Nanopore Measurements of Filamentous Viruses Reveal a Sub

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Nanopore Measurements of Filamentous Viruses Reveal a Sub-Nanometer-Scale Stagnant Fluid Layer Angus J. McMullen, Jay X. Tang, and Derek Stein ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.7b06767 • Publication Date (Web): 01 Nov 2017 Downloaded from http://pubs.acs.org on November 1, 2017

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Nanopore Measurements of Filamentous Viruses Reveal a Sub-Nanometer-Scale Stagnant Fluid Layer Angus J. McMullen, Jay X. Tang, and Derek Stein∗ Department of Physics, Brown University, Providence RI, USA E-mail: [email protected]

Abstract We report measurements and analyses of nanopore translocations by fd and M13, two related strains of filamentous virus that are identical except for their charge densities. The standard continuum theory of electrokinetics greatly overestimates the translocation speed and the conductance associated with counterions for both viruses. Furthermore, fd and M13 behave differently from one another, even translocating in opposite directions under certain conditions. This cannot be explained by Manningcondensed counterions or a number of other proposed models. Instead, we argue that these anomalous findings are consequences of the breakdown of the validity of continuum hydrodynamics at the scale of a few molecular layers. Next to a polyelectrolyte, there exists an extra-viscous, sub-nanometer-thin boundary layer that has a giant influence on the transport characteristics. We show that a stagnant boundary layer captures the essential hydrodynamics and extends the validity of the electrokinetic theory beyond the continuum limit. A stagnant layer with a thickness of about half a nanometer consistently improves predictions of the ionic current change induced by virus translocations and of the translocation velocity for both fd and M13 over a wide range of

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nanopore dimensions and salt concentrations.

Keywords solid-state nanopores, biosensing, nanoscale electrokinetics, electrophoresis, stagnant layer.

The translocation of a molecule through a voltage-biased nanopore is a nanoscale example of electrophoresis that plays a central role in emerging single-molecule bio-analyses including DNA sequencing, 1,2 protein sensing, 3 DNA-binding protein mapping, 4 and the detection of viruses. 5 Exceptional control can be exercised over the geometry of solid-state nanopores, and the nanopores themselves serve as detectors that are sensitive to the dynamics and the physico-chemical structure of single molecules passing through them. Nanopores are an excellent arena for studying electrophoresis and electrokinetics, as exemplified by a decade of fruitful investigations that have mainly focused on DNA translocations. 6–18 Electrokinetic models, which are the established theoretical framework for describing the coupled dynamics of fluids and ions in electrolyte solutions, 19 describe the phenomenology and the scaling behavior of DNA translocations well. Quantitatively, however, electrokinetic models of DNA translocations are surprisingly inaccurate. Consider as a reference Ghosal’s translocation model, which applies the conventional electrokinetic framework to a charged cylinder, representing DNA, inside a charged cylindrical channel, representing a nanopore, in the presence of an electric field. 8,9 The predicted pulling force on the DNA is about twice as strong as the force measured in experiments. 20 The discrepancy is even wider for the translocation speed, v, whose predicted value exceeds the experimental one by an order of magnitude. 20 Additional problems with the theory arise when one looks at the current change caused by a DNA molecule’s presence inside the nanopore, ∆I. It is well-established that ∆I is negative in concentrated salt solutions 2


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because DNA sterically blocks a fraction of the ionic current through the nanopore, and positive in dilute solutions because the charged backbone of DNA entrains counterions that enhance the current. 21 But the conventional electrokinetic theory overestimates by a factor of three the critical salt concentration, C0s , where translocations switch from enhancing to blocking current. 22 Why are electrokinetic models of translocations so inaccurate? The problem might lie with the description of the electrostatics provided by the mean-field, Poisson-Boltzmann equation. This possibility is implicitly recognized in many studies which introduce an ad hoc theoretical fix based on Manning counterion condensation. 23 One postulates that the counterions whose electrostatic interaction energy with the DNA molecule exceeds the thermal energy, kB T , bind tightly to the molecule, become immobilized, and renormalize the linear charge density. This results in a lower pulling force, a lower translocation velocity, a lower C0s , and better agreement between experiment and theory. 20,21,24 Although Manning counterion condensation 23 is commonly invoked to justify that hypothesis, Manning’s theory does not actually predict that the condensed counterions are immobile. In fact, on the contrary, Manning assumed that the condensed counterions were mobile along the polyelectrolyte surface in order to model the polarizability of polyelectrolytes. 25,26 Zukoski and Saville proposed a different theoretical model called the “dynamic Stern layer” model to reconcile measurements of the conductance and electrophoretic mobility of latex beads in solution, 27–29 and that model could in principle be applied to polyelectrolyte translocations. 30 But the dynamic Stern layer model is based on counterions in the Stern layer, meaning those counterions located closest to the charged surface, having an anomalously high mobility. That model therefore predicts a higher, not a lower C0s than predicted by the conventional electrokinetic theory. We finally note that a number of models have been developed that account for ion crowding effects which can lead to problems with the Poisson-Boltzmann equation; 31 however, those effects only become significant at electrode surfaces where high potentials are applied, rather than at the surface of a typical charged biopolymer.

