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

The substrate contact area is the square bottom face, so the total traction force magnitude is stress times area:By Newton's third law, the force exerted by the substrate on the biological cell has the same magnitude and opposite direction to the cell's force on the substrate.

First convert the dynamic viscosity to SI units:Using the stipulated effective-size version of Stokes's law with $a=20\times 10^{-6}\mathrm{m}$, the drag coefficient isThe speed is $v=10^{-7}\mathrm{m}{\mathrm{s}}^{-1}$, henceThe drag acts opposite the velocity. Its magnitude is about $1.06\times 10^5$ times smaller than the substrate force in (a); balancing surrounding-fluid drag alone therefore cannot account for the much larger substrate traction. The calculation uses the size parameter exactly as supplied. An exact drag calculation for a cube moving close to a wall would need geometry and boundary corrections beyond this stipulated approximation.

Take $x=0$ at chamber A and $x=L$ at chamber B. At steady state, the diffusion equation with constant positive diffusion coefficient givesThe two Dirichlet boundary conditions give $A=c_0$ and $B=-c_0/L$. ThusThe concentration gradient is constant, $dc/dx=-c_0/L$. By Fick's first law, the diffusive flux is $J=-Ddc/dx=D{c}_0/L$, directed from A towards B. The profile is linear independently of $D$, while the flux and the time needed to approach it depend on $D$.

This result assumes no significant bulk flow, chemoattractant loss to the walls, degradation or cellular uptake. Experimentally, a uniform-concentration control with matched reservoir concentrations tests for directional cell migration caused by flow or apparatus asymmetry; reversing the reservoir concentrations tests whether the directional response follows the chemical gradient. Keeping both reservoirs well mixed and measuring the concentration profile checks the assumed boundary concentrations and whether uptake has distorted it. These controls distinguish the assumed passive diffusion field from other directional cues; the derived profile alone does not establish the mechanism of chemotaxis.

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.

For the same concentration gradient, the front-to-back difference scales with cell length. A much smaller bacterial cell samples a correspondingly smaller spatial signal, while stochastic arrival and binding of individual chemoattractant molecules make receptor measurements noisy. For illustration, a $2\mu \mathrm{m}$ cell in the $0.5$ cm capillary would experience only $0.04\mathrm{nM}$ across its length, compared with $0.4\mathrm{nM}$ for the longer cell. This illustrates a size penalty; it does not assume that bacterial and human detection thresholds are identical.

Many swimming bacteria, including Escherichia coli, use temporal sensing in bacterial chemotaxis: they compare recent receptor stimulation with an adapted baseline as they move. Increasing attractant suppresses reorientation and prolongs runs; decreasing attractant increases tumbling. Repeated runs and reorientations therefore generate a biased random walk up the gradient, without requiring a simultaneous comparison between the front and back. Chemotaxis feedback regulation supplies the adaptation needed to remain sensitive to changes rather than responding only to absolute concentration.

A discriminating experiment changes concentration rapidly and uniformly around the bacteria: a response then occurs without a spatial difference across the cell. The classic temporal-step experiments found increased tumbling after attractant decreases and smoother swimming after increases.^[ref] In a new experiment, compare up-steps and down-steps with constant-concentration and sham-mixing controls, holding temperature and medium composition fixed. Measure run durations and tumble frequency, rather than only average swimming speed. Such a test supports temporal comparison but does not by itself identify every molecular component or show that all bacterial species use the same mechanism.