and there cannot be any effects due to hydrodynamic interactions). In any case,
this result already shows that the relation τ / N
2 [81, 82] (derived on the assumption that the problem can be reduced to a quasi-one-dimensional problem of
diffusion over an entropic barrier, taking the numbers of monomers at the trans
side as “reaction coordinate”) cannot hold in general. Indeed, Dubbeldam
et al. [77, 78] have shown that different power laws can be obtained by a model
in which ordinary diffusion is replaced by fractional Brownian motion, which leads
to anomalous diffusion.
When one studies translocation driven by adsorption for temperatures where in
equilibrium the chain is adsorbed on the trans side in a “pancake configuration,”
one finds that a finite number of monomers adsorbed on the trans side (of order 10)
suffices to pull the remaining chain through the hole in the membrane in almost all
cases. The relaxation time was found to scale like τ / N
1.7 , but is not clear whether
this exponent can be theoretically explained [74, 75].
The standard model for forced translocation uses a biasing force on any monomer that has entered the pore, driving it from the cis to the trans side (see [83] and
references therein). Estimates for the various exponents describing the translocation
dynamics have been given [83], but in this study (as well as in other work that
can be found in the literature) it is not clear whether the asymptotic scaling regime
really has been reached or whether one observes “effective exponents” due to
slow crossovers. In addition, a rather fundamental problem [83, 84] is a strong
conformational asymmetry of the part of the chain that is still on the cis side and the
part that is already on the trans side: the former part is under stretch because parts of
the chain very distant (along the contour) from the pore have not yet relaxed.
Translocation happens too fast for the chain configurational degrees of freedom
to reach local equilibrium. So, the radius of the cis part is relatively too large.
Conversely, the configuration of the trans part is somewhat too dense and,
hence, the radius is too small [83, 84]. Although some aspects of this phenomenon
have been discussed by Sakaue [85, 86], we feel that a complete theory of
translocation that properly incorporates all these out-of-equilibrium effects still
needs to be developed.
In the wake of these developments, our studies of translocation dynamics
have been focused on the generic case of unbiased translocation [77, 79, 80] in
the absence of driving force, f ¼ 0, and on the case when the chain threading
through a pore is driven by applied force [78, 87–89]. In our original efforts to
capture the essence of the problem, we employed diverse methods (scaling theory,
fractional calculus, Monte Carlo and molecular dynamics simulations). We found
that the relevant dynamic variable, the transported number of polymer segments s
(t), displayed an anomalous diffusive behavior both with and without an external
driving force being present [77–80]. A closed analytic expression for the
probability, W(s, t), of finding s translocated segments at time t in terms of chain
length N and applied drag force f was derived from the fractional Fokker–Planck
equation and shown to provide analytic results for the time variation of the
statistical moments hs(t)i and hs
2 (t)i. It was shown that in the absence of driving
force, the time τ needed for a macromolecule of length N to thread from the cis into
Mechanical Properties of Single Molecules and Polymer Aggregates
23
this result already shows that the relation τ / N
2 [81, 82] (derived on the assumption that the problem can be reduced to a quasi-one-dimensional problem of
diffusion over an entropic barrier, taking the numbers of monomers at the trans
side as “reaction coordinate”) cannot hold in general. Indeed, Dubbeldam
et al. [77, 78] have shown that different power laws can be obtained by a model
in which ordinary diffusion is replaced by fractional Brownian motion, which leads
to anomalous diffusion.
When one studies translocation driven by adsorption for temperatures where in
equilibrium the chain is adsorbed on the trans side in a “pancake configuration,”
one finds that a finite number of monomers adsorbed on the trans side (of order 10)
suffices to pull the remaining chain through the hole in the membrane in almost all
cases. The relaxation time was found to scale like τ / N
1.7 , but is not clear whether
this exponent can be theoretically explained [74, 75].
The standard model for forced translocation uses a biasing force on any monomer that has entered the pore, driving it from the cis to the trans side (see [83] and
references therein). Estimates for the various exponents describing the translocation
dynamics have been given [83], but in this study (as well as in other work that
can be found in the literature) it is not clear whether the asymptotic scaling regime
really has been reached or whether one observes “effective exponents” due to
slow crossovers. In addition, a rather fundamental problem [83, 84] is a strong
conformational asymmetry of the part of the chain that is still on the cis side and the
part that is already on the trans side: the former part is under stretch because parts of
the chain very distant (along the contour) from the pore have not yet relaxed.
Translocation happens too fast for the chain configurational degrees of freedom
to reach local equilibrium. So, the radius of the cis part is relatively too large.
Conversely, the configuration of the trans part is somewhat too dense and,
hence, the radius is too small [83, 84]. Although some aspects of this phenomenon
have been discussed by Sakaue [85, 86], we feel that a complete theory of
translocation that properly incorporates all these out-of-equilibrium effects still
needs to be developed.
In the wake of these developments, our studies of translocation dynamics
have been focused on the generic case of unbiased translocation [77, 79, 80] in
the absence of driving force, f ¼ 0, and on the case when the chain threading
through a pore is driven by applied force [78, 87–89]. In our original efforts to
capture the essence of the problem, we employed diverse methods (scaling theory,
fractional calculus, Monte Carlo and molecular dynamics simulations). We found
that the relevant dynamic variable, the transported number of polymer segments s
(t), displayed an anomalous diffusive behavior both with and without an external
driving force being present [77–80]. A closed analytic expression for the
probability, W(s, t), of finding s translocated segments at time t in terms of chain
length N and applied drag force f was derived from the fractional Fokker–Planck
equation and shown to provide analytic results for the time variation of the
statistical moments hs(t)i and hs
2 (t)i. It was shown that in the absence of driving
force, the time τ needed for a macromolecule of length N to thread from the cis into
Mechanical Properties of Single Molecules and Polymer Aggregates
23