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Page 1: Cooperative translocation dynamics of biopolymer chains through nanopores in a membrane: Slow dynamics limit

DOI 10.1140/epje/i2010-10663-6

Regular Article

Eur. Phys. J. E 33, 251–258 (2010) THE EUROPEANPHYSICAL JOURNAL E

Cooperative translocation dynamics of biopolymer chainsthrough nanopores in a membrane: Slow dynamics limit

Hai-Jun Wang1,2,3,a, Fang Gu1, Xiao-Zhong Hong4, and Xin-Wu Ba1

1 College of Chemistry and Environment Science, Hebei University, Baoding, 071002, China2 International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang, 110016, China3 Key Laboratory of Medicinal Chemistry and Molecular Diagnosis, Ministry of Education, Hebei University, Baoding, 071002,

China4 College of Physics Science and Technology, Hebei University, Baoding, 071002, China

Received 11 February 2010 and Received in final form 11 July 2010Published online: 31 October 2010 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2010

Abstract. The cooperative translocation dynamics of two complementary single-stranded DNA chainsthrough two nanopores located in a membrane is investigated theoretically. The translocation process isconsidered to be quasi-equilibrium, and then under the limit of slow dynamics the average translocationtimes are numerically presented under different conditions. It is shown that the effects of the chemical po-tential gradient, the recombination energy and the distance between the two nanopores on the cooperativetranslocation are significant. The present model predicts that the cooperative translocation of such twochains can shorten the translocation time, reduce the backward motion and thus improve the translocationefficiency.

1 Introduction

The translocation of a biopolymer through a nanopore ina membrane plays a significant role in some elementaryprocesses of life. Salient examples include passage of pro-teins across cellular membranes, phage infection and RNAtransport through nuclear pore complexes [1,2]. The poly-mer translocation has attracted intense attention not onlybecause it involved many biological phenomena, but alsofor its technological applications, such as controlled drugdelivery and gene therapy. Meanwhile, it also becomes animportant topic of the polymer dynamics [3]. During thepast decade, many experiments [4–11] on the translocationof biopolymers across a channel under an applied electri-cal field have been proposed. Correspondingly, a consid-erable amount of theoretical and computational modelshave been developed. In the relevant studies, the interestsare focused on the mean first-passage time (MFPT) τ andits scaling with chain length N [12–46].

Sung and Park [12] treated polymer translocation asa normal diffusion process across an entropy barrier. Al-ternatively, Muthukumar [13] applied the nucleation the-ory to study the MFPT of a translocated polymer chain.For the unbiased translocation (in the absence of exter-nal driving force), they found that τ ∝ N2

Dtr, where the

diffusive coefficient Dtr is taken to scale as Dtr ∝ N−1

a e-mail: [email protected]

by Sung and Park, and a constant by Muthukumar, re-spectively. Both the two mean-field models are based onthe assumption that during the translocation process, thepolymer segments on the two sides of the membrane are inequilibrium or quasi-equilibrium. In this way, the translo-cation dynamics is discussed in terms of the “transloca-tion” coordinate and the equilibrium free energy [14–21].Such translocation dynamics is now called the Browniandynamics.

In 2001, Kardar and coworkers [22] proposed ananomalous dynamics of polymer translocation. Theyfound that for a Gaussian chain, τ scales approximatelyin the same manner as the Rouse time τR [3], but witha larger prefactor. This implies that such a translocatingpolymer chain governed by the Rouse dynamics has lit-tle time to equilibrate during the translocation process.Since then the anomalous dynamics has inspired new re-search interests [23–39]. Typically, the Brownian dynam-ics simulation of Tian and Smith [25] showed that thenucleation model may provide an excellent descriptionof the dependence of τ on chain length N and chemi-cal potential gradient Δμ; Luo and coworkers [26–31] ob-tained very valuable results on the biopolymer’s translo-cation by using the bond fluctuation model and Langevindynamics simulation; A lattice-based Monte Carlo sim-ulation performed by Panja and coworkers [33–35] con-cluded that the polymer translocation is anomalous un-til the Rouse time τR, and beyond τR which is Browniandynamics.

