Quantized Current Blockade andHydrodynamic Correlations inBiopolymer Translocation throughNanopores: Evidence from MultiscaleSimulationsMassimo Bernaschi,† Simone Melchionna,‡ Sauro Succi,† Maria Fyta,*,§ andEfthimios Kaxiras§

Istituto Applicazioni Calcolo, CNR, Viale del Policlinico 137, 00161 Roma, Italy,INFM-SOFT, Department of Physics, UniVersita di Roma La Sapienza, P. le A. Moro2, 00185 Rome, Italy, Department of Physics and School of Engineering and AppliedSciences, HarVard UniVersity, Cambridge, Massachusetts

Received December 12, 2007; Revised Manuscript Received February 1, 2008

ABSTRACT

We present a detailed description of biopolymer translocation through a nanopore in the presence of a solvent using an innovative multiscalemethodology that treats the biopolymer at the microscopic scale as combined with a self-consistent mesoscopic description for the solventfluid dynamics. We report evidence for quantized current blockade depending on the folding configuration and offer detailed information onthe role of hydrodynamic correlations in speeding up the translocation process.

Biopolymer translocation through nanoscale pores holds thepromise of efficient and improved sensing for many applica-tions in biotechnology and possibly ultrafast DNA sequenc-ing.1–3 Recent advances in fabrication of solid-state nano-pores4,5 have spurred detailed experimental studies of thetranslocation process, with DNA as the prototypical biopoly-mer of interest.6 Computer simulations that can account forthe complexity of the biomolecule motion as it undergoestranslocation, as well as its interaction with the environment(the nanopore and the solvent), are crucial in elucidatingcurrent experiments7,8 and possibly inspiring new ones. Here,we study the dynamical, statistical, and synergistic featuresof the translocation process of a biopolymer through ananopore by a multiscale method based on moleculardynamics for the biopolymer motion and mesoscopic latticeBoltzmann dynamics for the solvent. We report evidence forquantized current blockade depending on the folding con-figuration (single- or multifile translocation) in good agree-ment with recent experimental observations.7 Our simulationsshow the significance of hydrodynamic correlations inspeeding up the translocation process.

Nanopores are an essential element of cells and mem-branes, controlling the passage of molecules and regulatingmany biological processes such as viral infection by phagesand interbacterial DNA transduction.9 The last two decadeshave witnessed the emergence of artificial solid-state nano-pores as potential devices for sensing biomolecules throughnovel means.6 One of the most intriguing possibilities isultrafast sequencing of DNA by measuring the electronicsignal as the biomolecule translocates through a nanoporedecorated with electrodes.3 While this goal still remainselusive, a number of detailed studies on DNA translocationthrough nanopores have been reported recently.7,8 Theseexperiments typically measure the blockade of the ion currentthrough the nanopore during the time it takes the moleculeto translocate, which provides statistical information aboutthe biomolecule motion during the process.

Numerical simulation of the translocation process providesa wealth of information complementary to experiments butis hindered by the very large number of particles involvedin the full process: these include all the atoms that constitutethe biomolecule, the molecules and ions that constitute thesolvent, and the atoms that are part of the solid membranein the nanopore region. The spatial and temporal extent ofthe full system on atomic scales is far beyond what can behandled by direct computational methods without introducing

* Corresponding author. E-mail: mfyta@physics.harvard.edu.† Istituto Applicazioni Calcolo.‡ INFM-SOFT, Department of Physics, Universita di Roma La Sapienza.§ Department of Physics and School of Engineering and Applied Sciences,

Harvard University.

NANOLETTERS

2008Vol. 8, No. 41115-1119

10.1021/nl073251f CCC: $40.75 2008 American Chemical SocietyPublished on Web 02/27/2008

major approximations. Some universal features of translo-cation have been analyzed by means of suitably simplifiedstatistical schemes,10 and nonhydrodynamic coarse-grainedor microscopic models11–13 or other mesoscopic approaches.14

Many atomic degrees of freedom, and especially those ofthe solvent and the membrane wall, are uninteresting fromthe biological point of view. The problem naturally calls fora multiscale computational approach that can elucidate theinteresting experimental measurements while coarse-grainingthe less important degrees of freedom.

