polymer translocation in solid-state nanopores: dependence...

Chemical Physics 425 (2013) 1–13

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Chemical Physics

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Polymer translocation in solid-state nanopores: Dependence onhydrodynamic interactions and polymer configuration

0301-0104/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.chemphys.2013.07.016

⇑ Corresponding author. Address: Georgia Institute of Technology, School ofChemical & Biomolecular Engineering, 311 Ferst Drive NW Atlanta, GA 30332-0100,United States. Tel.: +1 (404) 894 4826.

E-mail addresses: [email protected] (C.M. Edmonds), [email protected] (P.J. Hesketh), [email protected] (S. Nair).

Christopher M. Edmonds a, Peter J. Hesketh b, Sankar Nair c,⇑a Interdisciplinary Bioengineering Graduate Program, Georgia Institute of Technology, Atlanta, GA 30332, United Statesb Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United Statesc School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United States

a r t i c l e i n f o

Article history:Received 20 December 2012In final form 29 July 2013Available online 3 August 2013

Keywords:NanoporesTranslocationBrownian dynamicsHydrodynamic interactions

a b s t r a c t

We present a Brownian dynamics investigation of 3-D Rouse and Zimm polymer translocation throughsolid-state nanopores. We obtain different scaling exponents a for both polymers using two initialconfigurations: minimum energy, and ‘steady-state’. For forced translocation, Rouse polymers (no hydro-dynamic interactions), shows a large dependence of a on initial configuration and voltage. Higher volt-ages result in crowding at the nanopore exit and reduced a. When the radius of gyration is inequilibrium at the beginning and end of translocation, a = 1 + t where t is the Flory exponent. For Zimmpolymers (including hydrodynamic interactions), crowding is reduced and a = 2t. Increased pore diam-eter does not affect a at moderate voltages that reduce diffusion effects. For unforced translocation usingnarrow pores, both polymers give a = 1 + 2t. Due to increased polymer–pore interactions in the narrowpore, hydrodynamic drag effects are reduced, resulting in identical scaling.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

One of the most significant advances in the single-moleculeanalysis of polymers is the use of nanopore devices to obtain infor-mation on individual biopolymer chains as they translocatethrough the nanopore. Such nanopores typically have diametersin the 1–10 nm range and have been fabricated using ion channelproteins [1–5] or solid-state materials such as silicon oxide [6–9]and silicon nitride [10–19]. Nanopore devices are one of the poten-tial ‘next-generation’ single-molecule analysis devices, due thepossibility of obtaining both high resolution and throughput [20].One example of the use of nanopore devices is in the determinationof DNA chain lengths (Supporting Information, Figure S1). BecauseDNA possesses an inherent negative charge, it can be driventhrough a nanopore with the aid of an applied voltage. Whenplaced in an aqueous electrolyte solution, a current flows throughthe nanopore. At the beginning of the experiment, when the DNA ison the ‘cis’ side of the nanopore (corresponding to the electrodewith negative voltage) and not blocking the pore, the current isat its maximum value. When the DNA chain begins to threadthrough the nanopore, a large fraction of the electrolyte ions willbe blocked and hence the current decreases to a minimum value.

Once the DNA has fully translocated and reaches the ‘trans’ sideof the nanopore, the ionic current returns to its original maximumvalue. Based upon the duration of current blockage, theoretically,the length of the DNA chain can be determined. This experiment,which is orders of magnitude faster than conventional gel electro-phoresis [20], is referred to as a translocation time measurement[3,20,21]. It should be noted that currently, due to the interactionsbetween the DNA and the nanopore, translocation times can havevery large distributions making it difficult to know the exact lengthof the DNA chain [21]. However, it may be possible to controlDNA-nanopore interactions by changing the nanopore surfacecomposition with either atomic layer deposition [15,22] or coatingthe surface with an organic material [23] thus reducing the sto-chasticity of the translocation process. Further research is neededto find the perfect nanopore composition that will provide opti-mum results.

Unfortunately, the underlying mechanisms of biopolymertranslocation through a nanopore are far from well-understood.Several authors, including ourselves, have studied a number of as-pects of biopolymer translocation with coarse-grained dynamicalsimulation techniques [24–51]. One primary issue, that has yet tobe completely resolved, is the scaling law behavior of translocationtime vs. chain length [24]. In fact, as we will discuss, the scalingexponent is heavily dependent upon applied force, pore diameter,and initial polymer configuration. In the first studies involvingunforced polymer translocation, Sung and Park [52] and Muthuku-mar [53], using a derived free energy equation involving polymer

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translocation through a narrow hole, found the translocation timescales as s � N2 where N is the number of monomers in the chain.Chuang et al. [25] later found an inconsistency in this scaling law inrelation to self-avoiding polymers, in which the radius of gyrationscales as Rg � Nt, where t (the Flory exponent) is 0.588 in threedimensions [54]. By estimating the distance a polymer travels dur-ing the translocation process as Rg and noting that the center-of-mass diffusivity of a Rouse polymer (i.e. no hydrodynamic interac-tions) is Do/N (where Do is the diffusion coefficient of a singlemonomer), Chuang et al. [25] estimated the unforced translocationtime scaling law as s � (Rg)2/(Do/N) � N1+2t, which is the samescaling behavior of the Rouse relaxation time [54], estimated asthe time required for a polymer to diffuse its radius of gyration[25,33]. Thus, for self-avoiding polymers, a scaling exponent ofs � N2 would indicate translocation being much faster than thepolymer relaxation time, which is not possible. Hence, s � N1+2t

could be seen as a better estimate. Unfortunately, as Chuanget al. [25] points out, one would assume that a polymer would dif-fuse through a pore much slower than in the bulk. As a result, thisscaling exponent could be seen as a lower bound. Panja et al. [26]modified the expression to s � Nt+2 to account for memory effects,due to a local change in monomer concentration on both sides ofthe pore during the translocation process, which was also observedby Dubbeldam et al. [50] and Gautheir et al. [55]. In addition, deHaan and Slater [46] found the scaling exponent is heavily depen-dent on the pore diameter varying from s � N1+2t � N2.2 for adiameter of r up to a value of s � N2.93 for a diameter of 10r, wherer is the diameter of each monomer. This increase in scaling expo-nent is due to the fact that for pore diameters larger than 1.5r, themonomers do not translocate in a single-file fashion but rather thepolymer folds inside the nanopore during the translocationprocess.

When translocation is aided with an applied force, the scalinglaws will change. For example, Kantor and Kardar [27] derived ascaling law expression for a long polymer chain traversing a shortpore with an applied force F, viz. s � Rg/(F/N) � Nt+1/F. A limitationof this scaling law is the assumption that the polymer is in equilib-rium throughout the translocation process. This may not always becorrect, especially in the presence of high driving forces. Vockset al. [28] derived a new scaling law, s � N(1+2t)/(1+t)/F, includingthe memory affects due to local tension in the polymer chain whena monomer translocates from one side of the pore to the other. An-other factor that greatly affects scaling law behavior is the appliedforce strength. In some previous simulations it was found that thescaling exponent increased with increasing force [29–31], while inothers the scaling exponent decreases with increasing force [32–34,56].

