Ion Transport in Confined Geometries below the Nanoscale: Access

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Ion Transport in Confined Geometries below the Nanoscale. Access Resistance Dominates Protein Channel Conductance in Diluted Solutions Antonio Alcaraz, M. Lidón López, María Queralt-Martín, and Vicente M. Aguilella ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.7b05529 • Publication Date (Web): 20 Sep 2017 Downloaded from http://pubs.acs.org on September 20, 2017

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Ion Transport in Confined Geometries below the Nanoscale. Access Resistance Dominates Protein Channel Conductance in Diluted Solutions

Antonio Alcaraza, M. Lidón Lópeza, María Queralt-Martína,b and Vicente M. Aguilellaa * a

Laboratory of Molecular Biophysics, Department of Physics, Universitat Jaume I, Av. Vicent Sos Baynat s/n

12071 Castellón, Spain. b

Program in Physical Biology, Eunice Kennedy Shriver NICHD, National Institutes of Health, Bethesda, MD

20892, USA (current affiliation). * Correspondence: [email protected]; Tel.: +34-964-72-8045

Abstract Synthetic nanopores and mesoscopic protein channels have common traits like the importance of electrostatic interactions between the permeating ions and the nanochannel. Ion transport at the nanoscale occurs under confinement conditions so that the usual assumptions made in microfluidics are challenged, among others, by interfacial effects such as access resistance (AR). Here we show that a sound interpretation of electrophysiological measurements in terms of channel ion selective properties requires the consideration of interfacial effects, up to the point that they dominate protein channel conductance in diluted solutions. We measure AR in a large ion channel, the bacterial porin OmpF, by means of singlechannel conductance measurements in electrolyte solutions containing varying concentrations of high molecular weight PEG, sterically excluded from the pore. Comparison of experiments performed in charged and neutral planar membranes shows that lipid surface charges modify the ion distribution and determine the value of AR, indicating that lipid molecules are more than passive scaffolds even in the case of large transmembrane proteins. We also found that AR may reach up to 80% of the total channel conductance in diluted solutions, where electrophysiological recordings register essentially the AR of the system and depend marginally on the pore characteristics. These findings may have implications for several low aspect ratio biological channels that perform their physiological function in a low ionic strength and macromolecule crowded environment, just the two conditions enhancing the AR contribution.

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Keywords Access Resistance | Ion Channels | PEG | nanofluidics | Ion transport | single-molecule electrophysiology

Graphical Table of Contents


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Small-scale biological and soft-matter channels often combine high affinity to particular species with the ability to exclude other undesired permeants.1–6 For instance, aquaporins show unusually high water permeability: three orders of magnitude larger than that expected from their pore size.2 Also, potassium channels have a permeability ratio for K+ over Na+ above 100:14 and calcium channels select for Ca2+ over Na+ with a ratio over 1000:1.7 Such peculiar transport features and others related have stimulated the design of biomimetic nanofluidic devices used in engineering applications like DNA sequencing, desalination, molecular separation, and energy conversion.8–15 In parallel, much effort has been made to elucidate how pore dimensions affect the transport mechanisms albeit many questions remain unanswered yet:2,8 a variety of specific phenomena found in the nanometer regime cannot be understood simply downsizing the concepts working well at the microscale.2 Compared to microfluidics, several factors make the difference. On the one side, the validity of the classical continuum treatments can be challenged by the molecular nature of the fluid, especially as regards the finite size of ions and the shielding effects.16 On the other side, transport at the nanoscale typically occurs under confinement conditions. The constrained geometry of nanochannels enhances the importance of interfacial effects,17,18 yields entropic effects due to obstacles and irregularities of the boundaries19 and allows the direct electrostatic interactions between the permeating ions and both the membrane and protein charges.1 Protein channels are ideal candidates to provide additional insights into these issues due to a number of reasons. Firstly, because their actual dimensions are just in the nanoscale or even below it, in contrast to synthetic nanopores whose typical diameters are tens or hundreds of nanometers2 (although pores below the nanoscale have been fabricated in the last years).20–23 Fortunately, the structure of many protein channels has been solved at atomic resolution during the last decades. Thus, the connection between the actual structure and the experiments can be attempted using models with different levels of complexity1,16 and, accordingly, the validity of continuum models can be probed with all-atom molecular dynamics (MD) simulations.16 Taking advantage of such detailed information, it is possible to assess the importance of geometrical effects such as the shape of the aqueous pore or the precise location of the charges.1,24 Furthermore, the reduced dimensions of protein channels yield much lower permeabilities than most synthetic nanopores.2 This implies that electroosmotic effects are almost negligible at the price that access resistance (AR) (also known as interfacing resistance) becomes noteworthy. In most biological channels, the AR of the two channel apertures contributes between 10% and 30% of the overall channel resistance,18,25 being then crucial to establish a proper relationship between the protein structure and the electrophysiological measurements. These AR estimations are made using Hall’s equation,26 which was originally developed for a hole embedded in a neutral membrane,17 However, it remains experimentally


