Rate of Adaptation in Large Sexual Populations

Abstract

Adaptation often involves the acquisition of a large number of genomic changes that arise as mutations in single individuals. In asexual populations, combinations of mutations can fix only when they arise in the same lineage, but for populations in which genetic information is exchanged, beneficial mutations can arise in different individuals and be combined later. In large populations, when the product of the population size N and the total beneficial mutation rate U_b is large, many new beneficial alleles can be segregating in the population simultaneously. We calculate the rate of adaptation, v, in several models of such sexual populations and show that v is linear in NU_b only in sufficiently small populations. In large populations, v increases much more slowly as log NU_b. The prefactor of this logarithm, however, increases as the square of the recombination rate. This acceleration of adaptation by recombination implies a strong evolutionary advantage of sex.

IN asexual populations, beneficial mutations arising on different genotypes compete against each other and in large populations most of the beneficial mutations are lost because they arise on mediocre genetic backgrounds or acquire further beneficial mutations less rapidly than their peers—the combined effects of clonal interference and multiple mutations (Gerrish and Lenski 1998; Desai and Fisher 2007). Exchange of genetic material between individuals allows the combination of beneficial variants that arose in different lineages and can thereby speed up the process of adaptation (Fisher 1930; Muller 1932). Indeed, most life forms engage in some form of recombination, e.g., lateral gene transfer or competence for picking up DNA in bacteria, facultative sexual reproduction in yeast and plants, or obligate sexual reproduction in most animals. Some benefits of recombination for the rate of adaptation have recently been demonstrated experimentally in Caenorhabditis reinhardtii (Colegrave 2002), Escherichia coli (Cooper 2007), and Saccharomyces cerevisiae (Goddard et al. 2005); for a review of older experiments, see Rice (2002).

Yet the benefits of sex become less obvious when one considers its disadvantageous effects: recombination can separate well-adapted combinations of alleles and sexual reproduction is more costly than asexual reproduction due to resources spent for mating and, in some cases, the necessity of males. The latter—in animals often termed the twofold cost of sex—implies that sexual populations can be unstable to the invasion of asexual variants. As a result, the pros and cons of sex have been the subject of many decades of debate in the theoretical literature (Crow and Kimura 1965; Maynard Smith 1968; Felsenstein 1974; Barton 1995a; Barton and Charlesworth 1998), and several different potentially beneficial aspects of sex have been identified, including the pruning of detrimental mutations (Peck 1994; Rice 1998) and host–parasite coevolution or otherwise changing environments (Charlesworth 1993; Ladle et al. 1993; Bürger 1999; Waxman and Peck 1999; Gandon and Otto 2007; Callahan et al. 2009). In the opposite situation of relatively static populations, it has been proposed that recombination is favored in the presence of negative epistasis (Feldman et al. 1980; Kondrashov 1984, 1988)—a situation when the combined detrimental effect of two unfavorable alleles is greater than the sum of the individual effects. While this may sometimes be a significant effect, most populations, especially microbes, are likely to be under continuing selection and the benefits of sex for speeding up adaptation are likely to dominate.

The Fisher–Muller hypothesis is that sex speeds up adaptation by combining beneficial variants. Moreover, it has been demonstrated by Hill and Robertson (1966) that linkage decreases the efficacy of selection. This detrimental effect of linkage, known as the “Hill–Robertson effect,” causes selection for higher recombination rates, which has been shown by analyzing recombination modifier alleles at a locus linked to two competing segregating loci (Otto and Barton 1997; Iles et al. 2003; Barton and Otto 2005; Roze and Barton 2006; Martin et al. 2006). Hitchhiking of the allele that increases the recombination rates with the sweeping linked loci results in effective selection for increased recombination.

Experiments and simulation studies suggest that the Hill–Roberston effect is more pronounced and selection for recombination modifiers is stronger in large populations with many sweeping loci (Felsenstein 1974; Colegrave 2002; Iles et al. 2003). However, the quantitative understanding of the effect of recombination in large populations is limited. Rouzine and Coffin have studied the role of recombination in the context of evolution of drug resistance in human immunodeficiency virus, finding that recombination of standing variation speeds up adaptation by producing anomalously fit individuals at the high fitness edge of the distribution (Rouzine and Coffin 2005; Gheorghiu-Svirschevski et al. 2007). The effects of epistatic interactions between polymorphisms and recombination on the dynamics of selection have recently been analyzed by Neher and Shraiman (2009). Yet none of these works consider the effects of new beneficial mutations. In the absence of new mutations (and in the absence of heterozygous advantage that can maintain polymorphisms) the fitness soon saturates as most alleles become extinct and standing variation disappears. Thus the crucial point that must be addressed is the balance between selection and recombination of existing variation and the injection of additional variation by new mutations.

Here, we study the dynamics of continual evolution via new mutations, selection, and recombination using several models of recombination. Our primary models most naturally apply when periods of asexual reproduction occur between matings, so that they approximate the life style of facultatively outcrossing species such as S. cerevisiae, some plants, and C. elegans, which reproduce asexually most of the time but undergo extensive recombination when outcrossing. The models enable us to study analytically the explicit dependence of the rate of adaptation and of the dynamics of the beneficial alleles on the important parameters such as the outcrossing rate and population size. In an independent study N. H. Barton and J. Coe (personal communication) calculate the rate of adaptation for obligate sexual organisms using several different multilocus models of recombination, including the free recombination model studied here. The relation of our work to theirs, as well as to that of Cohen et al. (2005, 2006) who have also studied the effects of recombination with multiple new mutations, is commented on in the discussion.

When deleterious mutations can be neglected, the rate of adaptation is the product of the rate of production of favorable mutations NU_b (N being the population size and U_b the genomewide beneficial mutation rate), the magnitude of their effect, and their fixation probability. The fixation probability is dominated by the probability that the allele becomes established, i.e., that it rises to high enough numbers in the population that it is very unlikely to die out by further stochastic fluctuations. In a homogeneous population a single beneficial mutation with selective advantage s has a probability of establishment and eventual fixation of (in discrete generation models, P_e≈2s) (Moran 1959). In a heterogeneous population, however, a novel beneficial mutation can arise on different genetic backgrounds and its establishment probability will thus vary, being greater if it arises in a well-adapted individual. But even well-adapted genotypes soon fall behind due to sweeps of other beneficial mutations and combinations. To avoid extinction, descendants of the novel mutation thus have to move to fitter genetic backgrounds via recombination in outcrossing events (Rice 2002). As a result the establishment probability decreases as the rate of average fitness gain, v, in the population increases. But the rate of average fitness gain or, equivalently, the rate of adaptation itself depends on the establishment probability. These two quantities therefore have to be determined self-consistently.