One proposed explanation for these differing observations isthat during forced translocation, the polymer is driven out ofequilibrium [32,33,35,36]. At first, as was demonstrated in previ-ous simulation studies, extreme monomer crowding on the transside of the nanopore [29–34,36], a clear indication that the poly-mer has not had ample time to equilibrate once it has passedthrough the nanopore, was thought to be responsible for scalinglaws differing from the value predicted by Kantor and Kardar[27]. However, in more recent studies involving tension propaga-tion theory [31,56–65], it has been proposed that non-equilibriumeffects are solely based on changes to the polymer on the cis sideof the nanopore rather than any trans side effects. As discussed byLehtola et al. [30], in the presence of a moderate driving force, thetranslocation time process can be thought of as a force balancebetween the applied driving force and the drag force due to themonomers on the cis side moving towards the nanopore. Whenthe force used to drive the polymer through the nanopore is ap-plied to monomers inside the nanopore, a tension in the chain iscreated. This tension propagates along the backbone of the chain

creating a ‘‘tension front’’ or boundary in which monomers influ-enced by the tension move towards the nanopore, and thus con-tribute to the overall drag force, while the other monomersbeyond the front do not. This tension in the chain, dependingupon the strength of the applied force, will alter the initial equi-librium shape of the polymer. Weak forces (N�t < F < 1) will resultin a ‘‘Trumpet’’ shape, moderate forces (1 < F < Nt) will result in a‘‘Stem–Flower’’ (or ‘‘Stem-Trumpet’’) shape, whereas strong forces(F > Nt) will result in ‘‘Strong stretching’’ (or ‘‘Stem’’) shape[31,59]. These changes in polymer shape are potential reasonsfor scaling law deviations.

One of the goals of tension propagation theory is to predict themovement of the tension front as a function of time during thetranslocation process. Using the conservation of mass relatingthe tension front and the number of monomers experiencingthe chain backbone tension, Saito and Sakaue [58,59] and Dubbel-dam et al. [31], predicted that the total translocation time is thesum of three individual time components with different scalinglaws. The first component, sini, is the time that it takes to createan initial blob state before monomer translocation. This term, inboth the research of Dubbeldam et al. [31] and Saito and Sakaue[59], has been hypothesized to be force, not length dependentand, in recent Brownian Dynamics Tension propagation theory[56], has been questioned to even exist. Hence, we will omit itfrom our discussions here. The second component, s1, is the timerequired for the tension in the chain (generated by the pullingforce) to propagate to the end of the polymer. This term domi-nates for longer chains. Once the tension reaches the end of thechain, the polymer then moves with a constant velocity for a timeperiod s2, which is the dominant term in short chains. For mod-erate to strong forces, the range at which most simulations andexperiments are performed at [59], Dubbeldam et al. [31] con-cluded the translocation time s = s1 + s2, where s1 � N1+t/F ands2 � N2t/F. In addition, they also proposed a scaling law transitionfrom s � N2t to s � N1+t as the applied force is increased, therebyindicating a lower bound exponent of a = 2t, also proposed byVocks et al. [28]. Slight differences were obtained for these scal-ing laws in the research of Saito and Sakaue [59]. For example,for moderately applied driving forces, the second translocationtime component was found to scale as s1 � Na/F, wherea = ((z � 1)(1 + t) � (1 � t))/(z � 1). For a Rouse polymer,z = (1 + 2t)/t, which results in s1 � N1.43/F, which is smaller thanthe values obtained by Kantor and Kardar [27] and Dubbeldamet al. [31]. On the other hand, for strong forces, Saito and Sakaue[59] obtained s1 � N1+t/F, agreeing with the previous results. Fi-nally, for both moderate and strong forces, Saito and Sakaue[59] obtained the third time component to scale as s2 � N2t/F,which agrees with the results obtained from Dubbeldam et al.[31].

Most recently, using the same mechanisms described in the ten-sion propagation theory discussed above, Ikonen and coworkers[56,62,63], beginning with the energy balance equations initiallyderived by Sung and Park [52] and Muthukumar [53], developed amethod for computing the Brownian dynamics motion of the trans-location coordinate (length of the chain that has translocated to thetrans side of the nanopore) in the high damping limit known as theBrownian Dynamics Tension Propagation (BDTP) theory. As shownin previous simulation results [30,42,66,67], the velocity of a poly-mer translocating through a nanopore is not constant, but rathervaries with time. Using this observation, instead of assuming a con-stant drag coefficient throughout the translocation time simulation,Ikonen and coworkers [56,62,63] instead assumed a drag coefficientthat varied in time. Interestingly, from their results, not only didthey find good agreement between their predictions and resultsfrom MD simulations, but they also discovered that the transloca-tion time scaling exponent is dependent upon length, only

C.M. Edmonds et al. / Chemical Physics 425 (2013) 1–13 3

converting to the value predicted by Kantor and Kardar [27](a = 1 + t) in the limit of very long chain lengths. This important dis-covery implies that there is no universal scaling exponent and ex-plains why there is such discrepancy in the literature. In addition,Lehtola et al. [30] investigated how the initial polymer configurationaffects the scaling law behavior, by simulating a polymer chain withan initial configuration of monomers in a straight line. They ob-served a scaling exponent of a = 2, far different than the scalingexponent predicted by Kantor and Kardar [27]. These findings indi-cate that, not only the applied force and the length of the chain, butalso the initial polymer configuration, affects the scaling exponentsstrongly.

As mentioned before, the formulations discussed above all as-sume polymer translocation in the absence of hydrodynamic inter-actions (HI) as modeled by a Rouse polymer. In other words, thediffusion of one monomer does not affect the diffusion of anotherand, as a result, the center-of-mass diffusion coefficient scales asD � N�1 and the polymer relaxation time scales as sR � N1+2t

[54]. On the other hand, when hydrodynamic interactions areintroduced – as modeled by a Zimm polymer – the diffusion ofeach monomer is affected by every other monomer in the chainthrough solvent interactions, resulting in a center-of-mass diffu-sion coefficient scaling law D � Rg � N�t and a Zimm polymerrelaxation time scaling law of sZ � N3t [54]. The assumption ofRouse behavior is likely valid inside a nanopore as long as minimalfolding occurs during the translocation process or if very littlewater is present inside the pore as would be the case for a very nar-row nanopore. However, because many polymers such as double-stranded DNA (ds-DNA) behave as Zimm polymers in bulk solution[68–70], it would seem that assuming Rouse behavior wouldunderestimate the diffusivity of the polymer, especially in the caseof studies involving unforced translocation through a nanopore. Tocomplicate matters, hydrodynamic interactions are long ranged inbulk solution [71], but have shown to be screened for polymersmoving near a wall or inside a channel [25,48,71]. Hence, the effectof hydrodynamic interactions in translocation time simulationstudies is not trivial and should not be omitted in any thoroughinvestigation.

