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Molecular Clock Models

Overview

Molecular clock models describe how genetic divergence accumulates over time along the branches of a phylogenetic tree. They are essential for calibrating node times and estimating evolutionary rates. The PRISTINE framework implements two main clock models:

  • ConditionalErrorClock: a binomial noise model based on JC69 substitution probabilities.
  • ContinuousAdditiveRelaxedClock: a gamma-distributed model for rate heterogeneity.

1. Conditional Error Clock

This model assumes that the observed distance \(d\) between two sequences is a noisy estimate of the true expected distance, which follows a Jukes-Cantor 1969 (JC69) model:

\[ p(l) = \frac{K - 1}{K} \left(1 - \exp\left(-\frac{K}{K-1} l\right)\right) \]

The likelihood is modeled as a binomial draw:

\[ \log \Pr(D \mid l) = \log \binom{L}{L p(d)} + L p(d) \log p(l) + L (1 - p(d)) \log(1 - p(l)) \]

where: - \(L\) is the number of sites - \(p(d)\) is the empirical JC69 probability for observed distance \(d\) - \(p(l)\) is the expected JC69 probability at branch length \(l = r \cdot t\) (rate × duration)

This model is used when the observed distance is fixed, and the goal is to estimate branch durations【124†source】.

2. Continuous Additive Relaxed Clock (cARC)

This model assumes that branch-specific substitution rates are gamma-distributed:

  • Mean rate: \(\mu = \exp(\text{log\_rate})\)
  • Dispersion: \(\phi\)
  • Shape parameter: \(\alpha = \mu \cdot L \cdot t \cdot r\)
  • Rate parameter: \(\beta = \frac{1}{1 + \phi}\)

Then the likelihood of observing a distance \(d\) is:

\[ \log p(d \mid t) = \log \text{Gamma}(d \cdot L; \alpha, \beta) \]

To ensure numerical stability, a barrier penalty is applied to the gamma shape:

\[ \text{barrier}(x) = \text{softplus}(-x, \text{strength}) \cdot \text{strength} \]

This model generalizes the strict clock and allows rates to vary across lineages【124†source】.

Simulation

The simulate() method in ContinuousAdditiveRelaxedClock draws distances:

\[ d \sim \text{Gamma}(\alpha, \beta), \quad d = \text{distance} / L \]

This supports stochastic modeling of genetic divergence for inference or synthetic data generation.

API

ConditionalErrorClock

  • log_likelihood(durations, distances): binomial JC69 likelihood
  • rate(): returns \(\exp(\text{log\_rate})\)

ContinuousAdditiveRelaxedClock

  • log_likelihood(durations, distances): gamma log-likelihood
  • simulate(durations): draw distances from gamma
  • rate(): mean substitution rate

Initialization Utilities

  • new_conditional_error_clock(K, L): creates a ConditionalErrorClock
  • new_continuous_additive_relaxed_clock(L): creates a ContinuousAdditiveRelaxedClock with default parameters【124†source】

References

  • Jukes, T. H., & Cantor, C. R. (1969). Evolution of protein molecules.
  • Drummond, A. J., & Suchard, M. A. (2010). Bayesian random local clocks.
  • Yang, Z. (2014). Molecular Evolution: A Statistical Approach. Oxford University Press.

Source

Defined in molecularclock.py【124†source】.