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Te ratio m=(md), in logarithmic scale. All numerical benefits are averaged more than 100 simulation runs, along with the heatmap is interpolated. Solid lines: bounds from the interval in Eq. 14. Dashed line: value of d above which a nonsubdivided population crosses the MedChemExpress KKL-35 valley quicker than an isolated deme. Dotted line: value of d above which an isolated deme is inside the tunneling regime. Dash-dotted line: value of d above which the non-subdivided population is in the tunneling regime. Parameter values: d 0:1, m 5|ten{6 , s 0:3; d and m are varied. B. Similar heatmap for the ratio tm =tns of the average valley crossing time tm of a metapopulation to that tns of a non-subdivided population (with K 500). Solid line: predicted value of d, from Eq. 8, for which the largest speedup by subdivision is expected. doi:10.1371/journal.pcbi.1003778.gBoth Eq. 19 and Eq. 20 show that increasing the number D of demes decreases the range where the highest speedup by subdivision is reached. This is because having more subpopulations makes the spreading of the beneficial mutation slower. In addition, we find that the bound on R is proportional to 1=m. Hence, despite this bound, the interval where subdivision most accelerates plateau crossing can span several orders of magnitude, given the small values of the actual mutation probabilities m in nature.Effect of varying the degree of subdivision of a metapopulationAn interesting question raised by our results regards the optimal degree of subdivision. Given a certain total metapopulation size, into how many demes should it be subdivided in order to obtain the highest speedup possible We PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/20170650 first attack this question using our analytical results, and then we present simulation results, which allow for going beyond the best scenario and its associated parameter window. Let us consider a metapopulation of given total size N ND. Our analytical results show that increasing subdivision, i.e. increasing the number D of subpopulations at constant N , leads to stronger speedups of valley crossing (see Eqs. 4 and 7, with N N =D). However, Eqs. 16 and 18, and the previous paragraph, show that when D is increased, the parameter range where the speedup by subdivision tends to the best-scenario value becomes smaller and smaller. Eventually, this parameter range ceases to exist altogether: this occurs when R becomes of order 1 and below. This sheds light on an interesting trade-off in the degree of subdivision D, between the magnitude of the optimal speedup gained by subdivision and the width of the parameter range over which the actual speedup is close to this optimal value. This effect can be observed qualitatively in Fig. 3A, where the valley crossing time tm of a metapopulation with fixed total size is shown versus the migration-to-mutation rate ratio, m=(md), for different values of D: when D is increased, the minimum becomes deeper but less broad. In addition, Eqs. 15 and 17 show that when D is increased, the lower bound of the interval where the speedup by subdivision tends to the best-scenario value decreases, as D log D for plateaus (Eq. 17) and even more rapidly for deep valleys (Eq. 15). Qualitatively, this is because spreading of the beneficial mutation gets longer when D increases. Conversely, the upper bound of this parameter range is independent of D for deep valleys (Eq. 15), and grows only logarithmically with D for plateaus (Eq. 17). Hence, when D is increased, the center of the interval where the actual speedup is close to t.