Strong effects of heterosis on the evolution of dispersal rates

Authors


Denis Roze, Station Biologique de Roscoff, Place Georges Teissier, BP 74, 29682 Roscoff Cedex, France.
Tel.: +33 2 98 29 23 20; fax: +33 2 98 29 23 36; e-mail: roze@sb-roscoff.fr, francois.rousset@isem.univ-montp2.fr

Abstract

When dispersal is limited, crosses between different regions may generate progeny of higher fitness than crosses within regions, due to the fact that individuals from the same region are more likely to share the same recessive deleterious alleles. This phenomenon (termed heterosis) generates a selective force favouring dispersal; however, the importance of heterosis on dispersal evolution has been a subject for debate. In this paper, we use computer simulations representing deleterious mutations occurring over a whole genome (of arbitrary map length R) to explore the magnitude of heterosis, and its effect on the evolution of dispersal. These results show that heterosis may have important effects on dispersal when it is in the upper range of values observed in natural populations, which occurs in our simulations when the genomic deleterious mutation rate U is also in the upper range of observed values. Comparing the results with extrapolations from an analytical two-locus model indicates that the effect of heterosis is mainly driven by pairwise associations between the locus affecting dispersal and selected loci when U is not too high (roughly, U < 0.5), whereas higher order associations become important for higher values of U.

Introduction

Because many organisms have limited dispersal abilities, it is very common to observe that individuals sampled from neighbouring locations tend to be more similar genetically than individuals sampled from more distant locations. Such spatial structure has important consequences: in particular, it increases the degree of competition among related individuals (provided that competition occurs locally), and it increases homozygosity, because mating occurs among individuals from the same region, which tend to share the same alleles. Furthermore, all populations are subject to recurrent deleterious mutations, which may occur at relatively high rates in multicellular organisms (Baer et al., 2007). When dispersal is limited, individuals from different regions will tend to carry different sets of deleterious alleles; in that case, crosses among individuals from the same region will tend to produce offspring carrying some deleterious alleles in the homozygous state, whereas crosses among individuals from different regions will produce more heterozygous offspring. Because deleterious alleles are often recessive, offspring produced by the second type of cross will thus tend to be fitter, on average. This increase in fitness is called heterosis; the term ‘inbreeding depression’ is also used to describe the same phenomenon but seen from the opposite perspective (decrease in fitness of crosses from related individuals, compared with crosses between more distantly related individuals).

Several experimental studies have shown that heterosis can be important. In particular, following the work by Price & Waser (1979), many studies involving crosses between plants located at different distances have tested the idea of an ‘optimal outcrossing distance’. According to this idea, under limited dispersal the fitness of offspring should be the highest when parents are separated by an intermediate distance, rather than by a short or a large distance. Indeed, offspring should suffer from inbreeding depression when parents are too close to each other (because they tend to be related, and carry the same recessive deleterious alleles); conversely, offspring may suffer from ‘outbreeding depression’ when parents are too far apart (which may be due to adaptations to different local conditions, or to incompatibilities between the parental genomes). Some studies have provided empirical evidence for such an optimal distance, and thus for heterosis when crosses involve parents separated by an intermediate (rather than a short) distance (e.g. Price & Waser, 1979; Waser & Price, 1989, 1994; Fischer & Matthies, 1997; Stacy, 2001; Willi & van Buskirk, 2005; Billingham et al., 2007). For example, Waser & Price (1989) showed that in scarlet gilia (Ipomopsis aggregata), the lifetime fitness of offspring whose parents are separated by 1 and 100 m is 47% and 68%, respectively, the fitness of offspring whose parents are 10 m apart. Other studies did not find support for such an optimal outcrossing distance (see references in Waser, 1993); nevertheless, in several cases it has been found that the fitness of offpring whose parents are near each other is 30–50% lower than the fitness of offspring whose parents are farther apart (Waser & Price, 1989; Dudash, 1990; Fenster, 1991; Waser & Price, 1994; Trame et al., 1995; Fischer & Matthies, 1997; Byers, 1998; Stacy, 2001; Billingham et al., 2007).

Another type of evidence for heterosis consists in comparisons of the fitnesses of offspring resulting from within- and between-population crosses (Levin & Bulinska-Radomska, 1988; van Treuren et al., 1993; Richards, 2000; Ebert et al., 2002; Paland & Schmid, 2003; Busch, 2006). Again, important reductions in the fitness of individuals produced by within-population crosses (compared with between-population crosses) have been observed in several cases, with values around 30%, 50% and up to 74% in the case of small, geographically isolated populations (Busch, 2006). However, these studies may not be representative of most cases as they involved small populations in which drift should be particularly strong, and in two cases heterosis was not observed in larger populations of the same species (Richards, 2000; Busch, 2006). Therefore, more experimental work is needed in order to obtain a better idea of the magnitude of heterosis in subdivided populations.

