Diffusion and Trapping of Single Particles in Pores with Combined

Article pubs.acs.org/JPCC

Diffusion and Trapping of Single Particles in Pores with Combined Pressure and Dynamic Voltage Matthew Schiel†,‡ and Zuzanna S. Siwy*,†,‡,§ †

Department of Physics and Astronomy, ‡Department of Biomedical Engineering, and §Department of Chemistry, University of California, Irvine, California 92697, United States S Supporting Information *

ABSTRACT: In this article we report resistive-pulse experiments with polystyrene particles whose transport through pores is controlled by modulating the driving voltage during the process of translocation. Balancing electric and hydrostatic forces acting on the particles allowed us to observe a random walk of single particles in a pore for tens of seconds and to quantify their diffusion coefficient using two methods. The first approach is based on the mean square displacement and requires passage of multiple particles for a range of diffusion times. The diffusion coefficient of individual particles was determined based on the variance of their local diffusion velocities. The developed methods for measuring the diffusion coefficient in pores are applicable to particles of different sizes, do not require fluorescence labeling, and are entirely based on ion current recordings. In addition, application of a modulating voltage signal together with rising edge triggers enabled transporting the same particle back and forth in the pore without letting the particle leave the pore. field to drive DNA molecules through single nanopores and by reversing the external electric field retrapped the same molecules multiple times, e.g. twelve retrapping events were recorded in as short a time as a quarter of a second.14 Trapping particles in pores and recording transport of particles at various voltages was also used to characterize their diffusion coefficient in a pore.18−25 It is known that particles when in a confinement are subjected to an increased drag force which slows down the translocations and leads to the decrease of their diffusion coefficient.19−22,24−29 A recent publication showed application of bright-field microscopy to perform 3D tracking of colloids in a microfluidic channel, characterize diffusion coefficient of individual particles, and even visualize spatial inhomogeneity of the diffusion coefficient value.24 None of the previously reported approaches however offers a possibility to characterize individual particles by their diffusion coefficient and electrokinetic velocity in the pore. It would also be useful to develop techniques to measure the transport parameters without imaging the particles but rather base the analysis entirely on the easily measurable transmembrane ion current signals. In addition, an ion current based method would be applicable in cases when direct tracking might not be possible e.g. in 3D nontransparent channels and to nanoparticles/molecules passing through nanopores. It would also be useful to develop a simple and inexpensive method to trap

I. INTRODUCTION For a few decades, single pores have been used as a basis for the detection of single molecules, viruses, particles, and cells.1−12 Each object passing through a pore causes a transient change of the system resistance detected as a transient change of the recorded current called the resistive pulse. The Coulter counter technique typically involves recording passage of many particles so that appropriate statistics of the translocation time and the pulse amplitude can be obtained. In case however of an unknown mixture, a different approach might have to be taken. Instead of statistics on an ensemble of particles/molecules, statistics based on one particle studied many times is needed. In order to do that, the same particle has to perform multiple translocations enabling statistics on the pulse duration and amplitude, or/and one particle has to stay in a pore for a prolonged period of time.8,13−15 Recapturing a molecule after its passage might also reveal if any structural changes of the molecule were induced by the interactions with the pore walls.14 Jossang et al. showed an elegant way of retrapping particles after they passed through a pore by the application of a triggered pressure reversal.8,13 White et al. used the pressurereversal method to observe passage of particles through a single glass pipet achieving retrapping probability close to 1.16 It was also proposed that applying a combination of pressure difference and electric field in the experiments of particle translocation could offer a larger control over the particle velocity and recapture probability compared to using electric field or pressure alone.17 Golovchenko et al. used an electric © 2014 American Chemical Society

Received: June 11, 2014 Revised: July 30, 2014 Published: August 1, 2014 19214

dx.doi.org/10.1021/jp505823r | J. Phys. Chem. C 2014, 118, 19214−19223

The Journal of Physical Chemistry C

Article

and connecting the trigger output of the Digidata 1322A to the trigger input of the arbitrary waveform generator. Data analysis was performed using Clampfit 9.2 and custom-written codes in Matlab. Particles detection was performed from a solution of 0.2 M KCl, pH 10 (Tris buffer), containing 0.1% (v/v) Tween 80. In all reported experiments the particles were present at the concentration of 109 particles per mL.

