An Inbreeding Model of Associative Overdominance During a Population Bottleneck

Nicolas Bierne,* Anne Tsitrone† and Patrice David† *Laboratoire Ge´nome, Populations, Interactions, Station Me´diterrane´enne de L’Environnement Littoral, 34200 Se`te, France and †CEFE-CNRS, 34293 Montpellier Cedex 5, France Manuscript received November 15, 1999 Accepted for publication April 17, 2000

ABSTRACT Associative overdominance, the fitness difference between heterozygotes and homozygotes at a neutral locus, is classically described using two categories of models: linkage disequilibrium in small populations or identity disequilibrium in infinite, partially selfing populations. In both cases, only equilibrium situations have been considered. In the present study, associative overdominance is related to the distribution of individual inbreeding levels (i.e., genomic autozygosity). Our model integrates the effects of physical linkage and variation in inbreeding history among individual pedigrees. Hence, linkage and identity disequilibrium, traditionally presented as alternatives, are summarized within a single framework. This allows studying nonequilibrium situations in which both occur simultaneously. The model is applied to the case of an infinite population undergoing a sustained population bottleneck. The effects of bottleneck size, mating system, marker gene diversity, deleterious genomic mutation parameters, and physical linkage are evaluated. Bottlenecks transiently generate much larger associative overdominance than observed in equilibrium finite populations and represent a plausible explanation of empirical results obtained, for instance, in marine species. Moreover, the main origin of associative overdominance is random variation in individual inbreeding whereas physical linkage has little effect.

ORRELATIONS between allozyme multilocus het- ing DNA markers (Bierne et al. 1998, 2000; Coltman C erozygosity (MLH, the number of heterozygous et al. 1998; Coulson et al. 1998; Pogson and Fevolden loci per individual) and fitness-related traits have been 1998). Although some noncoding DNA is functional, under study for decades. Positive correlations have been these markers seem unlikely to be anything other than reported for various organisms, especially marine bi- selectively neutral. Thus, these data favor the “associa- valves (Zouros 1987), salmonid fishes (Leary et al. tive” hypothesis. However, in the only direct comparison 1984), and pine trees (Bush et al. 1987; reviewed in between allozymes and DNA markers to date, Pogson Mitton and Grant 1984; Zouros and Foltz 1987; and Zouros (1994) reported a MLH-growth correlation David 1998). When detected, correlations usually ac- with seven allozyme loci in the scallop Placopecten magel- count for 3–6% of the observed phenotypic variance lanicus, while eight anonymous restriction fragment (Britten 1996; David 1998) and are often inconsistent length polymorphism (RFLP) loci failed to produce a across samples (Gaffney 1990; David and Jarne 1997). significant correlation in the same sample. However, if the magnitude of heterozygosity-fitness cor- The question of “direct” vs. “associative” hypothesis is relations (HFC) is now well known, it has proven ex- only a first step before resolving the basis of HFC. Under tremely difficult to distinguish between competing ex- the direct hypothesis, several genetic determinisms, planations of the phenomenon. As the neutral status of from single-locus true overdominance to more complex allozymes is questionable, the debate focused on two metabolic models involving epistasis (Koehn et al. 1988; hypotheses. The “direct overdominance” hypothesis Hawkins et al. 1989; Zouros and Mallet 1989; Clark (Koehn and Shumway 1982; Mitton 1993) treats en- 1991), could account for HFC. On the other hand, zymes as the causative agent of the correlation, whereas in a large, random mating population at equilibrium, in the “associative overdominance” hypothesis (Ohta heterozygosity at a few marker loci is poorly correlated 1971; Zouros et al. 1980), allozymes are neutral indica- with genomic heterozygosity (Chakraborty 1987). tors of genetic conditions responsible for the correla- Therefore, associative effects need particular population. Recently, HFC has been observed by using noncod- tion structures. Models of associative overdominance have considered two possible sources of correlations among loci. First, the effect of linkage disequilibrium (correlation of allelic states within gametes) has been Corresponding author: Nicolas Bierne, Laboratoire Geńome, Popula- studied in finite populations at mutation-selection-drift tions, Interactions, Station Me´diterraneénne de l’Environnement Lit- toral, 1 Quai de la Daurade, 34200 Se`te, France. equilibrium (Ohta and Kimura 1970, 1971; Ohta E-mail: [email protected] 1971, 1973; Zouros 1993; Pamilo and Pa`lsson 1998).

