in changing environments: Modifiers of mutation, recombination, and migration

Oana Carjaa,1, Uri Libermanb, and Marcus W. Feldmana,1

aDepartment of Biology, , Stanford, CA 94305-5020; and bSchool of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Contributed by Marcus W. Feldman, September 13, 2014 (sent for review June 20, 2014; reviewed by Lee Altenberg and Lilach Hadany)

The production and maintenance of genetic and phenotypic di- because, when the selective environment varies between positive versity under temporally fluctuating selection and the signatures and negative epistasis, the linkage disequilibrium is often of the of environmental changes in the patterns of this variation have opposite sign to the current epistasis. This lag between epistasis been important areas of focus in . On one hand, and linkage disequilibrium leads to a mismatch between hap- periods of constant selection pull the genetic makeup of popula- lotypes that are most fit and those that are more common, and tions toward local fitness optima. On the other, to cope with recombination should be favored because it breaks apart the cur- changes in the selection regime, populations may evolve mecha- rently maladapted allele combinations and combines alleles that nisms that create a diversity of genotypes. By tuning the rates at might constitute a fitter haplotype in the future (14, 17, 22, 23). which variability is produced—such as the rates of recombination, Environmental fluctuations can also affect the evolution of the . In bacteria, stress can increase the mutation rate mutation, or migration—populations may increase their long-term by inducing mutagenic mechanisms such as the SOS transcrip- adaptability. Here we use theoretical models to gain insight into tional response (24, 25) or contingency loci (26). Theoretical how the rates of these three evolutionary forces are shaped by studies have considered both plastic increases in the mutation fluctuating selection. We compare and contrast the evolution of rate in response to stress (27) and also nonplastic mutation rates recombination, mutation, and migration under similar patterns that evolve in synchrony with the rate of environmental change. of environmental change and show that these three sources of When selection fluctuates periodically and symmetrically be- phenotypic variation are surprisingly similar in their response to tween two states with different optimal genotypes, the mutation changing selection. We show that the shape, size, variance, and rate between allelic states should evolve to approximately 1=n, asymmetry of environmental fluctuation have different but pre- where n is the number of generations between temporal envi- dictable effects on evolutionary dynamics. ronmental changes (28, 29). However, numerical simulations showed that the stable mutation rate became zero as the variance in fluctuating selection | modifier genes | recombination rate | the length of selective cycles increased, or if there were asymmetries migration rate | mutation rate in selection pressure between the two environmental states (30). Variability in selection has also been shown to affect the nder constant selection, a large haploid population is ex- evolutionarily stable migration rate. When selection is hetero- pected to evolve toward a local fitness maximum. However, geneous in space but not in time and without variation in the U intensity of kin competition (31), migrants cannot displace lo- in natural populations selection may not be constant over time cally adapted individuals and there is selection against migration due, for example, to ecological changes, spatial variability, changes (32, 33). This has been shown in population genetic models of in environment, or even shifts in the genetic background (1). Under temporal and spatial heterogeneity in the direction and strength Significance of selection, a population may evolve mechanisms that create and maintain phenotypic diversity, thus increasing the long-term – Environmental variability is known to promote the evolution

