Simulation of polymer translocation throughprotein channelsM. Muthukumar* and C. Y. Kong
Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA 01003
Communicated by Richard S. Stein, University of Massachusetts, Amherst, MA, December 20, 2005 (received for review June 2, 2005)
A modeling algorithm is presented to compute simultaneouslypolymer conformations and ionic current, as single polymer mol-ecules undergo translocation through protein channels. Themethod is based on a combination of Langevin dynamics forcoarse-grained models of polymers and the Poisson–Nernst–Planckformalism for ionic current. For the illustrative example of ssDNApassing through the �-hemolysin pore, vivid details of conforma-tional fluctuations of the polymer inside the vestibule and �-barrelcompartments of the protein pore, and their consequent effects onthe translocation time and extent of blocked ionic current arepresented. In addition to yielding insights into several experimen-tally reported puzzles, our simulations offer experimental strate-gies to sequence polymers more efficiently.
Translocation of polymers through biological channels is verycomplex involving many machineries and is a fundamental step
in many life processes. Although several essential features oftranslocation are richly documented in systems such as mRNPcomplex through nuclear pores (1–3), a simple system has onlyrecently been identified for following the single-file passage of oneisolated polymer through one channel (4–11). In this system, thechannel is constituted by self-assembling heptamers of the Staph-ylococcus aureus �-hemolysin (�HL) protein. The channel is as-sembled in a phospholipid bilayer, which offers a physical barrier,and the channel has an opening diameter of �1.4 nm at thenarrowest constriction (12). A single-stranded polynucleotide, suchas poly(deoxyadenylate) and poly(deoxycytidylate), is pulledthrough the channel by an externally applied voltage gradient acrossthe channel in a solution of a strong electrolyte. The idea is that theionic current through the channel caused by the passage of smallions of the electrolyte is blocked to a certain extent during the eventof translocation of the polymer. It has been hoped that the extentand duration of the current blockade are unique signatures of theidentity of the polymer, both in terms of the polymer’s chemicalcharacteristics and physical length.
Even this simplest setup, where identical molecules undergotranslocation, has generated several puzzling results. The distribu-tion, P(�), of the duration � of blockade of ionic current Ib is verybroad and appears to exhibit at least two peaks. In addition, thereare several levels of ionic current blockade Ib for the same molecule.It is standard practice in experimental investigations to combine thehistograms of � and Ib (8). The resultant scatter plots always yieldtwo groups of data even for monodisperse homopolymers.
To gain insight into these puzzles, we have developed thefollowing simulation. It is complementary to a flurry of theoreticalactivity (13–21), based on entropic barrier dynamics (22), all ofwhich lead to a generic P(�) unlike in experiments. Although it isindeed desirable to perform the computation ab initio, the size ofthe system to be simulated is forbiddingly large to enable such acomputation. Therefore we restrict ourselves to generating a min-imal model of polymer translocation through the �HL channel, andour goal is to discover the most basic concepts relevant to amolecular understanding of the experimental results. Our minimalmodel incorporates enough chemical details of the polymer andchannel to evaluate the potential roles played by secondary struc-tures of the polymer, orientation of the chain (3� versus 5� end),binding sites inside the lumen of the channel, etc. In addition to
resolving the above puzzles, the details emerging from our simu-lations are so vivid that additional experimental protocols can beformulated to substantially narrow down P(�), and thus enable amuch faster sequencing scheme.
The key result of our simulations is that conformational entropyof the polymer plays a dominant role in dictating � and P(�). Thebreadth of P(�) can be directly attributed to the entropic trap arisingfrom the vestibule of the �HL pore. P(�) can be made dramaticallysharper, sufficient to facilitate sequencing of polymers, by deletingthe vestibule or dragging the polymer in a direction normal to thepore axis.
Simulation MethodThe system consists of essentially four parts: the heptameric �HLpore, a ssDNA molecule undergoing translocation, a phospholipidmembrane barrier carrying the pore, and an electrolyte solutionpermeating through the pore. The protein pore, �HL, is repre-sented by a united atom model where the residues of the pore arereplaced by beads. The coordinates of the residues are extractedfrom the Protein Data Bank. There can be many choices for thedescription of beads in terms of their sizes and shapes. In this articlewe present results for the simplest choice of the beads. In the
Conflict of interest statement: No conflicts declared.
