Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / D7 / e / Solution

The conserved total molecule amount fixes the final common concentration:

c_{\infty} = \frac{V _{c} c _{c} ( 0 ) + V _{n} c _{n} ( 0 )}{V _{c} + V _{n}} .

Solve

c_{c} - c_{n} = Δ c (t)

together with

V_{c} c_{c} + V_{n} c_{n} = (V_{c} + V_{n}) c_{\infty}

to obtain

c_{c} (t) = c_{\infty} + \frac{V _{n}}{V _{c} + V _{n}} Δ c (t), c_{n} (t) = c_{\infty} - \frac{V _{c}}{V _{c} + V _{n}} Δ c (t) .

Substituting the decaying concentration difference from part (d) supplies both explicit time dependences. The concentrations approach the volume-weighted initial average, which need not be the arithmetic average if the compartment volumes differ.

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