Ionic current inversion in pressure-driven polymer translocation through nanopores
IIonic current inversion in pressure-driven polymer translocation through nanopores
Sahin Buyukdagli , ∗ , Ralf Blossey † , and T. Ala-Nissila , ‡ Department of Physics, Bilkent University, Ankara 06800, Turkey Institut de Recherche Interdisciplinaire USR3078 CNRS and Universit´e Lille I,Parc de la Haute Borne, 52 Avenue de Halley, 59658 Villeneuve d’Ascq, France Department of Applied Physics and COMP Center of Excellence,Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, Finland Department of Physics, Brown University, Providence, Box 1843, RI 02912-1843, U.S.A. (Dated: February 27, 2018)We predict streaming current inversion with multivalent counterions in hydrodynamically drivenpolymer translocation events from a correlation-corrected charge transport theory including chargefluctuations around mean-field electrostatics. In the presence of multivalent counterions, electro-static many-body effects result in the reversal of the DNA charge. The attraction of anions to thecharge-inverted DNA molecule reverses the sign of the ionic current through the pore. Our theory al-lows for a comprehensive understanding of the complex features of the resulting streaming currents.The underlying mechanism is an efficient way to detect DNA charge reversal in pressure-driventranslocation experiments with multivalent cations.
PACS numbers: 05.20.Jj,61.20.Qg,77.22.-d
Coulombic interactions play a fundamental role in bio-logical systems as well as in various nanotechnologies cur-rently under development, from biopolymer analysis [1, 2]to nanofluidic transport [3–5]. Among them, the con-trolled translocation of biopolymers thorough nanoporeshas witnessed a rapid advancement recently [6–16]. Poly-mer translocation aims at probing the biopolymer se-quence via the characteristics of the ionic current thor-ough the pore. The improvement of this method requiresan accurate understanding of the electrohydrodynamicsat play. This challenge has not been fully met since pre-vious theoretical approaches either focused exclusively onentropic effects [17–21] or made use of mean-field (MF)electrostatics known to be inaccurate with multivalentions [22–24]. We have recently taken a step forward andshown that in electrophoretic
DNA translocation, DNAcharge inversion induced by multivalent ions reverses thedirection of the polymer without affecting the sign of theelectrophoretic current [25]. This result indicates thepossibility to use the charge correlation effect in orderto control the DNA translocation speed whose minimiza-tion is a required condition to improve the resolution ofthis method. In the present letter, we focus on pressure-driven translocation and show that the presence of mul-tivalent ions in the solution results in the inversion of thestreaming current thorough the pore, an effect absent inelectrophoretic polymer transport. Our prediction is sup-ported by previous nanofluidics experiments where ioniccurrent inversion had been observed in nanoslits confin-ing multivalent charges [26].Our translocating polymer model system depicted inFig. 1 consists of a charged liquid confined between ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] d −𝝈 𝒎 + -- + + + - + + + -- M em b ra n e L z 𝒂 r v −𝝈 𝒑 -- + + - M em b rane Reservoir, P m+ + - + m+ - + m+ -- - + - m+ - Reservoir, P FIG. 1: (Color online) Schematic representation of thenanopore. The cylindrical polyelectrolyte of radius a = 1nm and surface charge − σ p is confined to the cylindrical poreof radius d = 3 nm, length L = 340 nm, and wall charge − σ m .The pore connects the reservoirs at the hydrodynamic pres-sures P and P . Black circles denote the multivalent