HKY

HKY, short for Hasegawa–Kishino–Yano, is a nucleotide substitution model used in molecular evolution to describe DNA sequence changes along a phylogenetic tree. Introduced in 1985 by Hasegawa, Kishino, and Yano, it extends the simpler Felsenstein 1981 model (F81) by allowing different rates for transitions and transversions and by accommodating unequal base frequencies.

In HKY, base frequencies pi_A, pi_C, pi_G, pi_T (sum to 1) determine the stationary distribution. A single parameter kappa (κ) controls the relative rate of transitions (A↔G, C↔T) versus transversions (the other changes). The instantaneous rate matrix Q has off-diagonal elements q_ij = κ π_j for transitions and q_ij = π_j for transversions; diagonal elements ensure each row sums to zero. The model is time-reversible and assumes the process is stationary at equilibrium base frequencies.

HKY is commonly used in maximum likelihood and Bayesian phylogenetic inference. It is often preferred over simpler models when there is noticeable base composition bias or when transitions occur more frequently than transversions. It can be combined with rate heterogeneity across sites, like a gamma distribution, and with invariant sites.

Limitations: HKY assumes homogeneity across sites and lineages and a constant base composition over time; it may be inadequate for sequences with strong non-stationarity or compositional shift. Like other nucleotide models, it is a simplification of the true substitution process.