In this article we analyze several models via self-consistent calculations of the fixation probability of new mutations. For a given production rate of beneficial mutations NU_b, we find that interference between mutations is of minor importance if the recombination rate r exceeds . In this regime, the rate of adaption is v ≈ NU_bs² as found for sequential mutations or in the absence of linkage. At recombination rates below , however, v grows only logarithmically with log NU_b. We find this behavior in all our models and argue that it obtains more generally. The prefactor of the log NU_b increases with the square of the recombination rate, implying a strong benefit of recombination in large populations.

MODELS

We consider a population of haploid individuals with fitness (growth rate), X, determined by the additive effects of a large number of loci, each of which makes small contributions to the fitness. We assume selection is weak enough for the population dynamics to be described by a continuous time approximation, that the population size, N, is large enough that , and that a wide spectrum of fitnesses is present, characterized by the fitness variance, σ², of the population. Individuals divide stochastically with a Poisson rate , where is the mean fitness in the population, and they die, also stochastically, with rate 1 [that is, we use the death rate to set the unit of time and assume for convenience that ]. In addition to this asexual growth, individuals outcross with rate r. Within our models, outcrossing is an independent process decoupled from division (but this does not substantively affect our results).

The primary model of mating that we study is free recombination. In an outcrossing event two randomly chosen parents are replaced by two offspring and each parental allele is assigned at random to one or the other of the two offspring. This would be exactly correct if all loci were on different chromosomes and can be a reasonable approximation when the number of crossover sites is large, so pairs of substantially polymorphic loci are likely to be unlinked at each mating. At the end, we discuss briefly what happens when this approximation breaks down. When the number of polymorphic loci is large and their contributions to X are of comparable magnitude, the distribution of offspring fitness is well described by a Gaussian distributed around the value midway between the fitnesses of the two parents and with variance σ²/2 if loci are uncorrelated (Bulmer 1980): this is less than the σ² variance of the parental population. Note that σ² is proportional to the number of segregating alleles and represents the extent of genetic variation in the adapting population. It is not a fixed parameter of the model, but is calculated self-consistently as a function of the population size and the mutation and outcrossing rates.

In addition to the free recombination model described above, we study two other models. The first is a grossly simplified model of recombination in which a randomly chosen individual is replaced by an individual whose genome is assembled by choosing the alleles at each locus according to the allele frequencies in the entire population, independent of the “parents” (see also N. H. Barton and J. Coe, personal communication). In this case recombinant offspring have fitness distribution identical to the population distribution. It turns out that this communal recombination model, even if unrealistic, behaves similarly to the free recombination model while being much easier to analyze mathematically: this makes it a good source of insight as well as supporting the contention that the form of our results is more general than the particular models.

The free recombination model, and even more so the communal recombination model, overestimates the amount of gene reassortment during outcrossing events by assuming that all loci are simultaneously unlinked by recombination to the same extent, independent of their locations on the chromosomes. To study the effects of more persistent genetic linkage, we also study a third model in which only a single locus is exchanged with a mating partner in an outcrossing event or, equivalently, is picked up from DNA in the environment and randomly replaces the initial allele at the same locus. This model is reminiscent of lateral gene transfer among bacteria and related to, but not the same as, the model studied by Cohen et al. (2005). While this minimal recombination model preserves the linkage of all but one locus at a time, each locus is equally strongly linked to all other loci. Thus this model does not approximate the position-dependent crossing over of chromosomes.

The recombination processes in each of these models are characterized by a rate, r, and a function, K(X, Y, t), which is the distribution of offspring fitness Y, given a parent with fitness X mated with a random member of the population. Being the distribution of offspring fitness, the recombination “kernel” is normalized . Furthermore, since we ignore epistasis and assume that loci at intermediate frequencies are in linkage equilibrium, recombination leaves the fitness distribution P(X, t)dX of the population invariant . Within the free recombination model, each outcrossing event replaces two parents with two offspring. However, when following a rare allele, we can focus on the lineage containing this allele and ignore the fate of the other offspring. Matings between two individuals with the same rare allele are very infrequent and can be neglected. Since we are interested in the effects of recombination, we primarily focus on the limit .

Branching process and establishment probability:

The key element determining the rate of adaptation is the probability that a new beneficial mutation avoids extinction and establishes in the population. The establishment probability is the probability that the allele survives random drift and rises to a sufficiently large number so that its frequency in the population grows deterministically (and eventually fixates). This establishment occurs—if it does at all—when the population of the allele is large but its frequency in the population is still small. The fate of a new allele during the stochastic phase, when it exists only in a small fraction of individuals, can be described well by a branching process that accounts for stochastic birth, death, and, crucially, recombination events that move some of its descendants from one genetic background to another. The branching process takes place in a population whose mean fitness is steadily increasing due to beneficial mutations sweeping and fixing at other loci and in other lineages. Ignoring the short-term effect of mutations, the mean fitness, , increases with rate , where σ² is the (additive) variance of the fitness. The dynamics of a novel beneficial mutation linked to a spectrum of genomic backgrounds in a population adapting with rate v are illustrated in Figure 1. To establish, its descendants have to switch repeatedly to fitter genomic backgrounds. This general idea (see Rice 2002 for review) applies to the accumulation of beneficial as well as deleterious mutations.

