Theory of polymer translocation through a flickering nanopore under an alternating driving force
We develop a theory for polymer translocation driven by a time-dependent force through an oscillating nanopore. To this end, we extend the iso-flux tension propagation theory (IFTP) [Sarabadani et al., J. Chem. Phys., 2014, 141, 214907] for such a setup. We assume that the external driving force in the pore has a component oscillating in time, and the flickering pore is similarly described by an oscillating term in the pore friction. In addition to numerically solving the model, we derive analytical approximations that are in good agreement with the numerical simulations. Our results show that by controlling either the force or pore oscillations, the translocation process can be either sped up or slowed down depending on the frequency of the oscillations and the characteristic time scale of the process. We also show that while in the low and high frequency limits the translocation time follows the established scaling relation with respect to chain length , in the intermediate frequency regime small periodic fluctuations can have drastic effects on the dynamical scaling. The results can be easily generalized for non-periodic oscillations and elucidate the role of time dependent forces and pore oscillations in driven polymer translocation.
Polymer translocation through a nanopore, which has rapidly emerged as one of the most active research areas in biological and soft matter physics, plays an important role in many biological processes such as RNA transport through a nuclear membrane pore and -hemolysin channels salman_pnas_2001 (); kasi1996 (), virus injection into cells and transportation of protein through membrane channels albert_book (). It also has many technological applications such as rapid DNA sequencing kasi1996 (); meller2003 (); mellerPRL2001 (); tapioPRL2008DNAsequencing (); aksimentievNanolett2008 (); golestanianPRX2012 (), drug delivery meller2003 () and gene therapy, and has been motivation for many theoretical and experimental studies sung1996 (); muthu1999 (); meller2003 (); chuang2001 (); metzler2003 (); kantor2004 (); storm2005 (); grosberg2006 (); dubbeldam2007 (); sakaue2007 (); sakaue2008 (); luo2008 (); branton2008 (); luo2009 (); schadt2010 (); sakaue2010 (); Milchev_JPCM (); rowghanian2011 (); Muthukumar_book (); ikonen2012b (); ikonen2013 (); Tapio_review (); bhatta2009 (); lehtola2009 (); bhatta2010 (); lehtola2010 (); dubbeldam2012 (); saito2012 (); saito2012a (); saito2012b (); ikonen2012a (). So far most of the theoretical studies have been focused on the case of pore-driven translocation of a polymer chain through a nanopore with a constant radius and by a constant driving force, which is inherently a far-from-equilibrium phenomenon dubbeldam2012 (); saito2012 (); saito2012a (); saito2012b (); ikonen2012a (); ikonen2013 (); bhatta2009 (); bhatta2010 (); lehtola2009 (); lehtola2010 (). However, some theoretical and experimental work has also been done where the width of the pore changes during the translocation process. For example, the active translocation of a polymer chain through a flickering pore has been investigated by means of molecular dynamics simulations to show that translocation through the pore with an alternating width and sticky walls exhibits more efficient translocation as compared to the static pore golestanianPRL2011 (). There are some biological examples of this such as the nuclear pore complex, which plays an important role in nucleocytoplasmic transport in eukaryotes yamadaFlickeringPore (). Another example is the exchange of molecules between mitochondria and the rest of cell which is controlled by the twin-pore protein translocase (TIM22 complex) in the inner membrane of the mitocondria RehlingFlickeringPore (). On the experimental side, it has been shown that the translocation of DNA through a nanochannel can be modulated by dynamically changing the cross section of an elastomeric nanochannel device by applying mechanical stress HuhNature2007 (); AngeliLabonChip2011 (); fanzioSciRep2012 (). Moreover, the width of the nanopore can be tuned by using thermally driven nanoactuation of the polyNIPAM brushes inside the nanopore YameenSmall2009 ().