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unexplored how electric charges present in the system (pore + membrane) regulate this convergence resistance. We tackle here this issue by using the reconstitution of a protein channel, the bacterial porin OmpF from E. coli, into planar lipid bilayers, either neutral or charged. Lipid membranes are usually considered inert scaffolds not participating in the electrophysiological function of large β-barrel channels like OmpF, given the fact that these proteins span over the whole bilayer and do not bend the lipid leaflet.27 The possibility that lipid charges could control the pore AR in such wide pores opens a completely different scenario. In this manuscript, we show how to split the measurable channel conductance into two not directly measurable contributions, the one coming from the pore itself and the AR one. In concentrated solutions the total AR (considering the two channel openings) displays values similar to those found in the literature,25,28 regardless of the lipid charge. However, we find that at low concentration the lipid charge can increase the total AR contribution from 20% to almost 80% of the total resistance. In addition, the comparison between theory and experiments shows that in neutral lipids the AR could hide the contribution of protein charges in the low concentration regime, so that charged pores apparently behave like almost neutral ones, what might be very misleading to understand the role played by pore surface in ion conductance.

RESULTS AND DISCUSSION Ion conduction in a biological nanopore The bacterial porin OmpF is naturally located in the outer membrane of E. coli. Each one of its three monomers forms a large β-barrel pore. It belongs to the family of the so-called mesoscopic channels since pore dimensions are below the nanoscale, namely around 7-11 Å in diameter in the narrowest section, see the top and lateral view of the channel in Figure 1A. This channel is usually regarded as weakly selective, attending to its mild non-specific ion charge discrimination in moderate and concentrated solutions.29 However, it is an almost ideally selective channel at low salt concentration (Figure 1B). The current carried by cations nearly matches the total one, so that the corresponding transport number is t+ ∼1.30 Such cationic selectivity can be rationalized in terms of the protein charges. Actually, previous calculations have shown that the effective fixed-charge concentration inside the pore is around 200-300 mM.31


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Figure 1. OmpF charge distribution and selective properties. A) Top (membrane plane, left) and side view (right) representations of one of the three channel monomers from the atomic structure (PDB code 2OMF). Red and blue color denote negatively and positively charged regions of the pore, respectively. B) OmpF channel selectivity represented by the fraction of the current carried by positive ions (cation transport number). Values are taken from Reversal Potential experiments in neutral DPhPC bilayers in which the high KCl concentration on the side of protein addition (cis side) is varied (and plotted) while keeping constant the low salt concentration on the opposite side of the membrane (trans side). Three different series of measurements, with concentration ratios 1.5, 3 and 10 (as labeled) are shown.

The different mechanisms regulating the permeation of ions through nanochannels have been extensively investigated by experiments in which the concentration of the bathing solutions is varied. See

2

for a

review.


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1e+2

B

Silica channel

1e+1 1e+0 1e-1 h = 1015 nm h = 70 nm

1e-2 1e-5

1e-4

1e-3

1e-2

1e-1

Conductance / nS

A Conductance / nS

1 nm PETP nanopore 0,1

Neutral PETP Charged PETP

0,01

1e+0

0,001

KCl concentration / M

C Conductance / nS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10

0,01

0,1

1


OmpF channel

1

DPhPS DPhPC

0,1

0,001

0,01

0,1

1


Figure 2. Biphasic behavior of synthetic and biological nanochannels upon ionic concentration. A) Conductance of aqueous filled fused silica channels. The lines depict the conductance expected in each channel from the conductivity of the bulk solution. Adapted with permission from ref. 32. Copyrighted by the American Physical Society. B) Conductance of a single 1 nm radius PETP charged and neutral nanopore. Data taken from ref. 33. C) Salt concentration dependence of single-channel conductance of the protein channel OmpF when embedded in a neutral (triangles) and a negatively charged (circles) membrane; measurements were done at pH = 6.

