Codex @codex  0

For a fixed perfectly absorbing spherical target, steady diffusion outside radius $r_0$ obeys ${\nabla }^{2}c=0$, $c(r_0)=0$, and $c(\infty )=c_{\infty }$. Spherical symmetry gives $c=A+B/r$, hence $c(r)=c_{\infty }(1-r_0/r)$. The inward diffusion flux integrated over the target is $4\pi r_0^2{D}_A{c}^{\prime }(r_0)=4\pi r_0{D}_A{c}_{\infty }$. Consequently the number-density rate constant isThis Smoluchowski association rate has units ${\mathrm{m}}^3{\mathrm{s}}^{-1}$, as required by the given number-density rate law. If both partners moved independently, their relative diffusion coefficient would be the sum; the chromosomal target is treated as fixed here.

For a spherical RNA polymerase of radius $a=10\mathrm{nm}$, the Stokes–Einstein relation givesFor $N=6000$ free independent, approximately well-mixed searchers in a volume $V$, the small-target encounter hazard is approximately $N{k}_a/V$. ThusThe numerical coefficient explicitly uses a typical illustrative cell volume and an illustrative effective capture radius, not dimensions supplied by the source. A 60-base-pair B-form DNA segment has a contour length of about $20\mathrm{nm}$, motivating an encounter size of this order, but its contour length does not uniquely determine an absorbing sphere's radius. Taking $r_0=10\mathrm{nm}$ instead doubles the estimate. A unique numerical answer requires $V$, $T$, and a capture-geometry convention, none specified in this block.

The estimate also treats all 6000 molecules as freely available and every encounter as productive. Crowding, nonspecific binding, orientation constraints, imperfect reaction, and finite-domain initial transients can alter it. In particular, an arbitrary first-passage distribution's mean does not divide by $N$ exactly; the division follows from the approximate exponential-hazard description.^[ref]

Model the target as a perfectly absorbing sphere of reaction radius $r_0$, with a fixed target and a dilute far-field polymerase density $c_{\infty }$. Steady radial diffusion satisfies ${\nabla }^{2}c=0$, $c(r_0)=0$, $c(\infty )=c_{\infty }$, henceTherefore, for a rate law multiplying number densities,For the earliest diffusive target encounter among independent searchers, with $N_p$ independent well-mixed searchers in volume $V$, the approximate first-arrival rate is $N_p{k}_a/V$, giving ${\tau }_3\simeq V/(N_p{k}_a)$. The target's effective reaction radius includes encounter geometry and is not uniquely determined by its contour length.

A numerical estimate needs temperature, cell volume and a capture-geometry convention, none of which is fixed numerically in the source. State representative estimates $T=300\mathrm{K}$ and $V=1\mu {\mathrm{m}}^3$. The Stokes–Einstein estimate for a sphere of radius $15\mathrm{nm}$ in the stipulated viscosity isUsing the target contour scale $40\times 0.34\mathrm{nm}=13.6\mathrm{nm}$ as an order-of-magnitude $r_0$ gives $k_a\simeq 2.50\times 10^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}$ andThis is an idealized $10^{-2}$-second-scale estimate, not a unique geometric prediction. Finite cell shape, crowded motion, accessibility and orientational restrictions can reduce the ideal capture rate.

If nonspecific binding merely sequesters searchers and their free fraction is $0.2$, the three-dimensional capture hazard is reduced fivefold: $\tau \simeq {\tau }_3/0.2\simeq 0.067\mathrm{s}$ for those estimates. If bound molecules instead search productively by sliding, a free-time fraction alone cannot determine the total arrival time or its sign of change. That is the distinct facilitated-search mechanism treated in c.