The external driving force can also be time-dependent during the translocation process aksimentievNanolett2008 (); ikonen2012c (); golestanianJCP2012 (); fiasconaroPRE2015 (); StefureacOscillatingForce (); BatesBioPhysJ2003 (). As an example, polymer translocation has been studied for a time-dependent alternating driving force by means of Langevin dynamics simulations. It has been shown that when the polymer-pore interaction is attractive, resonant activation occurs at an optimal frequency of the alternating driving force ikonen2012c (). There are some biological applications for the alternating driving force case such as translocation of linear and -helical peptides through an -hemolysin pore in the presence of an AC field StefureacOscillatingForce (), and monitoring the escape of the DNA from an -hemolysin pore by using an alternating current signal lathropJACS2010OscillatingForce (). Moreover, using an alternating electric field in the nanopore has been suggested as a method for DNA sequencing aksimentievNanolett2008 ().
On the theoretical side, over the last few years a consistent theory of driven translocation has been developed based on the idea of tension propagation (TP) sakaue2007 (). Within the TP theory, the dynamics of driven translocation can be described by considering a tension force that propagates along the backbone of the chain. The dynamics is dominated by the drag of the cis side of the polymer chain, leading to asymptotic scaling of the average translocation time as a function of the chain length (number of monomers) ikonen2012a (); ikonen2013 (); ikonen2012b (); jalal2014 (), where the Flory exponent is exactly in two and approximately in three dimensions, respectively Edwardsbook (). In addition, the theory reveals that the pore friction plays an important role for chains of finite length, leading to a large correction-to-scaling term such that the translocation time actually scales as , where and are constants ikonen2013 (); jalal2014 (). This scaling form has most recently been theoretically derived using the iso-flux Brownian dynamics tension propagation (IFTP) theory of driven polymer translocation jalal2014 (); rowghanian2011 ().
In this paper, our main goal is to investigate the influence of an alternating external driving force and a flickering pore on the dynamics of the driven translocation process. To this end we employ the IFTP theory to theoretically study polymer translocation in the presence of a time dependent driving force through a flickering nanopore. Here, the time dependent driving force, , is incorporated into the theory directly, while the effect of the flickering pore is implemented by a time dependent pore friction coefficient . Using the IFTP theory modified in this manner allows us to explicitly derive the TP equations for the present setup. The formalism also allows us to investigate both the influence of thermal fluctuations from the solvent by introducing randomness into the effective driving force ikonen2012a (); ikonen2012b (); jalal2014 () and the influence of the distribution of the initial configurations of the chain on the cis side jalal2014 ().
The outline of the paper is as follows: In Sec. II we show how to model pore-driven polymer translocation in the context of the iso-flux Brownian dynamics tension propagation formalism when the pore friction is time dependent and the driving force is alternating. Secs. III.1 and III.2 present the finite-size scaling form for the translocation time in the presence of an alternating driving force and time dependent pore friction, respectively. Secs. IV and V are devoted to the results on the average translocation time, the waiting time distribution, and time evolution of the translocation coordinate, respectively. Conclusions and discussion are in Sec. VI.
For the sake of brevity, we use dimensionless units denoted by tilde as , with the units of length , time , velocity , force , monomer flux , and friction , where is the segment length, is the solvent friction per monomer, is the Boltzmann constant, and is the temperature of the system.
To elucidate the parameter values chosen in this paper, we compare them with those in real physical systems and also with those used in molecular dynamics (MD) simulations in reduced Lennard-Jones (LJ) units ikonen2012c (); jalal2014 (). Here, parameters without tilde such as external driving force, , pore friction, , and time, , are in LJ units. In MD simulations the scale of time, mass, and energy can be fixed by the LJ parameters , and which are the mass and diameter of each bead, and the strength of the interaction, respectively. We have chosen which is dimensionless and the time is scaled by . In our model the interaction strength is at room temperature (), the mass of each bead is about 936 amu, and the size of each bead corresponds to the single-stranded DNA Kuhn length, which is . Therefore, the LJ time scale is 32.1 ps. Here the thickness of the pore is set to . By assuming the effective charge of for each unit charge, three unit charges per bead brantonPRL2003 (); mellerbiophysj2004 (), and with a force scale of , an external driving force of corresponds to a voltage of across the pore ikonen2012c ().
