Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 14B (Mathematical Biology)

Write $E_n(t)=P(E,n)$ and $C_n(t)=P(ES,n)$. The master equation balances the transitions $E_n\to C_n$ at rate $k_1$, $C_n\to E_n$ at rate $k_1^{\prime }$, and $C_n\to E_{n+1}$ at rate $k_2$:for $n\ge 0$, with $C_{-1}=0$.

For zero products, put $A=k_1+k_1^{\prime }+k_2$ andThe initial conditions are $E_0(0)=1$ and $C_0(0)=0$. Eliminating $E_0$ givesThereforeand using $E_0=({\dot{C}}_0+(k_1^{\prime }+k_2)C_0)/k_1$ gives

Before the first product forms, the system is in either $E_0$ or $C_0$. Product formation is possible only through $C_0\to E_1$, so the first-passage-time density is the outgoing probability fluxEquivalently, the survival function is $E_0(\tau )+C_0(\tau )$ and its negative derivative is this density.

Integrating the density, and using ${\lambda }_{+}+{\lambda }_{-}=A$ and ${\lambda }_{+}{\lambda }_{-}=k_1{k}_2$, givesThe same result follows from first-step analysis: the enzyme waits on average $1/k_1$ to bind, and each complex either produces or returns to the initial enzyme state.