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252 The European Physical Journal E

m-L

2Db

I1 m-L

m+L

II

2Db

m+L

s

I2

II I1

I2 s

s

s

(a)

(b)

Fig. 1. Schematic view of two ssDNA chains’ transloca-tion process from regions I1 and I2 to region II through twonanopores. (a) No recombination occurs. (b) The recombina-tion occurs which might be not complete because of thermalfluctuation.

Recently, the hydrodynamic interaction (HI) in sim-ulations of polymer translocation has been explicitly in-cluded [40–46]. Fyta and coworkers used the multiscaleapproach to simulate the biased translocation of polymerchain [40–42]. They showed that the polymer transloca-tion is accelerated because of the hydrodynamics corre-lation. This conclusion is then confirmed by Izmitli andcoworkers by using a lattice Boltzmann method [43]. Gau-thier and Slater presented the molecular dynamics simu-lations for an unbiased translocation and concluded thatthe HI plays a minor role as the pore size is comparableto the segment size [44], and they also stated the validityof the quasi-equilibrium hypothesis for smaller pore ra-dius. Furthermore, Gauthier and Slater developed a newMonte Carlo algorithm for the biased translocation by us-ing the quasi-equilibrium hypothesis [45,46]. This methodreproduced the results observed by Storm and cowork-ers [10,11], and also confirmed the results of Panja andcoworkers [35]. Based on these conclusions, they pointedout that the quasi-equilibrium model can be used to studythe translocation problems [45,46].

These rich studies mentioned above deepen the under-standing on the translocation dynamics of polymer chain.In fact, polymer translocation is an intrinsic many-bodyproblem and hence it is a rather complicated physical pro-cess. As a result, its dynamics can be influenced by variousfactors such as the external driving field, membrane thick-ness, solvent property and pore-polymer interactions. Todate, the scaling exponents of MFPT on the chain lengthobtained in relevant theories, experiments and simulationshave not yet reached a consensus. Many problems on thepolymer translocation are still open.

In this paper, for two biopolymer chains (s in regionI1 and s in region I2) with the same chain segments N(each of size b), we consider their cooperative translocationthrough two nanopores of distance 2Db located in a mem-brane, as shown in fig. 1. The cooperative translocationcan take place because of interactions between them, such

as hydrogen bonding and electrostatic attraction whenthey translocate from regions I1 and I2 to region II, re-spectively. Such a design is motivated by the fact that thetranslocation dynamics of the structured polymer chainhas attracted intense attentions [47–49]. Our particularinterest is that s and s are just two complementary single-stranded DNA (ssDNA) chains such that the transportedparts of the two complementary ssDNA chains can re-combine in region II through the “base-pairing” betweenthem. The recombination plays, in essence, the role of anexternal driving force in the translocation process thoughit is merely a probability event.

In fact, the translocation of ssDNA and RNA underdifferent conditions has been the focus of this field, whichplays a critical role in numerous biological phenomena andvarious potential technological applications. Experimen-tally, the response of different bases on the ionic currentis different from each other [4], by which the base sequencecan be determined. In general, for an unstructured poly-mer chain, the translocation dynamics can be influencedby the external driving field, pore-polymer interactions,solvent effect, hydrodynamics interaction and the size ofnanopore, and so on. However, for the typical structuredchains of ssDNA and RNA, they can fold into the sec-ondary structures because of base-pairing, thereby thecharacteristic translocation dynamics could be found. Forexample, Bundschuh and Gerland [48] have investigatedthe coupled dynamics of RNA folding and translocation,and they concluded that the non-native base pairs cansignificantly speed up translocation. More recently, Bund-schuh, Gerland and coworkers [49] proposed that, in theabsence of an external voltage, the scaling of translocationtime of RNA can be drastically affected by its secondarystructure.