\Under the assumption that, in the presence of hydrodynamicinteractions, the translocation process is governed by a force bal-ance between a drag force of a polymer ‘blob’ with size equal toits Rg on the cis side of the nanopore and the driving force to facil-itate translocation of the polymer through the pore, Storm et al.[8] arrived at a translocation scaling law of s � N2t, which wasalso obtained by Sakaue [58]. Fyta et al. [37] also investigatedZimm polymer translocation by writing an energy balance equa-tion for the system equating the kinetic energy to the potentialenergy of the system where the potential energy consisted ofthe following terms: the change in energy due to the increaseand/or decrease in size of the polymer ‘blobs’ on both sides ofthe nanopore, the change in energy due to the hydrodynamicdrag caused by the fluid, and the energy provided by the appliedforce used to drive the polymer through the nanopore. Interest-ingly, Fyta et al. [37] also derived the same scaling relationshipof s � N2t. Unlike the derivation by Storm et al. [8], which onlystudied at the effects of the polymer on the cis side of the nano-pore, the derivation by Fyta et al. [37] included effects on bothsides of the nanopore, which could be viewed as a more accuratemodel. Just as was done with the Rouse polymer model, Vockset al. [28] derived a new scaling law, which also includes memoryaffects due to local tension in the polymer chain, and founds � N3t/(1+t). Later, Saito and Sakaue [59] obtained a differentscaling law for polymer translocation with hydrodynamic interac-tions using the tension propagation theory discussed above. Asmentioned earlier, for very long chains, the s1 time componentdominates with a scaling relationship given by: s1 � Na/F, where

a = ((z � 1)(1 + t) � (1 � t))/(z � 1), for moderate driving forces.For a Zimm polymer, z = 3, which results in s1 � N1.38/F, whichis different from the values given above. However, the scalinglaw for short chains obtained by Saito and Sakaue [59] was foundto be s2 � N2t/F, which agrees very well with the results givenabove. Just as before with Rouse polymers, Ikonen et al. [63],usingBDTP theory, also found the scaling exponent for Zimm polymersis also dependent upon chain length, and, interestingly, convergesto approximately the same value of a = 1 + t [27] in the limit oflarge N, although much slower than for Rouse polymers. Hence,just as for Rouse polymers, a universal scaling law for Zimm poly-mers may not exist.

In this paper we study the effects of hydrodynamic interac-tions, using a computationally efficient algorithm for Browniandynamics [72], on the translocation time vs. chain length scalinglaw for pores of different diameters. We also determine the effectof applied voltage strength on the translocation time vs. chainlength scaling law for both the Rouse and Zimm polymer models.We also investigate the scaling law behavior further via two dif-ferent initial polymer configurations: (1) a ‘minimum energy’configuration, and (2) a ‘steady-state’ configuration. The differ-ence in these two configurations is that Rg scales as �Nt for con-figuration (2), whereas it is much smaller for configuration (1).Due to the often implemented [29–37,39,42–49,51] repulsiveWeeks–Chandler–Andersen (WCA) [73] potential energy functionused for non-adjacent monomers in the polymer chain, Rg for thetwo configurations are very different. Finally, we also investigatethe effects of hydrodynamic interactions for polymer transloca-tion in the absence of an applied force, i.e. unforced translocation.Specifically, we study the effect of polymer–pore interactions onthe scaling exponent a. We find that when the pore diameter isvery small, polymer–pore interactions become dominant, slowingdown the translocation process, and thus weakening the hydro-dynamic drag caused by the polymer ‘blob’ outside the nanopore.As the pore diameter is increased, the polymer–pore interactionsare reduced resulting in a decrease in the translocation time andan increase in a.

2. Simulation methodology and models

2.1. Nanopore model

As in our previous work, we employ an atomistically detailednanopore model made from silicon nitride as described by us ear-lier [34]. In this work, we first consider a very narrow and shortnanopore (diameter 0.96 nm and thickness 0.5 nm). In later studiesinvolving hydrodynamic interactions, we increase the diameter to1.5, 2.0, and 2.5 nm. For unforced studies, we also further reducethe diameter to 0.60 nm to observe how the scaling laws are af-fected by increased polymer–pore interactions. Included in thesimulation volume is a reservoir of water both above (cis) and be-low (trans) the nanopore. The length of the simulation box in thedirection of translocation is 60 nm. Finally, periodic boundary con-ditions were employed in our simulations as well.

2.2. Simulation methods

Unlike investigations which study hydrodynamic interactionsusing either lattice Boltzmann techniques [37,38], stochastic rota-tion dynamics (SRD) [29,39], or dissipative particle dynamics(DPD) [40], our simulation methodology uses a recently devel-oped method – referred to as a truncated expansion ansatz(TEA) [72] – in three dimensions. The development of the TEAalgorithm begins with the equation derived by Ermak and


McCammon [74] used to study Brownian dynamics with hydro-dynamic interactions:

DriðDtÞ ¼X

j

DijFj

kBTDt þ

Xj

@Dij

@rjDt þ RiðDtÞ ð1Þ

where Dri(Dt) is the monomer displacement over coordinate i, Dij

are the components of the 3N � 3N diffusion tensor, Fj is the sumof all forces acting on each monomer, kB is Boltzmann’s constant,T is the system temperature, and Ri(Dt) is the random thermal dis-placement with mean and covariance given by:

hRiðDtÞi ¼ 0; hRiðDtÞRjðDtÞi ¼ 2DijDt ð2Þ

The hydrodynamic interactions are described by the Rotne–Prager–Yamakawa (RPY) tensor [75]:

Dii ¼kBT

6pgaI ð3Þ

Dij ¼kBT

8pgrijIþ rij � rij� �

þ 2a2

3r2ij

I� 3rij � rij� �" #

for i–j and

rij P 2a ð4Þ

Dij ¼kBT

6pga1� 9

32rij

a

� �Iþ 3

32rij

arij � rij

� �for i–j and

rij < 2a ð5Þ

where i and j are the indices of two monomers, a is the hydrody-namic radius of each monomer, g is the solvent viscosity and I isthe identity matrix. Using the above tensor, the second term inEq. (1) vanishes. As stated in the fluctuation–dissipation theorem[76], there is a relationship between the viscous drag and randomthermal collisions. The viscous drag force is dictated by the3N � 3N diffusion tensor D in Eqs. (3)–(5). The terms in the randomthermal displacement Ri(Dt), can be expressed as the product of a3N � 3N tensor, B, and a Gaussian random variable with zero meanand variance Dt [72,77]. To satisfy the fluctuation dissipation theo-rem, the following relationship must hold:

D ¼ BBT ð6Þ

One issue that limits the application of hydrodynamic interac-tions is the enormous computational expense in obtaining B fromD. Two widely used methods are Cholesky factorization [74] andChebyshev polynomial approximation [78], which are both expen-sive and scale as O(N3)and O(N2.25) respectively. The TEA algorithm,on the other hand, scales as O(N2), has been shown to have highaccuracy [77], and is being used in other simulation studies[41,79] as well as included in recently released Brownian dynamicssimulation packages [80,81].