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Here we explore a different possibility, that electrokinetic models are inaccurate because the validity of continuum hydrodynamics, on which they rest, breaks down near the surface of the polyelectrolyte. It is a common finding in experiments and molecular dynamics (MD) simulations alike that the viscosity of a fluid rises significantly within a few molecular layers of a solid surface relative to the bulk viscosity. This has been found in experiments using a surface force apparatus, 32,33 in MD simulations of uncharged systems, 34 in MD simulations of electrokinetics, 35–39 and most recently in MD simulations of DNA inside a nanopore. 22,40 The atomic-scale corrugation of the surface and electro-friction from solvated ions interacting with the discrete charges on the surface can each give rise to such a viscous boundary layer. Recently, Bonthuis and Netz showed that by including a sub-nanometer-thin layer of elevated viscosity near a surface the validity of electrokinetic models could be extended beyond the continuum limit; 41 their approach reconciled the electrokinetic theory with measurements of streaming currents and ionic conductance in nanofluidic channels. We expect that a thin, viscous boundary layer similarly affects the hydrodynamics in the vicinity of a translocating polyelectrolyte, with large consequences for the translocation dynamics and the ionic current signals. To test our hypothesis, we studied nanopore translocations by the filamentous virus strains fd and M13. These viruses have identical lengths (L = 880 nm) and diameters (d = 6.6 nm) but different charge densities. fd has a uniform negative linear charge density of −10e/nm, where e is the elementary charge. The protein coat subunit of M13, on the other hand, has one fewer charge than fd (−3e instead of −4e 42 ), and consequently a charge density that is 75 % as high as that of fd. Studying these two polyelectrolytes enables us to investigate whether anomalous behavior is observed with different polyelectrolytes than DNA, and whether the viscous boundary layer model is superior to the immobile, Manningcondensed counterion picture. A high-viscosity layer near the polymer surface is expected to have a large influence on the translocation speed and ∆I of both viruses, while leaving observable differences between fd and M13. On the other hand, the Manning condensation

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hypothesis, as it has been used to explain nanopore measurements, predicts that counterions will condense on both virus types until their effective linear charge densities obtain the same value, leading to indistinguishable behavior by fd and M13 in translocation experiments. Our measurements show that fd and M13 translocations, much like DNA, are much slower and change the current differently than the conventional electrokinetic model predicts. Furthermore, fd and M13 behave quite differently from one another; under some conditions, fd and M13 even translocate the same nanopore in opposite directions. We compare our results with the standard electrokinetic model by Ghosal and with a modified model that includes a thin, stagnant fluid layer near the polyelectrolyte surface. While the stagnant layer is a highly simplified description of the viscosity profile near the surface, it is able to resolve a number of puzzling experimental discrepancies in nanopore translocation experiments. It explains why counterions of M13 conduct less current than those of fd, and why they both conduct less than the bulk prediction. It also explains why the mobility of a virus in a voltage-biased nanopore is significantly lower than predicted by conventional electrokinetic models, and different for fd than for M13. Finally, measurements of ∆I induced by DNA translocations are consistent with the existence of a stagnant layer of similar thickness to that found around the two viruses, which suggests that a viscous boundary layer is indeed a general phenomenon. This work seeks to build a fundamental understanding of the determinants of the dynamics and electrical signatures of translocating molecules, which is also relevant to nanopore technology because the utility of nanopores rests on their ability to distinguish the ionic current signals and translocation dynamics of different molecular species. These findings also advance our understanding of electrophoresis, which is widely used to separate proteins and DNA molecules in gels.

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(a) fd virus: -4e per subunit

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(b) M13 virus: -3e per subunit A

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2.3 2.0 1

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Figure 1: (Left) Schematic showing the filamentous viruses fd (a) and M13 (b) translocating voltage-biased nanopores. (Right) Examples of the ionic current signals generated by fd (c) and M13 (d). The signals were recorded using a 19 nm diameter nanopore with a 150 mM KCl buffer at pH 8 and an applied voltage of 100 mV.

