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The salt-gradient approach for regulating capture-totranslocation dynamics of DNA with nanochannel sensors Yuhui He, Makusu Tsutsui, Ralph H. Scheicher, Xiangshui Miao, and Masateru Taniguchi ACS Sens., Just Accepted Manuscript • DOI: 10.1021/acssensors.6b00176 • Publication Date (Web): 18 May 2016 Downloaded from http://pubs.acs.org on May 19, 2016

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The Salt-gradient Approach for Regulating Capture-to-Translocation Dynamics of DNA with Nanochannel Sensors Yuhui He,† Makusu Tsutsui,∗,‡ Ralph H. Scheicher,¶ Xiang Shui Miao,∗,† and Masateru Taniguchi‡ School of Optical and Electronic Information, Huazhong University of Science and Technology, LuoYu Road, Wuhan 430074, China, The Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan, and Division of Materials Theory, Department of Physics and Astronomy, Angström Laboratory, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden E-mail: [email protected]; [email protected]

KEYWORDS: Salt gradient, DNA manipulating, DNA capture, Nanochannel, Electrokinetic modeling Abstract Understanding the physical mechanisms that govern the ion and fluidic transport in saltconcentration-biased nanochannel/nanopore systems is essential for the potential applications ∗

To whom correspondence should be addressed of Optical and Electronic Information, Huazhong University of Science and Technology, LuoYu Road, Wuhan 430074, China ‡ The Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan ¶ Division of Materials Theory, Department of Physics and Astronomy, Angström Laboratory, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden † School

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in bioanalysis. One central challenge is to interpret the observed four-stage change from osmosis to the reverse one with increasing salt gradient. Here we provide a unified model that outlines the intriguing role of two competing factors, the exclusion- and diffusion-induced electrical potentials. We demonstrate theoretically a direction control of a hydrodynamic flow via the salt-gradient. Based on this, we also propose a salt-gradient approach for regulating DNA motion in nanochannels that enables voltage-free single-molecule capture with a significantly low translocation speed. The present method would be used as a useful protocol to overcome the key hurdle of tailoring the capture-to-translocation dynamics of polynucleotides for nanopore sequencing.


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Tremendous processes have been made in the past decades on exploring the transport properties in nanoscale channels. 1,2 One unique feature of nanochannels is the ion selectivity as shown in Fig.1a: counterions are attracted by those charges on the nanochannel surface and hence nanometer thick electrical double layers (EDLs) are induced around the channel wall. While negligible in traditional microchannels, these charged layers take a significant proportion of the overall liquid inside nanochannels. In this manner the channels become exclusive for either cations or anions, and now the motion of charged solvent can be manipulated by electrical means. By exploiting such coupling between electrical and fluidic transport, important applications including biomolecule sensing, 3–5 water desalination 6 and energy conversion 7–9 have been under rapid development. Recently by imposing different concentrations of salt to the two ends of charged nanochannels, a bunch of interesting transport phenomena have been reported. 9–13 As shown in Fig.1b, due to the larger concentration Cmax in the left chamber (z < −L/2) the counterions at that end of the channel are more densely piled up, while at the opposite Cmin end the layer of screening ions is much thicker. Hence a nonuniformity of ion distribution along channel axis arises. Besides, the cations and anions may have obviously different capacities of diffusion, thus adding to the axial inhomogeneity of ion densities. Then, an electrical potential ∆V would be built up self-consistently to balance this ionic inhomogeneity. In line with the self-built electrical field, an electrical body force ~fe would emerge on the charged solution and thereby an electrical osmotic flow (EOF) would be stimulated. At first glance, the larger the added salt concentration difference, the stronger the electrical bias and thus the associated EOF. However, the experiments showed much more complicated behaviors. 10,11,13 One central discovery is the transition of fluidics from osmosis to the reverse one with increasing salt gradient. Yang et al 10,11 showed a four-stage variation of EOF: at first an osmotic flow was boosted given small LiCl concentration bias Cmax /Cmin across the channel (region I in Fig.2b); as the imposed salt gradient was enhanced, the osmotic flow first increased and then turned to decrease (region II in Fig.2b); when the salt gradient became larger than a critical value, the fluidics shifted to be reverse osmosis (RO, region III in Fig.2b); if the salt gradient still kept


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increasing, once again the magnitude of RO began to decrease (region IV in Fig.2b):

O ↑ ⇒ O ↓ ⇒ RO ↑ ⇒ RO ↓

(1)

In summary, rather than monotone increasing there existed a transition of osmotic flow to the reversed one. Furthermore, within either osmosis or reverse osmosis there was U-turn like dependence of the fluidics on the imposed salt gradient. Li et al’s experiments also demonstrated a sign-reverse of the self-built electrical potential ∆V as the MgCl2 salt gradient increased. 13 The above anomalous nanofluidics on one side caused substantial difficulty for theoretical understanding, while on the other side it might open new horizons in several application fields. We are going to demonstrate that one most promising topic may be the nanopore-based genome sequencing protocol. Nowadays the nanopore sequencing has been under intensive studies and one central requirement is to delicately manipulate the DNA polymer motion. 3,14 The orthodox method is an electrical approach: by imposing a longitudinal voltage through the nanopore, the anionic DNA polymers would be driven electrophoretically towards the pore entrance; once captured into the pore, the DNA strands are expected to thread through with a sufficiently slow speed so that each nucleotide can be read out by electrical or optical means. 4,14–16 Recently, by carefully selecting the operating region slow and smooth DNA penetrating through small-diameter nanopores (R < 5 nm) was reported, 17 indicating a promising approach towards the goal of controlling DNA motion. Other strategies such as adding viscous solvent 18 and using gate manipulation 19,20 were also developed. Here we point out that the above regulating of biomolecule motion could also be achieved by a salt gradient. As seen in Fig.3a the DNA capture movement can be realized via the osmotic flow through the salt-concentration-biased system; then by tuning the magnitude of the imposed salt gradient, the fluidics and DNA translocation through the nanopore can be manipulated as shown in Fig.3b. We stress that a prominent benefit would be gained by adopting such a salt-gradient-driven approach. That is, the decoupling of DNA motion control and nucleotide identification. One major


