Propagation and Separation of Charged Colloids by Cylindrical

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Propagation and Separation of Charged Colloids by Cylindrical Passivated Gel Electrophoresis Dimitri Bikos† and Thomas G. Mason*,†,‡ †

Department of Chemistry and Biochemistry, University of CaliforniaLos Angeles, Los Angeles, California 90095, United States Department of Physics and Astronomy, University of CaliforniaLos Angeles, Los Angeles, California 90095, United States

‡

ABSTRACT: We explore the electrophoretic propagation of charged colloidal objects, monodisperse anionically stabilized polystyrene spheres, in large-pore agarose gels that have been passivated using polyethylene glycol (PEG) when a radial electric field is applied in a cylindrical geometry. By contrast to standard Cartesian gel-electrophoresis geometries, in a cylindrical geometry, charged particles that start at a ring well near the central axis propagate outward more rapidly initially and then slow down as they move further away from the axis. By building a full-ring cylindrical gel electrophoresis chamber and taking movies of scattered light from propagating nanospheres undergoing electrophoresis, we experimentally demonstrate that the ring-like front of monodisperse nanospheres propagates stably in PEG-passivated agarose gels and that the measured ring radius as a function of time agrees with a simple model that incorporates the electric field of a cylindrical geometry. Moreover, we show that this cylindrical geometry offers a potential advantage when performing electrophoretic separations of objects that have widely different sizes: smaller objects can still be retained in a cylindrical gel that has a limited size over long electrophoretic run times required for separating larger objects.

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INTRODUCTION Gel electrophoresis (GE) is an extremely useful technique for characterizing and separating biological macromolecules.1 Many different forms of electrophoresis exist; these include various degrees of dimensionality2,3 as well as applying timevarying electric fields.4,5 Increasingly, GE is being used for the same purposes in applications involving larger-scale colloidal objects, such as viruses,6−8 nanoparticles,9−18 and nanospheres.19,20 Gels that can be commonly used for biomolecular separations do not typically function well for large-scale colloidal objects. In most cases, to perform electrophoresis on larger nanoscale objects having dimensions greater than ≈ 10 nm, the characteristic pore size of the gel must be expanded by reducing the gel’s concentration so that the objects can propagate through the pores of the gel, and often a passivation agent must be introduced to inhibit attractive binding of the nanoscale objects to the gel.20 In particular, large-pore agarose gels that are passivated by smaller nonionic polyethylene glycol (PEG) having a molecular weight of about 103 appear to be well-suited for performing gel electrophoresis on anionic sulfate-stabilized polystyrene (SSPS) nanospheres having radii up to about 160 nm.20 The steadystate velocities of SSPS nanospheres have been measured to be largely independent of the concentration of the PEG passivation agent, which is not true for anionic passivation agents, such as sodium dodecyl sulfate. Moreover, the propagation velocity vp as a function of sphere radius a, when SSPS nanospheres are subjected to a constant voltage in a © XXXX American Chemical Society

standard 1D horizontal gel apparatus, has been shown to decrease linearly as vp ∼ (a* − a) for a < a* and is zero for a ≥ a*, indicating that the effective viscosity of the passivated gel medium becomes infinite for spheres having a sufficiently large size.20 For a certain concentration of agarose, buffer, and PEG, a* has been measured to be about 180 nm,20 so that very large nanoscale objects can still be effectively separated using passivated GE. The size-dependence of the effective viscosity of the nanoscale objects in the passivated gels is the essential aspect that facilitates separations of differently sized objects into different spatial regions of the gel after applying a constant voltage over a certain duration. While significant attention has been given to examining electrophoresis in higher spatial dimensionalities, including 2D3,21,22 and 3D,2,23 these have primarily been performed in Cartesian geometries, in which the fields are applied and molecules propagate along orthogonal axes. Significantly less attention has been given to electrophoresis in non-Cartesian geometries, such as cylindrical and spherical geometries. The nature of the spatial variations of the electric fields in these geometries, compared to a simple constant electric field along a typical direction in a Cartesian geometry, could potentially be advantageous when performing electrophoresis for certain Special Issue: William M. Gelbart Festschrift Received: February 29, 2016 Revised: April 21, 2016

