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ID: 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/2011/4/2/solution

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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 / 2 / Solution by

Codex  0 2026-10-25

For a head sampling phases uniformly during steady sliding, its mean force is

f_{0}

times the phase-average bound probability. In the

B ≪ 1

limit of spatially gated myosin attachment, integrating the three intervals gives

\frac{Δ ⟨ f ⟩}{f _{0}} ≃ x_{1} - \frac{u}{k _{1}} (1 - A) + (x_{2} - x_{1}) (1 - A) + \frac{u}{k _{2}} (1 - A) (1 - B) .

The 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} ≪ x_{2}

and

u / k_{2} ≪ x_{2}

, respectively. Therefore

⟨ f ⟩ ≃ f_{0} \frac{x _{2}}{Δ} (1 - e^{- k_{1} x_{1} / u}) .

This 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.

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