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

Summary A nucleic acid chain L nucleotides in length, with the specific base sequence B₁B₂.B_L, each B_i being A, G, C, or T, is defined by the L-dimensional vector B = (B₁, B₂, , B_L), the k^th position in the chain being occupied by the base B_k. Let P_BB' be the twelve given constant non-negative transition probabilities that in a specified position the base B is replaced by the base B in a single step, and let P_BB'^(XX) be the probability that the position goes from base B to B in X steps. An exact analytical expression for P_BB'^(XX) is derived. Assuming that each base mutates independently of the others, an exact expression is derived for the probability P_BB'^(XX) that the initial gene sequence B goes to a sequence B = (B₁, B₂, , B_L) after X = (X₁, X₂, , X_L) base replacements, where X_k is the number of single-step base replacements in the k^th 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.