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DOI: 10.1021/acs.jpcb.6b02165 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B kinds of separations. Initial experiments have explored molecular gel electrophoresis in an axial-cylindrical geometry,24 showing that molecular separations are also possible in this geometry, yet these measurements did not provide a detailed quantitative description of the propagation distances or velocities of the molecules as a function of time. Moreover, these molecular GE experiments employed a series of small wells, rather than a single ring-like well; such a ring-like well would more efficiently use all of the gel material and could also provide insight into the stability or instability of a ring-like propagating front of electrophoretic objects in a radially diverging electric field. Here, we use simple real-time optical tracking of light scattered from anionic nanospheres in PEG-passivated agarose gels to study quantitatively how a cylindrical electrophoresis geometry affects the propagation and separation of these nanospheres over time. Taking advantage of fused-deposition additive manufacturing, we have custom-built a cylindrical electrophoresis apparatus that contains a smaller ring-like inner anode and a larger ring-like outer cathode. By digitally analyzing movies of the propagating ring-like fronts (i.e. bands) of monodisperse anionic nanospheres, we show that these propagating nanospheres, which have been initially loaded into a ring well closer to a central-ring anode, progressively slow down as they move further away from the anode toward the cathode at larger radii. Using a simple model for the electric field strength, which in a cylindrical geometry is inversely proportional to the radius, we calculate expressions for the propagation distance and velocity of the nanospheres, and we show that our measurements are well described by this simple model. In addition, we also demonstrate an effective separation of anionic nanospheres and measure their timedependent propagation, which we display as a space−time plot. Branching of lines in this plot indicates that different nanosphere mobilities can effectively be resolved from others. This reveals that the cylindrical geometry may be particularly useful compared to the standard 1D Cartesian geometry when separating objects that have a wide dynamic range in electrophoretic mobilities and sizes.

Figure 1. Cylindrical gel electrophoresis apparatus. (a) Photograph; scale bar = 5 cm. (1) DC power supply: positive red cable at +Vapp is attached to the outer Pt ring-cathode, and negative black cable at 0 V is attached to the inner Pt ring-anode; (2) white light box providing side illumination; (3) outer plexiglas cylinder used as a mold for casting agarose gel-slabs; (4) inner plexiglas cylinder used as a gel-casting mold; (5) outer plexiglas reservoir cylinder; (6) inner anode plexiglas cylinder; (7) removable circular well-insert; (8) lower black plexiglas substrate. A digital camera (not shown) is mounted above the central axis and captures time-lapse movies. (b) Schematic top view showing the inner anode radius r1, the outer cathode radius r2, the well radius rw, and the radius r in the gel medium, all measured from the central cylindrical axis (not to scale). (c) Schematic cutaway side view showing positions of the following: lower plexiglas substrate, ringanode (black dots), ring-cathode (red dots), annular gel-slab (blue cross-hatch), and central and outer plexiglas cylinders. Electrodes are fitted to the grooves of 3D-printed mounting rings. A buffer solution (not shown) is filled to a level that is at least 3× the height of the gel slab.

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MATERIALS AND METHODS The cylindrical electrophoresis apparatus, shown in Figure 1a, is constructed out of the following components: 4-in. long concentric sections of extruded acrylic cylinders that have been solvent welded to a laser cut acrylic disk base (substrate: 13.97 cm diameter × 0.3175 cm thick), 3D-printed polylactic acid (PLA) grooved electrode mounts, and 30 gauge, 99.9% platinum wire. The outermost cylinder [Figure 1a(5)] (12.07 cm outer diameter (OD), 11.43 cm inner diameter (ID)) serves as the outer wall of the buffer reservoir against which the cathode electrode mount is attached. A central acrylic cylinder (0.635 cm OD, 0.318 cm ID) serves to hold the 3D-printed anode sleeve. The base disk [Figure 1a(8)] is chosen to be black acrylic plastic so as to provide a background against which the colloidal suspension samples, which scatter light and appear white under the incoherent diffusive illumination of a white light box [Figure 1a(2)], can be better visualized during the course of their propagation. Acrylic solvent welding using SciGrip Weld-On 3 (a mixture of methylene chloride, trichloroethylene, and methyl methacrylate monomer) provides a strong, water-tight bond between cylinders and base, preventing any leakage of up to 800 mL of buffer solution. Rough contact surfaces may not properly weld

