Consider the stochastic catalytic reaction

$$E \leftrightharpoons E S, \quad E S \rightarrow E+P$$

in which a single enzyme $E$ complexes reversibly to $E S$ (at forward rate $k_{1}$ and reverse rate $k_{1}^{\prime}$ ) and decomposes into product $P$ (at forward rate $k_{2}$ ), regenerating enzyme $E$. Assume there is sufficient substrate $S$ so that this catalytic cycle can continue indefinitely. Let $P(E, n)$ be the probability of the state with enzyme $E$ and $n$ products and $P(E S, n)$ the probability of the state with complex $E S$ and $n$ products, these states being mutually exclusive.

(i) Write down the master equation for the probabilities $P(E, n)$ and $P(E S, n)$ for $n \geqslant 0$

(ii) Assuming an initial state with zero products, solve the master equation for $P(E, 0)$ and $P(E S, 0)$.

(iii) Hence find the probability distribution $f(\tau)$ of the time $\tau$ taken to form the first product.

(iv) Obtain the mean of $\tau$.