Article pubs.acs.org/JPCB
Proximal Capture Dynamics for a Single Biological Nanopore Sensor Emmanuel D. Pederson,† Jonathan Barbalas,† Bryon S. Drown,† Michael J. Culbertson,† Lisa M. Keranen Burden,‡ John J. Kasianowicz,§ and Daniel L. Burden*,† †
Chemistry Department, Wheaton College, Wheaton, Illinois 60187, United States Biology Department, Wheaton College, Wheaton, Illinois 60187, United States § Semiconductor Electronics Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8120, United States ‡
S Supporting Information *
ABSTRACT: Single nanopore sensors enable capture and analysis of molecules that are driven to the pore entry from bulk solution. However, the distance between an analyte and the nanopore opening limits the detection efficiency. A theoretical basis for predicting particle capture rate is important for designing modified nanopore sensors, especially for those with covalently tethered reaction sites. Using the finite element method, we develop a soft-walled electrostatic block (SWEB) model for the alpha-hemolysin channel that produces a vector map of drift-producing forces on particles diffusing near the pore entrance. The maps are then coupled to a single-particle diffusion simulation to probe capture statistics and to track the trajectories of individual particles on the μs to ms time scales. The investigation enables evaluation of the interplay among the electrophoretic, electroosmotic, and thermal driving forces as a function of applied potential. The findings demonstrate how the complex drift-producing forces compete with diffusion over the nanoscale dimensions of the pore. The results also demonstrate the spatial and temporal limitations associated with nanopore detection and offer a basic theoretical framework to guide both the placement and kinetics of reaction sites located on, or near, the nanopore cap.
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Here, C produces a physical response (i.e., current fluctuation) when passing through the constriction region. Examples which employ this straightforward scheme include the detection of various DNA conjugates (biotin/streptavidin, antibodies, duplex formation, stable hairpins, etc.)23,24 and ions.25,26 Reversible chemical reactions place additional kinetic constraints on nanopore detection because the reaction product(s) are only temporarily present.
INTRODUCTION Single nanopores have broad analytical potential.1−3 Numerous synthetic and natural nanometer-scale pores have emerged as interrogating devices, including those created by ion-beam lithography 4 (in materials such as silicon nitride or graphene),5−10 and protein ion channels that self-assemble in lipid bilayers (e.g., alpha hemolysin,11 gramicidin,12 OmpG13). In general, analytes are probed by applying a voltage across a nanopore in an insulating partition that is bathed on both sides by aqueous electrolyte solutions. The small passageway electrically connects one chamber to the other, permitting the flow of ions. This flow can be momentarily blocked, or reduced, when species of the proper dimension and charge enter the pore. A stochastic, time-dependent current fluctuation results as analytes are drawn in from solution and transit the pore lumen. These electrical signatures have been used to identify a host of chemical species, including small ions,14,15 organic molecules,16 and polymers.17−21 Additionally, the patterns embedded within the ionic current fluctuations can reveal features of molecular structure, such as polymer folding, conformational distributions, or distributions of size.17,22,23 Many nanopore-based sensors detect analytes that exist in a stable form. That is, a chemically stable entity (C) is either added directly to solution on one side of the sensor, or a detectable species is created in solution as a product of an irreversible chemical reaction:
A+B⇄C
Products that exist only briefly effectively limit the concentration sensitivity of the sensor, because a reaction with a large reverse rate constant demands higher reactant concentrations to produce a detectable yield. To combat this limitation, reversible reaction centers can be engineered inside the pore, or held within close proximity to the nanopore. In highly reversible instances, the probability of detecting the reaction product is increased by spatially locating the product near, or at, the nanopore detection zone. A number of reversible reactions involving small molecules16 or proteins27 have been reported. We recently tethered gold nanoparticles to the cap of the alpha-hemolysin (αHL) nanopore as a means of optically inducing a temperature change in proximity to the pore.28 Using this concept, temperature modulation could be used to Received: May 24, 2015 Revised: July 19, 2015
A+B→C © XXXX American Chemical Society
A