In a sense, the base-pairing of ssDNA and RNA in thetranslocation process can be considered as a “chaperone-assisted” translocation (in which the reversible binding be-tween particles and polymer chain also affects the translo-cation), as proposed by Gelbart and coworkers [50] andAmbjornsson and Metzler [51]. For the two complemen-tary ssDNA chains under study, when the recombinationoccurs, the polymer translocation would turn into a bi-ased translocation from an unbiased translocation. Thecombination between them is closely related to the helix-coil transition, as early proposed by Poland and Scheraga(PS) [52] and a recent model of Karfi and coworkers [53].In terms of these methods, the cooperative translocationof two complementary ssDNA is presented.

The system under consideration is now investigatedbased on the slow dynamics. Here the meaning of slowdynamics is twofold. On the one hand, it includes theBrownian dynamics for an unstructured polymer chainsuch that during the translocation the polymer chain liesin the quasi-equilibrium state. This requires, of course,the validity of the quasi-equilibrium assumption, whichholds true only under the cases of small pore size, largefrictional force at the pore, weak driving force and highenergy barrier. On the other hand, for the translocationof a structured polymer chain of RNA with base-pairing,it is necessary to assume that the time scale of base-

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Hai-Jun Wang et al.: Cooperative translocation dynamics of biopolymer chains . . . 253

pairing is much shorter than that of translocation to makeuse of the free energy landscape to study the relevantproblems.

Note that, whether the slow dynamics can be fulfilleddepends strongly on the time scales of chain relaxation,chain translocation and base-pairing. Bundschuh and Ger-land [48] have ever discussed the relevant problem bymeans of the rates of translocation and base-pairing ofa base, k0 and kbp, respectively. They found that severaltypical experiments [2,4,10,54] are performed in the tran-sition region characterized by a range of 0.1 < k0

kbp< 10,

where k0 ∼ kbp ∼ 1μs−1. This provides a criterion to es-timate whether the adiabatic approximation breaks downor not. Therefore the slow dynamics needs k0 � kbp evenif the translocation time is much longer than the timescale of chain relaxation. In other words, the slow dynam-ics depends not only on the Brownian dynamics for anunstructured polymer chain but also on the criterion ofk0 � kbp because of the existence of base-pairing upontranslocating.

Based on the assumption of slow dynamics, the co-operative translocation of the two complementary ssDNAchains is studied by the Fokker-Planck equation [55]. Themain conclusion is that for large chain length N , the re-combination of transported parts of the two ssDNA chainsis definite, and further the cooperative translocation ofsuch two chains can shorten the translocation time, reducethe backward motion and thus in a sense can improve thetranslocation efficiency.

The article is organized as follows. In sect. 2, we derivethe recombination probability of the transported parts oftwo chains in terms of the configuration statistics of end-anchored polymer chain. In sect. 3, the average transloca-tion time is studied by the Fokker-Planck equation and thecorresponding numerical results are also presented underdifferent conditions for Gaussian chain. In terms of the nu-merical results, the effects of chemical potential gradient,the distance between the two nanopores and the recombi-nation energy on the cooperative translocation dynamicsare analyzed. In sect. 4, the conclusion is summarized andsome relevant problems are discussed.

2 The recombination probability of the twotransported chains

In the translocation scheme shown in fig. 1, regions I1and I2 are assumed to be large enough so that the un-transported parts of the two complementary chains arewell separated and their entropy changes due to separationplay a minor role. Meanwhile, the surrounding tempera-ture is considered to be lower than the melting tempera-ture of double-stranded DNA, so that the two chains canrecombine in region II. Moreover, as an approximation,the membrane is regarded as a thin and rigid wall, andthe interactions between the two chains and membranecan be neglected.

