• We are sorry, but NCBI web applications do not support your browser and may not function properly. More information
Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Evolution. Author manuscript; available in PMC Sep 1, 2010.
Published in final edited form as:
Evolution. Sep 2009; 63(9): 2350–2362.
Published online May 21, 2009. doi:  10.1111/j.1558-5646.2009.00729.x
PMCID: PMC2770902
NIHMSID: NIHMS125946

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 DNA-based 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 switch-induced 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 bet-hedging 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 DNA-based “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 (mopt) 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 [dbl greater-than sign]1 and NΩ [dbl greater-than sign]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 well-documented phenocopies and genocopies in a wide range of taxa (reviewed in West-Eberhard 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 reversibly-induced variation when adaptive and costs when maladaptive are represented in a model of evolution at a modifier locus with two alleles, M1 and M2, that cause reversible switching at rates m1 and m2, 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 long-term threat from irreversible mimics (appearing at rate μirr). 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 μirr.

Figure 1
Summary of model: M1 and M2 are haploid genotypes at the reversible switching modifier locus; circles indicate phenotype A, adaptive in environment E, and squares indicate phenotype B, adaptive in environment F. Colored squares are irreversibly switched. ...

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 [dbl greater-than sign]1 where s is the selective advantage of B in environment E, and (m + μirr)N ≥1 where m + μirr is the total rate of appearance of the B phenotype. Each environmental change event then leads to fixation of either the reversible Br or the irreversible Bi within the deme (Figure 1b). If it is the Bi 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 long-term reasons related to loss of function. For mathematical simplicity, we assume that demes fixed for Bi 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 M1 or M2 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 Nmk) 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).

Figure 2
A single M2 individual migrates to a deme fixed for M1. The deme becomes fixed for M2 with probability pfix2. The pfix2 box is a visual representation of Equation (1) that includes the probability q(1, i) that there are i M2 alleles when the environment ...

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 (mevolved) we expect natural selection to favor. We then investigate how mevolved shifts in response to changes in the rate of appearance of irreversible mimics (μirr). The model is fully general and applies to any system that includes both evolved reversible phenotypic switching and intrinsic irreversible mimics.

Figure 7
m2 = 0.000398761 is the optimum switching rate in both infinite and finite numbers of demes for N = 105, Ω = 10−4, mk = 10−7, s = 0.01, μirr = 2×10−6. We competed this m2 against a variety of alternative ...

2.1 Population genetic model within each deme

We use a modified version of the within-deme 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 M1, M2 at a modifier locus. M1 and M2 alleles cause reversible switching from A to Br with rates m1 and m2, respectively. (We can ignore backswitching from phenotype Br to A for the purposes of the within-deme model, since in environment E Br 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 Bi phenotype are assumed to appear at rate μirr. Note that phenotype Bi is functionally identical to Br according to the within-deme model; its long-term 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 Bi or Br 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 Br. The reverse direction from F to E is implicit in extinction of Bi 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 Br and Bi have fitness zero in the original environment E, but a selective advantage in the new environment. Relative fitnesses are fAE = 1 and fBE = 0 in the old environment, and fAF = 1 and fBF = 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 (m1 and m2) of A individuals. In environment E, immediate replacement of B individuals means that the fitnesses of the M1 and M2 individuals are (1 − m1μirr) and (1 − m2μirr), respectively.

The probability that a single M2 allele will fix in a population of M1 alleles is given by (Figure 2; King and Masel 2007):

pfix(m1,m2,μirr,N,Ω,s)=i=0Nq(1,i)p(i)
(1)

where q(1, i) is the probability that there are i M2 alleles at the time of the next environment change event from E to F, given that there is initially one M2 allele; and p(i) is the probability that an individual bearing the M2 allele, which has not undergone irreversible switching, becomes fixed given that there i M2 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 μirr. 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 M1 allele and demes dominated by the M2 allele be given by P0, P1 and P2, respectively. The model can now be represented by the coupled differential equations:

dP0dt=cP0P1+e1P1cP2P0+e2P2
(2)

dP1dt=cP1P0e1P1+g21P2P1g12P1P2
(3)

dP2dt=cP2P0e2P2+g12P1P2g21P2P1
(4)

where M1 and M2 demes go extinct at rates e1 and e2, empty demes are colonized by M1 and M2 at rates cP1 and cP2, and M1 demes switch genotypes to M2 at rate g12P2 and M2 demes to M1 at rate g21P1. An overview is shown in Figure 3.