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branch of nanopore sequencing research is to discriminate the target DNA nucleotides via tunneling conductance, which is measured by the transverse electrodes. 21–24 In the real experiments, a cross-talk between the transverse detection voltage and the longitudinal motion-driving voltage often caused severe problems to the device functioning. 5,25 Nonetheless, if the DNA motion is manipulated by a salt-gradient the entanglement between the disciplines of molecule motion control and molecule detection should be substantially relieved. In this work we first demonstrate through our electrokinetic modeling that the alternate dominating of the exclusion and diffusion potentials is responsible for those puzzling experimental observations. Then we discuss the potential of tuning salt concentration bias for DNA motion control. Strategies to optimize the biopolymer capture rate and to control the molecule translocation speed would be illustrated.

Exclusion and diffusion potentials Exclusion potential As shown in Fig.1b, the nonuniform concentration of the counterions along the channel axial direction would result in diffusion flux of net charges through the channel. Yet for the opencircuit configuration shown in the experiments, 10,11 an electrical field Ez would be established in a self-consistent manner to offset this counterion diffusion flow. Hence a cross-channel electrical bias ∆Vσ emerges, namely the exclusion potential. 26 Mathematically, the above situations can be described by the Teorell-Meyer-Sievers model 27,28 or by the space-charge model. 29–31 The MeyerSievers model is widely used since it is compact and accurate in most cases. However, as we are going to see, the concentration variation of ions along the channel-radial direction, which is the very factor causing behavior IV as shown in Fig.2b, is neglected by this model (See discussions in Supplementary Materials). Hence in this work we employ the space-charge model and further develop it as follows. The total electrical potential V (r, z) inside the nanochannel is separated into


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electrostatic φ (r, z) and electromotive V0 (z) based on the superposition principle: 26,32 V (r, z) = V0 (z) + φ (r, z)

(2)

Then the channel-radial component of electrical potential, φ (r, z), is estimated through PoissonBoltzmann equation 33 for monovalent ions like LiCl used in the experiments (Detailed derivation is provided in Supplementary Materials): sinh φ¯ (r, z) 1 ∂ ∂ φ¯ (r, z) r = r ∂r ∂r λD2 (z)

(3)

with boundary conditions

∂ φ¯ (r,z) ∂r r=R

=

∂ φ¯ (r,z) ∂r r=0

eσw ε kT

=0

(4)

(5)

In the above, φ¯ = eφ /kT , σw is the density of charges on the channel wall surface and λD = (ε kT /2e2C0 )1/2 is the Debye length. Here C0 (z) is the salt concentration along channel axis (r = 0). As seen in Fig.1b, λD is now increasing from Cmax end through the channel to Cmin end, since C0 keeps decreasing along the axis (Detailed derivation for C0 (z) is provided in Supplementary Materials) C0 (z) = −

(Cmax −Cmin )z Cmax +Cmin + L + π R/2 2

(6)

On the other hand, under the open-circuit situation the channel-axial component of electrical potential V0 (z) is determined through the requirement of zero electrical current in channel axial direction: e(−µ+

∂ V0 ∂ Λ+ ∂ V0 ∂ Λ− Λ+ − D+ ) + (−e)(µ− Λ− − D− )=0 ∂z ∂z ∂z ∂z

(7)

where Λ± is the line density of monovalent cations/anions along the axial direction Λ± (z) = 2π C0 (z)

Z R 0

exp[∓

eφ (r, z) ]rdr kT


(8)

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Since we focus on the counterion-induced exclusion effect first, we neglect the difference of motility between cations and anions and derive the following expression for the self-built electrical field (Details are provided in the Supplementary Materials):

∂ Vσ (z) Λ+ − Λ− kT ∂ lnC0 = −Ez (z) = − ∂z Λ+ + Λ− e ∂ z

(9)

In the above Vσ is the exclusion potential built by the nanochannel surface charge effect and Ez is the channel axial electrical field. We remind that Ez points from lower salt concentration end to the higher one (Cmax ← Cmin ), given negative surface charges on the channel wall (Λ+ > Λ− ). In this manner the diffusion current of excessive cations would be counteracted by the Ez -induced electrophoretic one, and hence the open-potential system keeps stable. Then considering the presence of excessive cations in the channel, an electrical body force pushing fluidics from the Cmin end to Cmax would emerge. An electroosmosis is thus boosted in the nanochannel systems (Cmax ← Cmin ). Quantitatively the flow is evaluated through Navier-Stokes equation as follows (Detailed derivation is provided in Supplementary Materials) Z 1 L/2 1 ∂ ∂ ∂ V0 r uz = − 2eC0 sinh(φ¯ (r, z)) dz η r ∂r ∂r L −L/2 ∂z

(10)