Results We measured filamentous virus translocations through solid-state nanopores in the straightforward manner reported previously 5 and illustrated in Fig. 1. Briefly, we first drilled a nanopore with a desired diameter in a 20 nm thick silicon nitride membrane using a transmission electron microscope and used it to divide two reservoirs filled with buffered salt solution. A voltage ∆V applied across these two reservoirs caused a current I to flow through the nanopore. The voltage bias also drew charged macromolecules through the nanopore. Translocations caused deviations in the baseline current that we characterized by their mean amplitude h∆Ii and their duration τ . Figure 1 shows a few typical translocation events of each virus type. From these measurements we compute the mean conductance change h∆Gi = h∆Ii∆V −1 and the mean electrophoretic mobility µ = Lτ −1 ∆V −1 of a translocating polyelectrolyte. The scatter plots in Figure 2 plot h∆Gi and µ for each translocation of a nanopore by a virus, for both fd and M13 viruses, for three nanopores with different radii, and for a range of salt concentrations. In these experiments ∆V = 100 mV. The data in Fig. 2 only represent the successful translocations through the nanopore, not the collisions between the virus and

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the nanopore that did not result in translocations. We classified events as translocation or collisions according to the procedure described previously in ref. 5 Briefly, collisions are events with shorter τ and lower h∆Ii than translocations, and consequently have a systematically lower event charge deficit. 5 We are only interested in the translocations at present, so we leave an analysis of the collisions to a future study. The clusters of events in Fig. 2 shift with C s and the nanopore diameter in ways that reveal some common trends in the dynamics and the induced conductance changes of both virus types. Increasing C s consistently and significantly increased h∆Gi and reduced µ. Increasing the nanopore diameter, on the other hand, consistently reduced µ, but only modestly over the D = 19 nm to D = 28 nm range tested. h∆Gi decreased modestly with increasing nanopore diameter; the trend is similar to the one observed in previous measurements on DNA. 12 Our translocation measurements also reveal significant and interesting differences between fd and M13. In particular, consider the dependence of the direction of translocation on the polarity of the applied voltage bias across the nanopore. Whereas fd always translocated toward the positive side, M13 was observed to translocate toward the positive side in the smallest nanopore, but toward the negative side in the largest nanopore. In the intermediate, D = 26 nm nanopore, the direction of M13’s translocation flipped from positive-seeking to negative-seeking as the C s increased from 50 mM to 100 mM. It is remarkable that under these identical conditions, fd and M13 actually translocated the same nanopore in opposite directions. M13 exhibited relatively broad distributions of the mobility, and we did not discover additional experimental evidence that would help us understand the cause. Figure 3(a) plots h∆Ii for fd and M13 translocations of a 28 nm diameter nanopore as a function of C s . As C s increased, both virus types blocked more current. For fd, h∆Ii decreased from about -50 pA at C s = 100 mM KCl to about -300 pA at C s = 300 mM KCl. M13 translocations tended to block roughly 30 pA more current than fd translocations for any given C s .

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-5 -50

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Figure 2: Scatter plots showing h∆Gi and the translocation mobility µ for each nanoporevirus interaction from experiments on fd (a, b, and c) and on M13 (d, e, and f). The diameter of the nanopore used in (a) and (d) was 19 nm; in (b) and (e) it was 26 nm; in (c) and (f) it was 28 nm. Colors indicate, according to the legend in (d), the salt concentration at which the data were measured. Figure 3(b) plots h∆Ii for double stranded DNA translocations of a 26 nm diameter nanopore as a function of C s . At low values of C s , dsDNA translocations enhance the measured current. As C s increased, DNA translocations block more current, just as with the two virus types above. h∆Ii decreased from about 50 pA at C s = 20 mM KCl to about -100 pA at C s = 1 M KCl. By interpolating the data, we estimate that h∆Ii flipped from positive to negative around C s = 250 nM KCl. These measurements are consistent with those of Smeets et al. 21 To study the dynamics, we determined the mean virus drift speed v by fitting the first passage time probability density to each distribution of τ , taking the lengths of both viruses to be 880 nm. 42 The distributions of τ were obtained by collapsing the translocation events in Fig. 1 onto the τ axis and binning them; the first passage time analysis was described previously. 5 We measured v for fd at C s = 200 mM using many nanopores with different 8


