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Survival of the flattest in the quasispecies model

Maxime Berger, Raphaël Cerf

TL;DR

This work rigorously analyzes the quasispecies model under mutation and selection, focusing on Moran–Kingman dynamics and the sharp-peak landscape, to derive exact formulas for the quasispecies distribution, mean fitness $\lambda$, and mean Hamming distance. It then extends these results to general landscapes, including finitely many peaks and plateau-type fitness, using lumping and PF-eigenvalue techniques to obtain explicit representations and limiting behaviours. A central theme is the error threshold and the survival of the flattest: the authors show, in long-chain regimes, how a flat but broad plateau can dominate over a higher but narrow peak, yielding the phenomenon of survival of the flattest, with precise conditions and limits given by $\lambda(a,\sigma)$ and $\max(\delta e^{-a},\lambda(a,\sigma))$. Theoretical results are supported by probabilistic interpretations, binomial/Poisson limits, and asymptotic analyses, providing a robust framework for assessing mutational strategies in viral populations and informing concepts such as lethal mutagenesis and quasispecies structure.

Abstract

Viruses present an amazing genetic variability. An ensemble of infecting viruses, also called a viral quasispecies, is a cloud of mutants centered around a specific genotype. The simplest model of evolution, whose equilibrium state is described by the quasispecies equation, is the Moran--Kingman model. For the sharp peak landscape, we perform several exact computations and we derive several exact formulas. We obtain also an exact formula for the quasispecies distribution, involving a series and the mean fitness. A very simple formula for the mean Hamming distance is derived, which is exact and which do not require a specific asymptotic expansion (like sending the length of the macromolecules to $\infty$ or the mutation probability to $0$). We try also to extend these formulas to a general fitness landscape. We obtain an equation involving the covariance of the fitness and the Hamming class number in the quasispecies distribution. With the help of these formulas, we discuss the phenomenon of the error threshold and the notion of quasispecies. We recover the limiting quasipecies distribution in the long chain regime. We go beyond the sharp peak landscape and we consider fitness landscapes having finitely many peaks and a plateau--type landscape. We finally prove rigorously within this framework the possible occurrence of the survival of the flattest, a phenomenon which has been previously discovered by Wilke, Wang, Ofria, Lenski and Adami and which has been investigated in several works.

Survival of the flattest in the quasispecies model

TL;DR

This work rigorously analyzes the quasispecies model under mutation and selection, focusing on Moran–Kingman dynamics and the sharp-peak landscape, to derive exact formulas for the quasispecies distribution, mean fitness , and mean Hamming distance. It then extends these results to general landscapes, including finitely many peaks and plateau-type fitness, using lumping and PF-eigenvalue techniques to obtain explicit representations and limiting behaviours. A central theme is the error threshold and the survival of the flattest: the authors show, in long-chain regimes, how a flat but broad plateau can dominate over a higher but narrow peak, yielding the phenomenon of survival of the flattest, with precise conditions and limits given by and . Theoretical results are supported by probabilistic interpretations, binomial/Poisson limits, and asymptotic analyses, providing a robust framework for assessing mutational strategies in viral populations and informing concepts such as lethal mutagenesis and quasispecies structure.

Abstract

Viruses present an amazing genetic variability. An ensemble of infecting viruses, also called a viral quasispecies, is a cloud of mutants centered around a specific genotype. The simplest model of evolution, whose equilibrium state is described by the quasispecies equation, is the Moran--Kingman model. For the sharp peak landscape, we perform several exact computations and we derive several exact formulas. We obtain also an exact formula for the quasispecies distribution, involving a series and the mean fitness. A very simple formula for the mean Hamming distance is derived, which is exact and which do not require a specific asymptotic expansion (like sending the length of the macromolecules to or the mutation probability to ). We try also to extend these formulas to a general fitness landscape. We obtain an equation involving the covariance of the fitness and the Hamming class number in the quasispecies distribution. With the help of these formulas, we discuss the phenomenon of the error threshold and the notion of quasispecies. We recover the limiting quasipecies distribution in the long chain regime. We go beyond the sharp peak landscape and we consider fitness landscapes having finitely many peaks and a plateau--type landscape. We finally prove rigorously within this framework the possible occurrence of the survival of the flattest, a phenomenon which has been previously discovered by Wilke, Wang, Ofria, Lenski and Adami and which has been investigated in several works.
Paper Structure (38 sections, 11 theorems, 226 equations, 1 figure)

This paper contains 38 sections, 11 theorems, 226 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The quasispecies equations
  3. The sharp peak landscape
  4. The error threshold
  5. Extension to a general landscape
  6. Finitely many peaks
  7. Plateau
  8. Survival of the flattest
  9. The sharp peak landscape
  10. A representation formula for the quasispecies
  11. Mutation dynamics
  12. The equation for the mean fitness
  13. An exact formula for the quasispecies
  14. The error threshold
  15. An upper bound on $\lambda$
  16. ...and 23 more sections

Key Result

Theorem 1.9

For the sharp peak landscape, we have the following dichotomy: $\bullet$ If $a>\ln \sigma$, then $\lambda \rightarrow 1$, whence $x(w^*)\rightarrow 0$. $\bullet$ If $a<\ln \sigma$, then $\lambda \rightarrow \sigma e^{-a}>1$, and

Figures (1)

  • Figure 1: Fractions of the different types as a function of $a$ for ${\sigma}=5$.

Theorems & Definitions (17)

  • Theorem 1.9: Error threshold
  • Theorem 1.11
  • Theorem 1.16: Error threshold for finitely many peaks
  • Theorem 1.19
  • Theorem 1.23
  • Theorem 1.26
  • Proposition 2.6
  • Lemma 3.5
  • proof
  • proof : Soft proof
  • ...and 7 more