Expected relative fitness and the adaptive topography of fluctuating selection

Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, . The expected evolution of p or is a product of genetic variability and the gradient of the long-run growth rate of the population, , with respect to p or . This shows that the expected evolution maximizes , the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, as a function of p or represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of . Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.     

Evolutionary tracking is determined by differential selection on demographic rates and density dependence

Recent ecological forecasts predict that ~25% of species worldwide will go extinct by 2050. However, these estimates are primarily based on environmental changes alone and fail to incorporate important biological mechanisms such as genetic adaptation via evolution. Thus, environmental change can affect population dynamics in ways that classical frameworks can neither describe nor predict. Furthermore, often due to a lack of data, forecasting models commonly describe changes in population demography by summarizing changes in fecundity and survival concurrently with the intrinsic growth rate (r). This has been shown to be an oversimplification as the environment may impose selective pressure on specific demographic rates (birth and death) rather than directly on r (the difference between the birth and death rates). This differential pressure may alter population response to density, in each demographic rate, further diluting the information combined to produce r. Thus, when we consider the potential for persistence via adaptive evolution, populations with the same r can have different abilities to persist amidst environmental change. Therefore, we cannot adequately forecast population response to climate change without accounting for demography and selection on density dependence. Using a continuous‐time Markov chain model to describe the stochastic dynamics of the logistic model of population growth and allow for trait evolution via mutations arising during birth events, we find persistence via evolutionary tracking more likely when environmental change alters birth rather than the death rate. Furthermore, species that evolve responses to changes in the strength of density dependence due to environmental change are less vulnerable to extinction than species that undergo selection independent of population density. By incorporating these key demographic considerations into our predictive models, we can better understand how species will respond to climate change.     

WHEN DO ADAPTIVE PLASTICITY AND GENETIC EVOLUTION PREVENT EXTINCTION OF A DENSITY‐REGULATED POPULATION?

We study the dynamics of evolutionary recovery after an abrupt environmental shift in a density‐regulated population with evolving plasticity. Maladaptation to the new environment initially causes the population to decline, until adaptive phenotypic plasticity and genetic evolution restore positive population growth rate. We assume that selection on a quantitative trait is density‐independent and that the initial cost of plasticity is much lower than the benefit of the initial plastic response. The initial partially adaptive plasticity reduces the effective magnitude of the environmental shift, whereas evolution of plasticity increases the rate of adaptation. Both effects greatly facilitate population persistence. In contrast, density dependence of population growth always hinders persistence. With θ‐logistic population regulation, a lower value of θ produces a faster initial population decline and a higher extinction risk.     

Hysteresis Can Grant Fitness in Stochastically Varying Environment

Although the existence of multiple stable phenotypes of living organisms enables random switching between phenotypes as well as non-random history dependent switching called hysteresis, only random switching has been considered in prior experimental and theoretical models of adaptation to variable environments. This work considers the possibility that hysteresis may also evolve together with random phenotype switching to maximize population growth. In addition to allowing the possibility that switching rates between different phenotypes may depend not only on a continuous environmental input variable, but also on the phenotype itself, the present work considers an opportunity cost of the switching events. This opportunity cost arises as a result of a lag phase experimentally observed after phenotype switching and stochastic behavior of the environmental input. It is shown that stochastic environmental variation results in maximal asymptotic growth rate when organisms display hysteresis for sufficiently slowly varying environmental input. At the same time, sinusoidal input does not cause evolution of memory suggesting that the connection between the lag phase, stochastic environmental variation and evolution of hysteresis is a result of a stochastic resonance type phenomenon.     

Adaptation of timing of life history traits and population dynamic responses to climate change in spatially structured populations