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0

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Figure 3: (a) Dependence of h∆Ii on C s for fd (blue) and M13 (red) measured using the same 28 nm diameter nanopore. The solid lines are two parameter fits of the stagnant layer model to the data (Afd = 0.33±0.005, δfd = 4.2±0.4 Å, AM13 = 0.26±0.02, and δM13 = 7.0±1.7 Å). (b) Dependence of h∆Ii on C s for dsDNA translocations of a 26 nm diameter nanopore. The solid lines are two parameter fits of the stagnant layer model to the data (ADNA = 0.27±0.04 and δDNA = 3.5 ± 0.5 Å). In both (a) and (b), error bars are the standard error of the mean, and dashed lines are predictions of the reference model with δ = 0. radii R. For each nanopore, we measured v for at least three voltages and fit a line passing through the origin to determine the translocation mobility for the ensemble, µ = v∆V −1 . Figure 4(a) plots the mobility as a function of R. With increasing R, the mobility of fd decreased from about 30 mm s−1 V−1 at R = 7 nm to about 10 mm s−1 V−1 near R = 25 nm. Figure 4(b) shows the salt concentration dependence of v, measured for fd and M13 in three different nanopores, with |∆V | = 100 mV. Although there was significant variability in the salt-dependent dynamics between the different nanopores, two important trends are clear. First, v decreased as C s increased for both fd and M13. Second, v was always lower for M13 than for fd under the same conditions. Importantly, v was frequently negative for M13, meaning that the translocations had to be recorded using the opposite voltage polarity to the usual one.

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Figure 4: (a) Dependence of the translocation mobility of fd on R with C s = 200 mM. Error bars are the standard error in the mobility. The solid line is a fit of the stagnant layer model to the data with σw = −20 mC m−2 . The dashed line is the prediction of the reference model (δ = 0) with σw = −20 mC m−2 . (b) Dependence of v on C s for fd (blue) and M13 (red). Error bars are the standard deviation found by bootstrap resampling. The plot combines data from nanopores with diameters of 26 nm (diamonds), 28 nm (circles), and 19 nm (triangles). The solid lines are predictions of the stagnant layer model using the mean nanopore diameter (24.3 nm), σw = −20 mC m−2 , and δ = 4.6 Å. The data were all measured with |∆V | = 100 mV.

Discussion Previous studies of nanopore translocation dynamics have focused almost exclusively on double-stranded DNA, which translocates more slowly and exhibits a crossover in the polarity of h∆Ii at a lower salt concentration C0s than predicted by the conventional electrokinetic model. If those anomalies stem from the breakdown of continuum fluid mechanics, they should be generic results for any translocating polyelectrolyte. In this section, we investigate that hypothesis by analyzing fd and M13 translocations in detail. To make quantitative comparisons with theory, we take as a reference the standard electrokinetic theory applied to a charged cylinder of radius a, representing the polyelectrolyte, inside a concentric cylindrical channel of radius R, representing the nanopore. 8,9 The nonlin-

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ear Poisson-Boltzmann equation describes the electrostatics in the electrolyte between the two cylinders ρe 2e2 C s =− sinh ∇ φ=− ǫǫ0 ǫǫ0 kB T 2

eφ , kB T

(1)

where φ is the electrochemical potential, ρe is the electric charge density arising from a local imbalance of anions and cations, ǫ is the dielectric constant of water (ǫ = 80 for water), and ǫ0 is the permittivity of free space. The motion of the ions is coupled to the motion of the fluid through the Stokes equation ~ = 0, η∇2~u + ρe E

(2)

~ is the axial electric field generated by the applied voltage, where ~u is the fluid velocity, E and η is the viscosity (η = 0.89 mN s m−1 for water at room temperature). The cylindrical symmetry guarantees that the polyelectrolyte moves in the axial zˆ-direction. The no-slip boundary condition applies at the surface of the polyelectrolyte and at the nanopore wall ~u(a) = vˆ z , ~u(R) = ~0.

(3)

The surface charge densities of the polyelectrolyte (σp ) and the nanopore wall (σw ) are related to φ(r) by Gauss’ Law σp ∂φ(r) σw ∂φ(r) =− , = . ∂r r=a ǫǫ0 ∂r r=R ǫǫ0

(4)

The fluid flow and electrochemical potential distributions can be obtained by first integrating eq. 2 to find u in terms of φ(r), then solving eq. 1 numerically to find φ(r). Figure 5 shows the predicted flow profile with a dashed line. The most direct probe of the viscosity near the polyelectrolyte surface is the salt concentration at which h∆Ii vanishes, C0s . The theoretical dependencies of h∆Ii on C s is plotted for fd and M13 in Fig. 3(a) and for DNA in Fig. 3(b), based in all cases on the conventional 11


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Stagnant layer δ A

lp

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Stagnant layer model fluid velocity profile

