The influence of small population size (N) on the genetic variance within ... and many loci, and predicted that variation between small populations in the.
VARIANCE IN QUANTITATIVE TRAITS DUE TO LINKED DOMINANT GENES AND VARIANCE IN HETEROZYGOSITY IN SMALL POPULATIONS P. J. AVERYl
AND
W. G. HILL
Institute of Animal Genetics, West Mains Road, Edinburgh EH9 3JN, Scotland Manuscript received November 11,1977 Revised copy received July 11,1978 ABSTRACT
The influence of small population size ( N ) on the genetic variance within and between randomly bred unselected lines, with selfing permitted, is investigated for a model of a quantitative trait determined by linked genes that show dominance within loci but are additive over loci. Formulae for within-line variance include terms in linkage disequilibrum, which occurs by chance in the lines and these are evaluated in terms of N , map length and gene number. -The expected variance within lines is increased by this disequilibrium, quite substantially if there are many loci, with most of the increase being between or within full-sib families and almost no change expected between half-sib families o r in the covariance of offspring and parent. If all loci are unlinked, there is no increase in variance within full-sib families. The variance between lines is little affected by disequilibrum generated by chance.Expressions for the variance between individuals i n heterozygosity over the whole genome are special cases of those for the variance due to linked dominated genes, and formulae are given and evaluated. The coefficient of variation of heterozygosity is at least I / d 3 3 and can be much higher for species with few chromosomes.
LINKAGE disequilibrium, which is generated by sampling in small populations and declines slowly when linkage is tight, can have a substantial effect on the variation in quantitative traits that are determined by many loci. BULMER (1976) and AVERY and HILL(1977) used a model of completely additive genes and many loci, and predicted that variation between small populations in the genetic variance within each population would be caused almost entirely by disequilibrium among genes linked on the same chromosome o r on different chromosomes. Disequilibrium generated by chance is equally likely to be in the coupling or repulsion phase and thus, for additive genes, does not change the mean level of variance within replicate populations (lines). With dominant genes, however, COMSTOCK and ROBINSON(1952) showed that the variance within lines would be increased by linkage disequilibrium, even if disequilibrium occurred randomly in coupling or repulsion phase. ROBERTSON(1952) studied changes in variance between and within lines for single recessive genes and, therefore, for many Present address: Department of Applied Statistics, University of Reading, Whiteknights, Readnig RG6 2AN, England. Genetics 91: 817-844 April, 1979.
818
P. J. AVERY A N D W. G . HILL
recessive genes assumed to segregate independently, and showed that variation within lines could rise in early generations of maintenance at small size, contrary to the case of additive genes. In this paper we extend his results to include linkage among genes showing dominance but not epistasis and so quantify and extend and ROBINSON(1952) and ROBINSONand the effects discussed by COMSTOCK COMSTOCK (1955). Formulae for changes in the mean levels of heterozygosity in small populations are well known following the work of WRIGHT(1977) ; the variance in heterozygosity has received much less attention. Such variation can come from variation in pedigree inbreeding, for example in a random mating line a few individuals may be progeny of selfed or sib matings, and from variation in length of sections of chromosomes identical by descent. SVED(1968) showed that the variance in heterozygosity could be expressed in terms of variation in linkage disequilibrium and considered the variance in heterozygosity to be a useful (1977) discussed the variance in measure of disequilibrium. Recently FRANKLIN heterozygosity in inbred lines maintained solely by selfing or by full-sib mating in which there is no variation in pedigree inbreeding. In this paper we extend and clarify SVED’S (1968) formulae and show that the variance in heterozygosity is also related to the genetic variance contributed by dominant genes, and that similar expressions are obtained f o r both. The following assumptions will be made. Generations are discrete and no selection, mutation or migration takes place. Individuals are monoecious diploids and mate at random, including random selfing. Replicate populations of N breeding individuals (i.e., lines) are taken from the same base population, which is assumed to be very large. The mating is assumed to follow the procedure given by SERANT and VILLARD (1972) in which a population of N zygotes is formed by random pairing of 2N gametes. An infinite gametic pool is then formed from those individuals taking into account crossing over, and 2N gametes are sampled from this pool to give the next generation. Two alleles per locus will be assumed. The analysis could be generalized to many alleles (cf.,AVERY and HILL1977), but the algebra would be very tedious. The results f o r variation in heterozygosity are directly valid for multiple alleles, and it is unlikely that general conclusions for other cases are substantially affected. The two alleles at locus i are denoted A , and Ai’, and let q2t be the frequency of A , in the infinite gametic pool, obtained from the individuals of generation t-1. This is also the frequency of Ai in the members of generation t-1. For pairs of genes, however, the chromosome frequency in parents and their gametic output differs, so let D*;jt-l be the linkage disequilibrium between loci i and i in the N individuals of generation t-1, and let D z j t be the corresponding linkage disequilibrium in the gametic pool formed from these parents, which would also be the value in an infinitely large number of progeny born in generation t. The N individuals that produce the progeny of generation t are a random sample from the progeny born in generation t-1. The variables are listed for reference in Table 1.