Changes in the seasonal timing of life history events are documented effects of climate change. We used a general model to study how dispersal and competitive interactions affect eco-evolutionary responses to changes in the temporal distribution of resources over the season. Specifically, we modeled adaptation of the timing of reproduction and population dynamic responses in two competing populations that disperse between two habitats characterized by an early and late resource peak. We investigated three scenarios of environmental change: (1) food peaks advance in both habitats, (2) in the late habitat only and (3) in the early habitat only. At low dispersal rates the evolutionarily stable timing of reproduction closely matched the local resource peak and the environmental change typically caused population decline. Larger dispersal rates rendered less intuitive eco-evolutionary population responses. First, dispersal caused mismatch between evolutionarily stable timing of reproduction and local resource peaks and as a result, reproductive output for subpopulations could increase as well as decrease when resource availability underwent temporal shifts. Second, population responses were contingent on competition between populations. This could accelerate population declines and cause extinctions or even reverse population trends from negative to positive compared to the low dispersal case. When dispersal rate was large and the early resource peak was advanced available niche space was reduced. Hence, even when a population survived the environmental change and obtained positive equilibrium population density, subsequent adaptation of competing populations could drive it to extinction due to convergent evolution and competitive exclusion. These results shed new light on the role of competition and dispersal for the evolution of timing of life history events and provide guidelines for understanding short and long-term population response to climate change.     

Potential rapid evolution of foot morphology in Italian plethodontid salamanders (Hydromantes strinatii) following the colonization of an artificial cave

How organisms respond to environmental change is a long‐standing question in evolutionary biology. Species invading novel habitats provide an opportunity to examine contemporary evolution in action and decipher the pace of evolutionary change over short timescales. Here, we characterized phenotypic evolution in the Italian plethodontid salamander, Hydromantes strinatii, following the recent colonization of an artificial cave by a forest floor population. When compared with a nearby and genetically related population in the natural forest floor and a nearby cave population, the artificial cave population displayed significant differences in overall foot shape, with more interdigital webbing relative to the other populations. Further, this population evolved significantly larger feet, which corresponded more closely to those found in other cave populations than to forest floor populations to which the cave population is closely related. Finally, we quantified the rate of evolution for both foot shape and foot area, and found that both traits displayed large and significant evolutionary rates, at levels corresponding to other classic cases of rapid evolution in vertebrates. Together, these findings reveal that the response to novel environmental pressures can be large and rapid and that the anatomical shifts observed in the artificial cave population of H. strinatii may represent a case of rapid evolution in response to novel environmental pressures.     

Inclusive fitness for traits affecting metapopulation demography

Defining computable analytical measures of the effects of selection in populations with demographic and environmental stochasticity is a long-standing problem. We derive an analytical measure which takes in account all consequences of the discrete nature of deme size. Expressions of this measure are detailed for infinite island models of population structure. As an illustration we consider the evolution of dispersal in populations made of small demes with environmental and demographic stochasticity. We confirm some results obtained from the analysis of models based on deterministic approximations. In particular, when there is an Allee effect, we show that evolution of the dispersal rate may lead the metapopulation to extinction. Thus, selection on the dispersal rate could restrict the distribution of species subject to Allee effects. This selection-driven extinction is prevented by kin selection when the environmental extinction rate is small.     

SLOWLY SWITCHING BETWEEN ENVIRONMENTS FACILITATES REVERSE EVOLUTION IN SMALL POPULATIONS

Natural populations must constantly adapt to ever‐changing environmental conditions. A particularly interesting question is whether such adaptations can be reversed by returning the population to an ancestral environment. Such evolutionary reversals have been observed in both natural and laboratory populations. However, the factors that determine the reversibility of evolution are still under debate. The time scales of environmental change vary over a wide range, but little is known about how the rate of environmental change influences the reversibility of evolution. Here, we demonstrate computationally that slowly switching between environments increases the reversibility of evolution for small populations that are subject to only modest clonal interference. For small populations, slow switching reduces the mean number of mutations acquired in a new environment and also increases the probability of reverse evolution at each of these “genetic distances.” As the population size increases, slow switching no longer reduces the genetic distance, thus decreasing the evolutionary reversibility. We confirm this effect using both a phenomenological model of clonal interference and also a Wright–Fisher stochastic simulation that incorporates genetic diversity. Our results suggest that the rate of environmental change is a key determinant of the reversibility of evolution, and provides testable hypotheses for experimental evolution.     

