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The evolution of reversible switches in the presence of irreversible mimics
Abstract
Reversible phenotypic switching can be caused by a number of different mechanisms including epigenetic inheritance systems and DNAbased contingency loci. Previous work has shown that reversible switching systems may be favored by natural selection. Many switches can be characterized as “on/off” where the “off” state constitutes a temporary and reversible loss of function. Loss of function phenotypes corresponding to the “off” state can be produced in many different ways, all yielding an identical fitness in the short term. In the long term, however, a switchinduced loss of function can be reversed, while many loss of function mutations, especially deletions, cannot. We refer to these loss of function mutations as “irreversible mimics” of the reversible switch. Here we develop a model where a reversible switch evolves in the presence of both irreversible mimics and metapopulation structure. We calculate that when the rate of appearance of irreversible mimics exceeds the migration rate, the evolved reversible switching rate will exceed the bethedging rate predicted by panmictic models.
1 Introduction
A variety of mechanisms allow for heritable, reversible phenotypic switching. These include most epigenetic inheritance systems (Rando and Verstrepen 2007), in which no change in DNA sequence occurs, as well DNAbased “contingency loci” (Moxon et al. 1994) that may be readily reversible through repeat contractions and expansions. These switches differ in three ways from classical mutations such as DNA point mutations and deletions. First, switching is easily reversible. Second, switching frequencies are typically higher than average mutation frequencies for most taxa (Drake 1999; Rando and Verstrepen 2007). Finally, switching may be preferentially induced at times when it is most likely to be beneficial. Here we neglect environmental induction and restrict our analysis to random reversible switching mechanisms. This simplifying assumption is conservative with respect to the evolution of switching mechanisms (Jablonka et al. 1995; Kussell and Leibler 2005; Wolf et al. 2005).
Phenotypic switching may sometimes be beneficial by producing adaptive phenotypes, and at other times be costly by producing phenotypes that are not adaptive. Several models have concluded that for organisms living in a fluctuating environment, mechanisms that enable reversible switching can evolve, with the optimal switching rate (m_{opt}) predicted to be equal to the frequency of environmental change events that make switching adaptive (Ω) (Lachmann and Jablonka 1996; Kussell and Leibler 2005; Kussell et al. 2005; Wolf et al. 2005; King and Masel 2007). Natural selection for reversible switching is strong enough to overcome genetic drift so long as Ns 1 and NΩ 1, where s is the selective advantage of phenotypic switching when the environment changes and N is the effective population size (King and Masel 2007).
Previous models have not, however, considered the possibility that the same phenotypes generated through evolved reversible switching mechanisms may also be generated by a loss of function mutation such as a deletion. Many phenotypic switches have a simple “on/off” form, where the “off” state represents a temporary and reversible loss of function. Phenotypic mimicry may be a common phenomenon, based on the prevalence of welldocumented phenocopies and genocopies in a wide range of taxa (reviewed in WestEberhard 2003), but the mechanism and inevitability of mimicry are particularly obvious when a phenotype is based on loss of function. Here we investigate the effect of the existence of loss of function irreversible mimics on the evolution of a reversible phenotypic switching system.
If, in a new environment, mutation alone gives rise to adaptations at a rate less than the optimal Ω, then previous models predict that a modifier allele facilitating more rapid phenotypic switching will invade the population. Indeed, many switching mechanisms found in nature have switching rates around 10^{−1} to 10^{−6} per generation (Jablonka and Lamb 1998; Rando and Verstrepen 2007), higher than typical mutation rates. However, for at least for one switching system, the yeast prion [PSI +], irreversible mimics of [PSI +] appear spontaneously even more often than [PSI +] (Lund and Cox 1981; Lancaster, Bardill, True, and Masel, in prep). Previous models predict how the evolution of phenotypic switching mechanisms is driven by the advantages of rapid switching. This approach cannot explain the evolution of the [PSI +] system, which switches less often than its mimics. Our model focuses on whether the property of reversibility rather than rapidity can explain the evolution of phenotypic switching in such cases.
