Population persistence under high mutation rate: from evolutionary rescue to lethal mutagenesis

Abstract

Populations may genetically adapt to severe stress that would otherwise cause their extirpation. Recent theoretical work, combining stochastic demography with Fisher’s Geometric Model of adaptation, has shown how evolutionary rescue becomes unlikely beyond some critical intensity of stress. Increasing mutation rates may however allow adaptation to more intense stress, raising concerns about the effectiveness of treatments against pathogens. This previous work assumes that populations are rescued by the rise of a single resistance mutation. However, even in asexual organisms, rescue can also stem from the accumulation of multiple mutations in a single genome. Here, we extend this model to study the rescue process in an asexual population where the mutation rate is sufficiently high so that such events may be common. We predict both the ultimate extinction probability of the population and the distribution of extinction times. We compare the accuracy of different approximations covering a large range of mutation rates. Moderate increase in mutation rates favors evolutionary rescue. However, larger increase leads to extinction by the accumulation of a large mutation load, a process called lethal mutagenesis. We discuss how these results could help design “evolution-proof” anti-pathogen treatments that even highly mutable strains could not overcome.

Authors contributions

Y.A. O.R. and G.M. initiated the idea for the study. YA, A.L, L.R and G.M derived the mathematical results. Y.A. performed the simulations. Y.A. O.R. and G.M. drafted the paper. All authors critically revised the manuscript, and gave approval of the final version for submission.

Introduction

Evolutionary rescue (ER) happens when a population confronted to severe stress avoids extinction by genetic adaptation. Understanding and predicting when and how evolutionary rescue occurs is critical in fields as diverse as conservation biology, invasion biology, emergence of new diseases and the management of resistance to treatment in pests and pathogens (see reviews in Gonzalez et al. 2013; Carlson et al. 2014; Alexander et al. 2014; Bell 2017). In all these situations, genetic variation, be it present before the onset of stress, or generated de novo after, is a key ingredient for evolutionary rescue, as expected theoretically (e.g. Gomulkiewicz and Holt 1995) and observed experimentally (e.g Ramsayer et al. 2013). Because mutation affects both standing and de novo genetic variation, it comes to no surprise that a number of evolutionary rescue models, combining stochastic evolution and demography, have predicted that higher mutation rates were associated with higher probability of evolutionary rescue (Orr and Unckless 2008, 2014; Martin et al. 2013; Anciaux et al. 2018). Few evolutionary rescue experiments have manipulated the mutation rate to test these predictions (reviewed in Bell 2017). For instance, Couce et al. (2015) found that two different mutator strains of bacteria with elevated rates of mutations evolved more than 100-fold resistance to antibiotic concentrations that caused the demise of control strains. Mutator alleles are indeed often found in antibiotic resistant strains causing serious health issues (Eliopoulos and Blázquez 2003), raising concern about pathogens escaping our control by evolving higher mutation rates (for theoretical predictions see Taddei et al. 1997; Greenspoon and Mideo 2017).

Yet, most mathematical models of evolutionary rescue assume that the population is rescued from extinction by the spread of a single mutant of large effect (Orr and Unckless 2008, 2014; Martin et al. 2013; Anciaux et al. 2018) and do not describe more polymorphic populations where several mutations of smaller effects can combine to allow population growth (see however the work of Uecker and Hermisson (2016) and Uecker (2017) where sexual reproduction allows to produce such rescue genotypes). The latter situation seems in particular to be common in the evolution of herbicide resistance, especially when the mutational target for resistance is large (Kreiner et al. 2018). Even in asexual organisms, when the mutation rate is high, evolutionary rescue may commonly result from the cumulative effect of multiple mutations accumulating stochastically over time in a given lineage. Such a mutation regime is particularly relevant in highly mutable viruses, mutator strains of bacterial (e.g. Springman et al. 2010) or cancer cells (e.g. Loeb 2001). Our aim here is to provide theoretical predictions for evolutionary rescue in such a regime with high mutation rates in asexual organisms, complementing existing theory on the subject.

Several complications arise when modelling evolutionary rescue in highly polymorphic populations with high mutation rates. First, the dynamics of allelic frequencies at different loci interact in asexuals. For example, the selective sweep of a given beneficial mutation is hindered by the co-segregation of other beneficial mutations (clonal interference, Gerrish and Lenski 1998). A theoretical study by Wilson et al. (2017) recently showed that, when evolutionary rescue is likely, it should most often be driven by soft selective sweeps, where multiple resistance mutations spread through the population simultaneously. Wilson et al. (2017) still assumed that each of these lineages carried a single mutation, each with the same effect on the population growth rate. When the mutational target is large, different lineages contributing to rescue are however likely to carry mutations with different fitness effects. Modelling the distribution of mutation effects (as in Martin et al. 2013; Anciaux et al. 2018) becomes then critical. Finally, when the mutation rate is high, multiple mutations may also accumulate on each lineage, either facilitating evolutionary rescue or impeding it, through their cumulative effect. Modelling both beneficial and deleterious mutations and, critically, the epistatic interactions between them, also becomes necessary.

