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.
Under canonical models in population genetics, mutation is seldom by design. The key contribution of quasispecies theory [1] (otherwise known as Eigen's model or the Eigen-Schuster model) is to reveal the full dynamical impact of mutation when it is very frequent, which is characteristic of asexual organisms such as bacteria but especially viruses [2,3]. Whereas selection alone may lead to a "survival of the fittest," quasispecies theory predicts that if mutation rates are high enough, less fit variants or "types" can even thrive because they are replenished through frequent mutation. The equilibrium population structure is called the quasispecies [4]. This provided a theoretical framework for understanding viral evolution and made a lasting influence in virology, where the term "quasispecies" is now used broadly to refer to the heterogeneous nature of viruses [5,6].
That less-fit mutants can persist did not originate from Eigen. Crow and Kimura have earlier studied a deterministic mutation-selection model [7]. But Eigen’s model is defined for arbitrary number of types, frequencies and mutation rates. Under this generalization, the mass of the population can be arbitrarily distributed among the fittest and its mutant relatives. In extreme cases, the quasispecies may be dominated by a single type (localization) such as in survival of the fittest or dispersed over many or all types (delocalization). A wellstudied example of the latter is the error catastrophe. Although there is no universal definition for it, it is often taken to be a (usually sharp) phase transition wherein the fittest type is lost from the population when its per-site mutation probability is above an “error threshold” [8,9]. Crossing this threshold, the population is totally delocalized from the peak of the fitness * cbpalpallatoc@up.edu.ph landscape (Fig. 1b). Vigorous theoretical effort has been spent understanding this threshold assuming mostly singlepeaked fitness landscapes [10][11][12], with extensions to drift [13][14][15], recombination [16,17], gene networks [18,19] and epistasis [20]. Complementary to the error threshold, a total localization threshold has also been obtained that determines when the population is localized entirely on a fitness peak (i.e. survival of the fittest) [21]. The analogy between quasispecies theory and Anderson localization has also been discussed in Ref. [22]. More recently, a noise-induced “fidelity catastrophe” has also been shown to occur wherein the population is always localized but switches dominant types over time [23].
Despite these investigations, our understanding of localization for error-prone replication remains incomplete even in the deterministic limit. The particular value of the error or the localization threshold depends on the choice of simplified landscape and mutation scheme. In particular, it has been shown that the error threshold does not generically exist [24][25][26][27][28]. Moreover, while error catastrophe and survival of the fittest carry important implications, they are extreme situations. Real error-prone populations should more commonly exist in an intermediate state of “quasilocalization,” where the population is diffused in the landscape around some dominant type, which might be the fittest. Indeed, this is the defining characteristic of real viral quasispecies [6,29], though we have poor analytical statements about this intermediate scenario.
We understand that quasilocalization surely represents a balance between mutation and selection effects, with neither dominating. But can we say something more concrete and quantitative, especially beyond single-peaked landscapes? As a step forward, we use information-theoretic bounds to study the degree of localization analytically. Recently, speed limits on the time-evolution of observables, which may or may not be biological, have been obtained without relying on the particular details of the system [30][31][32][33][34][35][36]. We take specifically the bounds in Ref. [35] based on decomposing the surprisal or stochastic entropy change rate. Adapting them to Eigen’s model, we derive an equilibrium relation for the degree of localization. This relation links what we are calling the localization factor, which is computed from the statistics of the dynamical parameters, and the Hill numbers, which are canonical diversity metrics in ecology, of the quasispecies.
This paper is organized as follows. In Sec. II, we introduce our formulation of quasispecies theory and important terminology of relevant quantities, including our definition of an “influx rate.” We also briefly review the speed limits in Ref. [35] based on decomposing the surprisal rate. We apply them to Eigen’s model, identifying localization and delocalization speeds. In Sec. III, we obtain mean approximations of these speeds. We then derive our main results, the equilibrium relation involving the localization factor and its critical value. In Sec. IV, we discuss th
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