Emmanuelle Porcher
email: porcher@mnhn.fr

John K Kelly

Pierre-Olivier Cheptou

Christopher G Eckert

Mark O Johnston

Susan Kalisz

The genetic consequences of fluctuating inbreeding depression and the evolution of plant selfing rates

Keywords: Mixed mating, genetic load, purging, environmental variation

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Over the last thirty years, theoretical studies of plant mating systems have considered the role of inbreeding depression as central for the evolution of self-fertilization [START_REF] Lloyd | Some reproductive factors affecting the selection of self-fertilization in plants[END_REF][START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetics models[END_REF]Charlesworth & Charlesworth, 1990). Inbreeding depression is the main evolutionary force that opposes the automatic 50% advantage of selfing [START_REF] Fisher | Average excess and average effect of a gene substitution[END_REF], it dictates the cost of seed discounting and determines the extent to which selfing causes reproductive assurance [START_REF] Lloyd | Self and cross fertilization in plants. II. The selection of self-fertilization[END_REF], so that the predicted outcome of mating system evolution critically depends on inbreeding depression values. Simple evolutionary models with a constant inbreeding depression, δ, predict a dichotomous outcome of evolution: complete selfing when δ < 0.5 and complete outcrossing when δ > 0.5. Importantly, incorporating purging, i.e. a decrease in inbreeding depression with increasing population selfing rate due to the elimination of recessive deleterious mutations by selection, does not change these conclusions [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetics models[END_REF]Charlesworth & Charlesworth, 1990; but see Uyenoyama & Waller, 1991a). Yet, the existence of natural populations with intermediate stable selfing rates is now widely acknowledged and a variety of theoretical models have been proposed to account for the maintenance of such mixed mating systems (reviewed in [START_REF] Goodwillie | The evolutionary enigma of mixed mating systems in plants: Occurrence, theoretical explanations, and empirical evidence[END_REF]. Although several models rely on ecology (e.g. pollination biology, [START_REF] Holsinger | Mass-action models of plant mating systems: the evolutionary stability of mixed mating systems[END_REF][START_REF] Johnston | Evolution of intermediate selfing rates in plants: pollination ecology versus deleterious mutations[END_REF][START_REF] Vallejo-Marin | On the evolutionary costs of selfincompatibility: Incomplete reproductive compensation due to pollen limitation[END_REF][START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF][START_REF] Johnston | Correlations among fertility components can maintain mixed mating in plants[END_REF] or population dynamics, [START_REF] Morgan | Plant population dynamics, pollinator foraging, and the selection of self-fertilization[END_REF] to explain mixed mating systems, numerous studies have elaborated on the prominent role of inbreeding depression by examining various genetic mechanisms (including overdominance, biparental inbreeding in isolated or structured populations, or decline in fitness with generations of selfing, [START_REF] Lloyd | Some reproductive factors affecting the selection of self-fertilization in plants[END_REF][START_REF] Uyenoyama | Inbreeding depression and the cost of meiosis: the evolution of selfing in population practicing biparental inbreeding[END_REF][START_REF] Holsinger | Inbreeding depression doesn't matter: The genetic basis of matingsystem evolution[END_REF]Uyenoyama & Waller, 1991b;[START_REF] Latta | Models for the evolution of selfing under alternative modes of inheritance[END_REF][START_REF] Ronfort | A stochastic model of selection on selfing rates in structured populations[END_REF][START_REF] Rausher | Stabilization of mixed-mating systems by differences in the magnitude of inbreeding depression for male and female fitness components[END_REF].

Recently studies have demonstrated that the magnitude of inbreeding depression is sensitive to environmental factors (reviewed in [START_REF] Armbruster | Inbreeding depression in benign and stressful environments[END_REF][START_REF] Cheptou | Population biology -The ecology of inbreeding depression[END_REF][START_REF] Willi | Inter-specific competitive stress does not affect the magnitude of inbreeding depression[END_REF]. Some theoretical models have shown that environmentally-induced variation in inbreeding depression is likely to influence the evolution of mating systems, particularly to favour mixed mating systems [START_REF] Cheptou | Can varying inbreeding depression select for intermediary selfing rates?[END_REF][START_REF] Cheptou | The cost of fluctuating inbreeding depression[END_REF]. However, these models do not incorporate a genetic basis of inbreeding depression and hence do not include the purging process that destabilizes mixed mating systems in most explicitly genetic models (e.g. [START_REF] Lande | High inbreeding depression, selective interference among loci, and the threshold selfing rate for purging recessive lethal mutations[END_REF][START_REF] Husband | Evolution of the magnitude and timing of inbreeding depression in plants[END_REF][START_REF] Kelly | Mutation-selection balance in mixed mating populations[END_REF].

In addition, the effectiveness of purging itself may depend on environmental fluctuations, as a result of environment-dependent inbreeding depression (e.g. [START_REF] Bijlsma | Environmental dependence of inbreeding depression and purging in Drosophila melanogaster[END_REF], a phenomenon that has received little theoretical attention so far. To properly assess the actual role of environmental fluctuations in inbreeding depression on the evolution of plant mating systems, we therefore need to (1) determine the genetic mechanisms underlying fluctuations in inbreeding depression and (2) examine the combined effects of fluctuating inbreeding depression and purging on the evolution of selfing.

Here, we use a common genetic model of inbreeding depression to examine one possible genetic mechanism of fluctuating inbreeding depression and to test whether such fluctuations can maintain mixed mating systems when purging occurs. Consistent with the bulk of experimental data (Charlesworth & Charlesworth, 1999), we assume that inbreeding depression is caused by partly recessive, deleterious mutations. Such a mutation can be characterized by its selection coefficient (S) and its dominance coefficient (h). From this genetic perspective, environmental variation in inbreeding depression could result from fluctuations in either S or h or both [START_REF] Bijlsma | Environmental dependence of inbreeding depression and purging in Drosophila melanogaster[END_REF][START_REF] Armbruster | Inbreeding depression in benign and stressful environments[END_REF], but little is known so far regarding the most likely source(s) of variation in inbreeding depression [START_REF] Cheptou | Population biology -The ecology of inbreeding depression[END_REF]. Numerous experimental observations of larger inbreeding depression in stressful environments (e.g. [START_REF] Bijlsma | Environmental dependence of inbreeding depression and purging in Drosophila melanogaster[END_REF]and reviews in Armbruster & Reed, 2005;[START_REF] Willi | Inter-specific competitive stress does not affect the magnitude of inbreeding depression[END_REF] suggest that environmental conditions may influence the selection coefficient of deleterious alleles, although other experiments comparing the mean fitness of Arabidopsis thaliana mutation accumulation lines across light and nutrient gradients failed to demonstrate any environment-specific effects of spontaneous mutations [START_REF] Chang | The contribution of spontaneous mutation to variation in environmental response in Arabidopsis thaliana: Responses to nutrients[END_REF][START_REF] Kavanaugh | The contribution of spontaneous mutation to variation in environmental responses of Arabidopsis thaliana: Responses to light[END_REF]. In the present study, we assume that fluctuations in inbreeding depression are caused by temporal changes in the selection coefficient of deleterious mutations. We apply an analytical approach assuming an infinite population size.

We address (1) the effect of temporal fluctuations of selection coefficients on the temporal variation in inbreeding depression, and compare this predicted variation to that observed in natural populations, (2) the effectiveness of purging under fluctuating selection, and (3) the evolution of mating systems. We specifically investigate whether intermediate selfing rates are evolutionarily stable when inbreeding depression is subjected to fluctuating selection and is allowed to co-evolve with the mating system.

The model

In the following theoretical approach, we define mixed mating systems as mating systems with a selfing rate r > 0 and r < 1, which is broader than the definition used in empirical studies (generally 0.2 < r < 0.8; reviewed in [START_REF] Goodwillie | The evolutionary enigma of mixed mating systems in plants: Occurrence, theoretical explanations, and empirical evidence[END_REF]), because we focus on genetically determined selfing rates, with no environmental effects and no measurement errors.