cationsI m + contained in the electrolyte mixture KCl + ICl m . the cylindrical polymer and pore with negative smearedcharge distributions − σ m and − σ p . The pore radius d = 3 nm and length L = 340 nm lie in the typicalrange of solid-state nanopores [13]. The polyelectrolyteradius is taken as the radius of double-stranded (ds)-DNA molecules a = 1 nm [25]. Driven by the pres-sure gradient ∆ P z = P − P > z -axis. We ne-glect any off-axis fluctuations. Moreover, we have re-cently found that dielectric discontinuities play a minorrole in solid-state pores with radius beyond the nanome-ter scale [25]. Thus, we assume that the whole systemhas the dielectric permittivity of water, ε w = 80. Theionic current through the pore is given by the number offlowing charges per unit time integrated over the cross- a r X i v : . [ c ond - m a t . s o f t ] F e b section of the channel,I str = 2 π e (cid:90) d ∗ a ∗ drr ρ c (r)u(r) . (1)In Eq. (1), e is the elementary charge, u ( r ) the liquidvelocity, and ρ c ( r ) the ionic charge density. Because thelength of the pore is much larger than its radius L (cid:29) d ,the liquid velocity and density are assumed to dependsolely on the radial distance. Furthermore, by introduc-ing the effective polymer radius a ∗ = a + a st and poreradius d ∗ = d − a st where the Stern layer a st = 2 ˚A cor-responds to the characteristic hydration radius of multi-valent cations [27, 28], we account for the stagnant ionlayer close to the charged nanopore and polyelectrolytesurfaces [29, 30].The correlation-corrected charge density in Eq. (1) iscomputed within the one-loop theory of electrolyte mix-tures in cylindrical pores [31], ρ c ( r ) = (cid:88) i q i n i ( r ) (cid:20) − q i φ ( r ) − q i δv ( r ) (cid:21) , (2)with the MF-level ionic number density n i ( r ) = ρ ib e − q i φ ( r ) θ ( r − a ) θ ( d − r ) . (3)The summation in Eq. (2) runs over the ionic species inthe solution, with each species i of valency q i and reser-voir concentration ρ ib . The external potential φ ( r ) de-termining in Eq. (3) the MF-level ion densities followsfrom the solution of the Poisson-Boltzmann (PB) equa-tion ∇ φ ( r ) + 4 π(cid:96) B (cid:88) i q i n i ( r ) = − π(cid:96) B σ ( r ) , (4)with the pore and polymer charge distribution function σ ( r ) = − σ m δ ( r − d ) − σ p δ ( r − a ). The one-loop potentialcorrection φ ( r ) and the ionic self-energy δv ( r ) includ-ing quadratic fluctuations around the MF potential areobtained from the relations φ ( r ) = − (cid:88) i q i (cid:90) d r (cid:48) v ( r , r (cid:48) ) n i ( r (cid:48) ) δv ( r (cid:48) ) (5) δv ( r ) = lim r (cid:48) → r (cid:8) v ( r , r (cid:48) ) − v bc ( r − r (cid:48) ) + (cid:96) B κ b (cid:9) . (6)The electrostatic propagator v ( r , r (cid:48) ) in Eqs. (5)-(6) is thesolution of the kernel equation [31] ∇ v ( r , r (cid:48) ) − π(cid:96) B (cid:88) i q i n i ( r ) v ( r , r (cid:48) ) = − π(cid:96) B δ ( r − r (cid:48) ) , (7)the Bjerrum length in water at temperature T = 300 Kis (cid:96) B = e / (4 πε w k B T ) (cid:39) κ b = 4 π(cid:96) B (cid:80) i q i ρ ib , and theCoulomb potential in an ion-free bulk solvent is givenby v bc ( r ) = (cid:96) B /r . ( a ) C R M g
S p d
S p m
Istr(pA) r m b ( M ) r = 0 . 1 M r = 0 . 0 1 M r -(r)/ r -b r ( n m ) r cum(r)/ ( ps pa ) ( c ) H y d r .E l e c t . ( b )
FIG. 2: (Color online) (a) Streaming current curves at thepressure gradient ∆ P z = 1 bar against the reservoir density ofthe multivalent counterion species given in the legend. Opencircles mark the charge reversal (CR) points. (b) Electrostatic(black curves) and hydrodynamic (black curves) cumulativecharge densities, and (c) Cl − densities in the KCl + SpdCl liquid at the reservoir concentrations ρ b = 0 .