Figure 5: Sketch of a virus inside a nanopore indicating, on the left half, the fluid flow profile in the reference case and, on the right half, the flow profile in the presence of a stagnant, interfacial fluid layer. electrokinetic model, which we calculated numerically as follows: The current density from Ohmic conduction through the fluid is jc = C + (r)µ+ E −C − (r)µ− E, where C +,− (r) and µ+,− are the concentration profiles and mobilities of the positive and negative ions, respectively. The advective current density is jA = C + (r)u(r) − C − (r)u(r). The total current is the sum of jc and ja , integrated over the available cross section inside the nanopore. h∆Ii is found by calculating the current flowing through an empty nanopore and subtracting that from the current flowing through a nanopore with a polyelectrolyte inside it. We have neglected the advective current from the motion of the polyelectrolyte because v is much smaller than speed of the ions. We did take account of the access resistance of the nanopore and the departure of its actual geometry from that of a perfect cylinder by applying a geometric correction factor that was derived by Dekker for similar nanopores. 21 (See Supplemental for details of the calculation of h∆Ii.) The slope of the curve in Fig. 3(a) reflects the shifting competition between two effects. We performed similar calculations to produce the dashed lines in Figure 3(b). The bulk electrolyte solution has a conductivity that is proportional to C s , and the presence of a polyelectrolyte inside the nanopore disrupts the flow of bulk ions by an amount proportional to the occupied volume. The charged surface of the polyelectrolyte, on the other hand, entrains a compensating density of counterions into the nanopore, and those mobile counterions contribute positively to the ionic current. 21,43 The most informative quantity to consider is C0s , where h∆Ii extrapolates to zero. At 12


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C0s , the reduction of current due to the steric exclusion of bulk ions precisely balances the increase in current from entrained counterions, regardless of the geometry. The theoretical curves cross h∆I = 0i around 180 mM for fd and 120 mM for M13, while the theoretical curve crosses around 800 mM for dsDNA translocations. Experimentally, h∆I = 0i extrapolates to zero at much lower salt concentrations, around 60 mM for fd, 30 mM for M13, and 250 mM for dsDNA. Therefore, the mobility of the counterions accompanying the polyelectrolyte into the nanopore must be lower than their mobility in bulk solution. Interestingly, the electrokinetic model overestimates C0s for fd and M13 by a similar factor to its overestimate of C0s for DNA; based on the bare charge density of double-stranded DNA (λ ≈ −6 e/nm), the model overestimates C0s by a factor of about three. 22 This similarity is striking, given the significant differences between the molecular structures of filamentous viruses and dsDNA. Turning now to the translocation dynamics, v is determined by the balance of axial forces on the polyelectrolyte. 8 The electric driving force is Fe = −2πaσp ∆V . The viscous drag force on the segment of the polyelectrolyte inside the nanopore is Fv = 2πaηu′ (a)lp ; note that Fv depends on the full hydrodynamic solution inside the nanopore, including the effects of electroosmotic flow and the motion of the polyelectrolyte. There is also a viscous drag force on the moving parts of the polyelectrolyte outside the nanopore; that contribution is often ignored in DNA translocation models, though it can be large. 11 Here, we include the viscous drag on the entire filamentous virus, which we model by the drag force on a rod moving axially in free solution, Fr = 2πηLv/ ln(L/2a). The ability to simply estimate the viscous drag on the polyelectrolyte outside the nanopore in this way is an advantage of comparing stiff polyelectrolytes like filamentous viruses to theoretical models. Long, semiflexible polymers like DNA, on the other hand, form a random coil outside the nanopore which makes it difficult to write down an expression for the drag on that part. Therefore, we do not attempt to compare the dynamics of DNA and viruses here. Following

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Ghosal’s approach, the balance of forces obtains the translocation velocity, given by

v=

ǫǫ0 φ(a)∆V − ηlp

−1 φ(R) L log R/a 1− 1+ . φ(a) lp log L/2a

(5)

In eq. 5, the first factor on the right hand side is the characteristic electrophoretic velocity of the polyelectrolyte in free solution, the second factor is a correction due to electroosmotic flow generated at the nanopore wall, and the third factor is a correction due to the drag on the parts of the polyelectrolyte outside the nanopore. To make quantitative predictions of v, we also need to know φ(R), the potential at the nanopore wall. To estimate this, we measured the conductance of a 22 nm diameter nanopore as a function of C s . The conductance at low C s is governed by the counterions at the nanopore wall, 44 and a fit of a simple model 21 to the data obtained σw ≈ −20±16 mC m−2 (see Supplemental for details); this value is similar to charge densities of −30 mC m−2 that have been reported previously. 14 For the calculations that follow, we took σw = −20 mC m−2 . Figure 4(a) shows the theoretical v as a function of R, alongside the experimental results. The theory and the measurements both show a gently decreasing trend in mobility with R. This is consistent with the decreasing trend in the nanopore pulling force that has been measured on DNA using optical tweezers. 20 The radius-dependent change in mobility reflects a shift in the competition between the electrophoretic driving force, which pulls the polelectrolyte toward the positive side, and the electroosmotic flow of fluid induced by the negatively charged surfaces of the nanopore, which carries the polyelectrolytes toward the negative side. As the size of the nanopore increases and its interior surface increasingly resembles a plane, the electroosmotic flow that it induces grows. fd consequently exhibits a decreasing mobility with R. M13, whose lower charge density evidently causes it to have a lower electrophoretic pulling force, also translocates more slowly as R increases, but then it actually translocates in reverse when the electroosmotic component of its velocity becomes larger than the electrophoretic component. Thus the effect of nanopore size on electroosmotic