DOI: 10.1021/acs.jpcb.5b04955 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B generate reaction products in a reversible fashion (e.g., temporarily melting double stranded DNA segments). Alternatively, locally enhanced temperatures could be used to momentarily increase the rate of chemical catalysis for an otherwise unfavorable reaction. Other propositions for sensor design with tethered reaction centers involve momentary binding of engineered ligands to proteins that contact the nanopore cap,29 or placing enzymes near the mouth of the nanopore to free nucleotides in a polymeric DNA strand.30 In order for schemes of this genre to be employed more commonly, deeper insight into the migration rates and capture dynamics for analytes located inside and outside the nanopore is needed. In this study, we use a 3D, cylindrically symmetric, softwalled electrostatic block (SWEB) model of the wild-type αHL nanopore to estimate the magnitude and direction of forces that arise on charged particles moving near the pore entry. The Poisson−Nernst−Plank and Navier−Stokes equations are solved via finite element analysis for the spatial and electrostatic features of the αHL nanopore. This yields a vector map of analyte drift both inside and outside the nanopore. The numerical vector map is then coupled with a Brownian dynamics computer simulation to track single-particle migration over large distances.31,32 Previous computational studies have examined various dynamic aspects of translocation through αHL.33−37 Most involve calculating many atom−atom interactions within a relatively small volume (∼103 nm3) and are limited to short periods of time (10−9−10−6 s). Other recent studies have sought to lessen the computational burden required for atomistic modeling by treating a fraction of mobile ions explicitly and modeling the surrounding protein and solvent implicitly.38−41 Many of these works seek to understand the specific mechanisms of signal generation within the channel in terms of short-range bonding, ionic interactions, and substrate shape, with a specific focus on the interaction of DNA within the nanopore. In this study, we develop an even further simplified model (Figure 1) that provides access to dramatically larger simulation volumes (∼1010 nm3) and long simulation times (∼10−3 s) but ignores specific, short-range protein/ substrate molecular and ionic interactions in an effort to track the motion of small nonpolymeric analytes prior to the signal generation event (i.e., capture). Importantly, the continuum theory used in the SWEB model matches available experimental data for ion and water flow through αHL reasonably well and also exhibits an electric field profile that mimics previous atomistic models (see the Supporting Information). Using this approach, we test four specific questions that relate random Brownian motion of analyte particles to systematic drift. First, we explore the relationship between particle distance from the nanopore and the applied electric field. Applied potentials of >300 mV significantly enhance capture probability for negatively charged particles. Second, because particle size likely plays a role in capture efficiency, we test a range of particle radii. The results suggest that it is more likely to capture big particles that can fit within the nanopore opening. Third, we investigate the relationship between negatively and positively charged particles that are released on the cap side of the nanopore. Although negative particles are readily captured, the electrostatic and electroosmotic flow conditions disfavor the capture of positive particles. Lastly, we evaluate the effects of salt concentration on the electric field intensity in the bulk and its corresponding impact on particle
Figure 1. Soft-wall electrostatic block (SWEB) model of αHL. Charged residues within the nanopore are modeled as rings of charge (a and b). Charged particles (c) are released from various locations (d, θ = 90°) above the nanopore cap, and the resulting diffusion trajectories are tracked through time. The capture point is located 5 nm below the pore entrance, just beyond the constriction region. In parts c and d, the applied transmembrane potential is +40 mV and the salt (KCl) concentration is 1 M on both sides of the bilayer. In part d, a mean distance of 1000 diffusion time steps (horizontal light blue arrow) is shown on the same logarithmic scale as the electrophoretic drift velocity (white arrows). Electroosmotic drift vectors (black arrows) are scaled up by 10 000 time steps for ease of visualization. At +40 mV, diffusive motion dominates particle behavior and electroosmotic flow contributes only minimally to drift.
capture probability. All of these results provide context for the design and use of novel reaction sites that produce capturable species within proximity to single-nanopore sensors.
