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In spatially gated myosin attachment, for a freshly available head encountering an isolated site, take $P=0$ before the attachment window. Converting time derivatives to position derivatives with $dx/dt=v$ gives $v{P}^{\prime }=k_{on}(1-P)$ in the binding window, $P^{\prime }=0$ in the protected bound region, and $v{P}^{\prime }=-k_{off}P$ after detachment becomes allowed. Continuity givesFor $v<k_{on}\Delta x$, the plateau exceeds $1-e^{-1}$ and approaches one for much slower motion. For $v>k_{on}\Delta x$, the binding probability is lower, becoming $k_{on}\Delta x/v$ at very high speed. The spatial detachment tail has length $v/k_{off}$, so faster sliding makes a longer low-probability tail. The following is an original calculation of those profiles; the force panel is derived in c.

Reset condition. Geometric separation $D\gg x_0$ does not alone guarantee detachment before the next site: independently available encounters also need $(D-x_0)k_{off}/v\gg 1$. Without that extra timescale separation the head may still be attached to an earlier site.

For completeness, the periodic steady-state correction can be written explicitly. Put $a=k_{on}/v$, $b=k_{off}/v$, $A_0=1-e^{-a\Delta x}$ andHere $Q_0$ is the probability of entering the new window still attached to an earlier site. In the window the new-site probability is $P(x)=1-e^{-ax}-Q_{0}a(e^{-bx}-e^{-ax})/(a-b)$; after it, use its value $p_b=P(\Delta x)$ in the plateau and tail above. For $a=b$, replace $a(e^{-bx}-e^{-ax})/(a-b)$ by $ax{e}^{-ax}$. This follows from the old-site probability $Q(x)=Q_0{e}^{-bx}$ and new attachment rate $a[1-P-Q]$. The isolated-site result is recovered as $Q_0\to 0$.

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.