Genet. Res., Camb. (1989), 53, pp. 63-70 With 3 text figures Printed in Great Britain 63 Effects of identity disequilibrium and linkage on quantitative variation in finite populations HIDENORI TACHIDA AND C. CLARK COCKERHAM Department of Statistics, North Carolina State University, Box 8203, Raleigh, NC 27695-8203, USA (Received 17 March 1988 and in revised form 15 July 1988) Summary Identity disequilibrium, ID, is the difference between joint identity by descent and the product of the separate probabilities of identity by descent for two loci. The effects of ID on the additive by additive (a * a) epistatic variance and joint dominance component between populations and in the additive, dominance and a * a variance within populations, including the effects on covariances of relatives within populations, were studied for finite monoecious populations. The effects are formulated in terms of three additive partitions, 7]b, i)a and i)A, of the total ID, each of which increases from zero to a maximum at some generation dependent upon linkage and population size and decreases thereafter. ijd is about four times the magnitude of the other two but none is of any consequence except for tight linkage and very small populations. For single-generation bottleneck populations only 7ja is not zero. With random mating of expanded populations 7jb remains constant and 9/a and rja go to zero at a rate dependent upon linkage, very fast with free recombination. The contributions of joint dominance to the genetic components of variance within and between populations are entirely a function of the ?/'s while those of a * a variance to the components are functions mainly of the coancestry coefficient and only modified by the TJ'S. The contributions of both to the covariances of half-sibs, full-sibs and parent-offspring follow the pattern expected from their contributions to the genetic components of variance within populations except for minor terms which most likely are of little importance. 1. Introduction variation in inbreeding coefficients of members of the Random mating finite populations, populations that population with varying pedigrees, and is further have been through a bottleneck and experimental enhanced by linkage (Weir & Cockerham, 19696). populations initiated with a few parents or inbred In random mating finite populations, initially in lines are generally treated as distinct entities. For linkage equilibrium, ID is the principal agent for the many characterizations including quantitative genetic variation in actual inbreeding or heterozygosity variation they all fit into the same framework, differing (Weir, Avery & Hill, 1980) and for the variation in only in details. In studying the quantitative genetic linkage disequilibrium (Weir & Hill, 1980). variation within and between finite populations it was We wish to extend the analysis to include the effects found that a joint dominance contribution for a pair of ID on additive by additive (a * a) variance in finite of loci was determined entirely by identity dis- populations. The a * a variance with self-fertilization equilibrium (Cockerham, 1984a). The question arises was studied in some detail with an emphasis on the as to the effects of identity disequilibrium on other effects of ID on the covariances of relatives (Cocker- joint effects of loci, epistatic variance. ham, 19846). In this case ID was entirely a function of Identity disequilibrium (ID) is the difference be- linkage and the relative effect of ID on the covariances tween the joint probability of identity by descent and of relatives decreased as the relatedness of the relatives the product of the separate probabilities of identity by increased. descent for two loci. It is never negative and occurs for To extend the analysis to finite populations we pedigree systems of mating for linked loci, increasing consider random mating monoecious populations as as recombination decreases (Weir & Cockerham, in Cockerham (1984 a). The initial population is in 1969 a). It occurs in finite random mating populations, linkage and Hardy-Weinberg equilibrium and gives even for unlinked loci when it is a measure of the rise to replicate finite populations of size TV each Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 25 Sep 2021 at 16:24:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0016672300027877 H. Tachida and C. C. Cockerham 64 generation. At some time the replicate populations are <S, and non-gametic, J/~, pairs of genes. The status of expanded, N = oo, and random mating is continued ^ changes over time. In terms of a = (1 +A)/2 where in each. The purpose of the expanded populations is (1— A)/2 = r, the recombination fraction, ^T+1 = to diagnose