Figure 3
Schematic of metapopulation model showing the transition rates between demes of each type. Demes may be empty (frequency P0), fixed for M1(P1), fixed for M2(P2), or fixed for an irreversible mimic. We approximate the extinction of a deme fixed for the ...

Environmental change events occur independently in each deme in the metapopulation. Adaptation to the new environment F can be mediated either by Bi or by Br. However, demes fixed for Bi 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 Br and Bi are initially selectively neutral relative to each other, fixation probabilities are proportional to their appearance rates. Since extinction corresponds to Bi fixation, extinction rates for demes of types M1 and M2 are given by:

e1=Ωμirr(m1+μirr)
(5)

e2=Ωμirr(m2+μirr)
(6)

The rate of migration is equal to the probability that an individual migrates (mk) 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:

c(P1+P2)=mkN(P1+P2)
(7)

For a deme to change type, an individual must migrate (mkN ) from a deme of opposite type (P1 or P2) and take over, i.e. fix, once it arrives (computed from equation (1)). The probability that a single M1 migrant fixes in an M2 deme is given by pfix1 =[p with hat]fix(m2, m1, μirr, N, Ω, s), and the probability that a single M2 migrant fixes in an M1 deme is given by pfix2 = [p with hat]fix(m1, m2, μirr, N, Ω, s), hence M1 and M2 demes change types at rates:

g12P2=mkNP2pfix2
(8)

g21P1=mkNP1pfix1
(9)

The total fraction of demes must be unity (Po + P1 + P2 = 1) and so the system can be reduced to the two dimensional system:

dP1dt=mkNP1(1P1P2)Ωμirr(m1+μirr)P1+mkNP1P2(pfix1pfix2)
(10)

dP2dt=mkNP2(1P1P2)Ωμirr(m2+μirr)P2+mkNP1P2(pfix2pfix1)
(11)

Equilibrium solutions to these equations can be found by standard techniques, details are in Appendix B. We look at max[P1, P2] 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 counter-fixation 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 long-term evolved switching rate, mevolved.

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 σ2 = 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 (mevolved) 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 within-deme 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 mkN < 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 mkN, we restrict to mkN > 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 [dbl greater-than sign]1 and (m + μirr)N ≥1.

3 Results

We computed the evolved switching rate, mevolved, for a metapopulation with a migration mk, deme size N, selection strength s, environmental change rate Ω, and irreversible mimic mutation rate μirr, 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 = 106 and s = 0.001. Here we examine how curves of mevolved versus μirr depend on the model parameters Ω and mk.

Figure 4
(a) (NΩ= 1000, Nmk=0.1, N = 106) The strength of selection, s, has no effect within the parameter range Ns [dbl greater-than sign] 1 for which our model holds, therefore we set s such that Ns=1000 throughout the rest of this paper. (b) (Ω= 0.001, m ...

Environmental change rate Ω

When irreversible mimics are rare (μirr small), mevolved ≈ Ω (Figure 5). This is the classic bet-hedging result described by Cohen (1966) in the absence of irreversible mimics. As μirr 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.

Figure 5
mevolved increases as the appearance of irreversible mimics at rate μirr becomes significant. Ns =1000, Nmk=0.1 and N = 106.