Evolutionary rescue under demographic and environmental stochasticity

Populations suffer two types of stochasticity: demographic stochasticity, from sampling error in offspring number, and environmental stochasticity, from temporal variation in the growth rate. By modelling evolution through phenotypic selection following an abrupt environmental change, we investigate how genetic and demographic dynamics, as well as effects on population survival of the genetic variance and of the strength of stabilizing selection, differ under the two types of stochasticity. We show that population survival probability declines sharply with stronger stabilizing selection under demographic stochasticity, but declines more continuously when environmental stochasticity is strengthened. However, the genetic variance that confers the highest population survival probability differs little under demographic and environmental stochasticity. Since the influence of demographic stochasticity is stronger when population size is smaller, a slow initial decline of genetic variance, which allows quicker evolution, is important for population persistence. In contrast, the influence of environmental stochasticity is population-size-independent, so higher initial fitness becomes important for survival under strong environmental stochasticity. The two types of stochasticity interact in a more than multiplicative way in reducing the population survival probability. Our work suggests the importance of explicitly distinguishing and measuring the forms of stochasticity during evolutionary rescue.     

Cryptic Evolution: Does Environmental Deterioration Have a Genetic Basis?

Cryptic evolution has been defined as adaptive evolutionary change being masked by concurrent environmental change. Empirical studies of cryptic evolution have usually invoked a changing climate and/or increasing population density as the form of detrimental environmental change experienced by a population undergoing cryptic evolution. However, Fisher (1958) emphasized that evolutionary change in itself is likely to be an important component of "environmental deterioration," a point restated by Cooke et al. (1990) in the context of intraspecific competition. In this form, environmental deterioration arises because a winning lineage has to compete against more winners in successive generations as the population evolves. This "evolutionary environmental deterioration" has different implications for the selection and evolution of traits influenced by resource competition than general environmental change. We reformulate Cooke's model as a quantitative genetic model to show that it is identical in form to more recent developments proposed by quantitative geneticists. This provides a statistical framework for discriminating between the alternative hypotheses of environmental change and environmental deterioration caused by evolutionary change. We also demonstrate that in systems where no phenotypic change has occurred, there are many reasonable biological processes that will generate patterns in predicted breeding values that are consistent with what has been interpreted as cryptic evolution, and care needs to be taken when interpreting these patterns. These processes include mutation, sib competition, and invisible fractions.     

Rapid evolution of a consumer stoichiometric trait destabilizes consumer–producer dynamics

Recent studies have shown that adaptive evolution can be rapid enough to affect contemporary ecological dynamics in nature (i.e. ‘rapid evolution’). These studies tend to focus on trait functions relating to interspecific interactions; however, the importance of rapid evolution of stoichiometric traits has been relatively overlooked. Various traits can affect the balance of elements (carbon, nitrogen, and phosphorus) of organisms, and rapid evolution of such stoichiometric traits will not only alter population and community dynamics but also influence ecosystem functions such as nutrient cycling. Multiple environmental changes may exert a selection pressure leading to adaptation of stoichiometrically important traits, such as an organism's growth rate. In this paper, we use theoretical approaches to explore the connections between rapid evolution and ecological stoichiometry at both the population and ecosystem level. First, we incorporate rapid evolution into an ecological stoichiometry model to investigate the effects of rapid evolution of a consumer's stoichiometric phosphorus:carbon ratio on consumer–producer population dynamics. We took two complementary approaches, an asexual clonal genotype model and a quantitative genetic model. Next, we extended these models to explicitly track nutrients in order to evaluate the effect of rapid evolution at the ecosystem level. Our model results indicate rapid evolution of the consumer stoichiometric trait can cause complex dynamics where rapid evolution destabilizes population dynamics and rescues the consumer population from extinction (evolutionary rescue). The model results also show that rapid evolution may influence the level of nutrients available in the environment and the flux of nutrients across trophic levels. Our study represents an important step for theoretical integration of rapid evolution and ecological stoichiometry.     

