Journal of Statistical Mechanics: Theory and Experiment
Volume 2008
April 2008
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Su-Chan Park and Joachim Krug J. Stat. Mech. (2008) P04014 doi:10.1088/1742-5468/2008/04/P04014
Evolution in random fitness landscapes: the infinite sites model
FREE ARTICLE Topical articles on Disorder, Fluctuations and UniversalitySu-Chan Park and Joachim Krug
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| Figure 1. Schematic of the WF model explained in the text. The population size is fixed at N = 5. Time direction is indicated by the arrow (five generations). Circles signify the individuals and different colors mean different genotypes which arise by mutation with probability U. The vertical location of individuals has no significance (the population is assumed to be well mixed, without spatial structure). Initially, all individuals have the same genotype (hence the same color). Line segments connecting individuals show who begets whom. Each individual can have only one parent but a parent can have many offspring. After five generations, the red mutation which arose in a single individual at time two is fixed in the population (see section 4.1 for further discussion of the fixation process). |
| Figure 2. Mean fitness of the WF model for finite and infinite populations. For finite populations, the mean fitness is also averaged over independent runs. The mutation probability U is set to 0.01 and the mutation distribution is g(w) = e–w. From bottom to top, the population size N increases (N = 103, 105, 107 and 109). The number of independent runs is 8 × 106 (N = 103), 8 × 105 (N = 105), 20 000 (N = 107), and 4000 (N = 109). Left panel shows the short-time simulation results and compares them to the infinite population calculation in section 3. Right panel depicts log–linear plots of the mean fitness for finite populations which clearly shows a logarithmic increase of the mean fitness at long times. |
| Figure 3. Log–log plots of mean fitness in the infinite population limit as a function of generation t for exponential (ν = 0 and β = 1 in (13); upper three datasets), Gaussian (ν = 0 and β = 2 in (13); middle three datasets) and Gumbel-type distributions (16) (lower three datasets). For all datasets, w0 and f0(w) are set to 1 and δ(w–1), respectively. For each case, the mutation probability is U = 0.5 (red), U = 0.1 (purple) and U = 0.01 (blue) from bottom to top. The lines are the approximate solutions (15) and (A.5) with the respective parameters. |
| Figure 4. Comparison of the exact numerical solution for the frequency distribution with the traveling wave equation. Left panel: frequency distributions at generation 100, 200 and 400 (from left to right) are shown for U = 0.5 (square) and U = 10–4 (circle). There is a slight mismatch due to the neglect of Ψt(w) but in general the traveling wave solution approximates the true distribution quite well. Right panel: similar study to the left panel with the Gaussian distribution. The data are collected at generation 500, 1000 and 2000. The Gaussian approximation is almost perfect. The frequency distribution for small w is Ug(w) for both panels as reasoned in the text (data not shown). |
| Figure 5. Left panel: semilogarithmic plots of the fitness versus dimensionless time τ = NUt for U = 10–5 (N = 103, 104, 105, 106), U = 10–2 and U = 0.5 (N = 103, 104, 105) obtained from simulations of the WF model. The mutation distribution and the initial frequency distribution are the same as those in figure 2. Right panel: the comparison of the simulation of the WF model to the DRP for N = 103 and U = 10–5 (τ = 10–2t). The difference in the fitness is barely observable. Inset: the same type of comparison for the variance κ2 (see (61)) of the fitness. |
| Figure 6. Comparison of the WF simulation with the record, mean field and the improved approximation scheme in section 4.4 with the stable value of CDRP = 8. The WF simulation results are obtained with the exponential mutation distribution (2) with N = 105 and U = 10–5 (τ = t). As explained in the text, mean field theory and record dynamics give lower and upper bounds on the true asymptotics. |
Figure 7. Log–log plots of  versus τ for the simulation results of the WF dynamics with the power law distribution (17) with
 (upper dataset), α = 2 (middle dataset) and α = 4 (lower dataset). The mutation probability is set to
U = 10–4 and the population sizes are 104 and 105 for and 2, and 105 and 106 for α = 4, respectively. For comparison, the record solution (B.10) (straight line segments above the simulation results) and the MF prediction (53) (straight line segments below the simulation results) are also depicted. |
Figure 8. The results of the approximation scheme for the mean
(CRP and
CDRP) and variance (κ2) including terms up to  for the RP (left panel) and for the DRP (right panel). Since we know the exact value of
CRP and κ2 for the record problem, in the left panel we compare the approximation to the exact values. |
| Figure 9. The cumulative frequency distribution for N = 105 and U = 0.5 at t = 102, 103 and 104 (from left to right). The data are collected from 64 000 independent simulations. The approximate expressions e(w) in (76) and ec(w) in (77) with ξ = 0.4725 are also drawn for comparison. Inset: the same but the abscissa is shifted by the amount m(τ) which is the solution of (73) with CDRP = 8. The datasets for t = 103 and t = 104 show a nice collapse. |
Figure 10. Plots of  versus τ for the datasets in the left panel of figure 5. All curves now collapse in the asymptotic regime. |
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