A Quantitative Model of the Relationship Between Phenotypic Variance and Heterozygosity at Marker Loci Under Partial Selfing

Copyright1999 by the Genetics Society of America

A Quantitative Model of the Relationship Between Phenotypic Variance

and Heterozygosity at Marker Loci Under Partial Selfing

Patrice David

CEFE-CNRS, 34293 Montpellier Cedex 5, France

Manuscript received March 16, 1999 Accepted for publication July 14, 1999

ABSTRACT

Negative relationships between allozyme heterozygosity and morphological variance have often been observed and interpreted as evidence for increased developmental stability in heterozygotes. However, inbreeding can also generate such relationships by decreasing heterozygosity at neutral loci and redistribut-ing genetic variance at the same time. I here provide a quantitative genetic model of this process by analogy with heterozygosity-fitness relationships. Inbreeding generates negative heterozygosity-variance relationships irrespective of the genetic architecture of the trait. This holds for fitness traits as well as neutral traits, the effect being stronger for fitness traits under directional dominance or overdominance. The order of magnitude of heterozygosity-variance regressions is compatible with empirical data even with very low inbreeding. Although developmental stability effects cannot be excluded, inbreeding is a parsimonious explanation that should be seriously considered to explain correlations between heterozygos-ity and both mean and variance of phenotypes in natural populations.

N

zygosity and morphological variance have been 1978;Chakraborty1987;Yezerinac et al. 1992), the

found in various species (Eanes 1978; Mitton 1978; role of genes other than the marker loci has not been

Zouroset al. 1980;MittonandKoehn1985;Yezeri- considered theoretically. This contrasts with

heterozy-nacet al. 1992;Davidet al. 1997) although not consis- gosity-fitness correlations, as various theoretical studies

tently (Handford 1980; McAndrewet al. 1982; Zink analyzed the apparent heterozygote advantage at

neu-et al. 1985; Booth et al. 1990; Livshits and Smouse tral markers due to correlations with fitness genes, i.e.,

1993;Bamshadet al. 1994; YampolskyandScheiner associative overdominance (Ohta1971;Ohtaand

Cock-1994;Gardner1995). With reference to concepts devel- erham 1974; Strobeck 1979; Charlesworth 1991;

oped byLerner(1954), these data have generally been Zouros 1993). Indirect effects on

heterozygosity-vari-interpreted as evidence for a higher developmental ho- ance correlations were not investigated as the relevant

meostasis in heterozygotes (Eanes1978;Mitton1978; models (Chakraborty and Ryman 1983;

Chakra-LivshitsandKobyliansky1985;MittonandKoehn borty1987) only considered infinite panmictic

popula-1985). Under this hypothesis, changes in phenotypic tions that lack correlations among loci. In this case

rela-variance among genotypes reflect changes in rates of tionships between phenotype and marker loci reduce

developmental error. Individual phenotypes are there- to their direct phenotypic contribution (Smouse1986).

fore considered as random variables with a fixed expec- Two factors promote genotypic correlations: (i) linkage

tation (the population mean) and different variances, desequilibrium due to small population size (Hilland

depending on heterozygosity. Alternatively, individuals Robertson 1968) and (ii) partial inbreeding, which

may have different expectations, their developmental _{generates identity desequilibria (i.e., heterozygosity}

cor-variance being constant or negligible. This was first pro- _{relations; Bennett and Binet 1956; Weirand}

Cock-posed byChakrabortyandRyman(1983), who mod- _{erham 1973). I here focus on partial inbreeding and}

eled a phenotype determined by additive alleles. In this _{its consequences on heterozygosity-variance and}

hetero-case, heterozygotes are phenotypically intermediate be- _{zygosity-mean correlations. I examine the effects of}

dif-tween homozygotes. As homozygotes include two geno- _{ferent genetic determinisms of the phenotype}

(additiv-types with different means, their phenotypic variance _{ity, dominance, and overdominance) and of the}

exceeds that of heterozygotes. With several diallelic loci, _{number of marker loci.}

this is amplified by a combinatory effect as many differ-ent genotypes are included in homozygotes for n loci

(ChakrabortyandRyman1983;Chakraborty1987). GENETIC VARIANCE AND MARKER GENOTYPES IN A PARTIALLY INBRED POPULATION

Distribution of phenotypic values among inbreeding

Address for correspondence: CEFE-CNRS, 1919 route de Mende, 34293

Montpellier Cedex 5, France. E-mail: [email protected] classes:I consider an infinite population at inbreeding

equilibrium. A proportion s of offspring are produced where the additive and dominance variance under ran-dom mating are VA5Ri2piqi[ai2di(pi2qi)]2and VD5

by selfing, whereas (12s) come from random matings.

Each individual belongs to a given inbreeding class k, Ri4p2iqi2d2i(FalconerandMackay1996), respectively.

Distribution of phenotypic values among genotypes: being the number of generations of selfing in his

pedi-gree, starting from the last outcrossing event (0# k# The genotype G also contains various inbreeding classes,

but the frequency Gk of class k within G differs from

∞). Class k has frequency (12s)sk_{assuming no selection.}

Although inbreeding depression usually occurs in par- the population frequency. For example, homozygous

genotypes contain high frequencies of inbred

individu-tially inbred populations (Charlesworthand

Charles-worth 1987), neutral frequencies computed using a als (large k). The mean and variance within G are there-fore

lower value s9(detailed inappendix a) provide a good

approximation of this case. s may be replaced by s9to _φ

G5

o

Gkφk (6)

account for selection in the expressions below.