Figure 1.—

A novel mutation needs to recombine onto fitter genetic backgrounds to become established and eventually fix. (A) The distribution in fitness of the population moves toward higher fitness with velocity v = σ². The new mutation, illustrated by the black bars, has to switch backgrounds by recombination to keep up with the moving wave of the population fitness distribution. (B) Initially, the novel mutation is present on a single genetic background with fitness X₀, struggling not to go extinct. Recombination can transfer the mutated allele onto a new background, e.g., from X₀ to X₁, and spawn a daughter clone that starts an independent struggle against extinction. The mutation establishes if at least one branch survives indefinitely. The complementary case of an unsuccessful mutation is shown: all branches die out. The probability of establishment, w(X, t), depends on the fitness X of the genome in which the mutation arose and is a solution to Equation 2.

The establishment probability at a time t − dt of descendants of a genome of fitness X, defined as w(X, t − dt), is simply related to that at time t,

(1)

(Barton 1995b), where D = 1 is the death rate and is the birth rate. After a division, either of the two offspring has a probability 1 − w of extinction: hence 2w − w² of at least one of these offspring fixing. For a low-frequency allele conferring additional fitness s on a genomic background with fitness X, we have .

In a sufficiently large population the adaptation process will proceed in a steady manner, leading to a fitness distribution of constant width translating toward higher fitness as a “traveling wave” (Tsimring et al. 1996) with the velocity set by the rate of increase of the mean fitness . We make the ansatz that the distribution of fitnesses of the population around its mean does not fluctuate substantially and that the distribution is close to Gaussian. These are analogous to “mean-field” approximations that must be justified a posteriori. We expect that such approximations will become valid for sufficiently large populations, but how this occurs and how large the population must be are not clear a priori: we discuss this below.

In the traveling-wave population, the establishment probability depends on time only via . Hence we measure fitness relative to , defining , and seek an otherwise time-independent solution of the form w(x) = w(X − vt) = w(x, t). [The properties of w(X, t) and K(X, Y, t) do not change by this shift of variables other than becoming time independent relative to a moving reference . We therefore use the same symbols for w(x) and K(x, y) in the moving frame.] Using ∂_tw(X − vt) = − v∂_xw(x), the establishment probability, w(x), then obeys

(2)

In many cases of interest, selection is important only on timescales much longer than the generation time. In that case x + s in the prefactor of the quadratic term is negligible compared to the inverse generation time, which is 1 in our units. Equation 2 then simplifies to

(3)

We have written this in a suggestive form. The left-hand side of Equation 3 defines the linear operator J acting on w(y). At very high recombination rates, we will obtain that w(x) ∼ (1 + 2x/r), which is almost independent of x for . In this limit, the J acting on w(y) vanishes and the population average establishment probability is just the solution to the right-hand side, giving simply w(x) ≈ s. This is the conventional result (obtained by the simple branching process) in the absence of linkage to the rest of the genome. More generally, the fixation probability of a new mutation that can arise in any individual is the population average of the x-dependent establishment probability over the approximately Gaussian distribution of the fitness, x:

(4)

Equation 3 has an important property. Its left-hand side is zero upon averaging with respect to the population distribution [as is readily confirmed by direct integration using v = σ² and , see above]. This property originates from the fact that in the deterministic limit (without the additional mutation, s), the population dynamics have P(X, t) = P(X − vt) = P(x) as a traveling-wave solution (Rouzine and Coffin 2005)—the initial rationale for assuming a Gaussian form. As a consequence, averaging Equation 3 yields a “solvability condition”

(5)

which, when combined with Equation 4, provides another expression for the establishment probability:

(6)

This equation together with Equation 3 describes the “surfing” of a beneficial allele (and far more often its drowning!)—the processes illustrated by Figure 1—under the assumption that the distribution of fitness in the population is sufficiently close to Gaussian. The latter holds when the large number of alleles at different loci are only weakly correlated: we justify this ansatz below.

Models of recombination:

The recombination kernel K(x, y) depends on the recombination model. For the free recombination model, the fitness of the offspring resulting from a mating of two parents with fitness x and z is again Gaussian distributed with mean (x + z)/2 and variance σ²/2. Averaging over the fitness z of the mate, which is Gaussian distributed with variance σ², results in the recombination kernel

(7)

In the communal recombination model, the fitness of the recombinant is a random sample from the population (assuming Gaussianity and linkage equilibrium). In that case, we have

(8)

i.e., the recombination kernel becomes independent of x and Equation 3 becomes mathematically much simpler.

Within the minimal recombination model, the probability per unit time of any particular locus being transferred is r and the sections are assumed small enough that they contain at most one segregating locus. From the point of view of a single mutation, there are two processes: either it can be transferred to another genome, which is effectively like the recombination process in the communal recombination model, or other sections can be transferred into its genome, gradually changing its fitness. With small sections transferred the fitness of the genome undergoes a random walk with bias toward the average fitness. The corresponding recombination operator is then

(9)

This form of the recombination operator is derived in appendix c. Note that for the minimal recombination model the recombination operator acting on P(x) is different from the adjoint operator acting on w(y).

RESULTS

Fixation probability and rate of adaption:

To calculate the rate of adaptation, we solved Equation 3 and obtained expressions for the average fixation probability P_e of a beneficial mutation, which is of the form , where and are the selective advantage of the beneficial mutation and the outcrossing rate rescaled by the to-be-determined width of the fitness distribution σ. The expression for P_e is used later to calculate σ² in a self-consistent manner. The derivation of the expressions for P_e in the different models are given in the following section. In the limit , our primary focus, we find for the free recombination model

(10)

with c a coefficient. [Note that in the limit of very small s, s < exp(−cr²/σ²), the expressions break down. This is unlikely to be relevant in practice.] At small r, the fixation probability decreases very rapidly with decreasing r. This stems from the fact that mutations in individuals from the high fitness tail of the Gaussian fitness distribution have an exponentially greater chance of fixing than those in the bulk. At large r, by contrast, the genetic background on which the mutation arises plays only a minor role, since the rate of switching background is larger than the selection differentials. While starting out on a fit background gives a mutation a slight advantage, mutations on any background have a significant chance of fixing. For large r, the result for P_e is therefore given by small perturbations of the result without background interference: P_e ≈ s.

