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

Take the membrane particle flux density to be positive from cytoplasm into the cell nucleus. With membrane permeability $\alpha$,For fixed compartment volumes, conservation of mass and the absence of exterior flux giveSubtract the two concentration derivatives:This linear ordinary differential equation has solutionIts relaxation time is $\boxed{T=V_c{V}_n/[\alpha A_n(V_c+V_n)]}$. Here $\alpha$ has dimensions length/time, so $\alpha A_n/V$ has dimensions inverse time. A positive initial concentration difference decays monotonically toward zero; a negative one approaches zero from below.