adaptability of the population (2 7). These mechanisms may in- EVOLUTION clude changes in the rates at which genetic variability is produced, of mechanisms that increase phenotypic diversity. The evolu- such as the rates of recombination, mutation, and migration (8). tion of recombination, mutation, and migration, which endow Understanding how population genetic dynamics are shaped a population with the needed genetic and phenotypic vari- by changing selection has constituted an important component of ability, has been a focus of study in population genetics for research in mathematical evolutionary theory over the past five more than five decades. Theoretical approaches have focused decades. These studies have addressed such issues as the re- on conditions for the evolution of recombination, mutation, lationship between fluctuations in selection and the dynamics of and migration when environments change periodically and evolution and how these fluctuations are reflected in the pattern with symmetric selection pressures. Here we extend these of genotypic frequency variation (9–12). models to incorporate random and asymmetric selection. We One important contributor to the pattern of genetic diversity is compare and contrast how fluctuating selection affects the recombination, which can affect variation by bringing together or stable rates of these three evolutionary forces and highlight breaking apart combinations of alleles. Recombination may ac- surprising similarities in their evolution under fluctuating se- celerate adaptation by expediting the removal of combinations of lection. This study offers insights into the role of environmen- deleterious alleles from the population, but also slow it down by tal duration, shape, and randomness in predicting the long- breaking apart favorable interactions among genes (13–18). The term evolutionary advantage of recombination, mutation, and prevalence of recombination in nature has stimulated theoretical migration. efforts to explore the evolution of the recombination rate under a wide variety of modeling assumptions (14–20). Most expla- Author contributions: O.C., U.L., and M.W.F. designed research; O.C., U.L., and M.W.F. nations of the advantage of sex and/or recombination involve performed research; O.C., U.L., and M.W.F. contributed new reagents/analytic tools; O.C., either allowing assembly of favored multilocus haplotypes from U.L., and M.W.F. analyzed data; and O.C., U.L., and M.W.F. wrote the paper. combinations of epistatically interacting mutations (19–21), or Reviewers: L.A., The KLI; and L.H., Tel Aviv University. adapting to a changing environment (14, 22, 23). The authors declare no conflict of interest. Charlesworth (14) showed that for a diploid genetic system, 1To whom correspondence may be addressed. Email: [email protected] or mfeldman@ when the sign of the linkage disequilibrium varies cyclically stanford.edu. in time, increased recombination may be favored if the period This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. of environmental fluctuation is strictly larger than two. This is 1073/pnas.1417664111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1417664111 PNAS | December 16, 2014 | vol. 111 | no. 50 | 17935–17940 Downloaded by guest on September 27, 2021 selection in multipatch environments with patch-dependent se- ordered as above and the modifier locus located on one side of lection of alleles (34–36), and in ecological analyses of the evo- the two major loci, let R be the recombination rate between the lution of dispersal (37–43). Migration is suppressed because, in modifier locus and the two major loci. We assume no interference the absence of temporal variation, it limits the ability of pop- between recombination events occurring in the two intervals ulations to adapt to local environmental conditions (44). With separating the two major loci and between the modifier locus and temporal variability in selection however, migration can increase the nearest major locus. the level of local adaptation (45) and higher rates of migration Assume two possible types of selection regimes T1 and T2, may be favored (46–50). such that the fitnesses, irrespective of the genotype at the M=m Here we use the framework of modifier theory to study how locus, can be represented as follows: environmental fluctuation affects the rates of recombination, Genotype AB Ab aB ab mutation, and migration. A modifier approach to study the evo-