This increase in mevolved can be interpreted as selection for demes with higher m that are better able to avoid extinction. Once μirr is sufficiently high, there is no mevolved that can avoid demes being dominated by Bi and therefore the entire metapopulation eventually goes extinct. This result is shown as white space at the right of the figure. The drop-off observed at high μirr, just before extinction, is a result of a high degree of non-transitivity, and as a result mevolved is not well-defined in this region (see Appendix C for details).

mevolved increases with population structure

A low migration rate mk indicates increased population structure in our model. With more population structure, reversible switching mevolved both rises above Ω for lower levels of μirr, and exceeds Ω by a larger margin for a given value of μirr (Figure 6). This is expected, since selection to avoid mimic-driven extinction acts at the deme level while individual-level selection favors mevolved ≈ Ω, and the extent of population structure affects the balance between the two. With high levels of migration, mevolved ≈ 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 (Nmk [dbl greater-than sign] 1), the metapopulation structure disappears and we have a single well-mixed population with mevolved ≈ Ω. This single population is of course highly vulnerable to one large extinction event.

Figure 6
Population structure, indicated by the migration rate mk, causes reversible switching to evolve to higher levels in order to outcompete mimics by a larger margin. N = 106, NΩ = 10 and Ns=1000.

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 long-term 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 (mevolved) is not necessarily equal to the rate of environmental change (Ω). mevolved increases significantly away from Ω as the irreversible mimic rate μirr increases. The critical μirr 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 Nmk. Our model considers the parameter range Nmk < 1 for which significant population structure exists.

Modeling assumptions

We have assumed a separation of timescales such that within-deme 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 (Nmk) 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 within-deme fixation event is instantaneous. This approximation might change the “effective” migration rate, perhaps even making it slightly different between the metapopulation and within-deme 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 short-term. Here we allow each an equal chance of taking over the population in the short-term. 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 μirr

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 individual-level simulation, drastically reducing the amount of computation time needed for a given set of parameters. Third, our model examines group-level 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 Within-deme 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 M2 alleles at the time of an environment change event, assuming that there is initially one M2 allele appearing through mutation. It is given in section 2.4 of King and Masel (2007) by the following tri-diagonal system of equations:

q(1,1)=1eΩ/N+eΩ/N(βj+1q(1,j+1)+(1αjβj)q(1,j))q(1,j)=eΩ/N(αj1q(1,j1)+βj+1q(1,j+1)+(1αjβj)q(1,j))forj=2,..,N2q(1,N1)=eΩ/N(αj1q(1,j1)+(1αjβj)q(1,j))q(1,N)=q(1,N1)eΩ/N1eΩ/NαN1
(12)

where αi and βi are the probabilities that the number of M2 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 αi and βi 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 M1 and M2 individuals in E now switch to B at rates m1 + μirr and m2 + μirr, respectively. As described in the main text, as some individuals immediately switch to the zero-fitness B phenotype, the fitness of phenotype A is reduced to (1 − m1μirr) and (1 − m2μirr) for the M1 and M2 genotypes, respectively. Following the first equation in section 2.4 of King and Masel (2007) (modified by the substitutions m1m1 + μirr and m2m2 + μirr), the probability that we transition from ii + 1 is given by the probability that an M2 individual is chosen to reproduce while an M1 individual is chosen to die:

αi=(1m2μirr)i(1m1μirr)(Ni)+(1m2μirr)ixNiN
(13)

Similarly, following the second equation in section 2.4 of King and Masel (2007), the probability that we transition from ii − 1 is given by the probability that an M1 individual is chosen to reproduce while an M2 individual is chosen to die:

βi=(1m1μirr)(Ni)(1m1μirr)(Ni)+(1m2μirr)ixiN
(14)

Model after environmental change

p(i) is the probability that a genotype with the M2 allele but no irreversible mimic becomes fixed, given that there are currently i M2 individuals in environmental F. Both M1 fixation and deme extinction are captured by the state i=0. M2 fixation may be due either to an M2 lineage with the adaptive phenotype Br sweeping the population or to an M2 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:

p(0)=0p(i)=rip(N)+rip(0)+aip(i1)+bip(i)+cip(i+1)+dip(0)fori=1,..,N1p(N)=1
(15)