The TEA algorithm updates the positions of each monomer bydecomposing Eq. (1) into a sum of two terms. The first term is sim-ply Eq. (1) with the random displacement term, Ri(Dt), removedand can be written as follows:

DrTerm 1i ðDtÞ ¼

Xj

DijFj

kBTDt ¼ DiiDt

kBTFeff

i ð7Þ

where:

Feffi ¼

Xj

Dij

DiiFj ð8Þ

Intuitively, this first term can be thought of as the displacement ofeach monomer due to the applied force terms, Fj, that are correctedfor hydrodynamic interactions resulting in Feff

i . The second term ac-counts for the displacements due to the random forces and is writ-ten as:

DrTerm 2i ðDtÞ ¼ DiiDt

kBTCi

Xj

bijDij

Diifj ¼

DiiDtkBT

f effi ð9Þ

where fj is a random force, in the absence of hydrodynamic interac-tions, with the following characteristics:

hfii ¼ 0; hfifji ¼2ðkBTÞ2

DiiDtdij ð10Þ

Similar to the first displacement term, fj is also corrected for hydro-dynamic interactions resulting in f eff

i :

f effi ¼ Ci

Xj

bijDij

Diifj ð11Þ

One of the assumptions of the TEA algorithm that allow for effi-cient computations is that the hydrodynamic interactions areweak, i.e. Dij� Dii. This assumption is valid for our simulationstudies because, due to the good solvent quality, the polymer willexperience high excluded volume interactions which results inminimal overlapping thus decreasing the effect of hydrodynamicinteractions. In the Supporting Information, we reintroduce theequations above, briefly describe the TEA algorithm and discusshow to compute the two coefficients, Ci and bij. The complete der-ivation of the TEA algorithm can be found elsewhere [72,77,81].

We should note that, whereas in our simulations we do includewater inside the nanopore, our model does not include hydrody-namic coupling between the polymer and pore. However, as willbe shown later, our simulation results agree very well with theo-retical predictions and experimental results, which could be anindication that hydrodynamic coupling effects are negligible forthe studies that we are interested in.

2.3. Calculation of forces

The biopolymer used in our studies is modeled as a freelyjointed chain [82] with each monomer represented by a singlebead with a diameter of 0.43 nm, approximately the same diame-ter as a single-stranded DNA (ss-DNA) monomer [83]. Adjacentmonomers are connected through a spring-like potential energyfunction that is comprised of two elements: an attractive finitelyextendable nonlinear elastic (FENE) [84] potential and a repulsiveWCA potential [73] (equations S9 and S10 in the Supporting Infor-mation document). In these potential functions, we choose param-eter values for the Lennard–Jones energy well depth e = epoly = kBT,r = rpoly = 0.43 nm (the bead diameter), the FENE spring constantkFENE = 7 epoly/r2

poly, and the maximum distance between beadsRo = 0.86 nm (twice the bead diameter [34]). These values resultin a minimum potential energy at a bead-to-bead distance ofapproximately 0.448 nm. The time step used in our simulationswas 0.05 psec. The WCA potential, with the same parametersabove, is used to model high-excluded-volume interactions (goodsolvent properties) between non-adjacent monomers. The poly-mer–nanopore interactions are determined using the repulsiveWCA potential with different r and e parameters depending uponwhich nanopore atom (Si or N) the polymer is interacting with. Theenergy well depth (e) for the polymer-N atom interaction wasdetermined empirically to be 0.1kBT through extensive trial simu-lations [34]. The well depth for the polymer–Si atom interactionwas set to a value 63% higher than for polymer-N atom interac-tions, as reported from experimental data [85]. The values ofrpoly-N and rpoly-Si were computed based on measured values ofr for Si and N [85], rpoly = 0.43 nm, and the Lorentz–Berthelot mix-ing rules [73]. Due to the high dielectric constant of the water thatis present on the cis and trans side of the nanopore, it is assumed, aswas done in previous studies [29,31,32,35,38,39,42–44], that theelectric field (and hence the force due to the applied voltage) is


non-zero only inside the pore. The force due to the applied voltagein the direction of translocation is F = qV/d, where q and d are thecharge and diameter of each monomer respectively and V is thevoltage drop across the pore. In our forced simulations, we useforces with values of 11, 30, and 186 pN. If it is assumed [4] thatthe charge on each monomer is 1e, these values correspond to volt-ages of 30, 80, and 500 mV. Our values are consistent with previousmeasurements using silicon nitride nanopores and ds-DNA [18].All computational results were obtained using a simulation tooldeveloped by us written in the Fortran programming language.

2.4. Equilibration procedure

To investigate how the initial polymer configuration affects thetranslocation time scaling laws, we begin the translocation processwith the polymer in one of the two different starting configura-tions. In configuration (1), the first monomer is placed inside poreand the remaining monomers are placed with random orientationsin the cis reservoir with center-to-center spacing of rpoly (0.43 nm).Next, we perform a Metropolis Monte Carlo (MMC) [86] procedurewith 50,000 trials to place the polymer in its minimum energy con-figuration. The translocation timer then begins and the monomersare permitted to move through the pore. In configuration (2), afterthe MMC procedure, the monomers in the cis reservoir are allowedto relax to a ‘steady-state’ radius of gyration for a certain time per-iod (based upon Fig. 1, see below) before translocation is allowed.In either configuration, the translocation time is defined as thetime required for all monomers to translocate from the cis reser-voir to the trans reservoir. Interestingly, because of the polymermodel defined by the FENE and WCA potentials, the ‘minimum en-ergy’ configuration of the polymer is not the same as the ‘steady-state’ configuration (which is commonly referred to as the ‘equilib-rium’ configuration).

3. Results and discussion

3.1. Simulations in bulk water

To verify the accuracy of the polymer model and the correctnessof the HI calculations, we first performed simulations in bulkwater, i.e. in the absence of a nanopore and driving force, for poly-mer models with and without hydrodynamic interactions. Wemeasured the radius of gyration, diffusion coefficient, and time re-quired for the polymer to reach its steady-state radius of gyrationstarting from the minimum energy configuration. Theoretically, fora polymer in a good solvent (high excluded volume) we should ob-tain hR2

gi � N2t � N1.18 where the Flory exponent t = 0.588 in 3-D[54]. As shown in Fig. 1a, our calculations give scaling exponentsfor the steady state radius of gyration only slightly higher than thistheoretical value for both Rouse (�6.5%) and Zimm (�8%) poly-mers, and are in good agreement with previous work using theTEA algorithm and a similar polymer model [77]. The center ofmass diffusion coefficient scaling exponent obtained for the Rousemodel agrees very well with the theoretical scaling of D � N�1 asshown in Fig. 1b [54]. A Zimm polymer, using the RPY tensor,should have a diffusion coefficient scale as D � N�t � N�0.588,which is in good agreement with our simulation results, other sim-ulation results using the TEA algorithm [72] and also measure-ments of ds-DNA [68–70]. Finally, the time required for thepolymer to reach its steady state radius of gyration was measuredfor both the Rouse polymer (sR � N2.19) and the Zimm model(sZ � N1.84) and found to agree very well with the theoretical relax-ation scaling [54] of sR � N1+2t � N2.18 and sZ � N3t � N1.76 andwith other simulations using the TEA algorithm for HI interactions[72].

3.2. Translocation time vs. chain length: minimum energyconfiguration

Fig. 2 shows the scaling of translocation time with N for threedifferent applied voltages, with and without HI, for a polymer ini-tially in configuration (1). For an applied voltage of 80 mV, in theabsence of hydrodynamic interactions, the translocation timescales as s � N1.35, which is in good agreement with our previoussimulation results [34] using the integration algorithm by Ermakand Buckholz [87], as well as other previous simulations results[32,33,35,37], results using the BDTP theory [62,63], and with theprediction of Vocks et al. [28] (s � N1.37). All translocation timevs. chain length studies presented in this paper resulted in a max-imum variation of a = ±0.01 using the standard error formulation[88].