We assume a neutral phenotypeφdetermined by L

and quantitative trait loci (QTL). Each QTL i has two alleles

Aiand Biwith respective frequencies piand qi. Its contri- s2G(φ)5 s2(within inbreeding classes)

butions to the phenotype are ai, di, and2aifor genotypes _{1 s}2_{(among classes)}

AiAi, AiBi, and BiBi, respectively, and are additive over

loci. Genotypes are assessed at a set of marker loci (ML) 5

o

k

Gks2k1

o

Gk(φk2 φG)2. (7)

different from the QTL. Linkage equilibrium is assumed

for all pairs of loci (QTL and ML) as a consequence of Using these general formulas, I now examine the

differences in phenotypic mean and variance among infinite population size. Throughout the article, I focus

on the simplest case of unlinked loci, as linkage merely genotypes at the ML and especially their relationship

to heterozygosity. I first consider the case of a single

reinforces the patterns (appendix d).

diallelic ML and then extend to multiple loci. Heterozygosity-variance and heterozygosity-mean

cor-The mean phenotype of a genotype G (Equation 6), relations appear if genotypes at QTL and ML are

corre-and the within-class component of its variance [first lated. In infinite, partially selfing populations,

correla-term of (7)], can be found from Equations 4 and 5, tions among loci result from the mixing of different

respectively, replacing fkby fG(the average inbreeding

inbreeding classes. Within each class, genotypes at

un-coefficient of genotype G) and f2

k by fG2 (the average

linked loci are not correlated. Therefore, phenotypic

squared inbreeding coefficient of G), defined as mean and variance within a genotype G at the ML are

determined by the relative proportions of the different

fG5

o

Gkfk (8a)

inbreeding classes within G.

Within class k, genotypic frequencies at QTL i are

f2

G 5

o

Gkfk2. (8b)

AiAi AiBi BiBi

p2

i 1 fkpiqi 2piqi(12fk) q2i 1 fkpiqi

, (1) The phenotypic mean of G is therefore

φG5φ02 fG

o

(2piqidi) (9)

where fk5121/2kis the usual inbreeding coefficient.

The mean and variance of phenotypic contributions

and the phenotypic variance, adding the among-class from QTL i within class k are therefore

to the within-class component, is

φki5 ai(pi2 qi)1diHki (2) _s2

G5 s201fG(

o

(4piqia2i)2VA2fGVD) and

1 s2

f(G)

o

o

(8pipjqiqjdidj), (10)

s2

ki52piqi(11fk)ai21 di2Hki(1 2Hki)

wheres2

f(G)5 fG22 (fG)2.

22aidi(pi2qi)Hki, (3)

One marker locus: Let us consider a ML with two

where Hki 5 2piqi(1 2 fk). As the contributions from

alleles M and m of respective frequencies P and Q. At

various QTL are independent within a given inbreeding _{equilibrium, the frequency of (Mm) is 2PQ[12}

s/(22

class, the phenotypic mean and variance of class k are _{s)] in the population (Hartl and Clark 1997) and}

simply φk5 Riφki andsk25Ris2ki, respectively. With ref- _2PQ(1/2)k _{within class k. Therefore the frequency of}

erence to their values for outbred individuals, φ0 and _{class k among Mm individuals is}

s2

0, and to classical terms of additive and dominance

variance,

[Mm]k5

2PQ(1/2)k

sk

(1 2s)

2PQ[12s/(22s)] (11)

φk5φ02 fk

o

(2piqidi) (4)

5 (s/2)k₍₁2 _s/2).

s2

k5 s202 fkVA 2fk2VD1fk

o

(4piqia2i), (5)

[MM]k5 sk(12 s)

12Q/2k

12Q[12s/(22 s)] (12)

and a similar expression for [mm]k, replacing Q by P in

(12). From these frequencies we obtain

fMm5s/(42s) (13)

f2

Mm5

s(41s)

(42 s)(82s) (14)

fmm5

s[21s1 2Q(12s)]

(42 s)[s12Q(12s)] (15)

f2

mm5

s[8(11Q)16s(22 Q)1s2₍₁ 2_2Q)]

(42 s)(82s)[s12Q(12s)] (16)

and similar expressions for MM, replacing Q by P in

(15) and (16). Clearly, f and f2_{are minimal for}

heterozy-gotes and maximal for the rarest homozygote.

Expressions (9) and (10) allow any kind of

determina-Figure1.—Ratio of phenotypic variances (pooled

homozy-tion of the phenotype by QTL. We focus on three cases _{gotes/heterozygotes) for additive QTL and one marker locus}

of interest: additivity, directional dominance, and sym- _{as a function of the allele frequency at that locus (P).}

metric overdominance.

Additivity: The genetic variance is entirely additive

when di50 for all QTL. In this case (8) gives Dominance: Dominance at multiple loci is modeled

by setting di5aifor all i (complete dominance). In this

φk5

o

ai(pi2qi)5 φ0. (17)

case the phenotypic mean of class k is a monotonic function of k:

Therefore the phenotypic mean does not vary among

inbreeding classes and/or genotypes. The phenotypic φk5 φ02 fk

o

i

(2piqiai). (21)

variance within G reduces to

An interesting case is directional dominance, when

s2

G5(11 fG)s20, (18)

the phenotype decreases following homozygosity for

re-whereas the population variance is cessive alleles. This may apply to fitness traits

(deleteri-ous recessives) or to any morphological trait subject to

loss-of-function mutations (Roff1997). In this case, all

s25 2

22 ss

2

0. (19)

aiare positive and the mean phenotype decreases with

inbreeding.