2 Model overview
To capture the key elements of the biology of reversible and irreversible switches we employed a two level model: (1) The benefits of reversiblyinduced variation when adaptive and costs when maladaptive are represented in a model of evolution at a modifier locus with two alleles, M_{1} and M_{2}, that cause reversible switching at rates m_{1} and m_{2}, respectively (Figure 1a). This stochastic model assumes an asexual haploid population and is based upon the finite population approach developed by King and Masel (2007). (2) This model is then nested within a deterministic metapopulation island model (similar to Levins 1969, 1970) to introduce the longterm threat from irreversible mimics (appearing at rate μ_{irr}). Irreversible mimics initially mediate adaptation, but in the long term cause demes fixed for the mimic to go extinct, since they are unable to switch back their phenotype when the environment switches back. Even if we relax the assumption of complete irreversibility, a delay in reversibility can be sufficient to cause such a population to be outcompeted by a rival reversibly switched population that is not handicapped by such a delay, again leading to the longterm extinction of the handicapped lineage. These two levels of the model constitute selection at the individual and deme levels, respectively. In order to avert extinction events caused by the appearance of mimics, evolution in a metapopulation may favor a higher rate of reversible switching than if mimics are not considered. The extent of the risk from mimics is captured in our model by their rate of appearance μ_{irr}.
Environmental change from environment E to environment F always leads to adaptation mediated by phenotype B. For this simplifying assumption to hold, we restrict our parameter space to Ns 1 where s is the selective advantage of B in environment E, and (m + μ_{irr})N ≥1 where m + μ_{irr} is the total rate of appearance of the B phenotype. Each environmental change event then leads to fixation of either the reversible B_{r} or the irreversible B_{i} within the deme (Figure 1b). If it is the B_{i} mimic that becomes fixed, the deme incurs a permanent handicap either by its inability to reverse its adaptation back to environment E, or for other longterm reasons related to loss of function. For mathematical simplicity, we assume that demes fixed for B_{i} go instantaneously extinct (Figure 1b).
We make the simplifying assumption that on the timescale of the metapopulation model, a deme is dominated either by an M_{1} or M_{2} allele, or it is empty. Random environmental change events within a deme occur independently with respect to other demes. Individuals in demes can (1) migrate and colonize empty demes (2) migrate and replace occupied demes of the opposite allele type, (3) go extinct, together with the rest of their deme, due to the fixation of an irreversible mimic phenotype. An empty deme is colonized when a migrant arrives (at rate Nm_{k}) from an occupied deme. Demes go extinct when environmental change (at rate Ω) leads to adaptation via an irreversible mimic. These give colonization and extinction rates conforming to the Levins model. In addition, demes switch type when a migrant allele becomes fixed and replaces the resident allele, a process that is not part of the original Levins model. The probability that such migration leads to replacement of the resident allele is computed by the population genetic model described in Section 2.1 and the replacement is approximated as instantaneous (see Equation (1) and Figure 2).
For a given pair of switching rates, we can compute the equilibrium between colonization, invasion and extinction to determine which of the two modifier alleles dominates more demes. Our analytical work assumes an infinite number of demes for mathematical tractability, but we also examine the effect of a finite number of demes for a limited number of test cases, as illustrated in Figure 7. Our model then explores a range of reversible switching rates to determine which rate (m_{evolved}) we expect natural selection to favor. We then investigate how m_{evolved} shifts in response to changes in the rate of appearance of irreversible mimics (μ_{irr}). The model is fully general and applies to any system that includes both evolved reversible phenotypic switching and intrinsic irreversible mimics.