particles in a pore without the necessity of using e.g. optical tweezers.22 In this article we report experimental methods to determine electrokinetic and diffusion characteristics for individual particles in a pore as well as to tune the amount of time the particles stay in the pore. The control over particle transport was achieved by modulating the driving voltage during the translocation when the particle was still in the pore. The experiments were performed with polystyrene particles passing through single 11 μm long pores in a polyethylene terephthalate (PET) film. The pores were prepared by the track-etching technique, which when applied to PET films leads to pores with undulating diameter along the pore axis.30,31 The pore topography is reflected in the pulse shape. Passage of particles through narrower parts of a pore causes a larger change of the transmembrane current compared to the case when the particles pass through wider pore segments. Each particle “follows” the same pore topography, thus all current pulses for a given pore look alike.31−34 Here we show that the ion current pulse substructure can be used as reference points for particle position along the pore axis. Balancing all forces acting on particles allowed us to observe a random walk of individual particles and estimate their diffusion coefficient using two methods. The first approach is based on the mean square displacement and requires analyzing multiple particles.35 The second method analyzes variance of diffusion velocities of a particle and allows one to determine diffusion coefficient for individual particles.36 The presented approaches are applicable to particles of different sizes, do not require fluorescence labeling or tracking, and are entirely based on ion current recordings.

III. RESULTS AND DISCUSSION Figure 1 shows example recordings of 410 nm carboxylated polystyrene particles passing through a polymer pore with an

Figure 1. Example passages of 410 nm polystyrene particles through a single 720 nm in diameter pore together with zoomed in ion current pulses vertically offset to facilitate comparison. The recordings were performed in 0.2 M KCl, pH 10, at −200 mV. The particles moved toward the positively biased electrode. The baseline current was 14.15 nA.

average opening diameter of 720 nm. Each current decrease corresponds to a translocation of one particle. Both particles and the pore walls are negatively charged, thus the particle transport occurs by a combination of electrophoresis and electroosmosis acting in the opposite directions. Since the particles always moved toward the positively biased electrode, the process of electrophoresis dominates the particle translocation in our system. The recorded resistive pulses feature modulations of the ion current signal in time, which as discussed before correspond to the modulation of the pore opening diameter (Figure 1).31,32 A repeatable pattern of ion current pulses was also observed by another group studying single molecules of ethylene glycol translocating an αhemolysin pore.38 Experiments of the particle detection were performed at high ionic strength of 0.2 M KCl to ensure the particle transport occurred already at voltages below 1 V.39 Low electric field in combination with a high aspect ratio of the pores made the translocation times tens of milliseconds long (Figure 1) and facilitated control of the particle transport as well as finding conditions at which diffusion could be studied. The long translocation times in our pores are in contrast to only a few-ms long dwell-times recorded in conically shaped glass pipettes.40 We realized diffusive transport of particles in a pore could be observed via modulations of ion current within a resistive pulse recorded when all forces acting on one particle were balanced. Changes of the recorded current in time would then result from the particle diffusively moving to the left or right along the pore axis and passing through regions with different opening diameters. With no transmembrane potential or pressure, any current change due to a diffusing particle would be however less than the noise of the system. Thus, a finite voltage has to be applied at all times to ensure a finite signal of the current; diffusive conditions can be then achieved by balancing the

