 Solution

ID: past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2024/paper-1/d7/d/solution

α

j_{m} = α (c_{c} - c_{n}) = α Δ c .

For fixed compartment volumes, conservation of mass and the absence of exterior flux give

V_{n} \frac{d c _{n}}{d t} = A_{n} α Δ c, V_{c} \frac{d c _{c}}{d t} = - A_{n} α Δ c .

\frac{d Δ c}{d t} = - α A_{n} (\frac{1}{V _{c}} + \frac{1}{V _{n}}) Δ c .

Δ c (t) = Δ c (0) exp [- α A_{n} (\frac{1}{V _{c}} + \frac{1}{V _{n}}) t] .

T = V_{c} V_{n} / [α A_{n} (V_{c} + V_{n})]

. Here

α

α A_{n} / V

has dimensions inverse time. A positive initial concentration difference decays monotonically toward zero; a negative one approaches zero from below.

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