The expressions for P_e presented above depend on the variance in fitness σ². In an evolving population the variance is not a free parameter. When the effects of mutation on the bulk of the fitness distribution can be neglected, as they can here, the variance is equal to the rate of adaptation, v. The rate of adaptation, in turn, is given by the product of the rate at which beneficial mutations enter the population NU_b, the magnitude of their effect s, and their probability of fixation:

(11)

The rate of adaptation, v, can therefore be obtained by solving self-consistently for σ in the above equations. Substituting our result for P_e and ignoring logarithmic factors in the arguments of large logarithms, we find, for the free recombination model,

(12)

Contrary to intuition, v is proportional to log NU_b rather than NU_b both for low r at fixed and at fixed r for sufficiently large populations sizes, N. This indicates that the interplay between mutations—especially their collective effects on fluctuations—is limiting the rate of adaptation (Gillespie 2001). As in the asexual case, because of interference between mutations, only a small fraction ∼log(NU_b)/NU_b of the beneficial mutations fix—the rest are wasted. However, this fraction increases with increasing rate of recombination leading to v increasing as ∼r²logNU_b, until it saturates at NU_bs², which is the limit of independently fixating mutations. In this high recombination limit, the rate of adaptation is limited simply by the supply of beneficial mutations NU_b. Very similar results for the dependence of v on r and N are obtained for the communal recombination model, differing only by coefficients inside logarithms and by correction terms.

In the minimal recombination model, for which only one locus is exchanged at a time, the behavior is slightly different. For the fixation probability, we find

(13)

In contrast to the other models for which recombination results in a macroscopic change of the genotype, the minimal recombination model changes only one locus at a time. This results in a slightly weaker dependence of P_e on the recombination rate for . Self-consisting the fitness variance as before determines the speed of adaptation to be

(14)

Surprisingly, this result in essentially independent of s for : the larger increase in the fitness per sweep is almost perfectly canceled by the decrease in establishment probability. Note that this model is defined with recombination rate r per locus so that the total number of recombinations in time 1/r is far more than in the other models. But the time for turnover of the genome and loss of linkage is of order 1/r and thus r is the useful quantity to compare with the other models.

Simulations:

In writing down Equation 3 for the establishment probability of a beneficial mutation, we have assumed that the distribution of fitness in the population is Gaussian and that correlations and fluctuations are negligible. Thus it is useful to compare the analytic results to individual-based simulations of an evolving population. In our simulations, we use a discrete generation scheme, where each individual produces a Poisson-distributed number of gametes with parameter . The population size, , is kept approximately constant with an average of N by adjusting the overall rate of replication through . Each individual is represented by a string of integers, where each bit represents one locus. Recombination, approximating the free recombination model, is implemented as follows: each generation, gametes are randomly placed into a pool of asexual gametes with probability 1 – r and into a pool of sexual gametes with probability r. The asexual gametes are placed unchanged into the next generation. The sexual gametes are paired at random and their genes reassorted to produce haploid offspring. Whenever one locus becomes monomorphic—via fixation or extinction of an allele—, one individual is chosen at random and a mutation is introduced at that specific locus. This allows us to make optimal use of the computational resources by keeping as many polymorphic loci as possible. However, this scheme renders the beneficial mutation rate, U_b, a dependent quantity, which, as shown in Figure 2, increases with L and decreases with r. The effective total rate for new beneficial mutations, NU_b, can be determined simply by measuring the average rate at which the new mutations are introduced (which, the way the simulations are done, is the sum of the extinction and fixation rates).

Figure 2.—

Fixation probabilities in recombining populations. (A) The mean fixation probability normalized to the value in the high recombination limit as a function of r for three different genome sizes L (with s = 0.002, N = 20,000). The effective rate of beneficial mutations NU_b is shown in the inset (see main text). The scaled fixation probability in the simulation (solid lines) is calculated as v/2NU_bs² and compared to the analytic results for the scaled establishment probability P_e(r, σ)/s (dashed lines). The latter are obtained through numerical solution of Equation 3, using σ² observed in simulations. The agreement between simulations and the analytic approximation improves with increasing L, i.e., increasing NU_b, as expected. (B) The scaled fixation probability as a function of the rescaled background fitness x/σ (relative to the mean). The solid lines are simulation results for w(x)/2s using L = 6400 and r = 0.512, 0.128, 0.064, and 0.032: the corresponding values of the key ratio r/σ, which determines the shape of w(x), are indicated. The dashed lines are predictions for w(x)/s obtained via numerical solutions of Equation 3. Note that the simulation data become noisy when the frequency of x in the population is ∼1/N.

Figure 2 shows the mean establishment probability as a function of the outcrossing rate r, for different values of L, which is roughly proportional to NU_b (see above). The establishment probability is small at small r but increases sharply and saturates at high r at P_e = 2s—the usual single-locus result. The upturn of P_e occurs at larger r for larger NU_b, in accord with the prediction that the high recombination limit is reached when r substantially exceeds σ. The agreement between the analytic predictions in the Gaussian ansatz (via numerical solution of Equation 3) and the simulation improves as NU_b increases, suggesting that, as we expect, the approximations used become valid for large populations. Note, however, that the corrections to the asymptotic results are quite large as the basic small parameter of the Gaussian ansatz is inversely proportional to log(NU_b). Figure 2B shows w(x), i.e., the establishment probability of a mutation arising on background x, measured in simulations together with the predictions obtained from numerical solution of Equation 3. At outcrossing rates much larger than σ, the fixation probability increases only slightly with the background fitness and all new mutations have a substantial chance—of order s—to establish. With decreasing r/σ, the establishment probability becomes a steeper function of the background fitness and only those mutations arising on high fitness backgrounds have a significant chance of establishment. Note that at r/σ ≈ 1, w(x) measured in simulations decays less rapidly at small x than the solution of Equation 3. These deviations are probably due to fluctuations of the high fitness edge and the width of the distribution, which are ignored in the analysis. However, as discussed below, such fluctuations decrease with increasing NU_b as long as .

