= 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
$$
\nabla^2c=0,\quad c(r_0)=0,\quad c(\infty)=C_A,\qquad c(r)=C_A\left(1-\frac{r_0}r\right).
$$
The inward capture rate per target is the <integral> of the <Fick's first law>:
$$
J=4\pi r_0^2D_A\left.\frac{dc}{dr}\right|_{r_0}=4\pi D_Ar_0C_A.
$$
For dilute independent targets of <number density> $C_B$, the volume association rate is $JC_B$, so
$$
\boxed{k_a=4\pi D_Ar_0.}
$$
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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