On Eigen's quasispecies model, two-valued fitness landscapes, and isometry groups acting on finite metric spaces

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, therefore the dimension of the problem is $2^N\times 2^N$, where $N$ is the sequence length. It is shown that if the fitness function is equal to $w+s$ 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 $N+1$. 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 $A$. Sufficient conditions for the so-called error threshold for sequences of orbits are given. 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 analogue 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.