Substitution Models

A substitution model for nucleotides is a continuous time Markov chain $\{X_t, t\ge0\}$ with 4 states $\{A,C,G,T\}$. It assumes a homogenous rate matrix $Q$ in which the off-diagonal values are the rates of mutations $i\rightarrow j$ for $i,j = A,C,G,T$ and $i\ne j$. For example, the Jukes-Cantor model assumes that all rates are equal to each other, i.e.,

$$Q = \begin{pmatrix}  -3u&u&u&u\\ u&-3u&u&u\\ u&u&-3u&u\\ u&u&u&-3u \end{pmatrix}$$

From Kolmogorov backward equations, the transition probability matrix $P(t)=\{P_{ij}(t),i,j=A,C,G,T\}$, in which $P_{ij}(t)$ is the probability that the nucleotide $i$ mutates to the nucleotide $j$ in time $t$, satisfies the differential equation

$$\frac{\partial P(t)}{\partial t} = QP(t)$$