Heterosis generates a selective pressure for dispersal, as it increases the fitness of offspring of migrant individuals, relative to the fitness of offspring of philopatric individuals. Several analytical models have compared the relative effect of this selective pressure with other forces that may affect the evolution of dispersal, such as dispersal costs and competition among kin (Gandon, 1999; Perrin & Mazalov, 2000, and references in Ronce, 2007). However, most of these models do not include an explicit genetic mechanism generating heterosis but treat it as a fixed fitness cost paid by philopatric individuals. Multilocus simulations using the island model have also been performed, where heterosis results from recurrent deleterious mutations occurring at a given number of unlinked loci. For example, Morgan (2002) considered 1024 loci at which deleterious mutations occur, and one locus controlling the dispersal rate; however, increased dispersal is always favoured in this model (because there is no cost to disperse), which makes it difficult to disentangle the relative effects of kin selection and heterosis on selection for dispersal. Other studies (Guillaume & Perrin, 2006; Ravignéet al., 2006) introduced a cost of dispersal, in which case the dispersal rate reaches an evolutionarily stable (ES) value which is lower than 1. The relative effect of heterosis on dispersal evolution can then be measured by comparing the ES dispersal rates obtained in the presence and absence of deleterious mutations. These studies found limited effects of heterosis: e.g. Guillaume and Perrin found that on average (over the range of parameter values considered), heterosis increases the ES dispersal rate by only one-third of its value in the absence of deleterious mutations. However, these results were based on scenarios leading to low heterosis (low genomic mutation rates and/or small population sizes, which leads to the fixation of deleterious alleles – once deleterious alleles are fixed globally, they do not contribute to heterosis anymore).

In a previous paper (Roze & Rousset, 2005), we developed an analytical model to study the interplay between kin selection and heterosis on the evolution of dispersal. We first considered a two-locus model, in which one locus affects the dispersal rate of individuals, whereas a second locus is subject to recurrent recessive deleterious mutations, generating heterosis. Three selective forces act on dispersal in this model: direct selection (dispersal is costly, migrants dying with a given probability), kin selection (dispersal reduces competition among kin) and heterosis. The model showed that although heterosis gives an advantage to migrants, it also acts indirectly against dispersal by decreasing the strength of kin selection, because heterosis increases the ‘effective migration rate’ (Ingvarsson & Whitlock, 2000) and thus decreases kinship within demes. We then extended this model to include a large number of selected loci, neglecting statistical associations among these loci. Multilocus simulations indicated that this could lead to accurate predictions of evolutionarily stable dispersal rates, at least in some cases. However, we did not explore extensively the effect of varying parameters (cost of dispersal, deme size, strength of selection against deleterious alleles, mutation rate, etc.) on the influence of heterosis on dispersal.

In this paper, we use multilocus simulations to explore the effect of heterosis on ES dispersal rates, over a range of parameter values. An important difference with previous simulation studies (that represented deleterious mutations occurring at ∼100 or 1000 unlinked loci) is that our model represents a larger number of loci and allows us to study the effect of linkage among loci (by changing the total map length R of the genome). We also represent larger populations, which prevents the fixation of deleterious alleles unless selection is very weak. In a first part of the paper, we explore the magnitude of heterosis generated by deleterious alleles for different values of deme size N, migration rate m, deleterious mutation rate U and map length R. In the second part, we study the joint evolution of heterosis and dispersal rate. These results show that heterosis can have an important effect (relative to kin selection) on evolutionarily stable dispersal rates, in particular when deme size is large and when the genomic deleterious mutation rate is high. The strength of selection against deleterious mutations has little effect on the results, whereas the effect of heterosis increases as mutations become more recessive. We compare these simulation results with predictions from the analytical model of Roze & Rousset, 2005 after correcting an error which affected the results for high dispersal costs (in particular, the model predicted that heterosis could reduce the ES dispersal rate at high dispersal costs, but this prediction is no longer found in the corrected model). These comparisons show that indirect selection at the locus affecting dispersal is mainly driven by pairwise associations with selected loci (at which deleterious mutations occur) when U is not too high (roughly, U < 0.5), whereas higher order associations may become important for higher values of U.

Heterosis

Several models have calculated the strength of heterosis due to a single selected locus subject to recurrent deleterious mutations in a subdivided population. Whitlock et al. (2000) used the infinite island model of population structure, and calculated the magnitude of heterosis by integrating numerically over Wright's distribution of allele frequency in a structured population (Wright, 1937). They extrapolated their results to many loci by assuming that each locus contributes multiplicatively to heterosis and found that heterosis can be substantial even with appreciable rates of gene flow among demes. They also found that heterosis is maximized for mutations of intermediate effect. Glémin et al. (2003) used the island model and the one-dimensional stepping stone (both with a finite number of demes), and used Kimura and Ohta's moment method to solve a diffusion equation (Ohta & Kimura, 1971) and obtain an expression for the mutation load, inbreeding depression and heterosis. This method assumes that deleterious mutations are not too recessive and that selection is strong enough that deleterious alleles remain at low frequencies in each deme. They found that heterosis is higher for less strongly selected mutations (but their method does not allow them to consider weak selection) and that inbreeding within demes reduces heterosis. In a previous paper (Roze & Rousset, 2004), we used a diffusion approach to compute heterosis in the finite island model, assuming that selection is weak relative to migration, and including self-fertilization at a given rate. We obtained approximations for the case of a very large number of demes, assuming that deleterious alleles remain at low frequencies in the whole metapopulation. We found that selfing decreases the magnitude of heterosis; furthermore, simulations confirmed that heterosis is maximized for intermediate selection coefficients.

Although these models give correct predictions of the magnitude of heterosis generated by a single selected locus (at least in the parameter range where the approximations used in the models are valid), it remains unclear to which extent they can be used to compute heterosis due to deleterious mutations occurring over a whole genome. Indeed, the assumption that each selected locus contributes multiplicatively to heterosis is not strictly correct in a spatially structured population, because several types of genetic associations will develop among loci: linkage disequilibria due to the Hill–Robertson effect (Hill & Robertson, 1966; Martin et al., 2006), and correlations in homozygosity among loci, which are generated by population structure even in the absence of selection (‘identity disequilibrium’, e.g. Vitalis & Couvet, 2001). Furthermore, these models do not allow one to predict the effect of recombination rates among selected loci on heterosis. In this section, we compare predictions derived from our previous model (Roze & Rousset, 2004) in the case of a very large number of demes (assuming multiplicative effects of the different loci) with simulations representing a very large number of loci, with arbitrary map length R.