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flow explains the most surprising observation in Fig. 2, that fd and M13 translocate in opposite directions under certain conditions. The salt concentration also has large effects on the electrophoretic and electroosmotic components of the velocity. As C s increases and the Debye screening length shrinks, both electrophoresis and electroosmosis decrease, leading to lower absolute translocation mobilities. The electroosmotic component decreases more slowly, probably because the surface charge density of the silicon nitride nanopore increases with C s due to its chemical origin, and we again observe a reversal in the direction of M13 beyond a critical C s . The closely balanced effects of electrophoresis and electroosmosis in the case of M13 might explain why its mobility was found to be broadly distributed, with many extremely slow translocation events: intuitively, when translocations are slow, the chances of M13 interacting and temporarily sticking to the nanopore increase, and this would tend to broaden the mobility distribution. Although the general trends above are well explained by the conventional electrokinetic picture, the measured mobility of fd is approximately a factor of 4 lower than the mobility predicted by the reference model (see dashed line in Fig. 4(a)). Computer simulations based on the conventional electrokinetic models can consider the nanopore geometry in greater detail, 45 but cannot resolve the large discrepancy between the predicted and measured mobility. One explanation for such slow translocation velocities that we have heard (but not read) is non-specific binding, i.e. “sticking” of the polyelectrolyte to the nanopore wall. The high reproducibility of fd ’s mobility and its subtle but clear dependence on R across many different nanopores is strong evidence against that hypothesis; it is hard to imagine how sticking could be so consistent and vary predictably with the nanopore radius. A better explanation, which is commonly given, is that Manning-condensed counterions become completely immobilized on the polyelectrolyte surface. Ghosal, for instance, took σp to be the charge density of DNA that is renormalized by Manning counterion condensation; that is the surface charge density one obtains by setting φ(a) = kB T /e, which is independent of the bare surface charge density, although the value can change with the salt concentration. According

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to this hypothesis, fd and M13 should have precisely the same mobility, but the data clearly show that they do not. Figures 2, 3, and 4(b) all show that fd and M13 behave in markedly different ways—sometimes even translocating in opposite directions under the same experimental conditions. Another theoretical approach that is sometimes taken in the literature is to assign an effective charge density to the polyelectrolyte (usually lower than the bare charge density) for which the linearized Poisson-Boltzmann equation correctly describes the long-distance electrostatic forces. 46 One could assign that same effective charge density to the polyelectrolyte for the purpose of calculating its electrophoretic mobility, which amounts to a variation of the Manning condensation picture. This does not improve the predictions of the dynamics, however; see Supplemental Information. Finally, the dynamic Stern layer model, which posits that the counterions closest to the charged surface are more mobile than in the rest of the double layer, is inconsistent with the fact that the conductance of the polyelectrolytes inside a nanopore is lower than predicted by the conventional model, not higher. Recent molecular dynamics simulations have shown that the mobility of counterions near a translocating DNA molecule decreases sharply over a sub-nanometer distance nearing the surface. 22,40 In the simulations, the reduced ionic mobility results in a lower C0s and a lower electrophoretic pulling force than predicted by the usual electrokinetic model, just as one observes in experiments. The additional drag can be attributed to interactions between the ions and the discrete charges on the polyelectrolyte, known as electrofriction, 47,48 and to the ordered hydration layer at the polyelectrolyte surface. These effects give rise to precisely the kind of high-viscosity boundary layer that Bonthuis and Netz accounted for within the continuum electrokinetic framework in order to extend the validity of its predictions in nanochannels. 41 We expect the same high-viscosity boundary layer to be present near stiff filamentous viruses, so we will adjust the electrokinetic model in a similar way. We incorporate into the continuum theoretical model a boundary layer of thickness δ where η rises above its bulk value. For simplicity, we take η → ∞ throughout the boundary

