Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 355 (Biological Physics and Fluid Dynamics)

For each Volvox colony, the excess gravitational force isFor a normal Stokeslet a distance $R$ below a flat stress-free interface, the image is an oppositely directed Stokeslet a distance $R$ above it. The real force creates no lateral velocity at the other colony because both centres have the same height. Their separation from the image is $(x,-2R)$, however, so each colony moves toward the other with speedBoth colonies move, and therefore

DefineThe dimensionless dynamics areThis gradient flow descends an even attractive potential with minimum $V(0)=-1$ and $V\to 0^{-}$ as $|\xi |\to \infty$.

For $\xi \gg 1$, $d\xi /d\tau \simeq -1/{\xi }^2$, so the approach from ${\xi }_0=x_0/(2R)$ takes ${\tau }_c\simeq {\xi }_0^3/3$. Restoring dimensions,With water viscosity $\mu \simeq 10^{-3}\mathrm{Pa}\mathrm{s}$, ${\rho }_w\simeq 10^3\mathrm{kg}{\mathrm{m}}^{-3}$, $ϵ=0.03$, $R=200\mu \mathrm{m}$, and $x_0=5R$, this givesBecause $x_0/(2R)=2.5$ is only moderately large, this is a far-field estimate rather than the exact collision time.

In the small-slope approximation the worm-like chain bending energy isThe clamped end obeys $h(0)=h_x(0)=0$, while a force-free and torque-free tip has the natural conditions $h_{xx}(L)=h_{xxx}(L)=0$. A static transverse tip force $f$ produces the cantilever compliance $h(L)/f=L^3/(3A)$. The equipartition theorem, or equivalently the static fluctuation--response relation, therefore gives the exact tip variance

For the dynamics, resistive-force theory gives the transverse drag per lengthat logarithmic accuracy and the stochastic beam equationLet ${\varphi }_n$ be clamped--free bending modes with $q_{n}L={\alpha }_n$, whereWriting $N_n={\int }_0^L{\varphi }_n^{2}dx$, independent thermally driven modes giveThe first mode contains almost all of the tip variance, so

For a microtubule, $A=k_{B}T{L}_p$. With $L=10\mu \mathrm{m}$ and $L_p=3\mathrm{mm}$,Taking water at room temperature and $a=0.5\mu \mathrm{m}$ gives ${\zeta }_{\perp }\simeq 3.6\times 10^{-3}\mathrm{Pa}\mathrm{s}$ andwith an order-one uncertainty from the logarithmic slender-body drag approximation.

Introduce ${\mathbf{e}}_i=(\cos {\theta }_i,\sin {\theta }_i)$ and ${\mathbf{e}}_i^{\perp }=(-\sin {\theta }_i,\cos {\theta }_i)$. The joint and tip positions aresoUsing the point drags ${\mathbf{F}}_A=-\zeta {\dot{\mathbf{r}}}_A$, ${\mathbf{F}}_B=-\zeta {\dot{\mathbf{r}}}_B$, and the follower force $\mathbf{\Gamma }=-\Gamma {\mathbf{e}}_2$ in the principle of virtual work gives the independent coefficients of $\delta {\theta }_1$ and $\delta {\theta }_2$:With scaled time $s=kt/(\zeta {\ell }^2)$ and dimensionless follower load $\Sigma =\Gamma \ell /k$, these are exactly the stated equations with primes denoting $d/ds$.

If the two links are constrained to remain collinear, their admissible virtual rotations satisfy $\delta {\theta }_1=\delta {\theta }_2$. Adding the two generalized equations and putting ${\theta }_1={\theta }_2=\theta$ givesThe drag factor five is the sum of the squared lever arms $1^2+2^2$. The follower force lies along the straight filament and has no moment, so it cannot affect this rigid rotational relaxation.

For unrestricted perturbations, linearization about the straight state givesFor modes proportional to $e^{\lambda s}$,and thereforeThe roots are negative and real below $\Sigma =2$, coalesce at $-1$, and then form a complex-conjugate pair. For $2<\Sigma <4$ they trace the unit circle from $-1$ to $+1$; they cross the imaginary axis at $\lambda =\pm i$ whenwhich is a Hopf bifurcation. Above $\Sigma =4$ they separate along the positive real axis.

Viscous drag and elastic spring forces alone have a symmetric positive mobility and a symmetric potential Hessian, so an overdamped gradient system has only real decay rates. The follower force is nonconservative: its linearized generalized-force matrix is nonsymmetric and cannot be derived from a potential. This broken variational structure permits complex eigenvalues and hence an oscillatory instability even though inertia is absent.