Inbreeding depression with a fluctuating selection environment

Inbreeding depression, the relative decrease in mean fitness of selfed vs. outcrossed individuals, is described by the model of [START_REF] Kondrashov | Deleterious mutations as an evolutionary factor. II. Facultative apomixis and selfing[END_REF], which models the evolution of the distribution of the number of homozygous and heterozygous deleterious mutations per individual in an infinite population with selfing rate r. Mutation occurs at an infinite number of loci, following a Poisson process with rate U per diploid zygote per generation. Each generation, the selection coefficient of mutations, S, is randomly sampled from a normal distribution truncated to the interval [0,1], with mean µS and variance 2 S σ before truncation.

This selection coefficient is applied to all mutations, which also have the same dominance coefficient, h. Mutations have multiplicative effects on fitness, so that the fitness of an individual carrying x homozygous mutations and y heterozygous mutations is given by (1 -

S) x (1 -hS) y .
Each generation, the modelled population undergoes mating, followed by mutation to deleterious alleles and selection. Recursion equations describing the distribution of number of homozygous and heterozygous mutations per individual are detailed in an online appendix and can also be found in [START_REF] Kondrashov | Deleterious mutations as an evolutionary factor. II. Facultative apomixis and selfing[END_REF] or Charlesworth et al. (1990). These equations were iterated for 1000 generations (C++ code for numerical iterations available upon request from E.P.), by which time the population had reached steady-state for all parameter combinations. To identify steady state, we considered the growth rate in the number of mutations across generations, Nt+1/Nt: the population was considered at steady state when the average growth rate over 50 generations did not differ significantly from 1 (one sample Student's t-test). Equations were subsequently iterated for another 1000 generations, to record fluctuations in inbreeding depression at (statistical) steady state. For each set of parameter values, we recorded (1) the mean and variance of the number of homozygous and heterozygous deleterious mutations per individual and (2) the mean and variance of inbreeding depression, measured over the last 1000 generations. Because inbreeding depression is constrained between 0 (in the present study) and 1, the variance in inbreeding depression depends on the mean (e.g. the variance must be 0 if the mean is 0 or 1). To account for this dependence on the mean, we compared the simulated variance in inbreeding depression to the maximum possible variance: 2 ( 1)

max ID ID σ µ µ = - [1]
where µID is the mean inbreeding depression.

The predicted mean and variance in inbreeding depression were compared to empirical means and variances in inbreeding depression, estimated in contrasting environments (compiled by [START_REF] Armbruster | Inbreeding depression in benign and stressful environments[END_REF]. Most of these studies measured inbreeding depression in only two environments, which were obviously not chosen at random, but instead likely represent extremes of a distribution (e.g. stressful vs. benign environments).

Therefore, it was not appropriate to estimate variance using the classical estimator

2 2 1 ( ) 1 i s x x n = - -∑
. We thus chose to calculate the sample variance, 2 2 1 ( )

ID i ID x x n σ = - ∑ :
due to small sample sizes and non-random sampling, this crude procedure provides an order of magnitude estimate for the actual variance in inbreeding depression.

Evolutionarily stable mating systems

We analyzed the joint evolution of selfing rate and inbreeding depression using an adaptive dynamics framework [START_REF] Dieckmann | Can adaptive dynamics invade?[END_REF]. The fitness of a rare mutant with selfing rate r′ is compared to that of the resident genotype, with selfing rate r. We assumed that the selfing rate is controlled by a single locus, with two alleles. Once the population with selfing rate r reached steady-state (after 1000 generations), a mutant allele with a selfing rate r′ different from the resident was introduced at a low frequency in linkage and identity equilibrium with deleterious mutations, and the recursion equations were numerically iterated for 1000 generations to detect invasion (or not) by the mutant.

We deduce stable selfing rates using Pairwise Invasibility Plots (PIP), in which regions of invasion (in grey) and non-invasion (in white) of a rare mutant are plotted against the selfing rate of the resident genotype, r and the selfing rate of the mutant, r′ (see Fig. 3).

Hence, regions located below the r = r′ line correspond to emergence of a mutant with a smaller selfing rate than the resident, and vice versa. If a mutant with selfing rate r′ > r can invade (i.e. the point with coordinates (r, r′) is located in a region of invasion), then evolution favours increased selfing rates and vice versa. Evolutionary equilibria occur at the intersection of the r = r′ line and a line separating regions of invasion and non-invasion. Details regarding the criteria to infer stability of equilibria can be found in [START_REF] Dieckmann | Can adaptive dynamics invade?[END_REF]. The equilibrium selfing rates discussed here, denoted by white circles on the figures, are evolutionarily stable (they cannot be invaded by neighbouring mutant selfing rates) and are evolutionary attractors or convergence stable (evolution by a series of small steps proceeds towards the equilibrium).

We refer to these equilibria as stable selfing rates.

Parameter values

We either chose parameter values according to experimental values or we used a wide range of biologically plausible values. The genomic mutation rate to deleterious mutations was U = 0.02, 0.2 or 1, which encompasses the range of experimental estimates among multicellular organisms (reviewed in [START_REF] García-Dorado | Rates and effects of deleterious mutations and their evolutionary consequences[END_REF]. The dominance coefficient of mutations was either h = 0.02, as in the only available experimental data for highly deleterious mutations [START_REF] Simmons | Mutations affecting fitness in Drosophila populations[END_REF] or h = 0.3, a more reasonable value for mildly deleterious mutations [START_REF] García-Dorado | Rates and effects of deleterious mutations and their evolutionary consequences[END_REF]. Exploratory analyses showed that the mean selection coefficient of deleterious mutations had little effect on the results presented here, which is consistent with previous theory showing that equilibrium inbreeding depression in a population depends little on the selection coefficient of deleterious mutations, and is much more influenced by their dominance coefficient (e.g. Charlesworth et al., 1990).

Hence, we considered a single, intermediate value for the average selection coefficient of deleterious mutations, µS = 0.5, so that this coefficient, which is constrained between 0 and 1, could be submitted to significant environmental variation. The initial variances were set to 2 S σ = 0.01 and 0.09, but the latter was actually smaller ( 2 S σ = 0.06) due to truncation of S values below 0 or above 1. Note that truncation did not bias the mean selection coefficient.

For comparative purposes, we also considered constant selection ( 2 S σ = 0), with S = 0.5. Finally, we considered different values of the selfing rate, across the range of r = 0 to 1.

Analytical approximation

To characterize the effect of purging on the selection of self-fertilization in a fluctuating environment, we compared the results of the Kondrashov model with stochastic variation in selection coefficients to [START_REF] Cheptou | The cost of fluctuating inbreeding depression[END_REF] analytical phenotypic approach.

Following [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetics models[END_REF], an approximate measure of the strength of selection on a rare genotype with selfing rate r′ is given by the expected fitness of such genotypes:

1 1 '(1 ) (1 ') (1 ) 2 2 w r r r δ = -+ -+ - [2]
where r is the mean selfing rate in the population and δ is the inbreeding depression.