II. MATERIALS AND METHODS Single Pores in Polyethylene Terephthalate (PET). Twelve μm thick films of polyethylene terephthalate (PET) were irradiated with single energetic heavy ions (e.g., 11.4 MeV/u Au and U ions) at the UNILAC linear accelerator of the GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany. The foils were then subjected to chemical etching in 0.5 M NaOH, 70 °C. Under these etching conditions, the pores are characterized by an undulating pore diameter along the pore axis and are symmetric i.e. the pore opening diameter at both entrances is similar.31 The mean pore diameter was estimated from the current−voltage measurements performed in 1 M KCl and relating the pore diameter with its conductance. Experiments presented in this manuscript were performed with 720 and 1100 nm in diameter single pores. The pore preparation led to membrane thinning, thus in calculations we consider the pore length to be 11 μm. Walls of pores in PET are known to carry a negative surface charge due to the presence of carboxyl groups.37 Particles. 410 nm in diameter carboxylated and 400 nm uncharged polystyrene particles were used in the experiments (Bangs Laboratories, Fisher, IN). Ion Current Recordings and Analysis. Electrochemical characterization of the pores was performed with amplifier Axopatch 200B and 1322A Digidata (Molecular Devices, Inc.). The square waveform was designed using a custom-written Matlab code to be applied with Agilent 33250A Arbitrary Waveform Generator and connected to one of the command voltage inputs of the Axopatch 200B. The waveforms were triggered from a high-pass filtered signal using a threshold trigger in the Clampex 9.2 software (Molecular Devices, Inc.) 19215



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Figure 2. Process of adjusting voltage across the membrane to counterbalance the hydrostatic pressure difference, Δp. (a) Electrolyte level in the chamber of the conductivity cell containing the particles was 0.3 cm higher compared to the electrolyte level in the other chamber filled with KCl. (b) Applied square voltage wave with tunable offset ΔV and duration is applied to counterbalance Δp. The recordings were performed in 0.2 M KCl, pH 10, using 410 nm particles passing through 720 nm in diameter pore. Red arrows mark the beginning of the current pulse thus particle entrance to the pore; the red star indicates the particle exit. When changing ΔV with ∼1 mV steps, conditions were found when ΔV (60 mV) balanced Δp (the bottom panel), and the particle exited the pore only when the driving voltage (−40 mV) was reapplied. For higher values of ΔV, the particle exited the pore within a few seconds. The insets indicate the directions of the particle motion. When the forces are balanced, the particles can move in both directions.

applied electric field with a pressure difference set by unequal electrolyte levels on both sides of the membrane. The experiments require driving single particles into the pore and balancing all forces before the translocation is completed. Fine-tuning of voltage can be performed easily, thus we started by setting a pressure gradient across the pore (Figure 2a). The level of electrolyte with the particles was 3 mm higher than the level of only KCl in the second chamber of the conductivity cell. Resistive-pulse experiments were then performed driving the particles through the pore with voltage of −40 mV. Ion current decrease caused by a single particle entering a pore was used as a signal to trigger a square wave voltage applied by a function generator with a set delay and constant amplitude of 100 mV. The wave offset originally zero was moved manually with ∼1 mV steps to reach a finite value of voltage, 60 mV in the example shown in Figure 2b, which counteracted the hydrostatic pressure. With the adjusted offset, the majority of particles did not leave the pore even after 40 s (Figure 3) and were driven out only by reapplying the driving voltage (−40 mV). This was expected based on the bulk diffusion coefficient of the 410 nm particles calculated from the Einstein−Smoluchowski equation, Dbulk = 1.1 × 10−12 m2/s, and the known reduction of particle diffusion coefficient in a pore.19−22,25,29 Assuming one-dimensional diffusion, the first passage time relation of L2/2Dbulk estimates the particles would need ∼60 s to diffuse through the whole length of the pore, L = 11 μm.35 In our experiments, the particles started to diffuse from a position inside the pore so that the distance from either end was smaller than L. The expected reduction of the particle diffusion coefficient in the pore would however explain why for the majority of particles even the duration of 40 s was insufficient to exit the pore by diffusion.

