Solution of classical evolutionary models in the limit when the diffusion approximation breaks down

The discrete time mathematical models of evolution (the discrete time Eigen model, the Moran model, and the Wright-Fisher model) have many applications in complex biological systems. The discrete time Eigen model rather realistically describes the serial passage experiments in biology. Nevertheless, the dynamics of the discrete time Eigen model is solved in this paper. The 90% of results in population genetics are connected with the diffusion approximation of the Wright-Fisher and Moran models. We considered the discrete time Eigen model of asexual virus evolution and the Wright-Fisher model from population genetics. We look at the logarithm of probabilities and apply the Hamilton-Jacobi equation for the models. We define exact dynamics for the population distribution for the discrete time Eigen model. For the Wright-Fisher model, we express the exact steady state solution and fixation probability via the solution of some nonlocal equation then give the series expansion of the solution via degrees of selection and mutation rates. The diffusion theories result in the zeroth order approximation in our approach. The numeric confirms that our method works in the case of strong selection, whereas the diffusion method fails there. Although the diffusion method is exact for the mean first arrival time, it provides incorrect approximation for the dynamics of the tail of distribution.

Figure 1

A comparison of the dynamics for the discrete time Eigen model with $L=100,r_i=exp(-2i),q=1-1/L,s={\sum }_i(1-2i/L)p_i\left(n\right)$ at the start of $p_0\left(0\right)=1$, and other $p$'s are equal to 0. The smooth line corresponds to the numerical solution of Eq. (1), and the solid circles are the analytical results by Eq. (8) from Ref. [23] for $x_0=1$.

Figure 2

A comparison of the steady state distribution of the Wright-Fisher model by Eq. (5) $L=100,u=0.1,v=0.2,s=0$. The smooth line is our result by Eqs. (13), (17), and (19), and the dashed line is calculated by the diffusion theory of Eq. (17). The solid circles are the results of the numerical solution of Eq. (5) after many iterations.

Figure 3

The ratio of the analytical results and the numerics $R\equiv (1-{\pi }_i^{\mathrm{theor}})/(1-{\pi }_i^{\mathrm{num}}),{\pi }_i$ is the fixation probability in the WF model by Eqs. (26) and (27) with $L=1000.{\pi }_i^{\mathrm{num}}$ is the numerical result, and ${\pi }_i^{\mathrm{theor}}$ is the analytical result. We consider the case of $i=100$. The selection is defined as $s=S/L$. The dashed line corresponds to the result given by the diffusion theory result Eq. (30), the solid dots are calculated by Eq. (44) of Ref. [26], and the smooth line is our analytical result of Eq. (41). Although at small selection $S$, all three analytical methods give accurate results $\left(R\approx 1\right)$, and for the higher values of $S$ our method is much more accurate than the diffusion method.