VARIANCE
819
TABLE 1 Symbols ai,di, -zi qi ai cy’( x27 539 x4
Dij
pij Qi j c 2.3.
n m ‘k
L N Y t Var ( W ) Var (FS) Var ( H S ) Var ( F S I H S ) Var ( WFS) Var ( B ) Var ( T )
H
*
Se NSe
I
P CV( 1 Cov(0P) Il
*
A2
B 6J
Genotypic values of genotypes Ai& A i A r i ,A’iA’i. Frequency of allele Ai a t locus i =ai (1-2qi)d; - ai (1-2qi)d‘$ = a i (1--2qi)d,/2 Frequency of chromosomes AiAj, AiA’j, A‘iAj, A’iA’j a t loci i and i - xl-qiqi. Linkage disequilibrium in infinite gametic pool
++
=qi(l-q,)
+
qj(l-qj)
= (1-29,) (l-2qj)D$j
Recombination fraction between loci i and i Number of loci Number of chromosomes Map length of chromosome k (morgans) fn
=h=1 Z 1,. Total map length Population size
= 1- 1 / ( 2 N )
Generation Total variance within lines Variance between full-sib families Variance between half-sib families Variance between full-sib within half-sib families Variance within full-sib families Variance between lines Total variance between and within lines Heterozygosity. Also used as subscript (e.g.,Var, ( W ) ) Value in N parents (superscript, e.g., D*ii) Value in the base population (superscript, e.g., 4i) Value for selfed matings (subscript, e.g., Varse (FS)) Value for nonselfed matings (subscript, e.g., Var,,,(PS)) Value due to individual loci (subscript, e.g., Var,(W)) Value due to pairs of loci (subscript, e.g., Var,(W)) Coefficient of variation Covariance of off spring and parent Average over all possible pairs of loci of E(Pij)/Fij Average over all possible pairs of loci of E(Qij)/Fj2 Average over all possible pairs of loci of E(Dij)/Fi, Mean genotypic value 2Ncij/(2N-l).