Single-locus model

Throughout the paper, we consider a population that follows the island model of population structure. Analytical expressions will be derived in the limiting case where the number of demes n tends to infinity, whereas in the simulations n will be finite. Each deme contains N diploid adults. Every generation, adult individuals produce a very large number of gametes and die. Gametes fuse randomly within each deme, and the juveniles produced migrate to another deme with probability m. Finally, N individuals are sampled randomly within each deme to form the next adult generation. We consider a single locus (that will be called locus B) at which recurrent deleterious mutations occur. Two alleles (denoted 0 and 1) segregate at this locus, and we suppose that adults of genotype 00, 01 and 11 have (relative) fecundities 1, 1 − hs and 1 −s, respectively, where fecundity simply means the number of gametes produced. Allele 1 is thus deleterious, and we assume a constant mutation rate u from allele 0 to 1 per generation. We call pB(ij) the frequency of allele 1 in individual j of deme i, pB(i) the frequency of 1 in deme i and pB the frequency of 1 in the whole population. Heterosis may be defined as:

image(1)

where fw is the average fecundity of offspring produced by random mating within demes and fb the average fecundity of offspring whose parents are sampled randomly from the whole population (e.g. Whitlock et al., 2000; Roze & Rousset, 2004).

In the supporting information (Appendix S1) we show that, to the first order in s (i.e. assuming weak selection),

image(2)

where inline image is the covariance in allelic state between two genes sampled with replacement from the same deme:

image

where E means the average over the whole population. At the neutral equilibrium, inline image equals FpBqB, where qB = 1 − pB, and where F is equivalent to FST in the neutral infinite island model (it is a function of N and m, given in Appendix S1). This gives us an expression for H at equilibrium, under weak selection:

image(3)

Furthermore, it is possible to obtain an expression for pB at mutation–selection equilibrium (e.g. Roze & Rousset, 2004); assuming that pB remains small, this expression takes the form u/(sT), where T is a function of N, m and h (given in Appendix S1). Therefore, we have:

image(4)

independent of s. Considering a genome made of L loci, with a deleterious mutation rate u per locus, and assuming for simplicity that all deleterious alleles have the same dominance coefficient, we can then obtain a simple expression for heterosis generated by mutations occurring over the whole genome (that we will denote inline image) if we assume multiplicative contributions of all loci (i.e. no association among loci):

image(5)

where U = uL is the mutation rate per haploid genome. Finally, assuming that m is small (of order ε) and N large (of order 1/ε), one obtains:

image(6)

We also obtained an expression for heterosis to the second order in s, i.e. for stronger selection (see Appendix S1). This requires computing the equilibrium value of the association inline image to the first order in s, i.e. the effect of deleterious mutations occurring at a given locus on FST at this locus. In general, FST is reduced compared with the neutral case, in particular when deleterious mutations are recessive; indeed in that case, migrants are advantaged because their offspring tend to be more heterozygous, which increases the effective migration rate (Ingvarsson & Whitlock, 2000). This effect of selection on FST tends to reduce heterosis. The calculations presented in Appendix S1 (using the methods developed in Roze & Rousset, 2008) lead to the following approximation for heterosis (for large N and small m):

image(7)

where

image(8)

In the following, we compare these approximations with multilocus simulation results.

Multilocus simulations

Our simulation program describes a population subdivided into a finite number n of demes. The genome of each individual consists of two homologous chromosomes. Each generation, the number of new mutations per chromosome is sampled according to a Poisson distribution with parameter U; the position of each new mutation along the chromosome is then sampled from a uniform distribution (the number of loci at which mutations can occur is thus effectively infinite). The relative fecundity of an individual (number of gametes produced) is given by

image

where nHe and nHo are the numbers of heterozygous and homozygous mutations in this individual respectively. Fecundity is assumed to be very large, so that the probability that a gamete contributing to the next generation comes from a given parent is equal to the relative fecundity of the parent. At meiosis, the number of crossovers is sampled according to a Poisson distribution with parameter R (genome map length); the density of crossovers is assumed to be constant along the whole genome. Details about the program are given in Appendix S2.

Figure 1 shows heterosis for different values of N, m, U and R. It shows that our approximations often give accurate predictions of the strength of heterosis, except for strong mutation rates (U = 1, right-hand side) and low recombination. This discrepancy is probably due to associations between selected loci, which are neglected in the analytical model (as we assume that all loci contribute multiplicatively to heterosis). Furthermore, Fig. 1 confirms that heterosis can reach appreciable levels even in the presence of substantial gene flow, in particular when U = 1 (e.g. in the lower right plot, inline image is about 30–40% when Nm is between 1 and 2). Last, it shows that the effect of the recombination rate between selected loci on heterosis depends on the degree of population structure: e.g. in the upper right plot, one can see that decreasing map length R from 10 to 1 reduces heterosis when N = 50 and 100 but increases heterosis when N = 20. A possible explanation is the following: when the recombination rate is low, mating between close relatives generates offspring of very low fitness (because these offspring tend to be homozygous at many loci). When population structure is strong, individuals from the same deme will be close relatives (drift within demes is strong); in that case, decreasing recombination increases heterosis. However, when population structure is less strong, the fact that mating among close relatives produce low-fitness offspring will maintain divergent lineages within the same deme (this is possible only if drift within demes is not too strong). Decreasing the recombination rate will increase this effect (as it decreases the fitness of offspring from matings within lineages). In the extreme case of no recombination, divergent chromosomes can be maintained by negative frequency dependence within the same deme (and individuals carrying two divergent chromosomes have relatively high fitness). This effect tends to reduce heterosis. Still, we can note that changing the recombination rate has little effect once it is above a given value: indeed, results for R = 10 and 100 are very similar.