Genetics 155: 1981–1990 (August 2000) 1982 N. Bierne, A. Tsitrone and P. David

Because strong linkage disequilibria are mainly re- Wj ϭ Wo Ϫ␤fj, (1) stricted to physically tightly linked loci (Hill and Rob- where W is the trait value of an individual j with inbreed- ertson 1968), authors referring to this approach usu- j ing coefficient f , W is the trait value for outbred individ- ally emphasized the role of physical linkage (but see j o uals, and ␤ is the inbreeding load (Morton et al. 1956; Ohta 1973). For this reason, these effects were catego- ␤Ͼ0 means inbreeding depression). In practice, there rized as local effects (David et al. 1995). Second, the is usually a variance in fitness among equally inbred effect of identity disequilibrium (two-locus correlation individuals (individuals with the same f ). However, the of homozygosity) has been investigated in large popula- j derivations below remain correct if W is replaced by tions with partial self-fertilization, at equilibrium (Ohta j W , the conditional expectation of W knowing f . and Cockerham 1974; Strobeck 1979; Charles- j j j The expected magnitude of associative overdomi- worth 1991; Zouros 1993; David 1999). Physical link- nance (AO, the difference between mean fitnesses of age only slightly affects identity disequilibrium (Weir heterozygotes and homozygotes at the marker locus) is and Cockerham 1973). Therefore, this was referred to as a general effect (David et al. 1995). Although the AO ϭ E(W )het Ϫ E(W )hom inbreeding model may well fit populations of self-fertil- ϭ␤ Ϫ ized plants, neither recurrent inbreeding nor perma- [E(f )hom E(f )het ], (2) nent small population size are likely to fit the population where E(W )het[E(f )het] and E(W )hom[E(f )hom] are the structure of marine organisms, where HFC was never- expected inbreeding coefficients (trait values) of het- theless often reported (David 1998). Other models of erozygotes and homozygotes at the marker, respectively. population structure, in which populations are not inevi- The expected inbreeding coefficient of heterozy- tably at equilibrium, have to be considered. gotes is The aims of this study are (i) to fill the gap between 1 E(f )het ϭ f Pr(f/het)df, (3) the two classical approaches, by showing that effects Ύ0 of finite population size can be expressed in terms of inbreeding coefficients (f ), just as in the case of system- where Pr(f/het) is the conditional probability that the atic inbreeding, and by reevaluating the role of physical inbreeding coefficient is f, knowing that the marker linkage in this context; and (ii) to relax the restrictive locus is in a heterozygous state. The reciprocal probabil- Ϫ assumption of population equilibrium, investigating as- ity, Pr(het/f ), is the well-known H0(1 f ), where H0 is sociative overdominance during a sustained population the probability for two non-IBD alleles at the marker bottleneck. locus to be nonidentical in state. Thus, using Bayes’ theorem, Pr(f ) Pr(het/f ) THEORY Pr(f/het) ϭ Pr(het) In this section, we first show how the degree of associa- Pr(f)H (1 Ϫ f ) tive overdominance can be related to inbreeding coef- ϭ 0 , (4) ficients in a general way. This result is then used to H0 (1 Ϫ E(f )) analyze the degree of associative overdominance in a where E(f ) is the unconditional expectation of f over population experiencing a sustained bottleneck, assum- all possible individuals in the population. ing that variation in fitness is due to deleterious muta- Noting ␴2(f ), the variance of f over all possible indi- tions. viduals in the population, The model: The rationale of the model is that homozygosity at a neutral locus correlates with genomic ho- 1 1 E(f )het ϭ f (1 Ϫ f ) Pr(f )df mozygosity through variation in individual inbreeding 1 Ϫ E(f ) Ύ0 coefficients (f ) and that inbreeding depression is the 1 source of fitness variation. By inbreeding coefficient, ϭ [E(f ) Ϫ␴2 (f ) Ϫ E 2(f )], (5) we mean the average autozygosity over all loci in an 1 Ϫ E(f ) individual, or, in other words, the proportion of loci which reduces to that are homozygous for two alleles identical by descent (IBD) within an individual. This proportion depends ␴2 (f ) E(f )het ϭ E(f ) Ϫ . (6) on the pedigree of the individual and on the physical 1 Ϫ E(f ) map (degree of linkage) of the loci (Weir et al. 1980). The definition of IBD for a pair of alleles is further The expected inbreeding coefficient of homozygotes detailed when considering the effects of particular pop- can be similarly derived as ulation dynamics (bottleneck). E(H) ␴2(f ) Consider that a fitness trait, W, is a linear function E(f )hom ϭ E(f ) ϩ , (7) 1 Ϫ E(H) 1 Ϫ E(f ) of the inbreeding coefficient (f ), as expected if deleterious mutations have nonepistatic effects, where E(H) ϭ H0(1 Ϫ E(f )). Marker Heterosis During a Bottleneck 1983

Using Equations 2, 6, and 7, the magnitude of associa- Wj ϭ WO Ϫ␤f0,j, (10) tive overdominance is where the inbreeding load is ␴2(f ) AO ϭ␤ . (8) 1 ΄(1 Ϫ E(H))(1 Ϫ E(f ))΅ ␤ Ϸ U Ϫ 1 (11) ΂2h ΃ AO is the product of the inbreeding load and a term (B in Charlesworth and Charlesworth 1999) and describing the magnitude of association between marker ≈ WO ϪU (Haldane 1937; Kimura et al. 1963). homozygosity and inbreeding coefficient. At this stage, Adding subsequent mutational load: To take into account no assumption has been used about population struc- mutations occurring after foundation, we introduce the ture or equilibrium and about the genetic basis of in- coefficient, fg,j, which denotes the proportion of loci in breeding depression (e.g., overdominance or deleteri- individual j for which the two alleles are copies of the ous mutations). same allele of generation g (1 Ͻ g Ͻ t). We derived AO at one marker locus. However, most The fitness function takes the form experimental studies describe the correlation between t MLH and a fitness trait. The value of this correlation Wj ϭ WO Ϫ␤f0,j Ϫ␤mut ͚ fg,j, (12) is derived in the appendix. gϭ1 Sustained population bottleneck under the deleteri- where ␤mut ϭ Us(1/2 Ϫ h) ϭ hs␤ is the inbreeding load ous genomic mutation model: Let us assume an infinite due to mutations occurring after foundation. random mating population experiencing a bottleneck The magnitude of associative overdominance for the of size N individuals at generation G0. We further assume fitness trait W ϭ ln(w)is that all individuals at G0 are unrelated [E(f ) ϭ 0 and ␴2 ϭ 1 (f ) 0] and that a deleterious mutation is present AO ϭ in only one copy in G0. The latter assumption is justified 1 Ϫ H0 (1 Ϫ E(f0)) by the very low equilibrium frequencies of deleterious ␴2(f ) t ␴2 (f ) alleles in the infinite population. For the neutral ϫ ␤ 0 ϩ␤ g . ΄ Ϫ mut ͚ Ϫ ΅ marker, H defined in the previous section is equivalent 1 E(f0) gϭ1 1 E(fg) 0 (13) to the initial genetic diversity at generation G0. The population size remains constant over t generations, This equation assumes no mutation at the marker after which AO is measured. Mutation at the marker locus. Indeed, even though the number of new muta- locus is neglected after the foundation event. tions affecting fitness loci in the whole genome may be Foundation load: As a first approximation, let us con- high, the frequency of mutations affecting any particu- sider that the variance in fitness is wholly due to the lar locus (in this case, the marker locus) in the first few segregation of mutations initially present at G0 (the ef- generations following the foundation can be neglected fect of new mutations will be considered in the next as a first approximation. section). The operational definition of IBD for a pair Purging selection: In the computation of inbreeding of alleles is their being two copies of the same allele of load coefficients, ␤ and ␤mut, purging selection is ne- generation G0. The fitness of an individual j will there- glected. Incorporating this process into analytical ex- fore depend on f0,j (the subscript 0 refers to the genera- pressions is a complex task and would require special tion of reference) and on the number and effect of attention (see Wang et al. 1999 for a simulation ap- mutations present in G0. proach of this problem). However, when selection is Using the classical deleterious genomic mutation not too strong and dominance is large enough, the model, the multilocus fitness is given multiplicatively by expected number of copies left by a mutant gene is roughly (1 Ϫ hs) per generation, assuming that such y z w ϭ (1 Ϫ s) (1 Ϫ hs) , (9) alleles are mostly in heterozygous state. Then, the expected number of copies at time t is (1 Ϫ hs)t, which where h is the dominance coefficient, s the selection represents the approximate rate of decrease of the in- coefficient against deleterious homozygotes, and y and breeding load. This can be incorporated into Equation z are the numbers of mutations in homozygous and 13: heterozygous states, respectively (Charlesworth et al. 1990). To obtain a linear function as in Equation 1, the 1 AO ϭ natural logarithm of fitness is taken as our fitness trait, 1 Ϫ H0 (1 Ϫ E(f0 ))

