Past exam of the biological physics course of the University of Oxford / 2017 / 5 / c / Solution

Write $A=V/D_3$ and $B=L/D_1$. The given mean-search scaling is $\tau =A/\lambda +B\lambda$. Three-dimensional relocation sees an effective capture length of order $\lambda$, giving a bulk encounter time $V/(D_3\lambda )$. A sliding excursion explores length of order $\lambda$ in time of order ${\lambda }^2/D_1$; sampling order $L/\lambda$ contour segments gives a total sliding cost of order $L\lambda /D_1$. Prefactors, correlations and repeated visits are coarse-grained in this scaling model.

Small sliding lengths make relocation inefficient; very long sliding wastes time revisiting a local DNA region rather than relocating to another. The optimum balances both contributions:Since $d^2\tau /d{\lambda }^2=2V/(D_3{\lambda }^3)>0$, it is the unique minimum for unrestricted positive $\lambda$. At the optimum, bulk and sliding times are equal. With $s=\lambda /{\lambda }_{\ast }$,

The model is meaningful only over an accessible range between molecular steps and the available chromosome length. An optimum outside that range is replaced by a boundary choice; the unphysical divergences at arbitrarily small or large $\lambda$ describe limits of the scaling expression. A common multiplicative searcher-number factor would not change the optimum, but an exact many-searcher first-arrival mean is not generally obtained by dividing an arbitrary single-searcher mean by their number without a waiting-time assumption.