 Solution

With total density one, the open probabilities for the individual gates are $m=S_{10}+S_{11}$ and $h=S_{01}+S_{11}$. Adding the relevant master equations cancels transitions of the other gate:These marginal ordinary differential equations hold even for correlated initial gates. Their product gives $S_{11}$ when the independence assumption in part (c) holds. Solving the first-order linear differential equations from the all-closed initial state givesPut $\lambda =\alpha +\beta =\gamma +\delta >0$. ThenFor positive opening rates this increases strictly from zero, so its half-equilibrium time is uniquely determined by $(1-e^{-\lambda t}{)}^2=1/2$:If an opening rate is zero, the all-open equilibrium density is zero and a nontrivial half-rise time is undefined.