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METHODS SWEB (Soft-Walled Electrostatic Block). The geometric contour of the heptameric nanopore is modeled as a cylindrically symmetric dielectric block (εp = 4) with an aqueous interior (εw = 80) embedded within a block lipid bilayer (εb = 2). The dimensions approximate the heptameric protein crystal structure.42 At pH 7, numerous charged residues line the inside (Figure 1a) and the top of the nanopore (Figure 1b). Charged amino acids (blue = positive, red = negative) are modeled as a ring of charge with 7e− (7·1.602 × 10−19 C) spread evenly over the ring surface area. Ring area is defined by the amino acid’s radial distance from the central lumen axis and the distribution of side-chain locations. In the case of Lys147, which resides at the narrowest constriction point of the channel (radius of 0.7 nm), the axial distribution of the seven positively charged amine groups forms a ribbon that is 0.4 nm tall. Thus, the charge density associated with the Lys147 ribbon is 7e/2πrh = 7·1.6 × 10−19 C/[2π (0.7 × 10−9 m) (0.4 × 10−9 m)] = 0.63 C/m 2 B
(1) DOI: 10.1021/acs.jpcb.5b04955 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
electroosmotic drift velocities for use in single-particle capture simulations. Simulated Diffusion with Drift. Diffusion is simulated using Brownian dynamics software developed for modeling single-molecule translation in fluorescence correlation spectroscopy (FCS) over long time scales.32,44 Routines from the single molecule diffusion simulator (SMDS) were adapted to implement diffusion with drift in the context of the geometric constraints typical of an αHL nanopore and the drift equations outlined above. The SMDS uses bulk physical parameters (such as the analyte diffusion constant, electrophoretic mobility, total transmembrane voltage, etc.) to perform a random walk that excludes movement in specified locations. No periodic boundary conditions are implemented, as each particle is allowed to translate until it is successfully captured, or a specified maximum run time is exceeded (1−1000 μs). Thus, many particles migrate well outside the volumetric boundary of the SWEB model (1 × 106 nm3). At this boundary, all driftproducing forces have effectively decayed to zero. Each diffusing species moves with a randomly determined cardinal direction and a step length determined by sampling from a Gaussian distribution with a mean of zero and a standard deviation, dxd, which is related to the characteristic diffusion coefficient by45
Electrostatic charges are added on, or near, the walls of the block to mimic the location of ionizable amino acid side chains (Arg, Lys, Glu, Asp). The following amino acids line the lumen interior and are included in the SWEB (ASP128, LYS131, ASP127, LYS147, GLU111, ASP227, ASP4, ASP108, ARG104, ARG56, ASP13, LYS8). Additionally, the following cap amino acids are located along the top surface and are also included in the SWEB (ARG236, LYS288, LYS240, LYS237, LYS51, LYS46, LYS21, ASP44, ASP285, GLU289, GLU287). The block walls form a barrier that prohibits ion and water permeation. Additionally, we implement spatially dependent diffusion constants and electrophoretic mobilities. These parameters change from their bulk values as particles approach the block wall, following the method of Simakov43 (see the Supporting Information). The block is said to possess a soft wall due to the slowing of particle motion near the water/ dielectric boundary. Spatially dependent electric field (Figure 1d, white arrows) and electroosmotic flow (c and d, black arrows) vectors are determined using the finite element method within the COMSOL multiphysics environment. The SWEB model produces vector maps in and around the nanopore by solving coupled differential equations with appropriate boundary conditions. Briefly, the flux of mobile electrolyte ions and water is determined by finding stationary solutions to the Poisson−Nernst−Plank (PNP) equation (eq 1) and the Navier−Stokes equation (eq 2) at spatial nodes that are defined by the COMSOL meshing algorithm. The continuity of the flux under steady state conditions requires the divergence of the PNP equation to be zero. The two equations are coupled through u (the fluid velocity parameter) and F (the volume force term). See the Supporting Information for details: 0 = ∇·[Di∇ci + μe, i zici∇V ] − u ·∇ci
