generalized_jukes_cantor - rasigadelab/mathwiki GitHub Wiki

The generalized Jukes-Cantor model

The generalized Jukes-Cantor model extends the original Jukes-Cantor (JC69) model to handle an arbitrary number of states $K$. This model is used in evolutionary biology to describe the substitution of discrete characters (such as nucleotides or amino acids) over time in a continuous-time Markov chain (CTMC) framework. The original JC69 model applies to four nucleotides (A, C, G, T), assuming equal rates of substitution between all pairs of nucleotides. The generalized model extends this to $K$ states and assumes symmetric substitution rates between any two states.

1. Generalized Jukes-Cantor Model Setup

Let the number of possible states be $K$ (where $K$ could represent nucleotides, amino acids, or other discrete characters). In the generalized Jukes-Cantor model:

  • Each state has an equal probability of transitioning to any other state.
  • The substitution rate from one state to another is the same across all state pairs.

The substitution process is governed by a transition rate matrix $Q$, where:

  • $q_{ij} = \beta$ for all $i \neq j$, representing the rate at which state $i$ transitions to state $j$.
  • $q_{ii} = -(K-1)\beta$, which ensures that the rows of the rate matrix sum to zero. This reflects the total rate of leaving state $i$.

Thus, the transition rate matrix $Q$ for $K$ states can be written as:

$$Q_{\text{Generalized JC}} = \begin{pmatrix} -(K-1)\beta & \beta & \beta & \dots & \beta \ \beta & -(K-1)\beta & \beta & \dots & \beta \ \vdots & \vdots & \vdots & \ddots & \vdots \ \beta & \beta & \beta & \dots & -(K-1)\beta \end{pmatrix}$$

This matrix describes the instantaneous rates of transitioning between states in the generalized Jukes-Cantor model.

2. Solving for the Transition Probability Matrix

The goal is to find the transition probability matrix $P(t)$, which gives the probability of transitioning from one state to another after time $t$. This matrix is the solution to the matrix differential equation:

$$\frac{dP(t)}{dt} = P(t) Q$$

The solution is given by the matrix exponential:

$$P(t) = e^{Qt}$$

For the generalized Jukes-Cantor model, since $Q$ has a symmetric structure, the matrix exponential can be computed in closed form.

3. Eigenvalues and Eigenvectors of $Q$

To compute the matrix exponential $e^{Qt}$, we first find the eigenvalues and eigenvectors of $Q$. The matrix $Q$ is symmetric, so it has a full set of real eigenvalues and orthogonal eigenvectors.

Eigenvalues

The eigenvalues of the matrix $Q$ are:

  • One eigenvalue is 0, corresponding to the stationary distribution where all states have equal probability.
  • The other $K-1$ eigenvalues are $-(K-1)\beta$.

Eigenvectors

The eigenvector corresponding to the eigenvalue 0 is the vector where all components are equal, representing the uniform stationary distribution. The eigenvectors corresponding to the eigenvalue $-(K-1)\beta$ are orthogonal to the uniform vector and represent the directions in which the process decays exponentially over time.

4. Closed-Form Solution for Transition Probabilities

Using the eigenvalue decomposition, we can derive the closed-form solution for the transition probabilities. The transition probability matrix $P(t)$ has the following structure:

$$P(t) = \frac{1}{K} \mathbf{1} \mathbf{1}^T + \left( I - \frac{1}{K} \mathbf{1} \mathbf{1}^T \right) e^{-(K-1)\beta t}$$

where:

  • $\mathbf{1}$ is the column vector of ones (dimension $K \times 1$).
  • $I$ is the identity matrix.
  • $\frac{1}{K} \mathbf{1} \mathbf{1}^T$ represents the stationary distribution, where all states are equally probable.
  • The term $e^{-(K-1)\beta t}$ represents the exponential decay in the probability of remaining in the same state.

This solution reflects two key behaviors:

  1. Stationary Distribution: As $t \to \infty$, the process converges to the uniform stationary distribution, where each state has probability $\frac{1}{K}$.
  2. Exponential Decay: Over time, the process moves away from the initial state and converges toward the stationary distribution at a rate proportional to $\beta$.

5. Transition Probabilities Between States

The specific probabilities of transitioning from state $i$ to state $j$ at time $t$ can be derived from the matrix $P(t)$. For the generalized Jukes-Cantor model, the probabilities are:

  • Probability of staying in the same state $i$ after time $t$:

    $$P_{ii}(t) = \frac{1}{K} + \left( 1 - \frac{1}{K} \right) e^{-(K-1)\beta t}$$

    This formula shows that the probability of staying in the same state decreases over time, eventually approaching $\frac{1}{K}$, the probability that corresponds to the stationary distribution.

  • Probability of transitioning to a different state $j \neq i$ after time $t$:

    $$P_{ij}(t) = \frac{1}{K} \left( 1 - e^{-(K-1)\beta t} \right)$$

    This formula shows that the probability of transitioning to any other state increases over time and converges to $\frac{1}{K}$, reflecting the uniform stationary distribution.

6. Long-Term Behavior (Stationary Distribution)

In the long run, as $t \to \infty$, the system reaches its stationary distribution, where the probability of being in any state is the same. The stationary distribution is:

$$\lim_{t \to \infty} P(t) = \frac{1}{K} \mathbf{1} \mathbf{1}^T$$

This reflects the fact that in the generalized Jukes-Cantor model, all states are equally probable at equilibrium.

7. Example: Generalized JC Model for $K = 4$ (Nucleotides)

For $K = 4$ (the case of nucleotides A, C, G, T), the generalized Jukes-Cantor model reduces to the original JC69 model. In this case:

  • The substitution rate between any two nucleotides is $\beta = \alpha/3$, where $\alpha$ is the overall rate of substitution.
  • The transition probabilities have the form:

$$P_{ii}(t) = \frac{1}{4} + \frac{3}{4} e^{-\alpha t}$$

$$P_{ij}(t) = \frac{1}{4} \left( 1 - e^{-\alpha t} \right)$$

This is the familiar result from the JC69 model, which describes how the probability of remaining in the same nucleotide decays over time, while the probability of transitioning to a different nucleotide increases.

8. Applications of the Generalized JC Model

The generalized Jukes-Cantor model is used in various fields, including:

  • Molecular Evolution: To model the substitution of nucleotides, amino acids, or other molecular sequences over evolutionary time.
  • Phylogenetic Inference: To estimate the evolutionary distances between species based on genetic data.
  • Markov Chain Models: In broader applications, to model systems with $K$ discrete states that transition between states over time.

Summary

  • The generalized Jukes-Cantor model extends the original JC69 model to $K$ states, assuming symmetric substitution rates between all pairs of states.
  • The transition rate matrix $Q$ describes the rates of transitioning between states, with equal off-diagonal elements $\beta$ and diagonal elements $-(K-1)\beta$.
  • The transition probability matrix $P(t)$ can be computed in closed form using the matrix exponential, with probabilities that describe how the system evolves over time.
  • As $t \to \infty$, the system converges to a uniform stationary distribution, where all states are equally probable.
  • The model has applications in molecular evolution, phylogenetics, and any context where transitions between discrete states are modeled using Markov chains.