W ϭ ln(w). 2 t 2 t ␴ (f0) g ␴ (fg) Neglecting purging selection (i.e., the selective elimi- ϫ ␤(1 Ϫ hs) ϩ␤mut ͚ (1 Ϫ hs) . ΄ 1 Ϫ E(f0) gϭ1 1 Ϫ E(fg)΅ nation of some deleterious alleles during the bottle- (14) neck), the expected W of an individual j is expressed as a function of the inbreeding coefﬁcient, Note that this is not a generally satisfactory approxi- 1984 N. Bierne, A. Tsitrone and P. David mation. However, it resulted reasonably accurately under the range of parameters studied.

METHODS FOR NUMERICAL ESTIMATION Mean and variance of the inbreeding coefficient during a sustained population bottleneck: The sustained bottleneck causes (i) an increase in the mean inbreeding coefficient [E(f )] and (ii) variation of the inbreeding coefficient among individuals due to random variation in pedigrees (Weir and Cockerham 1969; Weir et al. 1980). The increase in E(f ) is a well-known consequence of genetic drift (Malećot 1946). E(f ) depends on the effective population size and time since foundation. It is only slightly affected by the mating system (Malećot Figure 1.—Associative overdominance as a function of 1946) and is unaffected by linkage. In a random mating time. Differential contribution of mutations already present population with constant size N, the value of E(f ) after in the founding population and of new mutations to AO [the 0 difference between the average value of heterozygotes vs. ho- t generations is mozygotes for the fitness trait W ϭ ln(w) at the neutral marker t locus] and simulation results (1000). The population size is 1 Ϫt/2N E(f0) ϭ 1 Ϫ 1 Ϫ Ϸ 1 Ϫ e . (15) N ϭ 40. The mating system is monogamy. The genomic muta- ΂ 2N΃ tion rate is U ϭ 1. The dominance and selection coefficients are h ϭ 0.3 and s ϭ 0.05. The initial marker gene diversity is The evolution of the variance in inbreeding coeffi- H0 ϭ 0.9. Loci are unlinked. cients, ␴2(f ), during a sustained population bottleneck has been studied by Weir et al. (1980). Although there is no simple analytic expression, these authors provided tion, a Poisson-distributed (with mean U/2) random the transition matrix of two-locus descent measure vec- number of new mutations were uniformly distributed tors. This allows the computation of ␴2(f ) as a function along each chromosome. Simulations were performed of population size, time since foundation, mating sys- with or without selection. To account for selection, ran- tem, and degree of linkage (Weir et al. 1980). ␴2(f ) domly drawn offspring survived until reproduction with and E(f ) were numerically calculated in a Mathematica a probability proportional to their fitness (given by 3.0 program (Wolfram 1996). Equation 14 and the Equation 9); this probability was one in the absence of transition matrix method provide exact results under selection. The output values of the model were calcu- neutrality. Actually, selection may modify E(f ) and lated on offspring before selection and averaged over ␴2(f ) compared to neutral expectations. The validity of 1000 simulations. the neutral approximation is tested using simulations. Simulations: All programs were written in Turbo-Pas- cal. We simulated a single chromosome with a map RESULTS length of L Morgans. No constraint was applied to the General shape of AO as a function of time since number of loci in the genome. The position of the foundation: Figure 1 presents AO calculated with Equa- marker locus was generated from a uniform law at the tion 14, detailing the effects of the first term (mutational beginning of each simulation. To obtain the desired load already present at the foundation of the bottle- values of H0 (the initial genetic diversity at the marker neck) and of the second term (effect of subsequent locus), we started with H ϭ 1 (one different allele for mutational load). AO is maximal just after the bottle- each chromosome) and simulated neutral drift until H neck, when ␴2(f ) and E(H) are maximal and E(f ) mini- 2 decreased to H0. A random number of mutations were mal, and then decreases as ␴ (f ) and E(H) decrease then attributed to each chromosome, following a Pois- and E(f ) increases. Foundation load initially has a far son distribution with mean U/(hs). The mutations were more marked effect than mutations occurring after the uniformly distributed along the chromosomes. Three bottleneck. There is a good agreement between theoret- mating systems were modeled: monogamy (N/2 males ical values and simulated data with the deleterious pa- each mate with N/2 females), random mating (N mon- rameters used (h ϭ 0.3, s ϭ 0.05). However, purge is oecious individuals mate at random, including random underestimated when h Ͻ 0.3 and s Ͼ 0.15 (data not selfing), and random mating with selfing excluded. Re- shown). combination rate, r, was related to the map distance The diffusion approximation of Ohta (1971, 1973) between loci, d, using Haldane’s mapping function or gives AO ≈ 0.008 at mutation-selection-drift equilibrium r ϭ 0.5 for unlinked loci (Weir et al. 1980). Each genera- with the parameters used in Figure 1. This is the same Marker Heterosis During a Bottleneck 1985

