Article pubs.acs.org/ac
Controlling pH-Regulated Bionanoparticles Translocation through Nanopores with Polyelectrolyte Brushes Li-Hsien Yeh,†,‡ Mingkan Zhang,†,§ Sang W. Joo,∥ Shizhi Qian,*,§,∥ and Jyh-Ping Hsu*,⊥ ‡
Department of Chemical and Materials Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan Institute of Micro/Nanotechnology, Old Dominion University, Norfolk, Virginia 23529, United States ∥ School of Mechanical Engineering, Yeungnam University, Gyongsan 712-719, South Korea ⊥ Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan §
S Supporting Information *
ABSTRACT: A novel polyelectrolyte (PE)-modified nanopore, comprising a solid-state nanopore functionalized by a nonregulated PE brush layer connecting two large reservoirs, is proposed to regulate the electrokinetic translocation of a soft nanoparticle (NP), comprising a rigid core covered by a pH-regulated, zwitterionic, soft layer, through it. The type of NP considered mimics bionanoparticles such as proteins and biomolecules. We find that a significant enrichment of H+ occurs near the inlet of a charged solid-state nanopore, appreciably reducing the charge density of the NP as it approaches there, thereby lowering the NP translocation velocity and making it harder to thread the nanopore. This difficulty can be resolved by the proposed PE-modified nanopore, which raises effectively both the capture rate and the capture velocity of the soft NP and simultaneously reduces its translocation velocity through the nanopore so that both the sensing efficiency and the resolution are enhanced. The results gathered provide a conceptual framework for the interpretation of relevant experimental data and for the design of nanopore-based devices used in single biomolecules sensing and DNA sequencing.
S
olid-state nanopores,1 nanosized pores embedded in thin solid membranes, have emerged as single-molecule biosensors for both sensing and characterizing individual unlabeled proteins2−5 as well as DNA6−11 and DNA− protein11−13 complexes in the past decade. The original concept is based on discriminating the characteristic temporary changes in the trans-membrane ionic current by the translocation of a (bio)molecule electrophoretically driven through a nanopore by an applied electric field. Because various constituents of (bio)molecules carry different surface charges, each of them obstructs the nanopore to a different characteristic degree, resulting in different ionic current signals. To enhance the sensing resolution at high throughput, two crucial challenges need to be resolved, namely, to facilitate (bio)molecules more effectively captured into the nanopore8,14 and to slow down their translocation velocity inside.15−17 Our recent study revealed that these problems can be overcome simultaneously by using a novel polyelectrolyte (PE)-modified nanopore,18 comprising a solid-state nanopore functionalized by a PE brush layer. Similar concepts have been realized experimentally by coating tail-modified DNA,19 hairpin-loop (HPL) DNA,20,21 and fluid lipid bilayer22 on synthetic nanopores to enhance the performance of single biomolecule sensing. Most of biological entities, such as DNA, proteins, and amino acids, are charge-regulated. The charged conditions of these © 2012 American Chemical Society
entities depend highly on the degrees of the dissociation/ association reactions of their functional groups and, therefore, on the solution properties such as background salt concentration and pH. However, previous studies almost always assumed that the charge density of such an entity remains at a constant value.14,17,18,23−27 In addition, the ionic mass distribution near a charged surface was described by a Poisson−Boltzmann (PB) model,17,26,27 where the electric double layer near that surface is at equilibrium and without distortion. This model becomes inappropriate in nanoporebased systems because the overlapping of the EDL of a particle and that of the nanopore is usually significant during the translocation process. At the present stage, developing a more general and realistic model is highly desirable for both elaborating experimental observations and designing sensing devices. In this study, a continuum-based model, comprising a set of coupled multi-ion Poisson−Nernst−Planck (PNP) equations for ionic transport and Stokes and Brinkman equations for the associated hydrodynamic field, is developed for the first time to investigate the electrokinetic translocation of a bionanoparticle Received: September 6, 2012 Accepted: October 4, 2012 Published: October 4, 2012 9615
