Effects of Identity Disequilibrium and Linkage on Quantitative Variation in Finite Populations

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

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generation. At some time the replicate populations are

0040

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0-020 XO-9,20 0015

0010 /o-5,4 0005 U •• 0,20 \ 0-9,10jgV0w 2 0 0000 Sk&Zr-'^' P.1- . . 016

014 I t ;I \I 0-9.4 012 t I

010

0-08

006 \0-9,20 '/b-5,4' 004 0-9,100 \ .'0,4 002 0-5.100" 000 10 20 30 40 50 60 70 80 90 100 t

Fig. 1. Plots of 7jb, i/B and i)a against t for N =4, 20, 100 zero are labelled with values of A, N. and A = 0, 0-5, 0-9. Plots that can be distinguished from

unrelated but inbred individuals is (1 If X and Y are random members within our finite since 6 = F and 0 — F. populations the a*a variance in t?XY is 0+2y + Procedures were also established in Cockerham ^)f|a- We partition the total variance into that be- (19846) for evaluating the a*a variance in the tween populations, a\ = t>XY, and that within popu- covariance, ^xv, between individuals X and Y, in- lations

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Table 1. Coefficients of a* a variance, aj^, and joint dominance effect, 2hh', in the components of variance

l+20-2£-A 40-d-2y-A — 1-20 + 6 "as 2 l203A 4(dA)2 2hh'

populations. Jiang and Cockerham (unpublished) coefficients are given in terms of rfs. Only the portion have shown that (40-0-2y-A)o-|a becomes a part between populations, reflected by i)b, remains constant of means product of the inbreeding depressions, h and h', for contains) the two loci are also included in Table 1. It was found in Cockerham (1984 a) that the joint dominance contributions belonged in part in a\, which was substantiated in Cockerham & Tachida (1988) and in part in the dominance variance, a%,, within populations. The purpose of the inclusion in Table 1 is to demonstrate the behaviour of the coefficients in an expanded random mating population over time. The

0-20 018 ~~N 0-9,4 016 014 g 012 I 010 008 006 004

002 0-9,100 000 10 20 30 40 50 60 70 80 90 100 t Fig. 2. Percentage reduction in the coefficient 4(6 — A) due of A, N. to identity disequilibrium. Plots are labelled with values

One locus Two loci ibd* Gene arrangementj ibd** Gene arrangement ibd** Gene arrangement F (a | a') (ab | a'b') (al1 b)(a'Ib' | ) e (a) (a') (ab)(a'b') (al1 a')(bIb' I ) (ab)(a' 1 b') 4s (al (ab|a')(b') 44 (al1 a')(b)

t Genes are represented by a and by a' for two distinct genes at the A-locus and b and b' for two distinct genes at the B-locus. | Genes separated by | are on different gametes in the same individual and those separated by () are in different individuals. * Probability that the two genes a and a' are identical by descent. Symmetrical arrangements are omitted. ** Probability that the two gene pairs, a and a' and b and b', are jointly identical by descent. Symmetrical arrangements are omitted.

The descent measures are for the parental generation. a%,, 4(6 — A3 + f2 — f3) in cr\, and the remaining part 2 Note that the coefficients of Ga are the same as for of the coefficient for a w in o-?a,. The coefficients in additive variance, of Gaa are the same as for a * a ar\, were found by the same method of covariance be- variance and of i?d2hh' are the same as for the tween ancestor and descendant utilized in Cocker- dominance variance for these relatives in an infinite ham & Tachida (1988). With monoecy all descent equilibrium population. There are additional terms measures in a category are equal, e.g. f 1 = f 2 = involving cr\& and identity disequilibrium which are always negative for ^*o but negative for "

populations including selfing and there is a question where X, T is for the rth generation. After that we need as to the extent to which the results are modified by only to substitute 6, 02, f 3 and A5 for the initial values other mating structures such as separate sexes or of 6, d, y and A respectively to accommodate the finite monogamy. Weir, Avery & Hill (1980) studied the histories of other breeding structures. The identity 2 effects of mating structure on variation in inbreeding, disequilibriums are i/b = A5 — d , ^a = 2(f3 —A5) and and Weir & Hill (1980) studied the effects of mating tja = 02 — 2f 3 + A5 where i)b remains constant and i?a structure on variation in linkage disequilibrium. and 7jd go to zero at the same rate as before. Thus, the While they did not utilize the term 'identity dis- formulations in Table 1 need only to be adjusted for equilibrium ', their results can be expressed in terms of the initial values appropriate for the mating structure. various functions of identity disequilibrium. While the results of Weir & Hill (1980) and Weir, In Weir, Avery & Hill (1980) and Weir & Hill (1980) Avery & Hill (1980) are functions of ID their all measures are of non-identity by descent. For a formulations and standardizations preclude exact single locus P= 1-F and n= 1-0. The 10 two- comparisons with ours. The main differences among locus measures, required to accommodate the avoid- mating structures in Weir, Avery & Hill were related ance of self-fertilization and separate sexes with to the degree of avoidance of mating relatives and hierarchical or monogamous types of matings, are for were proportionately larger as linkage decreased. double non-identity by descent. We place a ~ on each To inquire directly into differences due to mating to denote double identity by descent and our descent structure 7jb, 7/a and ija are plotted against t for measures are listed in Table 2. monoecy, M, monoecy with the exclusion of selfing, To accommodate these other mating structures as Me, and monogamy, Mo, all with N = 4, in Fig. 3. We well as monoecy, the coefficients, Table 1, of 2hh' are chose N = 4 to exaggerate the differences in mating 2 A2 — F in < 1 — A2 in trw, 2(ff 2 —A4) in and structure. Even so the curves are very similar. There is 2 6l — 2f 2 — A2 + 2A4 in a v.. The coefficients of cr are a slight shift to the right for Me and Mo, both of which t X in 1-(92-2ff 1-A1 in avoid self-fertilization. Near the maximum MB is