We have developed a multiscale method for treating thedynamics of biopolymer translocation15 and performed anextensive set of numerical simulations, combining con-strained molecular dynamics (MD) for the polymer motionwith a lattice-Boltzmann (LB) treatment of the solventhydrodynamics.16 The biopolymer transits through a nanop-ore under the effect of a localized electric field applied acrossthe pore, mimicking the experimental setup.8 The simulationsprovide direct computational evidence of quantized currentblockade and confirm the experimentally surmised multiple-file translocation: the molecule passes through the pore in amultistranded fold configuration when the pore is sufficientlywide. The simulations offer detailed information aboutseveral experimentally difficult issues, in particular the roleof hydrodynamic correlations in speeding up the translocationprocess.

A three-dimensional box of size Nxh × Nyh × Nzh latticeunits, with h ) ∆x the spacing between lattice points,contains the solvent and the polymer. We take Nx ) 2Ny,Ny ) Nz; a separating wall is located in the midsection ofthe x direction, x ) hNx/2. We use Nx ) 100 and N0 ) 400,where N0 is the total number of beads in the polymer. Atthe center of the separating wall, a cylindrical hole of lengthlhole ) 10 h and diameter dp is opened. Three different poresizes (dp ) 5h, 9h, 17h) have been used in the currentsimulations. Translocation is induced by a constant electricfield acting along the x direction and confined to a cylindricalchannel of the same size as the hole, and length lp ) 12halong the streamwise (x) direction. All parameters aremeasured in units of the lattice Boltzmann time step andspacing, ∆t and ∆x, respectively, which are both set equalto 1. The MD time step is five times smaller than ∆t . Thepulling force associated with the electric field in theexperiments is qeE ) 0.01 and the temperature is kBT/m )10-4. The monomers interact through a Lennard-Jones 6–12potential with parameters σ ) 1.8 and ε ) ×10-4, and thebond length among the beads is set at b - 1.2. The solventis set at a density FLB, with a kinematic viscosity νLB and adrag coefficient γ ) 0.1.

We chose the separation d between the beads to be equalto the persistence length of double-stranded DNA, that is50 nm, and define the lattice spacing to be d/1.2 ) 40 nm.The hole diameter is 3∆x. The repulsive interaction betweenthe beads and the wall (with parameter σw ) 1.5∆x17) leavesan effective hole of size equal to ∼5 nm. Having set thevalue of ∆x, we choose the time step so that the kinematicviscosity is expressed as: νw ) νLB∆x2/∆t, with νw theviscosity of water (10-6 m2/s) and νLB the numerical value

of the viscosity in LB units; this procedure gives ∆t ∼ 160ps, with νLB ) 0.1. To ensure numerical stability, the relationγ∆t < 1 must be satisfied. Having established the value of∆t, we need to adjust the value of the drag coefficientaccordingly, γ < 6 × 109 s-1. This is significantly smallerthan an estimate of the friction based on Stoke’s law forDNA,18 which is equivalent to an underdamped system, oran artificially inflated bead mass. This approach is consistentwith the coarse graining of the time evolution in the coupledLB-MD scheme.

We focus on the fast translocation regime in which thetranslocation time, tx, is much smaller than the Zimm time,which is the typical relaxation time of the polymer towardits native (minimum energy, maximum entropy) configura-tion. This corresponds to the strong-field conditionqeEb/kT > 1. In this regime, simple one-dimensionalBrownian models,19 or Fokker–Planck representations, cannotapply because the various monomer units do not have timeto decorrelate before completing translocation. The ensembleof simulations is generated by different realizations of theinitial polymer configuration to account for the statisticalnature of the process. Initially, the polymer is generated bya three-dimensional random walk algorithm with differentrandom numbers for each polymer configuration and onebead chosen randomly constrained at the pore entrance. Thenthe polymer is allowed to relax for ∼104 molecular dynamicssteps without including the fluid solvent in the relaxationwhile keeping the bead at the pore entrance fixed. We defineas time zero (t ) 0), the time after the relaxation, when thefluid motion is also added, the pulling force begins to actand the translocation process is initiated; at this moment,the bead at the pore entrance is also allowed to move. Atthis stage, we do not include any electrostatic interactionswithin our model for reasons of computational simplicity.As far as the biopolymer motion in the bulk of the solventis concerned, this may actually be a good approximation ofexperimental conditions with high salt concentration, whichleads to strong screening of electrostatic interactions. Thesituation at the pore region may require more refinedtreatment, beyond the scope of the present work.

