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Mutation Rate Evolution in Replicator Dynamics

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Abstract

The mutation rate of an organism is itself evolvable. In stable environments, if faithful replication is costless, theory predicts that mutation rates will evolve to zero. However, positive mutation rates can evolve in novel or fluctuating environments, as analytical and empirical studies have shown. Previous work on this question has focused on environments that fluctuate independently of the evolving population. Here we consider fluctuations that arise from frequency-dependent selection in the evolving population itself. We investigate how the dynamics of competing traits can induce selective pressure on the rates of mutation between these traits. To address this question, we introduce a theoretical framework combining replicator dynamics and adaptive dynamics. We suppose that changes in mutation rates are rare, compared to changes in the traits under direct selection, so that the expected evolutionary trajectories of mutation rates can be obtained from analysis of pairwise competition between strains of different rates. Depending on the nature of frequency-dependent trait dynamics, we demonstrate three possible outcomes of this competition. First, if trait frequencies are at a mutation–selection equilibrium, lower mutation rates can displace higher ones. Second, if trait dynamics converge to a heteroclinic cycle—arising, for example, from “rock-paper-scissors” interactions—mutator strains succeed against non-mutators. Third, in cases where selection alone maintains all traits at positive frequencies, zero and nonzero mutation rates can coexist indefinitely. Our second result suggests that relatively high mutation rates may be observed for traits subject to cyclical frequency-dependent dynamics.

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Notes

  1. Perhaps contrary to intuition, the value of σ x is in general undefined on the individual fixed points of such a heteroclinic cycle. This is because, if v is a fixed point on this cycle, and if all fixed points are hyperbolic—the generic case—then σ x ({v}) will not converge according to the limit definition (1) (Gaunersdorfer 1992; Sigmund 1992). Thus, according to our definition, the singleton {v} is a non-measurable subset, as is any proper subset of the vertex set. However, this does not affect our current argument, which requires only that the entire vertex set is assigned probability one.

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Acknowledgements

We thank Yaneer Bar-Yam, David Fried, Glen R. Hall, Aaron Hoffman, Yoh Iwasa, Christopher J. Marx, Martin A. Nowak, Mike Todd, Mary Wahl, John Wakeley, C. Scott Wylie, and an anonymous referee for insightful discussions and comments. Financial support was provided by the National Science Foundation Graduate Research Fellowship Program (D.I.S.R.) and the Foundational Questions in Evolutionary Biology initiative of the John Templeton Foundation (B.A.).

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  1. Program for Evolutionary Dynamics, Harvard University, Cambridge, MA, 02138, USA

    Benjamin Allen

  2. Department of Mathematics, Emmanuel College, Boston, MA, 02115, USA

    Benjamin Allen

  3. Program for Evolutionary Dynamics, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, 02138, USA

    Daniel I. Scholes Rosenbloom

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Allen, B., Rosenbloom, D.I.S. Mutation Rate Evolution in Replicator Dynamics. Bull Math Biol 74, 2650–2675 (2012). https://doi.org/10.1007/s11538-012-9771-8

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