In region II, if only several matched bases of the twotransported parts are nearest neighbors to each other, they

would establish an instantaneous weak linking owing tohydrogen bonds between base pairs. Subsequently, therewould be two situations to take place: 1) If the corre-sponding bases along one chain are complementary withthose in another, then the two ssDNA chains would re-combine together, i.e., the base-pairing interactions occurwith a fraction determined by the temperature. 2) Other-wise, the weak linking would be broken by the fluctuationbecause no further stable linking can be established. Theformer is now defined as a recombination (partly renatu-ration), while the latter is thought of as an occasional en-counter. Obviously, the recombination of the transportedparts would result in the cooperative translocation of thetwo chains. Furthermore, the MFPT will be calculatedunder different conditions.

To begin with, we assume that each chain is in quasi-equilibrium state during the translocation process andcan be imagined as two independent end-anchored chainshanging from an impenetrable wall. Without loss of gen-erality, at a given time, one can assume that the segmentnumbers of chain s and s in region II are (m − L) and(m + L), respectively. Clearly, the quantity 2L measuresthe number of segment difference between the two chainsin region II, while (m − L) and (m + L) play the roles ofthe “translocation coordinate” of the two chains, respec-tively. Because of the complementarity, either chain canbe taken as the representative to study the cooperativeeffect in the translocation, for example, chain s.

With the known conditions, the free energy Fm of thechain s can be expressed as

βFm = (1 − γ2) ln(m − L) + (1 − γ1) ln(N − m + L)+(m − L)βΔμ, (1)

where β−1 ≡ kBT is the Boltzmann constant times theabsolute temperature, Δμ is the excess chemical potentialof a segment in region II relative to region I1. The chem-ical potential gradient Δμ can result from “chaperone”particles, weak driving voltage and some instable base-pairing between bases. In eq. (1), γ1 and γ2 are the criticalparameters describing the properties of an end-anchoredpolymer chain in regions I1 and II, and the values 0.5,0.69 and 1 correspond to the Gaussian, self-avoiding androdlike chains, respectively [56].

During the translocation process, once chains s and sin region II recombined with a fraction f , the correspond-ing free energy of chain s, for the case of L ≤ D, becomes

βF ′m = βFm − (γ′

2 − γ2) ln(m − L) − f(m − D)βε (2)

and for the case of L > D, which takes the following form:

βF ′m = βFm − (γ′

2 − γ2) ln(m − L) − f(m − L)βε, (3)

where ε (ε > 0) is the recombination energy, which mea-sures the energy difference between the bases-solvent andbases-bases [1] when the recombination takes place, andthe recombination fraction f means that the recombina-tion may be not complete. In eqs. (2) and (3), γ′

2 has asimilar interpretation as γ2, which is closely related to the

Page 4: Cooperative translocation dynamics of biopolymer chains through nanopores in a membrane: Slow dynamics limit

254 The European Physical Journal E

chain configurations and the persistence length after therecombination occurs, therefore γ′

2 > γ2 since recombina-tion is favored in energy but unfavored in entropy [56].It should be noted that the recombination implies thatm ≥ D and m > L from the geometrical consideration.

In fact, Fm and F ′m are free-energy barriers as a func-

tion of m, in which the terms with Δμ are due to thechemical potential gradient across the membrane, and thelogarithmic terms represent the entropic contributions dueto nanopore and the change in chain configuration inducedby the recombination. In contrast to the unbiased translo-cation of a single polymer chain where the entropic barrieris dominant [12,13], in the present model the competitionbetween entropic barrier and inter-chain interactions isinvolved. From eq. (2), one can find that the free-energybarrier lowers when recombination occurs. As a result, thiswould give rise to the reduction of the MFPT.