Figure 2.—Effect of initial marker gene diversity on AO. Curves are analytical results. Simulation results obtained for H0 ϭ 0.9 and H0 ϭ 0.5 are depicted by circles and triangles, respectively. N ϭ 20, parameters are as in Figure 1. order of magnitude as our results at generation 50 (AO ϭ 0.004). The discrepancy relies on slightly different assumptions in the two models. Ohta (1971, 1973) considered additive rather than multiplicative fitness across loci and a diallelic marker locus with fixed heterozygosity H ϭ 0.5, whereas marker heterozygosity was allowed to evolve from an initial value of H0 ϭ 0.9 in our model. Population size: As expected, smaller AO is observed in larger populations, in which ␴2(f ) is smaller. Indeed, at the second generation after foundation, when AO is Figure 3.—(A) Maximum correlation coefficient between maximal, AO is, respectively, 0.03, 0.02, and 0.01 for MLH (M ϭ 1, 5, and 10 loci) and fitness with generation time. N ϭ 20 (Figure 2), N ϭ 40 (Figure 1), and N ϭ 100. (B) Effect of the number of marker loci on the maximum Initial heterozygosity and number of markers: Figure correlation coefficient between MLH and fitness 10 generations after the foundation event. N ϭ 20, parameters are as 2 presents results for H0 ϭ 0.5 (e.g., allozyme) and H0 ϭ 0.9 (e.g., microsatellites). Initial heterozygosity has a in Figure 1. large effect, especially just after the bottleneck. Markers with larger initial heterozygosity will be better indicators mutation rate U or decreasing the dominance coeffi- of the inbreeding coefficient. Figure 3 shows that HFC cient h enhances the inbreeding load (Equation 11) increases with the number of marker loci, but quickly and subsequently AO. The effect of (ii) is reasonably saturates. accounted for by our approximation when the heterozy- Mating systems: Figure 4 presents results for three gous effects of mutations predominate (low s, large h). mating systems. The mating systems influence AO In this case, purge decreases AO by roughly a proportion through the magnitude of ␴2(f ), consistent with the hs per generation. When homozygous effects of muta- results of Weir et al. (1980). When selfing is allowed, tions become important (e.g., h Ͻ 0.2, s Ͼ 0.1), AO ␴2(f ) is enhanced. decreases faster than predicted by Equation 14 because Deleterious genomic mutation parameters: There are purging selection is underestimated (data not shown). no universal values for U, h, and s. For example, empiri- However, the purge effect never changes the order of cal orders of magnitude for U range from 0.01 to 1 magnitude of AO compared to analytical predictions. (Garcia-Dorado et al. 1999; Keightley and Eyre- The extreme situation is that of lethal mutations (say, Walker 1999; Lynch et al. 1999). In our model, the s ϭ 0.9–1 and h ϭ 0.03; see Wang et al. 1998). Lethals effect of mutation parameters is twofold: (i) they deter- greatly increase AO in the first few generations after the mine the inbreeding load ␤, to which AO is propor- bottleneck (large ␤), though the subsequent decrease in tional, all other things being equal, and (ii) they modify AO is very fast (strong purge). purging selection, which tends to decrease AO faster Finally, note that if AO scales with ␤, the latter does than the neutral expectation. Trivially, increasing the not influence the correlation coefficient between MLH 1986 N. Bierne, A. Tsitrone and P. David

Figure 4.—Effect of the mating system on AO. Curves are Figure 5.—Effect of linkage on AO. Curves are analytical analytical results. Simulation results obtained for monogamy, results. Simulation results obtained for unlinked loci, a ge- random mating, and random mating with selfing excluded nome size of 10 M, and a genome size of 1 M are depicted are depicted by circles, triangles, and squares, respectively. by circles, triangles, and solid squares, respectively. Simulation N ϭ 20, parameters are as in Figure 1. results without selection, obtained for a genome of 1 M, are depicted by open squares. N ϭ 20, parameters are as in Figure 1. and the fitness trait (see appendix), which, unlike AO, is not expressed in phenotypic units. Thus, purging with selection and linkage is therefore due to a decrease selection does not affect the correlation. in ␴2(f ). Linkage: Figure 5 presents results for unlinked loci and linked loci in a genome of either 1 or 10 M. Tight DISCUSSION linkage promotes strong AO, though this effect is re- stricted to very small genomes, in which deleterious Our aim was to derive associative overdominance as a mutations are highly concentrated (33.3 mutations in function of individual inbreeding coefficients (genomic 1 M). Another effect of linkage is to delay maximum autozygosity) and to apply it to the case of a sustained AO, which is reached a few generations (rather than population bottleneck. In this section, we first focus immediately) after the onset of the bottleneck. on how associative overdominance can be understood Results take into account two sorts of linkage: linkage under the inbreeding model, showing that the effects among fitness loci and linkage between the marker locus of all parameters may be related to inbreeding. Under and fitness loci. However, the latter is far more impor- this framework, we explain how bottlenecks enhance tant than the former. Indeed, if the marker locus is AO and reevaluate the role of physical linkage in small independent from the rest of the genome (consisting populations. Finally, the relationships between associa- in linked fitness loci), the AO obtained by simulation tive overdominance and selection are discussed. is roughly the same as for a completely unlinked system Associative overdominance and inbreeding: The de- of loci (data not shown). gree of associative overdominance depends on the joint Another effect of strong physical linkage (1 M) is to effect of several mechanisms, all related to inbreeding. introduce a departure from the analytical expectation Population structure and inbreeding variance: AO arises (Figure 5). The difference between analytical and simu- to the extent that population structure promotes correlation results disappears when selection is removed (Fig- lations between marker homozygosity and genomic in- ure 5) and is therefore entirely due to selection. How- breeding (autozygosity). Our approach shows that this ever, it does not rely on an inaccurate approximation association depends mainly on the variance in inbreed- of purging, as the discrepancy persists when the actual ing coefficients. Basically, if all individuals have the same inbreeding loads estimated at each generation in simu- inbreeding coefficient, as in a large random mating lations are plugged into Equation 14. Therefore, selec- population, no association can arise. Small population tion acts through a change in the distribution of in- size (instantaneous or at equilibrium) as well as recur- breeding coefficients [E(f ) and ␴2(f )] compared to the rent inbreeding in large populations are two cases of neutral distribution. E(f ) is little affected by selection population structure that enhance inbreeding variance. with linkage with the set of parameters used in Figure The first case involves random inbreeding (the random 5, as the maximum difference between neutral E(f ) variation in relatedness among pairing mates) and the and E(f ) from simulations is only 3% at generation 50. second case involves systematic inbreeding (the attribu- The departure between analytical and simulation results tion of a fixed proportion of matings to related mates; Marker Heterosis During a Bottleneck 1987