Fig. 2 also shows that the scaling exponent a increases withdecreasing voltage. This trend is in good agreement with previoussimulation results [32–34] as well as predictions with the BDTPmodel [56,62]. Based upon the findings in Refs. [25–28], the scalingexponent is larger for unforced translocation than for forced trans-location, indicating as the applied force is decreased a should in-crease. This trend agrees with our simulation results given inFig. 2. On the other hand, there are other simulation methodologiesthat predict the opposite trend [29–31]. We should point out thatwe did nothing in our simulations to prevent the polymer fromescaping out of the pore into the cis reservoir, as was done by Dub-beldam et al. [31] in which the radius of the first monomer was gi-ven a value larger than the pore diameter. As discussed by Ikonenet al. [56] implementing this ‘‘reflective boundary condition’’ couldbe responsible for the discrepancy in trends between scaling expo-nents and driving force.

A question exists as to why the scaling law for the non-HI sim-ulations does not match the value of a = 1.588 derived by Kantorand Kardar [27]. Fig. 3 shows hR2

gi of the polymer as a function ofN on the cis side before the translocation process has begun (timezero) and on the trans side at the conclusion of the translocationprocess. As mentioned earlier, one of our goals is to investigatehow the initial polymer configuration affects the translocationtime scaling law. Interestingly, because of the abrupt cutoff inthe WCA potential when the distance of the non-adjacent mono-mers is greater than 21/6r, the ‘equilibrium’ (or steady-state) poly-mer configuration is very different than the minimum energy state.For the calculations shown in Figs. 2 and 3, we place the polymer inits minimum energy state using a MMC procedure [86] for 50,000trials before the translocation process begins. As shown in Fig. 3, onthe cis side we obtain hR2

gi � N0.85 which is much smaller than thescaling for a polymer in a good solvent with high excluded volumeinteractions [54]: N2t � N1.18. In addition, as also shown in Fig. 3,on the trans side we obtain hR2

gi � N1.03 after the translocation pro-cess has ended, which is also much smaller than the theoreticalscaling. This is a clear indication of crowding of the polymer atthe exit of the nanopore after the translocation process has ended,observed during forced translocation [29–34,36]. In order to obtainthe scaling exponent predicted by Kantor and Kardar [27], thepolymer must be in equilibrium throughout the entire transloca-tion process. This does not occur in these simulations.

When the applied voltage is decreased to 30 mV, we find thathR2

gi � N1.11 on the trans side, indicating less crowding at the exitof the nanopore. As a result, the translocation time scalingexponent increases to a = 1.44, a value still different than the scal-ing exponent predicted by Kantor and Kardar [27]. Fig. 2b showssimulation data using the same polymer configuration with hydro-dynamic interactions (HI) included. Not only do the HI interactionsdecrease the translocation time [29,37], but the scaling exponent isreduced to a = 1.19 at 80 mV. This value is in good agreement withthe predictions in Refs. [8,37,58] wherein a = 2t = 2(0.588) = 1.18

Fig. 1. Scaling of the (a) average radius of gyration squared, (b) center of mass diffusion coefficient, and (c) time required for polymer to reach its steady-state radius ofgyration with number of monomers (N), for two different polymer models (blue – no HI, red – with HI), where m is the slope of each line. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of this article.)


and slightly higher than the prediction by Vocks et al. [28] (a = 3t/(1 + t) = 1.11). Our simulation values are also in good agreementwith the DPD simulations in Ref. [40] (a = 1.2) and only slightlylower than the results obtained from lattice Boltzmann techniques(a = 1.28) [37,38]. Just as in the non-HI polymer model, the scalingexponent a increases with decreasing voltage when hydrodynamic

interactions are included. Finally, we can see in Fig. 3, the radius ofgyration for the Zimm model after the translocation has completedscales as hR2

gi � N1.12 on the trans side for an applied voltage of80 mV, indicating less crowding at the exit of the nanopore thanfor the Rouse polymer with the same applied voltage. As statedearlier, the theoretical Zimm polymer relaxation time, which scales

Fig. 2. Average translocation time (500 trials) vs. chain length (N) for simulations(a) without HI and (b) with HI, for three voltages: 30, 80, and 500 mV, for initialconfiguration (1).

Fig. 3. hR2g i (100 trials) vs. N measured at time t = 0 (green), and after completion of

the translocation process for: No HI at 80 mV (blue), No HI at 30 mV (black), andwith HI at 80 mV (red). (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)


as sZ � N3t is much shorter than the Rouse polymer relaxationtime, which scales as sR � N1+2t [54]. Hence, our simulations showthat once the Zimm polymer exits the nanopore, due to secondarypolymer–solvent interactions, it begins to equilibrate to the stea-dy-state configuration much faster than the Rouse polymer.

3.3. Translocation time vs. chain length: ‘steady-state configuration’

To gain a better understanding of how the initial polymer con-figuration affects the scaling exponents, we perform another set ofsimulations with configuration (2), i.e. after the MMC procedure isperformed, we keep the first monomer inside the pore while theother monomers are free to move on the cis side of the pore for atime period determined by the values in Fig. 1c. We refer to thisas the ‘steady-state’ configuration. After the steady-state time ex-pires, the chain is then free to translocate. As shown in Fig. 4a,the scaling exponent a for the non-HI polymer has increased from1.35 to 1.44. This is in good agreement with the predictions of Saitoand Sakaue [58,59] (1.43), MD simulation results by Dubbeldamet al. [31] (1.47), and MD and LD simulations by Luo et al. [45](1.42 ± 0.01 and 1.41 ± 0.01 respectively). Again, the scaling expo-nent is smaller than the value predicted by Kantor and Kardar[27]. Once again, we measured hR2

gi on the cis side of the nanopore(after the steady-state time period but before the translocationprocess begins). As shown in Fig. 5a, hR2

gi � N1.20, which agrees verywell with the theoretical hR2

gi � N2t = N1.18 obtained for a polymerin a good solvent. However, as shown in Fig. 5b, at 80 mV we ob-tain hR2

gi � N1.08 on the trans side after translocation, once againindicating crowding at the exit. On the other hand, at 30 mV wesee that hR2

gi � N1.15 which is in good agreement for a polymer ina good solvent. As a result, the translocation time scaling exponentincreases to a = 1.52, which is in good agreement with the predic-tion by Kantor and Kardar [27] of a = 1 + t = 1.588. Hence, ourmodel indicates that in order to obtain the prediction by Kantorand Kardar [27], the polymer must be in its steady-state configura-tion throughout the translocation process. In addition, when com-paring results from Fig. 4a with results from Fig. 2a, we can see thatthe translocation time is larger for the ‘steady-state’ configurationpolymer than the minimum energy polymer. This is because, asshown in Fig. 3, the initial radius of gyration for the minimum en-ergy configuration is smaller than the radius of gyration for the‘steady state’ configuration given in Fig. 5, thus the polymer musttravel a longer distance which results in a longer translocationtime.

Fig. 4. Average translocation time (500 trials) vs. chain length (N) for simulations(a) without HI and (b) with HI, for three voltages: 30, 80, and 500 mV, for initialconfiguration (2).