lution of genetic systems was first introduced by Nei (51) and Environment T1 11+ s1 1 + s1 1 Feldman (52) in models for the evolution of recombination. They Environment T2 1 + s2 111+ s2 analyzed the consequences of indirect selection on recombination between genes under viability selection. This approach is based on Thus, if s1, s2 > 0, the genotypes that are fittest in one envi- the assumption that genetic variation exists in the rate of re- ronment are less fit in the other. Increased recombination should combination; such genetic variation has been demonstrated using be favored because, at every environmental change, recombinant heritability measurements and observed differences between sexes offspring are more fit than the nonrecombinant ones (also see or closely related species (53–55). Recombination and mutation refs. 14, 18, 22, and 23). have been shown to be extremely variable across a species’ ge- nome, with areas of low recombination or mutation, as well as Evolution of Mutation. The mutation model recapitulates that of hotspots of increased activity (25, 56, 57). The rate of migration Salathe et al. (30) and Liberman et al. (61), and we include it and ability to disperse are typically heritable and vary between here for comparison with recombination and migration. A major species and within environments for a single species (58, 59). locus A=a controls the fitness of the individual, whereas a second We aim to better understand the similarities and differences in locus M=m controls the mutation rate between alleles at the the dynamics of these three evolutionarily important forces un- major locus but is otherwise selectively neutral. The recom- der similar patterns of environmental change. Elucidating the bination rate between A=a and M=m is r. evolution of recombination, mutation, and migration under fluc- We study the evolution of the modifier locus M=m and determine tuating selection will lead to a more complete understanding of the stable mutation rate as a function of the pattern of fluctuation in both fundamental evolutionary processes and the pressures that selection. Alelles M and m at the modifier locus produce mutation have shaped genomic diversity. rates μM and μm, respectively. Assume two types of selection regimes T1 and T2, such that the fitnesses are as follows: Models of Neutral Modifiers in Changing Environments Genotype Aa Consider an infinite, randomly mating, haploid population and two types of biallelic loci: major loci, which control the pheno- Environment T1 11+ s1 type and fitness of this individual, and a modifier locus, which is Environment T2 1 + s2 1 selectively neutral. The modifier locus is assumed to control the force of interest, which can be the recombination rate between a If s1, s2 > 0, then alleles that are fitter in one environment are pair of major loci, the mutation rate between alleles at the major less fit in the other. At each generation, the population experi- loci, or the migration rate between two demes in a spatially ences random mating, selection, mutation, and recombination, in subdivided population. Under what conditions should evolution that order. increase these rates and how do these conditions depend on the pattern of temporal fluctuations in selection? Evolution of Migration. The population is divided into two demes We address these questions by studying the evolution of the Ex and Ey, with different selection regimes. There are two bial- modifier loci and by determining the evolutionarily stable rates lelic loci—a major locus A=a and a modifier locus M=m—where as functions of the pattern of fluctuation in selection experienced the major locus controls the fitness of an individual, whereas the by the population. As in ref. 52, we frame the problem in terms selectively neutral modifier locus controls the migration rate of the local stability of an equilibrium with one allele fixed at the between the two demes. We investigate the evolutionary dy- modifier locus to invasion by another allele introduced at low namics of the migration rate between Ex and Ey using an explicit frequency near this equilibrium. The case where a stable multi- population genetic model to track the frequencies of the four allele polymorphism exists at the modifier locus and a new genotypes AM, Am, aM, and am. At each generation, there is modifier allele arises near this equilibrium (60), and the selection recombination between A=a and M=m at rate r, and selection in regime fluctuates, will be treated elsewhere. each deme separately, after which individuals may migrate be- tween the two demes. Again, we frame the question in terms of Evolution of Recombination. Each individual is defined by three the local stability of the fixation equilibrium with only M present, biallelic loci: Two major loci A=a and B=b control the fitness of producing migration rate νM , to invasion by allele m, which the individual, whereas a third locus M=m is a modifier locus that produces migration rate νm. We assume νM and νm to be the controls the recombination rate between the two major loci but is same from Ex to Ey as from Ey to Ex. otherwise selectively neutral. Within each deme, the selection regime varies temporally, with T T T We study the evolution of the modifier locus M=m and de- two environmental states, 1 and 2.In 1, the fitnesses are termine the stable recombination rate as a function of fluctua- Deme Ex Ey tion in selection experienced by the population. This entails analysis of the stability of an equilibrium with only M present to Genotype AaAa invasion by an allele m, introduced near this equilibrium. To that Fitness 1 + s1 111+ s3 end, we track the frequencies of the eight genotypes MAB, MAb, MaB, Mab, mAB, mAb, maB, and mab. At each generation, the whereas in T2, they are population experiences random mating, recombination, and se- Deme E E lection, in that order. x y There are three possible recombination rates depending on Genotype Aa Aa MM Mm mm the genotype at the modifier locus: , , and produce Fitness 1 1 + s2 1 + s4 1 recombination rates r1, r2 and r3, respectively. With the three loci