2.1 Population genetic model within each deme
We use a modified version of the withindeme model introduced by King and Masel (2007) and shown in schematic form in Figure 1 and Figure 2. For the most part we follow their simplifying assumptions and notation (with the notable exception of using Ω to represent the rate of environmental change rather than Θ). Within each independent deme, the environment switches from state E to state F at rate Ω. The deme is of constant size N and consists of haploid individuals with one of two possible alleles M_{1}, M_{2} at a modifier locus. M_{1} and M_{2} alleles cause reversible switching from A to B_{r} with rates m_{1} and m_{2}, respectively. (We can ignore backswitching from phenotype B_{r} to A for the purposes of the withindeme model, since in environment E B_{r} individuals do not persist long enough to switch back, and in environment F reverse switching is initially both rare and unfavorable. If genetic assimilation is rapid, reverse switching may start to become relevant in environment F before M allele fixation is complete, but by this stage the relevant M allele will already have derived most of its benefit, and so we approximate fixation as complete before genetic assimilation.) In addition, alleles at one or more loci that cause the irreversible B_{i} phenotype are assumed to appear at rate μ_{irr}. Note that phenotype B_{i} is functionally identical to B_{r} according to the withindeme model; its longterm disadvantage is captured at the level of deme persistence within the metapopulation.
We assume that environmental change is rare relative to the timescale of fixation of B_{i} or B_{r} in response to each environmental change. In this way we can consider only the environment change from E to F and associated phenotypic switching from A to B_{r}. The reverse direction from F to E is implicit in extinction of B_{i} demes at the metapopulation level and in the repetitive nature of E to F environmental switching events.
We use a Moran model for the evolutionary process. At each time step one individual is chosen uniformly at random to die, and one individual is chosen to reproduce according to its fitness. Reversible or irreversible switching may occur at the moment of reproduction. We ignore rare cases where both occur simultaneously. At each time step the environment changes with probability e^{−Ω}^{/N}. This represents environmental change at rate Ω per generation, corrected for the fact that one generation corresponds to N time steps in the Moran model. Again following King and Masel (2007) we assume that phenotypes B_{r} and B_{i} have fitness zero in the original environment E, but a selective advantage in the new environment. Relative fitnesses are f_{AE} = 1 and f_{BE} = 0 in the old environment, and f_{AF} = 1 and f_{BF} = 1 + s in the new environment, where s is the selective advantage. Newborn B individuals in environment E are immediately replaced. The population in E therefore contains no B individuals, but two types (m_{1} and m_{2}) of A individuals. In environment E, immediate replacement of B individuals means that the fitnesses of the M_{1} and M_{2} individuals are (1 − m_{1} − μ_{irr}) and (1 − m_{2} − μ_{irr}), respectively.
The probability that a single M_{2} allele will fix in a population of M_{1} alleles is given by (Figure 2; King and Masel 2007):
where q(1, i) is the probability that there are i M_{2} alleles at the time of the next environment change event from E to F, given that there is initially one M_{2} allele; and p(i) is the probability that an individual bearing the M_{2} allele, which has not undergone irreversible switching, becomes fixed given that there i M_{2} individuals at the moment of environmental change. Intuitively, q can be seen as representing the disadvantages of frequent switching in the old environment (E), while p represents the advantages of frequent switching in the new environment (F). The equations are based on the approach of King and Masel (2007), suitably modified to take into account the irreversible switching rate μ_{irr}. For details of the calculations of q and p, see Appendix A.
2.2 Metapopulation model
Our metapopulation model is based on those by Levins (1969, 1970) and assumes uniform migration among an infinite number of demes. Following migration between demes fixed for different M alleles, Equation (1) from Section 2.1 determines the probability that the single migrant will displace the resident (see Figure 2). The fixation process is approximated as instantaneous.
Let the fraction of empty demes, demes dominated by the M_{1} allele and demes dominated by the M_{2} allele be given by P_{0}, P_{1} and P_{2}, respectively. The model can now be represented by the coupled differential equations:
where M_{1} and M_{2} demes go extinct at rates e_{1} and e_{2}, empty demes are colonized by M_{1} and M_{2} at rates cP_{1} and cP_{2}, and M_{1} demes switch genotypes to M_{2} at rate g_{12}P_{2} and M_{2} demes to M_{1} at rate g_{21}P_{1}. An overview is shown in Figure 3.