Fig. 5. hR2g i (100 trials) vs. N on (a) the cis side after the steady-state time has

expired (but before translocation begins) for non-HI (blue) and HI (red) polymers,and (b) on the trans side after complete translocation, for the non-HI case at 80 mV(blue), the non-HI case at 30 mV (black), and the HI case at 80 mV (red). (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)


In Figs. 2 and 4 we also notice that increasing the voltage from80 to 500 mV (more than six times) does not change the scalingsignificantly. Lowering the voltage to 30 mV, however, does in-crease the scaling exponent, agreeing with our earlier assessmentthat a should increase as the applied force decreases. Hence, fromour results, it appears that the lower bound on the scaling

exponent is determined by the initial polymer configuration,whereas the upper bound is set by the applied voltage. The roleof the initial polymer configuration was investigated by Lehtolaet al. [30]. They hypothesized that when the applied force is largethe translocation process is dictated by a force balance betweenthe applied driving force and the drag force felt by the monomersin the cis reservoir as they move towards the pore entrance. Lehtolaet al. [30] performed translocation time simulations using a poly-mer with an initially linear configuration and obtained a scalinglaw of a = 2, which is very different from any prediction or previoussimulation results. It was further stated that the diffusive motionof the monomers has no impact on the translocation time scaling.This would seem to explain why the scaling law changes very little

Fig. 6. Average translocation time (500 trials) vs. chain length (N) for varying porediameters for initial configuration (2) with hydrodynamic interactions with anapplied voltage of 80 mV.


from 80 to 500 mV in our results. In other words, a driving force of80 mV is large enough that the effects of diffusion are insignificantand the scaling is dictated by the initial polymer configuration.

Fig. 4b shows the effects of including HI using a steady-state ini-tial polymer configuration (2). Similar to the findings with the min-imum energy configuration, when comparing Fig. 4b and a, we cansee the Zimm polymer translocates faster than the Rouse polymerwhen starting with polymer configuration (2). Also shown inFig. 4b is a = 1.20 at 80 mV, which, interestingly, is not significantlydifferent from that obtained in the minimum energy configuration(1) given in Fig. 2b. In fact, the translocation times and scaling lawsfor all voltages for both configurations are approximately the same.This can be explained by noting the time required to equilibratefrom the minimum energy configuration to the steady-state con-figuration is much shorter for the Zimm polymer than the Rousepolymer, as shown in Fig. 1c. As a result, due to the secondary poly-mer–solvent interactions, Zimm polymers that begin in the mini-mum energy configuration immediately expand and approachthe equilibrium steady-state radius of gyration during the translo-cation process. In addition, we can see from both Fig. 3 and Fig. 5bthat the radius of gyration at the exit of the nanopore is onlyslightly different than the theoretical value for both the minimumenergy (�5%) and for the steady-state (�2%) configurations, indi-cating very little crowding at the exit of the nanopore. As stated be-fore, this reduction in crowding in the Zimm model as compared tothe Rouse is due to the Zimm model having a shorter relaxationtime as a result of secondary polymer–solvent interactions (i.e.hydrodynamic interactions) and, hence it can more quickly reachits steady-state radius of gyration after it exits the nanopore. Final-ly, just as in the case for the minimum energy configuration (1), wealso see for the steady-state configuration (2) that increasing theapplied voltage from 80 to 500 mV does not change a, whereasreducing the voltage increases a significantly.

3.4. Translocation time vs. chain length: effect of pore diameter (Zimmpolymer)

To gain a better understanding in how pore dimensions affectthe translocation process, we next performed translocation timevs. chain length simulations and varied the pore diameter asshown in Fig. 6. In each of these simulations, we used the stea-dy-state initial polymer configuration (2) and included hydrody-namic interactions as well. From Fig. 6 we can see that as thepore diameter increases the translocation time decreases as wellas the scaling exponent a slightly increases. Both of these sametrends were observed in our previous work in which we omittedhydrodynamic interactions [34]. One explanation for the decreasein translocation time is due to the decrease in polymer–pore inter-actions. In our initial simulations, we used a narrow pore (0.96 nm)with a highly repulsive potential energy function to ensure ‘single-file’ translocation of the polymer with no folding inside the pore. Inwider nanopores, the monomers inside the pore no longer experi-ence a strongly repulsive polymer–pore potential and can moreeasily translocate through the pore.

In addition, in forced translocation time simulations with mod-erately high applied voltages, as mentioned before, the effect of dif-fusion on the translocation process is negligible and hence, thescaling exponent a changes very little with an increase in porediameter. This is very different from unforced simulations in whichdiffusion is the primary mechanism for translocation through apore. In those simulations a reduction in polymer–pore interac-tions allow for polymers to diffuse more freely inside the poreincreasing the stochasticity of the process and, as a result, increas-ing the scaling law exponent a as shown in the work of de Haanand Slater [46].

3.5. Waiting time simulations

One way to observe the behavior of the polymer during translo-cation is by measuring the waiting time, defined as the time eachmonomer remains inside the pore during translocation. As shownin the Supporting Information Figure S2, we measured waitingtimes for the Minimum Energy configuration and for the ‘steady-state’ configuration for both Rouse (a) and Zimm (b) polymers,with N = 100. First, observing Figure S2a, we can see that the wait-ing time increases to a maximum value about three quarters downthe chain before reaching a minimum value at the end of the chain.This same behavior, observed in previous simulation methodolo-gies [30,42,56,62,66], indicates that the polymer does not translo-cate with a constant velocity. But rather, as described by Ikonenet al. [56,62] as the tension front propagates down the chain, moremonomers contribute to the overall drag force, thus slowing downthe translocation process. After the tension reaches the back of thechain, the drag force is now only determined by the number ofmonomers on the cis side of the nanopore. Since this number con-tinually decreases during the translocation process (as more mono-mers move from the cis side to the trans side of the nanopore) thedrag force continually decreases which results in the increase ofmonomer velocity, thus the waiting times of the monomers inthe back of the chain go down. Hence, the translocation processspeeds up until all monomers reach the trans side of the nanopore.The peak in the waiting time curve represents when the tensionfront has reached the back of the chain [56,62]. Interestingly, wesee the waiting times are much smaller for the minimum energyconfiguration than the ‘steady-state’ configuration. This could bedue to one of two reasons. First, because the minimum energy con-figuration has a much smaller radius of gyration, the polymer hasto travel a shorter distance than the polymer in the ‘steady-state’configuration, and thus, a shorter translocation time and waitingtimes are observed. This result is also observed when comparingtranslocation time simulations in Figs. 2 and 4a. A second reason


for the two different waiting times could be a result of smaller dragforces in the direction of translocation, observed in the minimumenergy configuration. As described earlier, at moderate drivingforces, the translocation time process can be thought of as a forcebalance between the driving force and the drag force of the mono-mers moving towards the pore. We can think of the drag force ashaving two components, one parallel to the direction of transloca-tion and one perpendicular. It would appear that a polymer with asmaller radius of gyration, like a curled ‘‘blob’’, would have lessdrag force in the direction parallel to translocation than a polymerwith a large radius of gyration with a long-drawn out configura-tion. Hence, the monomers in the minimum energy configurationhave a larger drag force in the perpendicular direction of transloca-tion and less drag force in the parallel direction of translocationthan does the ‘steady state’ polymer configuration.