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layer (i.e. a stagnant layer). This approximation corresponds to a highly simplified picture of the hydrodynamics, but one which facilitates the calculations considerably while nevertheless giving a reasonably good account of the electroosmotic flow velocity far from the polyelectrolyte, 41 which is what sets the observable translocation dynamics. The stagnant layer changes the boundary condition applied to ~u in the Stokes equation, shifting the no-slip plane away from the polyelectrolyte by δ, specifically u(r ≤ a + δ) = v. It also restricts ion transport to the region r ≥ a + δ. The other boundary conditions remain unchanged. The influence of these changes on the fluid flow profile is shown in Fig. 5. Within our model, δ is a fitting parameter, but one whose value should be consistent between different translocation measurements, and similar to the length scale over which viscosity enhancements have been observed with surface force apparatus measurements and molecular dynamics simulations. If this model is successful, a consistent value of δ ≤ 1 nm will improve predictions of v and ∆I for a variety of different polyelectrolytes. We begin by comparing the stagnant layer model to our measurements of h∆Ii. Figure 3(a) shows fits of the stagnant layer model to fd (blue) and M13 (red) data from a 28 nm diameter nanopore (solid lines). Figure 3(b) shows fits of the stagnant layer model to dsDNA (black solid line, see supplemental for more datasets). The theoretical curves were calculated as before, except with the new stagnant layer boundary conditions. The presence of the stagnant layer reduces the density of mobile counterions near the polyelectrolyte, thereby reducing C0s . Two parameters were varied in the weighted least-squares fits, δ and the multiplicative geometric correction factor A that accounts for details of the shape of the nanopore and the access resistance to it. 12,21 The fits shown in Fig. 3(a) obtained δfd = 4.3 ± 0.4 Å, δM13 = 7.0 ± 1.7 Å, and δDNA = 3.5 ± 0.5 Å; the uncertainties are the uncertainties in the fits. Measurements on three different nanopores (four for DNA) obtained mean stagnant layer thicknesses of δfd = 4.6 ± 1.4 Å, δM13 = 8.7 ± 3.2 Å, and δDNA = 3.2 ± 1.3 Å; the uncertainties include the uncertainties in each fit and the standard deviation of the δ values, all of which are assumed to be uncorrelated. The fits also typically obtained a geometric factor

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of A ≈ 0.3 for filamentous viruses. These comparisons show that a similar, sub-nanometer stagnant layer thickness explains the conductance change induced by different viruses in different nanopores, regardless of the nanopore size or other details about its structure or charge. The value of δ does not seem to track the Debye screening length, which varied by nearly an order of magnitude in the measurements on DNA. It is possible, however, that these estimates of δ suffer from systematic errors due to other, poorly-understood C s -dependent effects that become subsumed in the parameter A; for instance, access resistance at low salt concentrations is not yet properly understood. The mobility of fd and its systematic dependence on R seen in Fig. 4(a) provides another, independent opportunity to estimate δ. Adding a stagnant layer to the model of the translocation dynamics gives the following expression for v (see Supplemental):

v=

ǫǫ0 φ(a + δ)∆V − ηlp

1−

φ(R) φ(a + δ)

−1 L log R/(a + δ) 1+ . lp log L/2(a + δ)

(6)

We numerically solved the Poisson-Boltzmann equation to find φ(r) over a range of R, and then fit Eq. 6 to the data. Fig. 4(b) shows that Eq. 6 fits the data well. The fit obtained δfd = 4.6 Å, assuming σw = −20 mC m−2 . We note that the value of δfd is sensitive to σw ; the extreme estimates σw = −36 mC m−2 and σw = −4 mC m−2 yield δfd values of 20 Å and 2 Å, respectively. On the other hand, if we fix δfd = 4.2 Å — the value obtained from fitting the stagnant layer model to the C s -dependence of h∆Ii — and vary σw , we obtain σw = −22 mC m−2 , which is very close to our best independent estimate of σw . These results show that a 4–5 Å-thick stagnant layer explains both the R-dependent translocation dynamics of fd as well as the current change h∆Ii that it induces. The subtle dependence of the mobility on R arises from the changing electroosmotic flow generated at the charged nanopore wall and its interaction with the polyelectrolyte. 20 We note that it is possible to explain the slow translocation speed of a virus with a larger value of σw instead of a stagnant boundary layer. A higher σw results in a relative increase in the

