mechanisms for the translocation processes have been a subject of intense
discussion, e.g., electric fields [72], chemical potential gradients [73], selective
adsorption on one side of the membrane [74, 75], and related ratchet mechanisms
[76]. In addition, entropic barriers, segment crowding at the pore orifice, and,
most notably, the interplay between topological connectivity of the chain and
geometric constraints imposed by the pore, make the problem rather intricate and
complex. Moreover, since polymer translocation is such an ubiquitous phenomenon, it remains questionable whether a single universal scenario is operative
under all circumstances, so detailed studies of its various aspects are necessary.
It is convenient and customary to describe the translocation process by a single
variable s, called the translocation coordinate, which denotes the sequential number
of the monomer located in the pore at time t, and tells how much of the polymer has
passed meanwhile through the pore (cf. Fig. 15b). As a rule, once the initial monomer
has already threaded through the hole, among the principal quantities of interest are
the mean translocation time τ, and its dependence on polymer length N and on the
driving force magnitude f. In fact, τ is one of the few dynamic quantities that are
accessible in current experiments [67, 70, 71]. Assuming that τ / N
α f
À δ , one of the
objectives is to provide a plausible explanation for the observed values of the
exponents α and δ. Important information is obtained from the probability distribution function of translocation times, Q(t), and also from the time dependence of the
mean squared displacement (variance) of the translocation coordinate Δ(t)
hΔs
2
(t)i ¼ hs
2
i À hsi
2
. Both quantities play essential roles in numerous efforts to
elucidate a typical hallmark of the translocation process. Namely, its dynamics is
anomalous with hΔs
2
(t)i / t
β , where β < 1 (i.e., the dynamics is subdiffusive) for
the force-free (i.e., f ¼ 0) case and β > 1 (i.e., super-diffusive) for the force-driven
polymer translocation. This is well established at present [77–80].
There are different possibilities for how the forced translocation can be effected.
For example, Milchev at al. [74, 75] studied the possibility that the monomer–
membrane interaction is attractive on the trans side, while it is assumed to be
repulsive on the cis side. Assuming that a few monomers of a chain have already
passed through the pore and experience the favorable membrane–monomer interaction on the trans side, two questions that are asked are: (1) How likely is it that
the rest of the chain will follow from the cis to the trans side, depending on chain
length N and the distance T/T c À 1 from the adsorption transition that happens
on the trans side at T ¼ T c ? (2) How does the time needed for complete translocation depend on these parameters?
Milchev et al. [74, 75] found that it makes a big difference whether one
studies the case in which in equilibrium a chain is not yet absorbed on the trans
side (and still in a mushroom state when the chain gets through the pore) or whether
adsorption occurs. In the first case, the problem is similar to unbiased translocation
(which occurs by thermal fluctuations only [81, 82]), i.e., for any finite fraction of
monomers that have already passed to the trans side there is still a non-zero
probability that the whole chain returns to the cis side (and diffuses away). In
this case, the translocation time is found to scale as τ / N
2ν + 1
% N
2.2 , i.e., the
time is simply of the order of the Rouse time of a single chain in a good solvent
(note that the Monte Carlo modeling of Milchev et al. [74, 75] uses implicit solvent
22
R. Berger et al.
discussion, e.g., electric fields [72], chemical potential gradients [73], selective
adsorption on one side of the membrane [74, 75], and related ratchet mechanisms
[76]. In addition, entropic barriers, segment crowding at the pore orifice, and,
most notably, the interplay between topological connectivity of the chain and
geometric constraints imposed by the pore, make the problem rather intricate and
complex. Moreover, since polymer translocation is such an ubiquitous phenomenon, it remains questionable whether a single universal scenario is operative
under all circumstances, so detailed studies of its various aspects are necessary.
It is convenient and customary to describe the translocation process by a single
variable s, called the translocation coordinate, which denotes the sequential number
of the monomer located in the pore at time t, and tells how much of the polymer has
passed meanwhile through the pore (cf. Fig. 15b). As a rule, once the initial monomer
has already threaded through the hole, among the principal quantities of interest are
the mean translocation time τ, and its dependence on polymer length N and on the
driving force magnitude f. In fact, τ is one of the few dynamic quantities that are
accessible in current experiments [67, 70, 71]. Assuming that τ / N
α f
À δ , one of the
objectives is to provide a plausible explanation for the observed values of the
exponents α and δ. Important information is obtained from the probability distribution function of translocation times, Q(t), and also from the time dependence of the
mean squared displacement (variance) of the translocation coordinate Δ(t)
hΔs
2
(t)i ¼ hs
2
i À hsi
2
. Both quantities play essential roles in numerous efforts to
elucidate a typical hallmark of the translocation process. Namely, its dynamics is
anomalous with hΔs
2
(t)i / t
β , where β < 1 (i.e., the dynamics is subdiffusive) for
the force-free (i.e., f ¼ 0) case and β > 1 (i.e., super-diffusive) for the force-driven
polymer translocation. This is well established at present [77–80].
There are different possibilities for how the forced translocation can be effected.
For example, Milchev at al. [74, 75] studied the possibility that the monomer–
membrane interaction is attractive on the trans side, while it is assumed to be
repulsive on the cis side. Assuming that a few monomers of a chain have already
passed through the pore and experience the favorable membrane–monomer interaction on the trans side, two questions that are asked are: (1) How likely is it that
the rest of the chain will follow from the cis to the trans side, depending on chain
length N and the distance T/T c À 1 from the adsorption transition that happens
on the trans side at T ¼ T c ? (2) How does the time needed for complete translocation depend on these parameters?
Milchev et al. [74, 75] found that it makes a big difference whether one
studies the case in which in equilibrium a chain is not yet absorbed on the trans
side (and still in a mushroom state when the chain gets through the pore) or whether
adsorption occurs. In the first case, the problem is similar to unbiased translocation
(which occurs by thermal fluctuations only [81, 82]), i.e., for any finite fraction of
monomers that have already passed to the trans side there is still a non-zero
probability that the whole chain returns to the cis side (and diffuses away). In
this case, the translocation time is found to scale as τ / N
2ν + 1
% N
2.2 , i.e., the
time is simply of the order of the Rouse time of a single chain in a good solvent
(note that the Monte Carlo modeling of Milchev et al. [74, 75] uses implicit solvent
22
R. Berger et al.