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

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 [Formula: see text] 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 [Formula: see text], then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most [Formula: see text], 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 [Formula: see text] with p-adic metric.