Polymer translocation under time-dependent driving forces: resonant activation induced by attractive polymer-pore interactions.

Abstract

We study the driven translocation of polymers under time-dependent driving forces using N-particle Langevin dynamics simulations. We consider the force to be either sinusoidally oscillating in time or dichotomic noise with exponential correlation time, to mimic both plausible experimental setups and naturally occurring biological conditions. In addition, we consider both the case of purely repulsive polymer-pore interactions and the case with additional attractive polymer-pore interactions, typically occurring inside biological pores. We find that the nature of the interaction fundamentally affects the translocation dynamics. For the non-attractive pore, the translocation time crosses over to a fast translocation regime as the frequency of the driving force decreases. In the attractive pore case, because of a free energy well induced inside the pore, the translocation time can be a minimum at the optimal frequency of the force, the so-called resonant activation. In the latter case, we examine the effect of various physical parameters on the resonant activation, and explain our observations using simple theoretical arguments.

Full text

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Polymer translocation under time-dependent driving forces: ResonantPolymer translocation under time-dependent driving forces: Resonant
activation induced by attractive polymer-pore interactions activation induced by attractive polymer-pore interactions
Timo Ikonen, Jaeoh Shin, Wokyung Sung, and Tapio Ala-Nissila

Citation: J. Chem. Phys. 136136, 205104 (2012); doi: 10.1063/1.4722080
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THE JOURNAL OF CHEMICAL PHYSICS 136, 205104 (2012)
Polymer translocation under time-dependent driving forces: Resonant
activation induced by attractive polymer-pore interactions
Timo Ikonen,1Jaeoh Shin,2Wokyung Sung,2,a)and Tapio Ala-Nissila1,3
1Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto,
Espoo, Finland
2Department of Physics, Pohang University of Science and Technology, Pohang 790-784, South Korea
3Department of Physics, Box 1843, Brown University, Providence, Rhode Island 02912-1843, USA
(Received 10 February 2012; accepted 9 May 2012; published online 31 May 2012)
We study the driven translocation of polymers under time-dependent driving forces using N-particle
Langevin dynamics simulations. We consider the force to be either sinusoidally oscillating in time or
dichotomic noise with exponential correlation time, to mimic both plausible experimental setups and
naturally occurring biological conditions. In addition, we consider both the case of purely repulsive
polymer-pore interactions and the case with additional attractive polymer-pore interactions, typically
occurring inside biological pores. We find that the nature of the interaction fundamentally affects
the translocation dynamics. For the non-attractive pore, the translocation time crosses over to a fast
translocation regime as the frequency of the driving force decreases. In the attractive pore case, be-
causeofafreeenergywellinducedinsidethepore,thetranslocationtimecanbeaminimumattheop-
timalfrequencyoftheforce,theso-calledresonantactivation.Inthelattercase,weexaminetheeffect
of various physical parameters on the resonant activation, and explain our observations using simple
theoretical arguments. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4722080]
I. INTRODUCTION
Translocation of polymers across a nanopore is a ubiqui-
tousprocessinbiology,withexamplessuchasDNAandRNA
transport through nuclear pore complex, protein transport
through membrane channels, and virus injection into cells.1
Kasianowicz et al.2demonstrated in vitro that an electric
field can transport single-stranded (ss) nucleotides through an
α-hemolysin membrane channel and it is possible to charac-
terize individual molecules by measuring the ionic current
blockade when the chain moves through the pore. Later, Li
et al. showed3that also solid-state nanopores can be used
for similar experiments with a tunable size of the pore. To
further understanding the numerous biological processes and
examine the perspective of technological applications such
as sequencing and gene therapy, there have been extensive
experimental4–11and theoretical studies.12–37
One of the most important quantities of the process is
the translocation time and its dependence on the various sys-
tem parameters such as chain length, type of driving force,
pore width, etc. Even with the same chain lengths, recent
experiments4–7have shown that different nucleotides exhibit
unique patterns in, e.g., the translocation time distribution. In
particular, Meller et al.5,7have shown that in the translocation
can discriminate between polydeoxyadenylic acid (poly(dA))
and polydeoxycytidylic acid (poly(dC)) with the same chain
length. The translocation time of poly(dA) is found to be
longer with an exponential distribution while that of poly(dC)
is shorter with a narrow distribution. The origin of the dif-
ferent behavior for each nucleotide was attributed to different
a)Author to whom correspondence should be addressed. Electronic mail:
wsung@postech.ac.kr.
interaction between the polymer and the pore. Recent simula-
tion studies of Luo et al.22,25quantitatively support this idea.
Until now, most of the in vitro experimental as well as
theoretical studies of polymer translocation have used static
driving forces. However, it could be important to consider
time-dependent forces to understand the process in vivo. In a
cellular environment the driving forces can be time-dependent
due to the non-equilibrium fluctuations in the membrane po-
tential, fluid density, and ionic strength, etc. In the case of
translocation driven by a molecular motor,38depending on
the adenosine triphosphate (ATP) concentration the driving
force can also fluctuate. Motivated by these facts, Park and
Sung39studied the translocation of a rigid rod in the pres-
ence of a dichotomically fluctuating force. They found that
the system exhibits resonant activation,40where the translo-
cation time attains a minimum at an optimum flipping rate
of the dichotomic force that is comparable to the transloca-
tion rate in the absence of the force. Although the study gives
valuable insight on the effects of fluctuating forces in poly-
mer translocation, the study is somewhat limited, however,
as the flexibility of the chain is not considered and, the re-
flecting boundary condition which forbids the chain escape
to the cis side is in many cases artificial. Recent molecular
dynamics simulations41have shown that an alternating elec-
tric field in a nanopore exhibits a unique hysteresis in the
nucleotide’s dipole moment and in the chains back-and-forth
motion arising from the reorientation of the DNA bases in
the nanopore constriction. The authors suggest detection of
DNA sequences by measuring the potential or the change of
the DNA mobility in the pore. This study indicates that a
time-dependent driving force may be useful for technologi-
cal application as well. In addition, recent Langevin dynamics
study shows that the translocation time can be significantly
0021-9606/2012/136(20)/205104/13/$30.00© 2012 American Institute of Physics
136, 205104-1
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205104-2Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
shortened by oscillations of the pore width,42which accen-
tuates the importance of polymer-pore interactions in the
problem.