Equation (15) explicitly shows all the transition probabilities multiplied by the subsequent fixation probabilities. This includes those transitions which do not lead to M2 fixation, and which accordingly are multiplied by zero. We assume that the processes of fixation (after a Br destined for fixation appears) and extinction (after a Bi 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, ri represents the probability that an M2 with adaptive phenotype Br sweeps the population, hence jumps to the p(N ) = 1 absorbing state. In the second term, ri represents that an M2 with the irreversible mimic phenotype Bi sweeps the population, and hence jumps to a state where the deme eventually goes extinct. The probability that an M1 allele with either Br or Bi phenotype sweeps the population, and hences jumps to either M1 fixation or deme extinction, is given by di. 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 Br lineage does not subsequently acquire a Bi 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 bi of remaining in the p(i) state is given the sum of all other transition probabilities subtracted from 1, the coefficients ai, bi, ci, di, ri, ri in the above equations can be suitably modified from King and Masel (2007) to give

ai=1N2[i(N1)(1m1μirr)+i(Ni)(m1+μirr)(1y)(Ni1)/(N1)+i2(m2+μirr)(1y)(Ni)/(N1)]ci=1N2[i(Ni)(1m2μirr)+i(Ni)(m2+μirr)(1y)(i1)/(N1)+(Ni)2(m1+μirr)(1y)i/(N1)]di=(Ni)(m1+μirr)y/Nri=im2y/Nri=iμirry/Nbi=1aici(ri+ri)di=1aicii(m2+μirr)y/N(Ni)(m1+μirr)y/N
(16)

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

p(0)=0p(i)=ri+aip(i1)+bip(i)+cip(i+1)fori=1,..,N1p(N)=1
(17)

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 M1 or M2 alleles respectively, and Equation (21) represents co-existence, where each allele dominates a fraction of demes.

P1=P2=0
(18)
P1=1μirrΩmkN(m1+μirr),P2=0
(19)
P1=0,P2=1μirrΩmkN(m2+μirr)
(20)
P1=1mk2N2(pfix1pfix2)2(μirrmkN(pfix2pfix1)Ωm2+μirr+mkN(mkN(pfix1pfix2)+mirr(1m1+μirr+1m2+μirr)Ω))P2=(μirr(m2+μirr)mkN(pfix1pfix2)Ω+mkN(m1(m2mkN(pfix1pfix2)+Ω))+μirr(μirrmkN(pfix2pfix1)+m2(mkN(pfix2pfix1)+Ω))))/((m1+μirr)(m2+μirr)mk2N2(pfix1pfix2)2)
(21)

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 P1 and P2, and their sum P1 + P2, 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 mevolved

To compute mevolved, 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 mevolved is found in Algorithm 1. For each data point in our figures, we then averaged mevolved 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

Algorithm for computing mevolved. In all runs, number of steps S=500, number of “burn-in” steps, Sburnin=20. The variance was set to σ2 = 0.1, except for the region of non-transitivity (high μirr) where it is dynamically increased up to a maximum of σ2 = 1.0. The larger mutation steps are required in the non-transitive region in order to avoid being trapped in local optima. To save computation time, once the running average mevolved converged to within a tolerance 0.005 of the mevolved of previous iteration in step (3) for at least 10 consecutive iterations, the loop could be exited (pseudocode not shown). We used a modified version of the Golden search algorithm found in Numeric Recipes (Press et al. 1992). In a regular Golden search, functions are evaluated at 3 points, and the choice of the next point depends on their numerical ordering. In our case, no consistent fitness function at a single value of m is available, only evolutionary comparisons between two competing values. However, when 2 values are each compared to the same third reference value, a 3-way ordering can still be calculated and the Golden search algorithm applied. This initial Golden search estimate reduces the computation time needed to run the Monte Carlo algorithm.