dxd = (6D dt )1/2
The bulk diffusion constant is set to 3 × 10 cm /s, which is typical of small molecules in room-temperature solution. The diffusion constant decreases as particles approach the walls of the SWEB, as described above. The overall distance and direction a particle travels per unit time is calculated by summing drift and diffusion terms: ⎯→ ⎯⎯→ → ⎯ dx = v⎯→ (6) eo dt + ve dt + dxd
(2)
⎡ ⎤ 2 ρ(u ·∇)u = ∇· ⎢ − pI + μ(∇u + (∇u)T ) − μ(∇· u)I ⎥ + F ⎣ ⎦ 3
In eq 2, Di is the diffusion coefficient of the ith electrolyte species, ci is the buffer ion concentration, μe,i is the mobility of the buffer ions, zi is the charge state of the species, and V is the voltage. In eq 3, ρ is the solution density, p is the pressure, T is the viscous stress tensor, μ is the dynamic viscosity, and F equals the volume force (ρsE). The solution to these equations produces a map of the electric field (E⃗ ) and electroosmotic velocity (v⎯→ eo ) vectors. Electrophoretic drift velocity is determined by multiplying the electric field by the spatially dependent mobility, μe,i. → v⎯ = μ E ⃗ e, i
2
As long as the simulation step time (dt) is small, translational motion inside the nanopore can be modeled with high resolution. We found that a step time of 1 ps (i.e., step size standard deviation of 40 pm by bulk diffusion only) produces adequate spatial resolution given the size of the geometric features in αHL. This was verified by running a series of simulations with increasingly smaller step times and evaluating the convergence of the output. Step times below ∼10 ps gave self-similar results. With dt set to 1 ps and an applied potential of 8 V (maximum used in this study), the sum of the electrophoretic, electroosmotic, and Gaussian-sampled diffusion steps (assuming all three are in the same direction) is ∼0.16 nm (99.7% certainty). This step size can only arise in the centermost region of the pore where field strength is highest. Because this step size is relatively small compared to the αHL geometric features, a time step of 1 ps does not introduce significant error. Meshing and Look-up Table. Equations 1 and 2 are solved at spatial nodes defined by the COMSOL meshing algorithm. The algorithm creates a triangular grid with irregular spacing. The chosen node density accounts for the relative rate of change in the modeled fields. A typical 2D mesh cross section contains ∼10 000 nodes and is rotated symmetrically to create a 3D map consisting of 200 000−300 000 nodes. Each node contains information regarding the electric field, electroosmotic flow, diffusion constant, and mobility. This information is saved to a reference file and used with the simulated particle’s location to determine the proper direction and magnitude of
(3)
e
(5) −6
(4)
Bulk mobility values for small charged molecules range from 1.2 to 12 × 10−4 cm2/V s. We performed simulations using values over the entire range. However, the results presented here use a bulk value of 2.4 × 10−4 cm2/V s, which is typical of doubly and triply charged small molecules. Importantly, output from the SWEB model produces good agreement with experimentally measured ion currents, ionchannel fluid-flow estimates, and the electrostatic potential profile derived from all-atom calculations of αHL (see the Supporting Information). Results agree well over a range of supporting electrolyte concentrations (0.1−1.0 M). Correspondence to existing data validates the electrophoretic and C
DOI: 10.1021/acs.jpcb.5b04955 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B drift. Upon each particle step, the table is searched using a k-d Tree algorithm to find the node that most closely matches the particle location (see the Supporting Information for details). The look-up and translation routine was implemented in a distributed computing environment (∼200 nodes) using HTCondor for scheduling. Most of the data reported here represent trajectory durations of 10−6−10−5 s, which correspond to a maximum of 107 steps per particle. A few trials were run out to 1 ms to verify consistency. Analyses were typically compiled from data sets consisting of 10 000−50 000 simulated trajectories.
