= Pairwise molecular divergence
{title2=$d$}
\b[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.<A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences by Motoo Kimura>[Summary: evolutionary distances estimated from comparative nucleotide sequences.]{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 is
$$
\widehat d=-\frac12\ln(1-2P-Q)-\frac14\ln(1-2Q).
$$
For $P=0.01$ 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.<A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences by Motoo Kimura>[Summary and distance formula for transition/transversion differences.]{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.<A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences by Motoo Kimura>[Rate relation K/(2T).]{ref}
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