The trends displayed in the measurements shown in Figure 2A for fused silica nanoslits of similar surface charge but different height h32 are representative of a large variety of synthetic nanopores with very different aspect ratios.2 In the high concentration regime, the conductance increases linearly with concentration (as expected when the solution inside the channel is similar to the bulk one), whereas a saturation is found in the low concentration regime, commonly attributed to surface conduction. The competition between surface and bulk effects has been rationalized using the “Dukhin length” (LDu) in analogy to the Dukhin number used in colloid science. When LDu (defined as σ/2ec , being σ the pore surface charge density, e the electron charge and c the solution concentration) compares to the pore ACS Paragon Plus Environment

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radius, the surface conductance is expected to be significant. This is observed in the measurements shown in Figure 2A for concentrations below σ/2eh , i.e. when h ≤ LDu. The same trend has been reported in 1 nm radius polymer-etched nanopores

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as displayed in Figure 2B (green circles). This figure illustrates the

effect of surface charge on conductance for a given pore size. Hence, when negative charges in the membrane surface and within the pore are neutralized (blue triangles), conductance decreases monotonically with KCl concentration even in diluted solutions. Figure 2C shows the OmpF single-channel conductance as a function of KCl concentration measured in neutral (DPhPC) and charged (DPhPS) lipid membranes. Interestingly, for concentrated solutions, the picture seems quite similar to Figure 2A and 2B: concentrated solutions screen so effectively the pore and lipid charges that both curves overlap, indicating that conduction in this range is dominated by bulk effects. In diluted solutions the situation is different: conductance measurements in neutral membranes follow the trend expected from bulk behavior, whereas in charged membranes conductance attains a lower limit, as in nanopores shown in Figure 2A and 2B. In principle, the difference between the two series of measurements (in neutral and charged lipid membranes) plotted in Figure 2C could be due to the accumulation of counterions caused by the lipid charge near the channel entrances. Despite the apparent similarities, the datasets shown in Figure 2 A-C do not admit a common interpretation. As discussed in Figure 1B, the OmpF charged residues yield a remarkable pore charge that explains the almost ideal selectivity exhibited at low salt concentration in neutral membranes. Actually, when salt concentration drops below 0.1 M, Dukhin length becomes comparable to the radius of the channel constriction. This suggests that surface effects should yield a low limiting conductance independent on bulk salt concentration (as obtained for DPhPS, Figure 2C, circles). However, this reasoning is at odds with measurements performed in DPhPC over a wide range of low ion concentrations (Figure 2C, triangles): the surface conductance that explains charged nanopore conductance plateaus in Figures 2A and 2B seems insufficient for OmpF channel. It is intriguing why current saturation appears in experiments done with negatively charged membranes (Figure 2C, green circles), since it is known that lipid molecules do not take part in the OmpF pore inner structure (the protein spans the whole bilayer)34. Even more, one could wonder why lipid charges have an obvious impact on the pore conductance while protein charges seem to be invisible. The bulk/surface conduction explanation seems to be unsatisfactory in biological pores with dimensions below the nanoscale. To answer these questions one should consider not only the nanochannel but also entrance effects that become more significant the smaller the pore. In this connection, recent publications stress that the role of interfacing resistance could be decisive at low electrolyte concentration35, just the range where some features of Figure 2C may appear counterintuitive.


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The role of access resistance in biological nanopores The measured conductance in a synthetic nanopore or a biological ion channel does not depend exclusively on the number of charge carriers within the pore itself. The total resistance comprises the resistance of the pore itself as well as the AR in the two pore openings, that is:

R = Rp + 2Rac ,

(1)

being Rp the pore resistance. Hall26 developed the classical, widely used, expression of AR for a pore embedded in a neutral membrane (or in any inert substrate):