ANALYSIS OF ESTABLISHMENT PROBABILITY

We now turn to a derivation of the results given for the establishment probability in Equations 10 and 13, which requires solving Equation 3. We first study the case of applicable, as we shall see, for very large populations. We proceed by analyzing Equation 3 in different regimes of x. At large positive , the equation reduces to (x − r)w(x) ≈ w²(x) with solution w_>(x) ≈ x − r, as illustrated in Figure 3. In this regime, w(x) is independent of the recombination model and is simply given by the establishment probability of a mutation in the absence of any gains from recombination (but with the clonal growth rate reduced by r due to recombination). Establishment is driven by clonal expansion and contributions from recombination are negligible. (But we shall see that there are almost no individuals in the population with such high fitness.) In the opposite regime, at large negative x, w(x) is small and the quadratic term, as well as the perturbation sw(x), can be neglected. The resulting linear equation for w_<(x) valid for small x is

(15)

In this regime, the solution depends sensitively on the recombination model. This is intuitive, since the only—and very unlikely—way for a mutation at to fix is to recombine onto better backgrounds. We verify below for each model separately that the crossover from the linear regime, w_<(x), to the saturated behavior at large x, w_>(x), occurs rather sharply around . At intermediate σ < x < σΘ, the establishment probability w_<(x) increases steeply (while remaining small enough for the quadratic term to remain negligible). Individuals in this intermediate regime are much fitter than the average individual so that recombination usually leads to less fit offspring. Hence the recombination term is of secondary importance in this range and w_<(x) is governed by the first term in Equation 15. The solution to Equation 15 is therefore of the form , where φ(x) is a slowly varying function that depends on the recombination model. This behavior can be interpreted in terms of the dynamics of a genotype with initial fitness x. The genotype will expand clonally with rate x − r, giving rise to unrecombined descendants after t generations. Since each of these could give rise to a lineage that will fix, in this regime w(x) is proportional to , which increases rapidly with x. This is valid up to just below the crossover where the quadratic term, w(x)², starts to be important; see Figure 3.

Figure 3.—

Asymptotics of the establishment probability. The fitness distribution P(x) of the population is shown in black, and a sketch of the establishment probability, w(x), is shown in red for . At low x, w(x) is small and depends sensitively on the recombination model; at intermediate σ < x < σΘ, w(x) increases sharply as , modulated by a slowly varying function φ(x) that depends on the recombination model. At still larger x, beyond σΘ, the quadratic term in Equation 3 becomes important, forcing w(x) to saturate at x − r. The width of the crossover region is of the order of σ/Θ.

Note that the amplitude of w_<(x) is left undetermined by the homogeneous linear Equation 15 and hence the location Θ of the crossover is not fixed. To ensure that w(x) solves the complete Equation 3, we need to impose the solvability condition Equation 5 as an additional constraint. The solvability condition involves the first and second moment of w(x) with respect to the fitness distribution P(x). The first moment is dominated by small and intermediate x since P(x)w(x) decreases with x. The second moment, however, is dominated by a narrow range of width around the crossover point σΘ: for x≈σΘ, P(x)w_<(x)² increases rapidly with x, while P(x)w_>(x)² decreases rapidly. The solvability condition (5) then becomes

(16)

giving us a relation between P_e and Θ. To analyze the behavior of the various models it is convenient to rescale the rates and fixation probabilities as

(17)

Utilizing the transform,

(18)

turns out to be informative: note that the scaled fixation probability is p_e ≡ P_e/σ = Ω(z = 0). By integrating the rescaled Equation 3 over the kernel , we obtain an equation for Ω(z) of the form

(19)

which defines for each model a linear operator L acting on Ω(z) (J is the linear operator defined by the left-hand side of Equation 3). The integral over is again dominated by the crossover region and can be evaluated using and the (scaled) crossover width ∼Θ⁻¹:

(20)

The last step was obtained by substituting Equation 16. The condition that the solution joins smoothly to the saturated solution and hence grows slowly only for large χ translates into the condition that Ω(z) does not diverge at any fixed z: it should be an analytic function of z. We now examine separately the different models, the simplest first.

Communal recombination model:

In the communal recombination model, the genotypes of offspring are independent of their parental fitness, which makes this model particularly simple. It can, in fact, be solved exactly, as shown in appendix a, or, in the regimes of interest, by matched asymptotic expansions. But it is more instructive to proceed with the approximate but more general and asymptotically exact analysis outlined above. The equation for Ω(z) reads

(21)

which can be solved trivially. But in general it has a pole at . This pole has to be canceled, since we know that saturates at χ = Θ and Ω(z) cannot develop a singularity. Hence, we must have to eliminate the pole. Solving for Θ and substituting it into the solvability condition (16) yields

(22)

The last approximate equality is correct to leading order in .

Free recombination model:

In the free recombination model, the offspring obtains on average half of its genome from either parent. The parent carrying the new allele mates with a random member of the population: thus after recombination the average fitness of the genotype carrying the new allele is half as far from the population mean fitness as it was before recombination. As a result of this correlation between parents and offspring, the operator for the free recombination model is more complicated and couples Ω(z) to Ω(z/2),

(23)

where, as before, p_e = Ω(0). Neglecting the on the right-hand side (we need to consider only since ), we can analyze this as a power series in z, writing , finding

(24)

As the first part would yield ratios of successive terms that approach for large n and again induce a pole at , this has to be canceled by the second inhomogeneous term. The condition for convergence (up to well beyond the “almost pole” at ) is that for n → ∞, which requires that

(25)

The last approximate equality is accurate when and hence . Thus we must have

(26)

with the order-unity coefficient . We thus obtain p_e very similar to the communal recombination model,

(27)

Note that Ω(z) is approximately the Laplace transform of , which can be analyzed perturbatively for small ; see appendix b. This expansion in reveals the most probable—least unlikely—path of a mutation on a typical initial background to successively better backgrounds and establishment.

Minimal recombination model:

The minimal recombination model can be analyzed similarly: is now a differential operator, and we have

(28)

This can be explicitly integrated and the behavior for found to involve linear combinations of and e^zΘ. For , the condition that the solution matches correctly onto the nonlinearly saturated form for χ ≈ Θ can be shown to be that these two exponentials are almost the same. This yields the condition . In contrast to the other models, gives corrections only to Θ. The fixation probability is then found to be

(29)

which yields a different form for the speed of evolution:

(30)

High recombination rates:

In the limit of high recombination rate, the crossover to the saturated solution occurs far out in the “nose” (high fitness tail) of the population distribution—farther out than any individuals are likely to be. In this regime, the assumption that is dominated by the crossover region is no longer justified.