17936 | www.pnas.org/cgi/doi/10.1073/pnas.1417664111 Carja et al. Downloaded by guest on September 27, 2021 Thus, if s1, s2, s3, s4 > 0, allele A is favored in one temporal of fixation in M to invasion by m in the case τ1 = τ2 = 1, i.e., the state and allele a is favored in the other. This is an important environment changes every generation under the assumption assumption if the fitness regimes T1 and T2 are coordinated that all selection coefficients in each of the three models are the between the two demes. Otherwise it is not important which same. The mathematical analyses for modification of migration and environment is denoted as T1 or T2; all that matters is that en- recombination are in SI Appendix. As noted above, in a temporally vironmental change occurs independently in the two demes. If constant environment, the reduction principle holds. On the other the si are all equal, we call the selection symmetric. hand, with fluctuating selection of period 2, a higher rate will invade a population fixed on a given resident rate. The stable rate is 1 for Changing Environments: The General Simulation Model. Initially the migration and mutation and 1/2 for recombination (the biologically population is fixed on allele M at the modifier locus, which feasible maximum). Note that our results in the haploid case are determines a rate sampled randomly between 0 and 1 for different from those for a modifier of recombination in a diploid modifiers of mutation and migration and between 0 and 0.5 for model with period 2 (14). modifiers of recombination, and held constant thereafter. This We were able to obtain analytical solutions only under these population evolves for 1,000 generations or until it reaches a very rigid symmetry assumptions and for environmental period stable polymorphic equilibrium at the locus/loci under selection. of two. For the evolution of the three rates under more general We then introduce allele m at the modifier locus at a small conditions, simulations were used to find the stable rate. − frequency (10 4) near this equilibrium. The rate determined by m is the product of the resident rate of M and a number gen- Simulation Results erated from an exponential distribution with mean 1. This allows Periodic Environments. If the environment changes every genera- us to test invading rates that are more often close to the resident tion, we confirm the result found in our mathematical analysis rate, as well as those that are far from it. If these invading rates in SI Appendix. If the temporal environment changes deter- are larger than the biological maxima, then the maxima are used ministically every n generations (i.e., period 2n) in both demes, (i.e., 1 for mutation and migration, and 1/2 for recombination). the uninvadable rates are all decreasing functions of n (Fig. 1). In After 5,000 generations, we determine whether the newly in- all our simulations, the stable rates of mutation and migration troduced allele invaded the population; if the frequency of m is − are quantitatively similar, suggesting that these two forces lead to larger than its initial frequency of 10 4, we call it an invasion. In similar evolutionary dynamics. The stable rates of mutation and this case, the new allele had an induced selective advantage over migration are on the order of 1=n, in accord with previous the resident, and it should reach fixation if the population con- analyses (see refs. 28–30 and 63 for modifiers of mutation). This tinued to evolve. If allele m invaded, the next invasion trial is also shown in SI Appendix, Fig. S1, which plots the three stable begins with this invading rate as the resident rate. If there was no rates as functions of 1=n and displays the linear regression fit invasion, the resident allele determines the same resident rate as to our simulated data. Although the slopes corresponding to in the previous trial. We repeat the invasion steps described above. mutation and migration are very similar, the slope for the re- After at least 500 invasion trials and after the resident rate cannot combination rate is about twice these regression coefficients. be invaded in 50 consecutive trials, the final rate in the population Fig. 1 presents results for environmental waiting times n with is declared to be the evolutionarily stable rate. n ≥ 20, whereas the results for smaller n are shown in SI Ap- With periodic environmental change, the selection regime pendix, Fig. S2, where important differences in the three stable changes deterministically every n generations, making the envi- rates for small environmental periods can be seen. Whereas the ronmental period 2n. For random temporal variation, the waiting stable rate of recombination reaches the allowed maximum, 0.5, times between environmental changes have a gamma distribution, for all n < 6, mutation and migration exhibit different dynamics which is defined by two parameters: the shape α and the scale β, for small odd and even n. The maximal allowed rate is reached at with mean n = αβ and variance ψ = αβ2. This allows us to test n = 1 and n = 3 for migration; and n equal to 1, 3, 5, and 7 for a wide range of distributions between pure periodicity ðψ = 0Þ and mutation. This difference between odd and even n in the case of an exponential waiting time ðψ = 1Þ by fixing n while varying ψ.If mutation modification was also observed by Liberman et al. (61). ψ is strictly larger than zero (nonperiodic case), the final stable SI Appendix, Fig. S3 shows that these results are robust to dif-