01 M (dashedcurves) and 0 . σ m =0) contains a ds-DNA of charge density σ p = 0 . e/ nm , withthe bulk K + density ρ + b = 0 . The liquid velocity u ( r ) in Eq. (1) satisfies the Stokesequation with applied pressure η ∇ u ( r ) + ∆ P z L = 0 , (8)with the viscosity coefficient of water η = 8 . × − Pa s. Solving Eq. (8) in the pore with the hydro-dynamic boundary conditions u ( d ∗ ) = 0 and u ( a ∗ ) = v T where v T stands for the translocation velocity of the poly-mer, and taking into account that the viscous frictionforce F v = 2 πa ∗ ηu (cid:48) ( a ∗ ) vanishes in the stationary stateregime, the streaming current velocity follows in the formof a generalized Poisseuille profile, u ( r ) = ∆ P z ηL (cid:104) d ∗ − r + 2 a ∗ ln (cid:16) rd ∗ (cid:17)(cid:105) . (9)The charge density (2) and the velocity profile (9) com-plete the calculation of the streaming current in Eq. (1).We consider first a ds-DNA with charge density σ p =0 . e/ nm confined to a neutral pore ( σ m = 0) [32].Fig. 2(a) displays the streaming current of the elec-trolyte mixture KCl + ICl m against the reservoir den-sity of the multivalent cation species I m + specified inthe legend. One sees that for all multivalent counte-rion species, the ionic current is positive at dilute con-centrations. This limit corresponds qualitatively to theMF-transport regime driven by the attraction of cationsinto the pore by the negative charge of the translocat-ing DNA. With the increase of the magnesium densityin the KCl + MgCl liquid, the ionic current diminishesbut stays positive. Increasing the multivalent ion den-sity in the electrolyte mixtures with Spd or Spm molecules, the ionic current first vanishes and then be-comes negative above the respective reservoir concentra-tions ρ b ≈ × − M and ρ b ≈ × − M. Inthe density regime ρ mb (cid:38) . − . ρ cum ( r ) = 2 π (cid:82) ra d r (cid:48) r (cid:48) ρ c ( r (cid:48) ) and the hydrodynamic cu-mulative charge density ρ ∗ cum ( r ) = 2 π (cid:82) ra ∗ d r (cid:48) r (cid:48) ρ c ( r (cid:48) ) ofthe KCl + SpdCl liquid rescaled with the DNA charge.The hydrodynamic cumulative density accounts exclu-sively for the charges contributing to the streaming flow.We also report in Fig. 2(c) the local densities of Cl − ions.At the bulk concentration ρ b = 0 .
01 M, the net cumu-lative charge density is seen to exceed the DNA chargeat r (cid:38) .
25 nm. This is the signature of the DNA chargereversal induced by pronounced electrostatic correlationsbetween Spd counterions bound to DNA. One notesthat as a result of the charge-reversal effect, a weak Cl − excess ρ − ( r ) > ρ − b takes place between the nanopore andthe DNA molecule. However, because the anion attrac-tion to the charge-inverted DNA is not significant, thehydrodynamic flow charge is dominated by counterionsand stays positive at the corresponding Spd density,i.e. ρ cum ( r ) > a ∗ < r < d . By increasing the bulkspermidine concentration to ρ b = 0 . − adsorption into the pore (see Fig. 2(c)).This significant anion excess results in a negative hydro-dynamic charge density ρ cum ( d ) < ρ b ≈ . I str >0I str <0 I str >0 (M) p ( e / n m ) I str <0 +b = 0.1 M (c)(b)(a) Spd (M) p ( e / n m ) Mg r(nm) Hydr.Elect. c u m (r) / ( p a ) +b = 0.2 M +b = 0.1 M -1 M5.0 x10 -2 M10 -2 M10 -3 M p ( e / n m ) +b (M) FIG. 3: (Color online) Characteristic polyelectrolyte chargeagainst (a) Spd concentration (K + density fixed at 0 . + concentration curves (Spd densities dis-played next to each curve) separating phase domains withpositive current I str > str < liquid.The inset in (a) displays the phase diagram of the main plotfor the KCl + MgCl liquid. (c) Electrostatic (black curves)and hydrodynamic (black curves) cumulative charge densi-ties at the bulk K + concentration ρ + b = 0 . ρ + b = 0 . densities are σ p = 0 . e/ nm and ρ b = 0 .
01 M. Theremaining parameters are the same as in Fig. 2. ducing the charge-reversal effect driving the ionic currentinversion. This explains the minimum of the streamingcurrent curves in Fig. 2(a).These results show that although the DNA charge re-versal is a necessary condition for the occurrence of thecurrent inversion, the charge-reversal effect has to bestrong enough for the adsorbed anions to compensatethe contribution from cations to the streaming current.In particular with Mg ions in contact with the ds-DNAmolecule, the charge reversal that takes place at ρ b ≈ . × − M remains insufficient to invert the stream-ing current up to the highest bulk density ρ b = 0 .