Figure 1.

 Heterosis as a function of deme size N, for different values of m and U, and for s = 0.05, h = 0.1. Solid curves: first-order approximation for N large and m small (eqn 6); dashed curves: second-order approximation for N large and m small (eqns 7 and 8). Dots: simulation results for R = 1 (empty squares), R = 10 (filled circles) and R = 100 (empty circles). In the simulations, the number of demes is set to n = 1000 for N = 10 and 20, and to n = 500 for N = 50 and 100 (different values of n lead to similar results, as long as n is large enough). For low recombination (R = 1), we did not obtain simulation results for U = 1 and N = 10 when m = 0.05, and for U = 1, N = 10 and 20 when m = 0.02, because deleterious mutations accumulate and no equilibrium is reached.

Evolution of dispersal

Heterosis gives an advantage to migrant individuals, which translates into a selective pressure for increased dispersal. We now introduce a locus affecting the dispersal rate of individuals (‘migration modifier’ locus) in order to quantify this effect. We will compare simulation results with analytical extrapolations from a previous two-locus model (Roze & Rousset, 2005), in order to determine the relative effects of pairwise genetic associations between the modifier and the selected loci, and of higher order associations.

Mathematical model

The analytical model presented in Roze & Rousset, 2005 considers two loci: the first locus is a selected locus (generating heterosis), whereas the second locus controls the probability of dispersing at the juvenile stage. As in the previous model, mutations from allele 0 to 1 occur at rate u at the selected locus (called locus B), and we assume that 00, 01 and 11 individuals have fecundities 1, 1 − hs and 1 − s respectively. Two alleles (also denoted 0 and 1) segregate at the modifier locus (called locus A), and we assume that 00, 01 and 11 individuals migrate with probabilities m, m + ε/2 and m + ε (we assume additivity at the modifier locus for simplicity). We then call r the recombination rate between the two loci, N the number of adults per deme and introduce a cost of dispersing c (we assume that each dispersing juvenile dies with probability c). Allele frequency at the modifier locus changes during reproduction, due to genetic associations with the selected locus (individuals which migrate more tend to be more heterozygous at the selected locus and have a greater fecundity); it then changes during dispersal/population regulation, due to the cost of dispersal and to kin selection (because competition occurs at the level of the deme, dispersal reduces kin competition).

In Roze & Rousset, 2005, we show that the change in frequency of allele 1 at the migration modifier locus during reproduction is (to the first order in s):

image(9)

In this expression, DAB,B measures the genetic association between a gene sampled at the modifier locus, and the two genes of the same individual at the selected locus; this association is negative when ε > 0 (i.e. when allele 1 at locus A codes for increased dispersal), because individuals carrying allele 1 tend to be less homozygous at the selected locus. Then, inline image is the association between one gene sampled at the modifier locus, and the two genes at the selected locus, from a second individual sampled with replacement from the same deme. This association is also negative when ε > 0 (in demes where the frequency of allele 1 at locus A is higher, individuals tend to be less homozygous at the selected locus) but smaller than DAB,B in absolute value (therefore, inline image is negative).

The change in frequency of allele 1 at the modifier locus during dispersal, to the first order in ε, is given by (Roze & Rousset, 2005):

image(10)

where pA and qA are the frequencies of alleles 1 and 0 at locus A in the whole population, and where inline image is the association between two genes sampled at the modifier locus, from two individuals from the same deme, at the juvenile stage (before dispersal). The first term within the brackets (−cpAqA) represents the effect of direct selection against dispersal (due to the cost of dispersal), whereas the second term represents the effect of kin selection (due to positive relatedness within demes, i.e. positive inline image). When s = 0 (no heterosis), inline image can be replaced by its neutral equilibrium value (which is FpAqA), to obtain an expression for ΔMpA to the first order in ε. This expression takes the form εWIFpAqA, where WIF is a function of N, m and c. Solving WIF = 0 for m gives the ES dispersal rate in the absence of heterosis, assuming that evolution proceeds by mutations of small effects (e.g. Taylor, 1988; Gandon & Rousset, 1999, eq. 14).

When s ≠ 0, heterosis affects the evolution of dispersal through two terms: the term ΔSpA given in eqn 9, and the term inline image in eqn 10, which is affected by heterosis: indeed, heterosis increases the ‘effective migration rate’, thus reducing relatedness within demes (and therefore reducing the strength of kin selection for dispersal). Assuming that s and ε are small relative to r and m, we used a quasi-equilibrium argument to compute the associations DAB,B and inline image to the first order in ε, and the association inline image to the first order in s (Roze & Rousset, 2005). In Appendix S3, we correct an error that was made in this calculation (that stemmed from an error in deriving neutral two-locus associations, rather than in the quasi-equilibrium argument). Using these quasi-equilibrium expressions, the change in frequency at the modifier locus (over the whole life cycle) takes the form:

image(11)

where Wind is a complicated function of N, m, c and r (approximations for large N and small m are given in Appendix S3). From eqn 3, this may also be written as:

image(12)

where H measures heterosis generated by the selected locus. Equation 12 shows that, if we know the amount of heterosis generated by the selected locus, we do not need to know details about s, u and h to predict the effect of the selected locus on the change in frequency at the modifier locus (however, note that ΔpA depends on the recombination rate between the two loci, as Wind depends on r).

