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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.

A nucleotide transition is a nucleotide substitution between two purine-derived bases or between two pyrimidine-derived bases. In DNA, the possibilities are $A\leftrightarrow G$ and $C\leftrightarrow T$: adenine exchanges with guanine, or cytosine with thymine. The chemical class is preserved; the identity of the base changes.^[ref]

From any specified DNA base, there is exactly one possible transition target and two nucleotide transversion targets. If all three targets were equally likely, transitions would comprise $1/3$ of changes. This is a counting baseline, not a biological law about mutation rates. The Kimura two-parameter model allows the rates for these two classes to differ.^[ref]

The classification does not determine a mutation's effect on a protein. A transition can preserve an amino acid, change it or introduce a stop codon, depending on the affected codon.

A nucleotide transversion is a nucleotide substitution from a purine-derived base to a pyrimidine-derived base, or the reverse. In DNA, the unordered pairs are $A\leftrightarrow C$, $A\leftrightarrow T$, $G\leftrightarrow C$ and $G\leftrightarrow T$. A nucleotide transition instead stays within one of these chemical classes.^[ref]

Each starting base has two possible transversion targets and one transition target. Thus twelve directed changes split into eight transversions and four transitions. Equal counts of possible events do not imply equal observed frequencies: chemical mutation mechanisms and natural selection can affect those frequencies.

Suppose an illustrative sequence alignment contains twelve transition differences and eight transversion differences. Its observed transition-transversion ratio is $12/8=1.5$. That ratio summarizes the scored differences; repeated substitutions at the same site can make it differ from the ratio of underlying instantaneous rates. The two-class distinction motivated the Kimura two-parameter model.^[ref]

Pairwise molecular divergence measures the accumulated evolutionary change separating two homologous molecular sequences. For aligned DNA sequences, a model-based distance $d$ is commonly expressed as the estimated number of nucleotide substitutions per site along the path connecting the sequences. It differs from the observed fraction of mismatching sites: a site may have changed more than once, including changes that restore the starting base.^[ref]

For example, 20 mismatches among 1,000 usable aligned sites give an observed mismatch fraction $p=0.020$. That is a count of present-day differences. An ancestral A that changed to G and then back to A contributes two historical substitutions but no endpoint mismatch. Sequence alignment and an explicit nucleotide substitution model are therefore needed to turn observed differences into an evolutionary interpretation.

The Kimura two-parameter model distinguishes nucleotide transitions from nucleotide transversions. If $P$ and $Q$ are their observed fractions among compared sites, respectively, its corrected pairwise estimate isFor $P=0.01$ and $Q=0.01$, this gives approximately $0.02028$ substitutions per site, slightly above the observed mismatch fraction $P+Q=0.020$. The correction accounts for hidden changes under the model's assumptions; it is not a direct count of every event. Both logarithm arguments must be positive for this finite real-valued estimate.^[ref]

Under a constant, equal per-lineage molecular clock rate $r$ and a split $T$ time units ago, the two branches contribute $d=2rT$. A pairwise divergence $d=0.02$ at $r=10^{-9}$ substitutions per site per year corresponds to $T=10^7$ years in that idealized model. The factor of two counts both lineages. Without a rate calibration or another time constraint, distance alone does not identify elapsed time.^[ref]

An observed transition-transversion ratio compares the number of nucleotide transitions with the number of nucleotide transversions in a specified set of sequence alignment differences:Six transition differences and two transversion differences give $R=3$. If no transversion is observed, this ordinary finite ratio is undefined; the raw counts should remain explicit.

The instantaneous rate parameter can use a different convention. In the Kimura two-parameter model, let $\alpha$ be the rate to the one transition target and $\beta$ the rate to each of the two transversion targets from a base. ThenThus $\kappa =4$ corresponds to twice as many instantaneous transition events as total transversion events, not four times as many.^[ref]

Observed endpoint differences can hide repeated nucleotide substitutions. Under independent uniform endpoint bases, the four directed transition mismatches and eight transversion mismatches give a limiting ratio $1/2$, even when short-time event rates differ. Always state whether the ratio describes scored differences, total events or rates per possible target.