= Solution
There are four reversible transitions of the <ion channel>, each changing exactly one gate:
$$
\begin{aligned}
S_{00}&\mathrel{\mathop{\rightleftharpoons}^{\alpha}_{\beta}}S_{10},&
S_{01}&\mathrel{\mathop{\rightleftharpoons}^{\alpha}_{\beta}}S_{11},\\
S_{00}&\mathrel{\mathop{\rightleftharpoons}^{\gamma}_{\delta}}S_{01},&
S_{10}&\mathrel{\mathop{\rightleftharpoons}^{\gamma}_{\delta}}S_{11}.
\end{aligned}
$$
These are first-order <reaction-rate equations>: their <reaction rate constants> are per-gate transition rates. In the <continuous-time Markov chain> interpretation, simultaneous changes of both gates have <probability> of order $(dt)^2$, so there is no direct diagonal transition in its <infinitesimal generator matrix>.
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