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electroosmotic flow opposing the translocation. But the value of σw required to do that is considerably higher than our best independent estimate, and a higher σw cannot explain the low value of C0s . As a final test of the stagnant layer picture, Fig. 4(b) compares the model’s predictions with the C s -dependence of v at ∆V = 100 mV for fd and M13. For these calculations, we did not use any fitting parameters. Instead, we fixed σw = −20 mC m−2 and δds = δM13 = 4.6Å, based on our measurements of the nanopore conductivity and the R-dependent mobility of fd, respectively. The theoretical model captures the observed decrease in v with increasing C s . That trend is explained by counterions accumulating closer to the polyelectrolyte surface at high C s due to increased electrostatic screening, which increases the viscous drag on them. The magnitude of v is predicted to be higher for fd than it is for M13 at every C s , in agreement with our observations. The model quite accurately predicts the magnitude of v for fd. The predictions are slightly high for M13; a wider stagnant layer, around δM13 ≈ 6 Å, would give better agreement with the data. We speculate that a refined model of the viscosity profile, one which rises gradually from the bulk value to a high but finite value at the polelectrolyte surface, would give even better agreement with experiment. Such a profile would enlarge the differences in mobility and h∆Ii between fd and M13, because the counterions that are closest to the surface would contribute to the motion and the conductance, and it is there that the counterion densities of fd and M13 are most different. This concentration difference is a consequence of the nonlinearity of the Poisson-Boltzmann equation, Eq. 1.

Conclusion We probed the hydrodynamics within a few molecular layers of the surface of two filamentous viruses, identical except for their charge densities, by measuring their translocations through solid-state nanopores. The standard continuum theory of electrokinetics captures

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most of the important trends, but it greatly overestimates the translocation speed and the conductance associated with the counterions, just as the theory does for DNA. From our observations follows the inescapable conclusion that counterions close to a virus are less mobile than the same ions are in bulk fluid. This is a consequence of the end of continuum fluid mechanics’ validity and the existence of a thin boundary layer characterized by enhanced viscosity. Including a stagnant boundary layer is a simple way to capture the essential hydrodynamics in a theoretical model and extend the model’s validity and accuracy. A stagnant layer with a thickness of about half a nanometer consistently improves predictions of h∆Ii and v for both fd and M13 virus types over a wide range of nanopore dimensions and salt concentrations. Furthermore, the stagnant layer model captures the different behavior of fd and M13 — differences that are lost in the Manning-condensed counterion picture. These findings contribute to a deeper fundamental understanding of electrophoresis and other electrokinetic phenomena. Future work should focus on refining the description of the viscous boundary layer.

Methods We drilled nanopores through 20 nm thick membranes made of low stress LPCVD silicon nitride using a JEOL 2100F high-resolution transmission electron microscope (TEM). We also used the TEM to measure the diameter of the nanopore. The detailed fabrication procedure can be found in. 49 Before use, we cleaned each nanopore by boiling in a piranha solution for 15 minutes. The nanopore was then mounted in a custom made fluidic-cell. fd virus and M13 virus were grown and purified according to standard biological procedures, 50 using Xl1-Blue as the host E. coli strain. Virus samples were prepared in high concentrations in 100 mM NaCl, 20 mM Tris, pH 8.0 buffer, and then diluted to a 0.02 mg ml−1 (≈ 1 nM) virus concentration in the desired solution. We prepared solutions at pH 8 by diluting a stock concentration of 1 M KCl, 10 mM Tris, and 1 mM EDTA to the desired ionic

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strength using de-ionized water from a Milipore filtration system. We inserted Ag/AgCl electrodes in both chambers of the fluidic cell and connected them to an Axon Axopatch 200B current amplifier. The current amplifier maintained a constant potential difference while monitoring I. We conditioned the signal with an 8-pole Bessel filter with a cutoff frequency of 50 kHz, and then digitized it with a 250 kHz sampling rate. Before analysis, current recordings were further conditioned with an software based 8-pole low pass filter with a cutoff frequency of 10 kHz. We analyzed the current recordings with custom MATLAB (Mathworks) software to extract events and analyze them (described in detail in ref. 5 ). We separated signals corresponding to translocation attempts, or collisions between the virus and the nanopore, from those corresponding to end-to-end translocations via the event charge deficit, or ECD, 7 as well as a cutoff of ∆I. ECD is the integrated current change from the local baseline over the duration of the event. Signals with low ECD and low ∆I were classified as collisions. Full details of the analysis program can be found elsewhere . 5

Acknowledgement We thank Z. Dogic and P. Sharma and acknowledge support from NSF DMR-0820492 for virus samples. This work was supported by NSF DMR-1505878 and by the Brown University Institute for Molecular and Nanoscale Innovation.

Supporting Information Available The Supporting Information contains a full theoretical summary of the reference electrokinetic model and stagnant layer model, as well as a derivation of the predicted translocation velocity. It also presents conductance measurements used to estimate the surface charge of the nanopore. This material is available free of charge via the Internet at http://pubs.acs.org/. 21


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For table of contents only. Stagnant layer δ A

Fluid velocity profile without stagnant layer

Fluid velocity profile with stagnant layer

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