Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 2 / 7 / d

For a cell of length $\ell$ aligned with the capillary, the steady linear gradient produces the front-to-back concentration differenceAt the given initial length, $L=5\times 10^{-3}\mathrm{m}$, this is $100(20\times 10^{-6})/(5\times 10^{-3})=0.4\mathrm{nM}$, above the stated threshold. Requiring $\Delta c\ge 0.1\mathrm{nM}$ givesThe diffusion coefficient does not enter this steady-gradient threshold. It does constrain experimental waiting time: for fixed reservoir concentrations, the slowest decaying mode of the diffusion equation has timescale $L^2/({\pi }^{2}D)$. With the supplied $D$, this is about $7.04$ hours at $0.5$ cm and $4.69$ days at $2$ cm; several such times may be required for an accurate steady state. Thus the maximum from the detection threshold is usable only if the reservoirs and cells remain suitable while the profile develops.