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Review: The Jukes-Cantor Model (Evolution of Protein Molecules)

Citation

  • Jukes, T. H. & Cantor, C. R. (1969). Evolution of protein molecules. In H. N. Munro (ed.), Mammalian Protein Metabolism, Vol. 3, pp. 21–132. Academic Press.
  • DOI

Scope of this review

The primary source is a book chapter. Hifuku uses one narrow result from it, the one-parameter (JC69) substitution model and its nucleotide distance correction, so this review covers that result and its role rather than the whole chapter, contextualized by a modern secondary source in place of the chapter itself.

Jukes and Cantor introduced the simplest model of nucleotide substitution. Every substitution occurs at the same rate, so the four bases are fully exchangeable and the equilibrium base frequencies are each \( 1/4 \). The model has a single free parameter, the overall substitution rate.

Under this model, the expected number of substitutions per site \( d \), the quantity that behaves additively along a tree, relates to the observed proportion of differing sites \( p \) by

\[d = -\tfrac{3}{4}\,\ln\!\left(1 - \tfrac{4}{3}\,p\right).\]

The correction accounts for multiple and back substitutions at the same site, which make the raw proportion \( p \) an underestimate of the true number of changes. As \( p \) approaches \( 3/4 \), the argument of the logarithm approaches zero and \( d \) diverges: the sequences are saturated and the distance is no longer estimable.

According to PubMed-indexed literature, correction-based distance estimators of this kind, including the Jukes-Cantor correction, are placed in the context of modern protein and nucleotide distance estimation by Sonnhammer and Hollich (2005), who compare simple corrections against optimal-matrix methods (DOI).

Relevance to Hifuku

The Jukes-Cantor model appears in Hifuku in two places. As a substitution model it is the JC69 rate matrix (\( Q_{ii} = -1 \), \( Q_{ij} = 1/3 \) for \( i \neq j \)), available as jc69 and used in the Felsenstein likelihood as the simplest DNA model (Section 4). As a distance it is the DNA correction applied by nj_tree: the observed proportion of differing sites between two DNA sequences is turned into an additive distance by \( d = -\tfrac{3}{4}\ln(1 - \tfrac{4}{3}p) \), and pairs at or beyond \( p = 3/4 \) are flagged as saturated. That distance matrix builds the DNA neighbor-joining anchors that fix the chart. Felsenstein (1981) gives JC69 as the special case of the general reversible model (see Felsenstein 1981), and neighbor-joining uses the JC distance to build trees (see Saitou & Nei 1987). The protein counterpart is the Kimura (1983) correction (see Kimura 1983).