Past exam of the biological physics course of the University of Oxford / 2015 / 6 / c

The given facilitated target search time contains a rising term $L\lambda /D_1$ and a falling term $V/(D_3\lambda )$. Longer sliding explores more neighbouring DNA without another bulk encounter, reducing the cost of three-dimensional relocation; but it also spends longer revisiting local contour. Roughly $L/\lambda$ local scans, each taking time of order ${\lambda }^2/D_1$, give the first term. The second represents the bulk encounter cost for an effective capturable contour range of order $\lambda$, with a Smoluchowski association rate scaling as $D_3\lambda$. Numerical geometrical factors are already absorbed in the stipulated scaling formula.

Differentiating at fixed positive $L,V,D_1,D_3$ givesThe second derivative $2V/(D_3{\lambda }^3)>0$ proves the minimum is unique. The two contributions are equal there. The formal curve diverges at both zero and infinite $\lambda$; real sliding lengths are bounded, for example by the available contour, so an inaccessible optimum must be replaced by the nearest physical boundary. In dimensionless form, $\tau /{\tau }_{\min }=(u+u^{-1})/2$ with $u=\lambda /{\lambda }_{\ast }$. The right panel of the original illustration shows its minimum and the two contributing terms. The supplied expression does not include a factor of 6000; it should not silently be identified with the preceding earliest-encounter estimate.