of genotypic variance are given for stable equilibria when all loci have equal effects and equal frequencies. Roughly speaking in the general case, more overdominance tends to maintain stable equilibria, while more additive X additive epistasis tends to destroy stable equilibria. The author wishes to express his hearty thanks to Drs. C. Clark Cockerham and Downloaded by guest on September 25, 2021 VOL. 45, 1959 GENETICS: K. KOJIMA 989

R. C. Lewontin for stimulating discussions and valuable help in preparation of this manuscript. * Contribution No. 1040 of Journal Series. North Carolina Agricultural Experiment Station, Raleigh, North Carolina. This work has been supported in part by grants from the National Science Foundation and Rockefeller Foundation. 1 Cockerham, C. Clark, Genetics, 39, 859-882 (1954). 2 Fisher, R. A., The genetical theory of natural selection. (London: Oxford Press, 1930). 3Fisher, R. A., Annals of Eugenics 11, 53-63 (1941). 4Kempthorne, O., Proc. Roy. Soc. B 143, 103-113 (1954). rf Kimura, M., these PROCEEDINGS, 42, 336-340 (1956). 6 Lewontin, R. C., Genetics 43, 419-434 (1958). 7Wright, S., Proc. Sixth Intern. Congr. Genetics, 356-366 (1932). 8 Wright, S., Bul. Amer. Math. Soc. 48, 223-246 (1942). 9 Wright, S., Cold Spring Harbor Symposia on Quantitative Biology 20, 16-24 (1955).

STABLE EQUILIBRIA FOR THE OPTIMUM MODEL BY KEN-ICHI KoJIMA* DEPARTMENTS OF GENETICS AND EXPERIMENTAL STATISTICS, NORTH CAROLINA STATE COLLEGE Communicated by Sewall Wright, May 11, 1959 The optimum model proposed by Wright4 has been used to demonstrate theo- retical outcomes in a mendelian population when selecting for an intermediate phenotype.2 3 6 Whether or not this model leads to stable equilibria, however, has not been investigated except in a few special cases.5 The purpose of this paper is twofold: to illustrate the method' of finding stable points of equilibria with respect to mass selection when loci exhibit epistasis, and to indicate that such points do exist for the optimum model. Wright4 introduced the optimum model, Y = -(S - 2 where S was the underlying (genotypic) value on the primary scaleD was the optimum or most desired value on the primary scale and Y was the observed or pheno- typic value on the secondary scale. In a subsequent paper' he investigated the quasi-static evolution of a population in which the selective values of individuals were proportional to their values on the secondary scale. The contributions of the loci to the primary scale, S, were always considered to be additive among loci. Two cases of dominance, either no dominance or complete dominance at all loci were considered. In either case stable equilibria were not found. Actually, other levels of dominance (partial or overdominance) on the primary scale produce stable equilibrium points. To illustrate that this is so, suppose the primary value, S, is determined additively by the loci such that the jth locus contributes aj, hjaj, and - aj corresponding to the phases AjAj, Ajaj, and ala, of the locus. Without loss of generality a can always be considered to be positive. The value of h reflects dominance on the primary scale as follows: Downloaded by guest on September 25, 2021