If δ varies in time and if evolution of the selfing rate occurs much slower than the fluctuations in inbreeding depression, the conditions for invasion of a rare genotype with selfing rate r′ are given by its long-run growth rate [START_REF] Lande | Expected relative fitness and the adaptive topography of fluctuating selection[END_REF][START_REF] Lande | Adaptive topography of fluctuating selection in a Mendelian population[END_REF], i.e. the geometric mean of fitness over time or similarly by the expectation of the logarithm of the annual fitness, E[Log(w)] [START_REF] Kisdi | Life-histories with lottery competition in a stochastic environment -ESSs which do not prevail[END_REF]. Without any specific information on the distribution of δ, this quantity can be approximated using a second order approximation (Jensen approximation: [START_REF] Bulmer | Theoretical evolutionary ecology[END_REF] by:

2

1 [log( )] log[ ( )] var( ) 2 ( ) s E w E w w E w = ≈ - [3]
Assuming small effects of mutation on the selfing rate, the evolution of self-fertilization can be analysed by modelling the invasion of a rare genotype with selfing r′ in a population with selfing rate r. The success of invasion can be evaluated via ( ) D r , the selection gradient / ' s r ∂ ∂ at r: if the latter is positive (negative) mutants with slightly higher (lower) selfing rate may successfully invade. The selection gradient ( ) D r for an inbreeding depression distribution with mean δ µ and variance 2 δ σ is given by:

2 3 1 0.5 2 ( ) 1 (1 ) r r D r r r δ δ µ σ µ µ   -   -   = - - - [4]
(see Cheptou & Schoen, 2002 for details). A necessary condition for mixed selfing rates to be stable is thus that ( ) D r = 0 (no mutant can invade) for intermediate values of the selfing rate (0 < r < 1). Moreover, a sufficient and more stringent condition for mixed selfing rates to evolve is that boundaries (r = 0 and r = 1) are unstable, i.e.:

(1) 0 ) 0 ( > D (Increased selfing is favoured in a fully outcrossing population)

(2) 0 ) 1 ( < D (Increased outcrossing is favoured in a fully selfing population)

We incorporated purging in this analytical approach by making both δ µ and 2 δ σ functions of the population selfing rate, r. This and equation [4] yield the following conditions for maintenance of mixed mating:

(1) (0) 0.5

δ µ < [5] (2) [ ] [ ] 2 (1) 2 1 (1) ² 0.5 (1) δ δ δ σ µ µ > - - [6]
We tested whether the values generated by the Kondrashov model with fluctuating selection (mean and variance of inbreeding depression at r = 0 and r = 1) met these conditions for the maintenance of mixed mating.

Results

Effectiveness of purging under fluctuating inbreeding depression

Temporal variation in the selection coefficient of deleterious mutations appeared to have little effect on the extent of purging (Fig. 1). Regardless of the mean (not shown) and variance of selection coefficients, the relationship between the average inbreeding depression at equilibrium and the population selfing rate differed little between variable vs. constant selection, i.e. the occurrence of purging was unchanged as the variance in selection coefficients was increased. As expected, inbreeding depression was large in predominantly outcrossing populations, and decreased with increasing selfing rate. The decrease was progressive under moderate mutation rate (U = 0.02 or 0.2, Fig. 1) or large dominance coefficient (h = 0.3, Fig. 1), but sharper under higher mutation rates and with nearly recessive mutations (U = 1 and h = 0.02, Fig. 1). This pattern was unaffected by fluctuations in the selection coefficient of deleterious mutations, although the average inbreeding depression in a predominantly outcrossing population tended to be slightly lower under large fluctuations of selection than in a constant environment (Fig. 1, σS = 0.3 and h = 0.02).

Magnitude of variation in inbreeding depression

As

Comparison of simulated vs. observed variation in inbreeding depression

Overall, the empirical data generally exhibited more variation in inbreeding depression than simulations (Fig. 2

Evolution of self-fertilization under fluctuating inbreeding depression.

Evolutionarily stable selfing rates can be deduced from Pairwise Invasibility Plots (PIP, Fig. 3), where regions of invasion (in grey) and non-invasion (in white) of a rare mutant are plotted against the selfing rate of the resident genotype, r and that of the mutant, r′.

Evolutionarily stable selfing rates are located at the intersection of the r = r′ line and a line separating regions of invasion and non-invasion, and are characterized by the fact that no mutant can invade (i.e. the areas just above and below are white on the PIP). The main result of our study is that intermediate selfing rates were never evolutionarily stable (Fig. 3), even though large variances in the selection coefficient of deleterious mutations generated fluctuations in the magnitude of inbreeding depression. This was true regardless of the mutation rate, the dominance coefficient of deleterious mutation, as well as the mean and variance of the selection coefficient of mutations (and hence variance in inbreeding depression). The only stable mating systems were complete outcrossing with nearly codominant deleterious mutations (h = 0.3, Fig. 3) and complete outcrossing or complete selfing (depending on the initial conditions) with nearly recessive mutations (h = 0.02, Fig. 3).

In addition, the threshold for the selection of pure selfing or pure outcrossing did not depend on the variance in inbreeding depression. Thus, in our genetic model, fluctuations in inbreeding depression do not favour the maintenance of mixed mating.

This was confirmed by considering the modified version of [START_REF] Cheptou | The cost of fluctuating inbreeding depression[END_REF] analytical approach, accounting for purging via selfing rate dependent inbreeding depression. Under this model, condition (2) for the maintenance of stable mixed mating (namely D(1) < 0, invasion of a mutant with selfing rate < 1 in a completely selfing population) requires substantial variance in inbreeding depression in a completely selfing population where average inbreeding depression is generally low. Under the Kondrashov model, high variance is generated by a high mutation rate (e.g. U = 1, Fig. 1). However, this set of parameters also generates high average inbreeding depression (µδ close to 1) at r = 0, which invalidates condition (1) for the maintenance of mixed mating (namely D(0) > 0, invasion of a mutant with the selfing rate > 0 in a completely outcrossing population, µδ < 0.5). Hence, although some combinations of µδ and σδ fell within the 'mixed mating' window of Fig. 2, sufficient conditions for stable mixed mating (1) and (2) were never met.

Mixed mating can therefore never be maintained when inbreeding depression is modelled with partly recessive deleterious mutations in an infinite population (as in [START_REF] Kondrashov | Deleterious mutations as an evolutionary factor. II. Facultative apomixis and selfing[END_REF].

Discussion

In this study, we generated fluctuating inbreeding depression using a genetic model where environmental variation affects the selection coefficient. This allowed us to characterize purging under a fluctuating environment and analyse the evolution of plant mating systems in this context, specifically testing for conditions that allow the maintenance of mixed mating.

Fluctuations in selection coefficients have little effect on purging

Temporal variation in the selection coefficient of the deleterious mutations has very little effect on the average equilibrium inbreeding depression. This is consistent with single locus theory in randomly mating populations: [START_REF] Crow | An Introduction to Population Genetics Theory[END_REF] showed that with incompletely recessive (h > 0), mildly deleterious mutations (S >> u) mutant homozygotes are kept sufficiently rare by selection so that most selection occurs against mutant heterozyogtes.

As a result, the dynamics of the mutant allele in a constant environment are approximately a linear function of the selection coefficient, S, and can be written dq/dt ≈ u -hSq [START_REF] Crow | An Introduction to Population Genetics Theory[END_REF][START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetics models[END_REF], where q is the mutant allele frequency and u is the per locus mutation rate. With fluctuating selection, this yields E(dq/dt) = u -E(hS)q. At equilibrium (dq/dt = 0), q = u/E(hS). Hence, in randomly mating populations under fluctuating selection, inbreeding depression at a single locus depends solely on the mean selection coefficient (or dominance coefficient) of the deleterious mutation, not on its variance. Our results extend these predictions with a single locus to many loci, as well as non-random mating. Similarly, when selfing is enforced in a previously completely outcrossing population, fluctuations do not affect the time required for the population to reach the new equilibrium inbreeding depression (not shown), which is due to the fact that, under complete selfing, highly recessive deleterious mutations are quickly exposed to selection and a few generations of strong selection are enough to purge most of the load (e.g. [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetics models[END_REF].

