depending on chain length N and driving force strength f. Thus, one expects a
crossover from N
2ν
/f
1/ν to N
1 + ν /f, i.e., the translocation exponent α grows with
increasing force f from α % 1.18 to α % 1.59.
• For strong forces, the translocation time can be estimated as < τ >¼ C 1 τ 0 N
1þν
=
e f a þ C 2 τ 0 N
2ν
= e f a where the first term dominates under condition N
1 À ν
)
C 2 /C 1 . Therefore, the translocation scaling exponent grows from α ¼ 2v to
α ¼ 1 + v as the chain length N increases. This is in agreement with the results
of MD and MC simulations. A scaling relation very close to τ
h i / N
2ν
= e f a
has been found experimentally [91] in the case of translocation of long doublestranded DNA through a silicon oxide nanopore. In this experiment the translocation exponent was α ¼ 1.27, which is close to our theoretical prediction.
• Even if the mentioned approach appears to yield physically plausible
and qualitatively correct results, generally, the MD simulation findings yield
systematically smaller values for the translocation exponent, e.g., α % 1.11
and α % 1.47 for weak and strong forces, respectively. As we show below,
this shortcoming of the theory stems most probably from neglecting the role
of fluctuations during the translocation process, which is common to most
theoretical treatments so far.
In our most recent work, we investigated the impact of thermal fluctuations on
the driven translocation dynamics, theoretically and by means of extensive MD
simulation [88]. Indeed, the role of thermal fluctuations is by no means self-evident.
Our theoretical consideration is based on the Fokker–Planck equation (FPE)
Eq. (6), which has a nonlinear drift term and a diffusion term with a time-dependent
diffusion coefficient D t
ð Þ.
Our MD simulation reveals that the driven translocation process follows
a superdiffusive law with a running diffusion coefficient D t
ð Þ / t
γ , where γ < 1.
Therefore, although in the unbiased translocation case, the diffusion is Brownian
(or slightly subdiffusive), in the biased regime the process becomes superdiffusive, i.
e., the variance hΔs
2
i / t
θ , where 1 < θ < 2 (cf. Fig. 17b). Moreover, the exponent
θ increases with the growth of the driving force f, namely, θ ¼ 1.5 for f ¼ 1, and
θ ¼ 1.84 for f ¼ 10. This finding is then used in the numerical solution of the FPE,
which yields an important result: for comparatively small driving forces, fluctuations
facilitate the translocation dynamics. As a consequence, the exponent α, which
describes the scaling of the mean translocation time hτi with the length N of the
polymer, hτi / N
α
, is found to diminish. Thus, taking thermal fluctuations into
account, one can explain the systematic discrepancy between the theoretically
predicted duration of a driven translocation process, considered usually as a deterministic event, and measurements in computer simulations.
Finally, a related interesting problem occurs when a force acts on a chain end of a
flexible macromolecule, which is dragged into a nanotube with repulsive
walls (Fig. 18). Since in a nanotube of diameter D, the chain (under good solvent
conditions) forms a string of blobs, g / N
1/ν monomers per blob, and each blob costs
a free energy of k B T, there is a free energy cost of order N/g / ND
À 1/ν that needs to
be overcome by the force. Klushin et al. [92] showed, by a phenomenological scaling
26
R. Berger et al.
crossover from N
2ν
/f
1/ν to N
1 + ν /f, i.e., the translocation exponent α grows with
increasing force f from α % 1.18 to α % 1.59.
• For strong forces, the translocation time can be estimated as < τ >¼ C 1 τ 0 N
1þν
=
e f a þ C 2 τ 0 N
2ν
= e f a where the first term dominates under condition N
1 À ν
)
C 2 /C 1 . Therefore, the translocation scaling exponent grows from α ¼ 2v to
α ¼ 1 + v as the chain length N increases. This is in agreement with the results
of MD and MC simulations. A scaling relation very close to τ
h i / N
2ν
= e f a
has been found experimentally [91] in the case of translocation of long doublestranded DNA through a silicon oxide nanopore. In this experiment the translocation exponent was α ¼ 1.27, which is close to our theoretical prediction.
• Even if the mentioned approach appears to yield physically plausible
and qualitatively correct results, generally, the MD simulation findings yield
systematically smaller values for the translocation exponent, e.g., α % 1.11
and α % 1.47 for weak and strong forces, respectively. As we show below,
this shortcoming of the theory stems most probably from neglecting the role
of fluctuations during the translocation process, which is common to most
theoretical treatments so far.
In our most recent work, we investigated the impact of thermal fluctuations on
the driven translocation dynamics, theoretically and by means of extensive MD
simulation [88]. Indeed, the role of thermal fluctuations is by no means self-evident.
Our theoretical consideration is based on the Fokker–Planck equation (FPE)
Eq. (6), which has a nonlinear drift term and a diffusion term with a time-dependent
diffusion coefficient D t
ð Þ.
Our MD simulation reveals that the driven translocation process follows
a superdiffusive law with a running diffusion coefficient D t
ð Þ / t
γ , where γ < 1.
Therefore, although in the unbiased translocation case, the diffusion is Brownian
(or slightly subdiffusive), in the biased regime the process becomes superdiffusive, i.
e., the variance hΔs
2
i / t
θ , where 1 < θ < 2 (cf. Fig. 17b). Moreover, the exponent
θ increases with the growth of the driving force f, namely, θ ¼ 1.5 for f ¼ 1, and
θ ¼ 1.84 for f ¼ 10. This finding is then used in the numerical solution of the FPE,
which yields an important result: for comparatively small driving forces, fluctuations
facilitate the translocation dynamics. As a consequence, the exponent α, which
describes the scaling of the mean translocation time hτi with the length N of the
polymer, hτi / N
α
, is found to diminish. Thus, taking thermal fluctuations into
account, one can explain the systematic discrepancy between the theoretically
predicted duration of a driven translocation process, considered usually as a deterministic event, and measurements in computer simulations.
Finally, a related interesting problem occurs when a force acts on a chain end of a
flexible macromolecule, which is dragged into a nanotube with repulsive
walls (Fig. 18). Since in a nanotube of diameter D, the chain (under good solvent
conditions) forms a string of blobs, g / N
1/ν monomers per blob, and each blob costs
a free energy of k B T, there is a free energy cost of order N/g / ND
À 1/ν that needs to
be overcome by the force. Klushin et al. [92] showed, by a phenomenological scaling
26
R. Berger et al.