Malećot 1969). Within a small population, the mating it has been suggested that transient and local inbreeding system (opportunity of random selfing, monogamy...) can arise as a result of the fragmentation of these popu- affects random inbreeding variation, ␴2(f )(Weir et al. lations into small, transient reproductive groups (Blanc 1980). It is therefore an important parameter to explain and Bonhomme 1986; Avise 1994; Hedgecock 1994a; associative overdominance. Finally, because more vari- Li and Hedgecock 1998). This is consistent with the able markers are better indicators of inbreeding, they slight but significant genetic differentiation observed at will exhibit stronger AO. Increasing the number of small spatial scales in marine species (Johnson and marker loci allows a better approximation of the geno- Black 1984; Watts et al. 1990; Hedgecock 1994b; mic autozygosity and MLH-fitness correlation increase David et al. 1997). These patterns of differentiation with this number, with a saturation effect. are generally ephemeral and do not reflect permanent Fitness variation and inbreeding depression: The degree structure as in island models or isolation-by-distance of AO depends on variation in fitness within the popula- systems (Johnson and Black 1984; David et al. 1997). tion, which is caused here by inbreeding depression. The effect of transient population fragmentation is simi- Therefore, mutation parameters that generate a large lar to a few generations of bottleneck at the scale of a inbreeding load, such as large U or small h, promote local population. AO could therefore appear within lo- AO. However, inbreeding depression also depends on cal samples. Moreover, the predicted transient nature population structure. In small populations at equilib- of AO in such systems is also consistent with the observed rium, the standing variation is low as new mutations temporal and spatial irregularity of its occurrence either disappear or go to fixation quickly. The expected (Gaffney 1990; David and Jarne 1997; Pogson and depression is therefore usually very small (Bataillon Fevolden 1998). Artificial populations also provide ex- and Kirkpatrick 2000) and only very tight linkage can amples of the bottleneck effect. Indeed, Bierne et al. maintain polymorphism (Charlesworth et al. 1993; (2000) observed a significant correlation between heter- Pa`lsson and Pamilo 1999) and inbreeding depression. ozygosity at three microsatellite markers and growth in On the other hand, in very large populations, recessive the shrimp Penaeus stylirostris after 10 generations of deleterious mutations are maintained by the mutation- controlled bottleneck (N ϭ 20). Finally, marine bivalves selection balance and inbreeding depression is maximal display high genetic loads that have been interpreted (Bataillon and Kirkpatrick 2000). as a consequence of their high fecundity (Bierne et al. Effects of a population bottleneck: High inbreeding 1998). With such a load, significant AO arises with little depression associated with large inbreeding variance variation in inbreeding. and marker diversity promote AO. However, they can- Effect of linkage under the inbreeding model: Linkage not act simultaneously in populations at equilibrium, affects AO because it increases ␴2(f ). The variance in f unless there is some degree of systematic inbreeding can be partitioned into two components: an “among- (Charlesworth 1991; David 1999). Indeed, as ex- pedigree” component arising from the random varia- plained above, large random-mating populations at tion in pedigrees and a “within-pedigree” component equilibrium display high inbreeding depression and among possible individuals with identical pedigrees marker diversity but lack variation in inbreeding, (e.g., full sibs). Without linkage, individual inbreeding whereas the reverse is true for small populations at equi- coefficients mainly depend on the pedigree. Indeed, librium. In contrast, the initial stages of a sustained the pedigree gives the probability of autozygosity at a bottleneck are transient situations that maximize AO. locus, which is independent from that of other loci in During the first generations after the bottleneck, the the same individual. The proportion of autozygous loci, population retains high marker diversity and high in- f, varies little within pedigrees because it is averaged over breeding depression corresponding to the mutation- a large number of loci. Moreover, the within-pedigree selection equilibrium in the founding large population, variation in f is not correlated to homozygosity at the but displays high inbreeding variance because of the marker locus. Therefore, variance in f and correlation small instantaneous population size (Weir et al. 1980). with the marker locus come from the fact that different This effect progressively dies out until the equilibrium individuals have different pedigrees. With linkage, al- AO for a small population is recovered. leles at different loci do not segregate independently Bottlenecks are likely mechanisms by which AO can along the pedigree; rather, the genome is fragmented arise in natural populations. Indeed, equilibrium mod- into a finite number of chunks of chromosome. This els cannot apply to all the taxa in which AO has been creates correlations among autozygosities at different detected. Pine trees may well consist of large, partially loci within a pedigree, which greatly inflate the within- selfing populations at equilibrium (Ledig 1986; pedigree variance in f, and reinforces the correlation Strauss 1986). However, neither marine bivalves nor with homozygosity at the marker locus. salmonid fishes seem likely to fit equilibrium models. The quantitative importance of physical linkage in Moreover, their population sizes are orders of magni- promoting AO has been largely debated. Effects directly tude too high to predict detectable AO as a result of attributable to physical linkage in the vicinity of the finite population size at equilibrium. On the other hand, marker locus have been referred to as “local” effects 1988 N. Bierne, A. Tsitrone and P. David and contrasted to “general” effects affecting the whole to the neutral expectation. However, this effect is usually genome (David et al. 1995). Previous models either small in the first generations following the bottleneck. consider infinite populations with partial selfing and When selection and linkage act simultaneously, not only unlinked alleles or emphasize the role of linkage dis- inbreeding load is decreased but the evolution of in- equilibrium in small populations. Although physical breeding variance is greatly perturbed. This can substan- linkage is predicted to be relatively unimportant in the tially decrease AO compared to neutral expectations first case (Weir and Cockerham 1973; David 1999) (Figure 5), although the order of magnitude remains and not strictly necessary in the second (Ohta 1973), the same. On the whole, selection is predicted to have empiricists often refer to tightly linked fitness loci, i.e., relatively little impact on empirically detected AO for local effects, to explain the observed correlations (Pog- two reasons. First, AO is more likely to be detected in son and Fevolden 1998). Our results confirm that link- the initial phases of the bottleneck, when it is maximal age is not needed for AO and allow us to quantify its and selection has had no time to modify the standing relative contribution when present. This contribution mutation load and the distribution of inbreeding. Sec- appears to be quite low: Only in very small genomes ond, although physical linkage enhances the impact (e.g., 1 chromosome of 1 M) is AO substantially different of selection, this effect is diluted proportionally to the from that predicted using completely independent loci. haploid chromosome number (as described above). However, species usually have several chromosomes. As- suming a chromosome length of 1 M, the neutral CONCLUSION marker is only about four times more influenced by mutations localized on its chromosome (Figure 5) than Mating system and linkage should not be considered by independent mutations. This local effect may be as competing hypotheses to explain apparent heterozy- hardly detectable in empirical studies, as the marked gote advantage. Indeed, both act by increasing the vari- chromosome on average harbors a fraction 1/n of the ance in individual inbreeding levels. Associative over- total mutations (n being the haploid number of chromo- dominance is expected whenever population structure somes). For example, Bierne et al. (2000) detected sig- and/or dynamics enhance this variance. Bottlenecks nificant AO in the shrimp P. stylirostris. However, n ϭ provide such situations. Moreover, associative overdomi- 46 (Nakaura et al. 1988) and most of the genome is nance is high during bottlenecks because the genetic unlinked to any particular marker. variation in fitness inherited from the founding large Associative overdominance and natural selection: population has not yet been eroded by purging selection Two levels of selection may be considered: (i) indirect or random drift. However, the association between selection on the marker locus and (ii) direct selection marker loci and fitness genes is ephemeral and has little on deleterious mutations. effect on marker variation. More complex population How does AO influence the fixation rate at the marker locus? structures, such as metapopulations, still have to be in- Several studies of populations artificially maintained at vestigated. small numbers revealed that the decrease in heterozy- We are very much indebted to F. Bonhomme, K. Dawson, and P. gosity at presumably neutral markers is slower than the Jarne for constructive discussions. P. Keightley and two anonymous neutral expectation (Connor and Bellucci 1979; referees provided insightful remarks on the manuscript. We also thank Frankham et al. 1993; Rumball et al. 1994; Latter et K. Belkhir for his advice on Pascal programming and R. Vitalis for al. 1995). AO has been invoked to explain such an drawing our attention to relevant references. observation (Latter 1998) as well as the maintenance of neutral polymorphism in small populations (Ohta LITERATURE CITED 1971; Pamilo and Pa`lsson 1998; Pa`lsson and Pamilo 1999). However, we need to distinguish between two Avise, J. C., 1994 Molecular Markers, Natural History and Evolution. Chapman & Hall, New York. classes of phenomena, both referred to as AO. The first Bataillon, T. M., and M. Kirkpatrick, 2000 Inbreeding depres- is a statistical correlation between heterozygosity and sion due to midly deleterious mutations in finite populations: fitness (HFC) and is the one described by our model. size does matter. Genet. Res. 75: 75–81. Bierne, N., S. Launey, Y. Naciri-Graven and F. Bonhomme, 1998 As pointed out by Charlesworth (1991), it is impor- Early effect of inbreeding as revealed by microsatellite analyses tant to keep in mind that it is only an apparent selection on Ostrea edulis larvae. Genetics 148: 1893–1906. that does not necessarily influence the evolution of the Bierne, N., I. Beuzart, V. Vonau, F. Bonhomme, E. Be´dier et al., 2000 Microsatellite-associated heterosis in hatchery-propagated neutral marker. The second is really indirect balancing stocks of the shrimp Penaeus stylirostris. Aquaculture 184: 203–219. selection on neutral marker loci. In the conditions used Blanc, F., and F. Bonhomme, 1986 A genetic basis of species im- in our model, only physical linkage can indirectly—and provements in aquaculture: polymorphisme geńe´tique des populations naturelles de mollusques d’inte´reˆt aquicole, pp. 59–78 in weakly—select for neutral marker heterozygosity, Selection, Hybridization and Genetic Engineering in Aquaculture, Vol. whereas it is not necessary to generate HFC. I, edited by K. Tiews. H. Heenemann GmbH, Berlin. Selection on deleterious mutations: For fitness traits, in Britten, H. B., 1996 Meta-analyses of the association between multilocus heterozygosity and fitness. Evolution 50: 2158–2164. the case of unlinked loci, the main effect of selection Bush, R. M., P. E. Smouse and F. T. Ledig, 1987 The fitness conse- is to decrease the inbreeding load (purge) compared quences of multiple locus heterozygosity: the relationship be- Marker Heterosis During a Bottleneck 1989