A number of experimental approaches have demonstrated that inbreeding depression varies across environments and generally appears to be stronger in more stressful environments, generally defined as environments reducing mean fitness [START_REF] Armbruster | Inbreeding depression in benign and stressful environments[END_REF]. As a consequence, purging might be inefficient in relatively benign environments, where inbreeding depression is weak (e.g. [START_REF] Bijlsma | Environmental dependence of inbreeding depression and purging in Drosophila melanogaster[END_REF][START_REF] Keller | Inbreeding effects in wild populations[END_REF][START_REF] Armbruster | Inbreeding depression in benign and stressful environments[END_REF]. Such faster reduction in inbreeding depression was observed by [START_REF] Kristensen | Effects of inbreeding and environmental stress on fitness -using Drosophila buzzatii as a model organism[END_REF] and [START_REF] Swindell | Associations between environmental stress, selection history, and quantitative genetic variation in Drosophila melanogaster[END_REF] in inbred laboratory populations of Drosophila species submitted to stressful environments (high temperature, competition, chemical stress) vs. benign environments. Their Drosophila data are however not directly comparable to our theoretical results suggesting that variable selection has little effect on purging because in these experimental studies, environment-dependent purging was examined in two or more contrasting, but constant environments. In the variable environment simulated here, our results suggest that only five to ten generations of stress can increase inbreeding depression and are enough to purge most of the load (not shown), regardless of the average effects of mutations/stress in the environment, because the rate of purging is much faster than mutation accumulation [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetics models[END_REF].

Fluctuations in selection coefficients can generate substantial variation in inbreeding depression

Although the time-averaged inbreeding depression is little affected by fluctuating selection, the current inbreeding depression can exhibit substantial variation across environments. In the present study, we considered mutations that were moderately deleterious on average (µS = 0.5), but this intermediate selection coefficient was allowed to vary between 0 (neutral mutations) and 1 (lethal mutations) following a truncated normal distribution with variance between 0.01 and 0.06. As a result, the observed variance in inbreeding depression ranged between 5 x 10 -4 and 0.05, and was highest when the mean inbreeding depression was close to 0.5, which was generally observed at intermediate selfing rates (with nearly recessive deleterious mutations, h = 0.02, Fig. 1). Although empirical data on variation in inbreeding depression magnitude are scarce, several studies (e.g. [START_REF] Jiménez | An experimental study of inbreeding depression in a natural habitat[END_REF][START_REF] Koelewijn | Effects of different levels of inbreeding on progeny fitness in Plantago coronopus[END_REF]Cheptou et al., 2001;[START_REF] Haag | Strong inbreeding depression in a Daphnia metapopulation[END_REF] find that their study organisms exhibit more variation in inbreeding depression than any of our simulations (regardless of parameter values). This would suggest that varying S in time, as simulated here, might not be the only mechanism generating the observed variation in the experimental data. It is however important to keep in mind that experiments on inbreeding depression do not necessarily mimic natural conditions, and that the variance in inbreeding depression was generally estimated from two data points (i.e. two experimental conditions). Obviously there is a need for experimental studies that estimate inbreeding depression over wider ranges of environmental conditions. In addition, the magnitude of inbreeding depression in natural populations depends on the history of populations [START_REF] Kristensen | Effects of inbreeding and environmental stress on fitness -using Drosophila buzzatii as a model organism[END_REF][START_REF] Leimu | Effects of experimental inbreeding on herbivore resistance and plant fitness: the role of history of inbreeding, herbivory and abiotic factors[END_REF], so that it could be less sensitive to environmental variation than suggested by greenhouse experiments (e.g. [START_REF] Galloway | Inbreeding depression in an autotetraploid herb: a three cohort field study[END_REF]. Comparison of inbreeding depression variation in the wild (see for instance [START_REF] Dole | Inbreeding depression in two Mimulus taxa measured by multigenerational changes in the inbreeding coefficient[END_REF] would be more relevant.

Purging cancels out the role of fluctuating inbreeding depression in the maintenance of mixed mating systems

Based on a phenotypic model with no purging of deleterious mutations, [START_REF] Cheptou | The cost of fluctuating inbreeding depression[END_REF] predicted that intermediate selfing rates can be stable with fluctuating inbreeding depression under some combinations of mean and variance in inbreeding depression (see Fig. 2). Our results with an explicit genetic model of inbreeding depression contrast with their predictions: mixed mating systems were never evolutionarily stable, although some parameter combinations appear to generate a mean and variance of inbreeding depression consistent with stable mixed mating systems under Cheptou and Schoen's feedback that selects for selfing [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetics models[END_REF] and that overwhelms the negative feedback produced by fluctuating inbreeding depression [START_REF] Cheptou | The cost of fluctuating inbreeding depression[END_REF].

A heuristic understanding of why the genetic architecture of inbreeding depression can disrupt the maintenance of intermediate selfing rates was seen using an analytical approach.

By incorporating selfing rate-dependent mean and variance of inbreeding depression into [START_REF] Cheptou | The cost of fluctuating inbreeding depression[END_REF] analytical approach, we derived sufficient conditions for mixed mating: moderate mean inbreeding depression under complete outcrossing (µδ < 0.5) together with appreciable variance in inbreeding depression under complete selfing. In our simulations, these conditions were never met because the genetic architecture of inbreeding depression generates two contrasting situations that depend on the characteristics of the deleterious mutation. First, a low mutation rate (not shown) or nearly codominant deleterious mutations that can be eliminated by selection even in a heterozygous state (h = 0.3, Fig. 1) result in moderate inbreeding depression in a completely outcrossing population (condition (1)). However, this also generates a low inbreeding depression (and hence a low variance in inbreeding depression) under complete selfing, due to extensive purging, so that condition (2)

is not met. Alternatively, a high mutation rate to nearly recessive mutations (U = 1 and h = 0.02, Fig. 1) creates situations in which the mean inbreeding depression under complete selfing is close to 0.5, so that the variance in inbreeding depression can be large (condition (2)). However, with this characteristic of mutations, condition (1) is not met due to the absence of purging in a completely outcrossing population, generating large mean inbreeding depression.

Some limits of the model

Although fluctuating selection coefficients can generate substantial inbreeding depression, our approach suggests that when purging occurs, fluctuating inbreeding depression is very unlikely to maintain intermediate selfing rates. This probably depends strongly on several assumptions of our model. First, [START_REF] Kondrashov | Deleterious mutations as an evolutionary factor. II. Facultative apomixis and selfing[END_REF] approach assumes infinite population size, whereas in natural populations purging is a stochastic process [START_REF] Byers | Do plant populations purge their genetic load? Effects of population size and mating history on inbreeding depression[END_REF] that can be slowed down in small populations [START_REF] Glémin | How are deleterious mutations purged? Drift versus nonrandom mating[END_REF], but should also be reinforced by the possibility of biparental inbreeding in finite populations. The role of demographic stochasticity in the evolution of genetic load and mating systems has been addressed in a limited number of studies (e.g. [START_REF] Charlesworth | The effect of linkage and population-size on inbreeding depression due to mutational load[END_REF][START_REF] Wang | Dynamics of inbreeding depression due to deleterious mutations in small populations: mutation parameters and inbreeding rate[END_REF][START_REF] Bataillon | Inbreeding depression due to mildly deleterious mutations in finite populations: size does matter[END_REF][START_REF] Theodorou | Inbreeding depression and heterosis in a subdivided population: influence of the mating system[END_REF][START_REF] Glémin | How are deleterious mutations purged? Drift versus nonrandom mating[END_REF][START_REF] Guillaume | Joint evolution of dispersal and inbreeding load[END_REF] but the interaction between demography, genetics and environmental variation remains to be explored. Second, we assumed that environmental variation affected the selection coefficient of a large number of strictly identical, unlinked deleterious mutations.

Although little is known regarding the actual mechanisms for environment-dependent inbreeding depression [START_REF] Armbruster | Inbreeding depression in benign and stressful environments[END_REF], recent studies suggest the existence of environment-specific deleterious alleles (e.g. [START_REF] Vermeulen | Characterization of conditionally expressed mutants affecting age-specific Drosophila melanogaster: Lethal conditions and temperaturesensitive periods[END_REF], i.e. alleles that are usually neutral but become deleterious in specific (e.g. stressful) environments. Such environment specific alleles, which are rarely submitted to selection, may accumulate in the genome and eventually add up to environment-independent inbreeding depression in stressful environments, which can affect the evolution of mating systems. However, our model suggests that their effects are likely to remain minor if these alleles are expressed frequently enough, because purging occurs within a couple of generations under selection. Lastly, other genetic mechanisms such as variation in dominance or even purely additive gene action when the strength of stabilizing selection varies may create inbreeding depression variance under heterogeneous environments [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetics models[END_REF].

