= Solution
Take the membrane <particle flux> density to be positive from <cytoplasm> into the <cell nucleus>. With <membrane permeability> $\alpha$,
$$
j_m=\alpha(c_c-c_n)=\alpha\Delta c.
$$
For fixed compartment volumes, <conservation of mass> and the absence of exterior <flux> give
$$
V_n\frac{dc_n}{dt}=A_n\alpha\Delta c,\qquad
V_c\frac{dc_c}{dt}=-A_n\alpha\Delta c.
$$
Subtract the two <concentration> derivatives:
$$
\frac{d\Delta c}{dt}=-\alpha A_n\left(\frac1{V_c}+\frac1{V_n}\right)\Delta c.
$$
This <linear ordinary differential equation> has solution
$$
\boxed{\Delta c(t)=\Delta c(0)\exp\!\left[-\alpha A_n\left(\frac1{V_c}+\frac1{V_n}\right)t\right]}.
$$
Its relaxation time is $\boxed{T=V_cV_n/[\alpha A_n(V_c+V_n)]}$. 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.
Back to article page