In addition to polymer translocation, there are a few
simulation studies of different types of polymer transport in
the presence of time-dependent driving forces. Tessier and
Slater43considered polymer transport through a microchan-
nel in the presence of a periodic driving force, where they
foundthatthemobilitycanhaveamaximumatanoptimalfre-
quency. More recently, Pizzolato et al.44have studied the ef-
fects of sinusoidal driving force on the polymer barrier cross-
ing over a metastable potential, which is also subject to the
reflecting boundary condition. They found a similar resonant
behavior of the barrier crossing time. In addition, Fiasconaro
et al. have studied the one-dimensional (1D) polymer chain in
the presence of sinusoidal45and dichotomically fluctuating46
driving forces. They found that the sinusoidal driving force
mayinduceanoscillatingbehaviorofthetranslocationtime,45
whereas the dichotomic force does not.46
Despite the related work found in the literature, the ef-
fect of the polymer-pore interactions on the translocation of
biopolymers under time-dependent driving forces and in a
realistic geometry needs to be studied. In this work, the ef-
fectsoftime-dependentdrivingforcesonthetranslocationdy-
namics are investigated as a first step towards understanding
translocation both in vivo and in practical applications. We
consider both dichotomically fluctuating forces as an exam-
ple of in vivo non-equilibrium noise39,47and sinusoidal driv-
ing forces, which might be easier to implement experimen-
tally. We find that the polymer-pore attraction fundamentally
changes the behavior of the translocation time with respect to
the flipping rate of the dichotomic force or the angular fre-
quency of the sinusoidal force. For the non-attractive pore,
the translocation time has a cross-over to a fast translocation
regime at low flipping rates (frequencies), but does not have
a resonant minimum. For the attractive pore, we show that
the system exhibits resonant activation within a broad range
of physical parameters. We examine the effect of parame-
ters such as chain length, driving force, and polymer-pore in-
teraction strength on the resonance. The results suggest that
in vitro experiments with time-dependent driving force might
be useful to DNA sequencing.
II. MODEL AND METHOD
We consider the translocation of a self-avoiding chain
in two dimensions (2D). The polymer chain is modeled by
Lennard-Jones particles interconnected by finitely extensible
nonlinear elastic (FENE) springs. Excluded volume interac-
tion between monomers is given by the short-range repul-
sive Lennard-Jones potential: ULJ(r) = 4?[(σ
for r ≤ 21/6σ and 0 for r > 21/6σ. Here, r is the dis-
tance between monomers, σ is the diameter of the monomer,
and ? is the depth of the potential well. Neighboring
monomers arealsoconnected byFENEspringswithUFENE(r)
= −1
stant and R0is the maximum allowed separation between con-
secutive monomers. The geometry of the system is shown in
Fig. 1. The wall is constructed of immobile Lennard-Jones
r)12− (σ
r)6] + ?
2kR2
0ln(1 − r2/R2
0), where k is the FENE spring con-
FIG. 1. A schematic representation of the system. The polymer, placed ini-
tially on the cis side, is driven through the pore of length L = 5 and width
W = 3 by the time-dependent external force F + f(t).
beads of size σ. All monomer-wall particle pairs have the
same short-range repulsive LJ interaction as described above.
To investigate the effect of polymer-pore interactions, we con-
sider two main types of interactions between the monomers
andtheporeparticles:attractiveandnon-attractive.Inthecase
of non-attractive interactions, the pore particles are consid-
ered to be identical with the wall particles, having a purely
repulsive interaction with the monomers. In the case of the at-
tractive polymer-pore interactions, the cut-off distance of the
LJ potential between monomer-pore particles is increased to
2.5σ (with ULJconstant for r > 2.5σ), and the interaction
strength is characterized by ?pm. The interaction can be ei-
ther attractive or repulsive, depending on the distance of the
monomer from the pore particles.
In our simulations, the dynamics of each monomer is de-
scribed by the Langevin equation
m¨ ri= −∇(ULJ+ UFENE) + Fext− ξvi+ FR
where m is the monomer mass, ξ is the friction coefficient, vi
is the monomer velocity, and FR
correlations ?FR
is the Boltzmann constant and T is the temperature. In the
pore, the monomers experience an external driving force Fext
= [F + f(t)]ˆ x, where F is static (time-independent) force,
f(t) is the time-dependent force, and ˆ x is the unit vector along
the direction of the pore axis. In this work, we consider two
types of time-dependent forces f(t). The first is the dichotomic
noise, for which f(t) is either +Ador −Ad, and changes from
one value to the other with flipping rate ω. The dichotomic
f(t) has zero mean and is exponentially correlated: ?f(t)? = 0
and ?f(t)f(0)? = A2
consider the sinusoidal force given by f(t) = Asin(ωt + φ),
where A is the amplitude, ω is the angular frequency, and φ is
a constant phase.
We use the LJ parameters ?, σ, and m to fix the scales for
energy, length, and mass, respectively. The time scale is then
i,
(1)
iis the random force with
j(t?)? = 4ξkBTδi,jδ(t − t?), where kB
i(t) · FR
dexp(−2ωt). As a second example, we
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205104-3Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
given by tLJ= (mσ2/?)1/2. The dimensionless parameters in
our simulations are R0= 2, k = 7, ξ = 0.7, and kBT = 1.2.
In our model, the bead size corresponds to the Kuhn length
of a single-strand DNA, giving approximately σ ≈ 1.5 nm.
The bead mass is approximately 936 amu, and the interaction
strength ? corresponds to 3.39 × 10−21J at room temperature
(295 K). The Lennard-Jones time scale is then 32.1 ps. Where
appropriate, we will express our results also in terms of the
mean translocation time in the absence of f(t), τ0, which is
the physically relevant time scale in the system and also easy
to measure experimentally. The pore dimensions we set as L
= 5 and W = 3, as shown in Fig. 1. With the force scale of
2.3 pN, a static driving force of Fext= 1 then corresponds
to a voltage of 375 mV across the pore (assuming three unit
charges per bead and the effective charge 0.094 e for a unit
charge48). The equations of motion are integrated with the Er-
mak algorithm49,50with time-step typically ?t = 0.01, and
shorter when necessary.
Initially,the firstmonomer of the chain is held fixed atthe
pore (see Fig. 1) while the remaining monomers are allowed
to fluctuate until an equilibrium configuration is reached.
Then at time t = 0 the first monomer is released and the exter-
nal force is applied. For the dichotomic force, the initial force
is randomly selected from +Ad and −Ad with equal prob-
ability. Correspondingly, for the sinusoidal force, the phase
φ is randomly selected from a uniform distribution between
[0, 2π]. For small Fextand weak polymer-pore attraction, the
chain may slip out of the pore back to the cis side instead
of translocating to the trans side. In that case, the equilibra-
tion process is repeated and the simulation is begun anew. The
process is repeated until at least 2000 successful translocation
events are recorded. In addition to this standard procedure, it
is possible to impose a reflecting boundary condition that pre-
vents the first bead from slipping back to the cis side. In this
case, the simulation is run simply until a successful transloca-
tion occurs. It turns out that this boundary condition, although
widely used in translocation study, fundamentally changes the
translocation dynamics, as will be discussed in Sec. III. That
is why, unless otherwise indicated, all the results presented in
this work have been computed without the reflecting bound-
ary condition.