Remarkably, the ratio s2

Mm/s2 5(42 2s)/(42s) is

The phenotypic variance within class k is a function of the selfing rate only. As this ratio is always

,1, the phenotypic variance in heterozygotes is always _s2

k5 s201fk(

o

(4piqia2i)2 VA 2fkVD) less than the population variance. The variance ratio

between MM and Mm is _{5 s}2

01fk

o

(4piqia2i)(12 2q2i 2 fkpiqi). (22)

s2

MM/s2Mm5

(42s)24Q(12s)

(422s) 24Q(12s). (20) When inbreeding k increases, the contribution of

lo-cus i to the change in phenotypic variance is

propor-From (20), the variance of homozygotes exceeds that _{tional to fk(122qi}22_f

kpiqi), an increasing function of

of heterozygotes by a maximal factor of 1.5. The differ- _{k for pi . 0.5. When the phenotype is a fitness trait,}

ence is more pronounced for rare homozygotes as (20) _{recessive alleles are rare (pi ≈ 1), and the variance}

in-increases with Q. _{creases with inbreeding.}

The variance of heterozygotes is often compared to _{Phenotypic mean and variance for a genotype G are}

that of pooled homozygotes (MM1mm) (Eanes1978;

φG5 φ02 fG(

o

2aipiqi) (23)

Mitton1978;King 1985), which is simply a weighted

average ofs2

MM ands2mmas both have the same

pheno-and typic mean. The variance ratio between homozygotes

and heterozygotes is plotted in Figure 1. This ratio, for _s2

G5 s201 fG

o

[4a2

ipiqi(122q2i)]

a given s, varies little with the allelic frequency at the

ML and reaches its maximum value (11 s/2) at P5 ₂

(fG)2(

o

2aipiqi)21 8f2G

o

o

aiajpipjqiqj. (24)

As phenotypic means are negative linear functions

of fG, heterozygotes have the highest (and rare

homo-zygotes the lowest) phenotypic mean. This is the

well-known “associative overdominance” phenomenon (Ohta

andCockerham1974;Charlesworth1991). Pheno-typic means under partial selfing were previously

de-rived byHoule(1994), using the adaptive distance

for-malism invented by Smouse (1986). Mean fitness was

predicted to be a negative, linear function of adaptive distances (defined as 0, 1/P, and 1/Q for genotypes

Mm, MM, and mm, respectively). Houle’s (1994) ap-proximations included discrete inbreeding depression (the fitnesses of inbreds and outbreds are constants) and low selfing (negligible recurrent inbreeding). The present results allow for a variance in fitness among inbreds and among outbreds, arbitrary values of s, and

recurrent inbreeding. For small s, fGand adaptive

dis-tances are linearly related, in agreement with Houle

(1994).

Phenotypic variances also vary among genotypes. As-suming, for simplification, directional dominance, iden-tical allele frequencies (p, q), and phenotypic effect (a) for all QTL (24) becomes

s2

G5 s20   11

1

q(11 q)[fG(12 pq fG22q

2₎

1pq(L 21)s2

f(G)]

 

. (25) _{zygotes (MM}Figure2.—The ratio of phenotypic variance among homo-_{1 mm) to the variance among heterozygotes}

(Mm) at a marker locus, with directional dominance. The

With small frequencies of recessives (q , 0.5), the _{graphs are given for two values of q, the frequency of recessive}

alleles, and four values of L, the number of loci that contribute

variance increases with fG and s2f (G), both minimum

to phenotypic variation. The allele frequency at the marker

for heterozygotes (Mm), except for very large s. Figure

locus has little effect on variance ratios (data not shown) and

2 shows that for s,0.75, the variance among

homozy-is set at P50.5.

gotes exceeds that among heterozygotes, sometimes considerably (ratio of 3, with many QTL and/or rare recessive alleles).

s2

hom/s2het5

31(12 s)(124PQ)

11(12 s)(124PQ). (27)

So far, selection at QTL has been neglected. However, correlations between the phenotype and fitness have

two consequences: (i) The equilibrium frequency of This ratio stays between 2 and 3 (when P 5 0.5) in

deleterious alleles is small and (ii) it varies across in- agreement with Figure 2.

breeding classes, as purging selection occurs following The consequences of (ii) are complex. Although the

repeated selfing; (i) means that we need consider only computations assume constant allelic frequencies,

dele-the bottom part of Figure 2 (q50.01). In this case, the terious alleles decrease in frequency in selfed lineages.

homozygote-to-heterozygote ratio of phenotypic vari- However, this effect is small (appendix b).

ances takes high values (near 3) over a large range of Symmetric overdominance: Overdominance occurs

s and L. Indeed, assuming very small qi’s (relative to 1/ when di.|ai|. I focus on the simplest case of symmetric

L and s), the variance becomes proportional to the overdominance, i.e., ai50 for all i. Although this looks

average inbreeding level of G: unrealistic, the discrepancy with real situations may be

small for two reasons. First, very asymmetrical

overdomi-s2

G≈4fG

o

qia2i. (26) _{nant polymorphisms are usually instable with partial}

selfing (KimuraandOhta1971, pp. 190–196). Second,

This agrees withDavid’s (1997) prediction that the asymmetry results in extreme allele frequencies (near

variance in fitness traits increases linearly with adaptive 0 or 1) when a stable equilibrium exists. Such loci

con-distance for small s. Indeed, adaptive con-distances are lin- tribute little genetic variance.

early related to fGin this case. More generally (for any With symmetric overdominance, phenotypic means

s), the homozygote-to-heterozygote ratio in phenotypic decrease with inbreeding (Equation 4), and so do vari-ances:

s2

k5(1 2fk)(s021fkVD). (28) zygosity being retained in inbred lines than predicted

under neutrality. This introduces little change to the This is expected as heterozygotes for QTL are

progres-above conclusions (appendix c).

sively eliminated from increasingly inbred classes; only

Several marker loci:Most studies on

heterozygosity-homozygotes eventually remain, which have

homoge-variance relationships use several ML (typically 5–10; neous phenotypes.

