 Solution

ID: past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2007/a/2/ii/solution

Adding the last two equations gives

d (e + c) / d t = 0

. The initial conditions therefore imply

e + c = e_{0}

, or

e = e_{0} - c

. Substitute this in the first two balances:

\overset{s}{˙} = (k_{0} - k_{1} e_{0}) s + (k_{- 1} + k_{1} s) c, \overset{c}{˙} = k_{1} e_{0} s - (k_{1} s + k_{- 1} + k_{2}) c .

The enzyme inventory is conserved even while substrate can grow autocatalytically.

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