the effects of finite population history on a ^T + (l—a) JVT, i.e. gametic genes were gametic in quantitative genetic variation and of continued the previous generation with probability a and non- random mating to determine the permanency or gametic in the previous generation with probability transient behaviour of the effects. Obviously, popu- 1 —a. In contrast in an infinite population Jf7+X = JfT lations that have been through a bottleneck are since genes on two gametes trace back to genes on included in this framework. For the expanded popu- one gamete with probability zero. From the definitions 2 2 lations we partition the a*a variance into portions of 6, y and A, 0T+1 = a OT + 2a(\ -cc)yT + (\ -a) Ar; a within and between populations and further subdivide Yr+i = 7r + (l -a) AT;andAT+1 = AT. In terms of initial 2 the portion within populations into portions that values 6T = ay0 + 2a'(\ -of) y0 + (1 -O Ao; yT = T 7 behave as additive and a * a variances. The total ID is a y0 + (l — a )Ao; and AT = Ao. Thus, in time, all also partitioned and the effects of the partitions on the two-locus descent measures are the same, 0X = various portions of a * a variance are formulated for yOT = K = K the expanded population and for modifications that 2 Except for j)b = Ao — 6 , the identity disequilibriums occur with continued random mating. A parallel T go to zero in time: i/ar = a^ar_, = a ija0 and i/d7 = partitioning is also given for joint dominance effects. ^dr-i = a27Vao- The rate of approach to zero per The contributions of the joint dominance and a * a generation for each is a constant, 1 — a = r for rja and variance to the covariance of half-sibs, full-sibs and 2 1—a = r(2 —r) for 7/d. With free recombination the parent-offspring within populations are formulated. rates are fast, 1/2 for ?/a and 3/4 for t]A. To obtain some notion of the magnitude of ??'s in 2. ID measures relation to N, t and A some values are plotted against t in Fig. 1 for N = 4, 20, 100 and A = 0, 0-5, 0-9. Each The analysis is facilitated by measures of the prob- line in the figure that can be distinguished from zero ability of identity by descent. For two loci the double is labelled according to A, N values. identity by descent measures are for a pair of genes at Each of the i?'s is zero initially, builds up to a one locus and a pair of genes at the other locus. For maximum and eventually goes to zero at fixation. random mating monoecious populations only three There is considerable similarity in the shape of the are required (Cockerham, 1984a): 6 for genes on two curves among the T/'S. i/d reaches its maximum quickest gametes, y for genes on three gametes, and A for genes and »/a next. An increase in A increases the time on four gametes. The measures are the same for all required to reach the maximum. Note the difference in configurations within and between random indi- scale for •)/„ which is about four times that for the other viduals. Consequently, 6 is the joint inbreeding two. While linkage increases the T/'S A must be near coefficient F, and so on. Transition equations from 0-9 for the increase to be of much consequence. With which these two-locus descent measures can be A = 0 the i/'s are due to the variation among inbreeding obtained are given in Appendix A in Cockerham coefficients (Weir & Cockerham, 1969*) and of little (1984 a). consequence except for i/d in extremely small popu- One single-locus measure, the coancestry coefficient lations. In fact, with N ^ 20 it would appear safe to 8, which is also the inbreeding coefficient F, is needed, 1 ignore 7/b and i/a, particularly when viewed over many and dt= 1 —(1 —1/2A0 - pairs of loci where A on the average is probably very It was found useful (Cockerham, 1984a) in par- small, and in this case even 7?d will be very small. titioning the joint dominance contributions for two loci for finite populations to partition the total ID, if = 6 — 62, into three parts associated with different variance components. 3. Partitions of the a * a variance and joint dominance effects 7)f., = A — d2 : between population variance For a genotype formed from the union of gametes rj& — 2(y — A) : additive variance Ak Bk and A, B, the a • a effects in the model for the 7)a = 6 — 2y + A : dominance variance genotypic value may be symbolized as (ak bk) + (a, b,) + These sum to the total (akb,) + (a,bk) where overbars indicate non-gametic, The descent measures for the expanded population pairs of genes. In the initial equilibrium population will be determined by t, the number of generations for which these effects are defined each has mean zero before expansion, and A^, the number of individuals in and <f(ab)2 = a2^ where $ denotes expectation.