rate is computed as the average of the stable rates obtained in 10 ferent values of the symmetric selection coefficient. EVOLUTION different simulation runs. This uninvadable rate also maximizes mean fitness at equi- librium. We show this in SI Appendix, Fig. S4 for recombination Analytical Results modifiers, and in SI Appendix for migration modifiers; it was also In the case of mutation rates, Liberman et al. (61) showed that shown by Liberman et al. (61) for the evolution of mutation for period 2 and symmetric selection, increased mutation rate is rates. Although this mean fitness principle holds for migration always favored, and the mean fitness at equilibrium is an in- and mutation modifiers in diploid modifier models, it does not creasing function of the mutation rate. Also, with period 4, the hold for the modification of recombination (64). stable mutation rate is 1/2. Moreover, the critical points of the As recombination between the modifier locus and the gene(s) mean fitness with respect to the mutation rate are those that under selection increases, the strength of indirect selection cannot be invaded. Carja et al. (62) also showed that these on the modifier decreases because the linkage disequilibrium results hold with spatial heterogeneity in selection: For period 2 between these genes decreases. SI Appendix,Fig.S5shows and symmetric selection, a modifier increasing mutation can al- that the stable rates decline with increasing recombination ways invade in the population regardless of the migration rate. rate between the modifier and the major locus/loci. In earlier Furthermore, for higher environmental periods, analytic compu- studies of neutral modifiers of mutation and recombination, tation in Mathematica (Wolfram Mathematica) was used to show the indirect selection on the modifier locus was of the order of that the parameter that controls the evolution is mb = ν + μ − 2νμ, the square of the disequilibrium between the major and the where ν is the migration rate and μ the mutation rate. In this case, modifier loci (63). the maximum mean fitness at equilibrium is achieved for mutation rates for which this parameter mb is on the order of 1=n,where2n Random Waiting Times. If the waiting times between temporal is the environmental period. changes are random, the 1=n rule no longer holds for the evolution Here, using similar analyses, we study the stable recombination of mutation rates (30). Fig. 2 shows the stable rates of re- and migration rates as functions of the environmental period, i.e., combination, mutation, and migration when the expected time be- the length of the period of the fluctuating selection. A general fore an environmental change is 10 generations, and this waiting environmental cycle consists of τ1 + τ2 selection steps, the first τ1 time is sampled from a gamma distribution with variability pa- of type T1 selection followed by τ2 of type T2 selection. We are rameter ψ represented on the x axis. Environmental variability able to derive exact analytical conditions for the local stability ψ = 0 is the case of two periodically changing environments with

Carja et al. PNAS | December 16, 2014 | vol. 111 | no. 50 | 17937 Downloaded by guest on September 27, 2021 Appendix, Fig. S6 shows that the results of Fig. 2 are robust to 0.25 changes in the mean waiting time for environmental change.