Environmental change events occur independently in each deme in the metapopulation. Adaptation to the new environment F can be mediated either by B_{i} or by B_{r}. However, demes fixed for B_{i} will eventually go extinct, because irreversibility is now a liability. In our parameter range of interest and given enough time, one or the other B lineage will eventually fix. Since B_{r} and B_{i} are initially selectively neutral relative to each other, fixation probabilities are proportional to their appearance rates. Since extinction corresponds to B_{i} fixation, extinction rates for demes of types M_{1} and M_{2} are given by:
The rate of migration is equal to the probability that an individual migrates (m_{k}) multiplied by the number of individuals that might migrate, which is the size of each deme (N). Hence the rate that an empty deme is colonized is given by:
For a deme to change type, an individual must migrate (m_{k}N ) from a deme of opposite type (P_{1} or P_{2}) and take over, i.e. fix, once it arrives (computed from equation (1)). The probability that a single M_{1} migrant fixes in an M_{2} deme is given by p_{fix}_{1} =_{fix}(m_{2}, m_{1}, μ_{irr}, N, Ω, s), and the probability that a single M_{2} migrant fixes in an M_{1} deme is given by p_{fix}_{2} = _{fix}(m_{1}, m_{2}, μ_{irr}, N, Ω, s), hence M_{1} and M_{2} demes change types at rates:
The total fraction of demes must be unity (P_{o} + P_{1} + P_{2} = 1) and so the system can be reduced to the two dimensional system:
Equilibrium solutions to these equations can be found by standard techniques, details are in Appendix B. We look at max[_{1}, _{2}] to determine the “winner”.
2.3 Finding the “evolved” switching rate
We are interested in identifying the reversible switching rate that is favored by evolution at the modifier locus. A common approach is to define optimality as that which maximizes some measure of fitness, such as geometric mean fitness (Seger and Brockman 1987). A drawback of this technique is that it assumes infinite population sizes and does not deal with the case of weak selection that may exist in real populations (Philippi 1993; King and Masel 2007).
An alternative approach is to focus on pairwise comparisons, e.g., evolutionary stable strategies (ESS) (Maynard Smith and Price 1973; Maynard Smith 1982) or fixation versus counterfixation probabilities (Masel 2005; King and Masel 2007). In this approach, the optimal strategy is defined as that which beats all others in pairwise competition. When pairwise comparisons are nontransitive, this definition sometimes fails to imply a unique optimum, and unfortunately this problem arises for our model: see Appendix C for examples.
An alternative to pairwise comparisons is to consider K possible allele types with transition probabilities defined for each pair, based on the products of mutation rates and fixation probabilities (see, e.g., section 4.1 of King and Masel 2007). We can then calculate the stationary distribution of the system (Claussen and Traulsen 2005; Fudenberg et al. 2006) and summarize it according to an average longterm evolved switching rate, m_{evolved}.
By analogy to this approach, we assumed a mutational model where the reversible switching rate m is treated as a quantitative trait. A Monte Carlo simulation was then used. In each step, a single mutant was selected from a normal distribution (on the logarithmic scale), centered on m with variance σ^{2} = 0.1, and compared to the resident using our deterministic metapopulation model. The winner according to this deterministic comparison was retained. The final “evolved” switching rate (m_{evolved}) was computed as the average switching rate over time. Details of the algorithm can be found in Appendix C.
2.4 Parameter restrictions
We consider only biologically realistic and interesting parameter ranges, leading to four restrictions on the parameters. First, natural selection for switching is too weak to overcome genetic drift within a single finite deme unless Ω > 1/N (King and Masel 2007). Second, the population genetic equations of the withindeme dynamics are based on the assumption of a single founder allele being introduced to the deme at the time of environmental change. This sets an upper limit to migration m_{k}N < 1 in order to maintain the accuracy of our approximation. Higher levels of migration would in any case lead to a loss of the population structure that is of interest in the current model. Third, the metapopulation will go extinct if demes die at a greater rate than they colonize. Since the deme “death rate” is proportional to the rate of environmental change Ω and the “birth rate” is proportional to m_{k}N, we restrict to m_{k}N > Ω. Finally, as discussed in the model overview, since we assume all environmental change events lead to fixation of the B phenotype, we restrict our parameter space to Ns 1 and (m + μ_{irr})N ≥1.