In order to study the cooperative translocation dynam-ics, it is necessary to consider the “encountering” issue ofthe two chains anchored in the semi-infinite space. Forthis purpose, we first consider the terminal segments’ en-countering of two Gaussian chains with segments n1 andn2, respectively. Assuming that the two chains are end-anchored at the positions of (0,D, 0) and (0,−D, 0), theprobabilities of finding their terminal segments arriving at�r(x, y, z) for the cases of n1 � 1 and n2 � 1, as a three-dimensional generalization of previous results [57,58], w1

and w2 can be written as follows:

w1 =92π

x

(n1)2exp

{−3[x2 + (y − D)2 + z2]

2n1

},

w2 =92π

x

(n2)2exp

{−3[x2 + (y + D)2 + z2]

2n2

}, (4)

in which the rigid membrane is denoted by the plane ofx = 0. Thus the encountering probability of the terminalsegments of the two chains can be given by u(n1, n2,D) =∫

w1w2d�r, that is,

u(n1, n2,D)=π√

n1n2

n1+n2

[3

2π(n1+n2)

] 32

exp(− 6D2

n1+n2

).

(5)Substituting n1 = m − L and n2 = m + L in this equa-tion, the probability that the terminal segments of the twochains encounter in region II, um(L), can be obtained as

um(L) =π

2

(3

4πm

) 32

exp(−3D2

m

)√1 −

(L

m

)2

. (6)

While for the case of m < D, um(L) = 0.Note that the recombination of the two chains can start

with not only the encountering of their terminal segments,but also that of the corresponding matched segments ex-cept for the occasional encounter. Therefore the proba-bility that no recombination occurs should be written asqm =

∏mi=D(1 − ui(L)), while the recombination proba-

bility is (1 − qm). Moreover, since the translocation is adynamic process, the fact that recombination does not oc-cur until the transported segments of chain s are (m−L)

100 101 102 1030.0

0.2

0.4

0.6

0.8

1.0

1x103 2x103 3x1030.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

p m

m

P

N

D=10 D=15 D=20

Fig. 2. P and pm for different values of D for the case of L = 0.

implies that no recombination occurs when the index i isless than m. Therefore, pm, the conditional probability offinding the transported parts of the two chains recombin-ing, can be given by

pm = (1 − qm)m−1∏i=D

qi. (7)

From this equation, we can see that with the increasing ofm, the value of the term (1−qm) increases while the valueof the term

∏m−1i=D qi decreases, hence pm would undergo

a maximum in the translocation process. Furthermore, asfar as the whole chain is concerned, the recombinationprobability during the translocation process, P , can beexpressed as P =

∑Nm=1 pm.

The quantities P and pm play a central role in thecooperative translocation of the two chains because theydetermine whether the cooperative translocation can takeplace or not. In order to illustrate the dependence of Pand pm on the chain length and the distance between thetwo pores, the numerical calculations of P varying with N ,and pm varying with m are undertaken for D = 10, 15,20 and L = 0, 5, 10, respectively. For simplicity, only theresults for the case of L = 0 are shown in fig. 2, becausefor a given D there is little difference between the curvesof L = 0, 5 and 10.

As illustrated in fig. 2, the recombination of two chainsis shown to be definite during the translocation process aslong as the chain length is large (typically, N ∼ 1000 asD equals 10). Making a comparison between the curvesof D = 10, 15 and 20, one can find that the delay ten-dency of recombination is evident when D increases. Thisstems from the fact that for a given chain length, the re-combination probability decreases when D increases, asindicated in eqs. (6) and (7). In the inset of fig. 2, pm hasthe same tendency to delay with the increasing of D, andas expected pm indeed displays a maximum.

Because the two biopolymer chains are subject to thesame external conditions, the most probable case is thatL takes a small value. Throughout the paper we considerthe situation of L ≤ D. In this way, for a given value of D,

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Hai-Jun Wang et al.: Cooperative translocation dynamics of biopolymer chains . . . 255

little differences between the probability curves of L = 0, 5and 10 can be observed, therefore as representative, belowwe would only study the translocation dynamics under thecase of L = 0.