To analyze this high r regime, we can make use of the expansion of , which is equivalent to expanding in Hermite polynomials , where the . In the limit of , the second term in Equation 24 can be neglected for the first few coefficients and we have (for the communal recombination model we have ). The value of Ω₀ = p_e has to be determined by the solvability condition . From the orthogonality of the Hermite polynomials one finds that the right-hand side is simply . Hence, we find for the fixation probability the formal expression

(31)

The n! would cause the sum to diverge if it extended to infinity. But for large , this is a valid asymptotic series, which can be truncated at any finite number of terms. To zeroth order, one finds in both models , which is simply the result in a homogeneous population. Including the first two nontrivial correction terms, one finds

(32)

(Note that the divergence of the expansion for large n, for which this approach breaks down, is related to the singular dependence of p_e on for small discussed above.) For the minimal recombination model, the behavior for large r is similar and the expansion in inverse powers of can be analyzed; we do not carry this out here.

Range of validity of analysis:

Throughout the analysis, we have assumed that the fitness distribution of individuals in the population, , is Gaussian and also that of recombinant offspring. Crucially, for the analysis, we assumed that it remains Gaussian in the high-fitness nose of the distribution all the way to the crossover point Θ that controls the establishment probabilities. We need to justify this ansatz. First, as noted earlier, we observe that a Gaussian fitness distribution is the exact traveling-wave solution to the linear recombination model in the absence of fluctuations: the Gaussian approximation should thus be valid throughout the bulk of the distribution in the limit of very large populations. Second, in the absence of fluctuations (or epistatic interactions that we are ignoring in any case) the frequencies of alleles at different loci are independent. And third, if the establishment probabilities of different beneficial mutations are independent, then it can be shown that the resulting Poisson process of the establishments together with random combining of the alleles with their corresponding frequencies leads to a distribution of fitnesses whose logarithm averaged over the establishment times, , is exactly parabolic—corresponding to a Gaussian distribution. However, due to fluctuations and correlations, the distribution of fitnesses will be neither exactly Gaussian nor exactly time independent and we must check that the nonfluctuating Gaussian is a good enough approximation far enough out in the nose in the large N regimes of interest.

We first check that the sampling of the distribution due to the finite population size is sufficient. A population of size N samples a close-to-Gaussian distribution only out to ∼ ahead of the mean. But this implies that, with the fitnesses of individuals only weakly correlated, the crossover region near Θ is indeed well sampled by the population since

(33)

The last inequality is valid when the rate of beneficial mutations per genome per generation, U_b, is small as is surely always the case: there are then of order 1/U_b individuals in the population with fitnesses in the crucial crossover region of the establishment probabilities. Furthermore, the Gaussian shape of the fitness distribution will be a good approximation when the number of polymorphic loci that contribute substantially to the fitness variance is large. However, the total number of established polymorphic loci is dominated by low-frequency alleles. (The total number of polymorphic loci is much higher still, but almost all of these are not established and destined to soon go extinct.) Nevertheless, there are sufficiently many polymorphic sites with high enough frequencies that they contribute substantially to the fitness distribution. Since sweeps occur at rate v/s and since a sweeping allele is at intermediate frequencies for a few times 1/s generations, the number of loci, K, contributing substantially to the variance is of order v/s² ∼ (r/s)²log(NU_b). For these K loci are approximately in linkage equilibrium, giving rise to a Gaussian fitness distribution with corrections to parabolic log(P(x)) of order (x/σ)²/K. At the crossover point, σΘ, it can then be checked that the corrections to P(x) are small as long as . We thus expect that this is the condition for validity of the Gaussian ansatz from which our analytic predictions are obtained. A more detailed analysis of the effects of fluctuations, in particular in the crucial nose of the distribution, is left for future work.

DISCUSSION

We have analyzed in several simple models the dependence of the speed of adaptation on the rate of recombination and the population size, focusing on the particularly interesting behavior in the wide range of outcrossing rates or, equivalently, on population sizes . In the high recombination limit and moderate N the conventional analysis of independent fixations holds and the rate of adaptation (and concomitantly the variance of fitness) is proportional to the total production rate of beneficial mutations, NU_b. In contrast, for large populations (with recombination rates in the intermediate regime) we find adaptation rate v ∼ r²logNU_b. This change from linear to logarithmic dependence on NU_b indicates that the rate of adaptation is limited by interference among multiple simultaneously segregating beneficial mutations rather than by the supply of beneficial mutations. This reduction in the rate of adaptation due to linkage is, qualitatively, the Hill–Robertson effect (Hill and Robertson 1966). Most interestingly, while logarithmic in population size, the rate of adaptation increases with the rate of recombination as r². Hence our results confirm the heuristic arguments by Fisher and Muller and provide a quantitative framework for identifying conditions favoring sexual reproduction (Barton and Charlesworth 1998; Rice 2002).

The rate of adaptation is determined by the dynamics of the linkage between new beneficial alleles and the spectrum of fitnesses of the rest of the genome. This results in most new mutations being eliminated by their linkage to modestly fit genomes that rapidly lose out with respect to the steadily increasing average fitness driven by the anomalously fit genomes. Only those alleles that either arise on very fit genomes or are lucky enough to recombine to make a very fit genome will survive long enough for their frequency to grow deterministically and sweep through the population. The logarithmic dependence on population size is similar to that found for purely asexual evolution when multiple beneficial mutations are present in the population (Desai and Fisher 2007). But with r > s, recombination speeds up the adaptation by allowing new mutations that arise on modestly fit backgrounds to recombine to very fit backgrounds and thereby fix.

We have shown that the typical number of simultaneously segregating alleles at intermediate frequencies is on the order of K ∼ r²/s² log NU_b. For , the number of possible combinations of these sweeping loci therefore dramatically exceeds the population size. This implies that the limit of “infinite” population size, for which each genotype is well sampled, is unattainable at fixed recombination and beneficial mutation rate. On the contrary, sampling becomes sparser and the benefits of recombination more pronounced in larger populations. The population size dependence of the beneficial effects of recombination has been a subject of considerable theoretical debate (Crow and Kimura 1965; Maynard Smith 1968; Barton and Otto 2005). The increased advantage of sexual reproduction in large populations has been demonstrated in model simulations by Iles et al. (2003). It has also been observed experimentally by Colegrave (2002), who studied this phenomenon in an evolution experiment with C. reinhardtii.

