= Solution
For the <work of a spatially gated myosin head>, the mechanical <force> exerted by an attached head in the increasing-$x$ direction is $f(x)=-dU/dx=\kappa(x_0-x)$. It assists motion before $x_0$ and resists it afterwards. For one site encounter the expected work is
$$
W=\int_0^\infty P(x)\kappa(x_0-x)\,dx.
$$
In the narrow binding-window limit, retain the capture <probability> $p_b=1-e^{-k_{on}\Delta x/v}$ but neglect the attachment <displacement> compared with $x_0$. The positive work is $p_b\kappa x_0^2/2$. After $x_0$, use $s=x-x_0$ to obtain the negative contribution $-p_b\kappa\int_0^\infty se^{-k_{off}s/v}ds=-p_b\kappa(v/k_{off})^2$. Hence
$$
\boxed{W\simeq\kappa(1-e^{-k_{on}\Delta x/v})
\left[\frac{x_0^2}{2}-\left(\frac v{k_{off}}\right)^2\right].}
$$
An independent <energy> calculation gives the same factor: binding near zero loads <energy> $\kappa x_0^2/2$, while an <Exponential distribution>[exponentially distributed] post-threshold attachment time has $\langle t^2\rangle=2/k_{off}^2$, leaving mean <elastic energy> $\kappa v^2/k_{off}^2$ to be lost upon detachment. The factor of two in that <second moment> is essential.
One site passes every distance $D$, so average work per distance, or power divided by $v$, gives
$$
\boxed{\overline F(v)=\frac{W}{D}\simeq
\frac\kappa D(1-e^{-k_{on}\Delta x/v})
\left[\frac{x_0^2}{2}-\frac{v^2}{k_{off}^2}\right].}
$$
The <force> tends to $\kappa x_0^2/(2D)$ as $v\downarrow0$ and changes sign at $v=k_{off}x_0/\sqrt2$ in this idealized model. Above that speed forced <Facilitated target search>[sliding] is resisted on average. The formula uses the reset condition in <2/b>, as well as $\Delta x\ll x_0$; if reset fails, replace $p_b$ by the periodic capture <probability> to leading order in $\Delta x/x_0$. For arbitrary window width, the isolated-site work is exactly $\int_0^{\Delta x}(k_{on}/v)e^{-k_{on}x/v}[\kappa(x_0-x)^2/2-\kappa v^2/k_{off}^2]dx$. The loaded <Spring>[spring] <energy> must be supplied by the binding/power-stroke chemistry; this model does not specify an <ATP> efficiency. Real force-dependent transition rates and <energy> budgets limit extrapolation of the toy <Work of a spatially gated myosin head>[force-velocity] law.
Back to article page