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0000 016 0-9.M 014 'AT 0-9,Mo 012

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000 0 10 20 30 40 50

Fig. 3. Plots of tjb, rj^ and rja against t for monoecy, M, all with N = 4. Plots are labelled with values of A, X(X = monoecy with selfing excluded, Me, and monogamy, Mo, M, Me, Mo).

intermediate to M and Mo reflecting an increased ID Cockerham (1984 a), as a means of diagnosing the due to a higher frequency of full-sib mating than for effects of finite population history on quantitative Mo but a decreased ID due to no self-fertilization in genetic variation. Continued random mating of the comparison to M. The greatest percentage differences expanded populations was utilized to determine the are for small A but then the ij's are of little consequence. permanency or transient behaviour of the effects. The effects of ID. on 2hh' are considerably different from those on a * a variance. 6. Discussion The coefficients of 2hh' in the components in Table We have utilized expanded populations, which was 1 are entirely a function of the i/'s. Those, i/a and i/d, done in Cockerham & Tachida (1988) and implied in for components within populations go to zero with

continued random mating while the variation i/b2hh', Cockerham, C. C. (1984a). Covariances of relatives for between populations remains constant. The portion quantitative characters with drift. In Human Population Genetics: The Pittsburgh Symposium (ed. A. Chakravarti), 7/ 2hh' in o-\ contributes to permanent response in a t pp. 195-208. New York: Van Nostrand Reinhold. descendants due to selection (Cockerham & Tachida, Cockerham, C. C. (19846). Additive by additive variance 1988) but is reduced each generation of random with inbreeding and linkage. Genetics 108, 487-500. mating before selection is practised. Cockerham, C. C. & Tachida, H. (1988). Permanency of The coefficients of cr? in Table 1 are functions response to selection for quantitative characters in finite s populations. Proceedings of the National Academy of mainly of 6 and A. The main component of interest is 2 Sciences U.S.A. 85, 1563-1565. (dvv, + yOu. + yu0. + 1984 a). This ID dissipates at the rate r(2-r) with 2 2 2 AOu0 o-fa = [a. 6 + 2a(2 - a) y + (2 -a) A] means n random inbred lines or homozygous parents the 2 contains. For half-sibs let P be the common parent only ID is i)a = (n— l)/n which soon dissipates and Q and R be the other parents of H and S except for tight linkage. There are corresponding respectively. We expand the descent measures back to simplifications in the covariances of relatives. Bryant those in the parent. A parental gamete in S from P is et al. (1986) let their experimental single-generation parental or recombinant in P with probability a and bottleneck populations flush to normal size in about 1 — a respectively. The same holds for R. Then, for a five generations before estimating the covariance random parental gamete for S, 6HS = [a(#HP + #HR) between mid-parent and offspring which fairly well 1 + (1-a)(yHp + yHa)]2" . We further expand these insured that the estimates are affected little by identity measures to include the parent Q disequilibrium. Finite population history can modify the genetic 2 components a\., a- ,, and craa.within populations considerably, but linkage and concomitant identity disequilibrium play very minor roles in the mod- ifications, particularly after a few generations of random mating of the expanded populations. 2 = [a (0~Pp + 36) + 2a(l - a) (ypP + 3y) 2 2 Drs B. S. Weir and C. H. Langley provided helpful com- + (l-a) (APP + 3A)]2- . ments. Paper no. 11502 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh, N.C. We made use of the fact that distinct parents are 27695-7601. This investigation was supported in part by random members of that generation. Descent meas- Research Grant GM 11546 from the National Institute of ures involving the same individual require simple General Medical Sciences. 1 probabilistic arguments 6VV = (1 +§)2~ , ypp = 0 and 1 ApP = (1 + 0)2~ . For yHg, note that one gene of the References recombinant gamete in S comes from P and the other Bryant, E. H., McCommas, S. A. & Combs, L. M. (1986). gene from R. Then, by an expansion, yHg = yHPR, The effect of an experimental bottleneck upon quantitative where the subscripts P and R denote that one gene is genetic variation in the housefly. Genetics 114, 1191-1211. taken from P and R, respectively. Using an expansion

similar to the one used for a parental gamete in the calculation of 0HS, this can be further expanded, 1 f HS = [<*f QPR + (1 - a) AQP^ + ay ppA + (1 - a) Agp^" 1 For full-sibs let the two parents be P and Q. Then, = [«(7 + fpPB) + (1 - a) (A + APpft)]2- . utilizing the fact that any measure involving the same Again we made use of the fact that distinct parents are parent twice is the same for P or Q, random members of that generation. With simple a) (yPp + y) probabilistic arguments, descent measures involving 2 (l-a) (AP the same individual are calculated to be yPP]j = APpA 1 = (d + y)2~ . Because of the symmetry, yfts is the same f FS = f FS = + (1 - a) APPQ as yHg. Finally, applying the expansion on each recombinant gamete in H and S, we obtain Aft§ = AQPPR- TWO genes from P are at the same locus or With appropriate substitutions the a*a variance in different loci with probability 1/2 and independently the covariance of full-sibs is obtained, they come from the same gamete or different gametes 2 of P with probability 1/2. Therefore, making use of Ks = [2(0 - A) + (2 + A ) (1 - 26 + A + 7a)/8 the fact that distinct individuals are random members 2 of the parent generation, Asg = (0 + y + 2A)2~ . Com- 7* bining these and correcting for the contribution to The contributions of joint dominance effects to ' 'HS and ^pg were given in Cockerham (1984 a) ^vv. already calculated, we obtain the a * a variance in the covariance between half-sibs within populations,