if such a weakly viscous solvent incompletely fills larger gaps. To overcome this, we cut the acrylic cylinders carefully and sand all contact surfaces to be flat and smooth. Components are held gently against one another before solvent is introduced dropwise via the blunted needle of a syringe. This solvent rapidly advances into the gap joint via capillary action, dissolves the contact surfaces, and rapidly evaporates leaving a weld between the parts. Before use, we wait 48 h to achieve bonds that have full strength and prolonged water-tightness. Disk-like agarose slabs are formed in situ between an additional pair of cylinders that are not solvent welded to the base disk and function as a removable mold. The ends of an outer acrylic cylinder (10.16 cm OD, 9.53 cm ID) [Figure 1a(3)] and an inner acrylic cylinder (3.81 cm OD, 3.18 cm ID, or 1.27 cm OD, 0.953 cm ID) [Figure 1a(4)] are fitted with customized silicone gaskets, concentrically aligned with the gaskets against the base disk, and sealed/held in place with a 1.5 B

DOI: 10.1021/acs.jpcb.6b02165 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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at regular intervals until spheres exit the outer frontier of the gel slab. The trajectories of the advancing fronts of scattered light from propagating nanospheres in the PEG-passivated agarose gel are then analyzed using image processing software (ImageJ and Photoshop). An image math operation is employed to subtract a given frame in the movie from the background (i.e. an image taken prior to loading); this result is adjusted using levels to enhance contrast and then inverted so that white bands of scattered light appear as dark bands, to better reveal details in the figures.

kg disk-shaped weight. Gaskets are made using a Permatex Ultra Blue Multipurpose RTV Silicone Gasket Maker which is liberally applied to the roughened ends of the mold cylinders before being stood upright and gently pressed against a flat sheet of Teflon to cure for 12−24 h. After drying, dried silicone sticking out both inside and outside of the cylinders is trimmed with a hobby knife to produce a flat, water tight gasket that serves to temporarily seal the liquid agarose within the mold during gel-slab casting. After the agarose sets, the weight is lifted and the cylinders are gently removed to leave behind a disk-like agarose slab [Figure 1c]. Three components of the cylindrical electrophoresis apparatus are fabricated using a consumer-grade FlashForge Creator Pro fused deposition 3D printer. These include the anode mount [Figure 1c] which slides like a sleeve onto the central cylinder [Figure 1a(6)], the cathode mount [Figure 1c] which fits tightly against the inner wall [Figure 1a(5)], and the well mold [Figure 1a(6)] (larger 50 mm diameter version shown) which slides like a sleeve onto the inner cylinder mold [Figure 1a(4)] and is present during in situ agarose casting. Anode and cathode mounts are carefully designed and printed at 100 μm layer resolution to feature a groove 1.00 cm in height and 1.00 mm deep that snugly accommodates the platinum wire electrode rings [Figure 1c]. These grooves secure the electrodes in place and provide excellent alignment in the horizontal plane of the apparatus. Electrodes are constructed using 30 gauge (0.25 mm diameter) platinum wire (purity 99.9%). The well-mold sleeve fits snuggly around the inner cylinder mold and is held in place by friction, making its height adjustable to handle different well depths, to accommodate thinner or thicker gel slabs, and to align the bottom of the well horizontally in the plane with the inner and outer electrode rings. The well-mold is vented to prevent air pockets from interfering with the complete flow of agarose around the well mold. Inner and outer Pt electrodes are in full contact with buffer solution. Gels are cast from a hot aqueous solution containing SigmaAldrich Type I-A, low EEO agarose, PEG-1000 passivation agent, and sodium borate buffer (SBB), according to a microwave method as previously described.20 Surfactant-free polystyrene nanospheres are obtained from Life Technologies and suspended in D2O to volume fractions as dilute as possible (≈ 0.14 w/v %) while still remaining visible for purposes of imaging using scattered light. After the gel has set at ambient temperature (≈ 45 min after pouring), 800 mL of 5.0 mM sodium borate buffer at pH = 9.0 is carefully poured into the reservoir so as not to fracture the gel. Wide, rectangular strips of porous filter paper are cut to the height of the apparatus walls and introduced around the anode post and immediately within the cathode cylinder, thereby preventing bubbles from moving into the viewing region. Before loading the nanospheres, an image of the viewing area is taken to enable background subtraction in later image analysis. D2O-diluted aqueous dispersions of nanospheres are then introduced with a narrow-tip pipettor for ≈ 10 successive and equal infusions equally spaced around the circular ring well. A Whatman Biometra Model 125 Power Supply [Figure 1a(1)] operating at a 50 V setting is turned on, and light scattered from electrophoretically propagating particles is recorded with a Flea Camera (650 × 650 pixel acquisition area, monochrome, optical axis of lens nearly coaxial with axis of the cylindrical gel electrophoresis apparatus) using time lapse (1 frame every 15 s) with side illumination provided by a light box [Figure 1a(2)]

