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 $\Gamma$ acting transitively on $X$. We show that if the domain of the fitness function has a natural decomposition into the union of $t$ $G$-orbits, $G$ being a subgroup of $\Gamma$, then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most $t\cdot {\rm rk}_{\mathbf Z} R$, where $R$ is what we call the orbital ring. The general theory is illustrated by two examples, in both of which $X$ is taken to be the metric space of vertices of a regular polytope with the "edge" metric; namely, the case of a regular $m$-gon and of a hyperoctahedron are considered.