In order to provide evidence the forces on the particles were indeed sufficiently balanced; we estimated the magnitude of the particle velocity induced by the imposed hydrostatic pressure as well as the electrokinetic velocity due to applied voltage. A 3 mm difference in the electrolyte levels on both sides of the membrane corresponds to ΔP ∼ 30 Pa, and the average pressure induced velocity can be calculated from the Poiseuille relation41 vpressure =

ΔPR2 8ηL

(1)

where R is the pore radius, L is the pore length (11 μm), and η is the solution viscosity. The calculated vpressure for ΔP = 30 Pa, and R = 360 nm, is 45 μm/s. Due to the finite size of the particles as well as the high ratio of particle and pore diameters in the studied systems, the magnitude of the particle velocity is expected to be close to vpressure and to exhibit small radial variations. When voltage is applied across a pore with charged pore walls, the particles’ transport occurs by a combination of electroosmosis and electrophoresis. In case of a thin electrical double-layer, the net translocation velocity can be found as ε velectrokinetic = (ζparticle − ζpore)E η (2) where ζparticle and ζpore are zeta potentials of the particle and pore walls, respectively, ε = ε0εr, and E stands for electric field calculated as the applied voltage divided by the pore length.42 Due to a large aspect ratio of the pores used in the experiments, the access resistance constitutes only ∼5% of the geometrical resistance of the pore and was not taken into account in the calculations.43 Zeta potential of 410 nm particles in the 19216



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Figure 3. Comparison of diffusive and electrophoretic transport of single 410 nm particles in a 720 nm in diameter pore. (a) The experiment was performed in a configuration shown in Figure 2a with the difference in electrolyte levels in the conductivity cell chambers of 0.5 cm, using the following sequence of applied voltages. With a set time delay, Δtstart, after the particle entered the pore, the voltage waveform was applied setting the voltage across the membrane to a value that balanced the hydrostatic pressure difference (85 mV in this experiment). The particle was allowed to diffuse for a controlled period of time up to 40 s. The original voltage was subsequently turned on for 70 ms, and the particle exited the pore. The voltage was set to a small reverse value (125 mV) for 800 ms driving the same particle back through the pore in the opposite direction. The same particle was again driven to the right using −75 mV. Due to changes of the voltage value and polarity, the recorded range of ion current is wide thus to facilitate visual analysis of the recordings, there are three nonconsecutive ion current scales, separated by horizontal lines. (b) In order to estimate the diffusion displacement of each particle in the pore, three durations of time were monitored: (1) Δtstart, (2) Δtend − time needed to complete the translocation after the particle underwent diffusion in the pore; an example of recorded data for one 10 s long diffusion pause is shown. (3) Δttotal − time needed for the particle to electrophoretically pass through the pore at the same voltage at which Δtstart and Δtend were found. All three portions of the traces are plotted in terms of δI/I with the baseline due to the capacitive discharge corrected. (c) Example recordings of ion current with the diffusive pauses of 1 s, 10 s, and 40 s for the same consecutive voltage values as in (a). For each diffusive pause, recordings with three different particles are shown. All scale bars shown on the far right correspond to 250 pA with colors matching the three distinct recordings for three particles. Note that as shown in (a) there are three different current regimes shown.

presence of surfactants was measured before and is equal to ζparticle ∼ −25 mV.31 The value of ζpore ∼ −10 mV was estimated based on reported measurements of zeta potential of porous polymer membranes44−46 and known reduction of ζ by Tween 80.31 With zeta potential values as mentioned above and 60 mV applied voltage, velectrokinetic ∼ 40 μm/s, thus almost identical to vpressure ∼ 45 μm/s. Since velectrokinetic and vpressure have opposite directions, these calculations confirmed the hydrostatic and electric forces acting on particles in the pore were sufficiently balanced. Due to the undulating opening diameter of the pores, local variations of fluid velocity and electric field have to be considered as well. Using the approach described in ref 9, we calculated the variations of the pore opening diameter along the axis using the profile of resistive pulses recorded in the 720 nm pore. Based on calculations performed with 20 different pulses,

the standard deviation of the opening diameter does not exceed ∼30 nm. Thus, the ratio of openings one standard deviation above and below the mean diameter is 1.09. Since velocities in both hydrodynamic and electrokinetic modes scale with diameter squared, the velocities at different axial positions are not expected to change more than 20%. This simple calculation suggested that diffusion conditions can indeed be achieved in our system. The analysis however also indicates pores with significantly larger undulations of the opening diameter might not allow for sufficient balancing of forces, and it might not be possible to observe particle diffusion in these structures. All experiments reported here were performed with pressure gradients corresponding to 3−5 mm and voltages not exceeding 100 mV. The diffusion coefficient of particles in a pore was first found from the mean square displacement measured for diffusion 19217