  1. mopt:= use a modified Golden ratio search over the [0, 1] interval.
  2. mevolved:= 0 (initialize running average)
  3. for t:= 1, S + Sburnin
    1. x:= sample from the normal distribution, N(0, σ2), with mean 0 and variance σ2 (mutation step)
    2. m:= exmopt (mutation steps are normally distributed on the log scale)
    3. if Pmopt > Pm (determine whether m or mopt “wins” according to equation (22))
      • mopt:= mopt(keep current mopt)
        else
      • mopt:= m (new optimum is found)
    4. if t > Sburnin (don’t start recording average until burn in complete)
      • mevolved:= mevolved + (moptm evolved)/(tSburnin)(compute running average)

Similar to an ESS or to the criteria used by King and Masel (2007), an optimal reversible switching rate mopt could be defined as that in which the corresponding Mopt allele outcompetes any other possible allele M. In our model, this corresponds to the condition

Pmopt(m,mopt)>Pm(m,mopt)m[0,1]andmmopt
(22)

where Pmopt and Pm 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 Pma(mb, ma) > Pmb(mb, ma) and Pmb(mc, mb) > Pmc(mc, mb) then Pma(mc, ma) > Pmc(mc, ma) must hold. Typically high degrees of non-transitivity are found for switching rates that are close together and for higher values of μirr (Figure 8). When μirr is very high, there is no single well-defined optimum, or fitness, and Algorithm 1 results in a final mevolved that exhibits a “drop-off” from the peak value.

Figure 8
(a) Drop-off of mevolved at high μirr results from non-transitive relationships. (b) Transitivity holds for μirr = 10−6 as shown in a grid of values where m2 beating m1 is depicted as a black box and m1 beating m2, an “x” ...

D Algorithm for finite deme model

Algorithm 2

Computing P2, the time-weighted average of P2, in a finite metapopulation of D demes for a given set of parameters m1, m2, Ω, mk, N, μirr. (Number of steps S = 5, 000, number of“burn-in”steps, Sburnin = 1, 000).

  1. m2:= use modified Golden ratio search portion of Algorithm 1 in Appendix C
  2. initialize the discrete deme type D0, D1, D2 based on the infinite solution
  3. initialize the current proportions: P1:= D1/D, P2:= D2/D, P0:= 1 − (P 1 + P 2 )
  4. P2total:=0 (initialize weighted total)
  5. t:= 0.0 (initialize time)
  6. for s:= 1, S + Sburnin
    1. recompute the rates for the six possible events (based on equations (10) and(11).)
      r1:= mkNP1P0; r2:= mkNP2P0 (colonization by M1and M2, respectively)
      r3:= ΩμirrP1/(μirr + m1); r4:= Ωμirr P2/(μirr + m2) (extinction of M1 and M2, respectively)
      r5:= mkNP2pfix2P1; r6:= mkNP1pfix1P2(deme switch M1M2 and M2M1, respectively)
    2. R:=i=16ri (compute the total rate)
    3. Δt:= 1/R (choose a new timestep based on the total rate)
    4. u:=sample from the uniform distribution [0, R] interval
    5. choose event i if u falls within the ri proportion of the total rate interval, R
    6. if event 1 then D1:= D1 + 1, Do:= D o − 1 (an empty deme is colonized with M1)
      else if event 2 then D2:= D2 + 1, Do:= Do − 1 (an empty deme is colonized with M2)
      else if event 3 then D1:= D1 − 1, Do:= Do + 1 (deme of type M1 goes extinct)
      else if event 4 then D2:= D2 − 1, Do:= Do + 1 (deme of type M2 goes extinct)
      else if event 5 then D1:= D1 − 1, D2:= D2 + 1 (deme switches from M1 to M2 )
      else if event 6 then D1:= D1 + 1, D2:= D2 − 1 (deme switches from M2 to M1 )
    7. P1:= D1/D, P2:= D2/D, P0:= Do/D (recompute proportions)
    8. if s > Sburnin (don’t start recording average until burn in complete, save this time as tburnin)
      • P2total:=P2total+ΔtP2 (update the weighted total, weighting the current P2 by the length of the timestep)
      • P¯2:=P2total/(ttburnin+Δt) (compute new time-weighted average)
    9. t:= t + Δt (update current time)

