Quasilocalization under coupled mutation-selection dynamics

Abstract

When mutations are rampant, quasispecies theory or Eigen's model predicts that the fittest type in a population may not dominate. Beyond a critical mutation rate, the population may even be delocalized completely from the peak of the fitness landscape and the fittest is ironically lost. Extensive efforts have been made to understand this exceptional scenario. But in general, there is no simple prescription that predicts the eventual degree of localization for arbitrary fitness landscapes and mutation rates. Here, we derive a simple and general relation linking the quasispecies' Hill numbers, which are diversity metrics in ecology, and the ratio of an effective fitness variance to the mean mutation rate squared. This ratio, which we call the localization factor, emerges from mean approximations of decomposed surprisal or stochastic entropy change rates. On the side of application, the relation we obtained here defines a combination of Hill numbers that may complement other complexity or diversity measures for real viral quasispecies. Its advantage being that there is an underlying biological interpretation under Eigen's model.

Figures (12)

• Figure 1: (A) In Eigen's model, mutants are produced as a result of erroneous replication, coupling the replication rate $A_i$ (hence, selection) of a type $i$, shown above as a sequence of nucleotides, and its copy fidelity $Q_{ii}$ (hence, mutation). (B) Much of the theoretical work has focused on the error catastrophe (right) in which the population (closed black circles) is delocalized from the peak of the fitness landscape (purple curve). On the opposite extreme is survival of the fittest (left) where the entire population is on the peak. The middle diagram represents the case when mutation is seldom.
• Figure 2: (A) Cartoon of the spectrum of equilibrium frequencies (cerulean curve) against the influx landscape (black curve). In the localized regime, the population concentrates on the peak of the influx landscape. In the delocalized regime, the population disperses over all types. (B) The population dynamics can be represented as motion along an octant of an $N$-dimensional hypersphere (or equivalently, as motion along an $N$-simplex). Here, $N=3$. Each point on this hypersphere corresponds to a vector of type frequencies $\mathbf{p}(t) \in \mathbb{R}^N$ that represents the state of the population. The population moves on the hypersphere from $t=t_0$ to $t=t_f$ with a net velocity given by the surprisal rate $\dot{\sigma}(t)$, which can be decomposed into contributions of the influx rate ($\dot{\sigma}^s$(t)) and mutation rate ($\dot{\sigma}^\mu$(t)). (C) The norm of these surprisal rates yield the (de)localization speeds and the Cramér-Rao bound.
• Figure 3: (A) The degree of localization as measured by different metrics increases with $F$. Each point is the median value of the respective metric over $10^3$ realizations per given $F$. Notably, a population with larger $N$ will localize at a larger $F$. (B) Different $F$ lead to qualitatively different frequency dynamics. The left panel shows the time-evolution of frequencies at $F=1$, the middle at $F=1.5\times10^3$, and the right at $F=5.5\times10^5$, with $N=50$ in all cases. The frequencies are plotted against time normalized by the maximum time of integration $t_{max}$ (see Eq. \ref{['eq:tmax']} in main text).
• Figure 4: Accuracy of the equilibrium relation. See the main text for exact definition of relative and absolute errors. Top: The relation Eq. \ref{['eq:log_relation']} has a median relative error $e_\text{rel}$ within $10\%$ across regimes, with median errors below $6\%$ in the quasi- and delocalized regimes. The dotted lines and arrows (top of the plot) indicate the approximate onsets of the localized regimes for different $N$ that have the same but darker color scheme as the errors. Here, the onset of the localized regime is defined as when the median $R_{50}$ for a given $F$ drops to one. Bottom: The median absolute error $e_\text{abs}$ is shown for low values of $F$ as the relative error becomes undefined when $F \rightarrow 0$. Each median error in both panels is computed from $10^3$ realizations per given $F$. The $95\%$ confidence intervals are constructed from the interquartile range ($IQR$) of the computed errors: $\pm 1.57\times IQR/\sqrt{10^3}$.
• Figure 5: Time-evolution of pairs of localization and delocalization speeds for $F = 1.5\times 10^6$ (top panel), $F = 1.5\times 10^3$ (middle), and $F = 1.5$ (bottom) with $N = 50$ in all cases. If $v_s$ ($v_\mu$) starts out much larger, it stays larger than $v_\mu$ ($v_s$) until equilibrium. If the speeds have similar orders, they may switch in dominance (middle panel) over time but they will still have comparable magnitudes.