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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 isIn 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 $-p_b\kappa {\int }_0^{\infty }s{e}^{-k_{off}s/v}ds=-p_b\kappa (v/k_{off}{)}^2$. HenceAn independent energy calculation gives the same factor: binding near zero loads energy $\kappa x_0^2/2$, while an exponentially distributed post-threshold attachment time has $⟨t^2⟩=2/k_{off}^2$, 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 $v$, givesThe force tends to $\kappa x_0^2/(2D)$ as $v\downarrow 0$ and changes sign at $v=k_{off}{x}_0/\sqrt{2}$ in this idealized model. Above that speed forced sliding is resisted on average. The formula uses the reset condition in 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 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 force-velocity law.