We use Brownian dynamics (BD) in the overdamped limit as the basic theoretical framework ikonen2012a (); ikonen2012b (). The time evolution of the translocation coordinate which is the chain length on the trans side is governed by the BD equation that can be written as
where is the effective friction, the first term in the right hand side is the entropic force and is the surface exponent ( for self-avoiding chains in two and three dimensions, respectively, and for ideal chains), is the chain length or the total number of beads in the chain (here the contour length of the polymer chain is ), is the external driving force, is the total force, and the Gaussian white noise, , satisfies and .
As both the cis side subchain and the pore friction contribute to the effective friction , it can be written as . For a fully flexible self-avoiding polymer chain, the dynamical contribution of the trans side to the dynamics has been shown to be insignificant ikonen2012a (); ikonen2012b (); ikonen2013 (); dubbeldam2014 (); suhonen2014 () and it can be absorbed into the pore friction . Note that here the pore friction is assumed to be a function of time. We solve the dynamics of the cis side by using the TP equations. Following the arguments of Rowghanian et al. rowghanian2011 () we are able to derive the TP equations analytically. We assume that the monomer flux, , is constant in space on the mobile domain of the cis side and through the pore, but evolves in time. The tension front which is the boundary between the immobile and mobile domains, is located at the distance from the pore. The external driving force inside the mobile domain is mediated by the backbone of the chain from the pore located at to the last mobile monomer which is located at the tension front. Using the force-balance relation for the differential element located between and , the magnitude of the tension force at the distance from the pore can be obtained. One can integrate this force-balance relation over the distance from the entrance of the pore to , to obtain the tension force at as , where is the force at the entrance of the pore (see Appendix A of Ref. 52 for technical details). As the drag of all the preceding monomers diminishes, the mediated force closer to the the tension front becomes smaller and it vanishes at the tension front.
According to the blob theory, the shape of the mobile part of the chain can be classified into three different regimes. For a moderate external driving force, i.e. , the monomer number density close to the pore is greater than unity, and the chain shape resembles a trumpet-like shape with the narrow end closer to the pore. This regime is called trumpet (TR) which is depicted in Figs. 1a and b. For a stronger external driving force, , the force can only straighten a small part of the chain which is called the stem, and the part following it is called the flower. This regime is called stem-flower (SF) (see Fig. 1c). Finally, for a very strong driving force, i.e. , the force is strong enough to straighten the whole moving part of the chain, which is called the strong-stretching (SS) regime (see Fig. 1d). In the TR and SF regimes, the location of the tension front is at the farthest blob from the pore, while in the latter one, i.e. SS, the tension front is located at the farthest mobile bead of the chain from the pore as can be seen in Fig. 1.
The monomer flux can be written as a function of the total force and linear size of the mobile part by integrating the force balance equation over the whole mobile part:
The effective friction can be obtained as
Equations (1), (2) and (3) together with the knowledge of determine the full solution for . Therefore, the only equation remaining is the equation of motion for that must be derived separately for the propagation and post propagation stages. In the propagation stage, as shown in Fig. 1a, c and d, the tension has not reached the last monomer of the chain, i.e. , where is the number of total beads that have been already influenced by the tension force and is the number of dragged monomers in the cis side. Using the scaling relation of the end-to-end distance of the self-avoiding chain, , where is a constant prefactor and is the last monomer inside the mobile domain at the tension front, together with the blob theory and (see Appendix B of Ref. 52 for details) the equation of motion for the tension front in the propagation stage can be derived as
for the SS regime, and
for the TR and SF regimes. Here the subscript ”a” in and stands for the trumpet regime as and correspond to positive and negative values of , respectively. For the stem-flower regime ”a” denotes SF. The time derivative of is . and are given by
In the post propagation stage which is schematically shown in Fig. 1b for the TR regime, all monomers on the cis side have already been affected by the tension force. Therefore, the correct closure relation for the post propagation stage is . Therefore, one can derive the equation of motion for the tension front as
for the SS regime, while for the TR and SF regimes it can be written as
For more details, the interested reader is referred to the Appendix B of Ref. 52.
