Evolving mutation rate advances the invasion speed of a sexual species

Background Many species are shifting their ranges in response to global climate change. Range expansions are known to have profound effects on the genetic composition of populations. The evolution of dispersal during range expansion increases invasion speed, provided that a species can adapt sufficiently fast to novel local conditions. Genetic diversity at the expanding range border is however depleted due to iterated founder effects. The surprising ability of colonizing species to adapt to novel conditions while being subjected to genetic bottlenecks is termed ‘the genetic paradox of invasive species’. Mutational processes have been argued to provide an explanation for this paradox. Mutation rates can evolve, under conditions that favor an increased rate of adaptation, by hitchhiking on beneficial mutations through induced linkage disequilibrium. Here we argue that spatial sorting, iterated founder events, and population structure benefit the build-up and maintenance of such linkage disequilibrium. We investigate if the evolution of mutation rates could play a role in explaining the ‘genetic paradox of invasive species’ for a sexually reproducing species colonizing a landscape of gradually changing conditions. Results We use an individual-based model to show the evolutionary increase of mutation rates in sexual populations during range expansion, in coevolution with the dispersal probability. The observed evolution of mutation rate is adaptive and clearly advances invasion speed both through its effect on the evolution of dispersal probability, and the evolution of local adaptation. This also occurs under a variable temperature gradient, and under the assumption of 90% lethal mutations. Conclusions In this study we show novel consequences of the particular genetic properties of populations under spatial disequilibrium, i.e. the coevolution of dispersal probability and mutation rate, even in a sexual species and under realistic spatial gradients, resulting in faster invasions. The evolution of mutation rates can therefore be added to the list of possible explanations for the ‘genetic paradox of invasive species’. We conclude that range expansions and the evolution of mutation rates are in a positive feedback loop, with possibly far-reaching ecological consequences concerning invasiveness and the adaptability of species to novel environmental conditions. Electronic supplementary material The online version of this article (doi:10.1186/s12862-017-0998-8) contains supplementary material, which is available to authorized users.


Environment and fitness
Temperatures in the environment are denoted by τ E . The effect of temperature on survival probability is assumed to be of Gaussian shape: with µ E denoting the environmental mean temperature and s E the standard deviation of temperatures in the environment. Note that we assume that the curve scales to 1 to simplify further calculations, but could also be easily multiplied by an additional factur representing other temperature-independent survival components ( Fig. 1

Population
We further assume a population of individuals living in this environment, which is characterized by a distribution of trait values (i.e. individual optimal temperatures) that is also modeled as a Gaussian distribution with mean µ P and standard deviation s P : (2)

Mutations
We assume that mutations, depending on their strength, affect the variability of the trait distribution at the population level. Assuming that the effect of mutations itself is Gaussian this results in an update of the population-level standard deviation s P to s P :

Environmental change
The environment changes to a certain degree in the next generation, the strength of which is denoted by ∆ τ .
Here we assume no change in the variance, but only the mean of the distribution of temperature values in the environment (i.e. µ E = µ E + ∆ τ ).

Mean fitness computation
To calculate the fitness effect of the given mutation strength we first calculate the product of above functions (i.e. f (τ ) · g(τ )), which gives us the survival probability for all occurring trait values: We integrate h(τ ) over τ to calculate mean expected fitness ω: The graph in Fig. 3 shows survival probability (black line) and fitness distribution (dashed red line) at the population level. The highlighted area below the trait distribution indicates expected mean fitness (calculated via integration, eq. 5). The dashed red line shows the fitness distribution within the population. The highlighted area below the trait distribution indicates expected mean fitness (calculated via integration, eq. 5). Note that for reasons of clarity the original survival probability in response to temperatures before environmental change has been omitted. Parameter values as in Fig. 2.

Calculating optimal mutation rates
In order to calculate optimal mutation rates we simply create a vector of possible mutation rates for each strength of environmental change ∆ τ and compute their fitness effects using equation 5. We then pick the mutation rate (s m ), which maximizes the population's fitness expectations. Results are summarized in Figure

Conclusions
Both the individual-based simulations from the main text and this simple analytical approach lead to the result that a stronger change in the environmental conditions should favor higher mutation rates in order to maximize the populations' fitness expectations. The analytical approach directly demonstrates that the combination of a normally distributed environmental characteristic and a corresponding normally shaped response in survival are sufficient to predict selection pressure on mutation rate.