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times between 10 ms and 10 s.35 We set the time each particle was allowed to diffuse, and in the first sets of experiments with the 720 nm pore, 15 particles were examined for each diffusion pause; 7 additional particles were allowed to diffuse for 40 s. Diffusion displacement of each particle could be determined from the electrokinetic displacements before and after the diffusion pause and the total length of the pore. We therefore measured the time during which a particle passed through the pore before the voltage was adjusted to reach diffusion conditions (Δtstart was chosen to be ∼1/5 of the total translocation time found before starting the diffusion experiments) and the time needed to complete the translocation after the voltage was switched back on (Δtend) (Figure 3). During Δtstart and Δtend the particles were therefore moving electrokinetically; both durations were determined after removing the capacitance portion of the recording created when the voltage was switched by the applied square wave (Supporting Information).47 In order to find the electrokinetic velocity of each particle in the pore, we assumed the velocity was position independent and measured the time, Δttotal, the particle would need to translocate if the voltage was on all the time (−75 mV, the base of the square signal, Figure 3). After completing diffusion and driving a particle out of the pore, the same particle was driven back to the starting point using a reversed voltage and subsequently pulled once again at −75 mV (Figure 3a), producing a pulse of duration Δttotal. The diffusion displacement of the particle in the pore could then be calculated as the electrokinetic velocity multiplied by the difference between Δttotal and (Δtstart+ Δtend). The electrokinetic velocity at −75 mV equal to 150 μm/s was ∼800 times higher than the diffusion speed calculated from the bulk diffusion coefficient (60 s are needed to diffuse a distance of 11 μm in the bulk), confirming the dominant role of electric field in these conditions. The time Δttotal and consequently the electrokinetic velocity were characterized by small variations among different particles (Δttotal = 74 ± 4 ms at −75 mV), thus subsequent experiments were performed without measuring the parameter for each individual particle. This allowed us to speed up the analysis and to examine diffusion of a larger population of particles and a wider set of diffusion times needed for viable measurements of diffusion coefficients. The Supporting Information describes procedures that were used to match a fully electrokinetic translocation (from a library of 200 electrokinetic events) to each event containing a diffusion recording. Figure 4 shows mean square displacement (MSD) vs time for 410 nm particles diffusing in a different single-pore membrane; this pore had an opening of 1100 nm in diameter. Axial undulations of the pore diameter were of similar magnitude as in the previously analyzed 720 nm pore, thus variations of particle velocity along the axis were not expected to exceed 20%. Each MSD was calculated based on at least 40 different, individually diffusing particles and corrected for their mean displacement. A total of 348 of 410 nm particles was used to create the plot in Figure 4 (black squares). The slope of MSD vs time gave the diffusion coefficient of the particles equal to (1.8 ± 0.2) 10−13 m2/s thus ∼5 lower than the value of the bulk diffusion coefficient. Note the offset in the graph stems from the localization error, i.e. the finite precision in determination of the particle position, as reported in the literature before.48 In order to confirm the forces on the charged particles were sufficiently balanced, measurements of MSD in the same 1100

Figure 4. Mean square displacements of 410 nm (black squares) and 400 nm (green triangles) in diameter polystyrene particles in a 1100 nm in diameter pore as a function of diffusion time set as 10 ms, 2 s, 4 s, 6 s, 8 s, and 10 s. Black and green lines represent the linear fit to the shown points for 410 and 400 nm particles, respectively. The 410 nm in diameter particles were modified with carboxyl groups and passed through the pore in the direction of electrophoresis. The 400 nm polystyrene particles were uncharged, and their translocation occurred by electroosmosis.