Rac =

1 , 2 Dκ

(2)

being D the pore diameter and κ the solution conductivity. In the case of a neutral channel, the pore resistance could be written as Rp = 4L / (κπ D2 ) , so that an effective way of considering the AR is to increase the channel length by ∼0.8 times its diameter (i.e. π D/4).28 From Equations 1 and 2 it becomes apparent that the relevance of AR (quantified as 2Rac /Rp = πD/4L) is minimal, if any, for pores of high aspect ratio, i.e. when L >> D. This is common in most, although not all,36–38 synthetic nanopores. In contrast, biological nanopores span cell membranes whose typical thickness is around 3-5 nm and their mouth diameter and length are often comparable. The Voltage Dependent Anion Channel (VDAC) of the mitochondrial outer membrane is an example of a low aspect ratio pore. Its 3D atomic structure was resolved a few years ago and revealed that the aqueous pore is an almost cylindrical β-barrel structure whose D/L ratio is ca. 0.6,39 so that AR estimation should be ca. 50% of pore resistance. Recent measurements of VDAC conductance40 in concentrated solutions (∼0.5 M KCl) report 2Rac values around 20% of the channel total resistance. The discrepancy could be ascribed to the fact that VDAC is not a neutral channel, but it is selective to anions.28 Also, it is possible that Hall’s equation overestimates the AR. In any case, these measurements convey at least one clear message: AR deserves an in-depth analysis in large β-barrel channels because it is an essential part of the total resistance. Furthermore, detailed calculations of AR for a highly idealized pore made by Aguilella-Arzo et al.41 reported values considerably smaller (almost an order of magnitude) than those predicted by Equation 2 in the case of charged membranes. Deviations from Hall’s prediction underscore the fact the electric


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double layer created by a charged membrane locally enhances conductivity resulting in a decrease in AR. To our knowledge, no experimental validation of this fact has been reported up to date.

Measurement of access resistance in OmpF Having in mind that the two experimentally accessible quantities are the channel conductance G and the solution conductivity κ, we can rewrite Equation 1 as:

1 G = 1 Gp + ( D κ )

−1

(3)

In principle, this equation should work relatively well for neutral membranes, but not when the pore lies on a charged surface. As commented before, previous theoretical studies showed that Hall’s equation does not consider the excess of counterions originated by the charge of the lipid molecules41. To account for this effect, we assume that the local conductivity κ* near the pore is increased by a factor γ (whose meaning will be discussed later in the paper) with respect to bulk conductivity, being κ*= γ κ: Accordingly,

1 G = 1 G p + ( Dγ κ ) . −1

(4)

Our strategy to obtain separate estimations of the AR and the pore conductance is the following. We measure OmpF single-channel conductance in several solutions with the same effective KCl concentration but different concentrations of high molecular weight polyethylene glycol (PEG) polymers. PEGs have been used before to explore channel geometry.42,43 We chose large enough PEGs to ensure that they are sterically excluded from the aqueous pore of the channel (PEG 8000 and 10000 are about 5 times larger than the pore entrances).44 Under these conditions, the pore conductance Gp should remain unchanged because PEG concentration in the external bulk solution has no effect on the ion concentration inside the pore (See cartoon in Figure 3). Then, the inverse measured conductance 1/G should scale linearly with the inverse of bulk conductivity κ-1 (which we measure independently). Once we get the linear fitting of the data, the AR to the channel in a KCl solution is simply calculated by substracting 1/Gp from the value of 1/G in a PEG-free solution:

2Rac =1 GPEG− free −1 Gp .


(5)

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Figure 3. Cartoon depicting the procedure for measuring AR to the OmpF protein trimer. Keeping constant KCl concentration in bulk solution while increasing the concentration of PEG causes a decrease in solution conductivity and a concomitant increase in AR. Pore resistance Rp remains unaltered. Representative traces are shown for current recordings in 0.1 M KCl solutions without PEG (left panel) and with 10% wt PEG 8000 (right panel).

Note that ensuring that KCl bulk concentration is the same in all conductance measurements done with solutions of varying PEG concentrations is critical for the procedure. This issue is problematic because it is known40,45 that PEG binds some water molecules and thus increases the salt activity of the solutions. To account for this effect, we adjusted the bulk KCl concentration to get the same effective KCl activity (measured using a K+ selective electrode). Figure 4 shows different series of experiments done following the above-explained protocol to estimate AR. In Figure 4A we present measurements done with PEG8000 and PEG10000 in DPhPC membranes in 1 M KCl. Both sets give essentially the same results, showing that the size of the PEG (once it is assured that is large enough to be excluded from the pore) is not a critical issue. Taking the average of both curves we obtain Rac = 21 ± 5 MΩ. If we compare this value with the channel total resistance measured in absence of PEG, 230 MΩ, we see that the relative contribution of AR is about 20% (10% on each channel opening).