A POPULATION GENETIC THEORY OF CANALIZATION

Canalization is the suppression of phenotypic variation. Depending on the causes of phenotypic variation, one speaks either of genetic or environmental canalization. Genetic canalization describes insensitivity of a character to mutations, and the insensitivity to environmental factors is called environmental canalization. Genetic canalization is of interest because it influences the availability of heritable phenotypic variation to natural selection, and is thus potentially important in determining the pattern of phenotypic evolution. In this paper a number of population genetic models are considered of a quantitative character under stabilizing selection. The main purpose of this study is to define the population genetic conditions and constraints for the evolution of canalization. Environmental canalization is modeled as genotype specific environmental variance. It is shown that stabilizing selection favors genes that decrease environmental variance of quantitative characters. However, the theoretical limit of zero environmental variance has never been observed. Of the many ways to explain this fact, two are addressed by our model. It is shown that a “canalization limit” is reached if canalizing effects of mutations are correlated with direct effects on the same character. This canalization limit is predicted to be independent of the strength of stabilizing selection, which is inconsistent with recent experimental data (Sterns et al. 1995). The second model assumes that the canalizing genes have deleterious pleiotropic effects. If these deleterious effects are of the same magnitude as all the other mutations affecting fitness very strong stabilizing selection is required to allow the evolution of environmental canalization. Genetic canalization is modeled as an influence on the average effect of mutations at a locus of other genes. It is found that the selection for genetic canalization critically depends on the amount of genetic variation present in the population. The more genetic variation, the stronger the selection for canalizing effects. All factors that increase genetic variation favor the evolution of genetic canalization (large population size, high mutation rate, large number of genes). If genetic variation is maintained by mutation-selection balance, strong stabilizing selection can inhibit the evolution of genetic canalization. Strong stabilizing selection eliminates genetic variation to a level where selection for canalization does not work anymore. It is predicted that the most important characters (in terms of fitness) are not necessarily the most canalized ones, if they are under very strong stabilizing selection (k > 0.2V_e). The rate of decrease of mutational variance V_m is found to be less than 10% of the initial V_m. From this result it is concluded that characters with typical mutational variances of about 10^–3 V_e are in a metastable state where further evolution of genetic canalization is too slow to be of importance at a microevolutionary time scale. The implications for the explanation of macroevolutionary patterns are discussed.     

Evolution of density- and patch-size-dependent dispersal rates

Based on a marginal value approach, we derive a nonlinear expression for evolutionarily stable (ES) dispersal rates in a metapopulation with global dispersal. For the general case of density-dependent population growth, our analysis shows that individual dispersal rates should decrease with patch capacity and-beyond a certain threshold-increase with population density. We performed a number of spatially explicit, individual-based simulation experiments to test these predictions and to explore further the relevance of variation in the rate of population increase, density dependence, environmental fluctuations and dispersal mortality on the evolution of dispersal rates. They confirm the predictions of our analytical approach. In addition, they show that dispersal rates in metapopulations mostly depend on dispersal mortality and inter-patch variation in population density. The latter is dominantly driven by environmental fluctuations and the rate of population increase. These conclusions are not altered by the introduction of neighbourhood dispersal. With patch capacities in the order of 100 individuals, kin competition seems to be of negligible importance for ES dispersal rates except when overall dispersal rates are low.     

Evolutionary rescue can prevent extinction following environmental change

The ubiquity of global change and its impacts on biodiversity poses a clear and urgent challenge for evolutionary biologists. In many cases, environmental change is so widespread and rapid that individuals can neither accommodate to them physiologically nor migrate to a more favourable site. Extinction will ensue unless the population adapts fast enough to counter the rate of decline. According to theory, whether populations can be rescued by evolution depends upon several crucial variables: population size, the supply of genetic variation, and the degree of maladaptation to the new environment. Using techniques in experimental evolution we tested the conditions for evolutionary rescue (ER). Hundreds of yeast populations were exposed to normally lethal concentrations of salt in conditions, where the frequency of rescue mutations was estimated and population size was manipulated. In a striking match with theory, we show that ER is possible, and that the recovery of the population may occur within 25 generations. We observed a clear threshold in population size for ER whereby the ancestral population size must be sufficiently large to counter stochastic extinction and contain resistant individuals. These results demonstrate that rapid evolution is an important component of the response of small populations to environmental change.     

The timescale of environmental fluctuations determines the competitive advantages of phenotypic plasticity and rapid evolution

Organisms can respond to fluctuating environments by phenotypic plasticity and rapid evolution, both occurring on similar timescales to the environmental fluctuations. Because each adaptation mechanism has been independently studied, the effects of different adaptation mechanisms on ecological dynamics are not well understood. Here, using mathematical modeling, we compared the advantages of phenotypic plasticity and rapid evolution under conditions where the environment fluctuated between two states on various timescales. The results indicate that the advantages of phenotypic plasticity under environmental fluctuations on different timescales depend on the cost and the speed of plasticity. Both the speed of plastic adaptation and the cost of plasticity affect competition results, while the quantitative effects of them vary depending on the timescales. When the environment fluctuates on short timescales, the two populations with evolution and plasticity coexist, although the population with evolution is dominant. On moderate timescales, the two populations also coexist; however, the population with plasticity becomes dominant. On long timescales, whether the population with phenotypic plasticity or evolution is more advantageous depended on the cost of plasticity. Moreover, our results indicate that the mechanisms resulting in the dominance of the plastic population over the population with evolution are different depending on the timescales of environmental fluctuations. Therefore, the timescales of environmental fluctuations deserve more attention if we are to better understand the detailed competition results underlying phenotypic variation.     