tween heterozygosity and growth rate in pitch pine (Pinus rigida American Oyster; Crassostrea virginica (Gmelin). Mar. Biol. Lett. Mill.). Evolution 41: 787–798. 3: 35–42. Chakraborty, R., 1987 Biochemical heterozygosity and phenotypic Koehn, R. K., W. J. Diehl and T. M. Scott, 1988 The differential variability of polygenic traits. Heredity 59: 19–28. contribution by individual enzymes of glycolisis and protein catab- Charlesworth, B., and D. Charlesworth, 1999 The genetic basis olism to the relationship between heterozygosity and growth rate of inbreeding depression. Genet. Res. 74: 329–340. in the coot clam, Mulinia lateralis. Genetics 118: 121–130. Charlesworth, D., 1991 The apparent selection on neutral marker Latter, B. D., 1998 Mutant alleles of small effect are primarily loci in partially inbreeding populations. Genet. Res. 57: 159–175. responsible for the loss of fitness with slow inbreeding in Drosoph- Charlesworth, D., M. T. Morgan and B. Charlesworth, 1990 ila melanogaster. Genetics 148: 1143–1158. Inbreeding depression, genetic load and the evolution of out- Latter, B. D., J. C. Mulley, D. Reid and L. Pascoe, 1995 Reduced crossing rates in a multi-locus system with no linkage. Evolution genetic load revealed by slow inbreeding in Drosophila melanogas- 44: 1469–1489. ter. Genetics 139: 287–297. Charlesworth, D., M. T. Morgan and B. Charlesworth, 1993 Leary, R. F., F. W. Allendorf and K. L. Knudsen, 1984 Superior Mutation accumulation in finite outbreeding and inbreeding developmental stability of heterozygotes at enzyme loci in salmo- populations. Genet. Res. 61: 39–56. nid fishes. Am. Nat. 124: 540–551. Clark, A. G., 1991 Mutation-selection balance and metabolic con- Ledig, F. T., 1986 Heterozygosity, heterosis and fitness in outbreed- trol theory. Genetics 129: 909–923. ing plants, pp. 77–104 in Conservation Biology (the Science of Scarcity Coltman, D. W., W. Don Bowen and J. M. Wright, 1998 Birth and Diversity), edited by M. E. Soule´. Sinauer Associates, Sunder- weight and neonatal survival of harbour seal pups are positively land, MA. correlated with genetic variation measured by microsatellites. Li, G., and D. Hedgecock, 1998 Genetic heterogeneity, detected Proc. R. Soc. Lond. Ser. B 265: 803–809. by PCR-SSCP, among samples of larval Pacific oysters (Crassostrea Connor, J. L., and M. J. Bellucci, 1979 Natural selection resisting gigas), supports the hypothesis of large variance in reproductive inbreeding depression in captive wild house-mice. Anim. Behav. success. Can. J. Fish. Aquat. Sci. 55: 1025–1033. 33: 929–940. Lynch, M., J. Blanchard, D. Houle, T. Kibota, S. Schultz, L. Coulson, T. N., J. M. Pemberton, S. D. Albon, M. Beaumont, T. C. Vassilieva et al., 1999 Perspective: spontaneous deleterious Marshall et al., 1998 Microsatellites reveal heterosis in red mutation. Evolution 53: 645–663. deer. Proc. R. Soc. Lond. Ser. B 265: 489–495. Malećot, G., 1946 La consanguinite´ dans une population limiteé. David, P., 1998 Heterozygosity-fitness correlations: new perspectives C. R. Acad. Sci. 222: 841–843. on old problems. Heredity 80: 531–537. Malećot, G., 1969 Consanguinite´ panmictique et consanguinite´ David, P., 1999 A quantitative model of the relationship between syste´matique. Ann. Geńe´t. Se´l. Anim. 1: 237–242. phenotypic variance and heterozygosity at marker loci under Mitton, J. B., 1993 Theory and data pertinent to the relationship partial selfing. Genetics 153: 1463–1474. between heterozygosity and fitness, pp. 17–41 in The Natural His- David, P., and P. Jarne, 1997 Heterozygosity and growth in the tory of Inbreeding and Outbreeding—Theoretical and Empirical Perspec- marine bivalve Spisula ovalis: testing alternative hypotheses. tives, edited by N. W. Thornhill. University of Chicago Press, Genet. Res. 70: 215–223. Chicago. David, P., B. Delay, P. Berthou and P. Jarne, 1995 Alternative Mitton, J. B., and M. C. Grant, 1984 Associations among protein models for allozyme-associated heterosis in the marine bivalve heterozygosity, growth rate, and developmental homeostasis. Spisula ovalis. Genetics 139: 1719–1726. Annu. Rev. Ecol. Syst. 15: 479–499. David, P., M.-A. Perdieu, A.-F. Pernod and P. Jarne, 1997 Fine Morton, N. E., J. F. Crow and H. J. Muller, 1956 An estimate of grained spatial and temporal population genetic structure in the the mutational damage in man from data on consanguineous marine bivalve Spisula ovalis. Evolution 51: 1318–1322. marriages. Proc. Natl. Acad. Sci. USA 42: 855–863. Frankham, R., G. J. Smith and D. A. Briscoe, 1993 Effects on Nakaura, H. K., A. Machii, K. T. Wada, M. Awaji and S. J. Townsley, heterozygosity and reproductive fitness of inbreeding with and 1988 A check list of decapod chromosomes (Crustacea). Bull. without selection on fitness in Drosophila melanogaster. Theor. Natl. Res. Inst. Aquacult. 13: 1–9. Appl. Genet. 86: 1023–1027. Ohta, T., 1971 Associative overdominance caused by linked detri- Gaffney, P. M., 1990 Enzyme heterozygosity, growth rate and viabil- mental mutations. Genet. Res. 18: 277–286. ity in Mytilus edulis: Another look. Evolution 44: 204–210. Ohta, T., 1973 Effect of linkage on behavior of mutant genes in Garcia-Dorado, A., C. Lopez-Fanjul and A. Caballero, 1999 finite populations. Theor. Popul. Biol. 4: 145–162. Properties of spontaneous mutations affecting quantitative traits. Ohta, T., and C. C. Cockerham, 1974 Detrimental genes with par- Genet. Res. 74: 341–350. tial selfing and effects on a neutral locus. Genet. Res. 23: 191–200. Haldane, J. B. S., 1937 The effect of variation on fitness. Am. Nat. Ohta, T., and M. Kimura, 1970 Development of associative over- 71: 337–349. dominance through linkage disequilibrium in finite populations. Hawkins, A. J. S., B. L. Bayne, A. J. Day, J. Rusin and C. M. Worral, Genet. Res. 16: 165–177. 1989 Genotype-dependant interrelations between energy me- Ohta, T., and M. Kimura, 1971 Behavior of neutral mutants influ- tabolism, protein metabolism and fitness, pp. 283–292 in Repro- enced by associated overdominant loci in finite populations. Ge- duction, Genetics and Distributions of Marine Organisms, edited by netics 69: 247–260. J. S. Ryland and P. A. Tyler. Olsen & Olsen, Fredensborg, Pa`lsson, S., and P. Pamilo, 1999 The effects of deleterious muta- Denmark. tions on linked, neutral variation in small populations. Genetics Hedgecock, D., 1994a Does variance in reproductive success limit 153: 475–483. effective population sizes of marine organisms? pp. 122–134 in Pamilo, P., and S. Pa`lsson, 1998 Associative overdominance, heter- Genetics and Evolution of Aquatic Organisms, edited by A. R. Beau- ozygosity and fitness. Heredity 81: 381–389. mont. Chapman & Hall, London. Pogson, G. H., and S. E. Fevolden, 1998 DNA heterozygosity and Hedgecock, D., 1994b Temporal and spatial genetic structure of growth rate in the Atlantic cod Gadus morhua (L). Evolution 52: marine animal populations in the California current. Calif. Coop. 915–920. Ocean. Fish. Invest. Rep. 35: 73–81. Pogson, G. H., and E. Zouros, 1994 Allozyme and RFLP heterozygo- Hill, W. G., and A. Robertson, 1968 Linkage desequilibrium in sities as correlates of growth rate in the scallop Placopecten magellan- finite populations. Theor. Appl. Genet. 38: 226–231. icus: a test of the associative overdominance hypothesis. Genetics Johnson, M. S., and R. Black, 1984 Pattern beneath the chaos: the 137: 221–231. effect of recruitment on genetic patchiness in an intertidal limpet. Rumball, W., I. R. Franklin, R. Frankham and B. L. Sheldon, 1994 Evolution 38: 1371–1383. Decline in heterozygosity under full-sib and double first-cousin Keightley, P. D., and A. Eyre-Walker, 1999 Terumi Mukai and inbreeding in Drosophila melanogaster. Genetics 136: 1039–1049. the riddle of deleterious mutation rates. Genetics 153: 515–523. Strauss, S. H., 1986 Heterosis at allozyme loci under inbreeding Kimura, M., T. Maruyama and J. F. Crow, 1963 The mutation load and crossbreeding in Pinus Attenuata. Genetics 113: 115–134. in small populations. Genetics 48: 1303–1312. Strobeck, C., 1979 Partial selfing and linkage: the effect of a heter- Koehn, R. K., and S. E. Shumway, 1982 A genetic/physiological otic locus on a neutral locus. Genetics 92: 305–315. explanation for differential growth rate among individuals of the Wang, J., A Caballero, P. D. Keightley and W. G. Hill, 1998 Dy- 1990 N. Bierne, A. Tsitrone and P. David

