Molecular Quasi-Species in Hopfield Replication Landscapes

Abstract

Twenty years ago Eigen¹ proposed a model which, in the mathematical formulation given by Eigen and Schuster², links the dynamics of population genetic in biological systems with the dynamics of chemical reactions opening the field of molecular evolution. In its simplest realization the model describes a system of aperiodic polymers with a fixed number, N, of monomers which may be of v different classes. For RNA we would have v = 4 to represent the four nucleotides or v = 2 if we discern only between purines and pyrimidines. In this case, which is assumed all over this article, each polymer is represented by a binary sequence (s ₁,s ₁, …,s _N ), with s _i coded here as 1 or -1. The dynamics of the population follows from the (very far from equilibrium) chemical reactions describing the replication of these molecules in the presence of activated monomers and possibly of some catalyst, like in the flux reactor experiments for RNA replication in the presence of a replicase enzyme³. If we use I _k with k = 1 to 2^N) as a shorthand for each possible sequence, the replication reaction is:

$${I_k} + \left[ {activated monomers} \right] \to {I_k} + {I_j}$$

(1)

with a polymer I _k producing a new polymer I _j out of activated monomers, which are kept in a concentration high enough to have the reaction flowing only in one direction, at a rate denoted by W _jk .