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Philos Trans R Soc Lond B Biol Sci. 2013 January 19; 368(1610): 20120088.

PMCID: PMC3538454

Guillaume Martin,^{1,}^{2}^{} Robin Aguilée,^{1,}^{2} Johan Ramsayer,^{1,}^{2} Oliver Kaltz,^{1,}^{2} and Ophélie Ronce^{1,}^{2}

e-mail: rf.2ptnom-vinu@nitram.emualliug

One contribution of 15 to a Theme Issue ‘Evolutionary rescue in changing environments’.

Copyright © 2012 The Author(s) Published by the Royal Society. All rights reserved.

This article has been cited by other articles in PMC.

Evolutionary rescue occurs when a population genetically adapts to a new stressful environment that would otherwise cause its extinction. Forecasting the probability of persistence under stress, including emergence of drug resistance as a special case of interest, requires experimentally validated quantitative predictions. Here, we propose general analytical predictions, based on diffusion approximations, for the probability of evolutionary rescue. We assume a narrow genetic basis for adaptation to stress, as is often the case for drug resistance. First, we extend the rescue model of Orr & Unckless (*Am. Nat.* 2008 **172**, 160–169) to a broader demographic and genetic context, allowing the model to apply to empirical systems with variation among mutation effects on demography, overlapping generations and bottlenecks, all common features of microbial populations. Second, we confront our predictions of rescue probability with two datasets from experiments with *Saccharomyces cerevisiae* (yeast) and *Pseudomonas fluorescens* (bacterium). The tests show the qualitative agreement between the model and observed patterns, and illustrate how biologically relevant quantities, such as the *per capita* rate of rescue, can be estimated from fits of empirical data. Finally, we use the results of the model to suggest further, more quantitative, tests of evolutionary rescue theory.

Forecasts of future rates of species extinction are three to four orders of magnitude higher than known background rates of extinction in the fossil record [1]. Such forecasts of biodiversity loss have been criticized for not taking into account the capacity of organisms to adapt to their changing environment [2]. Evolutionary rescue describes the process by which a population, initially confronted with an environment causing its decline, is saved from extinction through genetic changes that recover growth. Emergence of resistance to chemotherapy (antibiotics, antivirals, pesticides, etc.) is also an important example of evolutionary rescue, well studied both empirically (reviewed in MacLean *et al*. [3]) and theoretically [4]. Several theoretical models have addressed the joint evolutionary and demographic processes leading to evolutionary rescue when the environment deteriorates gradually [5,6] or abruptly [6–8]. The very same process has also been modelled in more epidemiologically oriented models [4]. Rescue or demise depends on a race between population decline and adaptation: genotypes that adapt the population to the new environment must reach a substantial frequency before the population becomes extinct. These models predict that the probability of evolutionary rescue decreases with stress intensity and increases with initial population size or with the abundance of genetic variation available to fuel adaptation to the new conditions (reviewed in Bell [7]).

Forecasting extinction requires the development of a validated quantitative theory of evolutionary rescue. This includes, as a case of special interest, forecasting the emergence of resistance in diseases and pests affecting human health and economy. Current models of evolutionary rescue make different quantitative predictions about the probability of evolutionary rescue because of differences in assumptions regarding the genetic basis of adaptation to stress, and the stochastic process governing population dynamics. For instance, the pioneering model of Gomulkiewicz & Holt [8] used the infinitesimal model from quantitative genetics to describe the genetic basis for fitness, which is more appropriate when adaptation to stress is caused by a large pool of alleles at many loci, already present in the population at the onset of stress. Orr & Unckless [9] studied evolutionary rescue in the opposite case where adaptation is conveyed by a single genetic variant. A narrow genetic basis for adaptation has indeed been found in many cases of drug resistance [3,10]. They further compared (see also Ribeiro & Bonhoeffer [4]) the contribution from genetic variants, either present before the onset of stress (pre-existing mutation), or afterwards, during the period of population decline (*de novo* mutation). While the heuristic model by Gomulkiewicz & Holt [8] describes the deterministic growth of the rescued population above a threshold critical population size, the model by Orr & Unckless [9] explicitly describes the stochasticity inherent to the establishment of beneficial variants (see also [11]).

