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

TL;DR

This paper generalizes Eigen's quasispecies model to two-valued fitness landscapes on finite metric spaces with isometry group actions, deriving algebraic equations for mean fitness and analyzing error thresholds, with applications to antigenic variation.

Contribution

It introduces a new two-valued fitness landscape model for Eigen's quasispecies, incorporating isometry group actions and providing explicit algebraic formulas for mean fitness.

Findings

Mean population fitness as the largest root of an algebraic equation of degree at most N+1.

Explicit algebraic equations derived using spherical growth functions.

Sufficient conditions for error thresholds in sequences of orbits.

Abstract

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.