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The limitations of adaptive dynamics as a model of evolution

2005; Oxford University Press; Volume: 18; Issue: 5 Linguagem: Inglês

10.1111/j.1420-9101.2005.00943.x

1420-9101

N. H. BARTON, Jitka Polechová,

Evolutionary Game Theory and Cooperation

Adaptive dynamics describes the evolution of an asexual population through the successive substitution of mutations of small effect. Waxman & Gavrilets (2005) give an excellent overview of the method and its applications. In this note, we focus on the plausibility of the key assumption that mutations have small effects, and the consequences of relaxing that assumption. We argue that: (i) successful mutations often have large effects; (ii) such mutations generate a qualitatively different evolutionary pattern, which is inherently stochastic; and (iii) in models of competition for a continuous resource, selection becomes very weak once several phenotypes are established. This makes the effects of introducing new mutations unpredictable using the methods of adaptive dynamics. We should make clear at the outset that our criticism is of methods that rely on local analysis of fitness gradients (eqn 2 of Waxman & Gavrilets, 2005), and not of the broader idea that evolution can be understood by examining the invasion of successive mutations. We use the term 'adaptive dynamics' to refer to the former technique, and contrast it with a more general population genetic analysis of probabilities of invasion. The role of mutation in maintaining genetic variation and in shaping genetic systems has received much attention over the past 20 years (e.g. Kondrashov, 1988; Lynch et al., 1999). Kimura (1965) showed that if mutations have small effects relative to standing genetic variation at each locus, then when mutation balances selection, large numbers of alleles will segregate, with a Gaussian distribution of effects. Lande (1975,1977,1979–1981) extended this model to describe a variety of evolutionary phenomena. However, Turelli (1984) argued that mutation rates per locus are too low for the Kimura-Lande Gaussian approximation to apply to sexual organisms: if heritable variation is maintained by mutation-selection balance, it must be based on rare alleles of large effect. We are now starting to get evidence on the size of effects of 'quantitative trait loci' (QTL); in many cases, these are large (Barton & Keightley, 2002). Ascertainment bias can cause effects to be overestimated, but nevertheless, individual QTL have been found which alter phenotype by several environmental standard deviations . For example, differences between maize and its wild progenitor, teosinte, are for the most part due to one or two QTL (Doebley & Stec, 1991); similarly, differences in floral structure and colour between Mimulus (monkeyflower) species adapted to different pollinators depend on rather few genes (Bradshaw et al., 1998). Such evidence concerns the alleles responsible for response to artificial selection, or species differences. If genetic variation is maintained by balancing selection, or if species differences are due to multiple substitutions at individual loci, then individual mutations could still be of small effect, as assumed by adaptive dynamics: the alternative alleles maintained by balancing selection could be built up by successive substitutions. (For example, the alcohol dehydrogenase polymorphism in Drosophila melanogaster involves multiple interacting sites within the Adh gene, which presumably arose separately; Stam & Laurie, 1996.) We do have direct evidence on mutational effects, from experiments in which variation is accumulated in populations that are initially homogeneous. Across a wide range of traits and organisms, the rate of increase of genetic variance is Vm≈10−3VE (Lynch & Walsh, 1998). This high rate of mutational variance is hard to reconcile with low per-locus mutation rates of μ∼10−5 per generation; even if 100 loci affect each trait, the mutational variance must then be due to alleles with effect . The new genetic variance generated by mutation could only be due to alleles of small effect if unusually mutable loci are involved, or if very many loci affect each trait (Turelli, 1984; Lynch & Walsh, 1998; Barton & Keightley, 2002). Estimation of the distribution of mutational effects is difficult but what evidence there is suggests a highly leptokurtic distribution of effects, that is, a mixture of large and small effects (e.g. Davies et al., 1999; Estes et al., 2004). For our purposes, what matters is that there are some mutations of large effect, so that evolution can involve moderately large jumps rather than the infinitesimal changes required for the 'adaptive dynamics' approximation. Evidence on the size of mutational effects has