nm pore were also performed for uncharged 400 nm in diameter polystyrene particles. The particles passed through the pore by the process of electroosmosis. We expected the diffusion coefficients of the 410 carboxylated and 400 nm unmodified polystyrene particles to be very similar. The data in Figure 4 (green triangles) confirmed this is indeed the case, validating the method, and suggesting the possible residual electric force on the particles was minimal and did not interfere with determination of diffusion coefficient. The plot of MSD for the 400 nm particles was created based on the diffusion of 576 individually studied particles. Calculating diffusion coefficients of particles from mean square displacement is time-consuming and requires analysis of many particles, thus this approach could not be applied to unknown and potentially heterogeneous samples. We realized the pulses containing diffusion pauses gave direct information on a random walk of single particles and could be used to obtain the diffusion coefficient for individual particles, based on the variance of their local velocities, ⟨(ΔV)2⟩36 2D ⟨(ΔV )2 ⟩ = ⟨V 2⟩ − ⟨V ⟩2 = (3) Δt yielding ⟨(ΔV )2 ⟩Δt (4) 2 where Δt is the time step between data points. Below we describe how recordings in the diffusion and electrokinetic modes can be used to determine the variance ⟨(ΔV)2⟩. First we realized, the local velocity, Δx/Δt, of a diffusing particle along the pore axis, x, is related to the changes of the pore opening radius, r, in time, Δr/Δt, as traced by the diffusing particle, and the intrinsic changes of the pore topography, Δr/Δx, so that D=

Δr Δr Δx = Δt Δx Δt

(5)

where r is the pore opening radius at a given x. The profile of how the pore opening radius changes along the pore axis can be found from the relative ion current change, 19218



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Figure 5. Procedure to find distribution of local approximate derivatives (Δ(δIelectr/I))/Δx and (Δ(δIdif f)/I))/Δt for a single diffusing particle (eq 7). (a) Signal of ion current in time, expressed as δIelectr/I, when a particle passed through a pore electrophoretically, converted into δIelectr/I vs position along the pore axis, x. The fraction of the signal in green corresponds to the portion of the pore between the start and end distances, over which the particle diffusion was centered. (b) A diffusion trace of a particle in the pore in time, expressed as δI/Idiff; it is a 1 s portion of a 10 s diffusion recording. (c) A portion of the baseline i.e. ion current trace in time without a particle, also expressed as a relative current change. Colored horizontal bars in panels (a,b,c) represent the bins of δI/I values used to find histograms of differentials at each δI/I. One example of the histograms set for δI/I marked with a dotted line in (a) and (b) is shown in panels (d,e). (f) An example histogram found from the signal without a particle, shown in (c), which was used as a measure of the noise.

δI/I, found as (Ie−Ip)/Ip, where Ie and Ip stand for ion current in the absence and presence of the particle, respectively.9,31 Ion current traces I(t) in both electrophoretic and diffusion signals were first expressed as δI/Ielectr and δI/Idif f. The time trace of δI/Ielectr was subsequently changed into δI/Ielectr vs x, assuming position independent electrophoretic velocity, and that the pulse duration is equivalent to the pore length of 11 μm. As mentioned above, δI/Ielectr vs x gives the dependence of the pore opening radius, r, on x. Each value of δI/Ielectr corresponds to one pore radius, keeping in mind that different positions x could have the same radii and thus the same δI/I. Since δI/I is voltage independent, the same value of δI/I in both the diffusion and electrophoretic signals will correspond to the same pore radius and the same subset of positions along the pore axis. Thus, relectr(x) = rdiff(x), and consequently δIdiff δIelectr (x ) = (x(t )) I I

δIdiff

δIdiff

I

I

Δt

Δx

( ) = Δ( ) Δx

Δ

δIdiff

Δt

( ) = Δ( ) Δx

Δ

I

Δt

(7a)

δIelectr I

Δx

Δt

(7b)

Since δIdiff

( ) = Δ( )

Δ

I

Δx

δIelectr I

Δx

(7c)

