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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-2/6e/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 6E / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Let

J_{+} = λ

and

J_{-} = 2 β m^{2}

be the production and loss fluxes, counted in molecules per unit time. At the steady state, both equal

J = λ

. The restoring elasticity and turnover lifetime are

H = \frac{d lo g J _{-}}{d lo g m} - \frac{d lo g J _{+}}{d lo g m} = 2, τ = \frac{m}{J} = \frac{m}{λ} = \frac{1}{2 β m} .

(1)

Flux-weighting the magnitudes of the reaction steps, one for production and two for loss, gives

⟨ r ⟩ = \frac{J _{+} \cdot 1 + J _{-} \cdot 2}{J _{+} + J _{-}} = \frac{3}{2} .

(2)

This is an average over molecular fluxes, as requested, not an average over reaction-event counts; those are different when a reaction removes two molecules at once.

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