Solution of a gene divergence problem under arbitrary stable nucleotide transition probabilities

A nucleic acid chain L nucleotides in length, with the specific base sequence B1B2....BL, each Bi being A, G, C, or T, is defined by the L-dimensional vector B = (B1, B2, ..., BL), the kth position in the chain being occupied by the base Bk. Let pBB, be the twelve given constant nonnegative transition probabilities that in a specified position the base B is replaced by the base B' in a single step, and let P(X)BB, be the probability that the position goes from base B to B' in X steps. An exact analytical expression for P(X)BB' is derived. Assuming that each base mutates independently of the others, an exact expression is derived for the probability P(X)BB' that the initial gene sequence B goes to a sequence B' = (B'1, B'2; ..., B'L) after X = (X1, X2, ..., XL) base replacements, where Xk is the number of single step base replacements in the kth position. The resulting equations allow a more precise accounting for the effects of Darwinian natural selection in molecular evolution than does the idealized but biologically less accurate assumption that each of the four nucleotides is equally likely to mutate to and be fixed as one of the other three. Illustrative applications of the theory to some problems in biological evolution are given.