The right hand side of the above equation characterizes the axial-component of electrical body force ~fe , while η is the fluid viscosity and uz is the axial component of fluid velocity ~uz . Then, a no-slip boundary condition is used as uz |r=R = 0 and another is ∂ uz /∂ r|r=0 = 0. Here we remind that the diffusion of those counterions along the electrical body force The calculated exclusion potential and the resulted fluidic velocity are plotted with dark-yellow lines in Fig.2, where parameters are set as those in the experiments. 10,11 It indicates that the exclusion potential ∆Vσ at first increases with the LiCl concentration bias, and then begins to decrease when the salt gradient is larger than a critical value (Cmax /Cmin ≈ 3 mM/0.1 mM). Such behavior can be understood from the expression of exclusion-induced electrical field shown in Eq.9. When the salt concentrations at both ends are small, the channel becomes extremely cation7 ACS Paragon Plus Environment

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selective due to its weak capability of screening channel wall surface charges. As a result, the factor (Λ+ − Λ− )/(Λ+ + Λ− ) becomes almost 1. The exclusion potential is then estimated as

(∆Vσ )I ≈

kT Cmax ln( ) e Cmin

(11)

in this region. It indicates an increasing trend with larger Cmax . However, once the imposed salt concentrations become large, the amount of induced charges would be trivial compared to that of imposed salt. Now the factor (Λ+ − Λ− )/(Λ+ + Λ− ) becomes −2π Rσw /2π NA eR2C0 , and thus the exclusion electrical field is (Ez)II ≈

−σw kT ∂ C0 RNA eC02 e ∂ z

(12)

By utilizing the expression of C0 (z) shown in Eq.6, we find that the magnitude of Ez decreases with larger Cmax (See FigS4 in Supplementary Materials). Consequently ∆Vσ begins to reduce with larger Cmax in II region. Here the physical picture can be found in the inset of Fig.3a. As the salt concentration Cmax in the left chamber increases from 100 mM to 1 M, LiCl concentration at the right end of channel is also enhanced, as indicated by Eq.6. It results in globally decreased thickness of EDLs in the channel, as seen by comparing the shadowed regions with blue and red lines. From previous discussion, we are aware that the exclusion potential is triggered by the variation of EDL along channel axial direction. Thinner EDLs under larger Cmax lead to attenuation of such skin effect. Thus smaller ∆Vσ is resulted in. Then, since the amplitude of electrical body force fz is directly determined by the self-built electrical bias, we conclude that it is the first increasing and then decreasing trend of the exclusion potential ∆Vσ that leads to the U-turn behavior of osmotic flow shown by blue line in Fig.2b.

Diffusion potential Now let us turn to the discussion of the diffusion potential ∆VD . It is estimated by emphasizing the cation and anion motility difference while neglecting the channel-wall surface charge effect in


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Eq.7:

∂ VD (z) D+ − D− kT ∂ lnC0 = −Ez (z) = − ∂z D+ + D− e ∂ z

(13)

In the above VD is the diffusion potential caused by the different diffusion coefficients of the cations and anions. For LiCl solution used in the experiments, 10,11 the relation DLi = 0.525DCl indicates a diffusion-difference-induced electrical field pointing from Cmax to the Cmin ends. Accordingly, there would be EOF flowing from higher concentration end to the lower one (Cmax → Cmin ), which is reverse osmosis (RO). Fig.2b shows quantitatively the calculated ∆VD and the associated RO with red lines. The magnitude of ∆VD becomes saturated with increasing Cmax , which is as expected from the logarithmic function shown above. Yet the variation trend of RO shows an unexpected U-turn variation. In order to interpret the RO behavior, we plot in Fig.4 the radial distributions of electrical body force fz and that of fluidic velocity uz under different salt concentrations Cmax = 0.1 M and 1 M. From Eq.13 we are aware that the amplitude of electrical body force fz under Cmax = 1 M should be larger than that at Cmax = 0.1 M, since fz = Ez ρ . It seems to suggest larger amplitude of RO under higher Cmax . However, what actually matters here is the radial distribution of the force as shown in Fig.4a. Due to stronger capability of screening the channel-wall surface charges, those counterions under Cmax = 1 M are more concentrated to the close-to-wall layer. This is clearly demonstrated by comparing the variation of Debye lengths along the channel axis under the two difference salt gradients shown in the inset of Fig.4a. Consequently a much larger proportion of the electrical body force, fz = Ez ρ , is adhered to the channel wall for the Cmax = 1 M case. From our previous work, 34 we are aware that this means much weaker efficiency of driving EOF. Here we provide an instinctive understanding of the above physical picture: given similar tangential driving force against the wall, putting the force closer to the frictional wall surface would lead to poorer efficiency of stimulating fluid than putting it far away from the wall. Therefore, the fluidic speed under Cmax = 1 M is obviously smaller than that under Cmax = 0.1 M as shown in Fig.4b.


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Exclusion versus Diffusion After discussing the separate effects of exclusion potential ∆Vσ and diffusion potential ∆VD , now we consider the joint effects of these two factors. By comparing the blue and red curves in Fig.2a we find two crucial points which cause the four-stage change of fluidics shown in Fig.2b: while the magnitudes of ∆Vσ and ∆VD are in the same order, their peak/saturation values appear at obviously different amounts of the imposed salt gradient. The physical mechanism for the former point can be traced back in Eqs.9 and 13: The values of both factors (Λ+ − Λ− )/(Λ+ + Λ− ) and (D+ − D− )/(D+ + D− ) vary in the range from −1 to 1, and therefore ∆Vσ and ∆VD are well matched in strengths. By contrast, the physical mechanisms for the second point are more complicated. The basic pictures are that the origin of exclusion potential ∆Vσ is restricted to the inhomogeneous EDLs around the channel wall, as seen in Fig.1b; on the other hand, that of diffusion potential ∆VD is extended to the whole liquid volume inside the channel, as seen in Fig.1a. The former is like skin effect, while the latter like bulk one. Hence the strength of ∆Vσ strongly depends on the thickness of the skin, which is characterized by Debye length λD . From electrostatics, we are aware that the smaller the imposed salt concentrations at both ends of the channel, Cmax and Cmin , the thicker EDLs, and thus the stronger the exclusion effect within the nanochannel. In this manner, the exclusion potential dominates and the fluidics appear to be osmosis. On the other hand, the larger the imposed salt concentration, the thinner the EDL around the channel wall. As a result, the exclusion effect deteriorates while the diffusion effect starts to take charge. It explains why the peak of (u¯z)σ appears at smaller Cmax /Cmin while that of (u¯z )D at larger one. The overall electromotive field is then derived as D+ Λ+ − D− Λ− kT ∂ lnC0 ∂ V0 (z) = −Ez = − ∂z D+ Λ+ + D− Λ− e ∂ z

