Fig 1.
The eight possible configurations on a graph of size N = 3.
Red nodes show mutants and blue nodes indicate wild types. The process starts with one of the states containing a mutant and continuous until it reaches one of the absorbing states with three mutants or three wild-types. The arrows show possible transitions between states of the Markov chain at each time step.
Fig 2.
Circular graphs: The dependence of the mean conditional mutant fixation time (starting with one mutant) on standard deviation.
(a) N = 3. (b) N = 4. (c) N = 5. (d) N = 6. Results for three cases are presented: mutant and wild type fitness values are fully correlated, uncorrelated, and anti-correlated.
Fig 3.
Circular graphs: The effect of skewness of the fitness probability distribution.
(a) The concept of skewness is illustrated for the two-value distribution. For fixed mean (μ = 1) and three values of variance, the two possible fitness values, x1 and x2, eq (10), are plotted as functions of the skewness. (b) For the same three values of variance, the mean conditional fixation time in the N = 3 system is plotted as a function of skewness. The effect of correlations is added in (c) and (d), where the mean conditional fixation time is plotted as a function of skewness for (c) N = 4 and (d) N = 6, with μ = 1, σ2 = 0.3, and three correlation conditions: correlated, uncorrelated, and anti-correlated fitness values.
Fig 4.
Small complete graphs: The effects of correlation and skewness.
(a-c): The conditional mean mutant fixation time is plotted as a function of the standard deviation, for (a) N = 4, (b) N = 5, (c) N = 6. The effect of skewness of the fitness probability distribution for N = 6 is shown in (d), where the mean conditional fixation time is plotted as a function of skewness, with μ = 1, σ2 = 0.3. In all panels, results for three cases are presented: mutant and wild type fitness values are fully correlated, uncorrelated, and anti-correlated.
Fig 5.
Different fitness distributions for wild types and mutants: The mean conditional fixation time on a circle/complete graph with population size N = 3.
(a) The mutants have random fitness with average one and the wild-type individuals have a constant fitness 1. (b) The mutants have fixed fitness 1 and the wild types have a random fitness with average one. (c) The mutants have random fitness with average 1.2 and wild types have random fitness with average one. (d) The wild types have random fitness with average 1.2 and mutants have random fitness with average one.
Fig 6.
The effect of randomness on the fixation time in the case where only one of the types has random fitness values.
(a,b) Circular graph and (c,d) Complete graph (N = 6 and N = 7). The orange lines correspond to constant fitness of the mutant, and the blue lines to constant fitness of the wild-types. The vertical axis is the mean conditional fixation time, and the horizontal axis is σ.
Fig 7.
The effect of randomness on the fixation time for advantageous and disadvantageous mutants; N = 6.
(a) Circular graph, (b) complete graph (note that the two lines overlap). The vertical axis is the mean conditional fixation time, and the horizontal axis is σ.
Fig 8.
The conditional fixation time on two different networks (a) circles and (b) complete graphs for population sizes N = 5, 6, ⋯, 9 (exact stochastic simulations, dots, and analytical results obtained by the matrix method, lines) and N = 10, 12, 15, 20 (stochastic sampling simulations).
For each value of σ, 106 random configurations were used, and the calculations repeated 6 times, to obtain the standard deviation, presented as error bars.
Fig 9.
The average conditional fixation time for a mutant with random fitness on a graph with N = 9 nodes and different numbers of neighbors (different degrees z).
Lines represent the analytical results and each data point is averaged over 106 independent realizations.