Zouroset al. 1980;Yezerinacet al. 1992;Livshitsand The mean phenotype of genotype G (Equation 4,

Smouse1993;Bamshadet al. 1994;Davidet al. 1997).

replacing fkby fG) is maximal for heterozygotes (Mm)

As the number of possible genotypes increases rapidly and minimal for rare homozygotes. The variance is

(3M_{with M diallelic loci), multilocus genotypes are}

usu-ally pooled into heterozygosity classes. The phenotype

s2

G5 s202fGVA2fG2VD1 s2f(G)(

o

2piqidi)2. (29)

and its variance (or absolute deviation from the mean) are then regressed on multiple-locus heterozygosity Although the within-class component (three first terms)

(MLH). is maximized in heterozygotes, the among-class

compo-The general expression of phenotypic variance

nent (fourth term) is usually (for s, 0.8) minimized

(Equation 10) still applies for multilocus genotypes or in heterozygotes. Heterozygosity-variance relationships

MLH classes. Regressions of phenotypic mean and

vari-depend on L, the number of QTL. With L5 1, the

ance on MLH are completely determined by f(h) and

variance increases with heterozygosity. However, for

large L, the among-class component dominates the f2

(h), the average inbreeding coefficient and average

within-class one, and the variance decreases with hetero- squared inbreeding coefficient of class h, respectively.

zygosity. In the simple case of M ML with identical genetic

diver-Overdominance is mainly cited for fitness traits, and sity H0(5 2PQ), they turn out to be

selection must be taken into account. The first

conse-quence of overdominant selection is to keep allele fre- _f

(h)5

s u(h,s)

o

M2h

i50

Ci M2h

(22H0)i (2h1i₂

s)(2h1i11_2s) (31)

quencies constant. With symmetric overdominance, pi

and qiapproach 0.5 (VA≈0). Assuming constant

locus-and

specific effects (di5 d for all i’s), we obtain

s2

G5 s20[1 2( fG)2 1(L2 1)s2f(G)]. (30) _f(h)2 5 s

u(h,s)

o

M2h

i50

Ci M2h

The homozygote-to-heterozygote ratio of phenotypic

variances depends on both s and L (Figure 3). In the _· (22H0)i(2h1i111s)

(2h1i2_s)(2h1i112_s)(2h1i12 2_s), (32)

usual range of s (0–0.3), the ratio is very slightly ,1

with one QTL. However, with two or more QTL, it is

where u(h,s) 5 RM2h

i50CiM2h((22H0)i/(2h1i2 s)) and

.1 and becomes quite large when L increases.

Cp

n5 n!/p!(n2 p)!.

Purging selection does not occur with

overdomi-Equations 31 and 32 are decreasing functions of h. nance. However, overdominance results in more

hetero-As expected from single-locus results, the relationship between heterozygosity and phenotypic mean is absent in the additive case and positive in the dominant and overdominant cases. Heterozygosity-variance relation-ships are negative under additivity and under domi-nance (Figures 4 and 5). Under overdomidomi-nance (Figure 6), this relationship may be positive, negative, or present a maximum at intermediate MLH values, depending on

L and s. The most realistic situation for empirical studies

is low s and L @ 1, in which case the relationship is

predominantly negative.

In all cases the differences in variance are higher than in the single-locus case, because a larger range of f and

f2_{is explored. The orders of magnitude depend on the}

determinism. For low s (0.05–0.1), the variance ratio between extreme MLH classes is of the order 1–2 under additivity, 5–10 under overdominance, and 30–60 under dominance. In all cases, heterozygosity-mean and het-Figure3.—The ratio of phenotypic variance among homo- erozygosity-variance relationships are nonlinear. An

as-zygotes (MM 1 mm) to the variance among heterozygotes _{ymptote exists for high MLH as very heterozygous}

geno-(Mm), with symmetric overdominance. The graphs are given

types contain no inbreds, and their phenotypic mean

for four values of L, the number of loci that contribute to

and variance approach those of a random-mating

popu-phenotypic variation. The allele frequency at the marker locus

Figure 4.—Relationship between heterozygosity at 10 marker loci (each with H05 0.5) and phenotypic variance

under additivity. The phenotypic variance is in units ofs2 0,

the phenotypic variance in the absence of inbreeding.

DISCUSSION

Our aim was to derive the relationships between het-erozygosity at neutral marker loci and the mean and variance of arbitrary phenotypes in infinite, partially inbred populations. In this section, I focus on three questions. First, how do the results on phenotypic means compare with and/or extend previous results? Second,

Figure5.—The change in mean (A) and variance (B) of

what are the main conclusions regarding heterozygosity- _{a phenotype under dominance with increasing heterozygosity}

variance relationships? Third, what is their relevance to _{at 10 marker loci (each with H050.5). A is valid for arbitrary}

empirical results? number, locus-specific effects, and allele frequencies of QTL.