Relationship to other recent work:

The description of the spread of beneficial alleles in space as a traveling wave goes back to Fisher (1930). The notion that adaptation of a panmictic population can be described as a traveling wave in fitness was introduced by Kepler and Perelson (1995) and Tsimring et al. (1996). In these effectively deterministic models, the velocity of the pulse is determined by the size of the population through a modification of the deterministic solution at the high fitness edge—the nose or “front”—to approximate the crucial stochastic behavior near the nose (Brunet and Derrida 1997). These concepts were applied to recombining populations by Rouzine and Coffin (2005) and Gheorghiu-Svirschevski et al. (2007), who studied the rate of (transient) adaptation when selection acts on standing variation. Cohen et al. (2005, 2006) studied continuing evolution with a large supply of beneficial mutations available in a model that is related to our “minimal recombination” model. Both approaches focused on the overall distribution of fitnesses within the population and the primary role of recombination they considered was to maintain a near Gaussian shape of the fitness distribution, achieved by producing higher-fitness individuals and thereby advancing the nose. Some of the results of the approximate analytic treatments are related to ours, including the log N scaling of the adaptation speed in certain regimes. Yet the actual underlying dynamics implicit in the approximations used are very different from what we find here and so is the dependence on parameters.

The key feature of the adaptation with substantial rates of recombination is the stochastic dynamics of new mutations. The probability that a new beneficial mutation will sweep to fixation is determined by its establishment probability: the probability that it escapes stochastic extinction. The establishment probability depends very strongly on the distribution of fitnesses of the genetic backgrounds with which the new mutation can be linked. As the distribution of fitness depends on the velocity, the steady-state velocity must be determined by matching the rate of establishment of new alleles with the velocity of the deterministic traveling wave describing the fitness distribution in the population. The latter is driven by the continuous incorporation of a large number of new sweeping alleles that have successfully established at earlier times. At any time there is thus a broad distribution of frequencies of the beneficial alleles. The primary problem with the earlier analysis is that the distribution and dynamics of individual allele frequencies are not treated directly and the approximations implicitly made for their forms are not consistent with the basic processes.

In contrast with the asexual traveling wave for which a description in terms of a simple traveling wave is valid (Desai and Fisher 2007; Rouzine et al. 2008) and the diversity within the population can be ignored, with any amount of recombination, the diversity and distribution of allele frequencies are absolutely crucial. It matters a great deal whether the advance of the fitness wave occurs via small amounts of each of several new alleles or all from a single allele. This information is lost by treatments in terms of the fitness distribution alone. Note that in general this is also true for adaptation from standing variation: beneficial alleles initially at low frequencies can be driven extinct by their linkage to different backgrounds. If all are initially at sufficiently high frequencies to avoid this fate, then neither linkage nor recombination plays much of a role in the dynamics of the adaptation.

The models we have studied were inspired by facultatively mating organisms, in which outcrossing occurs at rate r. N. H. Barton and J. Coe (personal communication) have recently performed a related analysis for obligate sexual reproduction. In addition to a model with a linear genetic map (see below), they study the free and minimal recombination models, for which they find similar logarithmic dependence on the population size and mutation rate. Their discrete generation models with obligate mating do not reveal the dependence of the rate of adaptation on the outcrossing rate, one of the results of our analysis, but a similar behavior is implicit in their results.

Extensions and open questions:

In this article we focused on the effect of recombination with r > s in simple models of mating without chromosomal organization and without epistasis. We conclude by considering going beyond these simplifying limits.

We first consider decreasing the recombination rate. In comparing our analytic results on the free recombination model with the direct simulations we found good agreement at high recombination rates, which confirms the accuracy of the simplifying assumptions made in analyzing the model (i.e., Equation 3). At lower recombination rates we observed that our mean-field treatment of the recombination underestimates the rate of adaptation. This is due to the gradual appearance of “fat tails” in the distribution of fitness: specifically, the high fitness nose of the distribution decays more slowly than the Gaussian assumed in the analysis. The fluctuations in the time of establishment of the currently intermediate-frequency alleles become important. Some of the causes of these can be studied analytically. The primary effect is the smaller number of segregating loci—of order v/s² ∼ r²/s²—at low recombination rates. As the ratio r/s decreases further, the acquisition of further beneficial mutations near the nose of the distribution—which dominates the asexual evolution—starts to become important. Correlations between loci caused by this process and other sources will also play important roles.

The behavior of the leading edge of the fitness distribution is known to be the key factor in determining the speed of adaptation in the asexual limit of (Desai and Fisher 2007) and it is of critical importance in the regime. A correct treatment of this regime, connecting with the known results for asexual adaptation (Desai and Fisher 2007; Brunet et al. 2008; Rouzine et al. 2008), requires analyzing the diversity that is generated by the asexual process and the effects of small amounts of recombination on this. It is worth noting that within our approximations, for the low recombination regime with , the branching process analysis yields an adaptation speed for all three models of the form v ∼ s²log(NU_b)/log²(s/r), which is a similar form to the asexual result, . This suggests that in spite of the breakdown of the assumptions, the approximations may give reasonable results, although not asymptotically accurate ones, even for . But we leave this regime, which is particularly important for microbes with rare genetic exchange, for future investigations.

Our analysis has focused on the simple approximation of additive growth rate (equivalent to multiplicative fitnesses in a discrete-generation model). Some of the most interesting extensions of the present models would include epistasis—i.e., genetic interactions—which makes the effect of each allele explicitly dependent on its genetic background. This dependence can be very complex, resulting in low heritability of fitness, in the sense that the fitness of recombinant progeny may be only weakly correlated with the fitness of the parents. Remarkably, in the limit of very strong epistasis (Neher and Shraiman 2009) the establishment probability of an allele is described by a model that reduces to the communal recombination model described above. The speed of adaptation is, however, determined by a different self-consistency condition that will be presented elsewhere. In general, how to set up—never mind analyze!—instructive models of evolutionary dynamics with epistasis between many segregating loci is largely an open field.

