Past exam of the biological physics course of the University of Oxford / 2019 / 1 / c

The term $V/(D_3\lambda )$ is the bulk-diffusion search cost: increasing the effective length of DNA searched after each encounter enlarges the useful capture region and reduces this contribution. The term $L\lambda /D_1$ is the accumulated sliding cost. Approximately $L/\lambda$ DNA landing-and-sliding episodes, each taking order ${\lambda }^2/D_1$, give $L\lambda /D_1$. These are scaling estimates; geometric constants and correlated revisits are omitted by the supplied model.

Thus the facilitated target search is slow as $\lambda \to 0$, because too little sequence is examined per encounter, and slow at very large $\lambda$, because sliding and repeated local exploration take too long. A finite intermediate minimum balances 1D scanning against 3D relocation. The right panel of the figure in solution shows the two contributions and their sum. The model requires a useful range of sliding lengths below the full chromosome length, finite positive diffusion coefficients and effective relocation between episodes; it is not a claim that arbitrarily large $\lambda$ is physically available.

Write $A=V/D_3$ and $B=L/D_1$. Then $\tau =A/\lambda +B\lambda$, soThereforeAt the optimum, the two contributions are equal. Expressing the optimal sliding length directly in terms of the minimum time givesThese relations concern the optimal point; a general nonminimal time can correspond to two different sliding lengths. A geometric or kinetic restriction on the allowed sliding length can instead place the optimum at a boundary.