Fig. 1. United atom model. (a) The end and side views of �HL. The diametersat the center and mouth of the vestibule are 46 and 30 Å, respectively. (b) Theinternal wall of the nanotube is made of spherical beads on a curved hexag-onal lattice. The interaction between polymer and tube is taken as theLennard–Jones potential of Eq. 5 with � � 0.2 kcal�mol and � � 2.75 Å. (c)United atom representation of single-stranded poly(dC).
present united atom model, each residue is replaced by a sphericalbead and the diameter of each bead is 3 Å. A bead is assigned �1,0, or �1 charge according to the ionization of the residue. Allaspartic acid residues, glutamic acid residues, and C-terminalgroups are deprotonated; arginine, lysine, histidine, and N-terminalgroups are protonated. The rest of the residues are electricallyneutral. In view of experimental contexts (23), the protein pore isnot allowed to move.
Further, the protein pore is treated as a rigid structure with adielectric constant of 2 surrounded by a solution of dielectricconstant 80. The protein pore is embedded in a 48-Å-thick mem-brane of dielectric constant 2. The membrane surfaces are hardwalls. The protein pore is the only space to allow passage of thepolymer from the donor compartment to the recipient compart-ment. The symmetric axis of the pore is positioned along the x axis,as shown in Fig. 1a, which gives the geometry of the pore. Thevestibule of the pore is submerged in the donor compartment. The�-barrel of the protein begins at x � 0 and extends up to x � 48 Å.
We have used the side-chain incorporated model to represent asingle-stranded polynucleotide (20), because of its simplicity, fastcomputational speed, and past success. Each nucleotide is made upof three subunits (phosphate, sugar, and base), and each subunit istaken as a bead. The bead representing the phosphate moietycarries one negative charge. To distinguish the chain directionalityin terms of 3� versus 5� ends, we deliberately tilt the side chain (base)at a 65° angle as shown in Fig. 1c. The chain backbone is freelyjointed in view of the fact that the persistence length of ssDNA hasbeen measured in high salt concentrations to be roughly the lengthof 1–2 nt (24–26). Although the three subunits are of different sizesand masses, the results presented below are for the model of allsubunits having equal size and mass. The diameter of these beadsis taken to be 2.5 Å, and the equilibrium ‘‘bond length’’ betweenthese ‘‘atoms’’ is 2.5 Å. The mass of a nucleotide is distributedequally into the three subunits. For example, in the case ofpoly(dC), each subunit carries 96 Da.
Polymer conformations are evolved by using the velocityVerlet algorithm (27) to follow the dynamics of the ith bead,
md2r�i
dt2 � ��v� i � �� r�iU F� i� t� , [1]
where r�i, m, and � are the position vector, mass, and frictioncoefficient, respectively, of the ith bead. F� i(t) is the random forcefrom the solvent bath acting on the ith bead and is stipulated tosatisfy the fluctuation–dissipation theorem,
F� i�t��F� j�t�� � ij6kBT��t � t��, [2]
where t is the time and is the usual delta function. U in Eq. 1is the total potential energy acting on the ith bead and consistsof four contributions.
U � Ubond ULJ UDH UV. [3]
These correspond, respectively, to bond stretch, short range,screened Couloumb, and electric potential given by
Ubond � k� l � l0�2 [4]
ULJ � �LJ���
r�12
� 2��
r�6
� [5]
UDH�rij� �ZiZj
4��0�rexp(��r) [6]
UV�r�i� � ZiV�r�i� . [7]
Here, l is the bond length, l0 (� 2.5 Å) is the equilibrium bondlength, r is the distance between interacting beads, Zi is the electriccharge on the ith bead, �0 is the permittivity of the vacuum, and Vis electric potential. The values of the parameters are: k � 171kcal�mol�Å2, �LJ � 0.5 kcal�mol, and � � 2.5 Å, ��1 � 3 Å for 1M KCl. The dielectric constant � is inhomogeneous. � is 2 insidebeads and the membrane and is 80 everywhere else.
The ionic current I(t) at time t, because of the passage ofelectrolyte ions accompanying the polymer transport, is computedby using the Poisson–Nernst–Planck (PNP) formalism (28–30).Taking advantage of the fact that small ions relax much faster thana large polyelectrolyte molecule and that the concentration ofelectrolyte in the pertinent experiments is very high in comparisonwith monomer concentration, we assume that at every time step ofthe Langevin dynamics simulation of the polymer, the electrolyteions have relaxed to the steady state so that the polymer chain istaken only as a fixed charge distribution p (r, t) at this time.