III. RESULTS AND DISCUSSION
A. Non-attractive pore, dichotomic driving force
We begin by considering the purely repulsive polymer-
pore interactions, which is the most common case studied in
the literature. The strength of the Lennard-Jones interaction is
?pm= 1 with a cut-off distance of 21/6σ. First, we consider the
dichotomic driving force, with the results for the sinusoidal
force presented later in Sec. III B. We have chosen the numer-
ical values F = 0.3 and Ad= 0.2 for the dichotomic force and
N = 64 for the chain length, which are within the experimen-
tal regime. We have checked that within the experimentally
relevant force regime and at least for N ≤ 128 the qualitative
behavior remains the same.
The main results for the dichotomic force as a function
of the flipping rate ω are gathered in Fig. 2. As a function
0.0010.010.1110100
0.5
0.6
0.7
0.8
0.9
1
1.1
( )/
0.5
0.6
0.7
0.8
0.9
1
1.1
P ( ), P ( )
( )/
P ( )
P ( )
0
0
0
0
0
FIG. 2. The mean translocation time τ and the probabilities P0and Pτ(see
text) as a function of the flipping rate ω of the dichotomic force for the repul-
sive pore. N = 64, F = 0.3, Ad= 0.2, and τ0≈ 750 ± 4. The statistical error
is smaller than the symbol size.
of the flipping rate ω, we observe two distinct regions. In the
fast flipping regime, ω ? 1/τ0, the average translocation time
is τ(ω) ≈ τ0. Here, τ0is the translocation time in the ab-
sence of dichotomic forces, i.e., Ad= 0. In this limit, due to
the high flipping rate, f(t) changes its sign many times dur-
ing the course of the translocation and is averaged out to
zero over the whole process. Therefore, we have τ(ω) ≈ τ0
for ω → ∞. In principle, in this limit, the time-dependent
force becomes a rapidly fluctuating δ-correlated noise sim-
ilar to the thermal random force FR
side the pore, the modified correlation of the random force
is given as ?[f(t)ˆ x + FR(t)] · [f(0)ˆ x + FR(0)]? = (4ξkBT
+ A2
in the limit ω → ∞. Therefore, the effect of the dichotomic
force in this limit is vanishing, and we recover τ(ω) ≈ τ0,
as shown in Fig. 2. This result is also in agreement with
Refs. 44–46. As the flipping becomes slower, ω < 1/τ0, we
observe a cross-over to a faster translocation regime, with
τ(ω) < τ0. This result is in sharp contrast with Refs. 44–46,
where it was found that τ(ω) > τ0. In addition, we do
not find a global minimum of τ(ω) at any finite ω, unlike
Refs. 44 and 45.
To understand the behavior of τ(ω) at small ω, we need to
look at the probability of achieving a successful translocation.
Due to confinement within the pore, the chain experiences an
entropic free energy barrier,12,13,15as illustrated in Fig. 3. Be-
cause of fluctuations, there is a finite probability that the chain
slips back to the cis side instead of translocating to the trans
side. Therefore, the probability of translocation is less than
one and increases with increasing driving force (for details of
the translocation probability as a function of various system
parameters, see Ref. 22). Thus, within the set of successful
translocations, we expect to find a larger number events that
have positive f(t), as compared to those with negative f(t). We
characterize this dependence of the translocation probability
on the driving force by looking at the set of successful translo-
cations, from which we calculate the distribution of f(t) at the
beginning of translocation (t = 0) and at the final moment of
translocation (t = τ). The probabilities P0≡ P[f(0) > 0] and
i. For the monomers in-
d/ω)δ(t) = 4ξkBTδ(t), where last expression is obtained
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205104-4Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
00.20.40.60.81
s/N
-10
-8
-6
-4
-2
Free energy (arbitrary units)
Non-attractive pore
Attractive pore
Well due to attractive pore
FIG. 3. Illustration of the free energy of the polymer chain as a function
of the number of translocated monomers s. The dotted line indicates the
free energy for the non-attractive pore, which has no well structure. A re-
flecting boundary condition at s = 0 forms a free-energy well (blue shaded
area). Attractive polymer-pore interactions can also create a free-energy well
(schematically shaded red).
Pτ≡ P[f(τ) > 0] that the force f(t) is positive for t = 0 and
t = τ, respectively, are shown in Fig. 2. In the high-rate
regime, the flipping rate is too high for f(0) or f(τ) to be cor-
related with the chain dynamics, and therefore P0and Pτap-
proach 0.5. On the other hand, in the low-rate regime, the pos-
itive direction of f(0) is strongly favored (P0≈ 0.98). In ad-
dition, since the correlation time of the driving force is much
longer than τ, the driving force remains constant during the
whole translocation process with high probability, being ei-
ther Fext= F + Ador Fext= F − Ad. Therefore, in this limit,
the average translocation time is given by the weighted aver-
age
τ = P0τ++ (1 − P0)τ−.
(2)
Here, τ+ and τ− are the translocation times with the to-
tal force F + Adand F − Ad, respectively. Assuming that
the translocation time is inversely proportional to the driving
force, τ(f) ∼ f−1, Eq. (2) gives τ(ω) ≈ 0.65τ0in the low-ω
limit. This agrees well with the results in Fig. 2. Therefore,
the cross-over to the fast translocation regime (τ < τ0) as
the flipping becomes slower is simply explained by the fact
that for low flipping rate the chain is most likely to translo-
cate when f(0) > 0. This strong bias for selecting the initial
value f(0) induced by the entropic barrier is the crucial dif-
ference between this work and Refs. 44–46. In Refs. 44–46
this kind of selection does not occur because the translocation
probability is one, independent of the time-dependent driving
force. A similar effect can be obtained in our model by im-
posing a reflecting boundary condition that prevents the first
monomer from exiting the pore to the cis side. However, we
stress that this kind of boundary condition may not be realis-
tic for, e.g., the translocation of a ss-DNA molecule through a
pore, although it has been used in many studies.
Finally, we look at the distribution of translocation times.
In the high flipping rate limit, the distribution is very similar
to the zero amplitude case (see Fig. 4). In general, the distri-
bution at this limit is either Gaussian (for large enough F) or
(a)(b)
(c)(d)
FIG. 4. The distribution of translocation times for chain length N = 64 and
F = 0.3 under dichotomic driving force in the non-attractive pore. Panel (a)
shows the distribution for Ad= 0, while panels (b)–(d) show the distribution
for Ad= 0.2.
has an exponentially decaying tail (for small F). In the present
case, the distribution is almost Gaussian with a slightly elon-
gated tail, which differs greatly from the typical distributions
of thermally activated processes. Furthermore, as shown in
Fig. 3, there is no metastable well (pretransition state) within
which the chain attempting to escape would oscillate. Thus,
in this case, the resonant minimum of τ(ω) does not exist,
in contrast to Ref. 44, where the adopted external potential
has a pretransitional well. At lower flipping rates, the peak of
the distribution moves toward shorter translocation times, as
the trajectories having f(t) predominantly in the positive di-
rection are favored (signaled by increasing P0and Pτ). How-
ever,alsotheprobabilityoflongtranslocationtimesincreases.
These events correspond to the trajectories with negative f(t).
