Xation probabilities. For ns , there’s no difference among the valley along with the plateau, due to the fact genotype `1′ just isn’t involved. As above, Eqs. 26, 27 and 33 yield rk rk , with r defined in Eq. 36. Thus, Eq. 32 givesD{1 X (1{rj )(1{rD{j ) D(D{1) : D )(1{p’)p 2(1{r)(1{r j(D{j) jeffectively neutral, and the above discussion regarding the fitness plateau applies.) Then, p01 de{Nd 1 (see Eq. 1). To lowest (i.e. zeroth) order in p01 , Eq. 37 becomes ne D D , 2p10 2d 4ns83.5 Simplified expressions for deep valleys and for plateaus. In our Results section, we have shown that thewhere we have used the approximation p10 d, which holds for d 1 and Nd 1. This expression of ne coincides with Eq. 11, which is obtained in the Results section through a more intuitive argument that directly assumes d 1 and Nd 1. Hence, from Eq. 40, the upper bound U is U p12 1 s 1z , d 2d 2 5benefit of subdivision is highest when m=(md) is situated between a lower bound, L ns p01 , and an upper bound, U ne p12 , D 0L 9where we used the conditions d 1, Nd 1, s 1 and Ns 1 to simplify the expression of p12 . Meanwhile, from Eq. 39 and 41, the lower bound L takes the form p01 de{Nd D log D , D log D p02 s(see Eq. 14), where p12 denotes the probability of fixation of a single mutant with genotype `2′ in a background of `1′-mutants. Here we present simplified expressions for ns and ne , and hence of L and U, in particular parameter regimes. Throughout this section, we focus on the regime where Ns 1 but s 1, such that mutation `2′ is substantially, but not overwhelmingly, beneficial [28]. We then have p20 se{Ns 1 and PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/20170650 p02 s (see Eq. 1). To leading (i.e. zeroth) order in p20 , we obtain from Eq. 38 that X D{1 D{1 1 D log D , s j 1 j s6where, again, we used the conditions d 1, Nd 1, s 1 and Ns 1 to simplify the expressions of p01 and p02 . Combining Eqs. 46 and 45 yields Eq. 15.4 A buy GSK2838232 population connected by migration to smaller population islandsLet us consider a population of N individuals connected by migration to S smaller population islands with NvN individuals each. These islands of identical size are assumed to be in the sequential fixation regime. For the sake of simplicity, we consider that migration only occurs between the large population and the islands: a migration step is a random exchange of two individuals between the large population and one of the islands (chosen at random at each migration event), and the total migration rate is denoted by M. Here, we focus on the valley or plateau crossing time of the large population. We demonstrate that the evolution of a large population can be driven by that of satellite islands. In the optimal case, the crossing time of the large population is determined by that of the champion island, i.e., that which crosses the fitness valley or plateau fastest. We now determine the conditions under which this optimum is achieved, focusing on migration rates much smaller than division/death rates, M min (dNS,dN ), such that fixation or extinction of a mutant lineage in either the large population or an island is not significantly perturbed by migration. Again, migration should be rare enough for islands to remain effectively shielded from migration events while they have fixed the intermediate mutation, until the final beneficial mutation arises. Second, migration should also be frequent enough for the spreading time of the final beneficial mutation from the champion island to the large population to be negligible with.
