Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 71 / 2

Approximate the gap by steady Couette flow, neglecting edge effects and additional losses from membrane deformation. Its velocity gradient is $U/h$, so the shear stress is $\eta U/h$. The force on the gliding bacterium and the rate of viscous dissipation areThe same power follows by integrating $\eta (U/h{)}^2$ over gap volume $Ah$. In SI units, $A=10^{-10}{\mathrm{m}}^2$, $h=10^{-7}\mathrm{m}$, and $U=10^{-6}\mathrm{m}{\mathrm{s}}^{-1}$. ThusAn energy needs a duration or distance: $E=10^{-18}\Delta t\mathrm{J}$ for $\Delta t$ in seconds, equivalently $E=F_{d}s$ for travel distance $s$. Moving one micrometre takes one second and dissipates $10^{-18}\mathrm{J}$. This is the mechanical cost of the sheared film. A numerical biological efficiency would additionally require the motor's input power and the losses associated with its peristaltic actuation.

Let $a_c\ll R$ be the contact radius of the adhering neutrophil. Indentation is of order $a_c^2/R$, so the elastic strain is of order $a_c/R$ within a region of volume $a_c^3$. The elastic energy is consequently $G{a}_c^5/R^2$, while adhesion lowers the energy by order $J{a}_c^2$. Balancing these terms givesThis estimate requires weak adhesion, $J\ll GR$.

For an adhesive-cell rolling threshold, assume peeling the rear bonds dissipates work of order $J$ per unit contact area. Rolling through angle $d\theta$ sweeps contact area of order $a_{c}Rd\theta$, giving resisting moment $M_r\sim J{a}_{c}R$. A surrounding fluid of dynamic viscosity $\eta$ supplies a moment of order $\eta \dot{\gamma }{R}^3$. Equating moments yieldsA dimensionless numerical coefficient depends on the contact and near-wall Stokes flow. Fluid viscosity is indispensable to a rate estimate, even though the contact model alone specifies only $R,G,J$.

A perfectly reversible elastic adhesive contact does not lose net energy when an unchanged contact footprint translates. It therefore does not, by itself, justify a finite rolling threshold. The above estimate assumes adhesive hysteresis or irreversible bond peeling. If equilibrium adhesion is $J$ but the work lost during peeling and rebinding is $\Delta J$, retain $a_c\sim (J{R}^2/G{)}^{1/3}$ and replace $J$ by $\Delta J$ only in the resisting moment: ${\dot{\gamma }}_c\sim \Delta J{a}_c/(\eta R^2)$.