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Survival of the flattest in the quasispecies model

Part of: Applications - Dynamical systems and ergodic theory General theory of linear operators

Published online by Cambridge University Press:  02 December 2024

Maxime Berger  and
Raphaël Cerf
Maxime Berger*
Affiliation:
Université de Bourgogne
Raphaël Cerf*
Affiliation:
Université Paris-Saclay
*
*Postal address: 40 Rue René Cassin, 21850 Saint-Apollinaire, France. Email address: [email protected]
**Postal address: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France. Email address: [email protected]
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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 derive several exact formulas. We also obtain 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 does not require a specific asymptotic expansion (such as sending the length of the macromolecules to $\infty$ or the mutation probability to 0). With the help of these formulas, we present an original proof for the well-known phenomenon of the error threshold. We recover the limiting quasispecies distribution in the long-chain regime. 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. Going beyond the sharp-peak landscape, we consider fitness landscapes having finitely many peaks and a plateau-type landscape. Finally, within this framework, we prove rigorously the possible occurrence of the survival of the flattest, a phenomenon which was previously discovered by Wilke et al. (Nature 412, 2001) and which has been investigated in several works (see e.g. Codoñer et al. (PLOS Pathogens 2, 2006), Franklin et al. (Artificial Life 25, 2019), Sardanyés et al. (J. Theoret. Biol. 250, 2008), and Tejero et al. (BMC Evolutionary Biol. 11, 2011)).

Keywords

Eigen’s modelmutationselectionfitnessplateau

MSC classification

Primary: 92D15: Problems related to evolution
Secondary: 47A75: Eigenvalue problems 37N25: Dynamical systems in biology
Type
Original Article
Information
Advances in Applied Probability , Volume 57 , Issue 2 , June 2025 , pp. 566 - 606
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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