 Solution

ID: past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2008/b/5/iv/solution

Use the longer scale

T_{s} = (k_{1} E_{0})^{- 1} = T_{f} / ϵ

and

τ_{s} = t / T_{s} = ϵ τ_{f}

. The equations become

s_{τ_{s}} = - s + (s + K - λ) c, ϵ c_{τ_{s}} = s - (s + K) c .

At leading order the fast complex variable has equilibrated to the quasi-steady manifold. Therefore

c ≃ \frac{s}{s + K}, \frac{d s}{d τ _{s}} ≃ - \frac{λ s}{s + K} .

Matching from the initial layer gives

s (0) = 1

for this outer problem; its implicit solution is

s + K lo g s = 1 - λ τ_{s}

. The dimensional substrate loss is the Michaelis-Menten rate with maximum rate

k_{2} E_{0}

(k_{- 1} + k_{2}) / k_{1}

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