Interestingly, as shown in Figure S2b, the waiting time curve forthe Zimm polymer is essentially flat, indicating a constant velocitytranslocation. This same behavior was also predicted by Fyta et al.[37], whom stated that during translocation, due to the size of thepolymer on the cis side of the nanopore decreasing, the amount ofwork done by the fluid also decreases, whereas on the trans side ofthe nanopore the amount of work done by the fluid increases be-cause of the increase in size of the polymer. Hence, during translo-cation, the amount of work done by the fluid remains constant.Coupled with the fact that the work done by the electric field isalso constant, Fyta et al. [37] came to the conclusion that themonomers should translocate through the pore with the samevelocity, which agrees with our simulation results given in Fig-ure S2b. Finally, the waiting times are almost the same values forboth initial configurations for Zimm polymers, also reflected inthe translocation time simulations given in Figs. 2 and 4b.

3.6. Simulation results compared with measured values

We now summarize how our computational results compare toexperiments. Before we begin these comparisons, we address thequestion whether it is feasible to compare our coarse-grained sim-ulation results with those from experiments. Whereas some of oursimulation model parameters do not exactly match those existingin experiments, we do use parameters that realistically depict rela-tionships between polymer and pore dimensions used in transloca-tion time experiments with ds-DNA. For example, the simulationsperformed by Storm et al. [8,9] investigate the translocation timeof ds-DNA, which has an approximate diameter [8] of 2 nm, usinga 10 nm-diameter SiO2 nanopore. Hence, the ratio between thediameter of the ds-DNA and nanopore is approximately 5, whichis the approximately the same ratio as our polymer (0.43 nm) tothe nanopore of diameter 2.0 nm used in the simulation results gi-ven in Fig. 6. In addition, in the translocation time vs. chain lengthscaling law studies, Storm et al. [8,9] uses ds-DNA chain lengths,the shortest containing approximately 6.6 kbp (length per base�0.34 nm [89], total length �2250 nm) much longer than thelength of the nanopore used in these experiments (approximately20 nm). Similarly, our polymer chains, the smallest being 10 mono-mers (length per monomer = 0.43 nm, total length 4.3 nm) is alsomuch longer than the length of the nanopore used in our simula-tion studies (0.5 nm). Finally, as given in Fig. 1, with hydrodynamicinteractions implemented, our polymer model behaves as ds-DNAin bulk solution.

As shown in Fig. 6, the translocation time vs. chain lengthscaling law (a) is 1.21 which agrees very well with experimentsperformed by Storm et al. [8,9] (a = 1.26–1.27) using an appliedvoltage of 120 mV. Hence, because this scaling law is very differentfrom the values obtained without HI (a = 1.44) as shown in Fig. 4, itcan be concluded that HI interactions are required to accuratelymodel the physics involved in these translocation time

measurements. It was shown in previous experimental resultsds-DNA diffuses as D � N�t, where t is between 0.57 and 0.611[68–70]. As mentioned earlier, the translocation time waspredicted to scale as s � N2t [8,37,58], which would result in atranslocation time scaling exponent that could vary betweena = 1.14–1.22. This is in good agreement with our simulation resultsand only slightly lower than the values obtained by Storm et al.[8,9].

In a second example experiment, Wanunu et al. [17] used a4 nm wide, 10 nm thick Si3N4 nanopore to observe a = 1.40 fords-DNA chain lengths of 0.150–3.5 kbp at 300 mV. This is vastlydifferent from our simulation results that include HI, but similarto our simulation result without HI (a = 1.44). One hypothesis thatcould be made as to why hydrodynamic interactions are not re-quired for the accurate modeling of these experimental results isdue to higher polymer–pore interactions due to a smaller diameterpore. Due to the smaller diameter of the nanopore used in theirexperiments, polymer–pore interactions heavily influence thedynamics in the translocation process. This is also evident fromthe higher voltage required for translocation. In addition, unlikewhat is demonstrated from our previous simulation results [34]and experiments using very large nanopores (30 nm diameter)[13],both of which agree with predictions [27] of s � V�1, Wanunu et al.[17] obtained experimentally an exponential relationship fortranslocation time vs. voltage a further indicator how polymer–pore interactions greatly influence the translocation process.

3.7. Translocation time vs. chain length unforced simulations

Finally, we examine the translocation time scaling behavior inunforced translocation, for both HI and non-HI models. In thesesimulations, the middle monomer is initially placed in the centerof the pore and the two halves of the chain are placed in the cisand trans reservoirs respectively. The chain is then allowed to re-lax, using the values given in Fig. 1c, and then permitted to trans-locate in either direction. A successful translocation event occurswhen the chain has exited to either side of the nanopore. As shownin Fig. 7a, the translocation time for the HI case scales witha = 2.30. These results are in good agreement with previous worksthat explicitly include polymer–solvent interactions via MD(a = 2.27–2.30) [47–49], SRD (2.30) [39], and DPD (2.24) [40] meth-ods. Interestingly, the scaling law obtained is very similar to thescaling prediction for unforced translocation derived by Chuanget al. [25] in the absence of hydrodynamic interactions (s � N1+2t

� N2.176), a result also obtained by Panja et al. [26] who hypothe-sized that any relationship between the two is pure coincidence.On the other hand, as described by Guillouzic and Slater [47],due to the small diameter of the pore, the strong polymer–poreinteractions heavily influence the translocation process whereasthe hydrodynamic interactions have a negligible effect. This couldbe because the strong polymer–pore interactions slow down thepolymer velocity thus significantly reducing the hydrodynamicdrag. In addition, as discussed earlier and pointed out by Gautherand Slater [48], hydrodynamic interactions have shown to bescreened for polymers moving near a wall or inside a channel[71]. Hence, by observing the scaling exponents, we can see thisscreening effect playing a major role in these unforced transloca-tion time simulations.

In order to further test this hypothesis, we simulated the un-forced case for a smaller pore diameter of 0.60 nm. As shown inFig. 7b, not only is the translocation time significantly increased,but the scaling exponent (a = 2.14) resembles even more thescaling exponent derived by Chuang et al. [25] for unforcedtranslocation in the absence of HI. Hence, for very small pores,polymer–pore interactions become very large and the effects ofhydrodynamic interactions are greatly reduced.

Fig. 7. Average unforced translocation time (over 500 trials) vs. N for porediameters of (a) 0.96 nm, and (b) 0.60 nm; with and without HI effects.


In addition, with the 0.96 nm pore, we find a = 2.52 for the non-HI case. This is in good agreement with our previous simulation re-sults [34], the predictions by Panja et al. (2.59) [26], and those ofDubbeldam et al. (2.52) [50] and Gauthier et al. [55]. However,our scaling law is larger than the prediction of Chuang et al. [25]and other previous simulations in which a = 2.22–2.23 [45,51] or2.33 [39]. In fact, there does not seem to be a consensus aboutthe scaling law for unforced translocation [24]. In addition to the

previous simulations with a HI model by Gauthier and Slater[48], de Haan and Slater [46] performed unforced translocationsimulations using a non-HI polymer model and varying the porediameter. For a pore with diameter of 2r, which approximatelycorresponds to our 0.96 nm pore, they obtained a = 2.5, which isin good agreement with the scaling law that we obtained. We alsoperformed simulations using a very narrow nanopore (0.60 nmdiameter) and obtained a smaller value of a = 2.23 (Fig. 7b), whichis in good agreement with the prediction of Chuang et al. [25](a = 2.176) and simulations by de Haan and Slater [46] (2.19). Asis the case in the simulations using a Zimm polymer, the scalinglaw is greatly affected by the pore diameter (i.e. polymer–poreinteractions). Hence, if one of the computational goals is to obtaina complete set of scaling laws for translocation time vs. chainlength, it is important that the pore diameter and the magnitudeof polymer–pore interaction effects be taken into account. Finally,an interesting observation is that, as mentioned before, the un-forced translocation scaling law for a pore diameter of 0.96 nm isthe same as found in our previous study [34] which consideredthe polymer to initially be in a minimum energy configuration,whereas in the present study we allowed the polymer to relax toa steady-state configuration. Both cases result in the same scalinglaw in unforced translocation because, unlike in forced transloca-tion, the translocation time is much longer than the relaxationtime.

