Past exam of the flows, fluctuations and complexity and symmetry and relativity paper of Part B of the physics course of the University of Oxford / 2011 / 4

Take $u>0$. Despite the source's terminology, $P(x)$ is a dimensionless conditional bound-state probability at phase $x$, not a spatially normalized probability density; the phase-average bound fraction is ${\Delta }^{-1}{\int }_0^{\Delta }Pdx$. The given equation has advection and binding reactions but no diffusion term. Large $N$ permits the prescribed steady $u$ approximation.

For spatially gated myosin attachment, define $A=e^{-k_1{x}_1/u}$, $B=e^{-k_2(\Delta -x_2)/u}$, $p_0=P(0)$ and $p_b=P(x_1)$. Continuity and periodicity give $p_b=1-(1-p_0)A$, $p_0=B{p}_b$, hence $p_b=(1-A)/(1-AB)$. The full periodic steady solution isThe requested limit makes $B\ll 1$, so $p_0\simeq 0$ and $p_b\simeq 1-e^{-k_1{x}_1/u}$. The curve rises exponentially over the binding window, remains flat to $x_2$, then decays over length $u/k_2$ to an exponentially small value before the next period. The piecewise expression is continuous at both switching points and at the period boundary.

For a head sampling phases uniformly during steady sliding, its mean force is $f_0$ times the phase-average bound probability. In the $B\ll 1$ limit of spatially gated myosin attachment, integrating the three intervals givesThe first integral is bounded by $x_1{p}_b$, and the last by $(u/k_2)p_b$. Relative to the plateau contribution, they are negligible when $x_1\ll x_2$ and $u/k_2\ll x_2$, respectively. ThereforeThis is a constant-force spatially gated motor approximation: complete detachment between binding windows, a narrow attachment window, a short detachment tail and essentially fixed sliding velocity. In particular, $k_1{x}_1/u$ need not be small.

Put $V=k_1{x}_1$ and $F_{\ast }=N{f}_0{x}_2/\Delta$. The constant-force spatially gated motor producesThe limit is approached after the periodic steady state is established; at exactly $u=0$ arbitrary initially bound heads in inactive phase intervals need not equilibrate. The curve decreases from $F_{\max }$ and has formal large-$u$ tail $F_{\max }V/u$. Extrapolation of that tail beyond the small-$u$ detachment assumptions is not controlled.

Sign inconsistency in the printed force balance. For a positive viscous drag coefficient $\gamma$, the signed drag force is $-\gamma u$. The printed $F_{\mathrm{ext}}+F_m+\gamma u=0$ cannot give the requested positive no-load contraction speed: both last terms are positive when $u>0$. With $F_{\mathrm{ext}}=-F_{\max }/2$, the left side starts at $F_{\max }/2>0$ and may have zero, one or two positive roots depending on parameters; the requested usual load construction is not implied by that equation.

The physical linear-drag load curve for a spatially gated motor usesOn the same axes plot the positive drag magnitude $D(u)=\gamma u$ and the decreasing motor force $F_m(u)$. The no-load speed $u_0$ is their intersection. For opposing load $F_{\max }/2$, intersect $F_m(u)$ with the parallel raised line $D(u)+F_{\max }/2$; equivalently intersect $D(u)$ with $F_m(u)-F_{\max }/2$. Drop verticals to the speed axis. The loaded speed obeys $0<u_{1/2}<u_0$. Each intersection is unique because $F_m$ decreases and $D$ increases. The signed drag is also indicated below the axis in the sketch.