Predicting population genetic change in an autocorrelated random environment: Insights from a large automated experiment

Most natural environments exhibit a substantial component of random variation, with a degree of temporal autocorrelation that defines the color of environmental noise. Such environmental fluctuations cause random fluctuations in natural selection, affecting the predictability of evolution. But despite long-standing theoretical interest in population genetics in stochastic environments, there is a dearth of empirical estimation of underlying parameters of this theory. More importantly, it is still an open question whether evolution in fluctuating environments can be predicted indirectly using simpler measures, which combine environmental time series with population estimates in constant environments. Here we address these questions by using an automated experimental evolution approach. We used a liquid-handling robot to expose over a hundred lines of the micro-alga Dunaliella salina to randomly fluctuating salinity over a continuous range, with controlled mean, variance, and autocorrelation. We then tracked the frequencies of two competing strains through amplicon sequencing of nuclear and choloroplastic barcode sequences. We show that the magnitude of environmental fluctuations (determined by their variance), but also their predictability (determined by their autocorrelation), had large impacts on the average selection coefficient. The variance in frequency change, which quantifies randomness in population genetics, was substantially higher in a fluctuating environment. The reaction norm of selection coefficients against constant salinity yielded accurate predictions for the mean selection coefficient in a fluctuating environment. This selection reaction norm was in turn well predicted by environmental tolerance curves, with population growth rate against salinity. However, both the selection reaction norm and tolerance curves underestimated the variance in selection caused by random environmental fluctuations. Overall, our results provide exceptional insights into the prospects for understanding and predicting genetic evolution in randomly fluctuating environments.     

The Population and Evolutionary Dynamics of Homologous Gene Recombination in Bacteria

In bacteria, recombination is a rare event, not a part of the reproductive process. Nevertheless, recombination—broadly defined to include the acquisition of genes from external sources, i.e., horizontal gene transfer (HGT)—plays a central role as a source of variation for adaptive evolution in many species of bacteria. Much of niche expansion, resistance to antibiotics and other environmental stresses, virulence, and other characteristics that make bacteria interesting and problematic, is achieved through the expression of genes and genetic elements obtained from other populations of bacteria of the same and different species, as well as from eukaryotes and archaea. While recombination of homologous genes among members of the same species has played a central role in the development of the genetics and molecular biology of bacteria, the contribution of homologous gene recombination (HGR) to bacterial evolution is not at all clear. Also, not so clear are the selective pressures responsible for the evolution and maintenance of transformation, the only bacteria-encoded form of HGR. Using a semi-stochastic simulation of mutation, recombination, and selection within bacterial populations and competition between populations, we explore (1) the contribution of HGR to the rate of adaptive evolution in these populations and (2) the conditions under which HGR will provide a bacterial population a selective advantage over non-recombining or more slowly recombining populations. The results of our simulation indicate that, under broad conditions: (1) HGR occurring at rates in the range anticipated for bacteria like Streptococcus pneumoniae, Escherichia coli, Haemophilus influenzae, and Bacillus subtilis will accelerate the rate at which a population adapts to environmental conditions; (2) once established in a population, selection for this capacity to increase rates of adaptive evolution can maintain bacteria-encoded mechanisms of recombination and prevent invasion of non-recombining populations, even when recombination engenders a modest fitness cost; and (3) because of the density- and frequency-dependent nature of HGR in bacteria, this capacity to increase rates of adaptive evolution is not sufficient as a selective force to provide a recombining population a selective advantage when it is rare. Under realistic conditions, homologous gene recombination will increase the rate of adaptive evolution in bacterial populations and, once established, selection for higher rates of evolution will promote the maintenance of bacteria-encoded mechanisms for HGR. On the other hand, increasing rates of adaptive evolution by HGR is unlikely to be the sole or even a dominant selective pressure responsible for the original evolution of transformation.     