(14)

The corresponding averaged fluidic speed u¯z is calculated based on Eq.10 and results are plotted with black line in Fig.2b (σw is set to be −0.8 mC/m2 and fitting details are provided as FigS5 in Supplementary Materials). The experimental data is plotted by blue line with round symbols in


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the same figure. Although it does not fit perfectly the experimental result, our calculation reproduces those most important features shown in the experiment: 10,11 the four-stage change, the salt gradient values at which these changes occurred, and the magnitude of the fluidics (Based on our model the estimated value of fluid flow through a single nanochannel is about 2.6×10−2 µ mol/min (u¯z · π R2 = 10−5 m/s · 3.14 · (50×10−9 )2 m2 ), while the authors of the experiments indicated the total flow was around 3.8 µ mol/min (20 µ mol/min/cm2 · π /16 cm2 ). Thus our results suggest there are about 150 nanochannels on the track-etched polycarbonate membranes). We then attribute those small disagreements between experiment and our calculation to several nonideal factors: in the real process there existed a distribution of diameter values of the fabricated nanochannels, instead of exactly the same one; during the actual osmosis/reverse osmosis, the salt concentrations at the two ends of the reservoirs slowly varied rather than kept invariant; the differences in heights of the reservoirs resulted from osmosis/reverse osmosis would impose hydrostatic pressure to the system, functioning as an extra restoring force. Therefore, we call for further experimental studies with more ideal design to justify our proposed mechanisms. The experimental results shown in Fig5 of the reference 10 further identified the exclusion versus diffusion mechanism proposed by us. By using three types of monovalant electrolyte separately, it was reported that LiCl gradient caused strongest reverse osmosis in the nanochannel, NaCl one ranked second, while for the case of KCl it turned to be a small osmotic flow. In the framework of exclusion versus diffusion potentials, LiCl gradient would induce the largest diffusion potential, NaCl the second while KCl the negligible, since DLi = 0.53DCl , DNa = 0.68DCl and DK ≈ DCl . According to the physical pictures shown above (Eq.9 and 13), in the LiCl concentration biased nanochannels the exclusion potential was largely overwhelmed by the diffusion counterpart, and consequently a significant reverse osmosis was boosted; in the NaCl one the diffusion potential was smaller and thus the net electrical field for driving reverse osmosis became smaller; finally in the KCl case the diffusion potential turned trivial and now the fluid changed direction to be osmosis. We remind that the diffusio-osmotic effect 9 is not considered here in our exclusion versus diffusion picture. As shown in Supplementary Materials, the factor of diffusio-osmosis would


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cause monotonic increasing reverse osmosis and moreover, the magnitude of this RO would be several orders stronger than osmosis stimulated by the exclusion effect. Hence the fluid behavior indicated by diffusio-osmosis is against the experimental observations, and thus is not supposed to take the major role in determining the salt-gradient boosted nanofluids shown in the experiments. In the end of this section, we summarize that the experimentally reported four-stage change of fluidics under increasing LiCl concentration bias was caused by the first dominating of exclusion potential ∆Vσ and then that of diffusion one ∆VD . The above discussion further indicates that whether such drastic transition occurs depends on the relative strengths between ∆Vσ and ∆VD . For example, assuming that the diffusion coefficients of the imposed cations and anions are very close (D+ ≈ D− ), ∆VD would be negligible and therefore there will be no reverse of voltage or O-to-RO transition. As we are going to see, this is exactly what has been observed in the latest experiments by Li et al. 13

Divalent salt gradient system Recently, another experiment showed that whether the self-built potential ∆V changed sign strongly depended on the type of salt and on the imposed concentration at the diluter side Cmin . 13 As shown in Fig.2a of Reference, 13 by using KCl the measured electrical potential was always positive, and it simply kept increasing with the imposed salt gradient Cmax /Cmin . On the other hand, when the salt was substituted for MgCl2 a sign-change of ∆V was reported. Several types of theoretical models may be able to interpret the above phenomena. One is the charge inversion by those strongly correlated multivalent ions. 35 It was proposed that for multivalent ions the strong spacial correlations between the ions would dominate the electrostatics. Accordingly, one most counterintuitive effect was the over-screening of channel-wall surface charges by the multivalent ions. The resulted reversal of the polarity of charged surfaces did have been observed. 36–39 Thus the authors who reported ∆V -inversion in MgCl2 salt-gradient system ascribed the phenomena to the above strong-correlation factor. 13