In Figure 1a, we present the number of pore-resident beadsNr(t) as a function of time, for a narrow (dp ) 5h), midsized(dp ) 9h), and large (dp ) 17h) pore, with h the mesh spacingof the lattice Boltzmann simulation, for representative(fastest, slowest, and average speed) trajectories. Simulationsare repeated over an ensemble of 400 realizations of differentinitial conditions and for total polymer lengths up toN0 ) 400. Time is measured in units of t1

E, the time it wouldtake for the polymer to translocate if the monomers were toproceed in single-file configuration at the drift speed; thisspeed is given by υE ) qeE/γm, with qe and m the chargeand mass of the monomer, E the external electric field, andγ the hydrodynamic drag. This gives t1

E ) bN0/υE ) 12N0,and the number of monomers in the pore for single filetranslocation is N1 ) 10 for the parameters used here.

Figure 1a clearly shows the highly nonlinear dynamics ofthe translocation process: In the initial stage of the translo-cation, the nanopore gets populated, with the number of

1116 Nano Lett., Vol. 8, No. 4, 2008

resident monomers significantly overshooting the single-filevalue N1, the horizontal dashed lines at heights qN1 indicatingq-file (q ) 1, 2, 3, ) translocation. The range of q exploredby the translocation trajectories grows approximately withthe cross-section of the pore, going from q ∼ 2 for thesmallest pore dp ) 5h up to q ∼ 8 for the largest onedp ) 17h. Note that these values correspond to about halfthe maximum allowed q number, qmax ∼ dp/b. The fastestevents correspond to the largest q value observed, while theslowest events correspond to essentially q ) 1 throughoutthe translocation. It is also noteworthy that the translocationtime typically exceeds the single-file value, t1

E, except forthe fastest events; for the most probable events, q ∼ 2 forall pore sizes, indicating that conservative monomer–mono-mer interactions produce an effective slow down comparedto a single Langevin particle subject to a constant electricdrive and frictional drag γ.

Figure 1a also presents the current blockade in all threepores for the most probable event in each case, which is theevent with a translocation time close to the peak of thedistribution over all translocation times. The current block-ade is proportional to the number of monomers in the poreper unit area and appears to occur in well-defined steps(quantized). Specifically, these blockades are calculated fromthe difference between the area of the resident beads,π(σ/2)2, and the total area of the pore, π(dp/2)2. To investigatethe quantization of the current blockade, we monitored the

distribution of Nr(t) at various time frame intervals of 100steps. The resident monomers block the current across thechannel so that Nr(t) conveys a direct measure of the currentdrop associated with the biopolymer passage through thenanopore. The corresponding histograms P(Nr,t) for threepore sizes are shown in Figure 1b. At early times, thesehistograms exhibit a multipeaked structure, which is a clearsignature of multifile translocation. As time passes, themultiple peaks recede in favor of a single-peak distribution,close to the single-file value N1 ) 10. This was found to bea stable attractor for every simulated trajectory, indicatingthat the tail of the polymer always translocates as a singlefile.

Collecting all results for the average number of residentmonomers Nr as a function of the translocation time tx forthe three pore sizes studied, we find a simple relationship,shown in Figure 2. Experiments7 have reported that theaverage number of resident atoms Nr in each translocationevent varies approximately inversely with the duration oftranslocation tx:

∫0

txNr(t) dt ≡ Nrtx ∝ N0 ) const (1)

The single-file asymptote N1 ) 10 (q ) 1) at long times,t . t1

E, is evident. The short-time asymptote, reaching up to4 < q < 5, corresponds to ultrafast translocations (t < t1

E)occurring in the case of the large-diameter pore, dp ) 17h.These results are intuitively reasonable because large residentnumbers imply that more monomers cross the pore per unittime, hence the translocation becomes faster. The results alsosupport the notion of Nr(t) as a measure of the time-rate ofthe translocation, dNT/dt ∝ Nr, from which the inverseproportionality between Nr and tx is a direct consequence of∫0

tx [dNT/dt] dt ) N0 ) const. In this expression, NT(t) is thenumber of translocated monomers at time t.

