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Wrote the paper: CL JLS. Conceived and designed the theory and calculations: CL JLS. Performed the calculations: CL. Analyzed the results from theory and calculations: CL.

Replicators such as parasites invading a new host species, species invading a new ecological niche, or cancer cells invading a new tissue often must mutate to adapt to a new environment. It is often argued that a higher mutation rate will favor evolutionary invasion and escape from extinction. However, most mutations are deleterious, and even lethal. We study the probability that the lineage will survive and invade successfully as a function of the mutation rate when both the initial strain and an adaptive mutant strain are threatened by lethal mutations. We show that mutations are beneficial, i.e. a non-zero mutation rate increases survival compared to the limit of no mutations, if in the no-mutation limit the survival probability of the initial strain is smaller than the average survival probability of the strains which are one mutation away. The mutation rate that maximizes survival depends on the characteristics of both the initial strain and the adaptive mutant, but if one strain is closer to the threshold governing survival then its properties will have greater influence. These conclusions are robust for more realistic or mechanistic depictions of the fitness landscapes such as a more detailed viral life history, or non-lethal deleterious mutations.

When a population of replicators faces a sudden change in its environment, often its fitness will decrease so that it has to adapt or face extinction. Examples include when a parasite infects a new host species, when a species is introduced to a new ecological niche, when viruses or bacteria are challenged by an antimicrobial drug administered to their host, or when a cancerous cell invades a new tissue

However, it is known that most mutations are deleterious or even lethal. In the case of viruses, for example, site-directed mutagenesis experiments have shown that 20 to 40% of point mutations in various viruses are lethal

If a replicator does not mutate at all, it never adapts, and then cannot survive environmental changes. But if a replicator mutates too often, it also carries a deleterious mutational load. The concept of mutations as a double-edged sword has been explored in many situations. For example, Bull studied the mean number of adaptive mutants produced by a single episode of mutagenesis

To our knowledge, only two studies have looked at the probability of survival of a replicator's lineage when both deleterious and adaptive strains are within mutational range. Eshel

We provide a more complete and unified analysis of the influence of deleterious and lethal mutations on the phenomenon of evolutionary invasion and escape. We develop and analyze a general stochastic model for the survival probability of a replicator lineage that begins with an arbitrary fitness, and can acquire mutations that are adaptive, deleterious, or lethal. We derive simple, biologically intuitive rules to delineate when mutations are beneficial (i.e. when a positive mutation rate leads to greater survival probability than the limit of no mutations), and in this regime, we calculate the optimal mutation rate (i.e. the mutation rate maximizing the survival probability of the replicator lineage for the environmental change being studied). This model can encompass the earlier results of Eshel

We study asexual replicators, such as cancerous cells, viruses with negligible recombination, or bacteria with negligible horizontal transfer. For sexual replicators the effects of deleterious mutations are mitigated by recombination and exhibit a very different sensitivity to the mutation rate

We consider whether an initial population of replicators in a novel environment leads to establishment of a successful population or goes extinct. Situations where replicators jump to a new environment often involve only a few explorers, or else abrupt environmental change can sharply decrease the size of an extant population. Also any population is most likely to go extinct when there are the fewest replicators present. For all these reasons, we focus on the dynamics of a replicator population that begins at low abundance. When there are few replicators, the interactions between them are limited, and depletion of resources is negligible. Hence for calculating the survival probability we assume that the demographic fates of replicators are independent. Consequently, we use a branching process framework

To analyze the branching processes, we use generating functions which gather the information on the probabilities

A mutation in the genome may change fitness. In the context of adaptation to new conditions, mutations with a significant effect on replicator fitness (e.g. those that modify a reactive site on a protein for instance) can be distinguished from mutations with small effect (e.g. those that marginally alter the thermodynamic stability of a protein)

A mutation at any of the

We consider a replicator of strain

The extinction probabilities

Solving this system numerically shows the dependence of survival probability on mutation rate (black circles in

Exact solution (black circles), approximation for small

To investigate whether mutations are beneficial at all in a given scenario, we study whether a small amount of mutation leads to more survival than no mutations, i.e. whether the initial slope of the survival probability as a function of the mutation rate is positive (orange dot-dashed lines on

If

If the coefficient of

This result can be generalized to the following simple rule (appendix S1.1 in

When mutations are beneficial, there is a finite mutation rate that maximizes the probability of survival for a given environmental change scenario, which we refer to as the optimal mutation rate,