3 The average translocation time and itsnumerical results

Now we turn our attention to the discussion on the MFPT.Note that the recombination of the two chains also takestime, which is now neglected as an approximation. Ac-cording to the theory of nucleation and growth, the trans-port of the biopolymer chain through the barrier can bedescribed by (for details, see ref. [13])

∂tWm(t)=

∂m

[βkm

∂Fm

∂mWm(t) + km

∂mWm(t)

], (8)

where Wm(t) is the probability to find m segments in re-gion II at time t, and km is the rate constant to move them-th segment into region II.

From this equation, the MFPT without recombinationin the whole translocation process, τ0, can be given by [55]

τ0 =∫ N

0

k−1m exp(βFm)dm

∫ m

0

exp(−βFn)dn. (9)

If the recombination occurs at translocation coordinate m,the corresponding MFPT τm can be calculated as a sumof two parts by writing, τm = τ1 + τ2, in which

τ1 =∫ m

0

k−1m exp(βFx)dx

∫ x

0

exp(−βFy)dy,

τ2 =∫ N

m

k−1m exp(βF ′

x)dx

∫ x

0

exp(−βF ′y)dy, (10)

are the MFPTs before and after the recombination occurs,respectively. It should be noted that in obtaining eq. (10),the fact that the chains in each part can translocate backand forth has been taken into account [55]. Then the av-erage translocation time of the whole chain, τav, can beobtained as

τav = τ0(1 − P ) +N∑

m=D

τmpm. (11)

Following Muthukumar [13], we also assume that therate constant km is independent of m and takes a constantk. For the case of L ≤ D, the free energy given by eq. (2)would be used in the following calculations. Substitutingeqs. (1) and (2) into eqs. (9) and (10), then the numericalresults can be carried out. At first, by taking γ′

2 = 0.75,D = 10, L = 0 and βε = 1.25, the numerical calculationsof τav for a Gaussian chain are presented for different val-ues of βΔμ. The corresponding results of τav are plottedagainst the chain length N , as shown in fig. 3.

In fig. 3, curves labelled with open and solid symbolscorrespond to τav for the cases of f = 0 (no recombination

102 103103

104

105

106

107

108

D=10

βΔμ= 0.01 βΔμ= 0.00 βΔμ=-0.01 βΔμ=-0.05

τ av

N

Fig. 3. Double logarithmic plot of the average translocationtime τav (in units of k−1) against length N under differentconditions. The curves with open and solid symbols correspondto the cases of f = 0 and 0.6, respectively.

102 103103

104

105

106

107

108

f=0.6

D=10 D=15 D=20

βΔμ= 0.01 βΔμ= 0.00 βΔμ=-0.01 βΔμ=-0.05

N

τ av

Fig. 4. Double logarithmic plot of the average translocationtime τav of a Gaussian chain (in units of k−1) against N forD = 10, 15 and 20. Symbols with cross: D = 10; solid symbols:D = 15; open symbols: D = 20.

takes place) and f = 0.6, respectively. It can be foundthat the average translocation time τav shortens once therecombination was taken into account, no matter whetherthe value of βΔμ is positive or not. The results show thatthe cooperative translocation has significant influences onthe average translocation time.

In the system of interest, the parameter D is anotherimportant physical quantity because it affects the recom-bination probability as shown in fig. 2. Intuitively, the in-creasing of D would make the recombination probabilitydecrease, and further affect the translocation dynamics. Inorder to observe the effect of parameter D on τav, the nu-merical results of τav for the Gaussian chain are presented,in which D is taken to be 10, 15 and 20, respectively.The corresponding numerical results of τav under differ-ent chemical potential gradients are illustrated in fig. 4. Itcan be found that for the same chemical potential gradi-ent, the shortening tendency of τav against N would notbe obvious with the increasing of D. This implies that the

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256 The European Physical Journal E

200 400 600 800 10004000

6000

8000

10000

12000

D=10βΔμ=-0.05

f=0.0 f=0.2 f=0.4 f=0.6

τ av

N

Fig. 5. Plot of τav (in units of k−1) against N for βΔμ = −0.05and D = 10 for different recombination fractions f .

larger the value of D, the weaker the role it plays in the co-operative translocation. Such a result is easily understoodbecause a large parameter D implies that the recombina-tion of the two transported chains becomes more difficult.