As the flipping rate is further decreased, most of the success-
ful translocations occur with f(t) > 0. In the low-rate limit,
one retains two peaks, corresponding to Fext= F + Adand
Fext= F − Ad. For F = 0.3 and Ad= 0.2, only the former is
practically visible.
B. Non-attractive pore, periodic driving force
As a second case, we study the translocation through a
non-attractive pore under sinusoidally time-dependent driv-
ing force f(t) = Asin(ωt + φ). The average translocation time
0.010.11101001000
0.5
0.6
0.7
0.8
0.9
1
1.1
( )/
0.5
0.6
0.7
0.8
0.9
1
1.1
P ( ), P ( )
( )/
P ( )
P ( )
0
0
0
0
0
FIG. 5. The mean translocation time τ and the probabilities P0and Pτas a
function of the angular frequency ω for the periodic force and repulsive pore.
N = 64, F = A = 0.3, and τ0≈ 750 ± 4. The statistical error is smaller than
the symbol size.
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Page 6

205104-5Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
τ(ω) and the probabilities P0and Pτ are shown in Fig. 5 as
a function of the angular frequency ω. For comparison with
the dichotomic case, we use the parameter values N = 64,
F = 0.3, and A = 0.3. The time-averaged amplitude of the
time-dependent force is then ?|Asin(ωt)|?t = 2A/π ≈ 0.2,
which corresponds to the value of Ad used in Sec. III A.
In the low-frequency (ω ? 1/τ0) and high-frequency (ω
? 1/τ0) limits we obtain results similar to the dichotomic
force explained above: in the high-ω limit, τ(ω) ≈ τ0, and in
the opposite limit of small ω, τ(ω) < τ0. The average translo-
cation time is given by a relation analogous to Eq. (2):
?2π
Here p(φ) is the probability density of the initial phase φ
within the set of successful translocations and τ(φ) is the av-
erage translocation time corresponding to the driving force
F + Asin(φ). Similar to the dichotomic case, the distribution
p(φ) is uniform only in the high frequency limit, while in the
low frequency limit, values of φ giving f(0) > 0 are strongly
favored (cf. Fig. 5), as we shall see. This leads to larger aver-
age driving forces and consequently faster translocation.
In the intermediate regime (ω ≈ 1/τ0), the periodic time-
dependence of the driving force fundamentally affects the
translocation dynamics. Instead of a simple cross-over in
τ(ω), one gets a series of local minima and maxima. In ad-
dition, the probability Pτhas a local maximum in the vicinity
of a local minimum of τ. In many cases, these could be ar-
gued to indicate resonant activation.40However, in this case
they have a deterministic origin weighted with the distribu-
tion p(φ). To show this, we consider a coarse-grained model
for the translocated segments already studied in Refs. 12 and
13, with the entropic contributions therein neglected to make
it analytically solvable. This approximation is reasonable be-
cause apart from the short initial (and final) stage of transloca-
tion, the entropic force is small compared to the mean driving
force F. Our model is the 1D equation of motion for ?s(t)?,
the average number of translocated segments, under the sinu-
soidal driving force with a fixed value of φ,
τ =
1
2π
0
p(φ)τ(φ)dφ.
(3)
ξeffd?s(t)?
dt
= F [1 + sin(ωt + φ)],
(4)
where ξeffis the effective friction. Equation (4) can be an-
alytically solved for ?s(t)? with the initial condition s(0)
= 0. Specifically, we are interested in the time that it takes
for the system to evolve from s = 0 to s = N as a function
of the phase, τ(φ). We fix ξeffby setting the time-scale of the
model so that F/ξeff= N/τ0, giving ξeff≈ 17.6 for N = 64 and
F = A = 0.3. The integration of Eq. (4) yields
τ(φ) = τ0+1
This describes the approach of the translocation time τ to τ0
in the ω → ∞ limit, as well as the local oscillation in the
intermediate ω regimes. Once τ(φ) is obtained as a function
of φ as well as ω from Eq. (5), the translocation time averaged
over φ is found from Eq. (3).
In Fig. 6, we compare our model with the N-particle
Langevin dynamics simulations. First, the dotted line shows
ω[cos(ωτ + φ) − cos(φ)].
(5)
1 10 100
0.6
0.8
1.0
1.2
( )/
Langevin dynamics
Theory, uniform distribution
Theory
0
0
FIG. 6. Comparison between LD simulations (N = 64, F = A = 0.3) and
the theoretical toy model. Dotted line: toy model with uniformly distributed
φ; solid line: toy model with Boltzmann distributed φ (see text). The latter
shows good agreement with the LD results (circles).
the results for a uniformly distributed φ. In contrast to the LD
simulations, this curve shows a global minimum of translo-
cation time, and also a strong oscillating behavior as a func-
tion of ω. The behavior is very similar to the simple 1D chain
model driven by sinusoidal force studied in Ref. 45.
However, φ should not be chosen uniformly. In the prop-
erly formulated translocation problem, the chain has to over-
come the initial free energy barrier, which leads to non-
uniform distribution of φ. In the zero-frequency limit, the
translocation probability follows the Boltzmann distribution,
which depends exponentially on the height of the initial free
energy barrier (cf. Fig. 3). Hence, we put the distribution in
the form p(φ) ∼ exp[αsin(φ)]. Since we consider only the
processes that complete the translocation, α is a non-trivial
function of not only kBT, F, A but also ω. In our procedure,
the parameter α is obtained by fitting the integral?π
= (15ω + 1/2.6)−1, serves as an empirical interpolation be-
tween the Boltzmann distribution for ω ? 1/τ0and the uni-
form distribution of φ for ω ? 1/τ0. This α then gives the
distribution p(φ), over which the average of τ(φ) is taken by
MonteCarlointegrationtoeventuallyfindtheaveragetranslo-
cation time τ. The results of our model with this distribution
of φ are shown in Fig. 6 as a solid curve. The model repro-
duces the essential features of the full N-particle Langevin
dynamics simulation: the cross-over to fast translocation as
ω is decreased, and the global and local maxima of τ near ω
≈π/τ0.Thisexercise clearlyshows thatthelocalmaxima and
minima are a result of deterministic dynamics and the non-
uniform distribution of φ, and are not indications of resonant
activation.