Conclusion

Our model attempts to link theoretical genetic models of inbreeding depression with the empirical observations that inbreeding depression is sensitive to environmental conditions.

We extend Kondrashov's framework to heterogeneous environments and show that temporally varying selection on deleterious mutations maintains substantial variance in inbreeding depression. However, this mechanism cannot in itself maintain stable mixed selfing rates. Additional experimental and theoretical work is needed to elucidate the genetic basis of environmental variation in inbreeding depression and its consequences for the evolution of mating systems.

  expected, temporal fluctuations in the selection coefficient of deleterious mutations generated variation in inbreeding depression across generations. Variation in inbreeding depression increased with the selection coefficient, 2 S σ up to ca. 0.05. It was also affected by the population selfing rate and the purging process, with a peak in variance when the average inbreeding depression was closest to 0.5. Because mean and variance of inbreeding depression are not independent (equation [1]), the changes in variance under different population selfing rates were in great part attributable to changes in mean inbreeding depression. Depending on the mutation rate and dominance coefficient of deleterious mutations, variation in mean inbreeding depression explained 51-97% of the variance of inbreeding depression (these values were obtained by examining the correlation between the variance in the simulations and the maximum variance in inbreeding depression calculated from equation [1]).

, 2 IDσ

 2 up to 0.18 and 0.046, respectively). The parameter space of µδ (mean inbreeding depression) and 2 δ σ (variance in inbreeding depression) where the analytical model of[START_REF] Cheptou | The cost of fluctuating inbreeding depression[END_REF] predicts stable mixed mating systems(equation [4], D(r) = 0) are plotted on Figure2(shaded area). Interestingly, none of the empirical points fall within this parameter space. Despite relatively low variance in inbreeding depression generated by our theoretical approach, the simulations nonetheless generated some situations for which[START_REF] Cheptou | The cost of fluctuating inbreeding depression[END_REF] model predicts stable mixed mating: this was the case with U = 0.2, h = 0.02, σS = 0.3 and U = 1, h = 0.02, σS = 0.3 (combination of µδ and σδ² within the shaded area, Fig.2).

  approach (Fig 2, e.g. a high mutation rate to deleterious alleles (U = 1), nearly recessive mutations (h = 0.02) and substantial fluctuation in the selection coefficient of deleterious mutations). Evidently, it is purging in our model that makes mixed mating unstable. While the mean and variance are independent parameters in Cheptou and Schoen's model, these quantities intrinsically covary in an explicit genetic model. Purging creates a strong positive

Figure 1 :

 1 Figure 1: Equilibrium inbreeding depression under fluctuating selection.

Figure 2 :

 2 Figure 2: Predicted and empirically derived estimates of the mean and variance in inbreeding

Figure 3 :

 3 Figure 3: Evolutionarily stable selfing rates under fluctuating inbreeding depression. The average selection coefficient of deleterious mutations is µS = 0.5. Regions where the mutant

  Figure 1

Acknowledgements

We thank other members of the "Understanding the Paradox of Mixed Mating Systems" working group (Elizabeth Elle, Monica A. Geber, Carol Goodwillie, David A. Moeller, Richard H. Ree, Risa D. Sargent, Mario Vallejo-Marín and Alice A. Winn) for insightful discussions, and the National Evolutionary Synthesis Center (NESCENT) for supporting this working group. Part of the simulations were run using resources of the Computational Biology Service Unit from the Muséum National d'Histoire Naturelle, which was partially

funded by Saint Gobain 1

Supporting information Recursion equations for Kondrashov's model with fluctuating selection. I. General equations

Here we derive recursion equations for the model describing the dynamics of inbreeding depression due partly recessive deleterious mutations in an infinite population with selfing rate r undergoing fluctuating selection. The dominance coefficient of mutations is h and their selection coefficient, S, is sampled at random each generation in a truncated Gaussian distribution with mean µS and variance 2 S σ . Mutation occurs at an infinite number of loci, and affects the number of heterozygous loci only, i.e. it occurs exclusively at loci that do not already carry mutations.

Let ( ) q ij be the frequency of individuals carrying i mutations in the heterozygous sate and j mutations in the homozygous state. Similarly, f(i) is the frequency of gametes carrying i deleterious mutations.

a. Selfing

The probability that a plant carrying n1 heterozygous loci and n2 homozygous loci produces by selfing an offspring carrying i (0 ≤ i ≤ n1) heterozygous loci and j (n2

Hence, the frequency of plants with i heterozygous loci and j homozygous loci obtained by selfing is:

The probability that a (n1, n2) plant produces a gamete with i mutations (n2

Hence, the frequency of gametes carrying i mutations in the population is:

Because each mutation is unique, random mating in an infinite population never generates homozygous mutations. Therefore, the probability that an individual carrying n1 heterozygous and n2 homozygous mutations produces by outcrossing a zygote with i heterozygous mutations (and no mutation in the homozygous state) is:

Mutation and selection

Mutation follows a Poisson process, with mean number of new heterozygous deleterious mutations per genome of U per generation. The frequencies of zygotes after mutation are therefore:

The probability that a zygote with x homozygous mutations and y heterozygous mutations survives to maturity is (1 -S) x (1 -hS) y . The frequency of mature plants with i heterozygous loci and j homozygous loci in the next generation is then: (1 ) (1 ) ( )

∑∑

(A6) Equations (A1-A6) completely define the recursion system. For each run, the initial population contained no deleterious mutation. Fluctuating selection was modeled by sampling each generation the selection coefficient of deleterious mutations, S, in a Gaussian distribution with mean µS and variance 2 S σ (equations A5 and A6). Simulations were run until the system closely approached mutation-selection balance.

II. Addition of a locus controlling the selfing rate

We assume that selfing rate is controlled by a single locus (hereafter named 'modifier locus'), with two alleles: A is the resident and B is the mutant allele.

fAA(i,j), fAB(i,j), and fBB(i,j) are the frequencies of the three diploid genotypes, carrying i and j mutation in the heterozygous and homozygous state, respectively.

fA(i) and fB(i) are the frequencies of gametes carrying allele A (respectively B) and i mutations. a. Selfing ( ) G J q ij is the frequency of zygotes with genotype G at the modifier locus, carrying i heterozygous loci and j homozygous loci and originating from selfing of J genotypes. We have:

is the frequency of gametes with haploid genotype g at the modifier locus, carrying i mutations and originating from plants with diploid genotypes G.

( )

is the frequency of pollen grains with genotype g at the modifier locus and carrying i mutations.

Recursion equations for mating

Let rAA, rAB, and rBB be the selfing rates of genotypes AA, AB and BB, respectively. In the main text, rAA = r and rBB = r'.

Mutation and selection

The frequencies of zygotes after mutation are, for any genotype G:

A25) where w is the mean fitness of the population: '' '' '' AA AB BB 0 0

(1 ) (1 ) ( ( ) ( ) ( ))

∑∑

(A26) Equations (A7-26) completely define the recursion system. As before, fluctuating selection was modeled by sampling each generation the selection coefficient of deleterious mutations, S, in a Gaussian distribution with mean µS and variance 2 S σ (equations A5 and A6). For each run, a small frequency of a mutant allele, with selfing rate rBB, was introduced in linkage and identity equilibrium with deleterious mutations, into a population with selfing rate rAA at mutation-selection equilibrium. Simulations were run for 1000 generations to detect invasion (or not) by the mutant.