The phenotypic mean under random mating (φ0) has been The relationship between heterozygosity and

pheno-arbitrarily set to 0 and the graphs are in units of R(2piqiai)

typic mean in partially inbred populations:This model

(see Equation 23). B is represented for 100 QTL loci with

explicitly relates the phenotypic mean with fG, the aver- _{identical effects on phenotype and allele frequency q}5_0.01.

age inbreeding level of a genotype. This relationship is _{Graphs are in units of s}2

0, the phenotypic variance under

linear, resulting in a simple prediction: whenever a trait random mating. Range of s values as in Figure 4.

is negatively affected by inbreeding (through overdomi-nance, directional domioverdomi-nance, or both), phenotypic

means at ML rank as heterozygote.common

homozy-loci byDavid(1997)] relied on restrictive assumptions:

gote.rare homozygote. This has long been established

small s and discrete inbreeding depression, neglecting

by simulations (Charlesworth1991;Pamiloand

Pals-the variance arising from Pals-the detailed genetic

architec-son1998) or analytically (OhtaandCockerham1974;

ture of the trait. This study agrees with previous results,

Houle1994), and has been repeatedly suggested as the

as the fitness predictors derived byHoule(1994;

adap-probable origin of heterozygosity-fitness correlations

tive distances) andDavid(1997;gcoefficients) are

lin-(Houle1989;Davidet al. 1997). However, alternatives,

early related with fG in their domain of applicability.

based on direct overdominance (see Mitton and

However, fGis more general, accounting for recurrent

Grant1984) or multiple-locus dominance (Dengand

inbreeding and arbitrary genetic architecture.

Fu1998), have also been cited. The occurrence of

posi-One of the predictions obtained using fGis that

hetero-tive heterozygosity-fitness correlations, resulting from

zygosity-phenotype relationships are nonlinear. They partial inbreeding, has indeed been referred to as a

saturate because very heterozygous genotypes are almost

general effect (Davidet al. 1995), a special case of the

exclusively outbred. Although intuitively predictable,

associative overdominance hypothesis (Ohta 1971;

this feature is usually ignored as most empirical studies

Zouros1993). The other case is the local effect arising

use linear regressions. Of course, individuals with ex-from linkage disequilibrium in small populations (see

treme MLH are rare and very likely lost from a sample

David1998 for review and definitions).

of reasonable size. However, saturation at high MLH This study extends previous results, providing a

sim-values is visually perceptible in some instances (Strauss

ple method (the computation of fG) to predict

genotype-1986;Davidet al. 1997).

phenotype relationships under partial inbreeding.

pheno-types evenly distributed among classes have higher vari-ance. The relative importance of (i) and (ii) depends on the genetic architecture.

Very generally, heterozygotes appear to have lower variance than homozygotes in partially inbred popula-tions. This applies to phenotypes coded by additive or directionally dominant alleles as well as by overdomi-nant alleles in large regions of the parameter space including the most realistic situations. Additivity is ex-pected for neutral phenotypes, directional dominance and/or overdominance being more common for fitness

traits (Roff1997). Negative heterozygosity-variance

re-lationships have been predicted by Chakraborty

(1987) in large, random-mating populations, but only when the marker genes themselves have additive effects on the phenotype (a quite unlikely situation). In non-random mating populations, negative heterozygosity-variance relationships have been predicted for fitness

traits by David (1997) but with restrictive conditions

(see above), and without evaluating the influence of the genetic architecture, the number of genes involved, and the inbreeding rate.

The genetic architecture largely affects the relation-ships. Directional dominance results in large differences in variance among genotypes, one order of magni-tude higher than the differences under overdominance, which in turn are larger than under additivity. Although details of genetic architecture are usually unknown, a broad consequence is that the influence of heterozygos-ity on variance will be larger for traits highly correlated with fitness (mostly under dominant or overdominant gene action) than for traits loosely or not correlated with fitness.

Figure 6.—Heterozygosity at 10 marker loci (each with _{The number of phenotype-coding loci (QTL) affects}

H0 5 0.5) and phenotypic variance under symmetric over- _{the relative importance of the within- vs. among-class}

dominance for three numbers of overdominant QTL: (A)

components of variance. When many genes affect a trait,

L51; (B) L510; and (C) L5100. Range of s values as in

the stochastic variation in the overall sum of their effects

Figure 4.

is buffered, and all classes have low phenotypic vari-ance. Increasing the number of QTL (L) thus reinforces the among-class component of variance compared to typic variance in partially inbred populations:Until now,

the within-class component [see (25) and (30)]. This changes in phenotypic variance with inbreeding have

effect is nonexistent under additivity (the among-class been mostly envisaged in artificial, homogeneously

in-component is zero) and weak for dominant fitness traits

bred lines (Robertson 1952; Falconer andMackay

1996). Recently, it has been recognized that the pat- [because of low frequencies of recessives—see (25)].

tern of variation of mean and variance of fitness traits However, it is important under overdominance.

An-upon controlled inbreeding provides information on _{other consequence under overdominance is that when}

the underlying determinism such as selection coeffi- _{enough polymorphic marker loci are considered, the}

cients and dominance (Deng1998). This method was _{variance no longer decreases uniformly with}

heterozy-designed for either outcrossing or selfing populations _{gosity. However, with an average scoring effort (10}

poly-at mutpoly-ation-selection equilibrium. However, npoly-atural

morphic loci or less), this is perceptible only in largely populations are often heterogeneous because of partial

inbred populations. inbreeding. This generates genotype-phenotype

cor-Inbreeding rate also influences the patterns. When s relations based on the distribution of genotypes among

increases, the decrease in variance with heterozygosity various inbreeding levels. These distributions determine

is steepened under additivity and dominance. However, phenotypic variances, because (i) inbreeding classes

under overdominance, increasing s rather tends to shift have different variances (within-class component),

the maximum variance toward more heterozygous therefore genotypes concentrated in high-variance

classes (Figure 6), although heterozygosity-variance re-classes have higher variance and (ii) re-classes have

Moreover, the initial increase may be overlooked be-cause very homozygous classes are often pooled to-gether. Although higher s may yield variable outcomes, highly selfing species are rarely examined, being essen-tially devoid of heterozygosity and inbreeding

depres-sion (CharlesworthandCharlesworth1987).