Asymmetric Selection. With fitnesses that are more complicated than can be represented by a single parameter (for example, if A T 0.20 the selective advantage to allele of being in environment 2 is greater than the selective advantage to allele a of being in en- vironment T1, in the case of mutation modification), we observe a threshold phenomenon; if the selection coefficients are similar enough, the evolved rates are the same as in symmetric case. 0.15 Modifier of However, beyond a certain threshold difference in selection recombination coefficients, there is a sudden drop in the stable rate and nonzero mutation rates can no longer be stable. This threshold determines when migration nonzero recombination, mutation, and migration rates can be 0.10 maintained, as shown in Fig. 3, for both periodic and random environmental changes. This was observed previously for muta- Evolutionarily stable rate stable Evolutionarily tion modifiers (30) and is qualitatively robust to differences in environmental variability ψ and environmental mean waiting times (SI Appendix, Figs. S7 and S8). 0.05 Discussion We find interesting similarities in the evolutionary dynamics of modifiers of mutation, recombination, and migration. These 0.00 similarities may stem from the fact that all three forces can en- hance a population’s adaptation to a changing environment by 25 50 75 100 increasing the genetic diversity on which selection can act. In a n: environmental period 2n sense, they all allow for anticipation of selection change but they Fig. 1. Evolutionarily stable rates as function of the number of generations may also create a load. In that context mutation, recombination, before an environmental change n; symmetric selection and periodic envi- and migration can be considered bet-hedging strategies. How- ronmental change. All selection coefficients are equal to 0.1. Recombination ever, the type of selection acting in the these models is different rates between modifier and major loci are 0 (R = 0 for recombination from what is classically called bet-hedging because we consider modification and r = 0 for mutation and migration modification). The rate of an infinite population size with no demography (6). environmental change, n, equal to half of the environmental period, is on Our results extend the qualitative messages of previous theo- the x axis. The curves represent a fit to the data using a generalized additive retical work. In constant environments, the reduction principle model with penalized cubic regression splines. holds and the rates of mutation, migration, and recombination all evolve toward zero. When environments change periodically through time, these rates can evolve to nonzero values, each SI Appendix period 10 and recaptures the behavior seen in ,Figs.S2 decreasing with increasing environmental period. Previous work ψ and S3. For small , the three uninvadable rates decrease slightly in the evolution of mutation rates (28, 29) found that with sym- as symmetric selection intensity increases. As ψ becomes larger, metric selection, mutation evolves to be in synchrony with the rate the behavior of the system changes dramatically. We find that the of environmental change: order of 1=n, where 2n is the environ- stable rates of recombination and migration can be up to 2 orders mental period. Here we show that the rate of migration between of magnitude lower than in the periodic regimes and depend two demes with different selection pressures also follows this strongly on the selection coefficients. In fact, the decrease in the pattern and the stable rates of migration are quantitatively similar stable rate is steeper as selection becomes weaker (Fig. 2). SI to those of mutation. This is because mutation and migration both

AB C 0.125 0.125 0.25 ● ● ● ● ●● ●● ● ●●● ●●● ●● ●●● ● ●● ●●●● ●● ● ● ●● ●●● ● ● ●●● ●● ● ● ● ●●● ● ●● ● ●● ●●● ●●● ● ●●● ●● ● ●●● ●● ● ●● ●● ●● ●● ● ● ● ●● ●●●●● ● ●● ●●● ● ●●● ●●●● ●●● ● ●●● ●●● ●●●● ●● ●●●●● ●● ●●●●●● ● 0.100 ●●● ● ● ● ● 0.100 ● ● ● ●●● ●●●●● ●●●●● ●● ●● 0.20 ● ●● ●● ● ● ● ●● ●●● ●● ●●●● ●●●● ●● ● ●●● ●● ● ● ●● ● ●●● ● ●● ● ● ●●● ● ●● ●● ●● ● ●●●●● ● ●● ● ●● ● ● ● ● ● ● ●● ●● ●●● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ●●●●● ● ●● ● ● ● ●● ● ● ● ●● ●● ●●●●● ●● ●● ● ● ● ● ● ●● ●●● ●●● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●●●● ● ● ●●● ● ● ● ●●● ●●● ● ●●●●● ● ● ● ● ● ● ●● ●●●● ● ●●● ● ● ●● ● ●● ● ● ● ●● ●● ● ● ●●● ● ● ● ● ● ●●●● ● ● ●● ●●● ● ●● ● ● ● ● ● ●●●●● ● ● ● ● ● ●● ● ●●●● 0.075 ● ● ●●● ● ● ● ●●● ● ● 0.075 ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ● ●● 0.15 ● ●●●● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●●● ● ● ●● Selection coefficient ● ● ● ●● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● 0.05 ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ●●● ●● ● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ●● 0.07 ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● 0.1 ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●●● ●● ●● ● ● ● ● ● ●●● ● ● ●●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●●●● ●● ● ● ● ● ●● ● ● ● ● ● 0.050 ● ● ● ● ● ● ● 0.050 ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● 0.10 ● ●● ● ● ●● ●● ●● ● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●●● ● ● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● 0.4 ● ● ● ● rate mutation Stable ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● rate migration Stable ● ● ●● ● ● ● ● 0.5 ● ● ●●●●● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ●● ● 0.7 ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● Stable recombination rate Stable ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● 0.9 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● 0.025 ● ● ●● ● ● 0.025 ● ●● ● ● ● 0.05 ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ●●●● ●● ● ● ●● ●●● ● ● ● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ●●● ● ● ● ● ● ● ●● ●● ● ●● ● ●●● ●● ● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ●●●● ● ●●●● ●● ● ● ● ● ● ●● ●●● ●● ●●●●● ● ● ●● ● ●● ● ●●●● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ●●● ● ● ● ● ●●● ● ●● ●●●● ● ●●● ●● ● ● ● ●●● ● ● ●●● ● ● ● ●●●● ● ● ●● ● ●● ● ●●●● ● ● ●● ● ● ● ●● ● ●●● ●● ●● ● ● ● ● ● ●●● ● ● ●●●●●●●● ●● ● ● ● ●●● ●● ● ● ●● ●● ●● ●● ● ● ●●●● ●● ● ●●● ●●● ●●●●● ●●●● ● ● ● ● ● ● ● ●●●●●●●●● ●● ●● ● ●●● ●●● ●●●●● ●●● ● ● ● ● ● ● ● ●● ●●●●●●●●●● ●●●● ●●●●●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ●● ● ● 0.00 0.000 0.000

