Past exam of the biological physics course of the University of Oxford / 2019 / 1 / b / i / Solution

Treat the target as a stationary perfectly absorbing sphere of reaction radius

r_{0}

, surrounded by an effectively uniform reservoir of protein number density

C_{A}

. At steady state the radial diffusion equation gives

\nabla^{2} c = 0, c (r_{0}) = 0, c (\infty) = C_{A}, c (r) = C_{A} (1 - \frac{r _{0}}{r}) .

The inward capture rate per target is the integral of the Fick's first law:

J = 4 π r_{0}^{2} D_{A} \frac{d c}{d r}_{r_{0}} = 4 π D_{A} r_{0} C_{A} .

For dilute independent targets of number density

C_{B}

, the volume association rate is

J C_{B}

, so

k_{a} = 4 π D_{A} r_{0} .

This is the Smoluchowski association rate, in volume per time because the concentrations are number densities. If the target also diffuses, use the relative diffusion coefficient

D_{A} + D_{B}

. Orientation constraints, incomplete absorption and chromosome geometry can reduce the actual association rate relative to this ideal collision limit.

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