5M considered in Fig. 2(a). In the electrolyte mixtureswith spermidine and spermine molecules, the charge re-versal densities ρ b ≈ . × − M and ρ b ≈ − M are lower than the current inversion densities by sev-eral factors. These observations contradict the conclusionof Ref. [26] where the authors had argued a one-to-onemapping between the reversal of the membrane chargeand the sign reversal of the streaming current. In orderto determine the charge densities where the current in-version effect is expected in translocation experiments,we plotted in Fig. 3(a) the critical polymer charge ver-sus multivalent ion density curves where the streamingcurrent switches from positive to negative. The mainplot shows that in the KCl + SpdCl liquid, the criticalpolymer charge for current inversion drops with increas-ing spermidine density ( ρ b ↑ σ ∗ p ↓ ) up to the point ρ b ≈ . ρ b ↑ σ ∗ p ↑ ). The minimum of thiscurve at σ p ≈ . e/ nm corresponds to the lowest poly-mer charge below which it is impossible to observe ioniccurrent inversion regardless of the bulk spermidine con-centration. How is the phase diagram modified if onereplaces the spermidine molecules with magnesium ions?The inset of Fig. 3(a) indicates that in the KCl + MgCl liquid, the critical polymer charges for current inversionare about three times higher than in the KCl + SpdCl electrolyte. The main prediction of this diagram is thatwith Mg cations, it is impossible to observe the signreversal of the streaming current with translocating ds-DNA molecules whose smeared charge density σ p = 0 . e/ nm is located well below the minimum of the currentinversion curve.What is the effect of monovalent K + counterions onthe streaming current inversion? In Fig. 3(b), we plotthe critical σ ∗ p − ρ ∗ + b curves separating the phase domainswith positive and negative currents. One sees that at ρ b = 0 .
01 M (blue curve) and σ p = 0 . e/ nm , theincrease of the K + density from 0 . . + ions drive the system back to the MF charge trans-port regime. We emphasize that such an effect has in-deed been observed in pressure-driven nanofluidic exper-iments [26]. In the phase diagram, one also sees that thehigher is the polymer charge or the spermidine concentra-tion (increased in the clockwise direction), the higher isthe K + density needed to cancel the net current. In orderto better characterize the role of K + ions, in Fig. 3(c), weshow that the increase of the K + density from ρ + b = 0 .
1M to 0 . + ions suppress the negative ionic current by cancelling theDNA charge reversal.We finally characterize the effect of the finite porecharge on the reversal of the streaming current. First ofall, Fig. 4(a) shows that the ion current rises with the wallcharge up to σ m ≈ . e/ nm where it reaches a peak anddrops above this value. Then, one notes that depending -0.4-0.20.00.20.4 I s t r ( p A ) (b)(a) IV IIIIII = 10 -2 M = 4.5x10 -2 M = 10 -1 M p ( e / n m ) m ( e/nm ) I str >0 I str <0 FIG. 4: (Color online) (a) Streaming current thorough theDNA-blocked pore ( σ p = 0 . e/ nm ) and (b) characteristicpolyelectrolyte charges splitting the phase domains with pos-itive and negative current against the surface charge densityof the nanopore confining the KCl + SpdCl liquid. Spd densities corresponding to each curve in (a) and (b) are dis-played in the legend of (b). The remaining parameters arethe same as in Fig. 2. on the Spd concentration, the non-monotonical shapeof the streaming current curve may result in a single cur-rent inversion point ( ρ b = 0 .
01 M), two inversion points( ρ b = 0 .
045 M), or no inversion point ( ρ b = 0 . σ p − σ m curves splitting the positive and negative current regions.In this figure, the current inversion points of Fig. 4(a) cor-respond to the intersections between the curves and thehorizontal line marking the DNA charge. We split theblack curve in Fig. 4(b) into three segments. We foundthat on the segment I-II, the increase of the pore charge σ m has the main effect of bringing more K + ions into thepore. This explains the increase of the pore conductivity( σ m ↑ I str ↑ ) in Fig. 4(a). Furthermore, the branch II-IIIof the critical curve corresponds to the regime of stronglycharged nanopores where correlations result in the rever-sal of the nanopore charge even in the absence of thepolyelectrolyte. As a result, the attraction of Cl − ions tothe