Directed motion emerging from two coupled random processes: translocation of a chain through a membrane nanopore driven by binding proteins

Summary, in English

We investigate the translocation of a stiff polymer consisting of M monomers
through a nanopore in a membrane, in the presence of binding particles
(chaperones) that bind onto the polymer, and partially prevent backsliding of
the polymer through the pore. The process is characterized by the rates: k
for the polymer to make a diffusive jump through the pore, q for unbinding of
a chaperone, and the rate qκ for binding (with a binding strength κ); except
for the case of no binding κ = 0 the presence of the chaperones gives rise
to an effective force that drives the translocation process. In more detail, we
develop a dynamical description of the process in terms of a (2+1)-variable
master equation for the probability of having m monomers on the target side
of the membrane with n bound chaperones at time t. Emphasis is put on the
calculation of the mean first passage time as a function of total chain length M.
The transfer coefficients in the master equation are determined through detailed
balance, and depend on the relative chaperone size λ and binding strength κ,
as well as the two rate constants k and q. The ratio γ = q/k between the two
rates determines, together with κ and λ, three limiting cases, for which analytic
results are derived: (i) for the case of slow binding (γ κ → 0), the motion is
purely diffusive, and M2 for large M; (ii) for fast binding (γ κ → ∞) but
slow unbinding (γ → 0), the motion is, for small chaperones λ = 1, ratchetlike, and M; (iii) for the case of fast binding and unbinding dynamics
(γ → ∞ and γ κ → ∞), we perform the adiabatic elimination of the fast
variable n, and find that for a very long polymer M, but with a smaller
prefactor than for ratchet-like dynamics. We solve the general case numerically
as a function of the dimensionless parameters λ, κ and γ , and compare to the
three limiting cases.