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RESULTS AND DISCUSSION Three major factors govern particle capture dynamics for nanopore sensors: electrophoretic migration, electroosmotic (EO) flow, and Brownian motion. Each produces forces that combine to move molecules from the bulk into, or away from, the nanopore, depending on the polarity of the applied electric field and the charge of the particle. Once inside the nanopore, a particle moving in close proximity to specific functional groups might experience additional forces arising from hydrogen bonding or dipole interactions. However, these forces arise over comparatively short distances, are relatively weak, and are spatially localized. Thus, this study focuses on the global longdistance driving factors that impact particle motion over the vast majority of its transit from bulk solution to the point of capture. Electrophoretic forces arising from the applied voltage cause charged species to migrate parallel to the electric field lines. Alpha-hemolysin has an ∼1.4 nm diameter constriction along the central axis of the lumen which causes the field strength to change rapidly over just a few nanometers.46 In some circumstances, this change is sufficiently large to dominate the analyte motion. Bulk fluid transport takes place via electroosmotic flow and produces a net convective force affecting charged and uncharged species alike.47 Under typical applied voltages, the rates of water flow through αHL are on the order of a femtoliter per second. The combination of electrophoretic and EO forces creates a flow velocity profile that changes over the lumen cross section, and dissipates with distance from the pore entrance. Brownian motion disrupts this imposed directionality, enabling some particles to evade capture despite the influence of electrophoretic and EO forces. Figure 2 illustrates the vector components of the electric field and the EO flow for a selection of conditions featured in this study. Because particle capture dynamics from the bulk are a primary motivation of this study, the relative magnitude and direction of vectors proximal to the pore opening are featured in the figure. The intensity of the fields in this region is a function of supporting electrolyte concentration on both the cis and trans sides of the bilayer. We investigated capture dynamics for both 1 M NaCl and 1 M KCl solutions (Figure 2a and c), as well as the commonly used experimental salt gradient of 0.2 M cis/1 M trans NaCl (Figure 2b and c). Applied voltage ranged from −8 to +8 V relative to ground. In all cases, ground is defined on the cis (cap) side of the nanopore. Only ±150 mV potentials are shown in Figure 2. Positive potentials are employed for the detection of negatively charged particles, while negative potentials are utilized for positively charged species. In both scenarios, the applied voltage favors particle translocation via electrophoretic migration. Note also that EO flow reverses direction monotonically upon polarity change but that the electrophoretic field maintains a spatial complexity that
Figure 2. Comparison of the position-dependent drift vectors arising from the SWEB model with Poisson−Nernst−Plank and Navier− Stokes equations solved by the FEM in COMSOL. Electrophoretic vectors (white arrows) and electroosmotic vectors (black arrows) are shown on a logarithmic scale. The length of each vector is equivalent to 2500 simulation time steps (2500 dt = 2.5 ns). A mean diffusion step occurring in the middle of the pore (scaled up to 2500 dt) is depicted as a horizontal gray arrow. At +150 mV, drift velocities and diffusion steps are of comparable magnitudes. Here, electrophoretic and electroosmotic flow velocities sum to give a maximum of 1.2 m/s along the center axis of the pore. Positive potentials (a and b) are used for simulations with negatively charged particles. Negative potentials (c and d) are used for simulations with positively charged particles (see text).
