Journal of Mathematical Biology

, Volume 78, Issue 3, pp 837–878 | Cite as

Generalized quasispecies model on finite metric spaces: isometry groups and spectral properties of evolutionary matrices

  • Yuri S. Semenov
  • Artem S. NovozhilovEmail author


The quasispecies model introduced by Eigen in 1971 has close connections with the isometry group of the space of binary sequences relative to the Hamming distance metric. Generalizing this observation we introduce an abstract quasispecies model on a finite metric space X together with a group of isometries \(\Gamma \) acting transitively on X. We show that if the domain of the fitness function has a natural decomposition into the union of tG-orbits, G being a subgroup of \(\Gamma \), then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most \(t\cdot \mathrm{rk}_{\mathbf {Z}} R\), where R is the orbital ring that is defined in the text. The general theory is illustrated by three detailed examples. In the first two of them the space X is taken to be the metric space of vertices of a regular polytope with the natural “edge” metric, these are the cases of a regular m-gon and of a hyperoctahedron; the final example takes as X the quotient rings \(\mathbf {Z}/p^n\mathbf {Z}\) with p-adic metric.


Quasispecies model Finite metric space Dominant eigenvalue Mean population fitness Isometry group Regular polytope 

Mathematics Subject Classification

15A18 92D15 92D25 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Mathematics-1Russian University of TransportMoscowRussia
  2. 2.Department of MathematicsNorth Dakota State UniversityFargoUSA

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