Empirical validation of evolutionary rescue theory is still in its infancy. Experimental evolution offers a potentially powerful method for this validation [12]. In particular, rapid evolution in microcosms allows the study of replicated trajectories of adaptation to evaluate the probability of rescue versus extinction. This is rarely possible in natural populations where only a single realization of any of these stochastic outcomes is observable. Usually, however, extinction has been considered as a nuisance in experimental evolution. Only recently, several studies have tackled the challenge of describing the probability of evolutionary rescue, using fast-reproducing organisms in microcosms with various stresses causing decline, such as yeast adapting to saline conditions [13,14], virus adapting to high temperature [15], flour beetles adapting to a new host [16,17] or bacteria adapting to antibiotic stress [18]. The study of the emergence of resistance to chemotherapy in microbes also has a very long and fruitful history (reviewed in recent studies [3,19]), which relates directly to evolutionary rescue. Bell & Gonzalez [13] showed a clear threshold in population size below which rescue was very unlikely in *Saccharomyces cerevisiae* undergoing salt stress, as predicted by theory [9]. Ramsayer *et al.* [20] studied adaptation of *Pseudomonas fluorescens* to antibiotic stress; they found that rescue probability increased with the genetic diversity at the onset of stress [17], giving qualitative support to other facets of evolutionary rescue theory.

In spite of this progress, tightening the link between theory and experiments requires evolutionary rescue models to use empirically measurable parameters. Models also need to account for certain peculiarities of microbial populations. In microbes undergoing limited recombination, clonal interference reduces polymorphism. Thus, the assumption of an asexual population with a narrow genetic basis for rescue (as in Orr & Unckless [9]) is a good way to describe the genetics of rescue in these cases. Furthermore, microbes often display continuous growth with overlapping generations, whereas most existing theories on evolutionary rescue (including Orr & Unckless [9]) assume discrete-time geometric growth or decay (but see Knight *et al*. [21]). Generalizing rescue predictions to a larger set of life cycles is now needed. For example, microbial populations experience periods of progressive growth or decline, but also frequent bottlenecks both in nature and in the laboratory, where serial transfers are routinely used. How bottlenecks affect the probability and dynamics of rescue is not known. Finally, even if a single variant generates the rescue in any given population, different variants may become fixed in different replicate populations, because they are sampled from a spectrum of potential mutation effects. This variability of outcomes has so far been ignored in analytical models where a single resistance fitness class is considered [4,9].

The aim of this paper is to propose methods to foster a better quantitative link between experimental evolution and predictions of evolutionary rescue theory. First, we generalize the predictions by Orr & Unckless [9] to a broader demographic and genetic context, with a full description of the stochasticity of the rescue process. This allows us to apply the model to empirical systems with variable mutation effects on growth rates, overlapping generations and the inclusion of bottlenecks. We show that the results obtained by Orr & Unckless [9] generalize, at least approximately, to more general demographic assumptions, in the absence of density-dependence. Second, we use the results of the model to discuss possible empirical tests of rescue theory, either by (mostly qualitative) tests of some predictions on existing datasets, or by discussing empirical methods to test other aspects of evolutionary rescue. We also discuss how models and data on the stochastic dynamics of rescued populations could enhance our understanding of the underlying evolutionary processes.

We compare the predictions from our model with data from experiments on evolutionary rescue, using two species: *S. cerevisiae* [13] and *P. fluorescens* [20]. These experiments illustrate aspects of population dynamics and experimental protocols that we aim to address in our theoretical developments. They also allow us to evaluate the general qualitative relevance of our analytical predictions.

Bell & Gonzalez [13] studied the effect of population size on the probability of rescue in bottlenecked populations of yeast under salt stress. From a large mass culture, derived from a single clone in permissive conditions, replicate populations were diluted to a wide range of starting population sizes (10–10^{7} cells) and then exposed to a saline (NaCl) environment or a salt-free control. This allowed a test of the effect of the inoculum size on the probability of rescue. High concentrations of NaCl were used such that populations initially declined in the new environment as a result of the combined effects of reduced growth in the presence of salt and dilution during transfers. In the control experiment, the wild-type was grown and serially diluted in the absence of NaCl. Final population density was measured by optical density; populations were considered extinct when the optical density was equal to or below the mean value of control wells containing no cells.

In a recent study, Ramsayer *et al.* [20] investigated the dynamics of evolutionary rescue in naturally declining populations of *P. fluorescens* facing three bactericidal doses of streptomycin (50, 100 and 200 μg ml^{−1} plus a control). Unlike in Bell & Gonzalez [13], replicate populations started from high density (≈ 10^{8} cells) and without any further bottlenecking during the experiment. To investigate the impact of initial standing variation on evolutionary rescue, two types of starting populations were compared. For the ‘clonal’ population type, 10 single clones were grown to high stationary density, from each of which replicate populations were established and exposed to the antibiotic. For the ‘admixture’ populations, the starting population was a mixture from eight large populations that had been maintained in replicate vials over 14 serial transfers in a permissive environment. Replicate admixture populations were established from this mixture. Ten replicates per population origin and antibiotic dose were tested, as well as two control antibiotic-free populations. Population sizes were measured by plating constant volume samples and counting colony-forming units (CFUs) at times *t* = 0, 4, 9, 22, 30, 53 h after introduction of the antibiotic. Populations were considered extinct when no CFUs were detected in any of the samples at *t* = 53 h.