charge inverted nanopore wall decreases the stream-ing current ( σ m ↑ I str ↓ ) and switches the latter frompositive to negative in Fig. 4(a). Fig. 4(b) also predictsthat at the nanopore charges located on the interval III-IV, the ionic current inversion is expected to occur at twodifferent polyelectrolyte charge densities. This complexdependence of the streaming current on the nanopore andpolymer charge calls for experimental verification.To conclude, we have predicted streaming current in-version induced by translocating polyelectrolytes in thepresence of multivalent counterions. We have shown thatat physiological charge densities, streaming current re-versal upon ds-DNA penetration takes place only if themultivalent charges in the electrolyte are trivalent or ofhigher valency. We have also found that ionic currentinversion is cancelled by monovalent cations but favoredby the nanopore charge. Our predictions can be easily verified by current pressure-driven translocation experi-ments. The proposed current inversion mechanism mayalso find applications in lab-on-a-chip technologies.SB gratefully acknowledges support under the frenchANR blanc grant “Fluctuations in Structured CoulombFluids” during the realization of the first part of theproject. T. A-N. has been supported in part by theAcademy of Finland through its CoE program COMPgrant no. 251748 and by Aalto University’s energyefficiency program EXPECTS. [1] J. J. Kasianowicz, E. Brandin, D. Branton, and D. W.Deamer, Proc. Natl. Acad. Sci. U.S.A , 13770 (1996).[2] S. Huang et al., Nature Nanotech. , 868 (2010).[3] G. Miles, S. Cheley, O. Braha, and H. Bayley, Biochemis-tery , 8514 (2001).[4] L. Q. Gu, and H. Bayley., Biophys. J. , 1967 (2000).[5] D. Stein, F. H. J. van der Heyden, W. J. A. Koopmans,and C. Dekker, Proc. Natl. Acad. Sci. U.S.A , 15853(2006).[6] A. Meller, L. Nivon, and D. Branton, Phys. Rev. Lett. , 3435 (2001).[7] D. J. Bonthuis, J. Zhang, B. Hornblower, J. Math´e, B.I. Shklovskii, and A. Meller, Phys. Rev. Lett. , 128104(2006).[8] Y. Astier, O. Braha, and H. Bayley, J. Am. Chem. Soc. , 1705 (2006).[9] J. Clarke, H.-C. Wu, L. Jayasinghe, A. Patel, S. Reid,and H. Bayley, Nature Nanotech. , 265 (2009).[10] I. M. Derrington, T. Z. Butler, M. D. Collins, E. Manrao,M. Pavlenok, M. Niederweis, and J. H. Gundlach, Proc.Natl. Acad. Sci. U.S.A , 16060 (2010).[11] H. Chang, F. Kosari, G. Andreadakis, M. A. Alam, G.Vasmatzis, and R. Bashir, Nano Lett. , 1551 (2004).[12] A. J. Storm, J. H. Chen, H. W. Zandbergen, and C.Dekker, Phys. Rev. E , 051903 (2005).[13] R. M. M. Smeets, U. F. Keyser, D. Krapf, M.-Y. Wue,N. H. Dekker, and C. Dekker, Nano Lett. , 89 (2006).[14] H. Liu, et al., Science , 64 (2009).[15] M. Tsutsui, M.Taniguchi, K. Yokota, and T. Kawai, Na-ture Nanotech. , 286 (2010).[16] M. Firnkes, D. Pedone, J. Knezevic, M. D¨oblinger, andU. Rant, Nano Lett. , 2162 (2010).[17] W. Sung and P. J. Park, Phys. Rev. Lett. , 783 (1996).[18] T. Ikonen, A. Bhattacharya, T. Ala-Nissila, and W. Sung, Phys. Rev. E , 051803 (2012).[19] T. Ikonen, J. Shin, W. Sung, and T. Ala-Nissila, J. Chem.Phys. , 205104 (2012).[20] T. Ikonen, A. Bhattacharya, T. Ala-Nissila and W. Sung,EPL 103(3), 38001, (2013).[21] F. Farahpour, A. Maleknejad, F. Varnikc, and M. R.Ejtehadi, Soft Matter , 2750 (2013).[22] S. Ghosal, Phys. Rev. E , 041901 (2006).[23] S. Ghosal, Phys. Rev. Lett. , 238104 (2007).[24] D. Keijan, S. Weimin, Z. Haiyan, P. Xianglei, and H.Honggang, App. Phys. Lett. , 014101 (2009).[25] S. Buyukdagli and T. Ala-Nissila, Langmuir , 12907(2014).[26] F. H. J. van der Heyden, D. Stein, K. Besteman, S.G. Lemay, and C. Dekker, Phys. Rev. Lett. , 224502(2006).[27] H. Ohtaki and T. Radnai, Chem. Rev. , 1157 (1993).[28] In our numerical calculations, the Stern layer is takenfinite only if the corresponding surface carries a finitecharge.[29] L. Joly, C. Ybert, E. Trizac, and L. Bocquet, Phys. Rev.Lett. , 257805 (2004).[30] R. Qiao and N. R. Aluru, Phys. Rev. Lett. , 198301(2004).[31] S. Buyukdagli and T. Ala-Nissila, J. Chem. Phys. ,064701 (2014).[32] The corresponding smeared charge distribution of ds-DNA had been estimated in Ref. [25] by fitting the con-ductivity data of DNA-blocked pores measured in elec-trophoretic transport experiments [13].[33] C. Labbez, B. J¨onsson, M. Skarba, and M. Borkovec,Langmuir , 7209 (2009).[34] J. Hoffmann and D. Gillespie, Langmuir29