Past exam of the biological physics course of the University of Oxford / 2022 / 1 / c / Solution

Write $A=V/D_3$ and $B=L/D_1$. The search time is $A/\lambda +B\lambda$: it decreases from infinity at small positive $\lambda$, reaches one minimum, and increases without bound at large $\lambda$ within this idealized model.

During a facilitated diffusion search, roughly $L/\lambda$ separate DNA landing-and-sliding episodes sample a chromosome. A three-dimensional encounter with any chromosomal landing region has characteristic time $V/(D_{3}L)$, so the total bulk-search contribution is $(L/\lambda )V/(D_{3}L)=V/(D_3\lambda )$. Each one-dimensional diffusion excursion lasts order ${\lambda }^2/D_1$; multiplying by $L/\lambda$ gives $L\lambda /D_1$. Longer sliding therefore reduces the number of bulk relocations but increases time spent repeatedly exploring local DNA. Numerical constants and overlap corrections are already omitted in the given model.

Differentiating and using the positive second derivative givesAt ${\lambda }_{\ast }$ the two time contributions are equal. The optimum is meaningful only if ${\lambda }_{\ast }$ lies between the microscopic capture scale and the available chromosomal contour length.

A decrease in DNA concentration means a decrease in $L/V$, but its effect on the time is not determined by that ratio alone. At fixed chromosome length $L$, dilution by increasing $V$ increases $\tau$ at fixed $\lambda$ and increases ${\tau }_{\min }$ as $\sqrt{V}$; bulk relocation becomes slower. If instead $L$ decreases at fixed $V$, while the target is retained, the stated model predicts a smaller sliding contribution and a smaller optimized time, proportional to $\sqrt{L}$. In both cases ${\lambda }_{\ast }\propto (L/V{)}^{-1/2}$. This distinction prevents attributing a unique time trend to concentration without specifying how it changes.

The left panel normalizes the search law to $\tau /{\tau }_{\min }=(u+u^{-1})/2$, $u=\lambda /{\lambda }_{\ast }$. The other panels illustrate b and c.