3 Results
We computed the evolved switching rate, m_{evolved}, for a metapopulation with a migration m_{k}, deme size N, selection strength s, environmental change rate Ω, and irreversible mimic mutation rate μ_{irr}, using the algorithm from Appendix C. Within the parameter range restrictions described above, we found that s and N played little role (Figure 4), and we consequently focus on N = 10^{6} and s = 0.001. Here we examine how curves of m_{evolved} versus μ_{irr} depend on the model parameters Ω and m_{k}.
Environmental change rate Ω
When irreversible mimics are rare (μ_{irr} small), m_{evolved} ≈ Ω (Figure 5). This is the classic bethedging result described by Cohen (1966) in the absence of irreversible mimics. As μ_{irr} increases and the mimic appears more frequently, the reversible switching rate increases until it reaches a peak, then descends before the metapopulation goes extinct at very high irreversible mimic appearance rates. The curves for different values of Ω appear to parallel each other for most of the range, however lower Ω curves reach the peak slightly earlier than higher Ω.
This increase in m_{evolved} can be interpreted as selection for demes with higher m that are better able to avoid extinction. Once μ_{irr} is sufficiently high, there is no m_{evolved} that can avoid demes being dominated by B_{i} and therefore the entire metapopulation eventually goes extinct. This result is shown as white space at the right of the figure. The dropoff observed at high μ_{irr}, just before extinction, is a result of a high degree of nontransitivity, and as a result m_{evolved} is not welldefined in this region (see Appendix C for details).
m_{evolved} increases with population structure
A low migration rate m_{k} indicates increased population structure in our model. With more population structure, reversible switching m_{evolved} both rises above Ω for lower levels of μ_{irr}, and exceeds Ω by a larger margin for a given value of μ_{irr} (Figure 6). This is expected, since selection to avoid mimicdriven extinction acts at the deme level while individuallevel selection favors m_{evolved} ≈ Ω, and the extent of population structure affects the balance between the two. With high levels of migration, m_{evolved} ≈ Ω alleles can “outrun” extinction by continuing to colonize new demes, even as these demes suffer from frequent extinction. In the limit, when gene flow is very high (Nm_{k} 1), the metapopulation structure disappears and we have a single wellmixed population with m_{evolved} ≈ Ω. This single population is of course highly vulnerable to one large extinction event.
Modeling a finite number of demes is likely to weaken selection at the deme level by introducing random effects. We developed a finite deme version of the model (Appendix D) and in the limited number of test cases we examined, found similar results to the infinite deme model. In Figure 7 we show representative results for a transitive test case. Finite demes do not change the value of the optimum, and introduce only a modest amount of noise into the solution.
4 Discussion
Our model shows that a reversible switching system can evolve in the absence of environmental sensing despite the presence of irreversible mimics. Although mimics initially share the same adaptive phenotype, their irreversibility dooms them to extinction at the next environmental change event, allowing a longterm advantage that can be exploited by a reversible switching mechanism.
In contrast to previous work that neglects mimics (Lachmann and Jablonka 1996; Wolf et al. 2005; Kussell et al. 2005; Kussell and Leibler 2005; King and Masel 2007), we find that the evolved reversible switching rate (m_{evolved}) is not necessarily equal to the rate of environmental change (Ω). m_{evolved} increases significantly away from Ω as the irreversible mimic rate μ_{irr} increases. The critical μ_{irr} at which this departure from Ω occurs depends on the amount of gene flow between the demes in the metapopulation, captured in our model by the product Nm_{k}. Our model considers the parameter range Nm_{k} < 1 for which significant population structure exists.
Modeling assumptions
We have assumed a separation of timescales such that withindeme dynamics are instantaneous and so for the purposes of the metapopulation model, each deme is always dominated by one of the two possible genotypes. When the rate of environmental change Ω is small, this assumption is warranted as the transient dynamics of fixation and extinction will have completed by the time of the next environmental change.