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THEORY In a cylindrical geometry, the potential V depends logarithmically on the radius r.25 For a cylindrical electrophoresis configuration having an applied voltage Vapp at the outer ring electrode, where the positive terminal is connected, and zero voltage at the inner ring electrode, the radial dependence of the potential is described by V (r ) = Vapp[ln(r /r1)/ln(r2/r1)] = V *ln(r /r1)

(1)

where V* = V app/ln(r2/r1) is a rescaled voltage that incorporates a geometrical aspect ratio. Since E(r) = −dV(r)/ dr in a cylindrical geometry, the radial electric field E in a dielectric material having a relative permittivity εr between the anode and cathode varies inversely with the radius: E(r) = −V*/(εrr). Consequently, the radial electric force Fe acting on a nanoscale object having a constant total charge q is then Fe(r) = qE(r) = −qV*/(εrr), and it will cause negatively charged nanoscale objects to move outward in the positive radial direction. Neglecting transient inertial effects when the voltage is turned on, for nanospheres having radius a steadily propagating in a porous gel medium, when the field is applied, this electric force is balanced by an effective Stokes drag force, FSt = 6πηeffa(dr/dt), where ηeff is an effective viscosity of a sphere moving in a passivated gel that grows with a. This force balance leads to a simple differential equation for the radial velocity: vr = dr /dt = −qV */(6πεrηeff ar )

(2)

Solving this by separation, we obtain the following for the propagation distance r(t), assuming that the spheres are in a ring-like well located at r = rw at an initial time t = 0: r(t ) = rw[1 + (t /τ )]1/2

(3)

where we have defined a positive characteristic time scale τ to be τ = (3πεrηeff arw 2)/( −qV *)

(4)

Here, we are concerned with the propagation of negatively charged nanoscale spheres, so the denominator −qV* is a positive quantity. Taking the temporal derivative of eq 3, the instantaneous radial velocity of a sphere is vr(t ) = (rw /2τ )/[1 + (t /τ )]1/2

(5)

which decreases over time as the sphere propagates further away from the well’s radius. These calculations for individual spheres are also valid for highly dilute dispersions of spheres, in which sphere−sphere interactions can be neglected. While we have assumed that the geometry is purely cylindrical, in actuality the ring electrodes will create a more complex three-dimensional electric field, and this field will C

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The Journal of Physical Chemistry B resemble the ideal cylindrical solution only near the plane of the electrodes between the two rings. Consequently, the above equations only approximate the electric field in the real experimental geometry, yet they may serve reasonably well as a starting point for analyzing electrophoretic measurements of nanosphere propagation in ring-like passivated gel media. In a prior experimental work,20 the speed of the nanoparticles in well passivated gels was found to approach zero at a critical radius a* which depends on the concentration of the gel material, which sets the effective pore size. This implies that the effective viscosity of the medium increases with a as ηeff = ηeff,0/ (1 − a/a*), diverging as a approaches a*, where ηeff,0 is the effective viscosity in the limit of very small sphere size a ≪ a*. When substituted into eqs 4 and 5, this further implies that τ will also diverge as a approaches a* and that no propagation will occur for particles loaded into the well having a ≥ a*. For a PEG-passivated agarose gel having [Agarose] = 0.195%, [PEG1000] = 3.25 mM, 5.0 mM sodium borate buffer (pH = 9.0) at constant E = 1.6 × 10−4 statV/cm, a prior experiment determined that a* ≈ 180 nm for sulfate-stabilized polystyrene nanospheres.20 Next, we consider separations of polydisperse multimodal size distributions. At a given observation time, t, after voltage was applied, the ratio Sij, given by the propagation radius for spheres having radius ai normalized by the propagation radius for spheres having radius aj, where ai < aj, that have both started from the same well at r = rw, is Sij = {[1 + (t /τi)]/[1 + (t /τj)]}1/2

