Substitution Models (cont.)

In the general time reversible model, the rate matrix $Q$ contains 6 rates $(\alpha,\beta,\gamma,\omega,\epsilon,\phi)$ and 4 limiting probabilities $(\pi_A,\pi_C,\pi_G,\pi_G)$,

$$Q = \begin{pmatrix}  -&\pi_C\alpha&\pi_G\beta&\pi_T\gamma\\ \pi_A\alpha&-&\pi_C\omega&\pi_T\epsilon\\ \pi_A\beta&\pi_C\epsilon&-&\pi_T\phi\\ \pi_A\gamma&\pi_C\omega&\pi_G\phi&- \end{pmatrix}$$

The solution to this differential equation is

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