Emmanuelle Porcher
email: porcher@mnhn.fr

Russell Lande

Evaluating a simple approximation to modeling the joint evolution of self-fertilization and inbreeding depression RUNNING TITLE: Approximation for selfing rate evolution

Keywords: Mixed mating, selfing, plants, recessive lethal mutations, pollen limitation, pollen discounting

A comprehensive understanding of plant mating system evolution requires detailed genetic models for both the mating system and inbreeding depression, which are often intractable. A simple approximation assuming that the mating system evolves by small infrequent mutational steps has been proposed. We examine its accuracy by comparing the evolutionarily stable selfing rates it predicts to those obtained from an explicit genetic model of the selfing rate, when inbreeding depression is caused by partly recessive deleterious mutations at many loci. Both models also include pollen limitation and pollen discounting. The approximation produces reasonably accurate predictions with a low or moderate genomic mutation rate to deleterious alleles, on the order of U = 0.02 to 0.2. However, for high mutation rates, the predictions of the full genetic model differ substantially from those of the approximation, especially with nearly recessive lethal alleles. This occurs because when a modifier allele affecting the selfing rate is rare, homozygous modifiers are produced mainly by selfing, which enhances the opportunity for purging nearly recessive lethals and increases the marginal fitness of the allele modifying the selfing rate. Our results confirm that explicit genetic models of selfing rate and inbreeding depression are required to understand mating system evolution.

INTRODUCTION

Inbreeding depression, the relative decrease in fitness of inbred vs. outbred individuals, is a critical force in the evolution of mating systems, with complex evolutionary dynamics.

Numerous theoretical approaches have explored these dynamics, and sometimes its joint evolution with the mating system, when inbreeding depression is caused by overdominance (Uyenoyama and Waller 1991b), or by deleterious recessive alleles at one (e.g. Uyenoyama and Waller 1991a;[START_REF] Glemin | How are deleterious mutations purged? Drift versus nonrandom mating[END_REF] or many loci with [START_REF] Charlesworth | Multilocus models of inbreeding depression with synergistic selection and partial self-fertilization[END_REF] or without [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF][START_REF] Charlesworth | Inbreeding depression, genetic load, and the evolution of outcrossing rates in a multilocus system with no linkage[END_REF][START_REF] Lande | High inbreeding depression, selective interference among loci, and the threshold selfing rate for purging recessive lethal mutations[END_REF]) epistatic interactions or genetic linkage [START_REF] Charlesworth | The effect of linkage and population-size on inbreeding depression due to mutational load[END_REF]. All these approaches stress the critical influence of the genetic basis of inbreeding depression and its evolutionary dynamics on mating system evolution (see e.g. [START_REF] Porcher | The genetic consequences of fluctuating inbreeding depression and the evolution of plant selfing rates[END_REF] for an example of how the dynamics of inbreeding depression substantially modify the predictions of an ecological model of mating system evolution). Yet, most models that address the effect of ecological forces on mating system evolution have overlooked the dynamics of inbreeding depression, which is often considered fixed [START_REF] Goodwillie | The evolutionary enigma of mixed mating systems in plants: occurrence, theoretical explanations, and empirical evidence[END_REF]. Simplified models of mating system evolution abound because modeling the joint evolution of mating system and inbreeding depression requires detailed genetic models for both characters, which are often intractable.

One approximation to modeling the joint evolution of inbreeding depression and mating system was proposed by [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF] and later extended by [START_REF] Johnston | Evolution of intermediate selfing rates in plants: pollination ecology versus deleterious mutations[END_REF] to incorporate ecological mechanisms (seed and pollen discounting) in the evolution of plant mating systems. This approximation is used to find joint equilibria of the mating system and inbreeding depression, by examining the indirect selection gradient on small changes in the selfing rate, assuming the mating system evolves by infrequent small mutation steps. The mating system is assumed to undergo no direct selection, but evolves because of its influence on inbreeding depression. This approximation contains elements of Evolutionarily Stable 4 Strategies, as well as inclusive fitness by weighting selfed seed twice as much as outcrossed seed to account for the automatic advantage of selfing described by [START_REF] Fisher | Average excess and average effect of a gene substitution[END_REF]. It can incorporate any genetic model of inbreeding depression, as well as ecological mechanisms influencing the selfing rate [START_REF] Johnston | Evolution of intermediate selfing rates in plants: pollination ecology versus deleterious mutations[END_REF][START_REF] Johnston | Correlations among fertility components can maintain mixed mating in plants[END_REF][START_REF] Devaux | Pollination ecology and inbreeding depression control individual flowering phenologies and mixed mating[END_REF].

This approximation ignores genotypic associations, due to both linkage (gametic) disequilibrium and identity (zygotic) disequilibrium [START_REF] Haldane | The association of characters as a result of inbreeding and linkage[END_REF]Crow and Kimura 1970 eq. 3.9.3), between the mating system locus and viability loci controlling inbreeding depression. Such genotypic associations have been observed in nature [START_REF] Weber | Purging of inbreeding depression within a population of Oxalis alpina (Oxalidaceae)[END_REF]) and theory predicts that in some cases they can greatly influence the dynamics of invasion of a modifier of the selfing rate [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF][START_REF] Uyenoyama | Ecological and genetic factors directing the evolution of self-fertilization[END_REF]). Several models have quantified the impact of genotypic associations on the spread of selfing modifiers [START_REF] Holsinger | Inbreeding depression doesn't matter: the genetic basis of matingsystem evolution[END_REF][START_REF] Charlesworth | Inbreeding depression, genetic load, and the evolution of outcrossing rates in a multilocus system with no linkage[END_REF]Uyenoyama and Waller 1991a,b,c;[START_REF] Schultz | Individual variation in inbreeding depression: the roles of inbreeding history and mutation[END_REF], but intermediate selfing rates were never evolutionarily stable in these models, unless they assumed inbreeding depression due to overdominance (Uyenoyama and Waller 1991b), which has little support from experimental data [START_REF] Charlesworth | The genetics of inbreeding depression[END_REF]. As a result, no study has quantified the influence of genotypic associations between inbreeding depression and selfing modifiers on the evolutionarily stable selfing rates and the accuracy of an approximation assuming no genetic association. Although it has previously been thought that the approximation should be accurate for infrequent small mutations affecting the selfing rate [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF][START_REF] Johnston | Correlations among fertility components can maintain mixed mating in plants[END_REF], this was never demonstrated.

Here we compare this approximation with a full genetic model for the joint evolution of both the selfing rate and inbreeding depression. Both models also include pollen limitation and pollen discounting, two ecological mechanisms that influence the evolution of plant mating systems and create the conditions for the maintenance of intermediate selfing rates [START_REF] Holsinger | Mass-action models of plant mating systems: the evolutionary stability of mixed mating systems[END_REF][START_REF] Knight | Pollen limitation of plant reproduction: pattern and process[END_REF][START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF].

THE MODELS THE APPROXIMATION

We assume a large (effectively infinite) population with selfing rate r in which an initially rare modifier with selfing rate r appears. The resident and modifier genotypes may differ (1) in their total seed set T, due to pollen limitation, and (2) in the amount of pollen exported for outcrossing, P, due to pollen discounting (the decrease in pollen export caused by selffertilization, [START_REF] Harder | A clarification of pollen discounting and its joint effects with inbreeding depression on mating system evolution[END_REF]. The fitness of the modifier genotype affecting the selfing rate incorporates the automatic advantage of selfing [START_REF] Fisher | Average excess and average effect of a gene substitution[END_REF], by weighting selfed seed twice as much as outcrossed seed

) ( ) ( ) ( 2 ) 1 ( ) ( 2 ) 1 ( ) ( 0 0 1 r T r P r P w r r T w r r T w r w - + - + =
where 0 w and 1 w are the mean fitnesses of outcrossed and selfed progeny, respectively [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF]Johnston 1998, eq. 2a). The intensity of selection on the modifier with a small effect on the selfing rate is approximately proportional to the selection gradient:

= =       ∂ ∂ - + ∂ ∂ - + ∂ ∂ - + - = ∂ ∂ ) ( ) ( 2 1 ) ( ) ( 2 ) 1 ( ) ( ) ( ) 1 ( 2 1 ) ( 1 0 δ δ where 0 1 / 1 w w - = δ
is the inbreeding depression in the resident population assumed to be at mutation-selection equilibrium for the given selfing rate [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF].