As in the experiments described earlier, the scenario envisioned by our model is that of a single isolated asexual population confronted with a new environment causing its decline. An initial number of individuals *N*_{o} is introduced in this new environment (inoculum size). We model the probability that such a population initially decaying at rate *r _{o}* < 0 is saved from extinction by the occurrence and the establishment of genetic variants showing positive growth (

We consider the case of continuously growing/decaying populations with overlapping generations such as microbial populations, but our analytical results apply to a broader range of demographic dynamics. We use a Feller diffusion process [23], also called ‘continuous state branching process’ (see the electronic supplementary material, appendix S1), to approximate the demographic process. This approximation reduces all the complexity of the life cycle into two key parameters: the mean reproductive output *r* (growth rate or Malthusian fitness), and the variance of this output *σ* (for use of similar diffusion approximation in age-structured populations, see Shpak [24]). This diffusion approximation and its range of validity are more detailed in the electronic supplementary material, appendix S1. In short, our approach, based on diffusion, is general in that it can approximate many life histories, but is still limited (i) to density independent and (ii) ‘smooth’ demographic processes (without instantaneous jumps in population size), and (iii) to moderately growing or decaying genotypes (|*r*|/*σ* 1). Here, we will mostly illustrate our predictions in the case of a birth–death process, where cells divide by binary fission at rate *b* and die at rate *d*, at exponentially distributed time intervals. In that case, the growth rate is *r* = *b* − *d*, and the reproductive variance is *σ* = *b* + *d* = *r* + 2*d*. We also show consistency with the results of Orr & Unckless [9] who assumed a discrete-time model.

Mutations occur as a Poisson process with given rate per time units: the process of growth, mutation, etc., are measured in consistent units (hours, days, etc.). Note that, as mutations occur every division event, the per unit time mutation rate is proportional to the birth rate (see details in Martin & Gandon [25]). In all scenarios, rescue mutations all arise in the same constant genotypic background (the dominant ‘wild-type’), so that a constant mutation rate can be used, except when multiple mutational steps are considered (see details in the corresponding section and electronic supplementary material, appendix S1). Mutations create a set of alleles with an arbitrary distribution of effects on growth rates *r* and reproductive variance *σ*. We call ‘resistant’ a genetic variant that shows positive growth (*r* > 0), and ‘rescue’ a variant that is resistant and has established, meaning that it has escaped initial stochastic loss and increased to substantial population size. Each genetic variant is either already present in the population at the onset of stress (pre-existing mutations) or appears afterwards by mutation (*de novo* mutations). We derive predictions for the probability of rescue in both scenarios. We further consider different demographic scenarios likely to occur in evolutionary rescue experiments: either the population is exponentially decaying in the new environment [20], or it declines owing to the combined effects of bottlenecks and decreased growth [13]. We also examine two scenarios concerning pre-existing mutations: either variants were generated from a single clone in a previous phase of expansion in the permissive environment, as in the ‘clonal’ treatment of [20] and in Bell & Gonzalez [13], or they were segregating in a population at mutation–selection balance before exposure to stress, as could be assumed in the ‘admixture’ treatment of Ramsayer *et al*. [20].

In the main text, we present the most simple and testable approximations, whereas the appendices in the electronic supplementary material provide a thorough description of how these approximations are derived. All analytical derivations were obtained with Mathematica v. 8.0 [26].

The analytical predictions were tested in exact simulations of a birth–death process, i.e. with cell division and death occurring at exponentially distributed times. In our simulations, mutation produced a spectrum of birth rates *b*, with a fixed death rate *d*. The simulation algorithm is described in Martin & Gandon [25], with demographic dynamics simulated by the exact Gillespie stochastic simulation algorithm [27]. Mutation is birth-dependent (it occurs only in cells that divide), and mutants show continuous variation in birth rates, generated by a Gaussian phenotype-to-fitness landscape. The landscape model here was merely used to produce mutations with a continuous spectrum of effects. This algorithm was optimized and encoded in C; matrix operations and random number generation were performed using the GNU Scientific Library [28]. Analysis of the simulations was performed with R [29].