10

B

PEG 8000

1 M KCl

0.28

Inverse conductance / GΩ

A

1/G = 0.193 + 0.350 1/k R2 = 0.970

0.27 0.26 0.25

PEG 10000 1/G = 0.188 + 0.390 1/k R2 = 0.969

0.24 0.23 0.10

0.12

0.14

0.16

0.18

1/G = 1.134 + 0.083 1/k R2 = 0.979

20 mM KCl

4

DPhPC DPhPS

3 1/G = 0.679 + 0.468 1/k 2 R = 0.989

2

0.20

4

5

6

7 -1

-1

Inverse bulk conductivity / mS

Inverse bulk conductivity / mS

Figure 4. OmpF single-channel conductance in KCl+PEG solutions. A) Measurements in 1 M KCl and varying concentrations of PEG 8000 (triangles) and PEG 10000 (circles) in neutral DPhPC membranes. Solid lines correspond to fitting to Equation 3. B) Measurements in 20 mM KCl and varying concentrations of PEG 8000. Experiments were done in neutral DPhPC and charged DPhPS membranes, as labeled. The lowest conductivity corresponds to a PEG-free solution and the remaining ones to solutions with different PEG wt%. Solid lines correspond to fittings to Equation 3 (DPhPC) and Equation 4 (DPhPS).

Figure 4B shows OmpF single channel conductance measurements in 20 mM KCl solutions of variable PEG wt%. These measurements in diluted solutions yield a value of Rac = 914 ± 160 MΩ in neutral membranes and Rac = 177 ± 50 MΩ in charged ones. Astonishingly, just a change in the lipid charge makes the AR five times smaller. Figure 5A summarizes several AR calculations obtained from a series of conductance measurements carried out in neutral and charged membranes for a wide range of KCl concentrations (see Figures S1-S6 in Supporting Information for details of conductance measurements). 10

B

10

γ

A

DPhPC DPhPS

0.1

1

KCl conc. / M

0.1 DPhPC DPhPS Fit to Eq. 2 0.01

%2Rac

1 0.01

0.01

100 80

1

Rac / GΩ

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Inverse conductance / GΩ

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60 40 20 0

0.1


1

0.01

0.1

1



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Figure 5. (A) Access resistance of OmpF channel in neutral (DPhPC) and charged (DPhPS) membranes as a function of solution concentration. Solid line is a linear fit of measurements with DPhPC to Hall’s equation with pore radius as a fitting parameter. Inset: Gamma parameter as a function of concentration (pink squares). (B) Percentage of the total access resistance in relation to the overall channel resistance. Dashed lines are drawn only to guide the eye.

Figure 5A shows that the experiments performed in neutral membranes (DPhPC) follow qualitatively the behavior predicted by Hall’s equation (solid line in Figure 5A). The effective diameter of the channel can be obtained by fitting the Rac values to Equation 2. We obtain D ∼0.7 nm, which is almost identical to the reported size of the channel constriction although it is approximately half the value of the diameter in the channel entrances.27 Such underestimation of the channel aperture size, also found in similar approaches like40, could be a consequence of the fact that Hall’s equation systematically overestimates the value of the AR. Interestingly, measurements done in charged lipids (green circles) show a different trend, especially at low concentration where calculated values show a plateau that is incompatible with the increase anticipated by Hall’s equation. To shed some light on this issue we can focus on the parameter γ that appears in Equation 4. It can be calculated by taking the ratio between the slopes of the 1/G versus 1/κ curves in neutral and charged membranes D-1/(Dγ)-1 for each given KCl concentration. The inset of Figure 5A shows that calculated γ increases significantly with decreasing concentration. The factor γ can be rationalized using the Dukhin number or alternatively in the context of a Donnan description by assuming that the electric double layer created by the charged membrane partially excludes coions (Cl−) and accumulates counterions (K+) to preserve local electroneutrality. Then, γ would be:

γ =

κ* 2 ≈ 1 + ( ρ l 2c ) , κ

(6)

where ρl is the Donnan effective charge in this region. Fitting γ values obtained from experiments to Equation 6 gives ρl ∼250 mM, a value that could reasonably correspond to the charges of the polar heads of the lipid molecules.41 Accordingly, in diluted solutions, the local conductivity in the channel entrances is not dictated by the bulk concentration but by the excess of counterions accumulated by the lipid charges. This explains why AR in this range is approximately constant, in contrast with the predictions of Hall’s equation.