namics of inbreeding depression due to deleterious mutations 2 ␴ (Hk) ϭ H0(1 Ϫ E(f ))(1 Ϫ H0(1 Ϫ E(f ))). (A4) in small populations: mutation parameters and inbreeding rate. Genet. Res. 74: 165–178. From Equation 1, Wang, J., W. G. Hill, D. Charlesworth and B. Charlesworth, 1999 Dynamics of inbreeding depression due to deleterious ␴2(W) ϭ␤2␴2(f ). (A5) mutations in small populations: mutation parameters and inbreeding rate. Genet. Res. 74: 165–178. Watts, R. J., M. S. Johnson and R. Black, 1990 Effects of recruit- As mentioned previously, Equation 1 assumes that ment on genetic patchiness in the urchin Echinometra mathaei in W depends only on f values. However, equally inbred Western Australia. Mar. Biol. 105: 145–151. individuals may actually have different W because of Weir, B. S., and C. C. Cockerham, 1969 Group inbreeding with random distribution of mutations and environmental two linked loci. Genetics 63: 711–742. Weir, B. S., and C. C. Cockerham, 1973 Mixed self and random variation. This introduces a within-pedigree variance in 2 mating at two loci. Genet. Res. 21: 247–262. W, noted ␴ within(W), which has to be added to Equation Weir, B. S., P. J. Avery and W. G. Hill, 1980 Effect of mating A5. Within-pedigree variation is uncorrelated to hetero- structure on variation in inbreeding. Theor. Popul. Biol. 18: 396–429. zygosity at the marker locus and therefore does not Wolfram, S., 1996 The Mathematica Book, Ed. 3. Wolfram Media/ participate in the covariance term. Therefore, cov(Hk, Cambridge University Press, Cambridge, United Kingdom. W) is simply Zouros, E., 1987 On the relation between heterozygosity and heterosis: an evaluation of the evidence from marine mollusks. Iso- cov(H , W) ϭ E(H )(1 Ϫ E(H ))␤(E(f ) Ϫ E(f ) ) zymes 15: 255–270. k k k hom het Zouros, E., 1993 Associative overdominance: evaluating the effects 2 ␤E(Hk)␴ (f ) of inbreeding and linkage desequilibrium. Genetica 89: 35–46. ϭ . (A6) Zouros, E., and D. W. Foltz, 1987 The use of allelic isozyme varia- 1 Ϫ E(f ) tion for the study of heterosis. Isozymes 13: 1–59. Zouros, E., and A. L. Mallet, 1989 Genetic explanations of the The maximal correlation coefﬁcient is obtained in 2 growth/heterozygosity correlation in marine mollusks, pp. 317– the absence of within-pedigree variation (␴ within(W) ϭ 324 in Reproduction, Genetics and Distributions of Marine Organisms, 0) and can be computed by plugging Equations A4–A6 edited by J. S. Ryland and P. A. Tyler. Olsen & Olsen, Fredens- borg, Denmark. into Equation A2: Zouros, E., S. M. Singh and H. E. Miles, 1980 Growth rate in E(H ) ␴(f ) oysters: an overdominant phenotype and his possible explana- ␳ ϭ k tions. Evolution 42: 1332–1334. max (Hk, W) . (A7) Ί1 Ϫ E(Hk) 1 Ϫ E(f ) Communicating editor: P. D. Keightley 2 When ␴ within(W) Ͼ 0, the correlation coefﬁcient decreases to