been measured relative to the environmental variance VE, which gives a convenient yardstick; with high heritabilites, VE∼Vg. However, for adaptive dynamics what matters is the effect of mutation relative to the range of phenotypic evolution. For short-term evolution within species, this range is comparable with the standing genetic variance (think of the classic examples of Galapagos finches; Grant, 1986). As we will see, the timescale of evolution under the assumptions of adaptive dynamics is extremely long, so that one might think of the phenotypic range as being very broad, relative to within-species variation. However, simple models in which competition depends on just a single trait then seem implausible. The key argument for the evolutionary importance of substitutions of small effect is that large changes are likely to disrupt the delicate interaction between components of a complex function. Fisher (1930) quantified this argument with his geometrical model of stabilizing selection on multiple traits: mutations with small effects relative to the deviation from the optimum have a roughly 50% chance of increasing fitness, whereas mutations with effect more than twice the deviation must be deleterious. On this argument the probability that a mutation is favourable decreases with the square root of the number of traits, . This provides some justification for the adaptive dynamics approximation: even though mutations of large effect on the focal trait do occur, they are likely to have deleterious effects on other traits, and so will be eliminated by selection. However, Kimura (1983) pointed out that the probability of fixation of a mutation is proportional to its selective advantage, which introduces a bias that increases the typical effect of successful substitution. Orr (1998) has extended Kimura's (1983) argument to 'adaptive walks', which involve several substitutions. Although Fisher's argument still stands, in that the average effect of successful mutations decreases with , Orr has shown that the size of the longest step in an adaptive walk can be a substantial fraction of the distance from the optimum. Orr's (1998) analysis of adaptive walks is very close to adaptive dynamics, the main difference being that Orr does not explicitly include frequency dependence. To know whether Fisher's argument justifies the approximation of small mutational effects, one needs an explicit model of multiple trait evolution. Even with the assumption that phenotypes are clustered around one or a few discrete points, the evolution of a population under mutation and selection is not straightforward. Adaptive dynamics supposes that traits evolve at a rate proportional to the local fitness gradient, an assumption, which can be justified for sufficiently small mutational effects (Dieckmann & Law, 1996). In order to follow the evolution of an actual population, we need to understand exactly how favourable mutations get established. In each generation, 2Nμ mutations arise. The chance of establishment in large numbers of each mutation is twice the selection coefficient, and so the net rate of establishment is 4Nμs. (Strictly, this holds for the Wright–Fisher model; in general, the probability of establishment is proportional to s, with a coefficient that depends on the detailed model of reproduction). Now, suppose that the population is at equilibrium at x0, and is away from any branch point. Then s(x0 + ɛ)∼ɛs′(x0), where s′ is the fitness gradient at x0. Integrating over the distribution of mutation effects, g(ɛ), the net rate of substitution is 4Nμs′. The rate of change of the mean is , where is the mutational variance; g(ε) is assumed symmetrical. (This formula differs from Hill [1982] because only mutations with positive effect [s′ > 0] contribute, and from eqn 3.6 of Dieckmann & Law [1996], because it is given for a discrete-time diploid population rather than a continuous-time haploid model.) This argument provides a quantitative genetic justification for the procedures of adaptive dynamics, which is valid in the limit of very small Nμ, and very small mutational effects. Predictions of adaptive dynamics can in principle be extended to finite mutational effects, assuming that fitness changes sufficiently smoothly (Metz et al., 1996). However, running simulations with definite mutational effects reveals several difficulties. First, including fixation probability increases the typical size of substitutions, and makes the pattern much rougher (Fig. 1a). Second, an allele that is introduced near to a resident type may displace an allele that is some distance away. This can be seen in cases where a branch ends abruptly (Fig. 1a), and is reflected in a long tail of the distribution of sizes of phenotypic jumps (Fig. 2b). This occurs when an allele is