In order to calculate the differentials, the diffusion and electrokinetic signals were smoothed using a moving average function over 9 nearest points and downsampled by a factor of 10. The diffusion coefficient of an individual particle was calculated using its diffusion recording as well as the electrokinetic translocation that was matched before for the MSD approach (Supporting Information). We will first describe how the distribution of (Δ(δIdiff/I))/Δt was found for one 410 nm in diameter particle diffusing for 10 s in an 1100 nm in diameter pore (Figure 5). The trace of δI/Idiff was first corrected for the capacitance portion present after the voltage was switched to balance the hydrostatic pressure. Due to the intrinsic noise of the current recordings, we binned δI/ Idif f into ranges of 8 × 10−5 (pA/pA); for each current range, the differential (Δ(δIdif f/I))/Δt was calculated based on the two nearest points, and Δt = 1 ms (Figure 5b). A frequency histogram of (Δ(δIdiff/I))/Δt vs δI/Idiff was subsequently found (Figure 5e). Since the particle was diffusing back and forth, the number of observations for each bin was between a few tens to

(6)

It is important to note that for the diffusion recording, the position, x, is time dependent, since a diffusing particle can visit the same location, x, multiple times. In the electrokinetic trace, each position is visited only once and as mentioned above (δIelectr/I)(x) was found based on (δIelectr/I)(t). Equation 6 is valid independently of the mathematical form of the relation between the pulse amplitude and the pore/particle diameter. We assume a local pore radius and δI/I are conservative functions of x, thus they do not depend on the path the particles took to reach the position, x. Finding the derivative of δIdiff/I with respect to time, and approximating the derivatives using differentials, one gets the following formulas relating ion current change with local velocity of a diffusing particle, Δx/Δt 19219



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(Figure 6). The obtained values were ∼6 times lower compared to the diffusion coefficient in the 1100 nm pore and ∼30 smaller than Dbulk. The observed lowering of the diffusion coefficient for 410 nm particles in the two pores with average opening diameters of 720 and 1100 nm is in semiquantitative agreement with predictions published before, see eq 24 in ref 49 and Table 1 in ref 26. According to the presented theory, in systems in which the ratio of particle to pore diameters is equal to 0.4 (410 nm/ 1100 nm) and 0.6 (410 nm/720 nm), the diffusion coefficient is expected to diminish by a factor of 8 and 60, respectively. Long translocation times and application of a tunable voltage waveform facilitated achieving additional control over the particle translocation. Using a rising edge trigger with delay we were able to automatically drive a single particle back to its starting point outside the pore, recapture it, and repeatedly apply our designed waveform on the same particle (Figure 7a). Similarly, a falling edge trigger allowed us to recapture the particle at the opposite entrance of the pore and move it back to the original position (Figure 7b) where the full event could be retriggered repeatedly and automatically. The final approach allowed us to move the particle back and forth without reaching any of the pore entrances. This involved a manual design of the voltage signal applied during the entire particle translocation in both directions (Figure 7c). The maximum and minimum value of the designed voltage wave was ∼75 mV above and ∼75 mV below the diffusive point found using the method previously shown in Figure 2.

hundreds. For further calculations only bins containing at least 50 samples were considered. The same analysis was applied to an 8 s long portion of a trace without a particle to construct the histogram of (Δ(δInoise/I))/Δt due to the noise induced ion current fluctuations (Figure 5 c,f). Equation 3 uses variance of local velocities of a diffusing particle, thus the variance of (Δ(δIdif f/I))/Δt, denoted as σdif f 2, was found for each δI/I (Figure S1). We also found an average variance of (Δ(δInoise/I))/Δt, ⟨σnoise2⟩ (Figure S1). Since noise is additive and the distributions were considered Gaussian, variance of the particle motion is equal to (σdiff 2 − ⟨σnoise2⟩). In order to find the histogram of (Δ(δIelectr/I))/Δx at each δI/I (eq 7b) we used the same bins as those chosen for analyzing the diffusion trace (Figure 5a,d). We then averaged the squared values of (Δ(δIelectr/I))/Δx for each δI/I between the x position where the particle started diffusing and the position where the diffusion was ended (i.e., when the voltage was switched back). This allowed us to calculate the scale factor, βelectr2 (Figure S1), to be applied to σdiff 2. The corrected variance σdiff 2 − ⟨σnoise2⟩ was then divided by βelectr2 and inserted into eq 4, yielding the diffusion coefficient based on each δI/I. Diffusion coefficients from all considered δI/I (Figure S1) were arithmetically averaged in order to obtain one diffusion coefficient for each resistive pulse (Figure 6).

