Generalized Time-Reversible (GTR) Model¶

Overview¶

The GTR model is a flexible and widely used continuous-time Markov model (CTMM) for modeling sequence evolution in phylogenetics. It defines a stationary and time-reversible substitution process over a finite set of states (e.g., nucleotides or amino acids).

The GeneralizedTimeReversibleModel class in the PRISTINE framework provides a fully differentiable implementation of this model. It supports gradient-based estimation of evolutionary parameters directly from data and provides efficient routines for computing matrix exponentials needed in likelihood calculations.

This model is parameterized by: - A stationary distribution \(\boldsymbol{\pi}\) over the \(K\) states - A symmetric matrix of exchange rates \(R\) between state pairs \((i, j)\)

The rate matrix \(Q\) is then derived from these components.

Continuous-Time Markov Models (CTMM)¶

In a CTMM, the probability of transitioning from state \(i\) to \(j\) over time \(t\) is described by the matrix exponential of a rate matrix \(Q\):

\[ P(t) = \exp(Qt) \]

The entries of \(Q\) satisfy: - \(Q_{ij} \geq 0\) for \(i \neq j\) (substitution rates) - \(\sum_{j} Q_{ij} = 0\) for all \(i\) (rows sum to zero)

The diagonal entries are defined as:

\[ Q_{ii} = -\sum_{j \neq i} Q_{ij} \]

GTR Parameterization¶

The GTR model assumes time-reversibility, i.e., \(\pi_i Q_{ij} = \pi_j Q_{ji}\).

It is parameterized via: - \(\boldsymbol{\pi}\): stationary distribution (a probability vector) - \(R\): symmetric exchangeability matrix

For \(i \neq j\), the off-diagonal rate is:

\[ Q_{ij} = \pi_j R_{ij} \]

Diagonal entries are set to ensure rows sum to zero:

\[ Q_{ii} = -\sum_{j \neq i} Q_{ij} \]

This guarantees that \(\boldsymbol{\pi}\) is the stationary distribution of \(Q\).

Stationary Distribution¶

Instead of directly optimizing \(\pi\), the GTR model uses an unconstrained parameterization via logits \(\boldsymbol{\ell} \in \mathbb{R}^{K-1}\). The first component of \(\pi\) is pinned to act as reference:

\[ \pi_0 = \frac{1}{1 + \sum_{i=1}^{K-1} \exp(\ell_i)}, \quad \pi_i = \frac{\exp(\ell_i)}{1 + \sum_{j=1}^{K-1} \exp(\ell_j)}, \quad i = 1,\dots,K-1 \]

This ensures \(\pi\) is a valid probability distribution (non-negative and sums to one) while reducing redundancy.

Exchangeability Matrix \(R\)¶

Only the upper triangle (excluding diagonal) of \(R\) is parameterized. These values are exponentiated for positivity and symmetrically assigned:

\[ R_{ij} = R_{ji} = \exp(r_{ij}), \quad i < j \]

Then the full \(Q\) matrix is constructed as described above.

Implementation Details¶

The GeneralizedTimeReversibleModel class supports:

stationary_dist(): returns \(\boldsymbol{\pi}\)
rate_matrix_stationary_dist(): returns the full rate matrix \(Q\) and \(\boldsymbol{\pi}\)
average_rate(): computes the mean substitution rate \(\mu = -\sum_i \pi_i Q_{ii}\)
compute_batch_matrix_exp(durations): efficiently computes \(\exp(Qt)\) for a batch of durations

Internally, the model builds \(Q\) via:

Q = pi.unsqueeze(0) * R
Q.fill_diagonal_(0.0)
Q.diagonal().copy_(-Q.sum(dim=1))

which guarantees that \(Q\) is valid and satisfies detailed balance.

Practical Usage¶

This model is typically used with the Felsenstein pruning algorithm to compute likelihoods on phylogenetic trees. In the PRISTINE framework, it integrates with:

FelsensteinPruningAlgorithm for sequence likelihoods
SequenceSimulationVisitor for simulating sequence evolution
Molecular clock models for dating trees

The GTR model is essential for inferring both phylogenies and substitution dynamics from sequence data.

References¶

Yang, Z. (2014). Molecular Evolution: A Statistical Approach. Oxford University Press.
Felsenstein, J. (1981). Evolutionary trees from DNA sequences: a maximum likelihood approach. Journal of Molecular Evolution.

Source¶

Defined in gtr.py【26†source】.