␳max (Hk, W) ␳(Hk, W) ϭ . (A8) APPENDIX: THE CORRELATION COEFFICIENT √ ϩ␴2 ␤2 ␴2 BETWEEN HETEROZYGOSITY AT SEVERAL MARKER 1 within (W)/ (f ) LOCI AND A FITNESS TRAIT Several marker loci: The covariance between MLH We consider M unlinked marker loci with the same and the fitness trait can be calculated as genetic diversity H0 and calculate the correlation coeffi- M cient between MLH and the fitness trait W, as a function cov(MLH, W) ϭ ͚cov(Hk, W) ϭ M cov(H1, W) kϭ1 of time after the foundation event. The foundation load (A9) only is considered, and subsequent mutational load is neglected. and the variance in MLH is MLH can be calculated as 2 2 ␴ (MLH) ϭ M␴ (H) ϩ 2 ͚ cov(Hi, Hj). M 1ՅiϽjՅM MLH ϭ ͚Hk, (A1) (A10) kϭ1 Since two unlinked marker loci do not covary within where Hk is the indicator variable of heterozygosity at a given level of inbreeding, we obtain marker k (Hk takes the value 0 when the marker k is 2 2 homozygous and 1 when heterozygous). cov(Hi, Hj) ϭ H 0 ␴ (f ) (A11) One marker locus: The correlation coefficient be- i϶j tween Hk and W is and, in the absence of within-pedigree variation in W,

cov(Hk, W) ␳max(MLH, W) ␳(Hk, W) ϭ . (A2) ␴(H )␴(W) k ␴(f )√H M ϭ 0 . √ 2 Hk is a binomially distributed variable with mean (1 Ϫ E(f ))(1 Ϫ H0 (1 Ϫ E(f ))) ϩ (M Ϫ 1)H0␴ (f ) (A12) E(Hk) ϭ H0 (1 Ϫ E(f )) (A3) Within-pedigree variation may be accounted for as in and variance Equation A8.