IV. CONCLUSIONS The manuscript presents a method to tune transport of single particles through pores by modulating transmembrane potential during the translocation. Balancing forces acting on the particles allowed us to observe their random walk and to determine their diffusion coefficient along with electrokinetic velocity. Diffusion coefficient of single particles was measured as well using the variance of local diffusion velocities. The particles were trapped in single pores for a controlled period of time up to 40 s. The presented approaches are based entirely on ion current recordings and do not require fluorescence labeling or imaging of the particles. The developed experimental and modeling tools can be applied to any pore system directly and without any modification if the translocation time of a studied molecule/ particle is significantly longer than the time constant of the capacitive discharge of the membrane. Thus, it could be used for shorter pores as well if the passage of particles was slowed down. We have also shown the signal due to the capacitive discharge can be removed to allow the underlying modulations of ion current due to the particle motion to be analyzed. While in this study we did this in post-analysis, it should be possible to subtract the capacitive signal in real time and analyze particles with short translocation times. Our future analysis of particle diffusion in a pore will focus on the determination of the potential of mean-force of single pores 50 and position dependent translocation velocity. Including the pore geometry in the analysis will be important to understand the process of particle diffusion in the complex landscape of nonuniform electric potential and pressure differences along the axis of a pore with varying cross-section. This analysis is expected to result in more accurate determination of diffusion coefficient of particles. It will also be important to determine whether large undulations of pore

Figure 6. Diffusion coefficients of individual particles calculated based on the procedure described in Figure 5. The black dotted line indicates the diffusion coefficient of 410 and 400 nm particles in the 1100 nm pore found from the MSD approach (Figure 4). 410 nm particles were carboxylated; 400 nm particles were uncharged. As a comparison, the diffusion coefficient of 410 nm particles in a 720 nm pore is shown as well.

Obtained values of the diffusion coefficient of 10 individual 410 and 400 nm particles are in a good quantitative agreement with the values found based on the MSD approach and shown in Figure 6. Diffusion coefficient values of individual particles were very similar to each other even though the particles performed a random walk at different parts of the pore, characterized with different topography. This observation suggests the magnitude of the diffusion coefficient is not significantly affected by the details of the diameter undulations but rather by the average pore diameter. In addition, another pore with an average opening diameter of 1150 nm was prepared, in which the diffusion coefficient of the 410 nm carboxylated particles was found to be 1.9 ± 0.2 ·10−13 m2/s, thus very similar to the value obtained with the previous 1100 nm structure. The diffusion coefficient of individual 410 nm particles in the 720 nm in diameter pore (Figure 3) was calculated as well 19220



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Figure 7. Modulation of voltage signal during particle translocation allows one to drive the same particle through a pore multiple times. Rising edge trigger with delay combined with appropriately designed voltage wave recapture the particle after the translocation is completed (a) or move the particle back and forth along the pore axis without letting it leave the pore (c). Falling edge trigger enables automatic recapture at the opposite entrance of the pore (b). The experiments were performed with 410 nm in diameter carboxylated particles and 720 nm in diameter pore. Horizontal lines in (a)−(c) indicate nonconsecutive scales of ion current.

opening diameter influence balancing forces on the particles and achieving diffusion conditions.

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ACKNOWLEDGMENTS

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REFERENCES

Irradiation with swift heavy ions was performed at the GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany. This research was supported by the National Science Foundation (CHE 1306058). The authors are grateful for discussions with Profs. Michael Dennin, Peter Taborek, Thorsten Ritz, and Clare Yu.

ASSOCIATED CONTENT

S Supporting Information *

Description of the procedure to remove the capacitance portion of the ion current signal, together with matching an electrokinetic and diffusion pulses are presented. This material is available free of charge via the Internet at http://pubs.acs.org.

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Diffusion and Trapping of Single Particles in Pores with Combined

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