6 Evolutionary equilibrium selfing rates occur when the selection gradient is zero, which yields the level of inbreeding depression that exactly counterbalances all other constraints on the evolution of selfing, i.e. the automatic advantage, reproductive assurance in the presence of pollen limitation, and pollen discounting

r r r r T r r r P r r r T r =             ∂ ∂ + ∂ ∂ - + ∂ ∂ + = ) ( ln 1 ) ( ln ) 1 ( ) ( ln 1 2 1 ) ( δ .
(1)

The right-hand side of equation ( 1), hereafter referred to as the constraint function, can be compared to the inbreeding depression ) (r δ expected at equilibrium in a population with selfing rate r under any explicit genetic model for inbreeding depression. Equilibrium selfing rates occur at the intersections of the constraint function and ) (r δ (Fig. 1).

To model inbreeding depression, we use the Kondrashov model (1985), which describes the evolution of the distribution of number of partly recessive deleterious alleles per mature plant in the population when mutations occur at an infinite number of unlinked loci in an infinite population [START_REF] Lande | High inbreeding depression, selective interference among loci, and the threshold selfing rate for purging recessive lethal mutations[END_REF][START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF]. We considered separately two contrasting classes of deleterious mutations that are believed to cause inbreeding depression [START_REF] Charlesworth | The genetics of inbreeding depression[END_REF]: nearly recessive lethal mutations vs. partially recessive, mildly deleterious mutations. We also analyzed a model incorporating a constant 'background' inbreeding depression into a Kondrashov model with nearly recessive lethals, because the equilibrium inbreeding depression due to stabilizing selection on quantitative characters, or to nearly additive, mildly deleterious mutations, undergoes relatively little purging in response to an increased selfing rate [START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF]; and see Fig. S1 in Supporting Information).

MAINTENANCE OF MIXED MATING WITH POLLEN DISCOUNTING AND POLLEN LIMITATION

Pollen limitation and pollen discounting are described by the seed set T(r) and pollen export P(r) functions employing the mass-action model for pollination [START_REF] Holsinger | Mass-action models of plant mating systems: the evolutionary stability of mixed mating systems[END_REF][START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF]. All genotypes are expected to produce the same number of flowers and the same amount of pollen P T . A rare mating system modifier changes the selfing rate by altering the fraction α of pollen a genotype exports for outcrossing. Because the mating system modifier is rare, the amount of outcross pollen landing on the stigma of any plant is approximately

T o o P π α = P
, where α is the fraction of pollen exported by the resident genotype and π o the probability that pollen exported for outcrossing actually lands on a stigma. The amount of self pollen landing on the stigmas of the resident and modifier genotypes are

T s s P π α = P ) 1 ( - and T s s P π α = P ) 1 ( -
, respectively, where π s is the probability that pollen not exported actually lands on the stigma. Under competing selfing, the primary selfing rate r of the modifier genotype, after fertilization, is the ratio of self-pollen to total pollen landing on the stigma r = P s /(P s +P o ), which can also be written

π α + α α = P π α + P π α P π α = r T o T s T s - - - - 1 1 ) 1 ( ) 1 ( ) (α (2)
where π = π o /π s, the relative success of outcross vs. self pollen, quantifies the strength of pollen discounting [START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF].

The amount of pollen the modifier genotype exports for outcrossing is P = απ o P T , which can be written as a function of the selfing rate using equation (2),

) (1 / 1 r π α r = α - - , so that T o P r r r P π π α       - - = 1 1 ) ( . (3)
Under pollen limitation, the total seed set T of the modifier genotype depends on the amounts of self and outcross pollen landing on the stigma,

( ) [ ] π α + α P π = P P = T T o s o / ) 1 ( exp 1 ] [ exp 1 - - - - - -
which can also be written as a function of the selfing rate r using equation ( 2

), [ ] ) 1 /( exp 1 r α P π = T(r) T o - - - . (4)
From equation (4), one can see that π o P T , the amount of pollen exported by a completely outcrossing individual that reaches a stigma, can be used to quantify pollen limitation independently of the population mating system.

Noting that )] 1 ( 1 /[ ) 1 ( π r r = α - - -
and differentiating equations ( 3) and ( 4) we find 2 ) 1 (

) ( ln r π = r r P r r - - ∂ ∂ = and [ ] [ ] ) 1 ( 1 ) 1 )( ( ) ( 1 ) ( ln π - - - - ∂ ∂ = r r r T P π r T = r r T T o r r .
These are used to obtain the constraint function from equation (1). The intersection between the constraint function and inbreeding depression ) (r δ is then explored numerically to obtain the equilibrium selfing rates expected under the approximation (Fig. 1A-C). The stability of an equilibrium is found by comparing the values of the constraint function and ) (r δ around the equilibrium: equilibrium selfing rates are evolutionarily stable if an increase in r results in the inbreeding depression being larger than the constraint function (greater costs than benefits of selfing) and vice versa.

THE FULL GENETIC MODEL

The equilibrium selfing rates predicted by the approximation are compared to those obtained using the full genetic model of [START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF] describing joint evolution of both mating system and inbreeding depression in an infinite population. This uses the same pollination model, where the selfing rate, seed set and pollen export are controlled by the fraction α of pollen exported for outcrossing. The pollination model is coupled with the [START_REF] Kondrashov | Deleterious mutations as an evolutionary factor. 2. Facultative apomixis and selfing[END_REF] model, in which inbreeding depression is caused either by nearly recessive lethals (with or without an additional constant background inbreeding depression) or by nearly additive, mildly deleterious mutations. To model the evolution of the selfing rate, we added an unlinked locus modifying the mating system to the genetic model of inbreeding depression, which accounts for associations between deleterious mutations and alleles influencing the selfing rate. In a resident population at mutation-selection equilibrium, a modifier allele with a pollen export fraction α slightly different from the resident is introduced at a low frequency in linkage and identity equilibrium with deleterious mutations.

We examine the fate (invasion or not) of this initially rare modifier allele after 2,000 generations to find the evolutionarily stable selfing rates, i.e. resident selfing rates that cannot be invaded.

PARAMETER VALUES

We varied genetic and ecological parameters to generate a wide range of intermediate stable selfing rates that can be compared between the two theoretical approaches. The genomic mutation rate to deleterious mutations was U = 0.02, 0.2 or 1; the dominance coefficient of lethals was set to h = 0.02 (references in [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF][START_REF] Lande | High inbreeding depression, selective interference among loci, and the threshold selfing rate for purging recessive lethal mutations[END_REF]; mildly deleterious mutations were characterized by s = 0.05 and h = 0.4 (references in [START_REF] Halligan | Spontaneous mutation accumulation studies in evolutionary genetics[END_REF]. We also analyzed a model with such nearly recessive mutation to lethals and a constant background inbreeding depression of d = 0.25 [START_REF] Winn | Analysis of inbreeding depression in mixed-mating plants provides evidence for selective interference and stable mixed mating[END_REF].

The relative success of self vs. outcross pollen π was varied between 10 -4 (no pollen discounting) and 0.9999 (strong pollen discounting). We also considered a wide range for the strength of pollen limitation, from π o P T = 0.5 to π o P T =10 10 . These values correspond to seed sets of 0.4 and 1, respectively, for a completely outcrossing population. In the figures, we use a value of π o P T = 1.5 for moderate pollen limitation, corresponding to a seed set of 0.78 in a completely outcrossing population, which is representative of realistic values observed in natural populations [START_REF] Knight | Pollen limitation of plant reproduction: pattern and process[END_REF]).

In the full genetic model, we examined the spread of a rare modifier allele that increased or decreased the selfing rate compared to the resident population (see [START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF] for details). Because we were interested in the accuracy of the approximation when the selfing rate evolves by small steps, we considered a rare modifier that increased or decreased selfing by 10 -6 . The modifier allele was introduced at an initial frequency of 10 -8 in a resident population at mutation-selection equilibrium for lethals. The modifier genotypes were initially at frequencies expected for a population with inbreeding coefficient 1921,1969) and in linkage and identity equilibrium with lethal alleles. The recursion equations were numerically iterated for 2,000 generations to detect successful invasion of the modifier of the selfing rate. We verified that if the resident population is initially not at a stable equilibrium selfing rate, a modifier allele causing a small change in the selfing rate toward the equilibrium eventually becomes fixed.