Iii Scaling of translocation time for SS regime
iii.1 Static pore and oscillating force
To obtain analytical results, we assume that only the external driving force has contribution to the total force in the BD equation (1), i.e. we ignore both the entropic and thermal noise terms here. We choose the external driving force as a combination of a constant force and an oscillatory term as
where , and , and are the amplitude, frequency and initial phase of the force, respectively. For simplicity, we first consider a static pore with a constant pore friction, i.e. . Then Eq. (2) reduces to . In the strong-stretching regime considered here, the number of mobile monomers on the cis side is given by . Knowing the external driving force as a function of time and pore friction together with the conservation of mass, , the propagation time can be solved by integration of from to . The result is
In the post propagation stage, one sets the condition and integrates from to to obtain the post-propagation time, , as
Finally, the total translocation time is then given by
is the translocation time in the absence of the oscillatory term in the external driving force. Here we have used the same value of the amplitude as in Ref. 38.
In Fig. 2 we plot the translocation time from Eq. (13) normalized with Eq. (14), as a function of the normalized frequency for various values of in the SS regime ( and ). For comparison, we also show results from the full deterministic IFTP model (green diamonds) where the entropic force term has been included. It can be seen that the agreement with the analytic approximation in the SS regime is very good for a wide range of frequencies.
Using the explicit form of we can now examine Eq. (13) in different limits. First, in the high frequency limit, i.e. , the integral simply vanishes and the total translocation time in the presence of the oscillatory term is the same as the translocation time in the absence of it, i.e. , as can also be seen in the high-frequency limit of Fig. 2.
In the opposite limit of a very small frequency, , the integral gives
The expansion of to first order yields and thus
As can be seen from Eq. (13), the scaling form for the average translocation time is the same for both static and high frequency limits. In the low frequency limit, the scaling with respect to the chain length is also the same, but the scaling form with respect to the external driving force becomes .
Finally, if the amplitude of the oscillating term, , is much smaller than the static force, , then we can use a small amplitude approximation, where the value of the upper limit in the integral over time in Eq. (13) can be replaced by . The translocation time can then be written as
This equation predicts an oscillatory dependence of the translocation time on the driving frequency. In Fig. 2 we plot the small amplitude expansion result for the case where the amplitude ratio (magenta dash-dash-dotted line). For this relatively small ratio, the agreement with the full solution of the model (orange curve) is good for a wide range of frequencies.
iii.2 Flickering pore and oscillating force
We consider next the most general case of an oscillating force and a flickering pore in the SS regime. We again neglect both the entropic force and thermal noise, and take the oscillatory force to be the same as in the previous section. The flickering pore has the following time dependent pore friction:
where and , and are the amplitude, frequency and initial phase of the pore friction, respectively. Then, Eq. (2) reduces to . As in Sec. III.1, we solve the propagation time by integration of from to , and the post-propagation time by integration of from to . The total translocation time is then given by
Using the explicit form of , at the high frequency limit, , and for very long chains the integral vanishes and the whole translocation time in the presence of the oscillatory force is the same as the translocation time with a constant pore friction . We note, however, that for short chains the integral can be non-zero due to the interaction of the polymer beads with the pore golestanianPRL2011 ().
In the opposite limit of a low frequency the integral gives
As we can see in Eq. (LABEL:scaling_trans_time_SS_low_frequency_flickering), at the low frequency limit the pore behaves like a static pore with pore friction of .
Equations (19) and (LABEL:scaling_trans_time_SS_low_frequency_flickering) show that the scaling form for the average translocation time for a very long chain at the high and also at the low pore frequency limits is the same as in the case where only the driving force is alternating and the pore is static (as discussed in the previous subsection). For a short chain at both the high and the low pore frequency limits the scaling may be affected by the value of the integral .
When the driving force is constant, if the amplitude of the oscillating term, , is much smaller than the static pore friction, , then we can also use here a small amplitude approximation, where the value of the time in the second integral in Eq. (19) must be replaced by . Then, for , Eq. (19) can be cast into
where . The result of Eq. (22) has been checked and it matches very well with the result of the full deterministic model in a wide range of frequencies.