We used the simple island model approximation of metapopulation dynamics to simplify migration patterns. However, we saw similar qualitative effects so long as some population structure existed, with the exact quantity of gene flow (Nm_{k}) affecting the magnitude. A second assumption of our island model is that there are infinite demes. Modeling a finite number of demes is likely to weaken selection at the deme level by introducing random effects. We therefore also examined a finite deme version of the model and found no appreciable change in our results. Note that from the perspective of the metapopulation model, each withindeme fixation event is instantaneous. This approximation might change the “effective” migration rate, perhaps even making it slightly different between the metapopulation and withindeme models.
We assumed that reversible and irreversible switches are equally able to meet the challenge of environmental change in the short term. A previous model by Masel and Bergman (2003) addressed the presence of irreversible mimics indirectly by defining environmental change as that leading to extinction unless reversible switching occurred. This implicitly assumes that if both reversible and irreversible mimic phenotypes appear in the population, then the mimics always lose in direct competition even in the shortterm. Here we allow each an equal chance of taking over the population in the shortterm. The disadvantage associated with mimics is instead captured indirectly through longterm extinction at the deme level. This approach therefore captures one of the key advantages of reversibility. Note that it is also possible that mimics do better than reversible phenotypes in the short term. For example, reversible phenotypes may suffer a cost from prematurely switching back. This could be captured through an extension of our model, and would lead to a higher “effective” irreversible mutation rate μ_{irr}
Note that if reversible switching is not random but induced at an elevated rate by the environment when it is most likely to be needed, then the evolution of reversible switching mechanisms becomes even more likely (Jablonka et al. 1995; Kussell and Leibler 2005; Wolf et al. 2005). Our assumption that switching is random is therefore conservative with respect to the evolution of reversible switching. Metabolic requirements to maintain environmental sensors may mean, however, that induced switching also has a cost (Wolf et al. 2005; Kussell and Leibler 2005) and random switching can be favored over direct sensing of the environment when environmental change rates are low (Kussell and Leibler 2005; Wolf et al. 2005).
Our modeling approach has three chief strengths. First, we represent all the dynamics occurring within a deme stochastically: this allows us to model both finite deme sizes and rare stochastic events. Second, all computation is done without recourse to individuallevel simulation, drastically reducing the amount of computation time needed for a given set of parameters. Third, our model examines grouplevel benefits that reversible switching mechanisms can confer on a metapopulation.
Acknowledgments
We thank Christine Lamanna for her early work on this project, Oliver D. King for C code, Cortland Griswold, Oliver King, Grant Peterson and Jessica Garb for helpful discussions, and the National Institutes of Health for funding (R01 GM076041). J.M. is a Pew Scholar in the Biomedical Sciences and an Alfred P. Sloan Research Fellow.
A Withindeme model equations
Both q(1, i), representing the model before the environmental change, and p(i), representing the model after the environmental change, can be computed by solving tridiagonal systems of linear equations using standard techniques.