The difference between the sinusoidal and dichotomic
driving forces can also be identified in the translocation time
distributions. In the high and low frequency limits, one re-
covers distributions very similar to the dichotomic case. In
the intermediate frequency regime, on the other hand, the si-
nusoidal time-dependence shows as a periodic modulation of
the underlying distribution. Here, the distribution has multi-
ple peaks, which correspond to translocations occurring when
f(t) > 0 with high probability. Each peak corresponds to one
0p(φ)dφ
to the probability P0(ω) for each ω. The α, obtained as α(ω)
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Page 7

205104-6Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
(a)(b)
(c)(d)
(e)(f)
FIG. 7. The distribution of translocation times for chain length N = 64 and F = 0.3 under sinusoidal driving force in the non-attractive pore. Panel (a) shows
the distribution for A = 0, while panels (b)–(f) show the distribution for A = 0.3.
period T?≡ 2π/ω of the sinusoidal force, with the distance
between consecutive peaks being ?τ/τ0≈ 2π/ωτ0, as shown
in Fig. 7(b). Near ω ≈ π/τ0, where the average translocation
time achieves its maximum, the distribution shows two dis-
tinct peaks, corresponding to fast (τ < τ0) and slow (τ > τ0)
translocation (see Fig 7(d)). The leftmost peak corresponds to
events that occur roughly between T?/4 < τ < T?/2, with a
phase φ between 0 < φ < π/2. For these events, f(t) is positive
forthewholetranslocationprocess,resultinginfasterthanav-
erage translocation. The peak on the right, on the other hand,
corresponds to the events with 3T?/4 < τ < T? and π/2
< φ < π. Here, although f(t) starts positive, it quickly crosses
over to negative values. Typically, translocation occurs when
f(t) turns back to positive. Thus, the average f(t) during one
event is negative, giving longer than average translocation
time. As the frequency ω is decreased, the rightmost peak
becomes smaller as the phases φ corresponding to that peak
become less probable. As a result, the average translocation
time crosses over to the regime where τ < τ0.
C. Attractive pore, dichotomic driving force
We have shown above that for purely repulsive pore-
monomer interactions, the system does not exhibit resonant
activation. This is due to the absence of a proper free-energy
well, in which an attempt frequency of crossing the imminent
barrier is well defined. Introducing attractive interactions be-
tween the polymer and the pore modifies the free energy in
such a way that a well is formed (schematically shown in
Fig. 3), and translocation becomes a thermally activated bar-
rier crossing process.15,22,25Therefore, for the attractive pore,
we expect to find a resonance similar to that reported for
the polymer escape in Ref. 44. We start with the case of di-
chotomic driving force, which is somewhat more pedagogical
than the sinusoidal force case.
1. Dependence on the polymer-pore
interaction strength ?pm
First, we study the effect of the polymer-pore interaction
strength ?pmon the average translocation time τ. For the at-
tractive pore, we use the value 2.5σ for the cut-off distance of
the Lennard-Jones potential, which yields an attractive force
between the pore and the monomer at distances 21/6σ < r
< 2.5σ. The chain is driven by a dichotomically fluctuating
force with the flipping rate ω and correlations described in
0.0001 0.001 0.01 0.11 10100
0.94
0.96
0.98
1
1.02
0
1.04
1.06
( )/
0.5
0.55
0.6
0.65
0.7
P ( ), P ( )
( )/
P ( )
P ( )
0
0
0
0
FIG. 8. The mean translocation time τ and the probabilities P0and Pτ for
the dichotomic force and attractive pore. N = 32, F = 0.5, Ad= 0.2, ?pm
= 1, and τ0≈ 226.8 ± 0.6. The statistical error is smaller than the symbol
size.
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Page 8

205104-7Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
0.01 0.11101001000
0
0.5
1
1.5
2
2.5
3
( )/
0.5
0.6
0.7
0.8
0.9
P ( ), P ( )
( )/
P ( )
P ( )
0
0
0
0
0
FIG. 9. The mean translocation time τ and the probabilities P0and Pτ for
the dichotomic force and attractive pore. N = 32, F = 0.5, Ad= 0.2, ?pm
= 2.5, and τ0≈ 2202 ± 29. The statistical error is smaller than the symbol
size.
Sec. II. In Figs. 8 and 9, we show the average translocation
times τ(ω) for the chain length N = 32, with the polymer-
pore interaction strength ?pm= 1 and ?pm= 2.5, respectively.
Here, the static force F = 0.5 and the amplitude of dichotomic
force is Ad= 0.2. In the high flipping rate regime, ω ? 1/τ0,
τ(ω) ≈ τ0, as for the non-attractive pore. On the other hand,
for ω ? 1/τ0, the translocation time is τ > τ0. This behavior
is completely opposite to the non-attractive pore case. Nev-
ertheless, it can be explained by the same arguments. The
average translocation time is given by Eq. (2). However, for
sufficiently large ?pmthe selectivity with respect to the initial
driving force f(0) is fairly weak, because a strong attraction
between the pore and the polymer prevents the escape to the
cis side. For example, for ?pm= 1.0, P0≈ 0.63, as shown
in Fig. 8. Assuming inverse dependence of the translocation
time on the driving force, Eq. (2) gives τ ≈ 1.06τ0, which is
in agreement with the simulation results.
In the intermediate regime (ω ≈ 1/τ0), the translocation
time τ(ω) shows different behavior depending on the value
of ?pm. While for ?pm= 1, τ(ω) monotonically decreases as
ω increases, for ?pm= 2.5, τ(ω) has a minimum at an op-
timal flipping rate ωτ0≈ 1.8. Related to this, we obtain the
probabilities P0(ω) and Pτ(ω). For ?pm= 1, P0monotonically
increases as ω decreases, similar to the non-attractive case,
but only by approximately 0.1. For ?pm= 2.5, P0≈ 0.52,
almost independent of ω. On the other hand, Pτshows non-
monotonic behavior, having a maximum at ωτ0≈ 0.4 and
ωτ0≈ 1.0 for ?pm= 1 and 2.5, respectively. Typically, for
a barrier crossing problem, such a maximum is an indication
of resonant activation, and is accompanied by a minimum in
the crossing time.40,47However, out of the two cases, ?pm= 1
and ?pm= 2.5, only in the latter has a minimum in τ(ω). Fur-
thermore, for ?pm= 2.5, the flipping rates ω at the maximum
of Pτ(ω) and at the minimum of τ(ω) do not coincide. To
understand these results, we divide the translocation process
into three components:15,22,25(1) initial filling of the pore, (2)
transfer of the polymer from cis to trans side, and (3) empty-
ing of the pore, as shown in Fig. 10. The translocation time is
then τ = τ1+ τ2+ τ3, where τiis the time for the ith pro-
cess. In Fig. 11 we show τ1, 2(≡ τ1+ τ2) and τ3for ?pm
FIG. 10. The translocation process divided into three stages: (1) initial fill-
ing of the pore, (2) transfer of the polymer from the cis side to the trans side,
and (3) the final emptying of the pore. The corresponding times of the sub-
processes are τ1, τ2, and τ3, with the total translocation time τ = τ1+ τ2
+ τ3.
= 1 and ?pm = 2.5. For the larger ?pm, τ3 dominates the
translocation time. As ω increases, τ1, 2(ω) decreases grad-
ually, but τ3(ω) behaves non-monotonically. In addition, the
minimum of τ3coincides with the maximum of Pτ(for ?pm
= 1 this is barely observable). This indicates that Pτ(ω) and
τ3(ω) are highly correlated. Thus, the non-monotonic behav-
ior of τ(ω) occurs because the time-dependent force couples
to the pore emptying process, i.e., the crossing of the final
free-energy barrier (cf. Fig. 3). The coupling to the first two
processes is very weak, and does not significantly contribute
to the resonant activation. However, since τ1, 2slightly de-
creases as ω increases, the optimal flipping rate that yields
the minimum of translocation time τ is somewhat larger than
the rate at the minimum of τ3.
