Estimation of the number of amino acid substitutions per site when the substitution rate varies among sites

A general model for estimating the number of amino acid substitutions per site (d) from the fraction of identical residues between two sequences (q) is proposed. The well-known Poisson-correction formula q = e^−d corresponds to a site-independent and amino-acid-independent substitution rate. Equation q = (1 − e ^{−2 d})/2 d, derived for the case of substitution rates that are site-independent, but vary among amino acids, approximates closely the empirical method, suggested by Dayhoff et al. (1978). Equation q = 1/(1 + d) describes the case of substitution rates that are amino acid-independent but vary among sites. Lastly, equation q = [ln(1 + 2 d)]/2 d accounts for the general case where substitution rates can differ for both amino acids and sites.