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On the other hand, we remind that the space-charge model has been generalized for multivalent ion systems. 26,40,41 The basic framework still relies on mean field approximation for electrostatics, while the multivalence effect is considered in the expression for ionic current. In this section, we apply our modified space-charge model for MgCl2 solution, and demonstrate that it can reproduce all the reported phenomena in the experiments. The experimental observation of KCl-gradient-induced ∆V is quantitatively demonstrated by blue rhombuses while our electrokinetic simulation by blue line in Fig.5a. Here we attribute the KCl gradient caused phenomenon to two facts. The first factor is that K+ and Cl− have very similar mobilities. Therefore, the associated diffusion potential becomes negligible, which means ∆V0 ≈ ∆Vσ . Second, the salt gradient used in the experiments was in the range Cmax /Cmin ≤ 64 mM/1 mM, while the width of the fabricated SiO2 nanoslit was about 20 nm. 13 It indicates that even at Cmax end the thickness of EDLs as characterized by Debye length was always larger than 1.2 nm. At Cmin end EDL is about 10 nm thick. Hence the cation-selectivity of the nanochannel remained significant. From previous discussion we are aware that the electrical potential should be described by Eq.11. In this way a monotonic increasing ∆Vσ was resulted in. For the MgCl2 case, we propose our physical explanations by further generalizing the previous model to divalent ions in nanoslit system. By resorting once again to the non-electrical current condition, we obtain the following expression of ∆V0 for divalent salt MgCl2 : 2D+Λ+ − D− Λ− kT ∂ lnC0 ∂ V0 (z) = −Ez = ∂z 2D+Λ+ + D− Λ− e ∂ z

(15)

In the above Λ± are the line densities of Mg2+ /Cl− ions along the channel axial direction, and are calculated with similar approach as shown in Eqs.3 and 8 (Here we turn to Descartes coordinate and provide derivation details in the Supplementary materials). The simulation results are plotted with black line while the experimental results with black rhombuses in Fig.5a. The signinversion is clearly demonstrated. The dash-dark-yellow and dot-red lines show the corresponding exclusion and diffusion potentials respectively. The changes of the two component potentials, ∆Vσ


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and ∆VD , with increasing MgCl2 concentration bias explain the variation of the overall electrical potential ∆V : at smaller salt gradient Cmax /Cmin , the magnitude of exclusion potential exceeds that of diffusion one; at larger gradient, it turns to diffusion potential dominating. The authors of the experiments 13 further demonstrated that for MgCl2 by using different start concentrations Cmin , ∆V (Cmax /Cmin ) show notably different varying trends. As shown by pink rhombuses and blue up-triangles in Fig.5b when Cmin was too large (10 mM) or too small (0.1 mM), the induced voltage was either constantly negative (∆V < 0) or positive (∆V > 0). Here we indicate that these behaviors can be interpreted quite satisfactorily with the physical picture by our model. For Cmin as large as 10 mM the EDL thickness was restricted to about 3 nm while at the other end it was even smaller (∼ 1 nm); therefore the exclusion effect got overwhelmed by the diffusion effect (∆V < 0). On the other hand, for Cmin as small as 0.1 nm, EDL at the Cmin end could reach 30 nm while it was larger than 10 mM at the other end. Since the channel was extremely cation-selective, now the situation reversed to be exclusion potential dominating (∆V > 0). We would like to point out that in the above modelling and simulation the only fitting parameter is the density of surface charges on the SiO2 channel wall. By setting σw = −25 mC/m2 for MgCl2 and σw = −6 mC/m2 for KCl separately, simulation results show quantitative agreement with the experiments. 13 On one side, the order of the two fitted values agrees with previous estimations using SiO2 nanopores; 42,43 on the other side, there existed obvious disparity between the estimated quantities of σw for MgCl2 and for KCl. We speculate that the latter can be attributed to the different screening capability of Mg2+ and K+ ions in the Stern layers. It was reported that by imposing large concentration of monovalent salt KCl into silica nanochannels which had already been filled with MgCl2 , the charge inversion would be cancelled due to the stronger screening of K+ cations in the Stern layer. 37 Here the key words are the Stern layer where the counterion concentrations of K+ and Mg2+ can be quite different even given similar imposed doses C0 . We suppose that due to the stronger screening of naked wall charges by K+ ions in the Stern layer than by Mg2+ , the effective density of surface charges felt by those counterions in the diffuse layer became smaller for KCl electrolyte. It may provide a potential explanation why the estimated


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magnitude of σw in KCl environment was smaller than that in MgCl2 one. From the above discussion, we are aware that although both the strong-correlation and spacecharge models may explain the experimental results, the illustrated physical pictures are absolutely different. The former insists that the mean-field approximation employed by Poisson-Boltzmann theory (PB) is no longer valid for multivalent ions. Instead, there should be some correction item which represents the correlation effect, 44 or calculations based on density functional theory are required. 45 In contrary, the model established in this work maintains the usage of PB and an invariant quantity of wall surface charge density σw . The presence or absence of ∆V -inversion under various types of salt and concentrations is then attributed to the relative strengths between exclusion and diffusion potentials. Hence we call for further experiments to justify the real mechanisms from the above two models.

Manipulating DNA motion with salt-gradient The tuning of DNA motion through nanopore by salt gradient has been demonstrated experimentally. 12 Yet it was a mixed approach where the biopolymer movement was primarily controlled by the electrical bias, and the salt concentration bias served as an adjusting factor. As stated in the beginning, here we explore a salt concentration bias approach to regulating DNA motion without the usage of electrical voltage. Our proposal is demonstrated in Fig.3a and b: by exploiting the osmotic flow through the nanopore, the DNA capture process is achieved through the biomolecule convective motion; then after DNA has been trapped into the pore, the DNA translocation speed is expected to be manipulated through the tuning of salt concentration bias. For the nanopore sequencing protocol, whether our proposal is feasible in the real application depends on two keynote indexes of performance. One is the DNA capture rate accomplished by the salt gradient. Can it catch up with the electrical counterpart, or even outperform it? The other is the tuning range of DNA translocation speed. Can it be slowed down to the required value 1 nucleotide/0.1 ms 14 ? Below we present a quantitative investigation on these issues.


