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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 aver- age 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<j-l kr22)k1, Downloaded by guest on September 25, 2021 986 GENETICS: K. KOJIMA PROC. N. A. S. 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 condi- tions for an equilibrium to be stable under mass selection is to investigate the condi- tions 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 heterozy- gote 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.