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The 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.^[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, isFor 1,000 comparable sites with 40 transition differences and 20 transversion differences, $P=0.04$, $Q=0.02$ and $\hat{d}\approx 0.06289$, compared with an observed mismatch fraction $0.06$. The excess accounts for multiple changes that can be hidden at the endpoints under the model.^[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.

Molecular evolution is inherited change in molecules such as DNA and proteins across generations. Mutations introduce new variants; natural selection and genetic drift influence which variants persist. Comparing a nucleotide sequence or protein sequence between lineages therefore supplies evidence about their history as well as their present chemistry.

An important early example was the comparison of cytochrome c from different organisms. Emanuel Margoliash's 1963 report connected differences in protein primary structure to evolutionary relationships. The experimental input was the ordered amino acid residues of the proteins, rather than only their bulk composition or their shared function. Placing corresponding positions side by side made molecular differences countable.^[ref]

For a deliberately simplified example, suppose two already aligned sequences have $k=8$ differing positions among $n=100$ compared positions. Their observed difference fraction is $k/n=0.08$. That statistic is not itself a divergence time. If both lineages split $t$ time units ago and each accumulates substitutions at the same constant rate $r$ per position per unit time, their expected accumulated pairwise molecular divergence isThe factor two counts the two descendant branches. At low divergence, the observed difference fraction can approximate $d$; repeated changes at one position can conceal earlier changes, and unequal branch rates invalidate the equal-rate simplification. A molecular clock additionally needs a rate or time calibration to turn molecular differences into dates.

Sequence identity is a measured agreement between aligned positions. homology is a historical relationship of common descent; it is not a percentage. Orthology and paralogy further distinguish relationships associated with lineage splitting and gene duplication. These distinctions matter when a phylogenetic tree is inferred from a chosen gene rather than directly observed.

The neutral theory of molecular evolution provides a complementary population-level explanation: molecular change need not always be an adaptive improvement. Motoo Kimura's 1968 argument emphasized the potential importance of neutral changes and genetic drift, while leaving room for natural selection.^[ref]