Analytical predictions provide a framework to analyse experiments that study how various factors affect rescue probability. We evaluated the fit of our predictions on the effect of inoculum size *N _{o}* on rescue probability in the yeast salt-stress experiment [13]. We then compared the observed versus the expected outcome of different evolutionary histories on rescue probability, in naturally decaying

As only resistance mutations (*r* > 0) can achieve substantial frequency, we consider only the rate of mutation to resistance. Resistant mutants may appear in the ‘permissive’ environment (before the onset of stress), or in the stressful environment. We therefore denote the mutation rate to resistance in each condition by *u*_{P} and *u*_{S}, respectively. We denote by an overbar () the mean of any quantity *x* among resistant genotypes produced by mutation.

As we will see below, a striking similarity emerges from the various scenarios considered, where the number of mutations destined to rescue the population is, in general, Poisson distributed. This is valid, in general, only when the *per capita* rate of mutation to rescue is small. Furthermore, the rate of the Poisson distribution can always be written i.e. it is proportional to the number of individuals present at the onset of stress, *N _{o}*. The factor can thus be denoted a rate of rescue per inoculated individual. It will depend on the particular scenario. The derivations that follow will consist of showing this Poisson limit and providing closed form expressions for for each subcase, as a function of various biological parameters. The probability of an evolutionary rescue is the probability that at least one rescue mutation appears before extinction, and is given by one minus the probability of no rescue, i.e. the zero class of the Poisson (). Thus, in what follows, all expressions for rescue probability take the form

3.1

Intuitively, the simple relationship to *N _{o}* reflects the fact that we assume independence between the fate of different lineages present at the onset of stress (no density – dependence).

Table 1 summarizes the explicit expressions for in each scenario, with the corresponding figures showing the simulation checks.

*General probability of establishment*. A ‘rescue’ mutant is a resistant mutant lineage that has risen to substantial copy number such that it is almost certain to avoid ultimate extinction (establishment). The probability of establishment, for a resistant mutant lineage with given demographic parameters (*r*,*σ*), initially present in a single copy is approximately [23]

3.2

Quite intuitively, equation (3.2) predicts that the probability of establishment increases for mutants with large growth rate and small reproductive variance [24,30]. Throughout this paper, most simplifications are obtained when establishment is unlikely (*π _{f}* = 0(

Accuracy of the Feller diffusion approximation for establishment probability. The probability of establishment (avoiding ultimate extinction) when started from a single individual is shown as a function of the ratio of Feller parameters (*r*/*σ*): **...**

*Continuously decaying populations*. Let us first consider a clonal population consisting of a wild-type with negative growth rate *r _{o}* < 0 that decays continuously to extinction in the absence of any genetic change. In this case, we can compute the exact probability that, before extinction of the wild-type, it produces a resistance mutation that establishes. We must take into account variation of (

3.3

Here, is the rate of rescue mutations per wild-type individual under stress, namely the corresponding rate of resistance mutations weighted by the probability that they establish. Equation (3.3) is accurate (figure 2*b*) unless the wild-type shows almost no decay (see the electronic supplementary material, figure S2), a case of little interest as rescue is effectively certain then. In agreement with previous theory, equation (3.3) predicts that the probability of a rescue increases when the initial population size is large (figure 2), the mutation rate towards resistance is high, the wild-type population does not decline too fast (small |*r _{o}*|) and the mean establishment probability of resistant mutants is high.

Rescue probability from *de novo* mutations and standing variance. The probability of evolutionary rescue as a function of inoculum population size (*N*_{o}) is given for various values of the mutations rate to resistance *u*_{R}. The dots are the results of exact **...**

The mean establishment probability depends on the distribution of the ratio *r*/*σ* among resistant mutants (see equation (3.2) and electronic supplementary material, appendix S1). This suggests that information about the distribution of growth rates among resistant mutants is not sufficient to predict the probability of rescue. For precise quantitative tests, experiments should also seek to evaluate the extent of reproductive variance *σ* and its joint variation with *r* among resistant mutants. If most resistance mutations are only weakly growing (0 < *r* *σ*), and mutations affect only *r* but not *σ* (*σ* is then a numerical constant, still to be estimated), then where is the average growth rate among resistant mutants. This simplification can be justified, for example, for a birth–death process with a high turnover rate (high birth and death rates that compensate each other). In this case, demographic stochasticity is important as *r* = *b* − *d* *σ* = *b* + *d*. Mutations affecting both *b* and *d* contribute the same variance to *r* and *σ*, but the impact of this variance is smaller, relative to the mean value, on *r* than on *σ* . Finally, note that this simplified version of our equation (3.3) corresponds to eqn (3) of Orr & Unckless [9], who assumed a single allele () and a discrete generation model (Poisson offspring distribution, *σ* ≈ 1 for all genotypes, see electronic supplementary material, appendix S1).