The simulations reveal that solvent correlated motionmakes a substantial contribution to the translocation energet-ics. The role of hydrodynamic correlations is best highlightedby computing the work done by the moving fluid on thepolymer (we call this the synergy, WH) over the entiretranslocation process as compared to the case of a passivefluid at rest:

Figure 1. Number of resident beads with time for three differentpore sizes dp ) 5h, 9h, 17h (h ) lattice spacing) and N0 ) 400. (a)Fastest (minimum time, blue), slowest (maximum time, red), andaverage speed (most probable time, green) translocation events; theinsets show the current blockade for the duration of an event withaverage speed (green) with the current normalized to the open porevalue (1). (b) Histogram P(Nr,t of the distribution of Nr with time:short-time trajectories show multifile character, reaching up to q∼2–8 in the initial stage of the translocation depending on the poresize; long-time trajectories show little departure from the single-file configuration.

Figure 2. Scatter plot of the average resident number Nr vstranslocation time (in units of t1

E) for the ensemble of translocationevents for three values of the pore diameter, dp ) 5h, 9h, 17h andN0 ) 400.

Nano Lett., Vol. 8, No. 4, 2008 1117

WH(tx)) γ∫0

tx ∑i)1

N0

uf

i(t) · Vf

i(t) dt (2)

where υbi is the velocity of monomer i and ubi is the fluidvelocity at the position of monomer i. For the sake ofcomparison, it is also instructive to contrast WH with thecorresponding work done by the electric field

WE(tx)) qe ∫0

tx ∑i)1

Nr(t)

Ef· Vf

i(t) dt (3)

where the sum extends over the resident monomers onlybecause the electric field is applied at the pore region only.These statistically averaged values of WH and WE reveal anumber of interesting features (see Figure 3). First, WH isalways positive, clearly showing that hydrodynamic correla-tions provide a cooperatiVe background, as compared to thecase of a passive “ether” medium (ub ) 0). Second, weobserve that the WE has a much narrower distribution ofvalues than WH, reflecting the ordered structure of thebiopolymer as it passes through the nanopore as comparedto its off-pore morphology. It is useful to introduce the workdone by the electric field on molecules that translocate single-file and proceed through the pore at speed υE,W1

E ) qeEN1υEt1E ) bqeEN1N0. In the absence of any other

interaction, a q-file translocation at speed υE would completein a time tx(q) ) t1

E/q under an electric work qW1E. In the

present simulations, W1E ) 0.12N0, thereby W1

E ) 48 for N0

) 400. Interestingly, the distribution of WE values is highlypeaked at a value very close to W1

E . The observation that W∼ W1

E implies that qυx(q)tx(q) = υEt1E, and because the

simulations show that tx(q) > t1E, the conclusion is that υx(q)

< υE/q, indicating that collective motion of the monomersslows down the process.

A major asset of numerical simulations for the study oftranslocation processes is the direct access to visualizationof the morphology of the translocating chain. As an example,we show in Figure 4 a typical “snapshot” at a time whenabout 65% of the monomers have already passed throughthe pore of a translocating 2-folded chain of N0 ) 400 beads.In the same figure, we show for comparison an event forthe same length, but for single-file translocation (unfoldedchain) through a very narrow (dp ) 3h) and shallow

(lp ) 1) pore. In addition to the polymer conformation, weshow isocontours of the magnitude of the hydrodynamicsynergy density

wH(rf

; t)) γ ∑i∈ B( r

f)uf

i(t) · Vf

i(t) (4)

which is a local (in both space and time) version of the totalsynergy WH defined in eq 2 , with B (rb) a grid cell centeredaround location rb) (x,y,z). The contours of wH (rb) illustratethe cooperative nature of the hydrodynamic field, withregions of high comoving flow surrounding the translocatingpolymer and assisting its motion. This is suggestive of thenotion of an “effective” polymer, dressed with the hydro-dynamic synergy field, which acts as a self-consistentlubricant, helping the polymer to negotiate a faster passagethrough the nanopore.