To represent the effect of heterosis due to deleterious mutations occurring over a whole genome on the evolution of dispersal, we model a genome made of a single chromosome of genetic length R. The structure of the genome and effects of mutations are the same as in the multilocus model for heterosis, except that a migration modifier locus is located at the centre of the chromosome. The recombination rate between two loci located at a distance x is the probability that an odd number of crossovers occurs between these two loci, which is r(x) = [1 − exp(−2x)]/2. To calculate the change in frequency of the modifier, we replace r by r(x) in eqn 12, and integrate over x:

image(13)

For a given set of parameters, we compute numerically the term within brackets of eqn 13 for a range of values of m (using the NIntegrate function of Mathematica), in order to find the value of m for which ΔpA = 0, which corresponds to the ES dispersal rate (again, assuming that evolution proceeds by mutations of small effect). Importantly, this method assumes that each selected locus contributes independently to the change in frequency at the modifier locus (through associations such as DAB,B, whose effects are summed over all selected loci). Higher order associations (such as DABC,BC between the modifier and two selected loci) should also be generated by the effect of the modifier but are tedious to compute. Although these higher order associations would affect the change in frequency of the modifier through terms of lower order of magnitude (e.g. DABC,BC would be multiplied by s2 in the expression of ΔSpA, whereas DAB,B is multiplied by s), the number of higher order associations may be very large when deleterious alleles segregate at many loci; therefore, the relative effect of these higher order associations is difficult to predict a priori.

Quantitative results

Figure 2 shows the ES dispersal rate multiplied by deme size (Nm, y-axes) as a function of the value of heterosis once the ES dispersal rate has been reached (inline image, x-axes), for different values of the cost of dispersal c, deme size N and the deleterious mutation rate per haploid genome U (for h = 0.1 and R = 10). An equivalent figure, showing the dispersal rate m on y-axes, is given in Appendix S4. Curves correspond to the analytical prediction, using eqn 13 above to compute the ES dispersal rate for different values of U, from zero (no heterosis, left-hand sides of the curves) to one (right-hand sides of the curves). The value of heterosis at the ES dispersal rate is computed using eqn 5. Note that these predictions are independent of the strength of selection against deleterious alleles (s). As the deleterious mutation rate U increases (moving from left to right on the curves), the ES dispersal rate increases and heterosis at the ES dispersal rate also increases. Dots in Fig. 2 correspond to simulation results for s = 0.05 and for different combinations of N and c, each time for U = 0, 0.2, 0.4, 0.6, 0.8 and 1. The general setting of our simulation program is the same as described in the ‘Heterosis’ section above, except that we add a locus located at the centre of the chromosome that controls the migration rate: each individual migrates with a probability given by the average value of its two alleles at this migration modifier locus (these values range from zero to one). During the first 5000 generations, we fix the migration rate to the ES value in the absence of heterosis (equation 7 in Gandon & Rousset, 1999), in order to reach mutation/selection equilibrium. We then draw the value of each allele at the migration modifier locus in a uniform distribution (between zero and one): at this point there are thus multiple alleles at the modifier locus, coding for different dispersal rates. We also introduce mutation at this locus, at rate 10−3 per generation. Mutation changes the value of an allele to its previous value, plus a random number sampled from a uniform distribution between −0.1 to 0.1 (if the new value is lower than 0 or higher than 1, it is fixed to 0 or 1 respectively). Once variability is introduced at the modifier locus, we let the simulation run for 105 generations, and measure the average migration rate, average fecundity and heterosis every 10 generations; the equilibrium mean dispersal rate is obtained by averaging over the last 90 000 generations (error bars, computed using Hastings’ (1970) batching method are smaller than the size of symbols). For all parameter sets presented in Figs 3–6, in the cases where U = 1 we also ran simulations where the initial value of each modifier allele (after the first 5000 generations) was set to 0.05 or to 0.95 (instead of sampling randomly these initial values). These different initial conditions led to the same equilibrium migration rates. More details about the simulation program are given in Appendix S2. The table given in Appendix S4 provides more numerical results from the simulations, in particular the average fecundity of individuals and the average number of mutations per chromosome, for the sets of parameters corresponding to the different points in Fig. 2.

Figure 2.

 Evolutionarily stable dispersal rate multiplied by deme size (Nm, y-axes) and heterosis at the ES dispersal rate (inline image, x-axes), for different values of U, N and c. Curves: analytical model prediction (where U increases from 0 to 1 as one moves from left to right on the curves); dots: simulation results for U = 0, 0.2, 0.4, 0.6, 0.8 and 1 (from left to right). Dotted line, filled circles: N = 500, dashed-dotted line, empty circles: N = 100, dashed line, filled squares: N = 30, solid line, empty squares: N = 10. Other parameter values: h = 0.1, R = 10. In the simulations, s = 0.05, and the number of demes is set to n = 1000 when N = 10, n = 500 when N = 30 and 100, and n = 100 when N = 500 (setting n = 100 for all values of N leads to very similar quantitative results, but increases the variance for low values of N).

Figure 3.

 Same as Fig. 2 for h = 0.3 and s = 0.01 in the simulations.

Figure 4.

 Weak effect of the strength of selection against deleterious mutations (s) on evolutionarily stable dispersal rate. Same as Fig. 2 for N = 30 and c = 0.4, using different values of s in the simulations: s = 0.02 (empty circles), s = 0.05 (filled circles), s = 0.1 (empty squares), s = 0.5 (filled squares) and s = 0.9 (diamonds).

Figure 5.

 Effect of the dominance coefficient against deleterious mutations (h) on evolutionarily stable dispersal rates. (a) Same as Fig. 2 for N = 30 and c = 0.4, and for different values of h: h = 0 (empty circles), h = 0.05 (filled circles), h = 0.1 (empty squares), h = 0.2 (filled squares) and h = 0.3 (diamonds). (b) Empty circles: results for h = 0 (same as top); empty squares: half mutations with h = 0.05, half with h = 0.15; filled circles: half mutations with h = 0, half with h = 0.2.

