Lag-Induced Critical Transitions to Extinction in Replicating Systems

TL;DR

It is proposed that the pathway to collapse described in this article can be understood as lag-time-induced tipping (\tau-tipping) and suggested new antiviral strategies based on modulating replicase availability.

Abstract

Replicating systems sustained by error-prone enzymatic amplification can undergo critical transitions between persistence and extinction. In RNA viruses, such transitions are classically governed by mutation rates and fitness landscapes, giving rise to error thresholds and lethal mutagenesis. Motivated by experimental evidence that polymerase-targeting antivirals constrain replication, we analyze replicating systems with explicit delays in replication-enzyme availability. We identify a lag-induced (dynamical) critical transition driven by the loss of temporal coordination between genome translation and replication. At a fixed mutation rate and replicative fitness landscape, populations cross an extinction threshold solely due to time delays. Within the quasispecies framework, replication-translation timing emerges as an independent control parameter, defining a distinct dynamical route to extinction and suggesting new antiviral strategies based on modulating replicase availability. More generally, we propose that the pathway to collapse described in this article can be understood as lag-time-induced tipping (τ-tipping).

Figures (5)

• Figure 1: Critical transitions in error-prone self-replicating population, such as RNA viruses. (a) The error threshold marks mutant dominance over the master genome Eigen2002Sole2006. (b) Lethal mutagenesis causes extinction via mutation accumulation Loeb1999Bull2007. (c) Replication time--lag-induced lethality identified in this study. Our model tracks master and mutant genomes replicated by the RNA-dependent RNA polymerase (RdRp), with a time lag $\tau$ in its functional availability. Phase-space trajectories are computed from Eqs. \ref{['eq:DIviruses']} for different situations: early RdRp production ($\tau=0.65$, black) supports persistence, whereas long lags ($\tau=50$, blue) cause collapse. Panels (a,b) show mutation- and fitness-driven transitions, while (c) shows extinction driven solely by replication timing at fixed mutation rate and replication fitness landscape.
• Figure 2: (a) Phase diagram of the master genome equilibrium of eqs. \ref{['eq:DIviruses']} for $\tau=0$, showing persistence at low $\mu$ and large $\gamma$. The blue area corresponds to the full collapse of the system. (b) Phase portrait at $(\gamma,\mu)=(0.9,0.4)$ [asterisk in (a)] showing three trajectories with same initial condition for $\tau=0$ (blue), $\tau=25$ (red trajectory overlapped to the green one), and $\tau = 50$ (green). Increasing $\tau$ drives extinction. Black and gray dots denote stable equilibria and saddles.
• Figure 3: (a–c) Basins of attraction of eqs. \ref{['eq:DIviruses']} for $p(0)=0.1$ with $\gamma = 0.5$ and $\mu=0.2$, increasing $\tau=0$, $25$, and $50$, showing convergence times to extinction (blue) or to persistence (orange) for $\xi\in[-\tau,0]$. Increasing $\tau$ enlarges the extinction basin without changing mutation or fitness traits. (d) Time series from the asterisked initial conditions.
• Figure 4: Phase portraits of Eqs. \ref{['eq:simpler']} for $\tau=0$ with $\varepsilon=0.25$ (a) and $\varepsilon=0.33$ (b). Trajectories converge to $P_0$ or $P_+$; the basin of $P_0$ is shown in transparent blue (black circles denote attractors; yellow circle denote the saddle point $P_-$). Overlapped, we display the separatrix of the two attractors for different time lags, with: $\tau = 0$ (blue curve); $\tau = 1$ (red curve); $\tau = 5$ (green curve); $\tau = 10$ (brown curve); $\tau = 25$ (turquoise curve); and $\tau = 50$ (orange curve). (c) Basin size of $P_0$ versus $\tau$ for $\varepsilon=0.15$, $0.25$, $0.29$, $0.31,$ and $0.33$ (bottom to top).
• Figure 5: Lag-dependent spectral stability of the equilibria of eqs. (2) for five values of $\varepsilon<\varepsilon_c$. Shown is the rightmost real part eigenvalue, $\max\Re(\lambda)$. The equilibrium $P_0$ is $\tau$-independent and stable (horizontal lines); $P_-$ is unstable for all $\tau$, with its instability weakening at larger lags (positive curves); $P_+$ remains stable over the full delay range (dotted curves), with its attractive strength also weakening.