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Remarkably, the approach presented here does not only allow the estimation of the AR, but also the pore resistance (1/Gp is the intercept of the 1/G versus 1/κ curves). Therefore, we can calculate the relative contribution of AR to the total resistance, as shown in Figure 5B. In concentrated solutions, the total access resistance (2Rac) is about 20% of the total resistance. In charged membranes this proportion is almost stable regardless of electrolyte concentration. However, in neutral membranes the AR contribution increases with decreasing concentration so that in millimolar solutions of KCl, 2Rac is almost 80% of the total measured resistance. This underscores the crucial importance of the lipid charge. Electrophysiological recordings of the channel at low concentration in neutral membranes register essentially the AR of the system and depend marginally on the pore characteristics. If such feature were ignored, any potential structure-function relationship attempted in these conditions could be totally misguided. Note that even in concentrated solutions the AR contribution may be significant in biological channels with very low aspect ratio. A remarkable example is the 3.5 nm-diameter pore of the c-ring complex of the V-ATPase.46 Recent electrophysiology studies in neutral planar lipid membranes report a unitary conductance of ∼8.3 nS.47 The analysis made by the authors using a simplified expression similar to Equation 3 gives AR contributions of nearly 30% of the total channel resistance and channel dimensions fully consistent with the values obtained by cryoEM.46

Reinterpretation of conductance measurements considering access resistance Hitherto we have shown not only that AR can be a significant contribution to the overall channel conductance but also that its actual value can be strongly dependent on membrane charge, limiting the qualitative validity of Hall’s equation to neutral membranes. Interestingly, when we come back to the OmpF conductance measurements displayed in Figure 2C, we see that 2Rac obtained in the previous section is of the same order of magnitude as the difference between the channel total resistance measured in DPhPC and DPhPS membranes. To get more insight on the role of AR in neutral and charged membranes we made a rough calculation of the two contributions on the right-hand side of Equation 1: the channel proper resistance Rp and 2Rac. Rp was calculated assuming a cylindrical pore (each one of the three OmpF monomers) of 1 nm effective diameter and the conductivity of the pore solution was estimated as κp = (F2/RT)D± (cK + cCl ), where F, R and T have their usual meaning and D± is the ion diffusion coefficient (the same for K+ and Cl−) . The ion concentrations cK and cCl in the pore are linked to bulk concentration c through Donnan equilibria. We use the value of pore effective excess counterion concentration ρp = 200 mM obtained elsewhere.31 We calculate AR through Hall’s equation (OmpF average


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diameter is ∼1 nm),27 but we modify the conductivity of the solution at the pore entrance in the case of the channel embedded in a charged lipid membrane, as in Equation 4. The factor γ is also calculated by assuming that there is an excess counterion concentration near the pore that depends on the Donnan effective charge concentration in this region created by the lipid charges. Thus, we have:

 1 G = (π D 2κ 4 L ) 

(ρ

p

(

−1

 2c ) + 1  + 2 Dκ  2

( ρl

2c ) + 1 2

)

−1

(7)

Such treatment is formally analogous to introducing surface conductance via Dukhin length 2, but this choice allows us to separate the contributions of the protein charges (within the pore) and the lipid ones (on the membrane surface). The outcome of calculations is shown in Figure 6A.

B

Channel proper conductance

Conductance / nS

A 10 Conductance / nS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 DPhPS

Total conductance including AR

0,1 DPhPC 0,001

0,01

0,1


Gexp DPhPC

10

Gexp DPhPS Gpore w AR Gpore w/o AR

1

0,1

1

0,001

0,01

0,1

1


Figure 6. (A) Theoretical prediction of the channel conductance calculated from Equation 7. (B) Salt concentration dependence of single-channel conductance of the protein channel OmpF when embedded in a neutral (triangles) and negatively charged (circles) membrane. Lines correspond to Equation 7.