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nm−2,28 and the assumption of constant ρfix,w is valid if the PE layer is either highly charged (e.g., DNA-functionalized nanopore)21 or nonregulated (e.g., lipid bilayer-coated nanopore).22 For simplicity, the possible morphology deformation of the PE layer28,29 is neglected. Suppose that the soft NP comprises a rigid core and an ion-penetrable soft layer of thickness d; the latter carries zwitterionic functional groups AH and B capable of undergoing the following reactions:
through a PE brush-functionalized nanopore connecting two large reservoirs. In contrast to previous studies, where the surface of a particle is assumed to maintain at a constant charge density, we consider a pH-regulated, zwitterionic, soft nanoparticle (NP), as schematically shown in Figure 1a. We show
AH ↔ A− + H+, BH+ ↔ B + H+
(1)
Let KA and KB be the equilibrium constants of these reactions, KA = [A−][H+]/[AH] and KB = [B][H+]/[BH+] with [X] being the molar concentration of species X in the soft layer. If we let NA and NB be the total concentrations of AH and B, respectively, then NA = [A−] + [AH] and NB = [B] + [BH+]. According to eq 1, the charge density of the soft NP, ρp, is30 ρp = 1000F([BH+] − [A−])
Figure 1. (a) Translocation of a pH-regulated, zwitterionic, soft NP through a PE brush-functionalized nanopore filled with a salt solution containing four major ionic species: H+, K+, Cl−, and OH−. (b) Mechanisms involved in the present problem (not to scale). The application of the electric field E yields an electro-osmotic flow (EOF) in the negatively charged nanopore which reduces the particle velocity. Meanwhile, a counterions (cations) rich concentration polarization (CP) field is induced, enhancing both the capture rate and the capture velocity of the NP before funneling the nanopore. If the NP is negatively charged, it experiences an electrophoretic force FE in the opposite direction as that of E and a hydrodynamic drag FH in the same direction as that of EOF due to the movement of counterions inside the EDL (red dotted line). Eenhanced is the enhanced local electric field inside the nanopore.
⎛ N [H+] NA ⎜ B KB = 1000F ⎜ − + [H ] [H+] ⎜1 + + 1 ⎝ KB KA
⎞ ⎟ ⎟⎟ ⎠
(2)
where F is the Faraday constant. Typical example for the type of soft NP considered includes proteins, amino acids, and synthetic NPs.31,32 The two reservoirs are large enough so that the concentration of the jth ionic species at a point far away from the nanopore reaches its bulk value, Cj0 (mM). Suppose that the background salt in the liquid phase is KCl of concentration CKCl, and the background solution pH0 (= −log[H+]0) is adjusted by KOH and HCl with [H+]0 being the bulk molar concentration of H+. This implies that four major ionic species need be considered, namely, H+, K+, Cl−, and OH− (i.e., N = 4). Let C10, C20, C30, and C40 be the bulk concentrations of these ions, respectively. Due to electroneutrality, the following conditions apply:30 if pH0 ≥ 7, C10 = 10−pH0+3, C20 = CKCl − 10−pH0+3 + 10−(14−pH0)+3, C30 = CKCl, and C40 = 10−(14−pH0)+3; if pH0 < 7, C10 = 10(−pH0+3), C20 = CKCl, C30 = CKCl + 10(−pH0+3) − 10−(14−pH0)+3, and C40 = 10−(14−pH0)+3. The following verified continuum-based model,18,25,33−36 taking into account the presence of multiple ionic species, is employed to describe the present problem. (i) Multi-ion Poisson−Nernst−Planck (PNP) equations for the ionic mass transport
that the charged properties of the soft NP depend highly on both its position in the nanopore and the background solution properties such as pH and salt concentration. Referring to Figure 1b, because the proposed PE-modified nanopore is capable of enhancing both the capture rate and the capture velocity of the soft NP and simultaneously reducing its translocation velocity through the nanopore, the difficulty that a soft NP is unable to pass through the corresponding charged solid-state nanopore under certain conditions can be resolved.