near a branch point, and hence is under very weak selection; and when the effects of competition span a broad phenotypic range, so that selection is very weak (see below). Third, displacement of alleles takes a very long time, because selection on mutations of small effect, near to branch points, is very weak (typically, tens of thousands of generations in the example of Fig. 1a). The time for a substitution to be completed is in addition to the long waiting times between successful mutations (∼ 1/4Nμ) shown in Fig. 1. This makes it difficult to determine the set of alleles which will persist after introduction of a new mutation: the simple rules of adaptive dynamics only apply with extremely small mutational effects (and hence extremely slow evolution, at a rate proportional to the square of effects). Finally, because slightly deleterious alleles persist for a very long time, standing variation is only negligible when Nμ is very small indeed–such that the waiting time between mutations (Fig. 1) is much longer than the (long) time for each substitution. (a) 'Evolutionary branching' under the Roughgarden (1972) model of competition (). Mutations are distributed in a Gaussian distribution around existing phenotypes, with variance . Time is scaled relative to 4Nμ, and is plotted on a logarithmic scale. (b) The same, but with a Cauchy distribution of mutational effects (∼1/(), where as in a). These examples are not individual-based simulations, but rather, depict the limiting case of low mutation rates, which is assumed in 'adaptive dynamics'. At each step, the probability of fixation is calculated as a function of phenotype (2r(x)). A successful mutation is drawn from the (normalized) distribution 2r(x) where φ is the distribution of new mutations. The waiting time to this substitution is exponentially distributed, with rate equal to the integral over the above expression, multiplied by 2Nμ, where 2N is the number of genes at carrying capacity. The distribution of changes in phenotype, for the example of Fig. 1a (300 substitutions in total). (a) The difference Δ x between the new allele and its progenitor, in 41 cases where the new allele was added to the existing set. The SD is 0.054, which is substantially greater than the SD of mutations . (b) The difference between the new allele, and the allele that was lost, for 244 cases where the new allele displaced one existing allele. The SD is 0.33, largely due to 13% of cases where a distant allele was displaced. (c) In 5% of cases, two or three alleles were displaced by the new allele. Again, these were often distant from the mutant phenotype. Similar results to those shown here and in Fig. 1 were obtained with weaker competition ( = 0.5; results not shown). If the number of mutations entering the population in every generation is not small (as will be the case in abundant species), then several favourable mutations will be established at low frequency. While mutations of small effect may establish first, those of large effect will establish before those have become common and will increase more rapidly. Which allele first appears at high frequency depends on the size of the population (which determines the time taken to get from one copy up to intermediate frequency) and on the tails of the mutation distribution. With a leptokurtic distribution (consistent with the observations discussed above), the first type to emerge may be close to the fittest type, so that there is no appreciable correlation between the phenotypes of new and existing alleles (Fig. 1b). This fits the assumptions of ESS approaches rather than those of adaptive dynamics (see Abrams, 2001, for a discussion of the distinction between these approaches). Reality lies somewhere between the extreme assumptions that mutational effects are infinitesimally small, or that all possible phenotypes are immediately available. However, we will see in the next section that even when mutational effects are quite small, evolutionary dynamics cannot be reliably predicted from the fitness gradient. Roughgarden (1972) introduced a model in which competition for a continuously distributed resource depends on a corresponding quantitative trait. This was used by Dieckmann & Doebeli (1999), and Doebeli & Dieckmann (2000, 2003) to model sympatric speciation. In continuous time, fitness is r = 1−ψ*/K0, where ψ* is the effective density, which is an average over densities of neighbouring phenotypes, weighted by a Gaussian function with variance . The carrying capacity K(x) is also Gaussian, with variance σ2K; we set σ2K = 1. (Note that since is large for x >> σK, there is no selection at low density and extreme phenotypes (>2σK) are subject to very strong density-dependent selection). For , Roughgarden (1972) showed that there is an equilibrium