With this assumption, the self-consistent coupled PNP equationsfor the local charge density Ci(r, t) of the ith ionic species and theelectric potential V(r, t) for a given polymer conformation at t are
����Ci�r� ZiCi
kBT�V�r�� � 0. [8]
�0�����r��V�r�� � � f�r� � p�r, t� � �i
Ci�r� . [9]
Here f(r) is the local charge density arising from fixed charges inthe system, such as the charged beads of the protein pore. Asalready mentioned, �(r) is set to 2 for the beads of polymer and poreand for the membrane. It is 80 everywhere else. In addition, theexternally applied voltage gradient is accounted for by fixing theelectrostatic potential at the boundaries in the x direction. The boxfor the ionic current calculations is located at �100 Å X 75 Å,�50 Å Y 50 Å, and �50 Å Z 50 Å. We set V � 0 at X ��100 Å and V � V0 at X � 75 Å.
Eqs. 8 and 9 are solved by successive overrelaxation method witha grid spacing of 1 Å, following ref. 29. First, we start with V(X)because of the externally applied electrostatic potential gradientacross the membrane of dielectric constant of 2 and uniformconcentrations (C� and C�) of cations and anions of the dissolvedelectrolyte.
Then, the coordinates of beads of the pore and polymer are readin. Next, using the successive overrelaxation method, V, C�, and C�
are computed iteratively until they converge within the tolerancelevels of �V � 10�8 V and �C� � 10�7 M. In our simulations wehave considered 1 M of KCl as the electrolyte. Because C� is veryhigh in comparison with the counterions from the polymer, thelatter are ignored. From the convergent values of C� and V at each
Fig. 2. PNP and approximated V(X) across the protein pore.
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time unit of the Langevin simulation, the ionic current at this timet is given by
I�t� � �d2r�J� J�� , [10]
with
J��r, t� � �D�� �C��r, t� Z�C�
�BT�V�r, t�� . [11]
In evaluating the ionic current, we have taken J� at X � 48 Å andA at this exit point is 441 Å2. The integral in Eq. 10 is over the crosssection of the pore at the exit point. D� and D� are taken to be1.96 � 10�5 cm2�s�1 and 2.03 � 10�5 cm2�s�1, respectively, for K�
and Cl� ions. The magnitude of the applied voltage is V0 � 120 mV.The simulation is carried out as follows. First, a single polymer
chain is equilibrated in the absence of pore and externally appliedvoltage gradient, by considering only the connectivity, Lennard–Jones forces, and Debye–Huckel forces among various beads of thechain. Separately the electrostatic potential for the protein pore iscomputed for 1 M KCl solution in the steady state with a given valueof V0 (120 mV) and in the absence of the polymer. The potentialalong the central axis of the channel is given as dotted numericaldata in Fig. 2. The different gradients (slopes of the curve) atdifferent parts of the trajectory are readily conspicuous in Fig. 2.Because we are interested here in the details of the actual trans-location event, and not in the way the polymer arrives at the channelentrance, we have extended the range of the potential gradientinside the vestibule to the donor compartment. In addition, we haveignored the finer details of the potential gradient and replaced the
actual potential gradient by the approximated gradient consisting ofthree parts, namely, up to the �-barrel, inside the �-barrel, andoutside the �-barrel in the recipient compartment.
Next, the equilibrated polymer is inserted in the donor compart-ment such that the center of mass of the chain is at X � �100 Åin front of the mouth of the pore. The polymer is dragged towardthe pore by the electrical potential gradient across the membrane.For each time unit of the Langevin dynamics simulation, thecoordinates of the polymer beads are stored, and the ionic currentat this time is computed. When any one monomer of the polymerenters the mouth of the pore (at X � �50 Å) a clock for measuringthe translocation time starts. As the polymer progressively invadesthe lumen of the pore, the ionic current is calculated at each time
Fig. 3. Typical ionic current traces. (A) Long �. (B) Short �.
Fig. 4. Histogram of � for �HL (n � 45, V0 � 120 mV).
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unit. This is how the ionic current traces given below are con-structed. It is to be noted that �V entering the Langevin equationfor the polymer bead is the original value of the protein pore in theabsence of the polymer. When the last bead of the polymer exits theouter edge of the pore (at X � 48 Å), the clock stops with a readingof the translocation time �. By repeating the above sequence ofsimulations thousands of times we construct the histogram of �, andfor each of the entries of the histogram, we have a trace of ioniccurrent as the polymer traverses through the pore.
ResultsFirst, we give results on chain conformations, ionic current, andtranslocation time � as the polymer transits through the channel in
typical simulations. Then we present distribution functions ofblocked current and �, based on thousands of simulations. Fig. 3Arepresents a typical simulation result for the model chain ofsingle-stranded poly(dC) with n � 45 nt undergoing translocationthrough �HL under a potential difference of 120 mV in 1 M KClsolution. One unit of simulation time is arbitrarily taken to be 0.15�s, by matching the peak values of � in our simulations and theexperiments of refs. 8 and 9, and by noting that the averagetranslocation velocity is constant in the long polymer and highvoltage gradient limits (9).
