title: On the behavior of the leading eigenvalue of Eigen's evolutionary matrices.
abstract: We study general properties of the leading eigenvalue w¯(q) of Eigen's evolutionary matrices depending on the replication fidelity q. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for w¯(q),w¯(')(q),w¯('')(q) for q = 0, 0.5, 1 and prove that the absolute minimum of w¯(q), which always exists, belongs to the interval (0, 0.5]. For the specific case of a single peaked landscape we also find lower and upper bounds on w¯(q), which are used to estimate the critical mutation rate, after which the distribution of the types of individuals in the population becomes almost uniform. This estimate is used as a starting point to conjecture another estimate, valid for any fitness landscape, and which is checked by numerical calculations. The last estimate stresses the fact that the inverse dependence of the critical mutation rate on the sequence length is not a generally valid fact.
authors.affiliation: Applied Mathematics-1, Moscow State University of Railway Engineering, Moscow 127994, Russia.
authors.forename: Yuri S
authors.initials: YS
authors.lastname: Semenov
authors.affiliation: Applied Mathematics-1, Moscow State University of Railway Engineering, Moscow 127994, Russia; Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow 119992, Russia.
authors.forename: Alexander S
authors.initials: AS
authors.lastname: Bratus
authors.affiliation: Department of Mathematics, North Dakota State University, Fargo, ND 58108, USA. Electronic address: artem.novozhilov@ndsu.edu.
authors.forename: Artem S
authors.initials: AS
authors.lastname: Novozhilov
mesh_terms: D005075:Biological Evolution; D008962:Models, Theoretical
keywords: Dominant eigenvalue; Eigen model; Error threshold; Quasispecies model; Single peaked landscape