The overall channel conductance (including AR) in a neutral membrane decreases monotonically with concentration (almost linearly over a large concentration range), whereas the conductance of the pore itself reaches a constant value in diluted solutions. The channel conductance in a charged membrane is practically identical to the pore conductance because of the big reduction in AR produced by lipid charges near the pore entrance. The concentration dependence of the conductance of the pore itself is qualitatively similar to that observed in the nanoslits shown in Figure 2A. In fact, a calculation using this


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basic model and the pore size and surface charge density32 yields plateaus even for the smallest one, consistently with the fact that AR is two orders of magnitude smaller than the nanoslit resistance. We are aware that a higher resolution approach, like an all-atom MD simulation carried out involving PEGs and addressing the role of access resistance, would be highly desirable and definitely enlightening. Having in mind the complexity of similar approaches,15 such theoretical refinements are out of the scope of the present study. We use a lower resolution mean field-theory just to examine the scaling behavior of the conductance versus concentration relationship. With the obvious limitations of such a simplified model used for calculating channel conductance, Figure 6B displays a doubly interesting result. First, by using as fitting parameters the effective charge concentration near the membrane surface (ρl = 250 mM) and the ion diffusion coefficients within the pore (0.8 times their bulk value of 2×10-9 m2/s), calculations match single-channel measurements in neutral and charged membranes. Recall that the only difference between both calculated curves lies in the contribution of AR. Second, amazingly, AR helps to “hide” the channel charges and makes the charged channel look like a neutral one, i.e. with a conductance that scales with bulk concentration and does not saturate even when excess counterion concentration in the pore commensurates with bulk concentration. Furthermore, when the channel is embedded in a DPhPS membrane, the reduction of the AR makes the channel charges “visible” and the typical low limiting value in synthetic nanopores of diameter over 100 nm appears. To prove that the saturation of OmpF conductance at low concentration is really an effect of the pore charge we measured single-channel conductance at pH 4, which is approximately the isoelectric point of this protein.30,48 Figure 7A shows that in this case there are little differences between measurements done in neutral and charged membranes. Conductance is slightly higher in the case of DPhPS membranes but it keeps decreasing with bulk concentration in diluted solutions. The direct comparison of measurements performed in DPhPS at pH 6 and pH 4 (Figure 7B) shows clearly that in the latter case (neutral pore) the membrane charge does not induce a conductance saturation below 100 mM, as happens in the former case. These data indicate (in agreement with the model discussed before) that lipid charges would be just merely modulators of the AR but do not have almost any effect on the pore conductance.


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B

DPhPC pH 4 DPhPS pH 4

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10 DPhPS pH 6 DPhPS pH 4 1

0.1

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Figure 7. (A) Salt concentration dependence of OmpF single-channel conductance at pH = 4 (channel net charge virtually zero) when embedded in a neutral (blue diamonds) and negatively charged (pink triangles) membrane. (B) OmpF singlechannel conductance in a negatively charged membrane over a wide range of KCl concentrations at pH = 6 (green circles) and pH = 4 (pink triangles).

CONCLUSIONS Biological ion channels and synthetic nanopores apparently behave in a very similar way as regards conductance versus concentration experiments. However, each group is regulated by different underlying physicochemical mechanisms. Electroosmotic effects that are capital in abiotic pores are absent in less permeable and mildly selective biochannels. On the contrary, AR that is crucial in protein channels becomes negligible in wider nanochannels. Here we show that the conductive properties of nanopores below the nanoscale in diluted solutions cannot be fully understood just in terms of the charge and size of the pore itself but need considering the interfacing resistance and, consequently, the surface electrostatic properties of the substrate where the pore is. To this end, we propose here a method of measuring AR. It is based on single-channel measurements of conductance in solutions containing an electrolyte solution and varying concentrations of a high molecular weight neutral polymer, PEG, which is sterically excluded from the pore but modifies the bulk solution conductivity. The AR measurements in a biological nanopore in concentrated and diluted KCl solutions made from experiments using PEG 8000 and PEG 10000 are fully consistent with the classical expression of AR proposed by Hall.26 By comparing AR measurements in a protein channel embedded in charged and neutral phospholipid membranes, we have shown that lipid surface charges induce a large decrease in AR,