We further investigate this issue by inspecting the average(over an ensemble of 400 realizations) translocation time,⟨tx⟩ , as a function of the polymer length, with and withouthydrodynamics. The results are shown in Figure 5: hydro-dynamics consistently accelerates the translocation by roughly30% percent. More intuitively, hydrodynamics literallyrenormalizes the diameter of the pore: as is clearly visible

Figure 3. Statistical distribution of the work performed by thehydrodynamic and electric field during translocation events. Thevertical dotted line corresponds to the work W1

E done by the electricfield on polymers that translocate single file.

Figure 4. Left panel: a typical 2-folded polymer configuration (dp

) 9h), at time where 65% of the N0 ) 400 total beads have alreadytranslocated from right to left; colored contours show the magnitudeof the corresponding hydrodynamic synergy field (only five of thenine wall layers are shown). Right panel: a single-file translocationevent for a narrow and shallow pore (dp ) 3h,lp ) 1h) with 60% of the beads translocated and the correspondingmagnitude of the synergy.

Figure 5. Average translocation time as a function of polymerlength with (closed circles) and without (open squares) hydrody-namic interactions. Colors correspond to different pore diametersdp ) 5h (green solid line), 9h (red dotted line), and 17h (blue dashedline). Black triangles indicate the value of the single-file translo-cation time t1E for each value of N0 with hydrodynamics (all numbersare scaled to the value of t1

E for N0 ) 400).

1118 Nano Lett., Vol. 8, No. 4, 2008

in Figure 5, a pore of diameter dp ) 5h for a bare polymer(without the hydrodynamic field) is essentially equivalentto a pore of almost double diameter dp ) 9h for thehydrodynamically dressed polymer. To assess the degree ofcorrelation between the translocation dynamics of the“dressed” polymer versus the actual one, we have measuredthe translocated specific synergy (synergy per monomer),defined as

⟨wH(t) ⟩ ) γNT(t)

∑i)1

NT(t)

98f

i(t) · 98f

i(t) (5)

with NT(t) the number of translocated monomers at time t.Clearly, any implicit time-dependent functional dependenceof the form ⟨wH(t)⟩ ) ⟨wH(NT(t))⟩ would indicate that massand synergy translocate in a synchronized manner. We find

that the ratio ⟨ 98f

· 98f

⟩kT ∼ 5, reflecting the fact that the

solvent locally “follows” the monomer and providing ameasure of the relative importance of synergistic versusthermal forces. Our results show that ⟨wH(t)⟩ is essentiallyconstant throughout the translocation process. This impliesa direct proportionality between the translocated synergy andthe number of translocated beads and supports the notionthat the “dressed” and the actual polymer proceed in fullsynchronization across the nanopore.

In conclusion, by using a new multiscale methodologybased on the direct coupling of constrained moleculardynamics for the solute biopolymers with a lattice Boltzmanntreatment of solvent dynamics, we have been able to confirma number of experimental observations, such as a directrelation between quantized current blockades and multifoldedpolymer conformations during the translocation process. Inparticular, the simulations reveal an intimate connectionbetween polymer and hydrodynamic motion that promotesa cooperative background for the translocating molecule, thusresulting in a significant acceleration of the translocationprocess. Such an acceleration can also be interpreted as theoutcome of a renormalization of the actual polymer geometryinto an effective one, more conducive to translocation. Thisopens up exciting prospects for the development of optimizednanohydrodynamic devices based on the fine-tuning of

hydrodynamic correlations. As an example, one may envis-age multitranslocation chips, whereby multiple moleculeswould translocate in parallel across membranes with an arrayof pores. The optimization of such devices will requirecontrol of solvent-mediated molecule-molecule interactionsto minimize destructive interference between translocationevents.

Acknowledgment. M.B., S.M., and S.S. acknowledgesupport from the Initiative for Innovative Computing andthank the Physics Department of Harvard University for itshospitality. M.F. acknowledges support by the NanoscaleScience and Engineering Center of the National ScienceFoundation under NSF award number PHY-0117795.

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