 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/14b/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Integrating the density, and using

λ_{+} + λ_{-} = A

and

λ_{+} λ_{-} = k_{1} k_{2}

, gives

E [τ] = \frac{k _{1} k _{2}}{λ _{+} - λ _{-}} (\frac{1}{λ _{-}^{2}} - \frac{1}{λ _{+}^{2}}) = \frac{A}{k _{1} k _{2}} = \frac{k _{1} + k _{1}^{'} + k _{2}}{k _{1} k _{2}} .

The 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.

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