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Review: The Kimura (1983) Protein Distance (The Neutral Theory of Molecular Evolution)

Citation

  • Kimura, M. (1983). The Neutral Theory of Molecular Evolution. Cambridge University Press.
  • DOI

Scope of this review

The primary source is a book whose main subject is the neutral theory of molecular evolution. Hifuku uses one narrow, unrelated result from it, Kimura's approximate distance correction for amino acid sequences, so this review covers that formula and its role, not the neutral theory, and is contextualized by a secondary source in place of the book itself.

For amino acid sequences there is no single-parameter analogue of the Jukes-Cantor nucleotide model, because the twenty-state substitution process is not well described by one rate. Kimura (1983) gave a simple empirical correction that avoids fitting a full rate matrix. It relates the expected number of substitutions per site \( d \) to the observed proportion of differing residues \( p \) by

\[d = -\ln\!\left(1 - p - 0.2\,p^2\right).\]

The linear term is the leading correction and the quadratic term \( 0.2\,p^2 \) is an empirical adjustment for multiple substitutions at the same site, fitted so that the formula tracks the more elaborate corrections (such as a Dayhoff/PAM series) over a moderate range without needing a matrix. It is fast and model-free, accurate at low to moderate divergence, and diverges as \( p \) grows toward saturation.

According to PubMed-indexed literature, Sonnhammer and Hollich (2005) evaluate correction-based protein distance estimators, including this Kimura correction, against optimal-matrix methods, and report that the simple corrections are fast but less accurate than matrix-based estimators at large distances (DOI).

Relevance to Hifuku

Hifuku uses the Kimura (1983) correction to build protein neighbor-joining anchors. nj_tree turns the observed proportion of differing residues between two protein sequences into an additive distance by \( d = -\ln(1 - p - 0.2 p^2) \), flags saturated pairs, and neighbor-joins the resulting distance matrix. This is the protein counterpart of the Jukes-Cantor DNA correction (see Jukes & Cantor 1969): both turn an observed proportion of differences into an additive distance, so a protein alignment yields anchor trees on the same footing as a DNA alignment, and the chart is built the same way regardless of the sequence type (see Saitou & Nei 1987).