Soft Order in Physical Systems pp 129-132 | Cite as

# Polymers in a Random Environment and Molecular Quasi-Species

## Abstract

The quasi-species model was introduced in 1971 by Manfred Eigen^{1} to describe evolving populations of self-reproducing (RNA-like) molecules. It lies at the basis of the theory of the origin of biological organization, and in particular of the hypercycle theory, developed by Eigen and P. Schuster^{2}. It may be cast in the following form. Consider a population of self-reproducing molecules, whose structure may be described by a collection of *N* binary variables, *S* _{ i } = ±1, *i* = 1,2,⋯,*N*. Then the fraction *x* _{S} of molecules of structure **S** = (*S* _{1}, *S* _{2},⋯, *S* _{ N }) obeys the following evolution equation: where A(**S**) is the average number of offsprings that a molecule of structure **S** produces at the next generation (if one assumes infinite environmental carrying capacity) and *Q* _{SS}, is the conditional probability that the reproduction of a molecule of structure **S**′ effectively produces a molecule of structure S, and therefore represents the effects of mutations. A convenient expression for the matrix *Q* is given by where 0 < *q* < 1 is the probability of having one mutation per unit and per generation, and is the number of different units in the structures **S** and **S**′ respectively. The factor *Z*(*t*) = ∑_{S} A(**S**)*x* _{S} ensures the normalization of *x* _{S}(*t*) at any generation. In order to derive equation (1) one assumes that “generations” of the self-reproducing molecules are non-overlapping and that the number of molecules in the population is sufficiently large to neglect fluctuations in the *x* _{S}(*t*).

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