Figure 6.

 Effect of map length (R) on ES dispersal rates. (a) Same as Fig. 2 for N = 30 and c = 0.4, and for different values of R: R = 1 (solid, filled circles), R = 10 (dashed, empty circles), R = 100 (dotted, empty squares). (b) Effect of introducing a distance between the modifier locus and the nearest selected loci on both sides of the chromosome (for the same parameter values as above, R = 10). Solid, filled circles: no distance; dashed, empty circles: minimum recombination rate of 0.1 between the modifier and selected loci; dotted, empty squares: minimum recombination rate of 0.3 between the modifier and selected loci.

Figure 3 shows equivalent results for h = 0.3 and for using s = 0.01 in the simulations. Figures 2 and 3 show that heterosis may have important effects (relative to kin competition) on ES dispersal rates, especially when the deleterious mutation rate is high. For example, for U = 1 and for the parameter values used in Fig. 2 (s = 0.05 and h = 0.1), the ES dispersal rate is often two to eight times greater than the dispersal rate in the absence of heterosis, whereas for the parameter values used in Fig. 3 (s = 0.01 and h = 0.3), m is often 1.5 to twice greater when U = 1 than when U = 0. Interestingly, the relative effect of heterosis (compared with the effect of kin competition) is stronger when deme size is large: e.g. in Fig. 2, for c = 0.2 and U = 1, the ES dispersal rate is about three times the dispersal rate in the absence of heterosis when N = 10 and eight times the dispersal rate without heterosis when N = 500. Therefore, although the selective pressures on dispersal generated by heterosis and by kin competition both decrease as deme size increases, the relative effect of heterosis increases. However, the absolute effect of heterosis on m is, of course, small when N is large (as can be seen in Appendix S4), as ES dispersal rates are very small in this case. Finally, Fig. 2 shows that our analytical model (that only considers two-locus associations) gives correct predictions as long as the deleterious mutation rate is not too high (roughly, U < 0.5); discrepancies observed at higher mutation rates probably stem from the fact that the model ignores higher order associations (as discussed above). The analytical results are also not very good for high dispersal costs (c = 0.6); indeed, high values of c select for low dispersal rates, which increases the magnitude of higher order associations, and also violates the quasi-equilibrium assumption used in the model (which supposes that selection is weak relative to migration and recombination).

Equation 13 predicts that the ES dispersal rate should not depend on the strength of selection against deleterious mutations (s). Figure 4 shows simulation results for different values of s ranging from 0.02 to 0.9. Results for s = 0.02, 0.05 and 0.1 are very similar quantitatively (empty and filled circles, empty squares in Fig. 4), confirming that s has little effect on heterosis and ES dispersal rates as long as selection is mild. Increasing the value of s above 0.1 (strong selection) decreases both the dispersal rate and heterosis at the ES dispersal rate (filled squares and diamonds in Fig. 4, for s = 0.5 and 0.9), but dots still roughly fall along the same curve in Fig. 4. Figure 5A shows the effect of varying h (between 0 and 0.3): decreasing h increases the ES dispersal rate. Several points are worth noting about Fig. 5A: first, the different points align on the same curve, which confirms the model's prediction that one does not need to know details about s, U and h to predict the effects of heterosis on the ES dispersal rate (however, one has to know N and c). Second, for low dominance coefficients heterosis goes through a maximum, and then decreases, as one increases the deleterious mutation rate (as long as dispersal is free to evolve). Last, the analytical model performs poorly for low dominance coefficients (and relatively high mutation rates), which again indicates a relatively strong effect of higher order associations when h is low.

A strong assumption of our model is that all deleterious alleles have the same dominance coefficient. The effect of introducing variance in h is not obvious (one cannot simply average over ES dispersal rates obtained for the different values of h). Figure 5B shows the effect of a variance in h; here, we consider two simple situations where the average h is 0.1: in the first situation h = 0.05 for half the mutations, whereas h = 0.15 for the other half; in the second situation h=0 for half the mutations and h=0.2 for the other half. Figure 5B shows that introducing variance in h does not affect the relation between heterosis and the ES dispersal rate (dots still fall along the same curve as in Fig. 5A); furthermore, mutations with lower h have a stronger influence on the results (e.g. the ES dispersal rate in the second situation is higher than the average of ES rates obtained when h = 0 for all mutations, and when h = 0.2 for all mutations).

Figure 6A shows the effect of varying R (average number of crossovers per genome per meiosis). In general, we found that decreasing R increases the ES dispersal rate, but that changing R has little effect as long as it is not too small (results for R = 10 and 100 are often not very different). We also modified our simulation program in order to introduce a minimal genetic distance between the modifier and selected loci (in order to see if loci that are closely linked to the modifier have a disproportionately high effect on evolution at the modifier locus); however, we found that results are quite similar in the absence of such a minimal distance, and with minimal recombination rates 0.1 and 0.3 between the modifier and the closest selected loci on each side of the chromosome (Fig. 6B). Therefore, the effect of heterosis on the evolution of dispersal is not driven by selected loci that are closely linked to the modifier locus.