0.1110 100 1000
0
0.2
0.4
0.6
0.8
1
1.2
1.4
( )/ , ( )/
/ ( =1)
/ ( =1)
/ ( =2.5)
/ ( =2.5)
pm
30
0
pm
pm
pm
1,2
1,2
3
0
0
0
0
1,2
3
0
FIG. 11. The times τ1, 2 (dashed lines) and τ3 (solid lines) for ?pm = 1
(squares) and ?pm= 2.5 (circles). Here N = 32, F = 0.5, Ad= 0.2 for both
cases. While τ1, 2monotonically decreases as ω increases, τ3shows a reso-
nant minimum. The statistical error is smaller than the symbol size.
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Page 9

205104-8Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
(a)(b)
(c)(d)
(e)(f)
FIG. 12. The distribution of translocation times for the dichotomic force and attractive pore. N = 32, F = 0.5, and Ad= 0.2. The left column shows the
distributions for ?pm= 1 and the right column for ?pm= 2.5.
The translocation time distribution P(τ) also profoundly
dependsonthemagnitudeof?pm,asshowninFig.12.Theleft
column shows the case with ?pm= 1, while the right column
corresponds to ?pm= 2.5. The first row shows P(τ) in the
presence of static force F only (corresponding to very high
flipping rate ω). While for ?pm= 1, P(τ) is nearly Gaussian
centered at τ0, for ?pm= 2.5 the distribution is an exponen-
tial.This indicates thatstrongpolymer-pore interactions make
translocation an activated process, where the chain has to sur-
mount the final free-energy barrier before it can completely
translocate to the trans side (cf. Fig. 3). For ?pm= 1, as ω de-
creases, P(τ) gradually splits into two Gaussian distributions,
centered at
P(τ) is changed in a non-trivial way: at intermediate flipping
rate ω ≈ 1/τ0, the tail of P(τ) is shortened, but for lower ω,
P(τ) develops a long tail. The behavior of P(τ) at interme-
diate ω is closely related to the probability Pτ(ω) in Fig. 9.
Although f(0) is either positive or negative with similar prob-
ability, most of the successful translocations finish with f(τ)
= +Ad, which results in a shorter translocation time. This
is the reason for the small probability of long translocation
times. On the other hand, at very low ω, P(τ) becomes a com-
bination of two exponential distributions, each corresponding
to the translocation time with the driving force either F + Ad
or F − Ad, which results in sharp increase of τ(ω).
F
F+Adτ0and
F
F−Aτ0. For ?pm= 2.5, as ω decreases,
2. Dependence on the chain length N
In Sec. III C 1, we found that for large ?pm, the transloca-
tion time τ has a resonant minimum as a function of the flip-
ping rate ω. Here we study how this behavior changes with
chain length N. Figure 13 shows the translocation times τ(ω)
for 16 ≤ N ≤ 128, with ?pm= 2.5, F = 0.5, and Ad= 0.2. The
optimal flipping rate that yields the minimum of translocation
time is roughly independent of N (see the inset of Fig. 13).
Since the resonant behavior occurs during the last emptying
process, this indicates that τ3is independent or only weakly
depends onthechainlength.Thefreeenergybarrierofthelast
emptying process can be approximated as ?F = L(?pm− F/2
− g(N)).15,22,25Here the first term accounts for the polymer-
pore interactions, the second term for the potential energy dif-
ference across the membrane due to the driving force, and
0.11101001000
0.6
0.8
1
1.2
1.4
1.6
1.8
( )/
N=16
N=32
N=64
N=128
0
0.00010.0010.01
0.5
0.6
0.7
0.8
0.9
1.0
1.1
( )/0
0
FIG. 13. Translocation times for chain lengths 16 ≤ N ≤ 128 with the di-
chotomic force and attractive pore. F = 0.5, Ad= 0.2, and ?pm= 2.5. While
the optimal rescaled flipping rate (ωτ0) shows a slight dependence on N
(main figure), the unnormalized flipping rate (ω) is independent of N (inset).
The statistical error is smaller than the symbol size.
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Page 10

205104-9Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
the last term is due to the entropic free energy.12,13,15For the
pore-emptying process the entropic force g(N) is in the posi-
tive direction, slowly increasing with N and eventually satu-
rating for very long chains.25On the other hand, as shown in
Ref. 25 for the static driving force and N ≤ 200, τ1, 2approxi-
mately increases as τ1, 2∼ N1.5. The normalized translocation
time is
τ(ω)
=τ1,2(ω) + τ3(ω)
τ1,2(0) + τ3(0)
where τi(0) is time for the ith process in the absence of the
time-dependent driving force f(t). In the short chain limit,
τ1, 2,(ω) ? τ3(ω), so that the normalized translocation time
is
τ0
τ1, 2(ω) ? τ3(ω), giving
tions predict that for short chains, one should observe a strong
minimum inτ(ω),whereas forvery longchains,theminimum
should vanish. This trend can be observed in Fig. 13, where
the minimum of translocation time becomes less pronounced
as N increases. In addition, the optimal flipping rates are quite
independent of N. This is in contrast to the results of Ref. 39,
where the authors consider the translocation of a rigid rod in
the presence of a reflecting boundary condition at s = 0 (see
Fig. 3). In that case, all the segments of the polymer are sub-
ject to the external forces, making the translocation time very
sensitive to their minute changes and the optimal flipping rate
decreases with N. However, in the present case the number
of segments within the attractive pore remains small through-
out the translocation process. Thus, the effect of the external
forces becomes small as the chain gets longer.
τ0
=τ1,2(ω)/τ3(0) + τ3(ω)/τ3(0)
τ1,2(0)/τ3(0) + 1
,
(6)
τ(ω)
≈τ3(ω)
τ3(0). On the other hand, in the long chain limit,
τ(ω)
τ0
≈
τ1,2(ω)
τ1,2(0). These limiting situa-
3. Dependence on the driving force F
Next, we study the effect of changing the driving force
magnitude. We consider static driving forces between 0.5 ≤ F
≤ 4 with the amplitude Adfixed as Ad= 0.4F. The results are
shown in Fig. 14. One can see that the optimal flipping rate for
0.01 0.11 10100 1000
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
( )/
F=0.5
F=1.0
F=2.0
F=4.0
0
0
0.00001 0.00010.0010.010.1
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
( )/0
FIG. 14. Translocation times for driving forces 0.5 ≤ F ≤ 4 for the di-
chotomic force and attractive pore. Ad= 0.4F, ?pm= 2.5. In this case, the op-
timal rescaled flipping rate (ωτ0) is roughly independent of F (main figure),
while the unnormalized flipping rate (ω) strongly increases with F (inset).