3.8. Why minimum energy configuration?

One of the important questions that we sought to answer is theeffect of the initial polymer configuration on the translocation pro-cess, which has been hypothesized by some to heavily influencethe scaling exponent, a, observed in the literature [30,38]. Basedupon our findings above, we demonstrated that a does indeed dif-fer depending upon which initial polymer configuration (minimumenergy or ‘steady-state’) is used in the simulation. Of course, onemay ask why perform simulations using the minimum energy con-vention as the starting initial polymer configuration, as there maynot be a direct correlation between this configuration and experi-mental findings.

Whereas there may not be a direct relationship to experimentalfindings and the minimum energy convention that we use in oursimulations, we feel that this initial polymer configuration is a sci-entifically based theoretical configuration, unlike a chain of mono-mers in a totally random configuration. In addition, theoreticallythe minimum energy configuration, due to the WCA potential en-ergy function used for non-adjacent monomers and the initialplacement of monomers being rpoly (0.43 nm) apart, should resultin the smallest radius of gyration (Rg) possible for this polymermodel. Hence, one could argue that by comparing the minimumenergy configuration with the ‘steady-state’ configuration, we arecomparing the smallest and largest possible values of Rg usingthe WCA potential energy functions. This can be seen in our simu-lation results when comparing Rg in Fig. 3 and Fig. 5a.

Finally, by comparing the minimum energy to the ‘steady-state’configuration we may have identified a potential source of discrep-ancy between scaling exponents listed in the literature. As de-scribed above, the often used polymer model in these coarsegrained simulation methods is the WCA potential (equation S10in the supporting information section). Upon observation of thismodel, one can see that, due to the cutoff in the potential energyfunction at distances greater than 21/6r, the potential energy ofthe polymer in both the minimum energy configuration and the‘steady-state’ configuration are equal even though the radius ofgyration of each are very different. This would not be the case if,for example, the potential energy function was modeled with a full6–12 Lennard–Jones potential in which the minimum energy


configuration would also be the equilibrium configuration. Asshown in Figs. 2 and 4a, we obtain values of a found in the previ-ous simulation results using either the minimum energy configura-tion or the ‘steady-state’ configuration. It must be made clear thatwe are not suggesting that other findings in previous research areincorrect or proper equilibration procedures were not followed,our purpose is to point out a potential source of discrepancy inthe values of a could lie in this oddity of the WCA potential energyfunction.

Of course, one way to remove this potential discrepancy sourcewould be to simply add a bond-angle potential as was done byKong and Muthukumar [90]. However, because of the parabolicnature of this potential energy function and the requirement tokeep the bond stiff to ensure a polymer with high excluded volumeinteractions, resulting in a large spring constant, a small simulationtime step would be required thus limiting both the overall simula-tion time and the number of monomers in the simulation. One wayto ensure that this discrepancy does not occur is to follow the‘start-up’ procedures that we used to obtain the ‘steady-state’ poly-mer configuration in which we first assign random monomer posi-tions, then place the polymer in its minimum energy configuration,and finally equilibrate the polymer to its ‘steady-state’ configura-tion. We believe this initialization procedure could possibly reduceany source of inconsistency related to the initial polymer configu-rations in simulation studies.

4. Conclusions

We have investigated the translocation time vs. polymer chainlength scaling behavior, for both Rouse and Zimm polymers, usinga computationally efficient simulation methodology – the newlydeveloped TEA algorithm [72] – in both forced and unforced trans-location studies. For forced translocation, using Rouse polymers,we obtained different scaling exponents depending upon the initialpolymer configuration and the strength of the applied voltage. Wedemonstrate that if the radius of gyration of the initial polymerconfiguration deviates from its theoretical value in a good solvent(hR2

gi � N2t), the scaling exponent will also deviate from a = 1 + t aspredicted by Kantor and Kardar [27].

In addition, if the applied voltage is large enough that it causesthe translocation time to be much shorter than the polymer relax-ation time, the polymer will crowd the nanopore exit and alsocause the scaling law to differ as well. However, if the radius ofgyration of the polymer begins and ends at its theoretical value,we find that the scaling exponent is in accordance with the valueof a = 1 + t as predicted by Kantor and Kardar [27]. Because ofthe strongly repulsive nature of the WCA potential often used intranslocation simulations, the radius of gyration is vastly differentdepending upon how long the polymer is permitted to relax. As weshow in this paper, the radius of gyration for the minimum energyconfiguration is different from the ‘steady-state’ radius of gyration.Hence, care must be taken in defining the initial polymer configu-ration before a translocation time simulation is performed and alsoin comparing simulations results from different studies.

Furthermore, we found the scaling law a increases withdecreasing voltage, which is in good agreement with previous sim-ulation results [32–34] as well as predictions with the recentlydeveloped Brownian Dynamics Tension Propagation model[56,62]. Based upon the findings in Refs. [25–28], the scaling expo-nent is larger for unforced translocation than for forced transloca-tion, indicating as the applied force is decreased a should increase,which agrees with our findings given in Figs. 2 and 4. From theseresults, we observed the lower bound on the scaling exponentwas determined by the initial polymer configuration, whereasthe upper bound was set by the voltage.

In the presence of HI, as with Zimm polymers, we obtained scal-ing laws that agree very well with the predictions of a = 2t[8,37,58] and also with the measurements of Storm et al. [8,9].We also found that, since the relaxation time in the presence ofHI is much shorter than in the absence of HI, there was less crowd-ing at the exit of the nanopore. In addition, just as in our previousstudies of forced translocation of Rouse polymers through nanop-ores with increasing diameters [34], translocation of Zimm poly-mers also results in small changes in a and decreases intranslocation time due to the reduction in polymer–pore interac-tions. Just as in the case with Rouse polymers, we also found thescaling exponent increases with decreasing voltage using Zimmpolymers.

Finally, we performed unforced translocation simulations withand without HI for the two different pore diameters. When thepore width is very small, the polymer–pore interactions dominatethe translocation process, resulting in approximately the scalinglaw for Rouse polymers as predicted by Chuang et al. [25](a = 1 + 2t) for both polymer models. When the pore diameter isincreased, the polymer–pore interactions are decreased, and, dueto the increased stochasticity of the process, the scaling exponentalso increases.

Acknowledgments

This work was supported by the National Science Foundation(ECCS-0801829). We acknowledge Prof. D.S. Sholl (Georgia Tech)for helpful discussions.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.chemphys.2013.07.016.

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