Substitution Model - The Mathematics of Substitution Models

The Mathematics of Substitution Models

Stationary, neutral, independent, finite sites models (assuming a constant rate of evolution) have two parameters, an equilibrium vector of base (or character) frequencies and a rate matrix, Q, which describes the rate at which bases of one type change into bases of another type; element for i ≠ j is the rate at which base i goes to base j. The diagonals of the Q matrix are chosen so that the rows sum to zero:

The equilibrium row vector π must be annihilated by the rate matrix Q:

The transition matrix function is a function from the branch lengths (in some units of time, possibly in substitutions), to a matrix of conditional probabilities. It is denoted . The entry in the ith column and the jth row, is the probability, after time t, that there is a base j at a given position, conditional on there being a base i in that position at time 0. When the model is time reversible, this can be performed between any two sequences, even if one is not the ancestor of the other, if you know the total branch length between them.

The asymptotic properties of P_ij(t) are such that P_ij(0) = δ_ij, where δ_ij is the Kronecker delta function. That is, there is no change in base composition between a sequence and itself. At the other extreme, or, in other words, as time goes to infinity the probability of finding base j at a position given there was a base i at that position originally goes to the equilibrium probability that there is base j at that position, regardless of the original base. Furthermore, it follows that for all t.

The transition matrix can be computed from the rate matrix via matrix exponentiation:

where Qn is the matrix Q multiplied by itself enough times to give its nth power.

If Q is diagonalizable, the matrix exponential can be computed directly: let Q = U−1 Λ U be a diagonalization of Q, with

$\Lambda = \begin{pmatrix} \lambda_1 & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \lambda_4 \end{pmatrix}\,,$

where Λ is a diagonal matrix and where are the eigenvalues of Q, each repeated according to its multiplicity. Then

where the diagonal matrix eΛt is given by

$e^{\Lambda t} = \begin{pmatrix} e^{\lambda_1 t} & \ldots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & e^{\lambda_4 t} \end{pmatrix}\,.$

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