Model before environmental change
q(1, i) is the probability that there are i M_{2} alleles at the time of an environment change event, assuming that there is initially one M_{2} allele appearing through mutation. It is given in section 2.4 of King and Masel (2007) by the following tridiagonal system of equations:
where α_{i} and β_{i} are the probabilities that the number of M_{2} alleles increase and decrease from i to i + 1 and i − 1, respectively (α and β replace the λ and μ symbols from King and Masel). To incorporate the effect of irreversible mimics, the computations of α_{i} and β_{i} need to be modified from King and Masel (2007). In the old environment E irreversible mimics increase the rate at which phenotype A switches to phenotype B. This means that M_{1} and M_{2} individuals in E now switch to B at rates m_{1} + μ_{irr} and m_{2} + μ_{irr}, respectively. As described in the main text, as some individuals immediately switch to the zerofitness B phenotype, the fitness of phenotype A is reduced to (1 − m_{1} − μ_{irr}) and (1 − m_{2} − μ_{irr}) for the M_{1} and M_{2} genotypes, respectively. Following the first equation in section 2.4 of King and Masel (2007) (modified by the substitutions m_{1} → m_{1} + μ_{irr} and m_{2} → m_{2} + μ_{irr}), the probability that we transition from i → i + 1 is given by the probability that an M_{2} individual is chosen to reproduce while an M_{1} individual is chosen to die:
Similarly, following the second equation in section 2.4 of King and Masel (2007), the probability that we transition from i → i − 1 is given by the probability that an M_{1} individual is chosen to reproduce while an M_{2} individual is chosen to die:
Model after environmental change
p(i) is the probability that a genotype with the M_{2} allele but no irreversible mimic becomes fixed, given that there are currently i M_{2} individuals in environmental F. Both M_{1} fixation and deme extinction are captured by the state i=0. M_{2} fixation may be due either to an M_{2} lineage with the adaptive phenotype B_{r} sweeping the population or to an M_{2} lineage with the A phenotype taking over by drift, possibly before the environment ever changes. We modify p(i) from section 2.5 of King and Masel (2007) to the following tridiagonal system of equations:
Equation (15) explicitly shows all the transition probabilities multiplied by the subsequent fixation probabilities. This includes those transitions which do not lead to M_{2} fixation, and which accordingly are multiplied by zero. We assume that the processes of fixation (after a B_{r} destined for fixation appears) and extinction (after a B_{i} destined for fixation appears) are instantaneous and therefore model both processes by introducing “jump” moves into the Markov chain. Each of these processes thus becomes a single step in the Markov chain (see King and Masel (2007) section 2.5 for details). In the first term, r_{i} represents the probability that an M_{2} with adaptive phenotype B_{r} sweeps the population, hence jumps to the p(N ) = 1 absorbing state. In the second term, ${r}_{i}^{\prime}$ represents that an M_{2} with the irreversible mimic phenotype B_{i} sweeps the population, and hence jumps to a state where the deme eventually goes extinct. The probability that an M_{1} allele with either B_{r} or B_{i} phenotype sweeps the population, and hences jumps to either M_{1} fixation or deme extinction, is given by d_{i}. The approximation of “jump” moves was numerically tested by King and Masel (2007) and found not to affect results. Note that in the current work this approximation also means that an adaptive B_{r} lineage does not subsequently acquire a B_{i} mutation. The dynamics of such mutational degradation were explored by Masel et al. (2007), and this phenomenon is not a problem for the parameters considered here.
Noting that the transition probability b_{i} of remaining in the p(i) state is given the sum of all other transition probabilities subtracted from 1, the coefficients a_{i}, b_{i}, c_{i}, d_{i}, r_{i}, ${r}_{i}^{\prime}$ in the above equations can be suitably modified from King and Masel (2007) to give
where y = (1 − (1 + s) ^{−1})/(1 − (1 + s) ^{−}^{N}) is the probability that a B individual is destined for fixation (see King and Masel (2007) section 2.5). Substituting in the values of the p(N) and p(0) reduces the system to
B Metapopulation model equations
Using standard techniques, four possible equilibrium solutions of equations (3) and (4) can be found. Equation (18) represents extinction of all demes, Equation (19) and Equation (20) represent dominance of all occupied demes by the M_{1} or M_{2} alleles respectively, and Equation (21) represents coexistence, where each allele dominates a fraction of demes.
Note that not all solutions apply to all parameter values. For a given set of parameter values, the first constraint we applied is that both P_{1} and P_{2}, and their sum P_{1} + P_{2}, must be bounded within the [0, 1] interval since they represent fractions of the total number of demes in the metapopulation model. After this constraint was used to eliminate potential solutions, we evaluated the stability of the remaining solution(s) by checking the signs of the derivatives about the equilibrium point. If none of the solutions in Equations (19), (20) or (21) were appropriately bounded or stable, then Equation (18), which represents extinction of the entire metapopulation, was assumed to be in effect.