*Rescue in multiple steps*. The earlier-mentioned treatment deals only with rescue produced by a single mutation event. Rescue can also occur in multiple steps, e.g. with a first intermediate step destined to be lost (‘transient’ mutation), but that produces the final rescuer genotype before its extinction. This process (in two steps) is modelled in appendix S1 (§2). The resulting rate of two-step rescue scales with the total rate of *genomic* mutations (with arbitrary effect on *r*) up until extinction: (equation (S1.7)). The first ‘transient’ mutation is most likely to be one with *r* ≈ 0 (critical process; electronic supplementary material, appendix S1). Indeed, these mutants are the ones that wander the longest time before extinction, thus allowing them to produce secondary rescues. Non-resistant mutants (*r* < 0) all get quickly extinct, whereas resistant mutants (*r* > 0) either produce a single step rescue or get extinct early when still at low numbers. It is difficult to be more explicit without introducing a particular model describing the combined effect of multiple mutations on growth rates [31].

*Rescue with bottlenecks.* The process that we have described so far is one where the wild-type population decays at a roughly constant rate until extinction. However, extinction may also take place in populations with non-steady decay. We consider here the case where the wild-type alternates periods of positive growth with catastrophic drops in population size (bottlenecks). We model bottlenecks at regular time intervals (every *τ* time units) with a constant dilution factor 0 < *D* < 1 (the portion of the population that is kept). This is particularly relevant to microbe studies, where serial transfers are often used to refresh the medium, and it indeed corresponds to the protocol used by Bell & Gonzales [13]. It should also be a good approximation to the case of rescue in natural populations undergoing randomly distributed catastrophic drops in population size [32]. As explained in electronic supplementary material, appendix S2, our treatment is based on the approach proposed by Wahl & Gerrish [33], and complementary to theirs, by allowing for long-term decay or growth, and assuming high reproductive stochasticity during growth phases (* π_{f}* 1) (see also [34]).

In this context, the demographic processes are intrinsically discrete at the time-scale of cycles, although the underlying demography during the growth phase is continuous at smaller time-scale (measured in our basic time units). We denote *m*_{r} the growth rate of a given mutant, over a cycle, i.e. its Malthusian fitness at the time-scale of cycles. It must satisfy *m*_{r} = log(*D*) + *r τ*, where

3.4

This time, is the rate of rescue mutations per individual over a cycle, where is the mean probability, among random resistant mutants, of avoiding stochastic loss during the growth phase and over the following serial dilutions. Equation (3.4) is valid only if (i) the rescue mutants grow very rapidly between bottlenecks, which is necessarily the case when bottlenecks are severe (*D* * π_{f}*), and if (ii) the reproductive stochasticity during growth phases is substantial so that

The exact similarity of equations (3.3) and (3.4) is striking, given the key differences in the process under study. The only difference, apart from the change in time-scale, lies in the expression for the probability that a given resistance translates into a rescue: in equation (3.4), instead of in equation (3.3). This simply reflects the fact that mutants must avoid extinction both during the growth phase in which they appear, and over the longer time-scale of growth-dilution cycles (*m*_{r}). If mutations only affect *r* and not *σ*, we have , where is the mean Malthusian fitness, and CV* _{r}* is the coefficient of variation of

*General result*. The inoculum population may also contain a certain proportion of resistant genotypes before the onset of stress. The proportion will depend on the state of the population before stress, and the corresponding probability of rescue from pre-existing mutations is dependent on the scenario assumed for this state. In electronic supplementary material, appendix S1 (equation (S1.9)), we show that, provided the probability of establishment of resistant genotypes in the stressful environment is small (* π_{f}* 1), equation (3.1) applies, with a rate of rescue per inoculated individual

3.5

where *E*(*n*_{R}) is the expected number of pre-existing resistance mutations in the inoculum, at the onset of stress, and is the average establishment probability * π_{f}* (equation (3.2)) among them. Note that, in general, is distinct from the average among

*Mutation–selection equilibrium*. We assume that a population has reached mutation–selection equilibrium before the stress occurs. Then, the population is culled to *N _{o}* individuals at the onset of stress. Alternatively, the population can be assumed to be at mutation–selection–drift equilibrium at constant size