Previous evolutionary rescue theory predicts that higher mutation rate allows populations to withstand higher levels of stress (e.g. Anciaux et al. 2018). Yet, there are reasons to expect this prediction not to hold above some critical mutation rate. Indeed, increased mutation rates also build-up detrimental mutation loads. Mimicking this mutation load through a constant cost associated with the mutator genotypes, Greenspoon and Mideo (2017) found that the evolution of elevated mutation rates facilitated evolutionary rescue only in a limited range of situations. The variance load, depressing mean fitness despite ongoing adaptation, is a critical component of quantitative genetics models assuming a non-linear relationship between fitness and phenotypes and increases under higher mutation rates (e.g. Bürger and Krall 2004). Artificially increasing the mutation rate has even been proposed as a mean to weaken or even eliminate pathogen populations, by a process denoted lethal mutagenesis (Loeb et al. 1999). Models of lethal mutagenesis predict that extinction of the target population could be observed under biologically realistic sets of parameters (Bull et al. 2007; Martin and Gandon 2010; Wylie and Shakhnovich 2012). In these models, mean fitness dynamics and extinction stem from the deterministic effects of selection and mutation. Alternatively, Matuszewski et al. (2017) discuss the continuity between these models and models of mutational meltdown, where extinction is driven by the interaction of genetic drift and deleterious mutation. Lethal mutagenesis has been investigated empirically for treatment against viruses (Springman et al. 2010; Arias et al. 2014), bacteria (Bull and Wilke 2008) or cancer cells (Liu et al. 2015). In particular, the combination of antiviral treatments with mutagenic agents is investigated as a strategy to fight fast evolving viruses, such as influenza (Bank et al. 2016). It seems important to improve our ability to predict whether and when such mutagenic agents will increase treatment efficacy or, conversely, facilitate the evolution of resistance.

The population genetics of adaptation behind the rescue process, in isolated asexual populations, sketchily fall into two alternative regimes: rescue may stem (i) from single mutations of large effect (strong selection weak mutation ‘SSWM’ regime) or (ii) from multiple mutations of small effects (weak selection strong mutation ‘WSSM’ regime) (reviewed in Alexander et al. 2014). The SSWM regime of adaptation has been extensively investigated via “origin-fixation” models describing the average behavior of stochastic evolutionary dynamics (McCandlish and Stoltzfus 2014) whereas the WSSM regime has been widely analyzed via deterministic models of quantitative genetics (Lande 1976, 1980). Corresponding evolutionary rescue models further include a coupling of adaptation and demographic dynamics, and naturally fall into the same two regimes (discussed in Anciaux et al. 2018). The SSWM regime of evolutionary rescue is characterized by the fact that the first resistant lineage to establish (and thus cause rescue) is only at one mutational step from the dominant sensitive ‘wild-type’ lineage (e.g. Feder et al. 2016). Models that describe highly polymorphic dynamics (WSSM regimes) often use the infinitesimal model assumptions (many unlinked polymorphic loci), which does not apply to asexual populations. In the WSSM regime, the exact stochastic evolutionary dynamics become quickly intractable, and have often been studied by simulation (e.g. Boulding and Hay 2001). The latter models further often consider initial standing genetic variance as given and pay little attention to the effect of mutation rates in maintaining this variance. They often ignore de novo mutations after the onset of stress, on the argument of short timescales being most critical for evolutionary rescue (e.g. (Gomulkiewicz et al. 2010)). The dichotomy between SSWM and WSSM entails a somewhat simplistic view of adaptation regimes, at the two extremes of all possible mutation rates. These approximations reflect different views of population genetics, which both have received empirical support.

To make analytical progress in our understanding of the effect of mutation rates on the process of evolutionary rescue, we here built on two recent theoretical developments (Martin and Roques 2016; Anciaux et al. 2018). As in Anciaux et al. (2018), we study evolutionary rescue using Fisher’s (1930) geometric model (hereafter “FGM”) to model the distribution of mutation effects on fitness. The FGM (detailed in Methods) is a single peak phenotype-fitness landscape where fitness depends on the position, in phenotype space, of a given genotype relative to an optimum. In the context of ER, stress may affect this landscape in various ways (height, width or position of the peak). In this model, the distribution of mutation effects (both beneficial and deleterious) depend on the context, both genotypic (epistasis) and environmental (e.g. effect of stress). These key features of the FGM are qualitatively and sometimes even quantitatively consistent with a wealth of empirical observations (reviewed in Tenaillon 2014). Under the assumptions of FGM, rescue mutants become very rare as the intensity of stress increases, because they require very large mutational steps. As a consequence, Anciaux et al. (2018) predict that there is a narrow window of stress levels where the probability of rescue shifts from being very likely to very unlikely. They also predict that this critical level of stress, beyond which adaptation is unlikely, is increased by higher mutation rates (Anciaux et al. 2018). Yet, predictions of this model apply to the SSWM regime and may not hold for high mutation rates.

Here, we extend our previous analysis of evolutionary rescue over Fisher’s geometric model (Anciaux et al. 2018) to the more complex and polymorphic WSSM regime. To do so, we use the approach in Martin and Roques (2016) to model the non-equilibrium dynamics of fitness distributions, in large asexual populations with epistasis. In particular, Martin and Roques (2016) showed that, under the FGM, while the fitness dynamics are more complex at higher mutation rates, they are also more predictable and less prone to stochastic fluctuations, even in relatively small populations. To model evolutionary rescue, we still need to describe the demographic stochasticity associated with the extinction process. In the WSSM regime, we thus use a combination of two analytically tractable theories: a deterministic approximation to the dynamics of mean fitness (from Martin and Roques 2016) and a diffusion approximation to the stochastic dynamics of population sizes (from Bansaye and Simatos 2015).

Beyond a derivation of the probability of ultimate rescue or extinction, this approach further allows tracking the rescue process over time. As stated in Gomulkiewicz et al. (2017), transient dynamics (population size dynamics, distributions of extinction times) are of high interest for applications of evolutionary rescue theory, yet are not available from existing predictions, which focused mainly on ultimate outcomes. Gomulkiewicz et al. (2017) studied the distribution of extinction times for populations doomed to extinction, mostly in the absence of mutation (i.e. with a fixed arbitrary set of competing asexual genotypes at the onset of stress). The present work allows extending this analysis to include frequent de novo mutation, rescue events involving several mutational steps, epistasis, variable mutation effects depending on stress intensity and an explicit description of the dynamics of mutation load. Our approach captures the continuum from evolutionary rescue to lethal mutagenesis, as mutation rate increases. Interestingly, some parameter ranges prove to greatly limit evolutionary rescue at all mutation rates, i.e. in spite of the possible apparition of mutators.

Methods

I. General framework

The population is initially adapted to a non-stressful environment where its mean growth rate is positive. At the onset of stress, the population size is N₀ and the mean growth rate of the population shifts to a negative value due to the new stressful environment. Without evolution, the population is doomed to extinction. Evolutionary rescue occurs if at least one resistant lineage (with a positive growth rate in the new environment) establishes, in spite of demographic stochasticity. These resistant mutant lineages can either already be present in the population or arise de novo after the onset of stress. It is thus crucial to determine how the number and growth rates of such mutants depend on the new environmental conditions and on the parental genotypes already present in the population. We do so using the FGM detailed below.

Fitness landscape

In the FGM, a given phenotype is a vector in a phenotypic space of n dimensions that determine fitness (here the growth rate r). The phenotype of an individual with genotype i, is characterized by a vector of the breeding values (heritable components) for the n traits, and its growth rate is r_i. Formulated this way, the model thus accommodates micro-environmental effects on phenotypes (partial heritability), but not systematic (plastic) shifts in phenotype with stress. In a given environment, fitness decays as a quadratic function of the phenotypic distance to a single phenotypic optimum, where the growth rate r_max is maximal at a given absolute level (height of the fitness peak). We assume that each environment is associated with a single optimum and fitness peak. In the scenario investigated here, in the non-stressful environment, the population is close to the ‘ancestral’ optimum z_A. When the environment changes, it is assumed to determine a new optimum z_*. Without loss of generality, the height of the peak may also differ between the ancestral and new environments. However, we do require that the n dimensions that determine fitness remain the same (in nature and number) across environments. In the new environment, the growth rate of an individual with genotype i is given by:

This is an isotropic version of the FGM (all directions are equivalent for selection and mutation) where phenotypes are scaled by selective strength.

Stochastic demographic dynamics

We restrain our analysis to finite haploid asexual populations. Individuals have independent evolutionary and demographic fate (frequency or density dependence are ignored). Each genotype i has a growth rate r_i and a reproductive variance σ_i (r_i = r(z_i, z_*) in a given environment with optimum z_*), which define its stochastic demographic parameters in the context of a Feller diffusion approximation (Feller 1951), as in e.g. Martin et al. (2013), Gomulkiewicz et al. (2017) or Anciaux et al. (2018). For simplicity, we further assume here that the average stochastic variance in reproduction is constant over time: where the average is taken over the K_t genotypes present at time t. This can for example be accurate whenever the σ_i are roughly constant across genotypes i (discussed in Martin et al. 2013 and Anciaux et al. 2018).

Table 1:

Notations

II. Evolutionary dynamics

In this section, we describe the model of evolutionary dynamics over the fitness landscape (FGM of the previous section), which is embedded into the ER model. In the following, de novo mutations (appearing after the onset of stress) are denoted “DN” and mutations from standing genetic variation (mutants already present before the onset of stress) are denoted “SV”. Correspondingly, evolutionary rescue dynamics from an isogenic population, adapting only from de novo mutations, are labelled “DN” and dynamics from a polymorphic population, adapting from both de novo mutations and standing genetic variation, are labelled “DN + SV”.

1. Evolutionary dynamics from an isogenic population

The population is maladapted in the new stressful environment and its growth rate is –r_D, corresponding to a decay rate r_D > 0. Mutations arise every generation following a Poisson process with rate U per unit time per capita. For a given parent phenotype, each mutation creates a random perturbation dz on phenotype, which is unbiased and follows an isotropic multivariate Gaussian distribution, where l_n is the identity matrix in n dimensions and λ is the variance of mutational effects on traits, standardized by the strength of selection. Mutation effects add-up on phenotype, but not on fitness because r(.) is nonlinear (epistasis on fitness and not on phenotype).

In the WSSM regime, the mean growth rate of the population shows limited stochastic variation among replicates, even in reasonably small populations. We thus approximate the evolutionary process by a deterministic fitness trajectory, derived in the WSSM regime under the FGM (Martin and Roques 2016). This seemingly rough approximation can be justified a priori: most of the ER process is determined by the speed at which the population adapts at the very onset of stress. This early trajectory takes place when the population is still large and the adaptive process proves to be relatively deterministic, especially over this short timescale, provided that the mutation rate is high enough (WSSM: U ≫ U_c = n²λ/4). Both analytical arguments and simulations, detailed in (Martin and Roques 2016), showed that mean fitness trajectories are indeed close to the deterministic prediction (with limited variation among replicates), provided that U ≫ U_c (WSSM) and NU ≫ 1 (large mutational input). Here, we use the deterministic fitness trajectory corresponding to these conditions to approximate the growth rate trajectory of all replicate populations under stress.

Provided U ≫ U_c we thus approximate the trajectory of the mean growth rate of all replicate populations by its deterministic trajectory for the WSSM (Martin and Roques 2016): where sech(z) = 2/(e^z + e^−z) is the hyperbolic secant, tanh(z) = (e^z – e^−z)/(e^z + e^−z) is the hyperbolic tangent and is a composite parameter of the mutational parameters. Recall that r_D is the decay rate of the isogenic population and r_max is the maximum fitness that can be reached in the new environment (with r_D + r_max the fitness distance between the parent genotype’s fitness and the top of the fitness peak). The mean growth rate in Eq.[2] reaches a plateau at infinite time of r_max – n μ/2 corresponding to the maximal growth rate minus the mutational load.

2. Evotutionary dynamics from an initiatty potymorphic poputation (at mutation-setection batance)

The evolutionary dynamics of rescue from a polymorphic population is obtained by a similar approximation. We assume that the population is initially at mutation-selection balance in the nonstressful environment, with an arbitrary positive mean growth rate. The phenotypic distribution at the onset of stress is centered on a mean phenotype , which growth rate in the new environment is used to characterize the harshness of the stress imposed. For consistency with the isogenic population model above, we thus denote this growth rate , where r_D > 0 is the decay rate of the central genotype as in the previous section. Resistant genotypes may already be present in the population at the onset of stress (“SV”) or appear by de novo mutation (“DN”), or arise as combinations of these (multiple step rescue mutants). Using the same reasoning as in the previous subsection, we approximate the mean growth rate of all replicate populations by the deterministic trajectory for the WSSM, i.e. whenever U ≫ U_c (Martin and Roques 2016):

In a polymorphic population at mutation-selection balance, the presence of a mutational load implies that the mean growth rate of the population in Eq.[3] is lower than the mean growth rate of an isogenic population in the same environmental conditions, with the same central genotype . As in Eq.[2], the mean growth rate in Eq.[3] reaches a plateau at infinite time of r_max – n μ/2 corresponding to the maximal growth rate minus the mutational load.

III. Evolutionary rescue probability

The demographic dynamics of the whole population can be approximated by an inhomogeneous Feller diffusion (see Appendix I section I) with parameters (the mean growth rate) and (the reproductive variance averaged across segregating genotypes): may vary stochastically as genotype frequencies change under the effects of drift, selection and mutation. However, as explained in the previous subsection, we approximate each replicate’s fitness trajectory by its deterministic expectation under the WSSM regime: or , using the relevant cases from Eqs.[2] or [3]. Therefore, the model approximately reduces to a Feller diffusion with constant and time-inhomogeneous deterministic growth rate . Under these hypotheses, the demographic dynamics follow the stochastic differential equation: where B_t is a Weiner process (see Appendix I section I for more details). We can then use the results from Theorem 1 of Appendix II (see also Bansaye and Simatos 2015) on inhomogeneous Feller diffusions to derive the probability that the population is extinct before time t: where is given by Eqs.[2] or [3]. This expression can be evaluated numerically at any time t > 0. ER happens whenever evolution (change in ) allows to avoid extinction. Hence, the general form of the rescue probability is readily obtained as the complementary probability of the infinite time limit of Eq.[4], namely the probability of never getting extinct: P_R = 1 – P_E with P_E = P_ext(∞), the extinction probability after infinite time. Depending on the scenarios, this rescue probability can be computed explicitly, or approximated via Laplace approximations to the integral , in some parameter ranges (see Results and Appendix I).

IV. Individual-based stochastic simulations

The analytical predictions are tested against exact stochastic simulations of the population size and genetic composition of populations across discrete, non-overlapping generations. The simulation algorithm is described in Anciaux et al. (2018). Briefly, reproduction is Poisson distributed every generations with parameter e^r_i for genotype i, mutations occur according to a Poisson process with constant rate U per capita per generation. The phenotypic effects of the mutations are drawn from a multivariate normal distribution, with multiple mutations having additive effects on phenotype. Fitnesses are computed according to the FGM using Eq.[1]. With such a Poisson offspring distribution, the reproductive variance of genotype i is σ_i = 1 + r_i ≈ 1, assuming small growth rates r_i ≪ 1, in per-generation time units. So this particular demographic model satisfies our assumption of constant in spite of changes in the genotypic composition of the population: here . Note also that the analytical derivations (relying on a Feller diffusion (1951) approximation) approximately cover other demographic models (e.g. birth death models, see Martin et al. 2013; Gomulkiewicz et al. 2017), as long as they also satisfy constant.

Results

Rescue probability: general form

In spite of their difference in the population genetics underlying ER, the two scenarios with or without standing variance yield similar expressions for the probability of ER (as shown in Eqs.(A5) and (A11-A12) in Appendix I): where the particular form of the functions f(.) and h(.) depend on the chosen scenario. Here, cosh(z) = (e^z + e^−z)/2 is the hyperbolic cosinus and we introduce two scaled parameters: ϵ = n μ/(2 r_max) and y_D = r_D/r_max (recall that depends on both mutation rate and effect). The parameter y_D describes how fast the initial clone decays, compared to how fast the optimal genotype grows, it gives a scaled measure of the harshness of the stress imposed (see also Anciaux et al. 2018). The parameter ϵ is the ratio of the mutation load (at mutation-selection balance) and the maximal absolute growth rate that can be reached in the stress. Certain extinction by lethal mutagenesis occurs whenever the load is equal or larger than the maximal growth rate (i.e. ϵ ≥ 1). A small value of ϵ means that we are far from this certain extinction regime.

The quantity ω in Eq.[5] is akin to a ‘per lineage rate of rescue’. Analytical expressions for this rate are either unavailable (DN) or complicated (DN + SV, Eq.(A12) in Appendix I).

Weak selection, intermediate mutation approximation

To get a more direct insight into the impact of each parameter, we sought an approximate expression for ω, detailed in Appendix I section III and IV. This approximation applies with an intermediate mutation rate, where ER mainly depends on the early adaptation of the population to stress, and not on the ultimate mutation load (lethal mutagenesis). More precisely, it requires that the WSSM approximation be accurate while ϵ remains small, which implies a small λ and intermediate U: n² λ/4 ≪ U ≪ r_max. This is why we call this limit a ‘weak selection, intermediate mutation’ approximation.

Under this approximation, the per capita rate of rescue ω takes a roughly similar form for both scenarios with or without standing variance (detailed in Eqs.(A8) through (A14) in Appendix I):

In both cases, the function g(.) is positive and increases (roughly log-log linearly) with y_D = r_D/r_max. The accuracy of this approximation is illustrated in Supplementary Figures 2 and 3 and Figures 1 and 2. Note that, whenever the approximation applies, the ER probability is independent of dimensionality (n).

Figure 1:

ER probability against decay rate r_D for a population without standing variance (DN). Dots show the results of 10² simulations; thin plain lines: the SSWM approximation (Eq.[6] from Anciaux et al. 2018); thick plain lines: the WSSM approximation in Eq.[5]; dashed lines: the corresponding closed form expression in Eq.[6]. The shaded area corresponds to the extra contribution to ER from multiple mutants, compared to single mutants. All models and simulations are shown for a high mutation rate (blue) or a low mutation rate (brown), indicated in legend. Other parameters are N₀ = 10⁵, n = 4, r_max = 1 and λ = 5.10⁻³.

Figure 2:

ER probabitity against mutation rate U for a poputation without standing variance (DN). Same tegend as Figure 1. Att modets and simutations are shown for a high decay rate (btue) or a tow decay rate (brown), indicated in tegend. Other parameters are N₀ = 10⁵, n = 4, r_max = 0.5 and λ = 5.10-³.

Sharp decay in ER probability with increasing stress levels

A possible measure of stress intensity in ER is the rate of decay r_D of a population after the environmental change (see also Anciaux et al. 2018). However, stress might also affect other parameters of the FGM: the height of the fitness peak r_max, the mutation rate U or the variance of mutational effects: λ. We detail the two latter effects (which affect ER via the composite parameter ), in the next section, and focus here on r_max and r_D. As in Anciaux et al. (2018), increased r_D means both a faster decay (purely demographic effect of stress) and a larger shift in optimum (which affects the whole distribution of fitness effect of mutations), which both decrease the ER probability. On the contrary, increasing r_max increases the ER probability through two effects. First, the size of the phenotypic space of resistance increases with r_max (as in the SSWM regime, see Anciaux et al. 2018). Second a large r_max counterbalances a high mutational load n μ/2 as can be seen in Eq.[2] (this latter effect is only captured in the WSSM approximation).

Figure 1 illustrates how the ER probability drops sharply with the decay rate r_D, both in the SSWM regime (brown curves, U ≪ U_c, see legend) and in the WSSM regime (blue curves, U ≫ U_c). This qualitatively similar behavior is a priori due to common geometric constraints imposed by the FGM. The alternative approximations (SSWM versus WSSM) capture a different and complementary portion of the range of possible mutation rates: compare the blue (U = 10²U_C) vs. brown (U = 10⁻²U_C) curves and dots in Figure 1. Higher mutation rates allow withstanding higher stress levels (large r_D), but it is not their only effect, as we now detail.

Non-monotonous relationship between ER probability and mutational parameters

In the following section, we now investigate the effect of mutational parameters. Both the mutation rate U and the variance of mutational effects λ affect the system in a similar fashion through the composite parameter . At small t (in Eq.[4]), an increase in μ speeds-up the early adaptive process, thus favoring rescue but also increases the ultimate mutation load, favoring extinction by lethal mutagenesis. These antagonistic effects of μ create a non-monotonic relationship between the rescue probability and mutational parameters. This is illustrated in Figure 2, which also shows how Eq.[5] (thick lines) captures this effect, through the parameter ϵ = n μ/(2 r_max)). For low stress r_D, P_R is maximal and approximately equal to 1 over a range of mutation rates (plateau Figure 2, see also Supplementary Figure 6 for higher stress values). Beyond this range, the rescue probability in Eq.[5] drops to 0 at . U_max is the mutation rate beyond which certain extinction is enforced by lethal mutagenesis because the mutation load (μ n/2) is larger than the maximal growth rate that can be reached in the stress (r_max). Hence, even if ER allowed the population to invade the new environment, it could not generate a stable population once at mutation-selection balance.

Note that Eq.[6], which is only valid for intermediate μ, does not capture the decrease in P_R close to U = U_max. It does however capture the increase in P_R as mutation rate increases, far below U_max.

Evolutionary dynamics from an initially polymorphic population (at mutation-selection balance)

The results presented in the previous figures illustrate rescue from de novo mutations. In the presence of additional standing genetic variation, rescue mutants can arise from de novo mutants, from preexisting genotypes or from a combination of both. Figure 3 shows the qualitative similarity between the case with and without standing genetic variation, in their dependence on r_D and U, as observed in simulations and captured by Eqs.[5] and [6]. Indeed, Figure 3 confirms that the addition of standing genetic variation does not qualitatively modify the relationship between the rescue probability and stress intensity (r_D, Figure 3a) or mutational parameters (here U, Figure 3b). Note that the accuracy of Eq.[5] is lower for higher r_max, where the continuous time approximations become less accurate to capture discrete time simulations (see Supplementary Figure 4). In the next sections, for the sake of clarity and simplicity, we will mainly discuss the scenario of ER from de novo mutations only, as the qualitative behaviors are similar with an extra contribution from standing variance.

Figure 3:

ER probabitities from de novo mutants onty (btue: Eq.[5], btack dashed tine: Eq.[6]) or from both de novo and pre-existent mutants (green: Eq.[5], red dashed tine: Eq.[6]) against the stress tevet r_D (a) or the mutation rate U (b). Dots show the resutts of 10² simutations (started from 10 simutated poputations at mutation-setection batance for the DN + SV scenario). The shaded area corresponds to the contribution from the standing genetic variance to the rescue compared to de novo mutation. Other parameters are N₀ = 10⁵, r_max = 0.5, n = 4 and λ = 5.10^-3. U = 10 U_c in panets (a) and r_D = 1.8 in panet (b).

Mutation window for ER

In the previous subsections, we have shown that the ER probability drops sharply with increasing stress and is maximal over a finite range of mutation rates, which we denote “mutation window” for ER. The “width” (range of mutation rates) and “height” (maximum of ER probability over the range) of this window strongly depends on stress.

Width of the window

To characterize the mutation window, its upper and lower bounds must be defined. The lower bound of the window (denoted U_*) corresponds to the mutation rate at which the rescue probability rises to 1/2. Thus, this lower bound is only defined if the height of the window lies above or at 1/2 (max(P_R) ≥ 1/2). The upper bound is set to the mutation rate U_max beyond which certain extinction is enforced by lethal mutagenesis. The ER probability drops off very sharply close to U_max, so that, approximatively, ER is only likely within the mutation window U_* ≤ U ≤ U_max. These two bounds are derived in Appendix I Eq. (A17): where W(.) is the Lambert W function, which converges to W(z) ≈ log(z/log(z)) as z gets large (yielding the right hand side approximation above). Here, g(y_D) is the function of stress intensity given in Eq.[6], which describes how stress intensity (y_D = r_D/r_max) affects ER rates. Depending on the scenario, one uses g(y_D) = g_DN+SV(y_D) or g(y_D) = g_DN(y_D) in the presence or absence of standing variance, respectively (note that the window is always wider in the latter case).

Figure 4 illustrates, for an initially clonal population, the sharpness of the transition from ER being almost certain to being highly unlikely (P_R = 1 to P_R = 0) as a function of U and r_D. It also shows the accuracy of the approximation for U_* in Eq.[7] (dashed black line), compared to its numerical estimation from Eq.[5] (color gradient). Supplementary Figure 5 shows a similar result for a population starting with standing genetic variance.

Figure 4:

ER probabitities from de novo mutants onty (Eq.[5]) for different vatues of r_D and U. The cotor gradient gives the vatue of P_R (see tegend). The red straight tine corresponds to U = U_max (Eq.[7]) and the btack dashed tine corresponds to U = U_* (P_R = 1/2) (Eq.[7]). For a given U, the ER probabitity drops sharpty from P_R = 1 (light yettow) to P_R = 0 (blue), over a short increase in r_D. For a given r_D, P_R rises sharpty as U increases about U_* and then drops sharpty as U increases around U_max. Other parameters are r_max = 1, N₀ = 10⁴ and λ = 5.10^-3. n = 4 in panet (a) and n = 8 in panet (b).

The upper bound U_max is independent of initial conditions or stochasticity, as lethal mutagenesis depends on the deterministic equilibrium state of the population, once adapted to the stress. Therefore, U_max does not depend on the presence or absence of initial standing variance, the decay rate imposed by the environmental change (y_D) or the initial population size N₀. On the contrary, the lower bound U_* depends on these factors as it is determined by the capacity of the population to transiently adapt to the new conditions. It shows, however, little dependence on dimensionality (n), as is also apparent in Figure 4, by the accuracy of Eq.[7], which is independent of n. Overall, the width of the mutation window where ER is likely decreases with increasing stress r_D (Figure 4) and increases with initial population size N₀ (eq. [7]: U_* increases with N₀, and U_max is unchanged). It decreases with dimensionality n and increases with the maximum fitness in the new environment r_max, but only because the upper bound of the window, U_max (eq. [7]), decreases with these parameters.

Finally, note that we focused on the effect of variation in U here, but similar results could be obtained if λ was varied (as both parameters affect P_R as a product ). This is apparent in Eq.[7] where U and λ could be exchanged.

Height of the window and “Mutation proof extinction”

within the mutation window (U_* ≤ U ≤ U_max), the ER probability P_R rises above 50% and then drops back to zero. Yet, for more extreme stresses, it cannot even reach above 50% for any value of the mutation rate: the height of the mutation window lies below 1/2). When this height is low, extinction is ‘mutation proof’, in that it is highly likely whatever the mutation rate(s) U or the variance of mutational effects λ in the population. To illustrate this, we compute the maximum of the ER probability max(P_R) when U varies from U_c to U_max, by numerically evaluating Eq.[5] over this range. Supplementary Figure 6 shows detailed profiles of ER probabilities against mutation rates (illustrating how max(P_R) is found), in the presence or absence of initial standing variance. Figure 5 shows the maximum P_R attainable as a function of r_D and N₀: it drops (transition from yellow to blue areas) with increasing stress r_D and decreasing population size N₀. In this example, a large part of the combinations of the two parameters N₀ and r_D correspond to max(P_R) lower than 10% (blue area below the lower white dashed line in Figure 5). Therefore, for a given inoculum size N₀, there is always a threshold of stress level beyond which ER is nearly impossible, whatever the mutation rates in the population.

Figure 5:

Maximum ER probability reached as U is varied, for different values of r_D and N₀ for a population with no initial polymorphism. The color gradient gives the value of max(P_R) this time (see legend). The black dashed line gives the value of (Eq.[8]) and the white dashed lines the value of and . The maximum of the ER probability attainable (for all possible U) drops sharply over a short range of increasing r_D for a given N₀, or over a short range of decreasing N₀ for a given r_D. Other parameters are r_max = 0.5, n = 4, and λ = 5.10^-3.

A closed form approximation can be obtained to describe this transition (detailed in Appendix I section VII): denote the value of r_D at which max(P_R) = p for some p ∈ [0,1]. We obtain the following simple expression for the threshold of level of stress beyond which max(P_R) cannot exceed some level p, independently of μ:

Setting p ≪ 1 in Eq.[8] thus provides the stress level beyond which ER is very unlikely, whatever the mutation rate U or phenotypic variance λ. This means, in particular, that the evolution of higher mutation rates (via hypermutator strains) or higher phenotypic variance (larger λ) would not allow the population to avoid extinction, when confronted to this stress level. The validity of the heuristic in Eq.[8] is illustrated in Figure 5, where we see that the dashed lines ( with p = {0.1,0.5,0.9} see legend) accurately predict the transition from high to low values of max(P_R), computed numerically from Eq.[5].

This whole argument applies to both DN and DN + SV scenarios (by choosing δ accordingly in Eq. [8]). Interestingly, we can also see that populations initially at mutation-selection balance (standing genetic variation) can withstand stresses twice larger or maximal growth rates twice smaller than populations only adapting from de novo mutations (initially clonal).

Distribution of extinction times

From Eq.[4] we can derive the probability density φ(t) of this distribution, in either of the two scenarios considered (purely clonal population DN or population at mutation-selection balance DN + SV). We get: where the functions f(.) and h(.) depend on the scenario considered and is given explicitly in Eq.[5]. Figure 6 illustrates the accuracy of this result and how the distribution of extinction times varies with stress intensity r_D and mutation rate U. In spite of neglecting evolutionary stochasticity, Eq.[9] still captures the shape and scale of extinction time distributions, in the WSSM regime.

Figure 6:

Density of the extinction probabitity dynamics for different vatues of U (panet (a) and (c)) and r_D (panet (b)). The distributions of extinction times from simutations started with an isogenic poputation are shown by shaded histograms, with the corresponding theory (Eq.[9]) given by the ptain red tines. The cotor gradient corresponds to increasing tevets of r_D (panet (b)) or of U (tower range in panet (a) and higher range in panet (c), as indicated on the tegend). Each simutated distributions is drawn from 1000 extinct poputations among a varying number of repticates depending on the ER probabitity. The mutation rates covered in panet (a) are U = {0; 4 U_c; 8 U_c; 14 U_c} and in panet (c) are U = {2 U_max; 1.5 U_max; 1.1 U_max} and in both panet the decay rate is r_D = 0.28. The decay rate covered in panet (b) are {0.285; 0.33; 0.4; 0.5} and the mutation rate is U = 15 U_c. Other parameters are N₀ = 10⁵, n = 4, r_max = 0.1 and λ = 5.10^-3.

Figure 6b shows that decreasing the stress level (r_D) increases the mean persistence time of the population and also increases the variance of this duration. This behavior could be seen as a mere scaling: even in the absence of evolution the population takes longer time to get extinct with a smaller r_D. On the contrary, increasing the mutation rate, keeping it below the lethal mutagenesis threshold (0 ≤ U ≤ U_max, Figure 6a), increases the mean and the variance of the persistence time. This likely stems from subcritical mutations (0 > r > r_D, beneficial but not resistant) that can transiently invade the population, thus delaying its extinction. However, beyond the lethal mutagenesis threshold (U > U_max, Figure 6c), the trend is reversed: extinctions (which are always certain then) occur faster at high mutation rates (panel c). Therefore, even in those cases where ER probabilities are uninformative (P_R = 0), the distribution of extinction times conveys important information on the underlying adaptive or maladaptive dynamics.

Discussion

We investigated the effect of an abrupt environmental change on the persistence of asexual populations with a large mutational input of genetic variance (WSSM regime), adapting either from de novo mutations arising after the environmental change (DN scenario) or from both de novo and pre-existing mutations (DN + SV scenario). In a previous study (Anciaux et al. 2018), we studied evolutionary rescue when considering adaptation over a phenotype-fitness landscape (FGM), which implies pervasive epistasis between multiple mutations and imposes a relationship between the initial decay rate of the population, the proportion and growth rate of resistance alleles, and their selective cost in the ancestral (before the stress) environment. However, we assumed that rescue resulted from rare mutations with strong effects (SSWM). The key contributions of the present model, building on our previous work, are to (i) allow for the cumulative effect of multiple mutations and (ii) provide insights into the distribution of extinction times in the presence of an evolutionary response.

Single step (SSWM regime) vs. multiple step (WSSM) regimes in ER

In spite of its complexity, the ER process in high mutation rate regimes can readily be captured by simple analytic approximations (Eqs.[5] and [6]), which neglect evolutionary stochasticity and only account for demographic stochasticity. Overall, the SSWM and WSSM approximations roughly capture complementary domains of the mutation rate spectrum (Figures 1–3).

This approach shows how multiple mutations allow withstanding higher stress than what the single step approximation (SSWM in Anciaux et al. 2018) predicts (Figures 1 and 3). However, this is only true for intermediate mutation rates: a further increase in mutation rate ultimately shifts the system to a lethal mutagenesis regime (Figures 2 to 4). Indeed, the dependence between the ER probability and the mutation rate is not monotonic. The model shows an optimal mutation rate for the ER probability, at which the maximal ER probability may be less than 1 (depending on the stress, Figure 5). Beyond this rate, the ER probability drops down, to some point (U_max, Eq. [7]) where the mutation load is so large that absolute fitness is negative at mutation selection balance. This non-monotonic dependence reflects the continuum between ER and lethal mutagenesis along a gradient of mutation rates.

The strategy used here (Eq. [4]) to cope with multiple mutations could in principle be applied to many different models: in fact to any for which deterministic mean fitness trajectories are known (equivalent to eqs. [2] or [3]). In particular, it could be used to extend the two broad classes of ER models defined in (Alexander et al. 2014). First, the ‘origin-fixation’ models, which consider single step rescue with stochastic demography (e.g. Orr and Unckless 2008, 2014), could be extended to multiple step rescue, as was done for (Anciaux et al. 2018) in the case of the FGM. Second, the ‘quantitative genetics’ models of ER, which consider deterministic evolution and demography, could be extended to account for stochastic demography. As these models are also obtained in the same limit of weak selection strong mutation (and loose linkage between loci) they should a priori be easy to handle with the approximation in Eq. [4]. The DN + SV scenario here, shares strikingly similar properties (in its phenotypic evolution part) with models of sexual evolution that assume a constant genetic variance for traits (also maintained by recombination in the latter model, not only by mutation). We conjecture that our results would easily extend to this context.

Comparison to existing ER models

Some of our key previous findings (Anciaux et al. 2018), regarding how ER depends on the parameters of the fitness landscape (FGM here) are still valid in the more polymorphic WSSM regime. The main common features are (i) the sharp decrease of ER probabilities with stress (decay rate r_D), (ii) their log-linear increase with initial population size N₀, (iii) the fact that standing variance allows to withstand higher stress, (vi) the limited effect of dimensionality n (eq. [6]). The effect of initial population size (P_R = 1 – e^−N₀ω eq.[5]) is exactly the same as in previous models where ER stemmed from single mutants (SSWM regime Orr and Unckless 2008, 2014; Martin et al. 2013; Anciaux et al. 2018). This is expected of any model ignoring evolutionary and demographic interactions between individuals: each of the N₀ lineages initially present contributes independently to ER (with some rate ω per individual). Decay rate has a broadly similar (but quantitatively different) effect in previous ER models not based on a fitness landscape. The other parameters (r_max, n,λ) are not defined outside the FGM. More generally, the key implications of considering the FGM to model ER are detailed more thoroughly in Anciaux et al. (2018).