is dependent upon the position of electrostatic charge, the salt concentration, the applied potential, and the overall nanopore contour. We note that the dielectric properties of conventional lipid bilayers cannot withstand large applied potentials. To our knowledge, the largest transmembrane potential reported in conjunction with a miniaturized lipid membrane is ∼800 mV.48 However, hybrid nanopore devices that embed αHL inside a larger hole drilled in a solid state membrane can withstand up to 5 V.49 Alternatively, polymerized membranes could provide tolerances well into the single volt range. Thus, with further material developments, the extreme ends of the voltage range explored in our simulations is within experimental feasibility. Figure 3 shows a 2D cross section through the middle of the nanopore with a probability-density plot arising from numerous particle trajectories as a function of time and applied voltage for a negatively charged particle. Throughout this study, we evaluate particles released at various distances above the D
DOI: 10.1021/acs.jpcb.5b04955 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
Figure 3. Cross section of particle density as a function of elapsed time. Individual particles are released from d = 1 nm, θ = 90 deg. Number of iterations = 104. All panels show the particle cloud for a 1 M/1 M cis/trans NaCl supporting electrolyte concentration, a charge of 2e−, and a 0.2 nm radius. Soft walls in the SWEB model cause the particle density to increase near the block wall. The impact of electrostatic attraction and repulsion from charged amino acid residues can also be detected near the block wall. Figure 4. Capture probability (a) and mean time to capture (b) for negatively charged particles released from various points above the nanopore. (a) θ = 90°, d = 5 nm, particle radius = 0.2 nm, 1 M/1 M cis/trans NaCl; Inset (a): time-integrated capture probability and the cumulative capture maximum as a function of time and applied potential. (b) Triangles (d = 5 nm); circles (d = 1 nm); squares (d = 0 nm). Black: particle radii = 0.2 nm, 0.2 M/1 M cis/trans NaCl. Red: particle radii = 0.4 nm, 1 M/1 M cis/trans NaCl.
nanopore to simulate the sudden production of an analyte by a reversible chemical reaction. The data in Figure 3 represent 104 trials at each voltage with particles released 1 nm above the pore opening. As can be seen, the trajectory plume fills space symmetrically above the pore, particularly at low applied potentials. However, when a particle approaches the nanopore opening, drift arising from electrophoretic migration and EO flow becomes readily apparent. In the nanopore vestibule, the plume is stretched down the lumen axis. In this model, particles are considered to be successfully captured once they have passed below the constriction region (∼5 nm below the opening). For both the 400 mV and 1.0 V scenarios, detection events begin to occur within the first few ns after release. However, at low potentials, only a fraction of the total number of particles released gets detected. Many particles migrate away from the pore, against the drift imposed by electrophoretic and EO forces. The distribution of capture times obtained over a range of applied potentials and particle charge is shown in Figure 4. In Figure 4a, the particle is released along the central lumen axis, 5 nm above the opening. This distance reduces the overall likelihood of capture. An approximate log-normal distribution describes the curves. Larger voltages produce a higher probability of capture. This trend is consistent with closedform capture models derived for a less complex, uncharged cylinder that was published previously.50 Capture probability is determined simply by dividing the number of trajectories arriving at the capture point by the total number of repetitions performed. As anticipated, the distribution shifts toward shorter capture times as voltage increases. Integration of the curve provides a measure of the likelihood that particles will migrate to the capture point as a function of time. As can be seen in the inset to Figure 4a, a potential of 5 V and a time interval of ∼100 ns is required to efficiently capture particles (>70%) when they are released from a relatively distant location (e.g., d = 5 nm).
At this distance, drift due to electrophoretic and EO flow sums to 23 cm/s (230 nm/μs) along the extended center axis of the pore. This velocity is sufficient to overcome the general dispersive effects of diffusion and yield a relatively high capture probability. Typically, particles are tracked for up to 10 μs. After this time period, simulation runs are terminated because the particle diffuses sufficiently far away from the pore and essentially escapes to the bulk. We checked this assertion by running a few simulations out to 1 ms at moderately large potentials (1−5 V). In these cases, the cumulative capture probability curves did not significantly increase after 10 μs, even at the lowest applied potentials. Figure 4b shows the capture time required for negative particles with a positive applied voltage. As expected, we observe a significant decrease in the average capture time as the potential grows large. High potentials produce mean capture times that are >100-fold shorter than low potentials (e.g.,
