= Solution
Model the <polymerization Brownian ratchet> by a rigid tip and a Brownian load. Let $a$ be the elementary load advance, $\zeta_L$ its drag coefficient, and $D=k_BT/\zeta_L$ its <diffusion coefficient>. For a spherical load, $\zeta_L=6\pi\mu R_L$, but no load radius is specified, so the rate is naturally expressed in terms of $D$. Assume addition is effectively instantaneous once the gap $h$ reaches $a$, and removal is negligible. The gap reflects at zero, first arrival at $a$ causes insertion, and insertion resets it to zero.
With no opposing <force>, the mean <first-passage time> satisfies $DT''=-1$, $T'(0)=0$, $T(a)=0$. Integration gives $T(h)=(a^2-h^2)/(2D)$. One advance per passage gives
$$
\boxed{T(0)=\frac{a^2}{2D},\qquad V_0=\frac a{T(0)}=\frac{2D}{a}.}
$$
The advance-event rate is $2D/a^2$, while $V_0$ is a length per time.
A constant resisting <force> $F\ge0$ produces drift $-u_F=-DF/(k_BT)$. The <diffusion-limited Brownian ratchet with constant load> now satisfies
$$
DT''-u_FT'=-1,\qquad T'(0)=0,\quad T(a)=0.
$$
An integrating factor gives $T'(h)=[1-e^{u_Fh/D}]/u_F$. Integrating again from $h$ to $a$ gives $T(0)=D(e^z-1-z)/u_F^2$, where $z=Fa/(k_BT)$. Therefore
$$
\boxed{V(F)=\frac Da\frac{z^2}{e^z-1-z},\qquad \frac{V(F)}{V_0}=\frac{z^2}{2(e^z-1-z)}.}
$$
At weak load the ratio is $1-z/3+O(z^2)$; at large load $V\sim(D/a)z^2e^{-z}$. Replacing the whole <force> dependence by $e^{-z}$ would discard the first-passage kinetics of this regime.
For two staggered strands that alternately advance the leading tip by half a monomer, $a=\delta/2$ and $V_0=4D/\delta$. Two aligned strands, as schematically drawn, do not automatically have that half-step: an advance may be $a=\delta$, with the other strand filling the same layer afterward. Use the same geometrical step in the passage problem and in converting event rate to <velocity>. The limiting formula assumes negligible removal, rigid tips, an overdamped load with constant drag, and fast insertion compared with $a^2/D$. With appreciable removal or additional internal strand states, the renewal reset model must be replaced by a coupled gap/reaction process.
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