Literature Cited

  • Claussen JC, Traulsen A. Non-Gaussian fluctuations arising from finite populations: Exact results for the evolutionary Moran process. Phys Rev E Stat Nonlin Soft Matter Phys. 2005;71:025101. [PubMed]
  • Cohen D. Optimizing reproduction in a randomly varying environment. J Theor Biol. 1966;12:119–29. [PubMed]
  • Drake JW. The distribution of rates of spontaneous mutation over viruses, prokaryotes, and eukaryotes. Ann N Y Acad Sci. 1999;870:100–7. [PubMed]
  • Fudenberg D, Nowak MA, Taylor C, Imhof LA. Evolutionary game dynamics in finite populations with strong selection and weak mutation. Theor Popul Biol. 2006;70:352–63. [PMC free article] [PubMed]
  • Jablonka E, Lamb MJ. Epigenetic inheritance in evolution. J Evol Biol. 1998;11:159–183.
  • Jablonka E, Oborny B, Molnar I, Kisdi E, Hofbauer J, Czaran T. The adaptive advantage of phenotypic memory in changing environments. Philos Trans R Soc Lond B Biol Sci. 1995;350:133–41. [PubMed]
  • King OD, Masel J. The evolution of bet-hedging adaptations to rare scenarios. Theor Popul Biol. 2007 Dec;72(4):560–575. [PMC free article] [PubMed]
  • Kussell E, Kishony R, Balaban NQ, Leibler S. Bacterial persistence: a model of survival in changing environments. Genetics. 2005;169:1807–14. [PMC free article] [PubMed]
  • Kussell E, Leibler S. Phenotypic diversity, population growth, and information in fluctuating environments. Science. 2005;309:2075–8. [PubMed]
  • Lachmann M, Jablonka E. The inheritance of phenotypes: an adaptation to fluctuating environments. J Theor Biol. 1996;181:1–9. [PubMed]
  • Lancaster AK, Bardill JP, True HL, Masel J. The spontaneous appearance rates of both the yeast prion [PSI+] and of other [PSI+]-like phenotypes in prep.
  • Levins R. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America. 1969;15:237–240.
  • Levins R. Extinctions. Lect Notes Math. 1970;2:77–107.
  • Lund PM, Cox BS. Reversion analysis of [psi-] mutations in Saccharomyces cerevisiae. Genet Res. 1981;37:173–82. [PubMed]
  • Masel J. Evolutionary capacitance may be favored by natural selection. Genetics. 2005;170:1359–71. [PMC free article] [PubMed]
  • Masel J, Bergman A. The evolution of the evolvability properties of the yeast prion [PSI+] Evolution. 2003;57:1498–512. [PubMed]
  • Masel J, King OD, Maughan H. The loss of adaptive plasticity during long periods of environmental stasis. Am Nat. 2007;169:38–46. [PMC free article] [PubMed]
  • Maynard Smith J. Evolution and the Theory of Games. Cambridge: Cambridge University Press; 1982.
  • Maynard Smith J, Price G. The logic of animal conflict. Nature. 1973;246:15–18.
  • Moxon ER, Rainey PB, Nowak MA, Lenski RE. Adaptive evolution of highly mutable loci in pathogenic bacteria. Curr Biol. 1994;4:24–33. [PubMed]
  • Philippi T. Bet-hedging germination of desert annuals: beyond the first year. Am Nat. 1993;142:474–487. [PubMed]
  • Press W, Teukolsky S, Vetterling W, Flannery B. Numerical Recipes in C: The Art of Scientific Computing. 2. Cambridge University Press; Cambridge, UK: 1992.
  • Rando OJ, Verstrepen KJ. Timescales of genetic and epigenetic inheritance. Cell. 2007;128:655–68. [PubMed]
  • Seger J, Brockman HJ. What is bet-hedging? In: Harvey PH, Partridge L, editors. Oxford Surveys in Evolutionary Biology. Oxford University Press; 1987. pp. 182–211.
  • West-Eberhard MJ. Developmental Plasticity and Evolution. Oxford University Press; 2003.
  • Wolf DM, V, Vazirani V, Arkin AP. Diversity in times of adversity: probabilistic strategies in microbial survival games. J Theor Biol. 2005;234:227–53. [PubMed]

Formats:

Related citations in PubMed

See reviews...See all...

Cited by other articles in PMC

See all...

Links

Recent Activity

Your browsing activity is empty.

Activity recording is turned off.

Turn recording back on

See more...