Punctuated equilibrium and shock waves in molecular models of biological evolution

We consider the dynamics in infinite population evolution models with a general symmetric fitness landscape. We find shock waves, i.e., discontinuous transitions in the mean fitness, in evolution dynamics even with smooth fitness landscapes, which means that the search for the optimal evolution trajectory is more complicated. These shock waves appear in the case of positive epistasis and can be used to represent punctuated equilibria in biological evolution during long geological time scales. We find exact analytical solutions for discontinuous dynamics at the large-genome-length limit and derive optimal mutation rates for a fixed fitness landscape to send the population from the initial configuration to some final configuration in the fastest way.

Figure 1

The dynamics of $x^{*}\left(t^{*}\right)$ (mean overlap with the reference sequence) as function of time $t^{*})$ by Eqs. (12) and (14) for the symmetric case ${\gamma }_f={\gamma }_b=\gamma =1$ and (top to bottom at $c{t}^{*}=2)$ $\gamma /c=0.05,0.1,...,0.7,0.75$. The initial population has overlap $x_0=0.01$ with the reference sequence. For $t^{*}\to \infty$, we find $x^{*}=1-\gamma /c$.

Figure 2

The dynamics of $x^{*}\left(t^{*}\right)$ by Eqs. (12) and (14) for $x_0=0.01$, $\gamma /c=0.05$ and by Eq. (1) (dashed vertical lines from left to right) $L=2000,4000,...,12000$. For $L\to \infty$ the jump in $x^{*}$ takes place at $c{t}^{*}=c{t}_d^{*}=1.939$ and has the size $\Delta x^{*}=0.755$.

Figure 3

The critical line ${\gamma }_c/c$ vs $x_0$ at which $\Delta x^{*}=0$. The critical $c$ is defined from the system of equations $d{x}^{*}/d{t}^{*}=0,d^2{x}^{*}/d^2{t}^{*}=0$. For the $\gamma$ values below this curve, the most probable overlap $x^{*}$ undergoes a discontinuous transition at $t^{*}=t_d^{*}$ (see Fig.  2).

Figure 4

The relaxation from the original distribution with $x_0=0$ for the fitness function $f\left(m\right)=4exp\left[\right(m-1\left)\right],\gamma =0.1$. The dashed lines from left to right correspond to $L=1000,5000,10000$. The jump is at the point $d{t}^{*}/d{x}^{*}=0$ at $L=\infty$.

Figure 5

Relaxation from the original distribution with $x_0=0.01,c=20$ for the fitness function $f\left(m\right)={c\left|m\right|}^{1.8}$.

Figure 6

Relaxation from the original distribution with $x_0=0.01,c=10000$ for the fitness function $f\left(m\right)={c|1-m|}^{1.8}$.

Figure 7

Time period $t^{*}$ needed for the maximum of the overlap distribution to reach the values (right to left at $c{t}^{*}=2)$ $x^{*}=0.1,0.2,...,0.8$ as functions of $\gamma /c$. The initial population has overlap $x_0=0.01$ with the reference sequence. The dynamics can reach $x^{*}$ provided that $\gamma /c<1-x^{*}$.

Figure 8

The dynamics $p_0\left(t\right)$ for $L=20$, symmetric-mutation rate $\gamma =1$. Fitness $f\left(x\right)=2L{x}^a$ for $a=1,2,4$ (from bottom to up at $t=1.4)$. The single peak fitness corresponds to the choice $f\left(x\right)=0$ for $x<1$ and $f\left(1\right)=2L$.

Figure 9

The dynamics of $x^{*}$ as a function of time $t^{*}$ for unidirectional-mutation case with ${\gamma }_b=0,x_0=0$, (left to right) ${\gamma }_f/c=0.1,0.05$, and $0.01$. The numerical solution of the system (5) is given by the dashed vertical lines for (left to right) $L=1000,5000$, and $10000$.