title: On Eigen's Quasispecies Model, Two-Valued Fitness Landscapes, and Isometry Groups Acting on Finite Metric Spaces.
abstract: A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single- or sharply peaked landscape. A general, non-permutation invariant quasispecies model is studied, and therefore the dimension of the problem is [Formula: see text], where N is the sequence length. It is shown that if the fitness function is equal to [Formula: see text] on a G-orbit A and is equal to w elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most [Formula: see text]. Here G is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length N with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the G-orbit A. Motivated by the analysis of the two-valued fitness landscapes, an abstract generalization of Eigen's model is introduced such that the sequences are identified with the points of a finite metric space X together with a group of isometries acting transitively on X. In particular, a simplicial analog of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.
authors.affiliation: Applied Mathematics-1, Moscow State University of Railway Engineering, Moscow, Russia, 127994.
authors.forename: Yuri S
authors.initials: YS
authors.lastname: Semenov
authors.affiliation: Department of Mathematics, North Dakota State University, Fargo, ND, 58108, USA. artem.novozhilov@ndsu.edu.
authors.forename: Artem S
authors.initials: AS
authors.lastname: Novozhilov
mesh_terms: D056084:Genetic Fitness; D055641:Mathematical Concepts; D008957:Models, Genetic; D009154:Mutation; D012641:Selection, Genetic
keywords: Eigen’s quasispecies model; Finite metric space; Isometry group; Mean population fitness; Regular polytope; Single-peaked landscape