The statistical error is smaller than the symbol size.
the resonant minimum translocation time increases with in-
creasing F. In addition, the resonant minimum becomes shal-
lower, finally disappearing for large F and τ(ω) becomes a
monotonic function of ω. It is because the free energy bar-
rier of the last emptying process vanishes for large F. It is
of interest to study the critical driving force Fc, for which
τ(ω) changes from non-monotonic to monotonic. Fccan be
approximated from the condition ?F = 0, so that Fc= 2(?pm
− g(N)). For N = 32 and F = 0.5, τ(ω) becomes monotonic
for ?pm? 1 (cf. Fig. 8), giving the estimate g(N) ≈ 1. There-
fore, Fc≈ 3 for ?pm= 2.5 and N = 32. This estimate seems to
be reasonable as shown in Fig. 14. This result also shows that
the condition of the non-monotonic behavior of the translo-
cation time is determined by the competition of the polymer-
pore interaction ?pmand the driving force F.
4. Dependence on the driving force amplitude Ad
As the last case of the dichotomic force, we study the
effect of changing the dichotomic force amplitude Adwhile
keeping the static driving force F constant. The results for
the translocation time are shown in Fig. 15. With increasing
Ad, the resonant minimum becomes deeper and the resonance
flippingrateω graduallyincreases.ForverylargeAd,thereso-
nance disappears and the translocation time becomes a mono-
tonic function of ω, similar to the non-attractive pore case.
This shift in behavior is because the selectivity of initial sign
of f(0) becomes stronger for larger Ad. The transition to this
regime happens when the initial barrier (see Fig. 3) that pre-
vents the chain escape to the cis side becomes comparable
to the thermal energy and the escapes become frequent. For
the negative dichotomic force, f(0) = −Ad, the barrier can be
written in the form ?Fcis= ?pm+ (F − Ad)/2 − g(N). The
pore length L does not enter the relation because in the ini-
tial configuration, only the first bead is inside the pore. For
?pm= 2.5 and kBT = 1.2, the requirement ?Fcis≈ kBT gives
the estimate Ad≈ 1 for the transition from the non-monotonic
τ(ω) to the monotonic one. This estimate matches the data
in Fig. 15.
0.1110100100010000
0
0.5
1
1.5
( )/
A=0.2
A=0.4
A=0.8
A=1.2
A=1.6
0
0
FIG. 15. Translocation times for the dichotomic force and attractive pore for
amplitudes Ad∈ {0.2, 0.4, 0.8, 1.6}. F = 0.5, ?pm= 2.5. The statistical error
is smaller than the symbol size.
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Page 11

205104-10Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
D. Attractive pore, periodic driving force
For the attractive pore, the dichotomic and periodic driv-
ing forces give qualitatively very similar results. Also for the
periodic driving force, f(t) = Asin(ωt + φ), a resonant min-
imum of the average translocation time appears, in contrast
to the non-attractive pore case. In addition, the location (fre-
quency ω) and the depth of the minimum depend on the var-
ious parameters in essentially the same way as described in
Sec. III C, which indicates that the origin of the resonance is
the same: the time-dependent force being most co-operative
to translocation during the pore-emptying time τ3. However,
there are also some obvious differences. For the periodic driv-
ing force, oscillatory behavior similar to the one described in
Sec. III B emerges, in addition to the resonant activation. In
this section, we will briefly describe the essential differences
between the two driving schemes and examine some of the
implications of employing the periodic driving force.
1. Dependence on the polymer-pore
interaction strength ?pm
First, it is instructive to consider the dependence of the
translocation time τ(ω) on the strength of the polymer-pore
interaction strength ?pm. As shown in Fig. 16, for low inter-
action strengths, one recovers the transition from fast to slow
translocation with local minima and maxima in τ(ω), charac-
teristic of the non-attractive pore case. For larger ?pm, a res-
onant minimum in τ(ω) develops, similar to the dichotomic
driving force. However, the global maximum observed for the
non-attractive pore persists, although it is reduced to a local
maximum located within the resonance minimum. This local
maximum arises because of the interplay of the periodic forc-
ing and the non-uniform distribution of the phase φ, as we
will discuss below.
2. Dependence on the driving force amplitude A
To highlight the differences between the sinusoidal and
dichotomic driving forces, we look at how the translocation
1 10100 1000
0.6
0.7
0.8
0.9
1
1.1
1.2
( )/
=0.1
=0.5
=1.5
=2.0
=2.5
pm
0
pm
pm
pm
pm
0
10100
0.97
0.98
0.99
1.00
1.01
( )/
0
0
FIG. 16. Translocation time τ as a function of frequency ω for the periodic
driving force f(t) = Asin(ωt + φ) for 0.1 ≤ ?pm≤ 2.5. F = 0.5, A = 0.3, and
N = 32. The inset shows a magnification of the data for 0.1 ≤ ?pm≤ 1.5. The
statistical error is smaller than the symbol size.
0.01 0.11101001000
0
0.5
1
1.5
2
( )/0
Increasing A
0.511.522.53
A
0
2
4
6
8
10

min 0
0
FIG. 17. Translocation time τ as a function of frequency ω for the periodic
driving force f(t) = Asin(ωt + φ) for amplitudes A ∈ {0.3, 0.6, 0.8, 1.0, 1.2,
1.5, 1.8, 2.4, 3.0}. Other parameters are F = 0.5, ?pm= 2.0, and N = 32. The
inset shows the dependence of the frequency ωminof the global minimum
translocation time on the amplitude A. Here τ0≈ 500 ± 6. The statistical
error is smaller than the symbol size.
time τ(ω) changes with the driving force amplitude A. For
the dichotomic force, as Adis increased, one merely crosses
from the non-monotonic τ(ω) with the resonant minimum to
the monotonic τ(ω) characteristic of the non-attractive pore
case. The sinusoidal driving force, on the other hand, exhibits
much richer behavior. As shown in Fig. 17, as the ampli-
tude A is increased, the minimum becomes deeper and slowly
moves toward higher frequencies. In addition to the original
one, another (local) minimum appears at the low-frequency
end of the spectrum and travels down the τ(ω) curve as A
is increased. Eventually, the new minimum becomes a global
one. This produces a sudden transition in the frequency of
minimum translocation time, ωmin, as shown in the inset of
Fig. 17. Finally, at sufficiently large A, the new minimum
merges with the original one. To better understand this com-
plex behavior, we again divide the translocation time τ to the
three components τ1, τ2, and τ3. Looking at τ1, 2≡ τ1+ τ2
and τ3separately reveals that the original global minimum of
τ(ω)isassociatedwiththeporeemptyingtimeτ3,asshownin
Fig. 18. The additional minimum, on the other hand, is related
to the periodic back-and-forth movement of the chain, which
is visible as a non-monotonic behavior in τ1, 2(ω).