Previous simulation models (Guillaume & Perrin, 2006; Ravignéet al., 2006) found weaker effects of heterosis on ES dispersal rates but represented fewer selected loci and smaller populations. In order to study the effect of the number of loci at which deleterious mutations can occur, we modified our simulation model to represent a finite number L of selected loci, the mutation rate at each locus being U/L, whereas the recombination rate between any two adjacent loci is R/(L − 1) (see Appendix S2 for more details). Figure 7 shows simulation results for N = 10 and 30, for different numbers of demes n, and different numbers of mutable loci (L = 103, 104, 105 and 106). Figure 7 shows that the number of demes n has little effect once it is above a given value (e.g. in most cases, results are very similar for n = 100 and 500). For smaller values of n, the ES dispersal rate decreases as n decreases. The ES rate in the absence of heterosis also decreases as n becomes small (dotted curves in Fig. 7), although the effect is weak unless n is very small (Gandon & Rousset, 1999); this effect is partly due to competition between emigrant juveniles from the same parent and partly due to lower deme relatedness. More importantly, Fig. 7 shows that the number of mutable loci may have important effects on the ES dispersal rate: results for L = 105 and 106 are almost indistinguishable (results from the continuous chromosome model are also virtually the same – not shown), but L = 104 leads to a slightly lower dispersal rate and L = 103 to a much lower rate. Because the genomic deleterious mutation rate is kept constant (U = 1 for all results shown in Fig. 7), decreasing L inflates the per-locus mutation rate, which may lead to the fixation of deleterious alleles. However, among the results presented in Fig. 6, fixations happened only for N = 10 and n = 25 (left-most points of the left figure), although at a slow rate (fixations occur at a faster rate for lower values of n). Simulation results for higher values of n showed no fixation of deleterious allele, but yet a lower ES dispersal rate for L = 103 than for higher values of L. We also observed that for a given migration rate, L = 103 generates a lower amount of heterosis than higher values of L: e.g. for N = 30, n = 500 and U=1, at the ES dispersal rate in the absence of heterosis (m ≈ 0.041), heterosis for L = 103 is about one-third lower than heterosis for L = 105 (not shown). Similarly, heterosis generated by a single locus with deleterious mutation rate u = 10−3 is about one-third lower than heterosis generated by 100 loci, each with u = 10−5 (not shown). From eqns 3 and 5, heterosis is a function of Lp(1 − p), where p is the equilibrium frequency of the deleterious allele at each locus. When p is small, this is approximately Lp, and p is approximately u/(sT), where T is given by eqn 13. From this, one predicts that heterosis is a function of the genomic mutation rate U, and should be independent of L. As L decreases for a fixed U, u increases and p also increases. However, at some point p becomes too large for approximations neglecting terms in p2 to hold: p becomes smaller than u/(sT) and one cannot ignore the term −Lp2 in Lp(1 − p) anymore. In that case, heterosis starts to decrease as L decreases (due to the fact that p becomes lower than u/(sT), and due to the term in −Lp2). In more biological terms, a high per-locus mutation rate increases the probability that two individuals from different demes carry a mutation at the same locus, due to two independent mutation events at this locus. When the per-locus mutation rate is very small, however, identity in state is almost always due to identity by descent. This explains the lower value of heterosis and lower ES dispersal rate observed for L = 103 than for higher values of L.

Figure 7.

 Evolutionarily stable dispersal rate as a function of the number of demes n, and for different numbers of mutable loci L (simulation results). Filled circles: L = 106 (under empty circles), empty circles: L = 105, empty squares: L = 104, filled squares: L = 103. Note that solid lines simply connect simulation results, and do not correspond to analytical predictions. Dotted curve: analytical prediction for the ES dispersal rate in the absence of heterosis (equation 13 in Gandon & Rousset, 1999). Other parameter values: c = 0.4, U = 1, and other parameters as in Fig. 2. Because deleterious mutations reach fixation for N = 10 and n = 25 (albeit at a slow rate), the ES dispersal rates shown for these parameter values only represent transient dynamics, and the population would eventually reach the ES rate in the absence of heterosis once all loci are fixed for deleterious alleles.

Discussion

In this paper, we used theoretical models to study the joint evolution of dispersal and heterosis generated by recessive deleterious mutations occurring at a large number of loci, in a subdivided population. In the first part of the paper, we fixed the dispersal rate and measured the amount of heterosis maintained at equilibrium in the population. In accordance with previous predictions (Whitlock et al., 2000), we found that heterosis can be strong even in the presence of moderate gene flow (Nm = 1 or 2) among subpopulations. Our model allowed us to explore the effect of recombination rates among loci (by varying map length R), which showed that genetic associations among selected loci have a significant effect mainly when map length is small (R = 1): when R = 10 and 100, extrapolations from a single-locus model (neglecting genetic associations between loci) lead to accurate predictions. We have also seen that when R is small, the direction of the effect of R on heterosis may depend on the degree of population structure.

In the second part of the paper, we let dispersal evolve, and compared the effects of kin selection and heterosis on selection for dispersal. We found that heterosis may have substantial effects, in particular when U is high. According to current estimates (e.g. table 1 in Baer et al., 2007), U lies between 0.1 and 1 (and perhaps higher) in multicellular plants and animals in which it has been measured. Figures 2 and 3 indicate that heterosis may have important effects for such values of U. It is important to note, however, that these results assume fixed s and h (s = 0.05 and h = 0.1 in Fig. 2; s = 0.01 and h = 0.3 in Fig. 3). Although simulations indicate that s has little effect on the results while selection is mild (s between 0.01 and 0.1), heterosis and its effect on dispersal should decrease as selection becomes very weak, as mutations may fix in the whole population (fixed mutations do not generate heterosis). It has been argued that an important proportion of deleterious mutations may have very small effects, with s < 5 × 10−4 (e.g. García-Dorado et al., 2004). If this is the case, our results for high U may overestimate the magnitude of heterosis, and its effect on dispersal evolution (it is difficult to obtain simulation results for very low s and high U, as many mutations are segregating and the program is very slow). However, our results also show that the relation between heterosis and the ES dispersal rate does not depend much on the genetic architecture of heterosis (map length R has an effect, but mainly when it is small). Therefore, the relation between heterosis and dispersal rate shown in Figs 2–5 does not depend on values of s and h, and can be used to infer how observed values of heterosis should affect dispersal evolution.