(6)

Figure 2. Imaging of scattered light from propagating rings of three different monodisperse anionic polystyrene nanospheres (radii: a = 42, 55, and 70 nm), which had been mixed and loaded into a central ring well, as they separate in a PEG-passivated agarose gel (0.3% agarose, 5.0 mM PEG-1000, 5.0 mM SBB at pH = 9.0) at a time t after applying 50 V. Background-subtracted images are inverted for clarity. (a) t = 900 s; blue circle is placed inside the innermost light scattering ring to highlight the near-circularity and stability of the propagating rings; (b) t = 1800 s. Scale bar: 1 cm.

where τi and τj are given by eq 4 with ai and aj, respectively, replacing a. As Sij approaches one, which is true for short observation times t ≪ τi and t ≪ τj, there is effectively no separation efficiency. By contrast, in the limit as time becomes much larger than both τi and τj, Sij approaches (τj/τi)1/2. Thus, the separation ratio between different sizes is reduced in a cylindrical geometry, as compared to a standard Cartesian geometry, for which Sij would be fixed by the time-independent ratio of the propagation velocities of the two differently sized spheres in the passivated gel medium. Thus, the cylindrical geometry offers a potential advantage of separating polydisperse distributions having a large dynamic range in radii that have been loaded into the same ring-like well, because smaller spheres will propagate less slowly as they approach the end of the gel (i.e., as they begin to approach r2). By contrast, in a Cartesian geometry (i.e., rectangular slab), the smallest nanospheres would leave a gel that has the same limited length over the same experimental duration.

nanospheres that scatter the illuminating light, appear as propagating dark rings in these inverted images. As expected theoretically and in contrast to what has been previously observed when determining size distributions of nanospheres using a standard Cartesian rectangular slab geometry,19,20 the propagation distances at a given instant do not simply decrease linearly as a function of a. In addition, the intensities of the rings decrease as they propagate further away from the well, qualitatively consistent with the diverging radial geometry and conservation of matter. Thus, the cylindrical electrophoresis apparatus can be used to separate mixtures of differently sized monomodal nanospheres into distinct and narrow ring-like bands, provided that these nanospheres are small enough to propagate readily in the passivated gel. Some azimuthal variations in intensity are observed around the band, and these likely arise as a consequence of the inhomogeneity introduced by pipetting the mixture into the ring well at different discrete locations. By analyzing a complete movie of these backgroundsubtracted images of scattered light from the propagating rings of Figure 2, we digitally construct a space−time plot, as shown in Figure 3. This space−time plot is obtained by taking a narrow slice along the same radial direction from each image frame in the movie and then concatenating all of these slices together. Three distinct lines in r(t) clearly emerge over time,

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RESULTS AND DISCUSSION To measure the separation of a multimodal mixture of nanospheres having several different monodisperse sizes in the cylindrical electrophoresis apparatus, we make a mixture of three different anionic nanospheres with a1 = 42 nm (sulfatestabilized), a2 = 55 nm (carboxyl-stabilized), and a3 = 70 nm (sulfate-stabilized), and load this mixture into the ring well. We record a movie of the electrophoretic propagation of these mixed nanospheres in a PEG-passivated agarose gel having [Agarose] = 0.3 w/w %, [SBB] = 5.0 mM, [PEG1000] = 5.0 mM after applying a voltage of Vapp = 50 V. Electrophoretic separation of this mixture into three distinct ring-like bands is observed, as shown in the background-subtracted and inverted images of Figure 2a−2b. Thus, circular bands, which contain D