Experimental test and parametrization

To experimentally test the prediction, the assumptions of the WSSM regime a priori imply to use organisms with relatively high mutation rates, such as viruses, highly mutating strains of bacteria or possibly cancer cells. Empirically mutation effects and rates, at least in microbes, are typically scaled by growth rates. In our model, these scaled parameters are and u ≡ U/r_max. If r_max is approximately equal to a birth rate (i.e. optimal genotype have a small death rate) s is akin to a mean mutation effect per division, while u is akin to a mutation rate per division. Expressed in these parameters, the WSSM approximation (Eq.[5]) applies when , lethal mutagenesis occurs when and the intermediate mutation weak selection approximation (Eq.[6]) is valid when u_c ≪ u ≪ 1.

For a complete experimental test of the predictions, the parameters N₀, U, r_D, r_max, λ and n must be measured. The methods and challenges in estimating these parameters are discussed in Anciaux et al. 2018). Note however that (i) the results only depend on the product and that (ii) an error on the dimensionality may not be critical given its relatively small effect. Furthermore, if the model is valid, fitting an observed distribution of extinction times (Figure 6) might provide estimates of most parameters of the model. This would allow using not only the information from rescued populations but that from extinct ones. Such empirical test would require fine-scale time series of the population size or at least the extinction status over time.

Treatment against pathogens, hyper-mutators and lethal mutagenesis

Our model as well as previous ones all suggest that the effectiveness of a given treatment depends on the mutation rate of the organism. Polymorphism for mutation rate and invasion of hyper-mutator genotypes are thus potentially important issues for treatments against pathogens. However, our results suggest that a sufficiently strong stress could be effective in spite of hyper-mutator evolution. Indeed, because of the lethal mutagenesis effect (Figure 2), ER is only possible within a mutation rate window: a hyper-mutator would have to hit this window to be advantageous, and the width of the window narrows with increased stress (Figure 4). At sufficiently higher stress levels, ER is unlikely whatever the mutation rate (Figure 5), making these strong treatments robust to hyper-mutator evolution. Whether this pattern is confirmed empirically and whether the required treatment levels are then not too harmful for the treated subject remain open questions. Note also that the same line of argument could be used, not for mutation rate evolution, but for the evolution of phenotypic variance, as the end result depends on the product μ² = U λ.

Our model covers the continuum from stress induced extinction to extinction induced by lethal mutagenesis. The latter might be an option, especially for organisms for which no “stress treatment” exists, or whose high mutation rates (above U_* in our model) allows them to withstand even strong stresses. Our results indeed confirm that increasing the mutation rate in this context (above U_max) will allow to fully eliminate the population. The addition of a stressor might also help in the process, as has been suggested before (Pariente et al. 2001, 2003, 2005): indeed, the ER probability does drop faster with increasing U (as we approach U_max) in the presence of a strong stress (with high y_D), see Supplementary Figure 6.

Treatment duration issues

One may also wonder how long a treatment must last (be it by stress effect or by lethal mutagenesis) for it to be efficient. From the distribution of extinction times (Figure 6), it is possible to predict the duration of the treatment needed to get rid of, say, 99% of the pathogen populations. Strong stresses (dark blue histogram in Figure 6b) tend to both decrease the overall ER probability and shrink the distribution of times to extinction. This means they are more efficient overall and require shorter treatment durations, with less risk associated with imperfect treatment compliance. However, when the mutation rate increases towards U_max (treatment by lethal mutagenesis), although extinction becomes highly likely (P_R → 0, Figure 2), extinction times get more variable and the mode of the distribution is higher (Figure 6a). If the mutation rate increases beyond U_max, the opposite pattern is observed: the distribution of extinction times shrinks as U increases further away from U_max (Figure 6c). Hence, a treatment by lethal mutagenesis, even if guaranteeing total extinction after infinite time, may need to be applied for a long time to significantly decrease ER probability (if the resulting U is close to U_max). Thus, observed distributions of extinction times hold valuable information.

Limits of the model

Obviously, the different results and applications described above are limited by the hypotheses of the model. First, the model ignores density or frequency-dependent effects. Moreover, the approximation of the demographic dynamics by a Feller diffusion imposes that demographic variations remain smooth, which may be inaccurate, e.g. for some viruses showing occasional large burst events. The Feller diffusion can also fail to predict discrete time demography (used here in the simulations) at high growth rates per generation (r ≥ 1). This could be overcome by using a numerical analysis of discrete time inhomogeneous branching processes instead of the Feller diffusion. Second, the model assumes a WSSM regime, which excludes its application to organisms with lower mutation rates (relative to selection intensity U < U_c). This could however be handled by complementary models under the SSWM regime such as (Anciaux et al. 2018), under the same landscape assumptions. Finally, our results assume a sharp change in the environment, which does not reflect all forms of stresses, including in the context of sudden treatment (e.g. antibiotics can have complex pharmacokinetic patterns over time Regoes et al. 2004). In that respect, extensions to more complex ecological scenarios might be useful: some are a priori possible using the same broad modelling framework as used here.

Acknowledgments

This work benefited from discussions with S. Gandon, R. Gomulkiewicz, O. Tenaillon and F. Débarre. This work was supported by the French Ministry of Higher Education, Research and Innovation (MESRI allocation doctorale to Y.A.), Agence Nationale de la Recherche (ANR-13-ADAP-0016 “Silentadapt” to G.M. and ANR-13-ADAP-0006 “MeCC” to O.R., ANR-14-CE25-0013 “NONLOCAL” to L.R.). This is ISEM publication ISEM 2019-XXX.

Footnotes

• Conflict of Interest Statement: The authors declare no conflict of interests.

• Data Accessibility Statement: The authors state that there is no data to be archived.

Authors contributions

Y.A. O.R. and G.M. initiated the idea for the study. YA, A.L, L.R and G.M derived the mathematical results. Y.A. performed the simulations. Y.A. O.R. and G.M. drafted the paper. All authors critically revised the manuscript, and gave approval of the final version for submission.