Genotypic richness and the distribution of clonal size at equilibrium under an increasing rate of clonality
In terms of genotypic diversity, our results showed a clear, progressive, and even stepwise decrease with increasing rates of clonality (Figures 1 and S1).
When genotyping the entire population, the relationship between Rand c (Figure 3) does not follow a linear trend, such as\(R=1-c\), as might have been assumed in some previous studies. The relationship is best modelled by \(R=\sqrt{1-c^{2}}\)(N =100000: MAE =0.011 and RMSD = 0.020 for\(\overset{\overline{}}{R}=0.69\)). The same equation fits the simulation results regardless of the population size, with slightly larger deviations at smaller population sizes, as expected with an increasing strength of genetic drift (N =10000: MAE =0.013 and RMSD = 0.021 for \(\overset{\overline{}}{R}=0.70\);N =1000: MAE =0.029 and RMSD = 0.041 for\(\overset{\overline{}}{R}=0.71\)), but still providing an accurate approximation.
The curve describing the evolution of the parameter Pareto \(\beta\) in the power-law distribution of clonal sizes depending on the rate of clonality shows a slightly more complex pattern. The curve has the typical shape of a sum of two sigmoid curves with three sub-domains delimited by two inflection points (Figure 1). Very low levels of clonality (0<c <0.1) lead to maximum Pareto\(\beta\)-values, which depend on the population size (approximately 8 for N =100 individuals to 15 for N =100000 individuals). For these distinct initial values, the curves show an extremely similar shape regardless of population size, with a marked sigmoid shape of Pareto \(\beta\)-values declining from approximately 8 (value corresponding to high richness and evenness; Arnaud-Haond et al., 2007) at c =0.1 to nearly 0 for c =1. Interestingly, the value\(\beta\)=2 is reached for clonal rates of approximately 0.8 to 0.9 for all population sizes. Between clonal rates of 0.2 and 0.9, the decline in \(\beta\) is nearly linear and flat for all population sizes. ForN = 100000, the sum of two fitted sigmoid curves produces the following equation:
\(\beta=\frac{337335}{1+e^{16\times(c+0.65)}}+\frac{5}{1+e^{6.8\times(c-0.80)}}\)(MAE =0.30 and RMSD = 0.40 for\(\overset{\overline{}}{\beta}=4.66\));
for \(N=10000\),
\(\beta=\frac{506607}{1+e^{9.8\times(c+1.12)}}+\frac{4}{1+e^{8.3\times(c-0.81)}}\)(MAE =0.27 and RMSD = 0.36 for\(\overset{\overline{}}{\beta}=3.94\)); and
for \(N=1000\),
\(\beta=\frac{5.6}{1+e^{5\times(c-0.58)}}+\frac{3.8}{1+e^{50\times(c-0.19)}}\)(MAE =0.32 and RMSD = 0.40 for\(\overset{\overline{}}{\beta}=3.63\)).