Polymers in a Random Environment and Molecular Quasi-Species
The quasi-species model was introduced in 1971 by Manfred Eigen1 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. Schuster2. 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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