Another result from our multilocus simulations is that the evolution of dispersal limits the amount of heterosis that can be maintained in a metapopulation: in Fig. 4, and also in Fig. 2 for N = 10, heterosis goes through a maximum, and then decreases, as the deleterious mutation rate increases. According to Fig. 2, high amounts of heterosis (e.g. greater than 20%) are possible only if the cost of dispersal c is sufficiently large, and deme size N sufficiently small. It would be interesting to see how data from natural populations match these predictions.

Finally, Fig. 7 shows that results are insensitive to the number of selected loci, as long as it is sufficiently high (roughly, above 104). Lower values of L may decrease substantially the amount of heterosis maintained (for the same genomic mutation rate U), due to a high mutation rate per locus. What would be realistic values of L? The answer is not immediately obvious: should L be the number of genes, or the number of selected nucleotides in a genome, or yet something else? In any case, L must be higher than the number of genes, because: (i) a mutation may affect fitness even if a second mutation is already present in the same gene and (ii) a potentially large fraction of noncoding DNA may be under selection (e.g. Halligan & Keightley, 2006). Therefore, L = 105 or 106 may be more realistic than L = 103 or 104. This effect of the number of selected loci on heterosis (and on ES dispersal rates) partly explains the difference between our results and the simulation results obtained by Ravignéet al. (2006) and Guillaume & Perrin (2006)– who found weaker effects of heterosis on dispersal evolution: indeed, Ravigné et al. represented up to 80 loci, whereas Guillaume & Perrin (2006) modelled 1024 loci. Although these models considered unlinked loci (whereas loci are linked in our model), linkage does not greatly affect the results (e.g. in the case of Fig. 7, for N = 30 and n = 100, changing map length R from 10 to 1000 has only a very small effect on the quantitative results, for the different values of L– results not shown). Furthermore, these models considered small total population sizes, which also decreases the amount of heterosis maintained.

What are the biological implications of our results concerning heterosis and its effects on dispersal? In populations subdivided into small groups (e.g. N = 10), and when the deleterious mutation rate per genome is relatively high, the results shown in Fig. 2 and in Appendix S4 predict that significant levels of heterosis should be maintained once the ES dispersal rate has been reached. For example, if U > 0.2 (still for N = 10), our results predict that heterosis should be at least 5% when all deleterious mutations have a dominance coefficient h = 0.1 (introducing a variance in h would increase the amount of heterosis maintained). If N is low and heterosis is below 5%, then heterosis probably has only a weak effect on selection for dispersal (compared with the effect of kin competition). If heterosis is stronger, then it may be an important component of selection for dispersal, in particular when the cost of dispersal c is low: e.g. if c = 0.1 and heterosis is around 10–11%, then heterosis may be responsible for half of the dispersal rate in the population (i.e. the ES dispersal rate may be twice that if heterosis were absent). If c is high (e.g. c = 0.6), the same value of heterosis implies little effect of heterosis on ES dispersal rates (see Fig. 2): to achieve the same effect (half of the dispersal rate due to heterosis), heterosis should be around 40% (however, when c is high and N small, heterosis may easily reach such high values, as long as U is not small). According to our results, heterosis should have important effects on dispersal evolution in species where N is low and U high, such as in primates (Baer et al., 2007) and probably in various other terrestrial vertebrates. Large effects of heterosis might also be expected for low deme sizes when c is large (e.g. Fig. 2, c = 0.6), which may represent plant patches in an unfavourable matrix (e.g. urban populations of Crepis, Cheptou et al., 2008). Larger deme sizes (e.g. N = 100–500) lead to lower values of heterosis and lower ES dispersal rates. Although the absolute effect of heterosis on the ES dispersal rate is smaller in this case (Appendix S4), its relative effect (compared with the effect of kin selection) may be quite important (Fig. 2).

These results underline the need for obtaining additional measures of heterosis from natural populations, at a spatial scale corresponding to the dispersal distance of individuals (indeed, we predict that once the ES dispersal rate has been reached, significant heterosis should remain between interconnected populations, in particular if N is not too high). Estimates of dispersal costs are more difficult to obtain (Clobert et al., 2004). From a theoretical perspective, many questions remain to be explored. In particular, we used a simplistic model of population structure (the island model, with fixed deme size and infinite fecundity), and it is not clear how variation around this scenario would affect our predictions: in a model with isolation by distance, what would be the effect of heterosis on the mean dispersal distance? What if the dispersal rates of males and females can evolve independently? Moreover, how would demographic effects (such as local extinctions) change our results? Finally, we assumed a simple mating system (gametes fuse randomly within demes). Inbreeding avoidance within demes (in particular self-fertilization avoidance) may also be seen as a form of dispersal (e.g. pollen dispersal) evolving in response to heterosis, at a local scale (Ravignéet al., 2006). Selfing avoidance would probably reduce the effect of heterosis on dispersal between demes, but the magnitude of this effect should depend on the other parameters (particularly deme size and the cost of dispersal). Further work is thus needed to understand the interplay between the evolution of dispersal, mating systems and heterosis.

Acknowledgments

We thank Sylvain Gandon, Frédéric Guillaume, Sally Otto, Virginie Ravigné and four anonymous reviewers for helpful discussions and comments. This work was financed by a fellowship from Région Bretagne (ACOMB CycleVie) to DR. This is publication ISEM 09-032.

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