The relevance of the model to empirical studies in

natural populations:This study involves usual

assump-tions of quantitative genetics and inbreeding depression models. The assumption of homogeneity in locus-spe-cific effects is unrealistic. However, mixtures of different

Figure7.—Relationship between heterozygosity at nine

al-determinisms have not been studied because of the

lozyme loci and variance in growth, measured as t1/2, the age number of situations to explore (Equation 10 provides

at half the maximum size in years, in the marine bivalve Spisula

a general formula). Negative heterozygosity-variance

re-ovalis (seeDavidet al. 1997). Classes MLH50 and MLH5

lationships are expected in mixed situations, because _{1 have been pooled to increase sample sizes. Error bars are}

each determinism separately produces such relation- 95% confidence intervals (slightly underestimated due to the

nonnormality of distributions). The dotted line indicates

theo-ships. The assumption of linkage equilibrium is not

retical predictions under partial inbreeding and directional

significantly violated in large populations, such as

ma-dominance (Equation 25) with L5100 and q50.01. A value

rine bivalves or trees (often used in empirical studies; _{of s50.05 has been chosen because this species has separate}

cf.Britten1996), even if QTL may be physically linked. _{sexes and natural levels of inbreeding must be small. The}

The main difference between our model and actual number of marker loci has been set to the observed value (9)

and, for simplicity, all loci were assumed to have identical

situations lies with the environmental variance VE. The

heterozygosity (Equations 31 and 32) equal to the geometric

model considers only genetic variance, although all

mean of observed values (geometric rather than arithmetic

traits display various levels of nongenetic variation. VE _{was used as heterozygosities interact multiplicatively rather}

will reduce relative changes in variance among geno- _{than additively in Equations 31 and 32). The graph has been}

types, adding a constant to all. Furthermore, sampling scaled to the observed values by settings2

050.02. variance must also be considered. Extreme

heterozygos-ity classes have low sample sizes, hampering the

estima-tribution of genetic variance upon inbreeding are not tion of phenotypic variance. A robust procedure is to

easily distinguished. In practice, genotypic replicates, pool extreme heterozygosity classes up to reasonable

such as clonemates, are needed. Within-clone variance

sample sizes (King1985;Davidet al. 1997).

increased in inbred lines of the facultative parthenogens Despite these potential problems, negative

heterozy-Daphnia pulex and D. pulicaria (Deng1997), suggesting gosity-variance relationships have been repeatedly

ob-that inbreeding enhances developmental instability. Yet served in natural populations (see references in

Intro-YampolskyandScheiner(1994) found no relationship duction). Most of them have been interpreted as

between MLH and within-clone variance in D. magna, evidence for a direct role of enzyme heterozygosity in

either because natural inbreeding is too low or because maintaining developmental stability without reference

the results on pulex and pulicaria do not extend to magna. to a possible genetic variation of the traits. As shown by

The two sides of a given individual are often used as our model, partial inbreeding is a plausible alternative,

substitutes for clonemates (e.g.,Learyet al. 1983).

How-especially for fitness traits. This is illustrated by a data

ever, the overall evidence for negative correlations be-set on heterozygosity at nine enzyme loci and growth

tween heterozygosity and fluctuating asymmetry ap-in the bivalve Spisula ovalis (Figure 7, modified from

pears weak (Britten 1996). In conclusion, partial

Davidet al. 1997). Although the similarity between

ex-inbreeding can generate MLH-variance relationships by perimental and theoretical graphs obviously depends on

two nonexclusive ways: altering developmental stability an adequate choice of parameters, the observed pattern

and redistributing genetic variance. The former relies (shape of the heterozygosity-variance relationship,

or-on assumptior-ons whose generality and quantitative im-ders of magnitude) is consistent with the inbreeding

portance remain to be properly established. The latter hypothesis under dominance. In this instance,

overdom-always occurs, and, in a variety of realistic situations, inance or additivity cannot generate effects of the

ob-may produce detectable patterns qualitatively and quan-served magnitude.

titatively similar to empirical results obtained to date. Although the redistribution of genetic variance upon

I thank P. Jarne for helpful comments on the manuscript.

inbreeding is sufficient to explain large changes in va-riance with MLH, developmental stability could also vary. Even if direct effects of allozymes on

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the marine bivalve molluscs Mytilus edulis (L.) and M. galloprovin- APPENDIX A: APPROXIMATIVE DISTRIBUTION OF cialis (Lmk.) from South-West England. Biol. J. Linn. Soc. 54: _{INBREEDING LEVELS WITH SELECTION} 71–86.

Without selection, inbreeding classes 0, 1, . . . , k have

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morphologi-cal variation. Nature 286: 261–262. _{frequencies (1}2_{s), s(1}2_{s), . . . , s}k

(12s). However,

Hartl, D. L.,andA. G. Clark,1997 Principles of Population Genetics.

mean fitness wkusually decreases when inbreeding (k)

Sinauer Associates, Sunderland, MA.

increases. Therefore the equilibrium distribution of k

Hill, W. G.,andA. Robertson,1968 Linkage desequilibrium in

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gaster. Genetics 123: 789–801.

resembles a decrease in selfing rate s to a lower value

Houle, D.,1994 Adaptive distance and the genetic basis of heterosis.

Evolution 48: 1410–1417. _s9_{. Let us compute s}9 _{values that provide reasonably} Kimura, M.,andT. Ohta,1971 Theoretical Aspects of Population Genet- _{accurate distributions of k.}

ics. Princeton University Press, Princeton, NJ.

Let the frequency of class k be lk. Assuming w05 1,

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L.). Heredity 54: 289–296.

Leary, R. F., F. W. AllendorfandK. L. Knudsen,1983 Develop-mental stability and enzyme heterozygosity in rainbow trout.