As seen from figs. 3 and 4, it can be found thatthe effect of the chemical potential gradient on the τav

is significant: the larger of βΔμ, the longer the averagetranslocation time τav. This indicates that some factorsenabling βΔμ to lower would be favorable to the coop-erative translocation. Furthermore, it is shown that thetranslocation time is sensitive to the recombination frac-tion f when βΔμ is low, but not when βΔμ is relativelyhigh. For βΔμ = −0.05 and various f , the correspondingnumerical result is presented in fig. 5. This indicates thatonly under a proper chemical potential gradient the ef-fect of recombination fraction becomes obvious. One couldwonder why βε is larger than βΔμ, while its effect onthe average translocation time is not so significant as ex-pected. This is because the recombination is in essence aprobability event. The term with βε taking effect requiresthe recombination to occur, while the action of βΔμ al-ways exists in the translocation process. Therefore, as longas the recombination probability is small, the reduction ofthe average translocation time τav would not be obvious,which can be seen in figs. 3 and 4 when N is small and Dis large.

Because of the computational limitation, the presentcalculations are only performed under the condition ofβε = 1.25 (actually, βε ∼= 1.25–2.5 [1]) and the maxi-mum value of N is 1000, which allows the recombinationfraction f to increase up to 0.6. This means that, for re-alistic conditions the recombination would occur in thetranslocation process even for a large D (see fig. 2), andτav would shorten more remarkably than that performedunder the present conditions. Furthermore, for the case ofL = 5 and 10, the average translocation time τav is alsocalculated numerically. It is found that there is only a tinydifference between τav for L = 0, 5 and 10, which is consis-tent with the analysis on P and pm in sect. 2. So only thecurves corresponding to the case of L = 0 are presentedin fig. 3, for simplicity.

According to the above results, it can be found thatthere are several advantages in the cooperative transloca-tion of two complementary ssDNA chains. First, when therecombination takes place, both the translocation timesof the two chains would shorten as illustrated above. Sec-ond, the cooperative translocation reduces the possibili-ties of backward motions of the two chains. The reasonfor this is that the unzipping of double-stranded DNA(dsDNA) would undergo a free-energy barrier, which hasbeen demonstrated in an experiment on this issue [7].Third, the cooperative translocation shows the possibil-ity of studying polymer physics under such restriction. Inaddition, because the two translocating chains are com-plementary, it would provide some useful clues on therelevant experiments. Thus, in principle, the efficiency oftranslocation might be improved if such an experimentwere performed.

4 Results and discussion

In summary, in terms of the quasi-equilibrium hypothesis,we have investigated cooperative translocation of the twocomplementary ssDNA chains through two nanopores lo-cated in a membrane. In order to capture the essence ofthe cooperative translocation dynamics, the encounteringprobability of two end-anchored polymer chains is pre-sented, and then the recombination probability of the twochains during the translocation process is obtained. Therecombination of the transported parts of the two ssDNAchains is found to be definite for long polymer chains.In the slow dynamics limit, the cooperative transloca-tion dynamics is studied in terms of the free-energy land-scape. Furthermore, the average translocation times for aGaussian chain are numerically presented under differentconditions. The results indicate the importance of “base-pairing” interaction between the two complementary ss-DNA chains in the translocation.

According to the present theoretical model, the dis-tance between the two nanopores (characterized by 2D),the chemical potential gradient (characterized by βΔμ)and the recombination energy (characterized by βε) arefound to have significant effect on the average transloca-tion time: 1) For a given parameter D, τav increases withthe increasing of βΔμ. 2) For a given βΔμ, τav increaseswith the increasing of D. 3) τav decreases with the increas-ing of βε since the increasing of βε results in the loweringof the free-energy barrier. On the basis of these results, onemay conclude that the translocation efficiency can be im-proved in this model by designing the related parameters.