) 2 /( r r f - = (Wright

RESULTS AND DISCUSSION

The approximation produces relatively accurate equilibrium selfing rates under low to moderate genomic rates of mutation to lethals (Fig. 2A,D and, to a lesser extent, B,E) or with nearly additive, mildly deleterious mutations (Fig. S2). We checked that with no inbreeding depression, U = 0, both models predict identical equilibrium selfing rates. For example, with no pollen limitation, T(r) =1, equation (1

) simplifies to )] 1 ( 1 [ 2 / 1 ) ( r r e - - = π δ
and the evolutionarily stable selfing rate is r * = 1 -π [START_REF] Holsinger | Mass-action models of plant mating systems: the evolutionary stability of mixed mating systems[END_REF]. With low inbreeding depression, i.e. moderate genomic mutation rate to lethals (U = 0.02) or mildly deleterious mutations, and no pollen limitation, the stable selfing rates are also close to 1 -π for both models (Fig. 2A and Fig. S2A-C). As expected, higher pollen limitation generally favors larger stable selfing rates but again the approximation is relatively accurate with moderate mutation rates (Fig. 2D and Fig. S2D-E).

With higher genomic mutation rates to lethals, U = 0.2 and U = 1, the equilibria predicted by the approximation can differ greatly from those of the full genetic model (Fig. 2) and the discrepancy between the two models increases as U increases. For U = 0.2, the predictions of the approximation agree reasonably well with those of the full model for limited pollen discounting, especially with pollen limitation (Fig. 2B,E), but the discrepancy between the models increases with more intermediate equilibrium selfing rates, which are obtained in this case with increasing pollen discounting. For low pollen limitation and high pollen discounting the full model predicts mixed mating when the approximation predicts complete outcrossing (Fig. 2B,E). With a very high genomic mutation rate to lethals, U = 1, the approximation becomes completely inaccurate (Fig. 2C,F), predicting that the only evolutionarily stable mating system is complete outcrossing (and also complete selfing in the case of no pollen discounting), because the genetic inbreeding depression usually exceeds the constraint function (Fig. 1C). In contrast, the full genetic model predicts a number of evolutionarily stable intermediate selfing rates, most of which are close to 1 and depend little on pollen discounting and pollen limitation (see [START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF].

A discrepancy between the full genetic model and the approximation is also observed, although to a lesser extent, with high mutation rates to mildly deleterious mutations and moderate pollen limitation (U = 1, Fig. S2F). Here, the full genetic model sometimes predict complete selfing when the approximation predicts mixed mating systems with a large selfing rate (r > 0.6). The combination of both types of mutations is therefore likely to exacerbate the difference between both models. This was not tested here, because the Kondrashov model with more than one mutation type is computationally demanding. Instead, we modeled the combined effects of nearly recessive, highly deleterious mutations and nearly additive, mildly deleterious mutations by adding a constant background inbreeding depression (d = 0.25) to the Kondrashov model with lethals. The results remain qualitatively the same, with larger discrepancies under higher mutation rates to lethals (Fig. 2G-L), but the inability of the approximation to predict mixed mating appears at lower genomic mutation rates, as expected (e.g. U = 0.02, Fig. 2J).

The discrepancy between the approximation and the full genetic model is caused by genotypic associations between alleles at the modifier locus and the genetic basis of inbreeding depression [START_REF] Holsinger | Inbreeding depression doesn't matter: the genetic basis of matingsystem evolution[END_REF][START_REF] Charlesworth | Inbreeding depression, genetic load, and the evolution of outcrossing rates in a multilocus system with no linkage[END_REF]Uyenoyama and Waller 1991a,c). In a partially selfing population, rare genotypes with a modified selfing rate may differ from the resident population in their average history of inbreeding, which impacts their associated genetic load (or mean number of recessive deleterious alleles), whereas the approximation assumes identical genetic load regardless of the genotype at the modifier locus.

For example, when the modifier allele is rare, plants homozygous for the modifier are initially produced mostly by selfing and therefore partially purged of recessive deleterious alleles (Fig. 1D-F). Extensive simulation showed that genetic associations between a modifier of selfing and recessive lethal mutations build up over a few dozen generations. As a result, a modifier allele that eventually invades and becomes fixed may initially decrease in frequency [START_REF] Schultz | Individual variation in inbreeding depression: the roles of inbreeding history and mutation[END_REF]. Thus the simulations were run for 2,000 generations to detect successful invasion.

Differential purging of the load associated with modifier genotypes is transient; for a modifier of small effect destined for fixation the homozygous modifier gains genetic load as it becomes common [START_REF] Schultz | Individual variation in inbreeding depression: the roles of inbreeding history and mutation[END_REF]. The initial differential purging for a rare modifier depends little on the magnitude of the modifier effect, and also occurs for (neutral) mutations with no impact on the selfing rate [START_REF] Charlesworth | The apparent selection on neutral marker loci in partially inbreeding populations[END_REF]. However, it can strongly influence the fate of mating system modifiers even with very small effect, and alter the evolutionarily stable selfing rate and inbreeding depression by shifting the equilibrium selfing rate, especially when inbreeding depression is caused by nearly recessive lethals. Overall, this promotes enhancers of selfing under broader conditions than expected with the approximation, as demonstrated in earlier studies [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF][START_REF] Charlesworth | Inbreeding depression, genetic load, and the evolution of outcrossing rates in a multilocus system with no linkage[END_REF]Uyenoyama and Waller 1991c;[START_REF] Schultz | Individual variation in inbreeding depression: the roles of inbreeding history and mutation[END_REF]. We have shown here that the joint evolution of inbreeding depression and mating system, combined with common mechanisms of pollination ecology, also favors mixed mating over complete outcrossing more often than predicted by the approximation.

Conclusion -We show that the accuracy of the approximation depends primarily on the genomic mutation rate to lethals, for which few estimates are available. The best estimates, from Drosophila, indicate that U = 0.01-0.03 per genome per generation [START_REF] Fry | New estimates of the rates and effects of mildly deleterious mutation in Drosophila melanogaster[END_REF][START_REF] Charlesworth | Estimates of the genomic mutation rate for detrimental alleles in Drosophila melanogaster[END_REF]. Evidence suggests that annual plants may have comparable values of U (e.g. [START_REF] Jürgens | Genetic analysis of pattern formation in the Arabidopsis embryo[END_REF] in Arabidopsis thaliana). For such species, the approximation is accurate. However, mutation rates to lethals may be an order of magnitude higher (U = 0.2) in large perennial or partially asexual species due to mutation accumulation in somatic cell divisions, as plants lack a separate germ line (Lande et al. 1994 and references therein;[START_REF] Remington | Whole-genome characterization of embryonic stage inbreeding depression in a selfed loblolly pine family[END_REF]. The approximation also tends to underestimate stable selfing rates under high mutation rates to mildly deleterious, nearly additive alleles, although less so than with lethals. High mutation rates to mildly deleterious alleles, sometimes greater than 1, may also occur in multicellular organisms [START_REF] Halligan | Spontaneous mutation accumulation studies in evolutionary genetics[END_REF].With high U the approximation greatly underestimates the evolutionarily stable selfing rates, so that explicit genetic models of both selfing rate and inbreeding depression are required for a comprehensive understanding of mating system evolution.

FIGURE CAPTIONS Figure 1 .

 CAPTIONS1 FIGURE CAPTIONS

Figure 2 . 1 U

 21 Figure 2. Equilibrium selfing rates predicted by the approximation vs. the full genetic model,

Evolution

Acknowledgments

This work was partly funded by the French CNRS grant PICS #5273 to E.P., grants from the Balzan foundation and The Royal Society of London to R.L.