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THEORETICAL MODEL Figure 1a depicts the problem considered: a pH-regulated, zwitterionic, soft NP of radius a electrophoretically driven by an applied electric field E of strength E translocating from the cis reservoir to the trans reservoir along the axis of a cylindrical PEmodified nanopore. Both the nanopore and the reservoirs are filled with an incompressible, Newtonian, aqueous salt solution containing N kinds of ionic species. The cylindrical coordinates (r,θ,z), are adopted with the origin at the center of the nanopore. Because the present problem is θ-symmetric, only the (r,z) domain needs to be considered. E is in the z direction. The PE-modified nanopore consists of a solid, cylindrical membrane of length LN and radius RN, and PE brushes, referred to as the PE layer, are end-grafted to its surface. For simplicity, we assume that this layer is ion-penetrable, homogeneously structured, of uniform thickness Rw, and of constant fixed charge density ρfix,w ≅ (eZσPE/Rw) with e, Z, and σPE being the elementary charge, the valence of the dissociable groups per PE chain, and the density of the PE chain grafted to the solid membrane, respectively. Typically, σPE ranges from 0.1 to 0.6
−∇2 V =
ρe + hpρp + hw ρfix,w εf
(3)
⎛ ⎞ Dj ∇·Nj = ∇·⎜ucj − Dj∇cj − zj Fcj∇V ⎟ = 0, RT ⎝ ⎠ j = 1, 2, ..., N
(4)
Here, V is the electric potential; u = uer + νez is the fluid velocity with er and ez being, respectively, the unit vectors in the r- and z-directions; ρe = F∑Nj = 1zjcj is the space charge density of mobile ions; Nj, cj, Dj, and zj are the flux, concentration, diffusivity, and valence of the jth ionic species, respectively; εf, R, and T are the fluid permittivity, universal gas constant, and the absolute temperature, respectively. hp = 1 (hw = 1) for the region inside the soft layer of the soft NP (PE layer of the PE-modified nanopore), and hp = 0 (hw = 0) for the region outside these layers. 9616
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(ii) Modified Stokes-Brinkman equations for the flow field μ μ −∇p + μ∇2 u − ρe ∇V − hp 2 (u − Up) − hw 2 u = 0 λp λw
nanopore (λw) are 1 and 0.3 nm, respectively. These values are typical to biological or synthetic polymer layers (i.e., 0.1−10 nm).31 pH-Regulated Bionanoparticles. To simulate a pHregulated, zwitterionic, soft NP, we consider the typical case of amino acid with a = 5 nm, d = 1.5 nm, pKA = 2.5 (αcarboxyl), pKB = 8.5 (α-amino),39 and NA = NB = 0.6 M.31 The size of the soft NP assumed is typical to detected biomolecules such as proteins in nanopore-based sensing devices.3−5,22 The values of pKA and pKB assumed imply that the isoelectric point (IEP) of the soft NP is 5.5. As described by eq 2, ρp depends highly upon the local pH (= −log[H+]) inside the soft NP. Figure 2 illustrates the spatial distributions of ρp at various
(5) (6)
∇·u = 0
Here, p and μ are the hydrodynamic pressure and the fluid viscosity, respectively; Up = upez is the particle translocation velocity along the axis of the nanopore; λp = (μ/γp)1/2 and λw = (μ/γw)1/2 are the softness degree (or Brinkman screening length) of the soft layer of the soft NP and that of the PE layer of the PE-modified nanopore, respectively, with γp and γw being the hydrodynamic frictional coefficients of those layers, respectively. The boundary conditions associated with eqs 3−6 are based on the assumptions provided in the Supporting Information. Under a quasi-steady state condition, the translocation velocity of the soft NP, up, can be determined by a balance of the forces acting on it in the z-direction, FE + FH = 0. Here, FE and FH can be obtained by an integration of the Maxwell stress tensor and the hydrodynamic stress tensor over the soft NP surface, respectively.24,25 A more detailed description for the calculation of the relevant forces, numerical implementation, and code validation is given in the Supporting Information. Due to the conservation of the ionic current, I, its value through the nanopore can be evaluated by N
I=
∫S
F(∑ zj Nj) ·ndS j=1
(7)
where S can be chosen as any one of the end surfaces of the two reservoirs. The performance of a translocation process can be measured by the ionic current deviation, χ = (I − I0)/I0, where I0 is the base ionic current when the soft NP is far away from the nanopore. If χ < 0 (χ > 0), a current blockade (enhancement) occurs during the NP translocation process.23
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RESULTS AND DISCUSSION Numerical simulation is conducted to investigate the influence of the background salt concentration CKCl and the background solution pH0, the two key factors in the experiment, on the translocation behaviors of the soft NP considered. Unless otherwise specified, the following values are assumed:22 nanopore length LN = 18 nm, nanopore radius RN = 9.6 nm, thickness of PE layer Rw = 3.5 nm, and applied potential bias V0 = 0.5 V. Note that, for RN ≥ 3 nm, the present continuum model is sufficient to capture and elucidate the essential physics of the problem considered.17,18,26,27 Corry et al.37 also showed that the results based on the present PNP model agree well with those on Brownian dynamic simulations when RN exceeds 1 nm. Suppose that the charge density of the PE layer of the PE-modified nanopore, ρfix,w, is fixed at −4.57 × 106 C/m3, corresponding to Z = −1 and σPE = 0.1 nm−2. The scales of both reservoirs are LR = 200 nm and RR = 200 nm. The diffusivities of H+, K+, Cl−, and OH− are 9.31 × 10−9, 1.96 × 10−9, 2.03 × 10−9, and 5.30 × 10−9 m2/s, respectively.34 For simplicity, the possible variation in the ionic diffusivities inside a nanopore38 is neglected in the present study. Other physical parameters used are εf = 7.08 × 10−10F/m, R = 8.31 J/(K·mol), F = 96490 C/mol, μ = 1 × 10−3 pa·s, and T = 300 K. For illustration, we assume that the softness degrees of the soft layer of the soft NP (λp) and the PE layer in the PE-modified
Figure 2. Variation of charge density in the soft layer of soft NP, ρp (C/m3), for various combinations of the background salt concentration CKCl and solution pH0 at zp = −60 (a, d, and g); zp = −10 (b, e, and h); and zp = 0 nm (c, f, and i). (a−c) CKCl = 50 mM and pH0 = 8.5; (d−f) CKCl = 50 mM and pH0 = 7.5; (g−i) CKCl = 1000 mM and pH0 = 7.5.
particle positions zp for various combinations of the background salt concentration CKCl and pH0. The corresponding spatial distributions of H+ (or local pH) are presented in Figure S2 of the Supporting Information. As expected, if pH0 > 5.5, the soft NP is negatively charged (ρp < 0). In addition, |ρp| increases with increasing local pH value. This is because the lower the H+ concentration (high pH) the more the amounts of A− and B dissociated, respectively, from the functional groups HA and BH+ of the soft layer of the NP, yielding a more amount of negative charge. 9617
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uncharged (dashed line with circles) solid-state nanopore are also presented. Note that the net amount of charge carried by the PE-modified nanopore and that of the solid-state nanopore are the same, that is, ρfix,w × ΦPE = σw × Δsolid, with ΦPE and Δsolid being the volume of the PE layer in the former and the surface area of the solid membrane in the latter, respectively. As expected, |ρ̅p| increases with increasing pH0. However, the increase of |ρ̅p| with increasing CKCl is unexpected, which can be attributed to the excluded effect of counterions. In the present case, the soft NP is negatively charged and, therefore, both H+ and K+ are electrostatically attracted into its PE layer. However, as CKCl increases, some amount of H+ is expelled out from that layer by the increasing background K+, resulting in a higher local pH inside the soft NP (Figure S2 of the Supporting Information), and |ρ̅p| becomes higher, accordingly. It is interesting to note in Figure 3 that |ρ̅p| has a local minimum (maximum) as the soft NP enters (leaves) the charged soft or solid-state nanopore. This can be attributed to the ion CP, which yields an enrichment (a depletion) of H+ near the cis (trans) side of the nanopore. Because that effect is significant when the EDL of the nanopore is seriously overlapped, the difference between the highest and the lowest values of |ρ̅p| increases with decreasing salt concentration CKCl, as seen in Figure 3a. However, if the nanopore is uncharged (lines with circles), ion CP does not occur and, therefore, |ρ̅p| is nearly independent of the particle position. Note that, in this case, ρ̅p still changes slightly as the soft NP enters an uncharged nanopore. This arises from the deformation of the EDL of the soft NP due to the presence of the nanopore. Figure 3b also reveals that, at a fixed level of CKCl, the difference between the highest and lowest values of |ρ̅p| increases with the increasing pH0. This can be attributed to a more significant DLP effect occurring at a higher pH0, yielding a more asymmetric distribution in H+. Control of the pH-Regulated BioNP Translocation. In the conventional nanopore-based biomolecule sensing techniques, a negatively charged (cation-selective) solid-state nanopore is used.42,43 In this case, an EOF is present inside the nanopore and ion CP occurs near both of its openings. The former is capable of slowing down the particle translocation through the nanopore, thereby improving the read-out accuracy.17,24 The latter, however, results in a significant decrease in the charge density of a pH-regulated bioNP when it approaches the cis side of the nanopore (nanopore mouth), as shown in Figure 3. This reduces the electrical driving force acting on the particle, and it is hard to thread the charged solidstate nanopore, defined as the access barrier. As will be shown in the following, this barrier can be overcome by adopting the present PE-functionalized nanopore (PE-modified nanopore). Figure 4 depicts the translocation velocity of a pH-regulated soft NP, up (Figure 4a,b), and the corresponding ionic current deviation, χ, (Figure 4c,d), as a function of its location, zp, under various combinations of the background solution properties, CKCl and pH0. For comparison, both the present results for a PE-modified nanopore and the corresponding results for an uncharged and a charged solid-state nanopore are presented. Figure 4a,b clearly shows that, due to an enhanced electric field inside the nanopore, the up in the uncharged solidstate nanopore (|zp| ≤ 14 nm) is appreciably larger than that in the fluid reservoirs. In addition, the up in the PE-modified nanopore is significantly smaller than that in the uncharged solid-state nanopore. That is, the negatively charged PE layer engrafted to the solid-state nanopore is capable of reducing the
Figure 2 also shows that if the soft NP is far away from the PE-modified nanopore (e.g., zp = −60 nm), ρp is approximately uniform along the axial direction, implying that it is uninfluenced by the nanopore. However, as the soft NP enters the PE-modified nanopore (e.g., zp = 0 and −10 nm), ρp becomes highly nonuniform. In this case, the |ρp| near the bottom region of the soft NP is smaller than that near its top region. The behavior of ρp can be explained by that of H+ shown in Figure S2 of the Supporting Information. The nonuniform distribution in ρp can be attributed to the combined effects of the double-layer polarization (DLP)40,41 due to the motion of the soft NP and the ion concentration polarization (CP) occurring in the PE-modified nanopore. The former effect arises mainly from the convective motion of the ionic species inside the EDL of the soft NP, and it becomes appreciable when the electric field is strong and the thickness of EDL is comparable to the particle radius.40,41 The latter effect results from the selective transport of ions in the nanopore, and is significant when the overlapping of its EDL is serious.33,42 Due to the mismatch of the cross-sectional areas of the fluid reservoir and the nanopore, the electric field inside the latter (Eenhanced) is extremely high, leading to a significant DLP effect. As a result, counterions (cations in this study) migrate from the upper region of the soft NP toward its lower region, yielding a smaller local pH (see Figure S2 of the Supporting Information) and, therefore, smaller |ρp| near its lower region. In addition, because ion concentration polarization (CP) occurs in the negatively charged nanopore, a significant amount of cations (H+ and K+) migrate from the trans side of the nanopore toward its cis side,42 which also makes both the distributions of H+ and, accordingly, ρp highly asymmetric when the soft NP enters the nanopore. Note that, as CKCl gets high (EDL is thin), the overlapping of EDLs is less significant, and the distribution of ρp becomes less asymmetric. This can be attributed to that the effects of DLP and ion CP are less significant under present conditions. To further see the influence of the position of the soft NP zp on its charge density, we plot its volume-averaged charge density ρ̅p as a function of zp at various combinations of CKCl and pH0 in Figure 3. For comparison, the corresponding results of ρ̅p at CKCl = 200 mM (Figure 3a) and at pH0 = 8 (Figure 3b) for the cases of a charged (dashed line with triangles) and an
Figure 3. Volume-averaged charge density of the soft layer of the soft NP, ρ̅p, as a function of its particle position zp for various levels of background salt concentrations CKCl at the background solution pH0 = 7.5 (a) and for various levels of pH0 at CKCl = 50 mM (b). The blue regions (−17.5 nm ≤ zp ≤17.5 nm) highlight the region where the soft NP locates in the PE-modified nanopore; lines, present results; lines with discrete symbols, corresponding result at CKCl = 200 mM (a) and at pH0 = 8 (b). Circles and triangles denote results for an uncharged and a charged solid-state nanopore, respectively. 9618
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Figure 4. Particle translocation velocity up (a, b) and corresponding ionic current deviation χ (c, d) as a function of the particle position zp for various levels of background salt concentration CKCl at background pH0 = 7.5 (a, c) and for various levels of pH0 at CKCl = 50 mM (b, d). The blue regions (−17.5 nm ≤ zp ≤ 17.5 nm) highlight the region where the soft NP locates in the PE-modified nanopore; the curves in (c) and (d) show the results for the case where the soft NP is able to pass through the nanopore: lines, present results; lines with discrete symbols, corresponding results at CKCl = 200 mM (a, c) and those at pH0 = 8 (b, d). Circles and triangles denote results for an uncharged and a charged solid-state nanopore, respectively.
up inside it. For example, the maximum NP translocation velocity inside the uncharged solid-state nanopore is reduced from 0.0592 to 0.0272 (0.169 to 0.07) m/s when the PE layer is coated at CKCl = 200 mM in Figure 4a (pH0 = 8 in Figure 4b), suggesting that the sensing resolution of the NP translocation can be enhanced by about 54−58%, depending on the background solution properties chosen. This can be attributed to the presence of a significant EOF inside the PE-modified nanopore, as schematically depicted in Figure 1b. Note that, if the solid-state nanopore is charged, the soft NP is trapped (negative up) before entering it, which arises from the combined effects of EOF and the decrease in |ρ̅p| shown in Figure 3. As will be shown later, the trapping of the soft NP can be resolved by adopting the proposed PE-modified nanopore. Figure 4a,b reveals that, at low CKCl and pH0, up is negative before the soft NP funneling the PE-modified nanopore (blue region), implying that it is trapped there. The behavior of up at a low pH0 is because |ρp| increases with increasing pH0, as shown in Figure 3b, resulting in a greater electrophoretic driving force FE. One could raise pH0 so that the hydrodynamic drag acting on soft NP due to EOF can be overcome by FE, thereby threading the nanopore. Our result that a positive up inside a PE-modified nanopore increases with increasing pH0 is qualitatively consistent with the experimental observation, where the translocation time of the protein, streptavidin, through a nanopore with a fluid lipid bilayer decreases with increasing pH0 (Figure 5 of Yusko et al.22). The behavior of up at a relatively low CKCl is surprising and inconsistent with the previous study, where a DNA is simulated by a long, rigid nanorod having a constant surface charge density, was trapped before entering a PE-modified nanopore at a high salt
Figure 5. Flow field (a, d) and the spatial distribution of the net concentration of mobile ions (∑4j = 1zjcj; b, c) for the case of Figure 4a at zp = −10 nm. CKCl = 50 mM in (a) and (b) and CKCl = 1000 mM in (c) and (d). (a, d) Color bars reflect the magnitude of axial fluid velocity, and streamlines with arrows denote fluid velocity vector. (a− d) Dashed curves denote the outer boundaries of the soft layer of the soft NP (pink) and the PE layer of the PE-modified nanopore (black).
concentration.18 In general, the FE acting on a particle having a constant charge density decreases with increasing salt concentration.44 If this effect dominates, the particle translocation velocity should decrease with increasing salt concentration,17,24,25,45 and even change its sign from positive to negative when that concentration is sufficiently high.18 9619
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(Figure 4d), current blockade (enhancement) occurs as the soft NP enters (exits) the PE-modified nanopore. The current blockade is known to arise mainly from the physical blocking by NPs, as observed experimentally in many studies.3,6,8,10,16 The current enhancement was also observed in some experiments conducted in solid-state nanopores47,48 or chemically functionalized nanopores.21 Recently, that the current decreases first and then increases was observed experimentally by Kowalczyk and Dekker49 for the electrokinetic translocation of dsDNA through a solid-state nanopore at low salt concentration (i.e., 100 mM) and intermediate voltage bias (0.2 V ≤ V0 ≤ 0.6 V). In our case, the current signals are influenced by two competing factors as the soft NP passes through the PE-modified nanopore. (i) The ionic current decreases because an equivalent volume of salt solutions is vented out from the nanopore. (ii) The screened counterions (cations) carried by the negatively charged soft NP raises the ionic current. If the former (latter) dominates, current blockade (enhancement) occurs. It should be emphasized that although the chemical interactions between the sensing molecules and the PEmodified nanopore is neglected, our results qualitatively agree with the experimental observations of the translocation of biomolecules through the fluidic liquid bilayer-22 and the DNAfunctionalized nanopores.21 As shown in Figure 4, the translocation of the bioNP can be regulated by the proposed PE-modified nanopore without affecting its basic ionic current signature. In addition to the background solution properties (CKCl and pH0), the flow field in the PE-modified nanopore, and therefore, the translocation velocity of the soft NP can also be tuned by the PE layer structure (e.g., softness degrees) of the nanopore through nanofabricating. The influences of the softness degrees of the soft layer of the soft NP (λp) and the PE layer of the PE-modified nanopore (λw) on the translocation behaviors of the soft NP are illustrated in the Supporting Information. The influence of the background solution properties on the critical access velocity of the soft NP, up,c, which is capable of overcoming the access barrier to thread the nanopore, for two values of nanopore length is given in the Supporting Information.
Although the assumption of constant charge density might be appropriate for DNA,46 it is inappropriate in our case because the soft NP is of charge-regulated nature, as in the cases of most proteins and biomolecules. A negative value of up at low CKCl can be attributed to two competing factors. First, |ρp| decreases with decreasing CKCl, as discussed in Figure 3a. Second, the net concentration of mobile ions ∑4j = 1zjcj near the center region of the PE-modified nanopore outside its PE layer increases with decreasing CKCl, yielding a stronger EOF there. The second factor is illustrated in Figure 5, where both ∑4j = 1zjcj and the flow field for two levels of CKCl are plotted for the case where the soft NP enters the PE-modified nanopore (i.e., zp = −10 nm). As seen in Figure 5b,c, the lower the CKCl, the thicker the EDL and, therefore, the larger the ∑4j = 1zjcj near the center region of the PE-modified nanopore (|r| < 6.1 nm). As a result, the magnitude of the opposite EOF velocity there increases with decreasing CKCl, as seen in Figure 5a,d. At a relatively low CKCl, the opposite EOF is sufficiently strong to prevent the soft NP from translocating through the nanopore, that is, it is not capable of threading the PE-modified nanopore. It is interesting to note in Figure 5a that the direction of flow field in the fluidic reservoirs is opposite to that of the EOF inside the PE-modified nanopore. As shown in Figure S3 of the Supporting Information, this unique feature occurs only in the present novel PE-modified nanopore, but not in the conventional solid-state nanopore. This results from the ion CP phenomenon in the former is more significant than that in the latter, as seen in Figure S3b,c. The fluid in the cis reservoir flows toward the entrance of the PE-modified nanopore (see Figure S3a), which drags the negatively charged NP inside toward that entrance, thereby enhancing its capture rate. As suggested by He et al.,14 the capture rate of DNA molecules into the nanopore depends upon the voltage residue ratio Vcis/ V0, where Vcis is the residue voltage drop in the cis reservoir; in general, the larger the Vcis/V0 the higher is the capture rate. Under the conditions assumed, Vcis/V0 is 0.115 in the case of Figure S3d (solid-state nanopore) and 0.126 in the case of Figure S3a (PE-modified nanopore), implying that the capture rate is enhanced by about 10%. Figure 4a,b reveals that if CKCl is not too high, up shows a local maximum at up ≅ −17.5 nm, implying that the translocation velocity of the soft NP before entering the PE-modified nanopore, defined as its capture velocity, is enhanced so that it can overcome the access barrier and enter the nanopore. This can be attributed to a significant ion CP occurring in the PE-modified nanopore, which induces a counterions-rich (i.e., cations-rich in this case) electrostatic field near the nanopore mouth.8,18 A stronger interaction between the negatively charge NP and the cations-rich electrostatic field enhances the capture velocity. As can be seen in Figure 4b, the higher the pH0 the more significant that enhanced effect in the capture velocity is. This is because the higher the pH0 the larger the ρ̅p and, therefore, the stronger the electrostatic attraction between the soft NP and the counterions-rich electrostatic field aforementioned. In nanopore-based sensing techniques, the physicochemical properties of a single biomolecule can be probed by detecting the variation in the ionic current through the nanopore due to its translocation. The influence of the background solution CKCl and pH0 on χ is depicted in Figure 4c,d. As seen in Figure 4c, if CKCl is sufficiently high, except for a momentarily current enhancement occurring as the soft NP is about to exit the PEmodified nanopore at CKCl = 200 mM (dashed line), current blockade occurs, in general. On the other hand, if CKCl is low
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CONCLUSIONS We have theoretically studied, for the first time, the electrokinetic translocation of a soft nanoparticle (NP), comprising a rigid core and a pH-regulated, zwitterionic, soft layer, through a nanopore functionalized with polyelectrolyte (PE) brushes connecting two large fluid reservoirs. In contrast to the existing studies where a particle is assumed to maintain at a constant charge, a much more general model is considered in our case, where the charged conditions of the NP depend upon both the solution properties such as background salt concentration and pH and the presence of the nanopore. We find that, as the soft NP approaches a negatively charged solidstate nanopore, due to a significant ion concentration polarization (CP) occurring at both openings of the nanopore, the concentration of hydrogen ions is enhanced near the inlet of the nanopore, resulting in a significant decrease in the charge density of the soft NP. Therefore, it is hard for the soft NP to enter the nanopore, yielding a low sensing efficiency. This problem can be overcome by the proposed PE-modified nanopore, which comprises a solid-state nanopore engrafted with a nonregulated PE layer. The PE-modified nanopore is able to induce a unique flow field in the two reservoirs, the 9620
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direction of which is opposite to that of the electroosmotic flow (EOF) inside the nanopore, thereby facilitating the soft NP to migrate toward the nanopore, that is, raising its capture rate. In addition, the induced EOF inside the PE-modified nanopore reduces the translocation of the soft NP through the nanopore. These two effects suggest that the PE-modified nanopore proposed has the potential in single molecules sensing because it is capable of raising both the sensing efficiency and the resolution. Our results also demonstrate that the translocation behaviors of the soft NP depend highly upon the properties of the background solution, such as its salt concentration and pH, as well as the interaction between the NP and the nanopore. The simulation results gathered provide both necessary theoretical background and valuable information for the experimental regulation of single biomolecules translocating through a chemically functionalized nanopore. The present model can be applied to simulate the DNA translocation through a nanopore by replacing the spherical NP by a cylindrical one and using appropriate parameters.
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ASSOCIATED CONTENT
S Supporting Information *
Details of the boundary conditions, numerical implementation, code validation, forces acting on the soft NP, the distribution of pH in its PE layer, the enhanced positive flow field at the mouth of the PE-modified nanopore, and the influences of the softness degree of the soft NP and that of the PE-modified nanopore and the nanopore length on the translocation behaviors of the soft NP. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected];
[email protected]. Author Contributions †
These authors contributed equally to this work.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by the National Science Council of the Republic of China and the World Class University Grant No. R32-2008-000-20082-0 of the Ministry of Education, Science and Technology of Korea.
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