3.6

As for rescue from *de novo* mutations, the probability of rescue from standing variance increases with the size of the initial population. It increases with the rate of mutation towards resistance in the permissive environment and depends on the joint distribution of demographic rates of resistant mutants (*r*,*σ*) in the stressful environment, and of their cost *c* in the permissive environment. Equation (3.6) simplifies when the cost of a resistance, before stress, is independent of *r* or *σ* after the onset of stress. In this case, we can write and the rate of rescue becomes Denote the rate of rescue from *de novo* mutations (from equation (3.3)). If we further assume that the mutation rate towards resistance is the same in permissive and stressful environments (*u*_{P} = *u*_{S}), we can then express the rate of a rescue from standing variance, as a function of the rate of *de novo* rescue mutations :

3.7

(Recall that *c*_{H} is the harmonic mean of the cost, before stress, of those random *de novo* mutations that are resistant in the stress.) Mutations with a small cost before stress (and therefore at high frequency at the onset of stress) will dominate the harmonic mean *c*_{H} and therefore the rescue process. Figure 2*a* shows the accuracy of equation (3.7) versus exact simulations. As for *de novo* rescue, we retrieve the corresponding result (eqn (5) of Orr & Unckless [9]) for a single allele (*c*_{H} = *c*) and discrete demography, assuming a small * π_{f}* (see electronic supplementary material, appendix S1).

Because the total number of rescue events from *de novo* mutations or from standing variants are both approximately Poisson distributed, the probability that a given rescue event is from either type is simply given by the relative weight of each Poisson parameter. In the case of a population at equilibrium before stress, and assuming independence between cost and resistance (equation (3.7)), the probability that the rescue mutation appeared after the onset of stress (*de novo*) as opposed to being present before is

3.8

where the right-hand-side limit is when mutation rates are equal across environments. We can see here that the relative contribution of each type depends on the rate of decay |*r _{o}*| of the wild-type (i.e. the strength of the stress imposed). Provided the cost distribution is less affected by the stress than the decay rate (

*Growth from a single clone*. Our second scenario is intended to describe some microbe experiments where a single clone is grown to high density then put into a new stressful environment [13]. This may also be relevant to natural populations that have undergone a severe bottleneck and then grown back to a large population size before facing an abrupt environmental change. Under these conditions, the distribution of the number of resistance mutations at the end of the growth phase is given by models of fluctuation test experiments [35]. Although the distribution of the number of resistance mutations in this situation can be complex, its mean takes a simple form [35]. Assume that the population is grown from *n _{o}* clonal individuals to some constant level

3.9

Figure 2*c* confirms the accuracy of the above expression versus exact simulations. Again, things simplify if we assume that the environmental change does not impact mutation rates (*u*_{P} = *u*_{S}) in which case equation (3.10) can be expressed in terms of the rate of *de novo* rescue, by setting Contrary to the case of mutation–selection balance, the probability of rescue here does not depend on the fitness effects of resistant mutations in the permissive environment. As in equation (3.8), we find another simple rule for the probability of rescue from *de novo* versus pre-existing mutations:

3.10

As in equation (3.8), we expect that rescue from standing variance will be more likely in more stressful conditions (larger |*r _{o}*|), all else being equal.

*With bottlenecks*. If we consider rescue from standing variance in the context of bottlenecked populations, all the results provided above equations (3.5)–(3.10) still hold, simply replacing by its equivalent in bottlenecked populations and *r _{o}* by (equations (3.4) and (S2.6)).

Equation (3.1) provides a theoretically based means to compare different rescue experiments by estimating in each case: below, we show an empirical test of this equation and explain how such comparisons can be used.

Bell & Gonzalez [13] studied the effect of inoculum size on rescue probability, in bottlenecked populations of yeast in a saline environment. To do so, they grew a culture from a single clone to a constant large size *N*_{1} then diluted it to appropriate levels to study the effect of *N _{o}* on the probability of a rescue. In that case,

Figure 3 shows the result of the fit for both controls and stress treatments, with two replicate experiments in each case. The estimated parameters are given in table 2. The model fits the data well (goodness-of-fit tests based on deviance: *p* ≈ 1, table 2, the model is not significantly less fitted than the saturated model). Overall, while equation (3.1) is well supported by the available data, this prediction is somewhat too general to be strongly informative, being expected in all scenarios, and in controls. It does, however, suggest that each lineage derived from each inoculated individual has roughly independent mutational and demographic fates, which was a key assumption of the model. Interdependence between lineages may be at play in these experiments, but if so, here it has undetectable effect on the relationship between *N _{o}* and rescue probability. The parameters of this relationship have a clear biological interpretation: estimates the overall rate of rescue per individual inoculated (in the stress, table 1), whereas

Relationship between inoculum size and rescue probability in yeast. The probability of a rescue versus size of the inoculum *N*_{o} is shown for yeast populations faced with a salt stress plus dilutions or in a control experiment in standard yeast **...**

We hope that future studies will estimate the parameter This will provide the basis for comparisons across experiments.

In a recent experiment, Ramsayer *et al.* [20] studied the dynamics of evolutionary rescue in naturally declining populations of *P. fluorescens* facing three doses of streptomycin (50, 100 and 200 μg ml^{−1} plus a control). In addition, they studied the effect of the origin of the inoculum population, by comparing single ‘clonal’ populations and genetically diversified ‘admixture’ populations created from a mix of eight large cultures, putatively at evolutionary equilibrium (see §2). On this basis of protocol, *de novo* rescue should follow equation (3.3) in all treatments but the control, while the rescue from pre-existing mutations should follow equation (3.9) for the ‘clonal’ treatment and equation (3.6) or (3.7) for the ‘admixture’ treatment. In the latter case, we assume that stationary mutation–selection balance has been reached over 14 transfers prior to mixing, while the population had only undergone roughly 100 bacterial generations. This seems a reasonable approximation, and is surely the closest of the scenarios described above. There was no significant dose effect in this experiment, but a significantly higher rescue probability among the admixture treatment populations, relative to the clonal treatment [20]. These treatments differ only by the origin of the inoculum (their decay rates |*r _{o}*| were indeed similar, data not shown). The observed effect must therefore come from differences in the rate of rescue from pre-existing mutations.

The quantitative impact of each treatment on rescue probabilities can be evaluated by looking at the inferred rate of rescue per individual inoculated (from equation (3.1)), for each treatment. The initial population sizes were roughly equal in both treatments (*N _{o}* ≈ 3.18 × 10

This is by no means a conclusive test of the theory, more an example of how it can be used to estimate several key quantities. It has the key drawback (i) of ignoring the effect of the environment on birth rates and consequently on mutation rates (*u*_{S} = *u*_{P}) and (ii) of neglecting rescue in two steps, so that the probability of *de novo* rescue is underestimated *a priori*. Yet, these rough estimates do seem at least of a reasonable order of magnitude. The estimate for seems reasonable, based on the mean selection coefficients among random mutations, for *Escherichia coli* in optimal conditions [36] (). For a mutation rate to rescue of 10^{−}^{11} is very small, but it is in fact not unrealistic. The rate of resistance to antibiotic stress in bacteria depends on the dose, the antibiotic and the species/genotype considered, but it often lies within [10^{−}^{5} − 10^{−}^{9}]. Further, recalling that is reduced by a factor relative to the rate of resistance mutation, this rate may be of reasonable order. Overall, the observed discrepancy between the treatments is thus consistent with resistance costs of a few per cent and a very low resistance mutation rate.

In this paper, we have derived the probability of evolutionary rescue in various biological situations (summarized in table 1). The results are general for a density-independent demographic model with smooth variation (figure 1*c*); they allow for arbitrary variation in the demographic parameters (*r*,*σ*) among lineages produced by random mutation. We generalize several results already obtained by Orr & Unckless [9] in a discrete-time model of geometric growth and decay. All results were consistent with exact simulations where individuals either reproduced by binary fission or died (birth–death process). In general, this probability takes the form (equation (3.1) and figure 2), where is a rate of rescue mutation per inoculated individual. This is confirmed by empirical relationships between and *N _{o}* in the yeast (table 2 and figure 3). To go further in the comparison of these predictions with experimental data, we now discuss several methodological issues concerning both evolutionary rescue experiments and modelling. We hope that these comments stimulate further experiments and strengthen the quantitative validation of models in this field of research.

Measuring the probability of rescue is not that straightforward. First, one must choose the time period over which populations can be said to be either doomed or rescued. Such a decision can be better informed by measures of population size over time (rescue trajectories). One must also distinguish genetic rescue from plastic responses, which may be important [37] and will display different dynamics (detailed in Chevin *et al*. [38]).

