When evolution realizes large deviations of fitness: from speciation to dynamical phase transitions

TL;DR

By connecting Moran-type evolution with large-deviation trajectory theory, the paper shows that trajectory fitness $F=\int_0^t dt'\, f^0_{\mathcal C(t')}$ is controlled by a biased generator $\hat{\mathbb W}_s=\mathbb W+\mathbb f$ and its largest eigenvalue $\psi_s$ gives the scaled cumulant generating function. In the large-$N$ limit, the nonlinear Moran dynamics reproduce the linear biased dynamics, with the genotype abundances obeying $|x(t)\rangle=|\hat P_s(t)\rangle/\mathcal Z_s(t)$ and $\mathcal Z_s(t)=\langle e^{sF(t)}\rangle$. Varying $s$ induces dynamical phase transitions in the genome distribution, from unimodal to bimodal (sympatric speciation) or from smooth to discontinuous changes, and finite-$N$ effects yield coexistence or stochastic switching depending on population size. The framework further extends beyond fixed-population Moran models to non-constant populations with birth–death and interaction terms, establishing a general route to realize and analyze large deviations and dynamical transitions in evolutionary dynamics.

Abstract

We explore the connection between evolution and large-deviation theory. To do so, we study evolutionary dynamics in which individuals experience mutations, reproduction, and selection using variants of the Moran model. We show that, in the large population size limit, the impact of reproduction and selection amounts to realizing a large-deviation dynamics for the non-interacting random walk in which individuals simply explore the genome landscape due to mutations. This mapping, which holds at all times, allows us to recast transitions in the population genome distribution as dynamical phase transitions, which can then be studied using the toolbox of large-deviation theory. Finally, we show that the mapping extends beyond the class of Moran models.

Figures (2)

• Figure 1: (a): Mutation rates $w^{\pm}(m) = \gamma ( 1 \mp m)$ and selection landscape $f(m) = s \left[ ( m + \alpha (m-1) ( 1+m)^2)^2+ \delta m\right]$ represented for $s=1$ in the symmetric case (red, $\alpha=\delta=0$) and asymmetric case (yellow, $\alpha=0.2$ and $\delta=0.1$). (b): Landau free energy Eq. (\ref{['eq:F']}) in the symmetric case ($s=0$, pink and $s=0.6$, red) and asymmetric case (for $s=0.6$, yellow) . (c-d) Distribution $x_s^*(m)$ and its maxima $m^*$ in the symmetric case of panel (b). The theory predicts the sympatric speciation transition at $s_\mathrm{c}=0.3$. Histograms from simulations of the Moran model are shown for $s=0.2$ (blue star & histogram) and $s=0.6$ (orange star & histogram). The black line corresponds to the WKB prediction ($M_0\gg 1$) and the dots correspond to the right eigenvector $|R_s \rangle$ obtained from the exact diagonalization of $\hat{\mathbb W}_s$ for $M_0=8$. (e-f): Same as (c-d) for the asymmetric free energy in panel (b). The theory predicts a first-order transition at $s\simeq 0.52$ and we show data from numerical simulations of the Moran model for $s=0.5$ (blue) and $s=0.6$ (orange). All simulations are performed for $N = 6 \times 10^5$, $M_0 = 8$ and $\gamma=0.3$.
• Figure 2: (a-b): Evolution of the genotype distribution from Moran simulations for $M_0=8$ and $N=3.189 \times 10^3$ (a) or $N=1 \times 10^7$ (b). (c-d): Order parameter $\bar{m} = \sum_{i=1}^{N} \frac{|m_i|}{N}$ as a function of $N$ (left) that distinguishes coexistence ($\bar{m}=0$) and stochastic switching ($\bar{m}\sim O(1)$), for different values of $M_0$. The curves can be collapsed around the transition by rescaling $N$ with a characteristic population size $N_\mathrm{c} \sim \mathrm{e}^{\lambda M_0}$. We found a good collapse for $\lambda \simeq 1.3$ which compares surprisingly well with the predicted value $\lambda=1.31$.