Polymerization Brownian ratchet

A polymerization Brownian ratchet advances a filament by incorporating a monomer when thermal motion opens enough space between its tip and a resisting load. The irreversible chemical step rectifies the load's fluctuations. The elementary load advance $a$ depends on tip geometry; two staggered strands can give $a=\delta /2$, whereas two aligned strands do not automatically give this step. An effective addition rate must refer to the same elementary advance.

When load-gap equilibration is fast compared with chemical events, an opposing force gives an exponential gap distribution. The probability of room for an addition is $e^{-Fa/(k_{B}T)}$. With unloaded addition rate $u$ and removal rate $w$, assumed force-independent except for this steric gating, the velocity is the displayed expression. For $u>w$, stall occurs at $F_s=(k_{B}T/a)\log (u/w)$. Concentration is included in $u=k_{\mathrm{on}}c$ if the on-rate constant is bimolecular.

In the instantaneous-addition, negligible-removal ratchet, the gap reflects at zero and a first passage to $a$ triggers addition and resets the gap to zero. With load diffusion coefficient $D$ and opposing drift $DF/(k_{B}T)$, the mean passage time solves $D{T}^{\prime \prime }-DF{T}^{\prime }/(k_{B}T)=-1$, $T^{\prime }(0)=0$, $T(a)=0$. It is $T(0)=a^2(e^z-1-z)/(D{z}^2)$, giving the displayed speed and zero-load limit $2D/a$. The finite-load slowdown is not simply $e^{-z}$ in this diffusion-limited regime.