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calculate the radial velocities as a function of time, as shown in Figure 4b. Thus, the velocity decreases by about a factor of 3 over the duration of the measurement, and each ring expands more slowly as it grows radially. Using the theoretical model, which reasonably captures the above-mentioned measurements, we have estimated the radial propagation distances as a function of time for different radii of monodisperse SSPS nanospheres in PEG-passivated agarose gels, as shown in Figure 5. Here, we have chosen τ = 100 s for a Figure 3. Radial space−time plot of propagating bands of SSPS nanospheres made by taking thin slices from a movie of backgroundsubtracted and inverted images from Figure 2. A three-peaked trimodal size distribution of anionic nanospheres, containing radii a1 = 42 nm, a2 = 55 nm, and a3 = 70 nm, are loaded into a common well at rw ≈ 10 mm at time t = 0; the different sizes are separated into distinct rings as time progresses (gray lines from top to bottom: a1, a2, a3). Gel composition and applied voltage are the same as those in Figure 2.

and the shapes of these lines are clearly nonlinear. Based on other measurements in the cylindrical geometry taken using the individual nanosphere dispersions, we associate these lines with nanosphere radii of 42, 55, and 70 nm, from top to bottom. From this space−time diagram, we obtain detailed measurements of propagation radius r as a function of time t for all three nanosphere sizes, as shown in Figure 4a. Initially, the radii

Figure 5. Predicted radial propagation distances, r, of monodisperse anionic nanospheres in PEG-passivated agarose gels as a function of time t. Here, τ = 100 s for nanosphere radius a = 90 nm and the radial well distance is fixed as rw = 5 mm. The divergence of the effective viscosity is set at a* = 180 nm (see ref 20).

= 90 nm, and we use a* ≈ 180 nm as the divergence point in the size-dependent effective viscosity, corresponding to PEGpassivated gels that have been well explored over a wide range of sizes and conditions, at a somewhat lower agarose concentration of ≈ 0.2%.20 These predictions show that, by using a cylindrical separation geometry, smaller nanosphere sizes can still be physically retained in passivated gels having limited spatial dimensions over longer durations required to separate larger nanosphere sizes. This useful feature contrasts sharply with the standard 1D Cartesian electrophoresis geometry, for which the smaller nanospheres would propagate at the same rate independent of their location in the gel, thereby causing them to leave gels having a similarly limited spatial dimension over a similar duration, which is typically undesirable. Thus, the cylindrical electrophoresis geometry offers a distinct advantage for separating mixtures of nanoscale objects that have widely different electrophoretic mobilities or sizes.

Figure 4. Measured propagation and separation of three different sizes of monodisperse anionic nanospheres in PEG-passivated agarose gels into distinguishably different circular bands based on the space−time plot of Figure 3. (a) Measured average radius r for each radius: a1 = 42 nm (green diamonds), a2 = 55 nm (blue squares), a3 = 70 nm (black circles). Solid lines: fits using eq 3. (b) Radial propagation velocities vr of circular bands calculated using eq 5 and fit parameters from part (a). Line colors: same as those in part (a).

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CONCLUSION We have demonstrated that cylindrical gel electrophoresis can be used to create stably propagating ring-like fronts of charged monodisperse colloids in passivated large-pore gels. Because the electric field strength in a cylindrical geometry decreases as the colloids propagate away from the central axis, the rings initially expand more rapidly and this expansion progressively slows down. Our observations of this effect, obtained by imaging light scattered from rings of propagating polystyrene nanospheres, are consistent with a simple theoretical model that predicts the radial propagation distance of dilute monodisperse charged spheres as a function of time. Based on this, we anticipate that this model would also describe the cylindrical electrophoretic propagation of ring-like fronts of monodisperse charged molecules through smaller pore gels, too. While using a

increase more rapidly, but as these rings propagate to larger radii further away from the anode at longer times, then the increase in the radius becomes more gradual. We fit these measurements of r(t) using eq 3, yielding fit parameters rw = 9.8 ± 0.2 mm, consistent with the known well location for all three sizes, and τ1 = 361 ± 22 s for a1, τ2 = 416 ± 21 s for a2, and τ3 = 517 ± 21 s for a3. The correlation coefficients of all fits are greater than 0.99, indicating that eq 3 describes the observed propagation well. Using these fit parameters and eq 5, we E

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(9) Griess, G. A.; Guiseley, K. B.; Serwer, P. The Relationship of Agarose Gel Structure to the Sieving of Spheres During Agarose Gel Electrophoresis. