arXiv:0709.3850v1 [physics.chem-ph] 24 Sep 2007 aooe igebs artast h oei about in pore the state transits solid pair base with ∼ single experiments noise a typical between nanopores off In trade resolution. the and and is magnitude. faster challenge of This is orders technological many that A by [9]. ones method pore existing single than the sequencing cheaper with the transits a read where DNA provide be point the would can as the strand resolution to DNA base a it motivated of refining extent sequence large of base a possibility to the In- is by phenomenon recently. of the appeared in translocation theoret- have driven terest few electrically nanopores a the across of as on well polymers number 8] as A [1, 7] ones detected 6, [3]. ical 5, reliably polymer [4, be the studies can characterize experimental pore to the used transits and nanopore polymer single a a of translocation. as conductance the electrical charge, drive in a change can carries The potential polymer phenomenon electric the applied common If an a [2]. systems is biological membranes in cell in apertures h oewlspa nipratrl.Tetranslocation of The proximity role. the important an an here play in that walls particles pore except the charged unlike field not small electric is of applied physics electrophoresis underlying of The the counter-ions that and within electrolyte. co- from the the balance for in arising accounting a forces proper by with viscous pore determined and is electrical translocation nanopore. of of the speed across process translocation The the driven describing for electrically of proposed was model dynamic experimental the guiding in value of param- work. therefore controllable A is the the on eters how noise. depends determine the speed to change above problem translocation the the detectable of then be analysis because not theoretical DNA would the current down in re- sufficiently slow be to cannot pore duced the across voltage the hand h rnlcto fplmr cosnnmtrscale nanometer across polymers of translocation The na ale ae 1 hneot ae )ahydro- a A) Paper (henceforth [1] paper earlier an In 10 h ffc fsl ocnrto nteeetohrtcspee electrophoretic the on concentration salt of effect The − 8 e uhtosott ersle.O h other the On resolved. be to short too much – sec h osblt fauiomsraecag ntewlso th of walls the fixed on for charge that surface shown length uniform thereby of a is cylinder Deb a of finite of a possibility that of the to case geometry the pore Here the restricting length. Debye varyi small slowly vanishingly symmetric a axially an through polyelectrolyte al h lcrpoei pe sidpneto h Debye the of independent is speed electrophoretic the wall, ntesl ocnrto nyt h xetta the that extent the to only concentration salt the on eednei eywa.I ssonta h acltdtran 87.15.Tt calculated numbers: insensi PACS the this that reveal shown that is nanopores silicon It in weak. measurements very is dependence napeiu ae 1 yrdnmcmdlfrdetermining for model hydrodynamic a [1] paper previous a In 15Seia od vntn L60208 IL Evanston, Road, Sheridan 2145 eateto ehnclEngineering Mechanical of Department ζ ptnil ntesraeo h oylcrlt n nthe on and polyelectrolyte the of surface the on -potentials hog nanopore a through Dtd coe 4 2018) 24, October (Dated: otwsenUniversity, Northwestern adpGhosal Sandip oe sraoal ic h essec egho double of length persistence the since reasonable is model aiga velocity a at cating h yidia oe(fradius (of pore cylindrical the bevdeprmna eedneo h translocation concentration. salt the on of speed the dependance with whether experimental consistent determine is observed to model hydrodynamic is long proposed objective a the The to geometry pore. the cylindrical Debye paper restricting finite a this while for mech- thickness allowing In but the Layer A the on Paper based or in detected. proposed calculated concentration anism be is salt to speed with translocation small the all too at ei- was vary time variation not translocation probable did it con- most ther Remarkably, the KCl 1.0M. that with to found electrolyte mM was 50 an from for varying nanopore centration state solid a recently More the work. in experimental Smeets KCl) the M (1 in justified salt used was of electrolyte concentration layers high The Debye the of experi- thin nanopores. because with infinitely state agreement of solid close in assumption in [7] be measurements to mental translocation shown and calculated was layer The Debye speed radius. thin pore infinitely varying an slowly assuming symmet- by cylindrically pores for ric calculated explicitly was speed ebe yebli ihasals imtro 10 re- of actually diameter smallest experiment a the with in hyperboloid shape a pore sembles The lation. ftepleetoyeisd h oeb tagtrigid straight a by radius pore (of part rod the the cylindrical model inside valid us polyelectrolyte strictly Let the is cylinder. of long that in infinitely uniform assumption an for be an only to direction, field flow axial the the dimensions. assume these will with we pore Moreover, experiments, cylindrical a with consider calculation will we our comparing of purpose fietcldaee n length nanopore and cylindrical a diameter of that identical to of equivalent is pore the of Smeets nm. iue1sostegoer o u ipie calcu- simplified our for geometry the shows 1 Figure ζ ptnil eedo t n ute,this further, and it, on depend -potentials gnnpr a rsne ntelmtof limit the in presented was nanopore ng iiyt atconcentration. salt to tivity egh h rnlcto pe depends speed translocation The length. uhlre hntedaee.Further, diameter. the than larger much tal. et aooei ae noacut It account. into taken is nanopore e elyrtikesi osdrdwhile considered is thickness layer ye i ie r ossetwt recent with consistent are times sit hsclRve etr 20)Vl 8 238104 98, Vol. (2007) Letters Review Physical tal. et 1]hv ulse xeietldt on data experimental published have [10] h lcrpoei pe fa of speed electrophoretic the 1]rpr httebl conductance bulk the that report [10] v nteaildrcin( direction axial the in fapolyelectrolyte a of d a m hti oailwith co-axial is that nm) 1 = R 5 = L . 4n.Frthe For nm. 34 = m n translo- and nm) 1 pore x .Sc a Such ). . 2 2

v Electrolyte with boundary conditions ǫE ζ Polyelectrolyte f(a)= v 0 p (8) ∆V − µ x = 0 ǫE ζ f(R)= 0 w . (9) − µ Substrate The equation for f is readily integrated, giving us the x = L flow profile u(r):

2R u(r) φ ζw v ζw ζp ln(r/R) = − + + − (10) Electrolyte ue ζp ue ζp  ln(a/R)

2a where ue = ǫE0ζp/µ is a characteristic electrophoretic ve- locity. The potential φ is determined from the Poisson- FIG. 1: Geometry of the pore region. Boltzmann equation which is obtained on substituting the Boltzmann distribution on the right hand side of equation (2): stranded DNA (ds-DNA) is about 50 nm. In the absence of an applied pressure gradient (the inlet and outlet reser- 2 (∞) zkeφ voirs are both at atmospheric pressure), the flow veloc- ǫ φ = ezknk exp . (11) ∇ − − kBT  ity u = xˆ u(r) in the region between the polyelectrolyte Xk and the pore wall satisfies Stokes equation (the Reynolds ∞ number Re 10−4): Here zk is the valence and nk the far field concentra- ∼ tion of ion species k, e is the magnitude of the electronic 2 µ u + xˆρe(r) E0 = 0 (1) charge, kB is the Boltzmann constant and T the absolute ∇ temperature of the electrolyte. However, for the purpose where µ is the dynamic viscosity of the electrolyte, E0 of determining the translocation speed v, an explicit so- is the applied electric field in the pore and ρe(r) is the lution for φ will not be needed. The required velocity is electric charge distribution due to the co and counter- obtained from the condition that the total force on the ions as a function of the distance from the axis (r). On section of the polymer inside the pore is zero: account of axial symmetry of the cylindrical geometry considered, the movement of ions due to the current or Fe + Fv = 0 (12) bulk motion does not change the electric charge density where F and F are the electric and viscous forces per in the diffuse layer. Thus, ρe(r) is both the equilibrium e v unit length of the polymer. If λ is the charge per unit charge density (E0 = v = 0) as well as the charge density after a potential difference is applied across the cylinder length of the polymer then (E0 and v = 0). Poisson’s law relates ρe to the electric ′ 6 Fe = λE0 = 2πaǫE0φ (a) (13) potential φ: − ′ 2 by Gauss’s law. The viscous force, Fv = 2πaµu (a) can ǫ φ = ρe (2) ∇ − be calculated using (10): where ǫ is the dielectric constant of the electrolyte. Equa- ′ Fv φ (a) v ζw ζp 1 tions (1) and (2) must be solved with boundary condi- = a + + − . (14) tions 2πµue ζp ue ζp  ln (a/R)

φ(r = a) = ζp (3) On substituting equations (13) and (14) into (12) we get

φ(r = R) = ζw (4) ζw ǫE0ζp ǫE0ζw v = ue 1 = . (15) where ζp and ζw are the ζ-potentials at the surface of the  − ζp  µ − µ polyelectrolyte and the wall respectively. The classical condition of no-slip at the walls imply that Equation (15) for the translocation speed is the main re- sult of this paper. Surprisingly, equation (15) is identical u(r = a) = v (5) to the result we would have obtained if we had made u(r = R) = 0. (6) the assumption of infinitely thin Debye layers as we did in Paper A. To see this, observe that equation (15) to- Equations (1) and (2) imply that the function f = u gether with the Helmholtz-Smoluchowski slip boundary − ǫE0φ/µ satisfies the equation condition would imply a uniform flow u(r)= ǫζwE0/µ in the fluid which would satisfy the condition of− zero force 1 d df on the polyelectrolyte considered together with its Debye 2f = r = 0 (7) ∇ r dr  dr  layer. Equation (15) also follows directly (with ζw = 0) 3

2 classical Manning factor when the DNA is inside the pore over a wide range of salt concentrations. The electric field intensity is obtained by assuming that the entire voltage 1.5 drop of 120 mV occurs over the length of the equivalent 6 cylinder which is L = 34 nm, thus, E0 = 3.53 10 V/m. The ζ-potential at the SiO wall may− be obtained× 1 2 from the expression

ζw = a0 a1 log10 c (17) 0.5 − Translocation Time (ms) where c is the molar concentration of K+ ions. The func- tional form of the dependence on concentration follows 0 0 0.2 0.4 0.6 0.8 1 in the low counter-ion concentration limit from the non- KCl concentration (M) linear Gouy-Chapman model of the Debye layer in case of symmetric electrolytes. However, it has been shown FIG. 2: Translocation times for 16.5 µm long ds-DNA through to provide a good empirical fit to experimental data for a 10.2 nm diameter solid state nanopore. Solid line is calcu- counter-ion concentrations up to 1.0M [13]. For KCl on lated from equations (15), (16) and (17), the symbols are silica a0 0 and a1 30 mV. replotted from the data presented in Figure 4(b) (inset) of The translocation≈ ≈− velocity, v is calculated from equa- Smeets et al. [10] tion (15) for a range of concentrations from 0.01 to 1.01M. The corresponding translocation time for a Lp = 16.5 µm long DNA, t = Lp/v is shown as the solid line in from equation (12) for the translocation speed in Paper A Figure 2. A notable feature is the lack of sensitivity of the if one assumes a uniform cylinder for the pore shape. In translocation time to the salt concentration: it changes addition, it has the following very simple interpretation in by at most a factor of three when the salt concentration the limit of thin Debye layers: the part ǫE0ζw/µ is the ranges over two orders of magnitude. Taking into ac- electro-osmotic flow through the pore generated− by the count the considerable scatter in the experimental data applied field and ǫE0ζp/µ is simply the electrophoretic and the various approximations made in the theory, the speed of an object of arbitrary shape in a reference frame agreement between the two is quite reasonable, pointing fixed to the moving fluid in the nanopore [? ]. to the adequacy of the underlying hydrodynamic model. Equation (15) will now be compared to experimental The existence of a maximum in the translocation time data due to Smeets et al. referenced earlier [10]. In at a concentration of about 0.1 M KCl seems to be sup- the Debye-Huckel approximation the ζ potential of the ported by the data, although one cannot be completely polyelectrolyte, ζp is related to its linear charge density certain of this on account of the uncertainty in the data. λ through the formula (see Paper A) The principal uncertainties involved in applying the hy- drodynamic model to nanopores were discussed in Pa- λλD K0(a/λD) per A. Those same considerations apply to the current ζp = , (16) 2πaǫ K1(a/λD) calculations as well and need not be repeated here. It should also be kept in mind that although the motion of where λD is the Debye length and Kn are the modi- the polymer is treated as a unidirectional translation at fied Bessel functions of order n. For a univalent salt constant speed, the actual translocation takes place via a like KCl the Debye length (in nm) is given by [11] drift diffusion process as described by Lubensky and Nel- λD = 0.303/√c where c is the Molar concentration of son [8]. Here it is assumed, as is done in the classical the- the salt. The experimental data is in the range 0.05 to ory of Brownian motion of particles, that, the mean part 1.0 M so that λD ranges from 0.30 nm to 1.36 nm. Since of the motion of the polymer may be obtained through R = 5.1 nm and a = 1.0 nm, there is no significant the solution of a classical hydrodynamics problem that overlap between the Debye layers at the polyelectrolyte ignores the fluctuating forces. The hydrodynamic model and the wall for concentrations above 0.05 M, though or indeed any model that localizes the entire resistive for even smaller concentrations such effects may be ex- force at the pore region would predict a translocation pected. Thus, it is reasonable to use the expression (16) speed that is independent of polymer length. This is which is strictly true only for an isolated infinite rigid valid only for polymers that are not too long (see Paper rod in an unbounded electrolyte. The dielectric constant A). For very long polymers the resistive force has an en- −4 ǫ/ǫ0 = 80 and dynamic viscosity µ = 8.91 10 Pa tropic part as discussed by various authors [14, 15, 16]. s for the electrolyte are taken as those of water.× For St¨orm et al. [17] have suggested that the viscous drag on the linear charge density on the DNA we take 5.9 elec- the randomly coiled part of the polymer lying outside the tronic charges per nm reduced by the Manning factor of pore could also be significant. 4.2, thus λ = 2.25 10−10 C/m. This assumption is DNA translocation experiments that have been per- supported by recent− force× measurement experiments [12] formed to date can be divided into two classes; those that show that polyelectrolyte charge is reduced by the that use a natural protein nanopore (α-hemolysin) on a 4 lipid membrane [3, 4], and those that use a mechanical paper neither supports nor refutes the validity of these nanopore on a solid substrate made by specialized tech- alternate mechanisms for the 1.5 2.0 nm protein pores. niques [6, 18]. Although the principle is similar, these It does however show that for the− 5 10 nm solid state two types of nanopores differ with respect to some im- pores hydrodynamic resistance can− explain the experi- portant details. One essential physical difference is that mental data in the absence of any of the other mecha- the narrowest part of the α-hemolysin pore is about 1.5- nisms. 2.0 nm in diameter so that only single stranded DNA In conclusion, the hydrodynamic model introduced in or RNA is able to pass through it. When the pore is Paper A to calculate the average transition time of a blockaded by such a single strand the blockade is al- polyelectrolyte across a nanopore under an applied elec- most complete in that very few ions and probably none tric field was extended to treat the case of a finite Debye of the water is able to pass through the blocked pore. layer thickness, though the geometry was restricted to the Although solid state pores can be made with pore sizes simple case of a cylindrical pore. The predicted translo- approaching 1 nm, most of the experiments to date have cation times are found to be consistent with available been done with 5 10 nm diameter pores which can experimental data to within the uncertainties inherent in be made in a more− reliable and reproducible manner. the experiment and the theory. As a final remark, it is These larger diameter pores admit both single and dou- worth noting a few practical implications of the simple ble stranded DNA, and furthermore dsDNA can enter model presented here in relation to the problem of how the pore in a folded fashion, notwithstanding the relative one needs to tune the available parameters to make the rigidity of these polymers [7]. The main observable dif- translocation time as large as possible. First, Figure 2 ference in terms of translocations across the two kinds of shows that an optimal salt concentration exists for which pores is that the polymer passes through the solid state the translocation speed is a maximum, though the gain pores about two orders of magnitude faster than it does here is no more than a factor of 3. A better strategy is through α-hemolysin pores. It is important to stress that suggested by equation (15) which shows that v vanishes the analysis presented here applies to only the 5 10 if ζw = ζp. Physically this essentially amounts to bal- nm solid state nanopores. Although a similar hydrody-− ancing the electrophoretic migration of the DNA against namic model could be constructed to model the viscous an opposing electroosmotic flow generated at the wall. force arising out of the water in the vestibular part of In principle this could be achieved by using an alternate the α-hemolysin pore, such a model must of necessity substrate, a coating on the existing substrate or a physi- differ from the current one in the details of its formu- cal or chemical treatment of it that alters its ζ-potential. lation. Furthermore, the applicability of the continuum The object is to select a substrate such that ζw ζp equations for electrostatics and hydrodynamics would be and then “fine tune” the salt concentration to achieve≈ a questionable to a much greater degree than in the anal- closer match. As an example, Poly(methyl methacrylate) ysis presented in this paper. It has been suggested that (PMMA) is a commonly used substrate in microfluidic in order to explain the much slower translocation speed application for which a0 = 4.06 mV and a1 = 12.57 in protein pores, something other than hydrodynamics mV [20]. Using these values− in (17) and plotting− the re- is needed: perhaps an atomic level pore-polymer interac- sult together with equation (16) it is easily seen that the tion, an electrostatic self-energy barrier [19] or the energy two curves intersect at a salt concentration of about 0.6 cost associated with stripping hydration layers from the M. Operating near this molarity with a PMMA substrate polymer as it enters the pore. The results derived in this should result in significantly slower translocations.

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