= Kimura two-parameter model
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The \b[Kimura two-parameter model] is a <nucleotide substitution model> that allows one instantaneous rate for <nucleotide transitions> and another for <nucleotide transversions>. A transition exchanges two purines or two pyrimidines; a transversion exchanges a purine with a pyrimidine. In the model's symmetric form, each base has one transition neighbor with rate $\alpha$ and two transversion neighbors, each with rate $\beta$. Equal equilibrium base frequencies and constant rates across sites are important assumptions.<A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences by Motoo Kimura>[1980 model construction and two classes of nucleotide differences.]{ref}
For aligned sequences let $P$ be the fraction of sites differing by a transition and $Q$ the fraction differing by a transversion. The estimated <pairwise molecular divergence>, measured in substitutions per site, is
$$
\widehat d=-\frac12\ln(1-2P-Q)-\frac14\ln(1-2Q).
$$
For 1,000 comparable sites with 40 transition differences and 20 transversion differences, $P=0.04$, $Q=0.02$ and $\widehat d\approx0.06289$, compared with an observed mismatch fraction $0.06$. The excess accounts for multiple changes that can be hidden at the endpoints under the model.<A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences by Motoo Kimura>[Distance formula and comparison of transition/transversion differences.]{ref}
The logarithms require $1-2P-Q>0$ and $1-2Q>0$. Near their boundaries, small sampling changes can produce large distance changes; beyond them this finite real-valued correction is unavailable. The formula is a model-based inference, not a universal correction for arbitrarily divergent sequences. Unequal base composition or strong variation among sites can violate its assumptions.
The name refers to the two substitution-rate classes, not to two observed substitutions or two species. When rates are expressed per unit time, the total instantaneous departure rate from a base is $\alpha+2\beta$. Converting a fitted distance to elapsed time further requires a <molecular clock> rate calibration.
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