GENOTYPIC VARIANCE
Variance within lines: Let the effects on the quantitative trait of genotypes A A , AiAi’ and Ai’Ai’ be ai,di and --ai,respectively, with additivity of effects over loci as illustrated in Table 2. At some generation t with gene and chromosome frequencies as shown in Table 2, the mean genotypic value, p , as determined by these two loci is
820
P. J. AVERY A N D W. G . HILL
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821
VARIANCE
which simplifies and readily generalizes for n such loci to n
p = .az= 1
C-ai(l-2qi)
+ediqi(l-qi)]
,
(1)
and is independent of Dij, the linkage disequilib'riumbetween loci. The genotypic variance within the line, Var ( W ), contributed by the two loci is
+. + +
+
Var ( W ) = (ai+aj) [qiqi+Dij] . (,-ai-aj) [ (l,-qi) (l-qj) Dijl -p2, = 2 [ai+& (1-2qi)]2qi (l'-qi) 2[aj+dj (1-2qj)Izqj (l,-qj) 4di24i2(l-qi) 4dj29j2(l-qj) 4[ai+di(l-2gi)] [aj+dj(l-2qj)]Dij+8didjD2ij. (2)
+
+
+
When Dij = 0, the formulae of FALCONER (1960) are obtained. TO simplify the expressions, let ai = ai+& (1-2qi), where ai is the average effect of a gene substitution at the ith locus when there is no linkage disequilibrium. (With linkage disequilibrium, the correlation of frequencies among loci renders the average effect concept less useful.) Generalizing (2) to n loci,
nn
where xz denotes summation over all values of i and j between 1 and n with 2 (1/2)ai2qi(l--gi) aiajDij 2=1 *
Table 11 gives examples of exact and approximate values for Z, for t 1, results being exact at 1 = 1. The approximation breaks down more quickly with time as c is made smaller, but the fit, however, is very good for all time with c = 0.5 and up to N generations more generally. As N increased, the approximations remain valid for similar values of t / N . Cross of two completely inbred lines. If two completely inbred lines are crossed to form a n F, and these bred to form an F, population, population size being kept large, then in this F, population, which we shall take as our base population, Z i . = 1/16 ( l , O , (1-2~)')
(36)
8%
P. J. AVERY A N D W. G . HILL
TABLE 11 Exoct and approximate values of ZT = [E (Pi t ) , E (Q i for all i and i t=5
Exact
0.5 0.1 0.02 0.5 0.1 0.02 0.5 0.1 0.02
E (Dzi t ) ] with D, = 0
t),
t=20 Exact Approx.
1=10
Approx.
Exact
Approx.
0.5994 0.6015 0.6027 0.0193 0.0812
0.5987 0.5987 0.5987 0.0193 0.0799
0.0644
0.1293
0.1270 0.0028 0.0262 0.0432
0.1366 0.0017 0.0347 0.0814
000%
0.0263 0.0435
0.3596 0.3676 0.3754 0.0116
0.1295 0.1409 0.1623 0.00442 0.0273 0.0877 0.0006 0.0183 0.0721
0.3585 0.3585 0.3585 0.0116 0.0602 0.1242 0.0017 0.0335 0.0776
0.1285 0.1285 0.1285 0.0041 0.0230 0.0624 0.0006 0.0160 0.0575
-
If c = 0.5, Z, is the same as that given for D,,= 0 with pij= 1/16 and the same approximation holds. If c # 0.5, new approximations are needed. They can again be generated as for D i j = 0 by arbitrarily assigning T,, = T,, = T3?= 0. The terms ignored are now, however, 0(1/N2) rather than O(l/N3) and thus the approximation breaks down more quickly. However, from Table 12, the fit is seen to be good for at least N/2 generations.
-
The approximations in this case are:
E(Pajt)
y2'/16,
E(D2 ) = {T31[y2('-1) L I t
(37)
+T
I
T,,TB, (T:;1
E ( Q . . ) ='lt
- yS(t-lJ)/(T
-yY"
33
.,--
33
0
-+F-$
+ T&(1-2~)*}/16,
-U')]
Tf-1 - Tt-1
7'33
y?('-lJ
T
16
33 ( ' ' 1 33
T,* - T33
33
Tt-1
(38)
-yZ(t--l)
"2
- -TZ2-y2
3i39)
TABLE 12 Exact and approximate values of ZT where the base population is the F, of two completely inbred lines (N = 10) t=2
16E (Pa3 t 1
,'
16E( P , 1
(DZ, 3I) 16E( D Z at1) 16E(Q,j t ) 16.E(Qa* t )
t=lO
1=5
C
Evact
Approx.
Exact
Approx.
Exact
Approx.
0.1 0.02
0.8239 0 8292 0.3685 07111 0.1408 0.2510
0.8145 0.8145 0.3649 07043 01408 0.2510
0.6276 0.6548 0.1935 0.5172 0.1392 0.3504
0.5987 0.5987 0.1791 0.4769 0.1363 0.3428
0.3988 0.4628 0.0918 0.3366 0.0749 0.2985
0.3585 0.3585 0.0756 0.2571 0.0676 0.2641
0.1
0.02 0.1 0.02
_-