There are several key features evident from the typical trajectoryof Fig. 3A: (i) The open pore current of �135 pA is very close tothe experimental value. (ii) The polymer enters the mouth of thevestibule not necessarily as a single file. The presence of polymermonomers at the mouth of the vestibule is sufficient to provoke aresponse in the ionic current by providing a spatial blockade to theflow of electrolyte ions. (iii) As time progresses the chain gets intothe vestibule. Because of its large volume, the vestibule acts as anentropic trap and the chain segments linger in this cavity for somedefinite time duration. At this stage, the ionic current is Ib1 (�100pA). The fluctuations in I around its mean value reflect thedynamics of the polymer rattling inside the vestibule. (iv) One endof the chain eventually enters the �-barrel. At this time, I(t) isreduced precipitously to the blocked current level Ib2. The averagevalue of Ib2 is 17 pA. This very low value reflects the reduction inthe cross-sectional area of �-barrel for flow of electrolyte ions by thepresence of polymer segments. The fluctuations in Ib2 is solelycaused by the dynamics of the polymer. The ionic current is aboutIb2 until the last monomer exits the end of the �-barrel. � is theduration of all of the above events. (v) When the chain hascompleted its translocation, the ionic current returns to the openpore current after leaving a distinct signature for the presence ofsome monomers in the back of the exit location, perturbing the flowof electrolyte ions.
It is to be noted from Fig. 3A that the chain spends a significantamount of time inside the vestibule before entering the �-barrel.Although such trajectories are quite common, there are also moreprevalent trajectories where the dwell time inside the vestibule isshort as illustrated in Fig. 3B. The essential features of Fig. 3B arethe same five features discussed for Fig. 3A, except that the timespent inside the vestibule is shorter. As seen from the chainconformations presented in Fig. 3B, the shorter duration inside thevestibule is caused by the initial chain orientation in line with theaxis of the channel and the chain entering the vestibule mouthessentially as a single file.
Vivid details such as those in Fig. 3 are available for each of thethousands of simulations that were performed, based on which wehave constructed histograms for � and Ib. The normalized proba-bility P(�) based on 4,000 simulations is plotted against � in Fig. 4
Fig. 5. Histogram of Ib. A and B correspond, respectively, to Fig. 3 A and B.
Fig. 6. Typical ionic current trace for a nanotube.
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for n � 45, V0 � 120 mV, and 1 M KCl. The error bars are estimatedfrom the square root of counts per bin before normalizing thehistogram. It is evident that the distribution of � is very broad, andtwo dominant peaks at 50 and 75 �s may be identified. This resultis in agreement with our earlier report of two peaks based on asmaller number of simulations with shorter (n � 15) chains (20).Now we are in a position to go back to the trajectories of the chainsand study them for various events taking different translocationtimes. Indeed, an example of the trajectory taking 76 �s is Fig. 3A,where the vestibule plays the role of an entropic trap, delaying thetranslocation process. In contrast, by stochasticity associated withchain end orientation and thermal noise from the background, thechain end can enter the �-barrel sooner by essentially avoiding thevestibule’s entropic trap. This is the case with Fig. 3B where � (�23.7 �s) is much shorter, and this event belongs to the dominantpeak in the histogram. We validate this mechanism for the occur-rence of two peaks, by performing further simulations on nanotubeswhere we have cut out the vestibule part of the channel (see below).Now, only one peak is seen, suggesting the significant role of thevestibule as an entropic trap. However, the two peaks have recentlybeen attributed (31, 32) to different orientations (3� and 5� ends) ofthe polymer entering the �-barrel first. Yet analysis of our simu-lation data shows that there is no significant discrimination betweenthe 3� and 5� ends entering the �-barrel first in terms of translo-cation time. This finding argues for more work, in terms of bothexperiments (without the vestibule) and simulations (with explicitsolvent molecules).
As already noted, there are two levels of blocked ionic currentduring the translocation event. For the event corresponding toFig. 3A, the normalized histogram P(Ib) of blocked current Ibwithin the duration of �, given in Fig. 5A, exhibits two separatedistributions. The average values of these two peaks representwhether the blockade corresponds to the vestibule (weak block-ade) or the �-barrel (strong blockade). The widths of the peakscorrespond to the accompanying polymer dynamics. It must beremarked that the occurrence of two populations of blockedionic current has nothing to do with the actual value of �. Thehistogram of Fig. 5A corresponds to � � 76 �s. For the case of� � 23.7 �s corresponding to Fig. 3B, the histogram of Ib is givenin Fig. 5B. Here again, there are two distributions representingpolymer dynamics inside the vestibule and the �-barrel. How-ever, the weight of Ib population relating to the vestibule isweaker for the faster translocation events. These results empha-size the need to interpret the scatter plots of ref. 8 differently byhaving to go into individual events instead of clumping them alltogether.
