Nonequilibrium Phase Transitions Associated with DNA Replication

Thermodynamics governing the synthesis of DNA and RNA strands under a template is considered analytically and applied to the population dynamics of competing replicators. We find a nonequilibrium phase transition for high values of polymerase fidelity in a single replicator, where the two phases correspond to stationary states with higher elongation velocity and lower error rate than the other. At the critical point, the susceptibility linking velocity to thermodynamic force diverges. The overall behavior closely resembles the liquid-vapor phase transition in equilibrium. For a population of self-replicating macromolecules, Eigen’s error catastrophe transition precedes this thermodynamic phase transition during starvation. For a given thermodynamic force, the fitness of replicators increases with increasing polymerase fidelity above a threshold.

Figure 1

Dependence of stationary properties on $g$. (a) $\overline{v}$ for $m=4$ and $\eta =5$; (b) $ϵ$ for $m=4$ and $\eta =5$; (c) $F$ for $m=4$ and $\eta =5$; (d) convergence of $\overline{v}$ in numerical simulations for $m=2$, $\eta =10$, $\gamma ={10}^{-3}$, and $g=0.1$; and (e) convergence of $ϵ$ in simulations for $m=4$, $\eta =5$, $\gamma ={10}^{-2}$, and $g=0$. Size of time step in Monte Carlo was 0.01 in units of $k^{-1}$. Circles in (a)–(c) are from the numerical simulations.

Figure 2

(a) Velocity and (b) error rate as functions of $F$ for $m=4$ and $\eta =5$. The solid line in (b) corresponds to Eq. (22). The critical point is reached when ${\gamma }_c=1.83\times {10}^{-3}$.

Figure 3

Phase diagram for $m=4$ and $\eta =5$. Thick (red) line denotes the spinodal of $\mathrm{\text{high}}\mathrm{\text{−}}v/\mathrm{\text{low}}\mathrm{\text{−}}ϵ$ phase terminating at the critical point $(F_c,{\gamma }_c)=(0.210,1.83\times {10}^{-3})$. The gray scale indicates the error rate $ϵ$. Thin solid (black) lines show constant-$ϵ$ contours at $ϵ={10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, ${10}^{-1}$. Dotted (blue) lines show constant-$\overline{v}$ contours at $\overline{v}=0.1$, 0.2, 0.3 from left to right.