The combined effects of temporal autocorrelation and the costs of plasticity on the evolution of plasticity

Adaptive phenotypic plasticity is an important source of intraspecific variation, and for many plastic traits, the costs or factors limiting plasticity seem cryptic. However, there are several different factors that may constrain the evolution of plasticity, but few models have considered costs and limiting factors simultaneously. Here we use a simulation model to investigate how the optimal level of plasticity in a population depends on a fixed maintenance fitness cost for plasticity or an incremental fitness cost for producing a plastic response in combination with environmental unpredictability (environmental fluctuation speed) limiting plasticity. Our model identifies two mechanisms that act, almost separately, to constrain the evolution of plasticity: (i) the fitness cost of plasticity scaled by the nonplastic environmental tolerance, and (ii) the environmental fluctuation speed scaled by the rate of phenotypic change. That is, the evolution of plasticity is constrained by the high cost of plasticity in combination with high tolerance for environmental variation, or fast environmental changes in combination with slow plastic response. Qualitatively similar results are found when maintenance and incremental fitness costs of plasticity are incorporated, although a larger degree of plasticity is selected for with an incremental cost. Our model highlights that it is important to consider direct fitness costs and phenotypic limitations in relation to nonplastic environmental tolerance and environmental fluctuations, respectively, to understand what constrains the evolution of phenotypic plasticity.     

The nearly neutral and selection theories of molecular evolution under the fisher geometrical framework: substitution rate, population size, and complexity

The general theories of molecular evolution depend on relatively arbitrary assumptions about the relative distribution and rate of advantageous, deleterious, neutral, and nearly neutral mutations. The Fisher geometrical model (FGM) has been used to make distributions of mutations biologically interpretable. We explored an FGM-based molecular model to represent molecular evolutionary processes typically studied by nearly neutral and selection models, but in which distributions and relative rates of mutations with different selection coefficients are a consequence of biologically interpretable parameters, such as the average size of the phenotypic effect of mutations and the number of traits (complexity) of organisms. A variant of the FGM-based model that we called the static regime (SR) represents evolution as a nearly neutral process in which substitution rates are determined by a dynamic substitution process in which the population's phenotype remains around a suboptimum equilibrium fitness produced by a balance between slightly deleterious and slightly advantageous compensatory substitutions. As in previous nearly neutral models, the SR predicts a negative relationship between molecular evolutionary rate and population size; however, SR does not have the unrealistic properties of previous nearly neutral models such as the narrow window of selection strengths in which they work. In addition, the SR suggests that compensatory mutations cannot explain the high rate of fixations driven by positive selection currently found in DNA sequences, contrary to what has been previously suggested. We also developed a generalization of SR in which the optimum phenotype can change stochastically due to environmental or physiological shifts, which we called the variable regime (VR). VR models evolution as an interplay between adaptive processes and nearly neutral steady-state processes. When strong environmental fluctuations are incorporated, the process becomes a selection model in which evolutionary rate does not depend on population size, but is critically dependent on the complexity of organisms and mutation size. For SR as well as VR we found that key parameters of molecular evolution are linked by biological factors, and we showed that they cannot be fixed independently by arbitrary criteria, as has usually been assumed in previous molecular evolutionary models.     

The evolution of bet-hedging adaptations to rare scenarios

When faced with a variable environment, organisms may switch between different strategies according to some probabilistic rule. In an infinite population, evolution is expected to favor the rule that maximizes geometric mean fitness. If some environments are encountered only rarely, selection may not be strong enough for optimal switching probabilities to evolve. Here we calculate the evolution of switching probabilities in a finite population by analyzing fixation probabilities of alleles specifying switching rules. We calculate the conditions required for the evolution of phenotypic switching as a form of bet-hedging as a function of the population size N, the rate theta at which a rare environment is encountered, and the selective advantage s associated with switching in the rare environment. We consider a simplified model in which environmental switching and phenotypic switching are one-way processes, and mutation is symmetric and rare with respect to the timescale of fixation events. In this case, the approximate requirements for bet-hedging to be favored by a ratio of at least R are that sN>log(R) and thetaN>square root R .