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 / 1 / Solution

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.