Another important simplification in the free recombination model studied here is the random reassortment of the parental alleles, ignoring the physical arrangement of the genes. More realistic models would account for the linear arrangement of genes on the chromosomes such that chromosomal proximity implies low recombination rate. In this case, the number of independently transmitted loci in the event of mating is the product of the number of chromosomes and the crossovers per chromosome. When the number of substantially polymorphic loci is sufficiently large, the free recombination approximation will certainly break down. But in facultatively mating organisms where periods of asexual reproduction are interspersed by outcrossing events, much reassortment can occur. Indeed, some facultative outcrossers have high crossover rates [e.g., S. cerevisiae (Mancera et al. 2008)]. In this case the free recombination model can have a reasonable regime of validity. More generally, the fact that our three rather different models yield similar behavior for the adaptation rates at large population sizes suggests that the forms of the dependence on parameters—especially speed proportional to log(NU_b)—may be valid much more broadly. Arguments to be presented elsewhere suggest that the balance between the lengths of linked regions and the number of polymorphic loci in them can result in v ∼ rs log(NU_b) in some regimes. Significant progress in the analysis of the rate of adaptation with linear chromosomes has recently been made by Barton and Coe. They invoke a scaling argument and use a perturbative analysis of nearby pairs of segregating loci to derive an expression for the rate of adaptation. In this approximation, the rate of acquisition of beneficial mutations tends to an upper limit independent of the population size, selection coefficient, or mutation rate, being solely determined by the map length: in our notation this would be equivalent to v ≈ Crs with C a constant. Note that this is similar to the conjecture quoted above but without the log(NU_b) factor. To check whether the approximations are accurate with many concurrent sweeps it will be necessary to go beyond the perturbative analysis of Barton and Coe. Furthermore, the interplay between the effectively asexual evolution of short regions of the chromosome that are linked for long times and recombination between and within them needs to be understood and could well change the behavior qualitatively.

The challenges of understanding evolutionary dynamics in the presence of many beneficial alleles and recombination between linear chromosomes, and of understanding the effects of epistatic genetic interactions, provide many important open problems.

APPENDIX A: EXACT SOLUTION OF THE COMMUNAL RECOMBINATION MODEL

In the communal recombination model the genotype of recombinant offspring is assembled at random from the alleles segregating in the population and therefore independent of the fitness of the parents. The equation describing the establishment probability, Equation 3, therefore simplifies to

(A1)

where all rates, the fitness, and have been rescaled by the standard deviation of the fitness distribution, as in Equation 17. The quadratic term can be removed by substituting , which gives rise to the equation

(A2)

A second substitution of with maps Equation A2 onto the parabolic cylinder equation

(A3)

The solution with the correct asymptotic behavior is and has the integral representation

(A4)

(Abramowitz and Stegun 1964, Equation 19.5.1). From , we obtain by taking the log derivative . The asymptotics of in the different regimes are

(A5)

as found via the perturbative scheme in the main text. The fixation probability entered Equation A1 as a free parameter and has to be fixed such that , which results in a very similar condition for p_e as the solvability condition of the perturbative scheme used in the main text.

APPENDIX B: THE LOW RECOMBINATION LIMIT OF THE FREE RECOMBINATION MODEL

In the intermediate regime where the recombination term and the quadratic term in Equation 3 are both small, the fixation probability is of the form , where φ(χ) is a slowly varying function compared to the Gaussian growth term. Ignoring the quadratic term, the equation for φ(χ) reads

(B1)

Hence, the dominant contribution to the recombination term comes from . The function φ(χ), however, drops to zero rapidly beyond Θ, implying φ(χ) constant in the interval Θ/2 < χ < Θ.

To study the behavior of φ(χ) more systematically, it is useful to rearrange Equation 23 as

(B2)

where we assumed and such that in the denominator and can be neglected. Assuming small , this equation can be solved iteratively. The two terms on the right, however, have to be matched to cancel the pole at , which can be done by adjusting Θ for each order in the iterative solution. Starting with Ω⁽⁰⁾(z)= p_e, we have

(B3)

with . Iterating Equation B2, it is found that with , which is rapidly converging to the value of the crossover point found by power series expansion of Ω(z) in Equation 26. The solution to the kth order reads

(B4)

where all poles are canceled by zeros of the numerator. For small z, Ω(z) is related to the Laplace transform of the function φ(χ) in the variable :

(B5)

Since φ(χ) is essentially zero for χ > Θ, it is useful to change variables to ρ = Θ – χ and consider the Laplace transform on ρ ∈ [0, ∞],

(B6)

where we dropped the z² and terms. We can now back transform Ω^(k)(z) in Equation B4 into χ-space and obtain an approximation for φ(χ). The inverse transform of terms of the form is , with u(x) being the Heaviside function. The most important observation is that the delay τ = Θ(1 – 2^−j) is different for the different orders and that higher-order terms come in only below a cutoff set by this delay:

(B7)

Here, f_j(ρ) is polynomial in ρ multiplied by a slowly varying exponential (). This behavior of φ(χ) [and ] has a simple interpretation: for Θ/2^j < χ < Θ/2^j−1 the least unlikely way for a new mutation initially with a background fitness χ to fix is to recombine j times, each time getting closer to the front at Θ beyond which it can rise to a high level without further recombination.

APPENDIX C: MINIMAL RECOMBINATION MODEL

In the minimal recombination model, the allele at each locus is exchanged for a random allele from the population at rate r. Let the locus i of a particular individual be in state s_i = {0, 1} and assume the beneficial variant is present in the population at frequency p_i. The expected change in fitness upon exchange of locus i is therefore

(C1)

Similarly, the variance of the increment is given by

(C2)

where we have used . Assuming each locus undergoes exchange with rate r, the drift and diffusion coefficients of the fitness x are given by

(C3)

These diffusion and drift processes are represented by the second and third terms of Equation 9. The possibility that the novel mutation itself is exchanged into a new genome is described by the first term.