These approximations do not lead to a simple explicit expression for

We have assumed that the risk of lethal mutations is the same for both strains. However in real systems there may be epistatic interactions such that strains have different robustness. Furthermore, from our first analysis we cannot conclude whether the results depend on the lethal mutations threatening the initial or the mutant strains. To explore this, we study a model that has two strains of differing fitness, as in

Once again we determine the regime of beneficial mutations by considering the low mutation rate limit. In this limit, the survival probability of strain 1 depends on the characteristics of strain 2 only via

The optimal mutation rate depends on

Red lines: exact numerical solution showing combinations of

Often a significant increase in fitness requires more than one mutation _{1}, with

Panel (a) represents the mutational map. There are

We consider how increasing the mutation rate affects the survival probability for a population starting from a single replicator of each genotype. Mutations are always a burden for the fittest replicator

Our analysis so far has assumed that deleterious mutations are all lethal, but of course fitness can decrease without going to zero

In all cases the model follows the broad scheme laid out in _{1}) and a fitter strain (_{0} = 0 (lethal) if

Our replicator model is very general, and may need to be adapted to apply to specific systems. As an example, if we describe the dynamics of a virus within a host, a virion may have a very low probability

We assume that a virion of strain _{i}

It appears that this description adds two more parameters to our replicator model. However, it can be shown that

As in the general model, mutations are beneficial when

In principle, there could be situations where a mutation that dramatically increases the number of virions produced by an infected cell (

When the initial strain needs to mutate to survive (

Replicators facing the threat of extinction because of an environmental change may survive thanks to adaptive mutations. Most models of evolutionary invasion and escape have focused on adaptive mutation pathways only and have concluded that higher mutation rates lead to higher probability that a replicator population will survive an environmental change

We have investigated the conditions under which mutations are beneficial, i.e. when some non-zero level of mutation increases the survival probability compared to the limit of no mutations. A sufficient criterion for mutations to be beneficial is when, in the limit of no mutations, the average survival probability of the mutational neighbors is larger than the survival probability of the initial strain. However, this condition is not necessary: if an adaptive mutant is more than one mutational step away, its influence may be felt at higher mutation rates only. In this case a very low rate of mutations may be detrimental to survival, but the maximum survival probability may be obtained for a larger finite value of the mutation rate.

The optimal mutation rate is defined for our analysis as the mutation rate maximizing the survival probability of a replicator's lineage facing a given environmental change. We first studied a general model with one adaptive mutant and lethal mutations threatening both the initial strain and this adaptive mutant. The optimal mutation rate depends on the characteristics of both the initial strain and the adaptive mutant, but if one of the strains is threatened by more lethal mutations or if it is closer to the survival threshold, the optimal mutation rate depends most strongly on the parameters of this strain. More realistic depictions of the fitness landscape – such as a more mechanistic fitness model for viral infections, or deleterious instead of lethal mutations – do not qualitatively change these results.

Gavrilets has studied such a “holey” fitness landscape comprising both fitness peaks and fitness “holes”, but focused on the distribution of a fixed-size population on such a landscape

To the best of our knowledge, only two studies have presented a stochastic model of evolutionary escape where the dependence of the survival probability on the mutation rate is analyzed in the presence of both deleterious and adaptive mutations

In light of these findings, we return to the question of why so many emerging infectious diseases are RNA viruses. Our analysis has shown that their extremely rapid mutation rates are not necessarily a beneficial trait even if evolutionary adaptation is needed to avoid extinction in the new host species. It is possible that the mutation rates exhibited naturally by RNA viruses, while high, are not so high that they cause survival probabilities to decline markedly. This is difficult to judge in general, because even in our simplified model a quantitative estimate of survival probability requires, at minimum, knowledge of the fitnesses of different genotypes and the frequency of deleterious mutations. It is also possible that RNA viruses are common emerging infections for reasons unrelated to their mutation rate, for instance if there is a larger pool of candidate RNA viruses circulating in animal reservoirs to which human populations are exposed (though see

Finally, we place our findings in the context of research on the evolution of mutation rates. Under stable conditions the mutation rate is expected to be small

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The authors thank S.J. Schreiber and M. Park for providing useful comments on earlier drafts of this manuscript.