C Algorithm for computing m_{evolved}
To compute m_{evolved}, we employed a Monte Carlo approach by competing pairs of switching rates in a series of rounds. In each round, the switching rate that “won” the pairwise comparison by the criterion in equation (22) would progress to the next round. Another randomly chosen switching rate close to the original winner would then be competed against that previous winner. We then found the evolved switching rate by computing a running average of the winning switching rate. Pseudocode describing the computation of a single replicate of m_{evolved} is found in Algorithm 1. For each data point in our figures, we then averaged m_{evolved} over 10 replicates of this algorithm to minimize noise introduced through the Monte Carlo sampling process. Note that in a typical Monte Carlo simulation, moves that decrease fitness are accepted with some low probability, in order to escape local optima and sample the entire parameter space. In our simulations, we suffered the opposite problem of lack of stability, and so this part of the classic algorithm was not done.
Algorithm 1

Similar to an ESS or to the criteria used by King and Masel (2007), an optimal reversible switching rate m_{opt} could be defined as that in which the corresponding M_{opt} allele outcompetes any other possible allele M. In our model, this corresponds to the condition
where P_{mopt} and P_{m} are the fraction of demes which are fixed for the allele of the respective switching rates.
If there is transitivity, there exists a unique solution to (22). If not, there may be no solution. Transitivity means if P_{ma}(m_{b}, m_{a}) > P_{mb}(m_{b}, m_{a}) and P_{mb}(m_{c}, m_{b}) > P_{mc}(m_{c}, m_{b}) then P_{ma}(m_{c}, m_{a}) > P_{mc}(m_{c}, m_{a}) must hold. Typically high degrees of nontransitivity are found for switching rates that are close together and for higher values of μ_{irr} (Figure 8). When μ_{irr} is very high, there is no single welldefined optimum, or fitness, and Algorithm 1 results in a final m_{evolved} that exhibits a “dropoff” from the peak value.
D Algorithm for finite deme model
Algorithm 2

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 The evolution of bethedging adaptations to rare scenarios.[Theor Popul Biol. 2007]King OD, Masel J. Theor Popul Biol. 2007 Dec; 72(4):56075. Epub 2007 Aug 31.
 Experimental evolution of bet hedging.[Nature. 2009]Beaumont HJ, Gallie J, Kost C, Ferguson GC, Rainey PB. Nature. 2009 Nov 5; 462(7269):903.
 Comparing environmental and genetic variance as adaptive response to fluctuating selection.[Evolution. 2011]Svardal H, Rueffler C, Hermisson J. Evolution. 2011 Sep; 65(9):2492513. Epub 2011 May 12.
 Timescales of genetic and epigenetic inheritance.[Cell. 2007]Rando OJ, Verstrepen KJ. Cell. 2007 Feb 23; 128(4):65568.
 Why are phenotypic mutation rates much higher than genotypic mutation rates?[Genetics. 2006]Bürger R, Willensdorfer M, Nowak MA. Genetics. 2006 Jan; 172(1):197206. Epub 2005 Sep 2.
 Pernicious Pathogens or Expedient Elements of Inheritance: The Significance of Yeast Prions[PLoS Pathogens. ]Byers JS, Jarosz DF. PLoS Pathogens. 10(4)e1003992
 Q&A: Evolutionary capacitance[BMC Biology. ]Masel J. BMC Biology. 11103
 Prions are a common mechanism for phenotypic inheritance in wild yeasts[Nature. ]Halfmann R, Jarosz DF, Jones SK, Chang A, Lancaster AK, Lindquist S. Nature. 482(7385)363368
 The Spontaneous Appearance Rate of the Yeast Prion [PSI+] and Its Implications for the Evolution of the Evolvability Properties of the [PSI+] System[Genetics. 2010]Lancaster AK, Bardill JP, True HL, Masel J. Genetics. 2010 Feb; 184(2)393400
 Complex Adaptations Can Drive the Evolution of the Capacitor [PSI+], Even with Realistic Rates of Yeast Sex[PLoS Genetics. 2009]Griswold CK, Masel J. PLoS Genetics. 2009 Jun; 5(6)e1000517
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