Na-ture 301: 71–72.

1

l0

l1

… lk

…

2

t11

51 wt

1

(12s) (12s) … (12s) (12s)

sw1 0 0 0 0

0 … 0 0 0

0 0 swk 0 0

0 0 0 … 0

2

·

1

2

Livshits, G.,andE. Kobyliansky,1985 Lerner’s concept of devel-opmental homeostasis and the problem of heterozygosity level

in natural populations. Heredity 55: 341–353. _(A1)

Livshits, G.,andP. E. Smouse,1993 Relationship between

fluctuat-or mfluctuat-ore concisely,

ing asymmetry, morphological modality and heterozygosity in an elderly Israeli population. Genetica 89: 155–166.

McAndrew, B. J., R. D. WardandJ. A. Beardmore,1982 Lack of _L

t11 5 1

wt

S ·Lt, (A2)

relationship between morphological variance and enzyme hetero-zygosity in the plaice, Pleuronectes platessa. Heredity 48: 117–125.

Mitton, J. B.,1978 Relationship between heterozygosity for enzyme _{where w}

tis the mean fitness at time t. The population

loci and variation of morphological characters in natural

popula-eventually reaches a steady state, the increase in k

tions. Nature 273: 661–662.

elimination of inbred individuals. The equilibrium q(k) in inbreeding class k. The deficiency in

heterozy-mean fitness w∞ is the real eigenvalue of S and the gous (AB) genotypes within class k compared to

random-distribution of k at equilibrium is the corresponding mating expectations is noted f(k). Without selection,

unitary eigenvector L∞. Although w∞ cannot be com- q(k)5q for all k, and f(k)5121/2k_{. However, if the}

puted analytically, numerical values can be obtained, fitnesses of AA, AB, BB are 1, 1, 12z, respectively, and

restricting S to a reasonable dimension (say, kmax510) ignoring mutation, q(k) and f(k) change from class (k)

and setting values to the wk’s according to a selection to class (k1 1) (one generation of selection followed

model. L∞ can then be computed using lk,∞ 5 ((1 2 by selfing),

s)sk_/wk_∞11₎

p

The approximation should conserve the crucial char- _q(k11)5q(k) 211w(k)

w(k) (B1)

acteristics of the distribution of k for

heterozygosity-variance relationships, i.e., the mean and heterozygosity-variance of fk.

The former is f∞5 R∞k50lk,∞(1 21/2k) at equilibrium. f(k1 1)512[1 2f(k)] q(k)

2q(k1 1), (B2)

We need only the first few terms for numerical estima-tions.

where w(k)5 12 z[q(k)21_{f(k)p(k)q(k)] is the mean}

As the approximate distribution of k is constructed

fitness of class k. As expected, q(k) decreases when k

as the neutral distribution under a selfing rate s9, its

increases, and selection slows down the increase in f(k).

mean f is the classical s9/(22s9). This will be equal to

Although recurrent mutation must exist to maintain the

the true f∞provided s9 52f∞/(11f∞).

B allele, it only tends to restore neutral frequencies.

The accuracy of this approximation was tested under

Therefore, ignoring mutation leads only to

overestima-a simple multiplicoverestima-ative selection model: wk5(12 a)fk

tion of purging effects. (0, a ,1). s9was determined numerically as explained

Plugging (B1) and (B2) into (4) and (5) does not

above for a variety of s (0.05–0.8) anda (0–1) values.

yield a simple expression of phenotypic mean and

vari-Stronger inbreeding depression (a z 1) generates

ance within class k. I therefore obtained numerical

esti-larger reduction (s9 , s) in apparent inbreeding,

al-mates (iterating B1 and B2) and compared them to though the absolute value of s has little influence on

neutral predictions. The expected homozygote-to-het-this reduction (data not shown). The variance and third

erozygote ratio of phenotypic variances (s2

hom/s2het) was central moment of f for the approximate distribution

calculated numerically (neglecting inbreeding classes are compared to their numerical values under selection

with k.25) with and without selection. P was set at 0.5

in Table A1. The approximate distribution is generally

and q at 0.01. All combinations of three s (0.05, 0.1, very close to the actual one. The largest discrepancy

0.3) and three z (0.05, 0.1, 0.3) values were considered. is obtained in an unlikely situation (high inbreeding

The simulations without selection perfectly matched an-depression and high selfing rate).

alytical predictions (Equation 25). Selection reduced the ratio (s2

hom/s2het) by an amount ranging from 0.2%

(s5 0.05, z 5 0.05) to a maximum of 8.3% (s5 0.3,

APPENDIX B: THE CHANGE IN ALLELIC

FREQUENCIES AND HOMOZYGOSITY OF A z 5 0.3, an unlikely situation). The same parameter

DELETERIOUS ALLELE WITH INBREEDING _{combinations were tested with several QTL (L510 and} UNDER PURGING SELECTION

100) in linkage equilibrium, and the effect of

selec-tion on (s2

hom/s2het) always remained within the same

Consider a single locus with two alleles A, B of

respec-tive zygotic frequencies p, q in the population, and p(k), range (0 to28.5%). Therefore purging selection does

TABLE A1

Comparison of the variance and third central moment offbetween the actual situation (inbreeding sand selectiona) and the approximation (inbreedings9, no selection)

s: 0.1 0.1 0.5 0.5

a: 0.9 0.5 0.9 0.5

s9 0.0332 0.0716 0.2118 0.3855

f∞ 0.0169 0.0372 0.1185 0.2387

s2_{( f )actual 0.0083 0.0181 0.0512 0.0958}

s2_{( f )approx. 0.0084 0.0182 0.0551 0.1006}

Third central moment, actual 0.0039 0.0083 0.0174 0.0217 Third central moment, approx. 0.0041 0.0085 0.0213 0.0252

not considerably affect heterozygosity-variance relation- additively. Thus, relationships between genotypes and phenotypic means [(4) and (9)] stay unchanged, ships.

provided ML and QTL remain unlinked.