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DNA capture In the biopolymer diffusion limit, 12,46 the capture rate γC is estimated through the convection speed of DNA molecules

γC = 2π r2 urCDNA = u¯z π R2CDNA

(16)

In the above, r is the distance of the target DNA coil from the pore entrance, ur is the convection speed of the coil there, u¯z is the averaged speed of fluid through the nanochannel and CDNA is the imposed density of DNA coils. As a comparison, the rate of DNA capture by electrical bias approach, γE , is expressed as follows 12 (Derivation details are shown in Supplementary Materials):

γE = 2π r∗2 D

CDNA µπ R2CDNA = Vz r∗ L + π R/2

(17)

where Vz is the imposed cross-pore voltage, D and µ are the diffusion coefficient and mobility of DNA coils, and r∗ is the capture radius. The calculated fluidic speed u¯z in open-pore systems is shown in Fig.6a for various kinds of salt and starting concentrations Cmin . By comparing the results of two types of salt (real lines and dash lines), we find that at smaller Cmax LiCl and KCl boost fluidics with quite similar amplitudes while at larger Cmax KCl outperforms LiCl. These observations can be attributed to two facts. First, as shown in the inset of Fig.3a smaller Cmax means thicker EDL. As a result, the nanochannel becomes so cation-selective that the exclusion potential overwhelms the diffusion one. In the current case that is, the effect of motility difference between Li+ and K+ is absolutely suppressed under smaller Cmax , and thus the two types of salt show almost the same efficiency of stimulating EOF. However, at larger Cmax the much thinner EDL attenuates the exclusivity of the channel, and in this way the diffusion effect gradually governs. Since the mobility of Li+ ions is much smaller than that of Cl− , the diffusion potential of LiCl gradient is stronger than that of KCl. The overall electrical bias ∆V is then more severely attenuated by this opposite ∆VD for the LiCl gradient situation. Hence the fluidics with LiCl concentration bias experiences a larger decrease of the osmotic movement. Moreover, by employing the same type of salt we find that the smaller the starting concentration 16 ACS Paragon Plus Environment

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Cmin , the larger the amplitude of the osmotic flow. This is once again ascribed to the stronger exclusivity of nanochannel at smaller salt concentration. The above results demonstrate that similar capture rates would be attained by using two different types of salt, LiCl and KCl. Yet for the purpose of controlling DNA motion at the next stage, we recommend the usage of LiCl. As shown by the dash-circled region, by tuning LiCl gradient the osmotic speed as small as zero can be reached while it cannot be for KCl. Since decreasing DNA translocation speed to the order of 10−6 m/s is one critical requirement of nanopore sequencing, our results suggest that LiCl is more appropriate. Then, for the concern that whether the salt gradient approach is as competent as the conventional electrical means in trapping DNA molecules, we plot in Fig.6b the ratio of DNA capture rates gained by LiCl gradient over that by electrical bias γC /γE . Here the electrical voltage Vz is assumed to be 0.1 V as used in previous experiments. 12 The results indicate that by using smaller starting concentration Cmin , the biopolymer capture rate via LiCl gradient can reach 80% of the electrical counterpart. Therefore our theoretical studies convince the feasibility of salt gradient method for DNA capturing.

DNA translocation As shown in Fig.3b, the DNA strands inside the nanopore are approximated as a concentric cylinder due to the repulsion from the negative surface charges on the pore wall. 19,47 The electrostatic equation is basically the same as that shown in Eqs.2, 3 and 14, while one boundary condition now becomes as follows at the DNA surface

∂φ ∂r r=r0

=−

λDNA 2π r0 ε

(18)

where λDNA is the line charge density of DNA strands and r0 is the radius of the strands. Besides, the Navier-Stokes equation for fluidics as shown in Eq.10 still applies while the no-slip boundary


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condition at the DNA surface becomes

uz |r=r0 = uDNA

(19)

In the above, uDNA is the translocation speed of DNA strand within the pore. The force balance equation for DNA strands reads as follows: 2π r0 Lη ∂∂urz

r=r0

+

Z L/2

−L/2

Ez dzλDNA = 0

(20)

In the above the first term is the viscous force from the surrounding liquid, while the second is the electrical driving force from the self-built electrical potential ∆V . Our calculation results are shown in Fig.7. It demonstrates that by using LiCl concentration bias Cmax /Cmin = 480 mM/10 mM, DNA translocation speed as small as 2 × 10−6 m/s can be accomplished in a R = 10 nm and L = 50 nm nanopore. The figure further illustrates that the shape of fluidic speed, uz (r), is quite complex compared to the conventional situation where no salt gradient was applied. 19 Here we attribute the physical mechanism to the pore-axial distribution of the electrical driving field Ez as shown in Fig.7b. At the higher salt concentration end (z = −L/2) due to the thinner EDL, the exclusion effect is overwhelmed by the diffusion effect, thus leading to electrical field pointing from Cmax to Cmin end. On the other hand, at the lower concentration end (z = L/2) the situation reverses and now Ez points from cis to trans chamber. As shown by the arrows in the figure, the overall effect of Ez on those counterions within the pore-wall EDL is a net electrical body force fe pointing from Cmax to Cmin end, while on the anionic DNA the net force points in the opposite direction. This interprets the reverse-osmosis-like fluidic speed near the pore wall (uz > 0), while osmosis-like one near the DNA surface (uz < 0), as demonstrated in Fig.7a. The above results indicate that by deliberately tuning the salt concentration bias, the device is capable of working within the transition region between exclusion and diffusion dominating. In this way, a substantially decreased DNA translocation speed would be achieved. Before ending the discussion, we hope to stress that our electrokinetic analysis of manipu18 ACS Paragon Plus Environment

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lating DNA with salt concentration bias is based on mean-field theory. However for extremely small-diameter nanopores, the molecular scale effects may substantially affect the DNA and ion transport properties. Such effects are difficult to be investigated in the framework of the mean-field theory, while can be addressed by molecular dynamic (MD) simulation. 48,49 Another issue is the DNA conformation manipulating within the nanopore/nanochannel. Stretched DNA and its regular motion of passing through nanopore belong to the basic requirements of nanopore-based DNA sequencer, and MD simulations can provide useful guidance on this point. 50,51 Hence all-atom MD simulation is called for to accurately evaluate our proposed regulating method of DNA motion in small-diameter nanopores.