The intervention by the vestibule of �HL as an entropic trapfor the translocation of the polymer leads to the breadth of P(�).To further validate this observation, we have repeated our
simulations by replacing the �HL channel by a cylindrical tubeof diameter 18 Å and length 98 Å, as illustrated in Fig. 1b. Thetube is inserted inside the membrane of dielectric constant 2, andthe membrane thickness is the same as the tube length.
A typical trajectory of polymer translocation through thecylindrical tube is given in Fig. 6 for n � 45, V0 � 120 mV, and1 M KCl. Fig. 6 also exhibits all of the features described for Fig.3, except the part corresponding to the role of the vestibule,where the weaker blockade is prominently absent. Based on1,000 simulations, the histogram P(�) is given in Fig. 7, where theresults of Fig. 4 are included for comparison. It is clear that P(�)is very sharply peaked, and there is only one peak for thenanotube. Our simulations thus suggest that it is preferable tohave a nanotube or just the �-barrel for sequencing purposesinstead of �HL.
Sequencing ProtocolAs already mentioned, there is a perturbation in the ionic currentthrough the channel, as soon as a monomer is eclipsing the pathwayfor small ions in front of the channel (see Figs. 3 and 6). This finding
Fig. 7. Comparison of P(�) between �HL and nanotube.
Fig. 8. Proposed experiment. (A) Strategy. (B and C) The ionic current tracesfor two homopolymers (B) and two heteropolymers (C) (sequence given inInset). Each polymer has eight bases.
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suggests a novel experimental setup for sequencing polymers byusing measurements of ionic current through channels. The strategyis illustrated in Fig. 8A, where the circle represents the mouth of thepore through which electrolyte ions pass and create the measuredionic current under an applied voltage gradient. The pore can beeither �HL or a nanotube with prescribed diameters. Then apolymer is dragged across the front of the pore in the direction ofthe arrow in Fig. 8A. The physical size of the monomer at the porefront excludes small ions to pass by and consequently reduces themagnitude of the ionic current. When monomers with differentsizes are dragged normal to the ion flow, different levels of ioniccurrent are registered. When a polymer with eight dC units isdragged at the speed of 1.0 Å��s near the pore entrance, in thenormal direction to the current flow, the time dependence of ioniccurrent is given in Fig. 8 B and C. The saw-tooth nature of the traceis caused by the periodicity arising from the monomer entering theaperture, then generating maximum eclipse, and then exiting theaperture. If an octomer of dA is simulated under identical condi-tions (where the diameter of the side-chain bead is 5 Å instead of2.5 Å for dC and the bond length connecting the side-chain to thebackbone is 3.75 Å), the ionic trace is as given in Fig. 8 B and C. Inthese particular simulations, the pore diameter and length are takenas 14 and 50 Å, respectively, to detect the identity of only onemonomer. Depending on the size of the aperture, signals corre-sponding to dimers, trimers, etc., can be generated. In the presentillustration of the concept, we now drag octomers with differentsequences containing dA and dC. Depending on which monomersequence is being dragged, the ionic current will toggle back andforth between the values corresponding to pure dA and dC values.
There is only one unique ionic current trace for a given sequence,as illustrated by two distinctly different traces for two sequences.Although the differences in the ionic current from our simulationsare rather small, the current experimental status (33) is able todiscern such small differences. The strategy emerging from oursimulations will enable the precise sequencing of polymers byseveral orders of magnitude faster than the currently availabletechniques.
ConclusionsOur simulations using coarse-grained models of the polymer andthe channel with appropriate accounting of dielectric heteroge-neity reproduce all of the essential features observed experi-mentally with �HL. The vestibule is found to play a significantrole by providing an entropic trap and thereby slowing down theprocess. Another observation is that there is no direct correla-tion between translocation time and blocked current. For eachvalue of �, there are two dominant values of blocked current.Finally, an experimental setup is proposed that would enableprecise sequencing of polymers by several orders of magnitudefaster than current techniques.
In summary, our combination of Langevin dynamics simula-tions of polymer dynamics with the PNP formalism for ioniccurrent, in the general premise of united atom description, offersa computational platform for discovering generic physical prin-ciples behind polymer translocation through biological channels.
This research was supported by National Institutes of Health Grant1R01HG002776-01, National Science Foundation Grant DMR-0209256,and the American Chemical Society Petroleum Research Fund.
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