Nonequilibrium time dynamics of genetic evolution

Biological systems are typically highly open, nonequilibrium systems that are very challenging to understand from a statistical mechanics perspective. While statistical treatments of evolutionary biological systems have a long and rich history, examination of the time-dependent nonequilibrium dynamics has been less studied. In this paper we first derive a generalized master equation in the genotype space for diploid organisms incorporating the processes of selection, mutation, recombination, and reproduction. The master equation is defined in terms of continuous time and can handle an arbitrary number of gene loci and alleles and can be defined in terms of an absolute population or probabilities. We examine and analytically solve several prototypical cases which illustrate the interplay of the various processes and discuss the timescales of their evolution. The entropy production during the evolution towards steady state is calculated and we find that it agrees with predictions from nonequilibrium statistical mechanics where it is large when the population distribution evolves towards a more viable genotype. The stability of the nonequilibrium steady state is confirmed using the Glansdorff-Prigogine criterion.

Figure 1

Summary of the genetic evolutionary processes considered in this paper. (a) Processes captured by our master equation (19). Initially various genotypes exist within a fitness landscape, shown here as the negative viability $-{\omega }_{ij}$ (equivalent to the death rate). The population distribution tends to converge towards the fittest genotype, where steady state is reached. While approaching equilibrium, the processes of selection, mutation, recombination, gene flow, and genetic drift take place. Selection is driven by genotypes with lower viabilities having a higher death rate. The genotype depicted by the rightmost individual spontaneously appears in the population due to gene flow during the transition state and then drifts off at steady state. (b) Labeling convention given in this paper depicted on two homologous chromosomes (left), their constituent sister chromatids (center), and after undergoing recombination (right). The allele configuration on the two chromosomes of a diploid organism is given by (1). Each chromosome has $M$ loci, with the genotype of the $n\mathrm{th}$ locus being labeled by $i_n,j_n$. The variables take the values $i_n,j_n\in 1,\cdots ,N_n$, which correspond to the alleles $A_{i_n}^{\left(n\right)}{A}_{j_n}^{\left(n\right)}$.

Figure 2

Time evolution of the master equation for evolution of $M=2$ gene loci under various conditions. [(a), (b), and (c)] Evolution under random mating ($r=1$) with no selection or mutation (${\gamma }_{ij}=v_{ij}=0$) for two loci $M=2$ and three allele variations $N=3$ with a linear viability function. (a) The initial condition where the entire population has the genotype $A_1^{\left(1\right)}{A}_3^{\left(1\right)}{A}_1^{\left(2\right)}{A}_3^{\left(2\right)}$. (b) Steady-state probability ($t=20$) with no recombination $c_{nm}=0$. (c) Steady-state probability ($t=200$) with recombination $c_{11}=1$, otherwise $c_{nm}=0$. (d) Steady-state probability ($t=100$) with recombination $c_{11}=1$ and selection (${\gamma }_{ij}=1$) for two loci $M=2$ and three allele variations $N=3$ with the same viability function defined in (29) and parameters $a=-0.5$, $b=0.1$, $c=0.7$.

Figure 3

The effects of different parameters on a fitness landscape of one gene locus with two genotypes with high viability. [(a) and (b)] Evolution for pure selection for one locus $M=1$ and five allele variations $N=5$. (a) Viability distribution for simulations in Fig. 3 with parameters $a=0.8$, $b=0.1$, and $c=0$. (b) Time evolution of the variable $s=s_m=s_f$ from different initial conditions: (I) $p_{ij}(t=0)=1/N^2$; (II) $p_{ij}(t=0)=0.9{\delta }_{i,1}{\delta }_{j,1}+0.1{\delta }_{i,N}{\delta }_{j,N}$; and (III) $p_{ij}(t=0)={\delta }_{i,1}{\delta }_{j,1}$. Here ${\delta }_{i,j}$ is the Kronecker $\delta$. [(c)–(h)] Evolution for mutation for one locus $M=1$ and five allele variations $N=5$. [(c) and (d)] Probability distribution at times (c) $t=60$ and (d) $t=120$ starting from an initial configuration $p_{ij}(t=0)=0.4{\delta }_{i,1}{\delta }_{j,1}+0.2({\delta }_{i,1}{\delta }_{j,2}+{\delta }_{i,2}{\delta }_{j,1}+{\delta }_{i,2}{\delta }_{j,2})$. Identically, (d) is also the landscape obtained at $t=20$ for the time evolution under pure selection (${\gamma }_{ij}=1-{\omega }_{ij}$) and no mutation or reproduction ($v_{ij}=r=0$) with an initial condition $p_{ij}(t=0)=1/N^2$. After a time $t\sim 10$ the distribution becomes entirely dispersed at $i=j=1$, until a time $t=t_c\sim 70$ (where $t_c$ is the critical wait time), when the population of the most viable genotype $i=j=5$ starts to grow. After a time $t_c$, the most viable genotype becomes stable. The parameters used are $N=5$, $a=0.9$, $b=0.1$, $c=0$, $r=0.01$, $v_{ij}=v(1-{\delta }_{ij})$, and $v={10}^{-4}$. (e) The average value $s={\overline{s}}_m={\overline{s}}_f$ for the same parameters as above but with the mutation rates $u$ as marked. Panel (f) is the same as panel (e) but with the initial condition $p_{ij}(t=0)=0.4{\delta }_{i,1}{\delta }_{j,1}+0.2({\delta }_{i,1}{\delta }_{j,2}+{\delta }_{i,2}{\delta }_{j,1}+{\delta }_{i,5}{\delta }_{j,5})$, illustrating gene flow. Panel (g) shows the variances $V\left(s\right)=V\left(s_{\text{m}}\right)=V\left(s_{\text{f}}\right)$ for the same parameters as above but with the mutation rates $u$ as marked. Panel (h) is the same as panel (g) but with the initial condition given in panel (f).

Figure 4

Entropy and entropy production during the evolutionary process. (a) The entropy (38) for the processes in Figs. 2 and 2. Lines are marked according to the figure numbers. (b) The entropy (38) for the processes in Figs. 3 and 3. Lines are marked according to the mutation rate. (c) The total entropy production (41) for the process in Figs. 3 and 3. (d) The Glansdorff-Prigogine stability of the steady state (42) is shown using a graph of the second variation of the entropy production as a function of time for different viabilities. The steady-state probabilities are approximated by the probabilities at the end of the time evolution ${\overline{p}}_{\mathit{i}\mathit{j}}\approx p_{\mathit{i}\mathit{j}}\left(t_{\text{max}}\right)$. The curves for $v=0.001$ and $v=0.0001$ have been multiplied by the factors as labeled to fit in the plots.