The present studies are crude in the sense that the co-operative translocation is presented under the frameworkof the slow dynamics. The reason is mainly twofold. Onthe one hand, the calculations of the average transloca-tion time in the cooperative translocation depend on thevalidity of the quasi-equilibrium hypothesis, which holdstrue under the conditions of small pore size, large fric-tional force and weak driving force. On the other hand,the relevant discussions on the cooperative translocation

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Hai-Jun Wang et al.: Cooperative translocation dynamics of biopolymer chains . . . 257

dynamics are presented by using an oversimplified pic-ture, which requires that the translocation time be muchlonger than the time scale of base-pairing. In addition, theeffects of base sequence and HI were not involved. Thus,the present conclusions on the cooperative translocationare only qualitatively acceptable.

So far, the inconsistency between the Brownian dy-namics and the anomalous dynamics still exists, eventhough recent simulations [44–46] imply that the quasi-equilibrium model may be used to study the translo-cation problem. On this issue, various theoretical mod-els and simulations from different viewpoints and algo-rithms led to different conclusions. This may be due to,partly the complexity of translocation process itself, andpartly because this effect is not yet well understood. Asfar as the translocation of ssDNA or RNA is concerned,it has been demonstrated that the translocation dynam-ics of DNA depends strongly on the base-pore interac-tions and base sequence [5,6] and base-pairing [48,49]. Inparticular, a novel translocation dynamics of RNA hasbeen proposed because it can fold into secondary struc-tures. Thus the effect of base sequences and base-pairingon the translocation dynamics is worth paying more at-tention [26,39,48,49]. Furthermore, the zipping-unzippingdynamics of DNA [53], HI, solvent quality and pH valueof the system are also noteworthy.

In principle, the cooperative translocation might be re-alized in terms of the modern fabrication techniques [9–11]for producing the solid-state nanopore with desired size.Meanwhile, it is also possible to make an array of solid-state nanopores for the translocation experiment of sev-eral polymer chains. Then, it is desirable to perform anexperiment on the cooperative translocation of polymerchains. On designing the experiment, perhaps two factorsare especially important in order to observe the coopera-tive translocation of two polymer chains. One is the pa-rameter D and the other is what kind of polymer chainsto be used in experiments.

As shown in sect. 2, the recombination probability P isclosely related to the cooperative translocation. By meansof the numerical evaluations on P , one can find that for a

given chain length N , the condition of D ≤√

N8 should

be satisfied to contribute a significant recombination prob-ability during the translocation process. Experimentally,besides the two natural complementary ssDNA chains, thecooperative translocation can be realized by other possi-ble candidates. For example, two designed complementaryssDNA chains made of poly(dG-dC) or poly(dA-dT) andtwo polyelectrolyte chains with opposite charge in theirside groups may all be utilized in the relevant experiments.

Clearly, the theoretical studies on the cooperativetranslocation have to face the challenge. In the relevantmodels or the simulations, the problems in the coopera-tive translocation are more complicated than that in thetranslocation of a single polymer chain. In fact, it is therecombination probability that plays the most importantrole in the cooperative translocation dynamics, on whichthe parameter D has a significant effect. Then, as onestudies the scaling behavior of τav or other physical quan-

tities of the system, the factor D would be an interestingphysical quantity for consideration. The present work isjust a minor step towards the understanding of biopoly-mers’ cooperative translocation through nanopores andfor further modelling. It is expected that our results mayprovide useful clues on the relevant experiments.

We are grateful to two anonymous reviewers for their kindcomments and suggestions. We also acknowledge discussionswith Prof. Kaifu Luo of University of Science and Technologyof China. This work is supported by the NNSF of China underGrant Nos. 20873035 and 20574016. Author Hong also thanksthe support of NSF of Education Committee of Hebei provinceunder Grant No. 2007106.

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