Conclusion We have presented an electrokinetic modeling for the salt-concentration-biased nanochannel system. We have demonstrated that two competing mechanisms, namely the exclusion- and diffusioninduced electrical potentials, and their alternate dominating of ion and fluidic transport are responsible for those changes of fluid and voltages reported in the recent experiments. Based on the illustrated physical pictures, we have proposed and proven the feasibility of using a salt gradient approach to regulating DNA motion for the purpose of nanopore sequencing. Our quantitative study has demonstrated that two crucial indexes, which are the DNA capture rate and the translocation speed, could be satisfied based on this novel method.

Method We set up an electrokinetic model including Poisson-Boltzmann equation for electrostatics and ion distribution along the channel radial direction, Nernst-Plank equation for ion transport along the axial orientation and Navier-Stokes equation for fluidics. For the Poisson-Boltzmann equation, we remind that (1) During the numerical calculation we did not linearize the right-hand-side of the equation shown as Eq.3 in the context. The reason is that for nanopore or nanoslit system 19 ACS Paragon Plus Environment

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with diameters about 10 nm and surface charge density σw about 10 mC/m2 , the Poisson potential becomes much larger than several kT /e. We have found the conventional linearizing treatment invalid for this situation. Instead, we develop a self-adapt algorithm to numerically solve that equation. (2) It was supposed that the surface charge density σw , as one boundary condition shown in Eq.4, depended on the imposed electrolyte concentration C0 . 9 Yet in our modeling and calculation, we find that by assuming a constant σw the theoretical results show good agreement with the experimental observations. Hence we think that the dependence of σw on C0 might be a second order effect for the experimental cases studied in this work, and thus we neglect it. For the evaluation of flow rate, u¯z is the averaged fluidic speed calculated from u¯z = Q/π R2 = 2

RR 0

uz (r)rdr/R2.

Parameters we used: the density of charges on DNA strands λDNA = −0.48 × 10−9 C/m; the radius of DNA strands r0 = 1.1 Å; the electrophoretic mobility of DNA coils in the chambers: µ = 4.1−8 m2 /V·s.

Supporting Information Available The following file is available free of charge. Supplement Materials for the Salt-gradient Approach for Regulating Capture-to-Translocation Dy-namics of DNA with Nanochannel Sensors: Theoretical Derivations of 1) Poisson-Boltzmann equation along channel radial direction; 2) Analytic expression for salt gradient in the nanochannel; 3) Teorell-Meyer-Sievers model for monovalent and divalent solution; 4) Equation for nanofluidics under salt gradient; 5) ∆Vσ and ∆VD for monovalent and divalent ions; 6) The role of diffusio-osmosis; 7) DNA capture by electrical field; 8) Exclusive electrical field in II region.

Acknowledgement Y. He thanks financial supports by Key Technology Research and Innovation Program of Hubei Province (No.2015AEA075), and by Thousand Talent Program for Young Outstanding Scientists. A part of this work was supported by the Japan Society for the Promotion of Science (JSPS) KAK-ENHI Grant Number 16K13652. RHS Thanks the Swedish Research Council (VR) for financial support. 20 ACS Paragon Plus Environment

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References (1) Schoch, R. B.; Han, J.; Renaud, P. Transport Phenomena in Nanofluidics. Rev. Mod. Phys. 2008, 80, 839–883. (2) Haywood, D. G.; Saha-Shah, A.; Baker, L. A.; Jacobson, S. C. Fundamental Studies of Nanofluidics: Nanopores, Nanochannels, and Nanopipets. Anal. Chem. 2015, 87, 172–187, PMID: 25405581. (3) Taniguchi, M. Selective Multidetection Using Nanopores. Anal. Chem. 2015, 87, 188–199, PMID: 25387066. (4) Heng, J.; Aksimentiev, A.; Ho, C.; Marks, P.; Grinkova, Y.; Sligar, S.; Schulten, K.; Timp, G. The Electromechanics of DNA in a Synthetic Nanopore. Biophys. J. 2006, 90, 1098–1106. (5) Tsutsui, M.; Taniguchi, M.; Kawai, T. Transverse Field Effects on DNA-Sized Particle Dynamics. Nano Lett. 2009, 9, 1659–1662. (6) Kim, S. J.; Ko, S. H.; Kang, K. H.; Han, J. Direct Seawater Desalination by Ion Concentration Polarization. Nat. Nanotechnol. 2010, 5, 297–301. (7) van der Heyden, F. H. J.; Stein, D.; Dekker, C. Streaming Currents in a Single Nanofluidic Channel. Phys. Rev. Lett. 2005, 95, 116104. (8) van der Heyden, F. H. J.; Bonthuis, D. J.; Stein, D.; Meyer, C.; Dekker, C. Power Generation by Pressure-Driven Transport of Ions in Nanofluidic Channels. Nano Lett. 2007, 7, 1022– 1025. (9) Siria, A.; Poncharal, P.; Biance, A.-L.; Fulcrand, R.; Blase, X.; Purcell, S. T.; Bocquet, L. Giant Osmotic Energy Conversion Measured in a Single Transmembrane Boron Nitride Nanotube. Nature 2013, 494, 455–458. 21 ACS Paragon Plus Environment

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(48) Aksimentiev, A. Deciphering ionic current signatures of DNA transport through a nanopore. Nanoscale 2010, 2, 468–83. (49) Belkin, M.; Aksimentiev, A. Molecular Dynamics Simulation of DNA Capture and Transport in Heated Nanopores. ACS Applied Materials and Interfaces 2016, X–X, PMID: 26963065. (50) Sathe, C.; Girdhar, A.; Leburton, J.-P.; Schulten, K. Electronic detection of dsDNA transition from helical to zipper conformation using graphene nanopores. Nanotechnology 2014, 25, 445105. (51) Qiu, H.; Sarathy, A.; Leburton, J.-P.; Schulten, K. Intrinsic Stepwise Translocation of Stretched ssDNA in Graphene Nanopores. Nano Letters 2015, 15, 8322–8330, PMID: 26581231.


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(a)

+ + + + + + + + + ++ + + + + + ++ + + + + + + ++

+ RO

+ +

+

electrode

V electrode

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L

(b)

R

Cmax

V z

r

Cmin

Figure 1: (a) Schematic view and (b) Our modelling of salt-concentration-biased nanochannel system. Due to the larger concentration of imposed salt Cmax at z < −L/2 end, counterions for screening the channel-wall surface charges are piled more densely and thus the electrical double layers is thinner there; the situation then reverses for the smaller concentration Cmin at the other end. In (b) the arrows with different lengths indicate different magnitudes of cation and anion diffusion coefficients.


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Figure 2: (a) The calculated self-built potential ∆V and (b) averaged speed of fluid u¯z through a nanochannel under increasing LiCl concentration Cmax in the left chamber (black lines). The darkyellow dash lines show exclusion potential ∆Vσ and the corresponding fluidic speed u¯σ , while the red dot lines show diffusion potential ∆VD and the resulted speed u¯D . The related experimental results shown in Fig.3 of Ref. 10 are plotted with blue lines with round symbols. Here in order to match the experiments we plot −u¯z so that −u¯z > 0 denotes osmotic flow while −u¯z < 0 for reverse osmosis. σw is set to be −0.8 mC/m2 while other parameters are from the experiments: R = 50 nm and L = 6 µ m.


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Figure 3: Schematic view of salt-gradient-driven (a) DNA capture into and then (b) translocation through negatively charged nanopore. The cis chamber is imposed with Cmin while trans with Cmax . (a) During DNA capture stage the exclusion potential ∆Vσ dominates so that the self-built electrical field points from cis to trans chamber and osmotic flow is boosted. (b) During DNA translocation stage the salt gradient is tuned so that ∆Vσ is weakened for the purpose of slowing down DNA threading-through speed.


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(a)

(b)

Figure 4: (a) The calculated z-component of the electrical body force fe and (b) the fluidic speed uz along the channel radial direction under different LiCl gradients. Blue lines are for Cmax /Cmin = 0.1M/0.1 mM while red lines for Cmax /Cmin = 1 M/0.1 mM. The numbers in (a) characterize the percentages of net charges in the respective regions. Inset of (a) shows the variation of Debye lengths along the channel axial direction. The shadowed areas correspond to the electrical double layer regions under different Cmax . Parameters are the same as those in Fig.2.


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ACS Sensors

Figure 5: (a) The self-built electrical potentials ∆V as functions of the imposed salt gradient Cmax /Cmin when Cmin = 1 mM. The black-round symbols and blue-rhombus ones are experimental data for MgCl2 and KCl salt respectively. 13 The black and blue real lines are our calculation results for the situations where σw are set as −25 mC/m2 and −6 mC/m2 separately. The dark-yellow dash line plots the exclusion potential ∆Vσ for MgCl2 , while the red dot line represents the diffusion one ∆VD . (b) The self-built electrical potentials ∆V as functions of the imposed salt gradient Cmax /Cmin under various Cmin when using MgCl2 . The orange, black and blue lines are for Cmin = 0.1 mM, 1 mM and 10 mM respectively. Our calculation results are plotted with real lines, while the experimental ones are with symbols. Parameters are from the experiments: Y = 20 nm and L = 116 µ m.


ACS Sensors

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(a)

(b)

Figure 6: (a) The averaged fluidic speed inside a R = 10 nm and L = 50 nm nanopore as a function of the salt concentration gradient under various Cmin : 0.1 mM (blue lines), 1 mM (dark-yellow lines) and 10 mM (red lines). The real lines are for LiCl while dash lines are for KCl. (b) The ratio of salt-gradient-driven capture rate over the electrical counterpart γC /γE as a function of the salt gradient. The density of pore-wall surface charge is taken from previous results σw = −50 mC/m2 . 19,47


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ACS Sensors

Figure 7: (a) The calculated fluidic speed along the pore radial direction with DNA inside (real line) and without it (dash line), under LiCl gradient Cmax /Cmin = 480 mM/10 mM. Inset shows magnified view of the velocity near the DNA surface (r = r0 = 1.1 nm) and we remind that the fluidic speed there is actually the translocation speed of DNA molecule. (b) The corresponding electrical driving field Ez along the pore axial direction. Inset shows schematically the local electrical body force on the liquid and the molecule. Other parameters are the same as those in Fig.6.


ACS Sensors

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Salt-Gradient Approach for Regulating Capture-to ... - ACS Publications

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