Iv Average translocation time
Next we consider the average translocation time, , in all the different regimes using the full IFTP theory. To this end, we numerically solve the IFTP Eqs. (1), (2), (3), (4), (5), (7) and (8) to obtain the total translocation time. We study the behavior of the translocation time first for a time-dependent force, time-dependent pore friction, and finally when both the force and the pore friction are functions of time.
In Fig. 3 the normalized translocation time, , has been plotted as a function of normalized frequency, for various values of the initial phases of and . The pore is static with the pore friction , chain length is , and the time dependent external driving force is given by , where and . Here, is the frequency related to the whole translocation process when the pore is not flickering, i.e. and the external driving force is constant, i.e. . For (black solid curve) at low frequencies the magnitude of the translocation time is the same as of the static force case. At this low frequency limit the translocation process occurs during the first half of the period of the oscillating force. During this first half of the cycle the value of the force increases from its static value to its maximum and back to the static value at , being always greater than its static value. Therefore, the translocation time gradually decreases for small frequencies, and has a minimum at . For frequencies higher than , the chain starts experiencing the second half of the cycle where the value of the force is always smaller than its static limit, and thus there is a first maximum at . For higher frequencies, i.e. , the chain again experiences the following half of the cycle, i.e. (), where again the value of the force is greater than its static limit and the translocation time increases. With increasing frequency the translocation time displays oscillations between minima and maxima with a decreasing amplitude upon approaching the high-frequency limit, where the rapidly oscillating force component averages to zero, as discussed in Sec. III. For comparison, we have also solved the full stochastic IFTP model for the initial phase , with results shown in orange circles in Fig. 3. In this case we take into account both thermal noise and the initial shape distribution of the chain as explained in detail in Ref. 52. The results are in very good agreement with the deterministically solved equations.
Next, we consider the case of a flickering pore with a constant drive. In Fig. 4 the normalized translocation time, , has been plotted as a function of normalized frequency of the flickering pore, , for a constant driving force, , chain length of , and time dependent pore friction with static pore friction and amplitude , for various values of the initial phases of and . As we can see for , in the low frequency limit the value of the translocation time reduces to its static value, where . As the pore oscillation frequency increases the value of the pore friction also increases, which means that the pore width, , decreases (see Fig. 1a). Therefore, the effective interaction between the pore and the beads becomes stronger and the result is that the value of the translocation time grows and reaches its maximum at . At this frequency the translocation process occurs during the first half period of the pore motion. During this first half of the cycle the radius of the pore decreases from its static value to its minimum and back to the static limit at , being always smaller than the radius of the static pore. Therefore, the translocation time gradually increases for low frequencies, and then it has a maximum at . For frequencies higher than , the chain experiences the second half of the cycle where the radius of the pore is larger than the radius of the static pore, and thus there is a strong minimum at . For even higher frequencies, i.e. , the chain experiences the next half of the cycle, i.e. (), where the radius of the pore is again smaller than its static value and the translocation time increases. This alternation of high and low pore friction creates successive minima and maxima in a way similar to the case of the time-dependent drive, except that increasing the pore friction leads to increasing translocation time, while increasing the drive leads to more rapid translocation.
Finally, we consider the combination of an oscillating drive and pore. The fact that increasing the drive or increasing friction work in the opposite directions with respect to the translocation time suggests that it could be minimized by out-of-phase combination of the two forces. In Fig. 5a we present the normalized translocation time as a function of either the normalized pore frequency or force frequency, for mixed initial phases and , and , and , and and . Here, the pore is flickering and the pore friction is given by , where and while the oscillating external driving force is where and . The pore frequency is equal to the force frequency during the translocation process here. For fixed values of , , and , the minimum value of the normalized translocation time, , occurs at very low frequency limit when the force phase is and pore phase is , because the driving force has its maximum value and the pore friction has its minimum value. Then the normalized translocation time gets its maximum value, , when and pore phase is