• Doyle Kirkeby posted an update 8 months, 1 week ago

    This minimum crossing time, which we denote by tc , is also known as the smallest (or initial) order statistic of your deme crossing time amongst a sample of size D [53]. Let us denote by p(t) the probability density function of valley or plateau crossing time to get a single deme, and let us introduce 1=(Dr01 ), because the second step will no longer be negligible (see [52] for a Digitoxin biological activity discussion of comparable issues). The crossover to this regime could be determined by computing the average from the distribution in Eq. 23 and comparing it to 1=(Dr01 ).P(t)p(t’)dt’ (it satisfies P(t) 1{C(t) where C(t) is thet3 Number of migration events for extinction or spreading in a metapopulationIn our Results section, we have derived an interval of the ratio of migration rate to mutation rate over which subdivision most reduces valley or plateau crossing time (see Eq. 14). The upper bound involves ne , the average number of migration events required for the `1′-mutants to be wiped out by migration, starting from a state where one deme has fixed genotype `1′, while all other demes have genotype `0′. Similarly, the lower bound involves ns , the average number of migration events required for the `2’mutants to spread by migration to the whole metapopulation, starting from a state where one deme has fixed genotype `2′, while all other demes have genotype `0′. In our Results section, we have provided intuitive derivations of the simple expressions of ns and ne , valid for Ns 1 and s 1, Nd 1 and d 1 (see Eq. 15). However, it is important to derive more general expressions, especially since subdivision generically most accelerates valley crossing in the intermediate regime where Nd 1 (see Results, Eq. 8). Here, we derive general analytical expressions for ne and ns , both for fitness plateaus and for fitness valleys. These more general expressions are those used for numerical calculations of the bounds in our examples. Throughout this section, we consider a metapopulation of D demes composed of N individuals each, and we assume that individual demes are in the sequential fixation regime (see Results). 3.1 A finite Markov chain. In order to determine ns and ne , we study the evolution of the number i [ ,D of demes that have fixed the mutant genotype (`1′ for the calculation of ne ; `2′ for that of ns ), while other demes have genotype `0′. Given that the value of i just before a migration step fully determines the probabilities of the outcomes of this migration step, and given that i 0 and i D are absorbing states, the number i evolves according to a finite Markov chain, each step being a migration event. We next express the transition matrix of this Markov chain. The only migration events that can affect i are those that exchange individuals from two demes with different genotypes. Let us call these migration events “relevant”. The probability pr of a i migration event being relevant corresponds to the probability that this migration affects one of the i mutant populations and one of the D{i `0′ populations: pr 2i(D{i)= (D{1). We only focus i on the final outcome of a migration event, after fixation or extinction of each of the two migrants’ lineages has occurred.