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in accordance with predictions from the computation in an ideal pore.41 This effect is measurable, particularly in diluted solutions. Finally, it can be straightforwardly explained in terms of the excess of ions in the electric double layer, which increases local conductivity and lowers AR. We use a simplified model, with a minimum of fitting parameters, to dissect the two contributions to the measured channel conductance: those coming from the pore itself and from the concentration polarization at the pore entrances. Successful comparison with measurements reveals that AR masks the pore surface charges in diluted solutions. In fact, measured channel conductance decreases with bulk concentration over a wide concentration range irrespective of the presence of net surface charge in the pore (pH 6) or its absence (pH 4). This finding is particularly relevant to recent studies of low-aspect-ratio pores. Although AR is only a moderate contributor to the total resistance in concentrated solutions, it is essential to account for it at low ion concentrations. In this range, the channel charges could pass functionally unnoticed if AR is ignored. Biological channels perform their physiological function in the cellular environment, which has two common features: relatively low ionic strength and macromolecular crowding. Both factors make the AR contribution to channel conductance important. Therefore, our findings are relevant to in vivo conductive properties of protein channels. Indirectly, by choosing a biological pore of nanometer size we show that continuum theoretical approaches provide a reasonable description of subnanometric biological channels. When such treatments cannot reproduce experimental findings in larger nanochannels this failure should not be attributed to the inadequacy of the continuum hypothesis, but possibly to physical effects missing in the model. To conclude, note that the distinctive feature of ion permeation at the nanoscale is that the molecular nature of both the fluid and the channel itself becomes evident. Biological ion channels provide an excellent opportunity to explore the limits of the conventional description of ion transport based on microfluidics by analyzing structure-function correlations at atomic detail. Considering that our findings are based on model-independent general statements not dependent on any structural feature of OmpF channel, the impact of our results goes beyond the ion channel biophysics and allows answering questions relevant to low-aspect ratio synthetic structures whose functioning is strongly influenced, if not dominated, by access resistance. This aspect is of great importance to develop ionic circuits making use of biomimetic nanopores as molecular Coulter counters, sensors and nanofluidic electronic devices.20,21,36,49 METHODS Wild-type OmpF, kindly provided by Dr S. M. Bezrukov (NIH, Bethesda, USA), was isolated and purified from an E. coli culture. Planar membranes were formed by the apposition of monolayers across orifices


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with diameters of 70–100 μm on a 15 μm-thick Teflon partition using diphytanoyl phosphatidylcholine (DPhPC) or diphytanoyl phosphatidylserine (DPhPS) lipids (Avanti polar lipids, Inc., Alabaster, AL). The orifices were pre-treated with a 1% solution of hexadecane in pentane. An electric potential was applied using Ag/AgCl electrodes in 2 M KCl, 1.5% agarose bridges assembled within standard 250 ml pipette tips. The potential was defined as positive when it was higher on the side of the protein addition (the cis side of the membrane chamber), whereas the trans side was set to ground. An Axopatch 200B amplifier (Molecular Devices, Sunnyvale, CA) in the voltage-clamp mode was used to measure the current and applied potential. Except otherwise noticed, the signal was digitized at 50 kHz sampling frequency after 10 kHz 8-pole in-line Bessel filtering. The chamber and the head stage were isolated from external noise sources with a double metal screen (Amuneal Manufacturing Corp., Philadelphia, PA). The pH was adjusted by adding HCl or KOH and controlled during the experiments with a GLP22 pH meter (Crison). Except where noted, measurements were done at T = (23 ± 1.5) °C. Electrolyte solutions containing PEG were prepared by adding PEG directly to the stock electrolyte solutions. KCl concentration was adjusted so that KCl activity remained constant in the series of experiments with different PEG wt %. K+ activity was measured with an Ion Selective Electrode (Crison).

SUPPORTING INFORMATION Supporting Information Available online: OmpF single-channel conductance in DPhPC and DPhPS membranes. Measurements in the range 30 mM-1 M KCl and varying concentrations of PEG 8000.

ACKNOWLEDGMENTS We acknowledge financial support from the Ministry of Economy and Competitiveness of Spain (projects no. FIS2013-40473-P and FIS2016-75257-P AEI/FEDER), and Universitat Jaume I (project no. P1.1B201528).

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