We have shown how rescue probabilities in different conditions could be used to infer key parameters of the process (rate of rescue mutations *u** and harmonic mean of resistance cost *c*_{H}, demographic stochasticity *r*_{c}/*σ*). In order to perform such inferences, key parameters must be measured in rescue experiments: the inoculum size (*N _{o}*), the rate of decay of the wild-type under stress (), the dilution factor and dilution times imposed for bottlenecked populations (

More powerful tests of our theory would be possible by measuring these parameters by independent methods and comparing the two inferences. Measures of mutation rates to resistance prior to stress (*u*_{P}) are possible using fluctuation tests (e.g. following the methods in recent studies [35,39]). However, quantitatively precise estimates may prove difficult to obtain: some estimates are biased by resistance cost *c* [35] and persistence probability * π_{f}* (‘plating efficiency’ [39]), which must then be estimated too. The distribution of the cost of resistance (hence its harmonic mean

What would also be required to fully test our predictions is a measure of the joint distribution of *r* and *σ* among a set of resistance mutations, in order to estimate or (i.e. the mean establishment probability among resistant mutants). This would ideally be carried out in high-throughput, with many replicate populations (and thus possibly many different mutants) jointly estimated. Measuring demographic stochasticity in microbial cells requires measuring the growth and death rate of cells (*σ* = *r* + *d*). Measuring death rates may be tricky, but the growth rate *r* of any given mutant can be obtained from time-series of plate counts [20] or of macroscopic measurements such as optical density [13]. However, plate counts are time consuming and thus limit the feasibility of high-resolution time-series, whereas optical density may not accurately estimate the cell density in environments causing high mortality (typical of rescue experiments), because dead cells can contribute to optical density for some time after death. Alternative high-throughput methods exist that measure both the rate of division and of death in bacterial populations, using fluorescence and luminescence signals from genetically marked strains [40]. This would allow estimation of joint distribution of (*r*,*σ*) among a set of screened resistant mutants, and thus parametrization of the predictions proposed in table 1.

Observed empirical dynamics of rescue could yield valuable tests of several of our model's assumptions, the first being the assumption of exponential decay, easily testable from the decay dynamics of doomed populations. These dynamics, which can be obtained from plate counts [20], or fluoro-luminometric measurements, could then potentially shed light on the relative contribution of different types of rescues (*de novo* single versus two steps versus pre-existing mutations). A full stochastic description of these dynamics, in the present analytical framework, will be the scope of future work.

In our data analysis, we focused on the effect of population size or evolutionary history of the population before stress. Other factors could be studied as well, and compared with the present predictions. The effect of the rate of decay could be studied by varying the intensity of stress, although one would have to keep in mind that not only |*r _{o}*| varies in this case, but also and . Note also that according to whether stress decreases the birth rate or increases the death rate, it may affect (or not, respectively) the mutation rates to resistance

Our model makes simplifying assumptions in order to analytically model a fully stochastic rescue process. A first assumption is that lineages are independent of each other, both genetically (no genetic exchange or sex) or demographically (no density-dependence). In natural populations, these assumptions may not hold due to the complexities of the environment, in particular when population dynamics depend on some key resource dynamics (e.g. host cells in the case of viruses [4]). Including density-dependence in this stochastic framework may be possible using the logistic stochastic process proposed in Lambert [42], or the diffusion analysis in Uecker & Hermisson [22]. The effect of sex should also be studied in these stochastic models. However, these assumptions may also not be so limiting, at least in some natural contexts: (i) when the genetic basis for the rescue is a single gene the system may behave almost as an asexual, (ii) when densities in the stress are low, density-independent growth may be driving the demographic dynamics. All these issues are a matter of experimental testing of course (for an example of rescue experiments with a sexual species, see Agashe and co-workers [16,17]).

A second key assumption of our model is that the environment changes abruptly from one where the population thrives to one where it starts to decline, the rate of decline remaining constant thereafter. A treatment of a more continuous environmental change is also necessary and would introduce time inhomogeneous processes, which are more challenging mathematically. Predictions from such extensions can be tested in controlled experiments where the stress increases/decreases at a desired rate.

Overall, we have presented a framework to address quantitative aspects of the process of evolutionary rescue, by comparing data from experimental evolution with simple yet relatively general predictions from stochastic process theory. We believe that an ongoing exchange between experimenters and theoreticians can help to better understand and predict the process and outcome of evolutionary rescue.

We thank Luis Miguel Chevin and David Waxman and three anonymous referees for their help and suggestions to improve the manuscript. Amaury Lambert pointed out two key results on Feller diffusions: the exponential distribution of the asymptotic process and the link between the cumulated size to extinction and the first passage time of a Brownian motion, via Lamperti transforms. Andrew Gonzalez kindly provided the data on the yeast rescue experiments. This work was funded by ANR EVORANGE (ANR-09-PEXT-011), G.M. and O.R. were funded by RTRA BIOFIS (INRA 065609) and G.M. was funded by PEPII (INSB-INEE-INSMI) from CNRS. This is publication ISEM 2012-130.

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