in which a Gaussian distribution of phenotypes, with variance , matches the availability of resources (i.e. ψ = K0). In the adaptive dynamic analysis, a population that starts with a single type evolves through a series of branchings, so that ultimately the continuous equilibrium distribution is approached (Fig. 1). However, in the Roughgarden model, as the number of phenotypes increases, selection becomes much weaker, because the resource distribution can be closely matched. Thus, the rate of evolution slows down drastically (note the logarithmic scales in Fig. 1). Several adaptive dynamic studies (cited above) have focused on sympatric and parapatric speciation. Adaptive dynamics itself cannot model the origin of biological species, because the method does not allow for sexual reproduction, or its converse, reproductive isolation. However, adaptive dynamics can model the clustering of asexual phenotypes, and it can identify circumstances when strong disruptive selection evolves. (For example, Geritz & Kisdi, 2000, show that disruptive selection arises in a similar way in a clonal and in a sexual diploid Levene's model. However, their analysis of the latter involves standard population genetic methods, rather than just the fitness gradient techniques which we criticize here.) A key insight is that populations may evolve towards branch points, where they experience strong disruptive selection. In the Roughgarden (1972) model, this is a transient state, because once more phenotypes evolve, the resource distribution is closely matched, and selection becomes weak. Nevertheless, it is possible in principle that transient disruptive selection could give an opportunity for speciation. First, consider the clustering of a strictly asexual population. Although the ultimate equilibrium for an infinite population is a continuous distribution of phenotypes, evolution of more than a few phenotypes is extremely slow. (In addition to the very long waiting times between mutations shown in Fig. 1, the time for a substitution to complete is also extremely long). In a finite population, random drift will ensure that only a few clusters of phenotypes coexist. The evolution of effectively neutral types through the space of possible phenotypes, via mutations of small effect, leads to widely spaced clusters. This stochastic process was described, by Felsenstein (1975) for the analogous case where individuals diffuse through a geographic distribution. (Young et al., 2001, rediscovered this phenomenon, and gave illustrative simulations). In other models, it may be impossible for any continuous distribution to match the resource distribution. Indeed, Gyllenberg & Meszéna (2005) show that continuous solutions such as that in Roughgarden's model are structurally unstable, and that the generic equilibrium is of discrete types that cannot be invaded by any other phenotype. For example, this occurs with the logistic model r = r0 − z2 − ψ*; with r0 = 1 and , there is an equilibrium at which two distinct phenotypes coexist. This is also the case for Levene's model (e.g. Metz et al., 1996; Kisdi & Geritz, 1999). Clearly, there may also be constraints on what phenotypic distributions are possible. In either case, discrete clusters can be maintained at equilibrium, and may experience either stabilizing or disruptive selection. Sexual populations cannot in general be modeled by the adaptive dynamic approximation, beyond the first branch point. Under the Roughgarden (1972) model, we expect rapid evolution of a broad Gaussian distribution, in which all phenotypes have equal fitness, and there is no selection for reproductive isolation. Only if there is some constraint on the range of possible phenotypes (as in the population genetic simulations of Dieckmann & Doebeli, 1999 and Doebeli & Dieckmann, 2003), or if there is a nonGaussian (e.g. bimodal) resource distribution, will there be the strong disruptive selection that drives speciation. This makes the Roughgarden (1972) scheme unattractive as a model of speciation, because (contrary to the claim of Dieckmann & Doebeli, 1999), disruptive selection does not emerge naturally from a continuous unimodal model. Adaptive dynamics is a useful technique for understanding the qualitative behaviour of an evolving population. It requires that the population clusters around discrete phenotypes, with either strictly asexual reproduction, or if there is sex, with only a single type. However, adaptive dynamics is not plausible as an actual model of evolution under mutation and selection. If mutational effects are as small as is assumed, then evolution is extraordinarily slow; conversely, even moderate mutational effects violate the assumptions required by adaptive dynamics. We are grateful to both reviewers for their helpful comments.