Charles-worth (1991) predicted slight differences (a few

APPENDIX C: THE CHANGE IN HOMOZYGOSITY _{percent) in associative overdominance between} un-AT SYMMETRICALLY OVERDOMINANT LOCI

linked and completely linked (l 51) QTL, probably

WITH INBREEDING

because QTL effects were considered multiplicative

Allele frequencies at symmetrically overdominant loci _{instead of additive and purging selection was not}

are p5q51_⁄

2, irrespective of inbreeding. Without selec- _{neglected. Linkage among QTL affects the}

within-tion, heterozygosity is halved following every generation _{class component of variance more seriously,}

of selfing. Under overdominance, the decrease in

het-s2

G(linked)5 s2G(unlinked) (D3)

erozygosity will be slower as heterozygotes have higher

viability than homozygotes (say, 11z for genotype AB ₁₈

_o

i

o

pipjqiqjdidjhij,G,

and 1 for AA and BB),

where

f(k1 1)512[1 2f(k)](11z)

2w(k) , (C1) _h

ij,G5

o

Gkhijk (D4)

where w(k)5 11 z[12 f (k)]/2 and f(k) is the zygotic

ands2

G(unlinked) is given by (10).

heterozygote deficiency within class k. As inappendix

From D3, linkage among QTL has no effect on

b, numerical simulations (up to k5 25) were used to

purely additive phenotypes (di50). However, it does

estimate the influence of selection as (C1) does not

affect heterozygosity-variance relationships under

di-yield a tractable expression of (s2

hom/s2het). Simulations

rectional dominance or overdominance. The exact using (C1) and parallel simulations using neutral f

distri-expressions for thehij,G’s, obtained from D4, D2, 11,

butions were performed for all combinations of three

and 12, are complicated. First-order approximations

s (0.05, 0.1, 0.3) and z (0.05, 0.1, 0.3) values, using 1, 2,

are

5, 10, and 50 QTL and P50.5. The neutral simulations

perfectly matched analytical predictions from (30). The

hij,Mm5 l2ijs/8 10(s2) (D5)

other results (not shown) indicate that (30) tends to

exaggerate absolute changes in variance between het- _and

erozygotes and homozygotes. Thus, (s2

hom/s2het) is

under-hij,MM5[(22Q)/(12Q)]l2ijs/810(s2). (D6)

estimated when ,1 (one QTL) and is overestimated

when.1 (several QTL). However, the effect of selection

Therefore,hij,Gis positive and more than twice as

on (s2

hom/s2het) is always small (from11.3% for L5 1, _{large in homozygotes than in heterozygotes. It will}

s50.3, z50.3; to26.5% for L550, s50.1, z50.3).

increase the difference between them. For fitness traits under directional dominance, this increase may

be negligible as qiand qjare small. However, under

APPENDIX D: THE EFFECT OF PHYSICAL LINKAGE

overdominance, with several QTL, the variance is

Physical linkage increases identity disequilibria in _{mainly sensitive to s}2

f(G), which is of order s/8.

selfed lineages (Charlesworth1991), although it does _{Therefore, tight linkage among QTL (l}

ij ≈ 1) may

not generate linkage disequilibria in infinite popula- _{appreciably increase the negative}

heterozygosity-vari-tions. Two cases are envisaged: (1) linkage among QTL _{ance correlation.}

and (2) linkage between ML and QTL. _{2. I now consider linkage between ML and QTL.}

Con-sider an indicator variable IG, which equals 1 for

indi-1. Heterozygosities at linked QTL are correlated within

viduals of genotype G and 0 otherwise. Because of a given inbreeding class. Consequently, phenotypic

physical linkage, IG and phenotypic values covary

contributions of loci i and j covary within class k by

within class k. This is quantified by two parameters, an amount

aG,k5covk(IG,φ)/p(G|k) andbG,k5covk(IG,φ2)/p(G|k),

Covk(φi,φj)54pipjqiqjdidjhijk. (D1) _{where cov}

kmeans the covariance and p(G|k) the

fre-quency of G within class k. Introducing covariance

The identity disequilibriumhijkdepends on

inbreed-terms into (6) and (7) yields

ing (k) and linkagelij(51 minus twice the

recombi-nation fraction between i and j): _φ

G5φG,unl1 aG (D7)

and

hijk5

1

11 l2

ij

4

2

k

2

1

4

2

k

(D2)

s2

G(φ)5 s2G,unl(φ)1 bG1 a2G2 2aGφG, (D8)

(WeirandCockerham1973). Covariation of QTL

(Equa-tions 6 and 7),aG5RkGkaG,k, andbG5RkGkbG,k. For ships are not affected when the phenotype is additive

a single QTL, we obtain (d 5 0). However, under dominance, the average

phenotypic difference between heterozygotes and

homozygotes increases. For small s and P5 Q51_⁄

2,

aMm5

4pqd(22 s)sl2

(4 2s)[42s(11 l2_)] (D9)

the increase is approximately proportional to (1 1

l2_{) (always},_{2), as inCharlesworth’s (1991)}

nu-and

merical results. Similar effects are observed under

bMm 5daMm, (D10) _{overdominance. Moreover, under dominance or}

overdominance, linkage also increases differences in

lbeing the linkage between the ML and the QTL.

variance between heterozygotes and homozygotes

Values for pooled homozygotes (MM 1 mm) are

(the last term of D8 being negative for the former

identical except for an opposite sign fora.