Let us first examine the pore emptying time τ3, since that
shares many similarities with the dichotomic force case. In
both cases, the minimum of τ3(ω) occurs due to resonant ac-
tivation. At the corresponding resonant frequency, the prob-
ability Pτ also reaches a maximum (not shown), similar to
the dichotomic force case. As shown in Fig. 18, for the sinu-
soidal force, there is also a small local maximum in τ3(ω) at
ωτ0≈ 15. Surprisingly, here Pτ also has a local maximum,
which should indicate efficient crossing of the final barrier.
However, instead of the expected decrease in τ3, one sees a
slight increase. The reason is that there is a special mismatch
between the frequency ω and the translocation time τ1, 2so
that the period T?≡ 2π/ω ≈ τ1,2. In other words, typically it
takes the chain one period of f(t) just to traverse from its ini-
tial position to the configuration where it may try to surmount
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Page 12

205104-11Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
0.010.11 10100 1000
0.5
1
1.5
2
( )/
/
/
/
3
0
0
0
0
1,2
0
FIG. 18. The translocation time τ and its components τ1, 2and τ3as a func-
tion of ω, showing that the leftmost minimum in τ(ω) is associated with τ1, 2.
Parameter values used are F = 0.5, A = 1.0, ?pm= 2.0, and N = 32. The sta-
tistical error is smaller than the symbol size.
the final free-energy barrier (cf. Fig. 3). Thus, the chain es-
sentially misses the first opportunity to cross the final bar-
rier, which slightly increases τ3. This can be also seen as
a suppressed first peak in the translocation time distribution
of Fig. 19(b).
Finally, let us look at the translocation times τ1and τ2.
The combined time τ1, 2shows features similar to the non-
attractive pore case, where the periodic driving force pro-
duces a series of alternating minima and maxima. In the case
of the attractive pore, the selection over the initial phase is
weaker because of the free-energy barrier that prevents the
escape to the cis side. Consequently, the qualitative behav-
ior of the τ1, 2(ω) curve is closer to the model of Eq. (4)
with uniformly distributed phase (see Fig. 6). Essentially,
the local maximum in τ1, 2is produced by the interplay of
ω-dependence of the distribution of φ and the periodicity of
the driving force. Close to the resonant minimum, ωτ0 ≈
5, the distribution is bimodal, as shown in Fig. 19. Similar
to the non-attractive pore case, the first peak corresponds to
the events that occur within the first half-period of the force
f(t), i.e., 0 < τ < T?/2, and whose initial phase is typically
−π/4 < φ < π/2. Therefore, for these trajectories, f(t) > 0
for most of the process, and translocation occurs faster than
average. In contrast, the second peak corresponds to π/2 < φ
< 7π/4 and T?/2 < τ < T?. As the frequency ω decreases,
the second peak moves further to the right (Figs. 19(d) and
19(e)). This increases the average translocation time. At the
same time, the area under the peak decreases, because with
decreasing ω, the distribution of φ becomes less uniform, fa-
voring φ belonging to the first peak. This tends to decrease τ.
The combination of these two factors creates the maximum
of τ1, 2. For larger A, the selection over φ is stronger, so the
second factor starts to dominate already at relatively high fre-
quencies. Conversely, for small A, the second peak in P(τ)
persists for even very small ω. This explains why the maxi-
mum occurs at lower frequencies for small A, and moves to-
wards the high-frequency end as A is increased.
IV. SUMMARY
In this work, we have studied the translocation of poly-
mers under a time-dependent driving force using Langevin
dynamics simulations. In particular, we have extracted the de-
pendence of the average translocation time on the flipping
rate ω of the dichotomic driving force and the correspond-
ing dependence on the angular frequency ω for the sinusoidal
driving force. We have also examined the influence of various
other physical parameters on the translocation dynamics.
(a)(b)
(e)(f)
(c)(d)
FIG. 19. Distribution of translocation times for the periodic driving force with N = 32, ?pm= 2.0, F = 0.5, A = 0.0 (panel (a)), A = 1.0 (panels (b)–(f)).
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Page 13

205104-12Ikonen et al.J. Chem. Phys. 136, 205104 (2012)
We find that the interactions between the polymer and
the pore play a fundamental role in the dynamics of the sys-
tem. For the non-attractive interactions, the translocation time
shows a cross-over to a faster translocation regime at low flip-
ping rates. For the sinusoidal force, in addition to the cross-
over, we observe a series of local minima and maxima, pro-
duced by the periodicity of the driving signal. However, in
this case we do not observe a global minimum of the translo-
cation time for any finite ω. On the other hand, with attrac-
tive polymer-pore interactions, which represent naturally oc-
curring biological pores such as the α-hemolysin, the sit-
uation is very different. In this case, the translocation be-
comes a thermally activated process due to the attraction be-
tween the pore and the polymer. Optimal modulation of time-
dependence driving force induces a resonant activation, man-
ifesting as a global minimum in the translocation time at fi-
nite ω. We also find that, although the details of this reso-
nance depend on various system parameters, in general the
resonance is quite robust and occurs for both the dichotomic
and sinusoidal driving force. Typically the resonant flipping
rate (angular frequency) is found in the neighborhood of
ω ≈ 1/τ0, with τ0being the translocation time without the
time-dependent component of the driving force. For an ex-
perimentally typical translocation time of the order of 100
μs,5this corresponds to the rate (frequency) in the kilohertz
regime.
Theoretically, the occurrence of the resonance relies on
the existence of a free energy well, from which the polymer
escapes via thermal activation. In practice, to observe the res-
onant behavior, one should choose the physical parameters so
that one has: (1) strong enough polymer-pore interactions, (2)
relatively short chain length, and (3) small enough static driv-
ing force. For example, a poly(dA)100chain driven through an
α-hemolysin pore should display the resonance for pore volt-
ages of roughly V ? 1000 mV. For V ≈ 200 mV, we would
expect to find the resonance in the neighborhood of ω ≈ 1–
10 kHz at room temperature. Furthermore, in the case of the
dichotomic driving force, one should also have a relatively
small amplitude of the time-dependent force, whereas for the
sinusoidal force, even significantly larger amplitudes can still
produce the resonance. In the latter case, a more complicated
behavior emerges, as the driving force not only assists translo-
cationduringthepore-emptyingtimeτ3,butalsosignificantly
alters to the overall motion of the chain. Our findings suggest
that time-dependent driving forces may play a fundamental
part in polymer translocation in biological systems, and may
also be useful in practical applications such as sorting and se-
quencing of DNA molecules.
ACKNOWLEDGMENTS
This work has been supported in part by the Academy of
Finland through its COMP Center of Excellence and Trans-
poly Consortium grant, and through the General Individual
Research Program as well as BK 21 Program administered
by the Korea Research Foundation. T.I. acknowledges the fi-
nancial support of the Finnish Doctoral Programme in Com-
putational Sciences (FICS) and the Finnish Foundation for
Technology Promotion (TES). The authors also wish to thank
CSC, the Finnish IT Center for Science, for allocation of com-
puter resources.
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