Review: Some Probabilistic and Statistical Problems in the Analysis of DNA Sequences

Citation

• Tavaré, S. (1986). Some probabilistic and statistical problems in the analysis of DNA sequences. Lectures on Mathematics in the Life Sciences, 17, 57–86. American Mathematical Society.
• WorldCat

Abstract

The chapter concentrates on statistical aspects of estimating substitution rates and divergence times from DNA sequence data. A new estimation method is suggested and applied to serum albumin and alpha-fetoprotein data. The Jukes-Cantor, Kimura, and general reversible substitution models are reviewed and unified in a common continuous-time Markov chain framework.

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This book chapter is the canonical reference for the General Time-Reversible (GTR) substitution model used in Hifuku's likelihood kernels. Tavaré works in Section II ("Stochastic Models of Substitutions," pp. 57–63) to build up the theory from first principles, starting with the basic properties of a continuous-time Markov chain on four character states and arriving at the most general reversible rate matrix.

A continuous-time Markov chain on nucleotide states has an instantaneous rate matrix \( Q = (q_{ij}) \), where off-diagonal entries are non-negative rates and diagonal entries are determined by the row-sum condition \( Q\mathbf{1} = \mathbf{0} \). The fundamental transition probability matrix (Eq. 2.7, p. 59) is the matrix exponential:

\[P_t = e^{Qt} := \sum_{n=0}^{\infty} Q^n \frac{t^n}{n!}\]

where \( P_{ij}(t) = P[X(t) = j \mid X(0) = i] \) is the probability of transitioning from state \( i \) to state \( j \) in evolutionary time \( t \). Every substitution model in Hifuku produces a \( Q \) matrix; the computation of \( P(t) \) then follows via eigendecomposition (see Section 4.2 of the theory document).

The chapter reviews how simpler models arise as special cases. Jukes-Cantor (Example 2.1, p. 60) sets all off-diagonal rates equal and stationary frequencies to 1/4. Kimura's two-parameter model (Example 2.2, p. 61) distinguishes transitions from transversions. Felsenstein's 1981 model (Example 2.4, p. 62) allows arbitrary stationary frequencies but a single exchangeability rate. GTR generalizes all of these.

The key reversibility condition (Eq. 2.11, p. 62) is:

\[\pi_i\,q_{ij} = \pi_j\,q_{ji}, \quad 1 \le i, j \le 4\]

This detailed balance condition is what allows the symmetrization trick used to compute \( P(t) \) efficiently via eigendecomposition rather than general matrix exponentiation, and it guarantees Felsenstein's Pulley Principle (root placement does not affect the likelihood).

The GTR rate matrix itself is given in Example 2.5, Eq. (2.13), p. 63:

\[Q = \begin{pmatrix} \cdot & x_1 & x_2 & x_3 \\ \pi_1 x_1/\pi_2 & \cdot & x_4 & x_5 \\ \pi_1 x_2/\pi_3 & \pi_2 x_4/\pi_3 & \cdot & x_6 \\ \pi_1 x_3/\pi_4 & \pi_2 x_5/\pi_4 & \pi_3 x_6/\pi_4 & \cdot \end{pmatrix}\]

with \( x_i \geq 0 \) and diagonal entries determined by Eq. (2.8). This gives 6 free exchangeability parameters \( x_1,\ldots,x_6 \) plus 3 free stationary frequencies (the fourth is constrained by summation to 1), for 9 parameters in total. In Hifuku and most modern phylogenetics software, the off-diagonal entries are written in the equivalent form \( q_{ij} = r_{ij}\,\pi_j \) where \( r_{ij} = r_{ji} \) are the exchangeability parameters.

In-document navigation

Equation	Location
Transition probability matrix	Eq. (2.7), p. 59
Rate matrix conditions	Eq. (2.8), p. 59
Stationarity condition	Eq. (2.9), p. 60
Detailed balance (reversibility)	Eq. (2.11), p. 62
GTR rate matrix	Example 2.5, Eq. (2.13), p. 63