Mutation Rate Evolution in Replicator Dynamics
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Mutation Rate Evolution in Replicator Dynamics Benjamin Allen · Daniel I. Scholes Rosenbloom
Received: 29 November 2011 / Accepted: 16 August 2012 / Published online: 1 September 2012 © Society for Mathematical Biology 2012
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
B. Allen () Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA e-mail: [email protected] B. Allen Department of Mathematics, Emmanuel College, Boston, MA 02115, USA D.I.S. Rosenbloom Program for Evolutionary Dynamics, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA
Mutation Rate Evolution in Replicator Dynamics
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second result suggests that relatively high mutation rates may be observed for traits subject to cyclical frequency-dependent dynamics. Keywords Mutation rate · Evolution · Replicator dynamics · Adaptive dynamics · Evolvability
1 Introduction The mutation rate of an organism is subject to heritable variation, and therefore evolves (Weber 1996; Kirschner and Gerhart 1998; Radman et al. 1999; Sniegowski et al. 2000; Earl and Deem 2004; Aharoni et al. 2005; Pigliucci and Box 2008; Wagner 2008). In populations that are well-adapted to a static environment, most mutations are neutral or deleterious, and mutation rates are therefore expected to evolve to levels as low as feasible, given the constraints and costs associated with faithful replication (Kimura 1967; Leigh 1970, 1973; Sniegowski et al. 2000). However, theoretical (Leigh 19
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