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Environmental variability Ψ Environmental variability Ψ Environmental variability Ψ

Fig. 2. Evolutionarily stable rates as function of the environmental variability ψ; symmetric selection and random environmental change. All selection coefficients are equal and as presented in the legend. Recombination rate between modifier and major loci is 0.1. Mean waiting time between environ- mental changes is equal to 10. The variability parameter ψ is presented on the x axis. The curves represent a fit to the data using a generalized additive model with penalized cubic regression splines. The three panels present the results for evolutionarily stable rates of recombination in A,mutationinB,and migration in C.

17938 | www.pnas.org/cgi/doi/10.1073/pnas.1417664111 Carja et al. Downloaded by guest on September 27, 2021 AB●●●●●●●●●●●●●●●●●●●●●● ● ● ●●●●● ● ● ● ● ●●●● ● ● ●● ● ● ●●●● 0.20 ●● ● ● ● ●● ●● ● ●● ●● ● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● 0.20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.15 ● ● ● ● ● ● 0.15

● ● Modifier of ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ●●●●●●●●●●●●●●●●●●●● ●● ● ●●● ●●● recombination ●●●●●●●●●●● 0.10 ● ● ● ● ●●● ● ●● ●●● ●●●●●● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●● ●●●● ● ● ● ● ● ● ● ● ●●● ●● ●●●●●●●●● ●●● ● ● ● ● ● mutation ●●●●● ●●● ● ● ● ● ● ●●●● ●● ●● ● ●●● ●● ●●● ● ● 0.10 ●● ● ● ● migration ●● ●● ●● ●● ●● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● Evolutionarily stable rate stable Evolutionarily

Evolutionarily stable rate stable Evolutionarily 0.05 ●

0.05 ● ●

●

● ● ●● ● 0.00 ● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● 0.00 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Difference in selection Difference in selection

Fig. 3. Evolutionarily stable rates as functions of asymmetry in selection. For recombination and mutation, s2 = 0.5 and the difference in selection s2 − s1 is on the x axis. For migration, s2 = s4 = 0.5 and s1 = s3 with the difference in selection s2 − s1 on the x axis. Recombination rates are R = 0.1 for recombination modification and r = 0.1 for mutation and migration modification. (A) Periodic environment, rate 10 (period 20). (B) Mean waiting time between environ- mental changes is equal to 10 and variability parameter ψ = 0.1. The dots represent averages over 10 runs of the simulation. The curves represent a fit to the data using a generalized additive model with penalized cubic regression splines.

