Establishment of New Mutations in Changing Environments

Abstract

Understanding adaptation in changing environments is an important topic in evolutionary genetics, especially in the light of climatic and environmental change. In this work, we study one of the most fundamental aspects of the genetics of adaptation in changing environments: the establishment of new beneficial mutations. We use the framework of time-dependent branching processes to derive simple approximations for the establishment probability of new mutations assuming that temporal changes in the offspring distribution are small. This approach allows us to generalize Haldane's classic result for the fixation probability in a constant environment to arbitrary patterns of temporal change in selection coefficients. Under weak selection, the only aspect of temporal variation that enters the probability of establishment is a weighted average of selection coefficients. These weights quantify how much earlier generations contribute to determining the establishment probability compared to later generations. We apply our results to several biologically interesting cases such as selection coefficients that change in consistent, periodic, and random ways and to changing population sizes. Comparison with exact results shows that the approximation is very accurate.T HE process of adaptation depends on the establishment of new beneficial mutations. To be successful, it is not sufficient that mutations simply enter a population; they have to survive an initial phase of strong random genetic drift during which they can be lost. Once a beneficial mutation rises to a sufficiently large number of copies, the strength of drift becomes negligible and selection drives them to fixation or maintains them at intermediate frequency. The probability that mutations survive loss has been called the probability of fixation, the probability of establishment, or the probability of invasion, depending on the context. This probability plays an important role in determining the rate of adaptation of populations (Gillespie 2000;Orr 2000).Fixation probabilities of beneficial, neutral, or deleterious mutations have been at the core of population genetics theory since the early days of the field. Traditionally, two alternative frameworks have been used: branching processes (Fisher 1922) and diffusion approximations (Kimura 1962). Branching processes allow for the derivation of simple analytical results. Simulations show that these results are accurate, provided the population size is sufficiently large such that N s 1. The disadvantage of this approach is that it is limited to beneficial and initially rare mutations. Diffusion approximations on the other hand are more powerful and flexible. One can study deleterious or neutral mutations of arbitrary initial frequency. The downside is that it is often impossible to obtain simple analytical results and the underlying assumptions are often less intuitive than in branching processes.Using branching processes, Haldane (1927) derived his famous result that the probability of fixation of a benefici...

Evolutionary rescue by beneficial mutations in environments that change in space and time

“…A more general version of equations (2.3) and (2.4) can be found in Peischl & Kirkpatrick [46]. There are two immediate implications.…”

“…). Then, the fixation probability is again given by equation (2.2) where n t is the number of mutant copies alive at generation t [46]. Equation (2.6) shows that the PGF that determines mutant survival is the geometric mean of the PGFs for the individual copies of the mutation.…”

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