act to introduce the fitter alleles into the changed environment. should be inversely proportional to population size (73). However, We also show that the rate of recombination follows the same even with large numbers of individuals, a drift-based advantage to qualitative pattern as the rates of mutation and migration. recombination may occur as long as there is spatial structure (71) Previous work has mostly focused on periodic, symmetric en- or selection acts on a large number of loci (73). The interactions vironmental changes. The total genetic load differs between between the kinds of fluctuations in fitness studied here and the randomly changing environments and periodically fluctuating effects of drift remain to be explored. environments and, therefore, evolution under random and pe- A major theoretical challenge is to understand how the dif- riodically fluctuating selection is expected to be different (65). ferent forces discussed here combine to influence the evolution Recombination, mutation and migration are surprisingly similar of genetic and phenotypic diversity; in natural populations re- in their evolutionary response to randomness in waiting time combination, mutation, and migration may together counteract before an environmental change, as measured by ψ; increase in ψ the stresses induced by a change in selection. An understanding leads to a decrease in all three evolutionary rates. We hypoth- of the interplay between the types of diversity introduced by the esize that as ψ increases, the longer stretches of constant selec- three different forces is needed in theoretical population ge-

tion that favor reduction play a greater role in the dynamics of netics (50). For example, it has been shown that, with migration EVOLUTION the system. A decreased role for short environmental durations among populations, spatial heterogeneity in selection can gen- would explain why the optimal rates of these three forces are erate positive or negative linkage disequilibrium and select for small when the environmental fluctuation is random. Moreover, recombination even in the absence of genetic drift or temporal the stable rates increase with selection intensity; when selection heterogeneity (17). This is because natural selection varying is weak, the evolutionary dynamics track environmental shifts across space maintains local differences in gene frequencies, slowly and the occasional very long runs of the same environ- and with migration these differences in frequency can generate ment pose a disadvantage to high recombination, mutation, or patterns of linkage disequilibria within a deme that create migration rates. As selection pressure increases, the system is better a selective pressure for increased recombination. In fact, in- able to respond to the more rapid shifts in the environment, which cluding migration extends the range of epistasis over which leadstoanincreaseinthethreestablerates. recombination can be favored (17). In our study of mutation An important limitation of our models is the assumption of modification in two populations connected by migration and infinite population size: In finite populations under a constant undergoing symmetric periodic selection, the parameter that environment, drift may be important in shaping the stable rates controls the evolution of the system is a nonlinear function of the of the three forces (66–68). A better understanding of the in- mutation and migration rates and its stable rate is inversely terplay of population size and environmental change in the proportional to the environmental period (62). Mutation and evolution of recombination, mutation, and migration is needed recombination have also been studied together in models for in both experimental and theoretical settings. Previous work on evolution of recombination in constant environments; if the evolution of dispersal suggested that, with drift, migration will be major loci are at an equilibrium between selection against selected as a way to avoid kin competition (69, 70). Similarly, deleterious alleles and mutation toward them, recombination random genetic drift can affect the evolution of recombination may increase in the population if the linkage disequilibrium is by generating negative disequilibria among selected loci (71) negative (15, 21). such that beneficial alleles at one locus become associated with Overall, we find that the three essentially different evolu- deleterious alleles at other loci. Modifiers that increase recom- tionary forces are surprisingly similar in how they respond to bination could help regenerate good combinations of alleles, and fluctuating selection, as each of them constitutes a form of ge- will rise in frequency along with these alleles (67, 68, 72). In large netic bet-hedging and endows a population with the genetic populations, it has been proposed that selection on recombination diversity that can accommodate changes in their selective

Carja et al. PNAS | December 16, 2014 | vol. 111 | no. 50 | 17939 Downloaded by guest on September 27, 2021 environments. Our models show that knowing the difference in help without knowing the extent of asymmetry in selection selection between environmental regimes is not sufficient to coefficients. Some of these complications have been addressed predict the long-term advantage of these three forces of evolu- by Altenberg (35, 50). Future work needs to consider the inter- tionary change. Knowledge of the duration, shape, and ran- actions across multiple dimensions in the parameter space to domness of the environmental regimes is also essential. increase our understanding of how recombination, migration, and Similarly, knowing how the environment fluctuates is of little mutation contribute to adaptation in fluctuating environments.

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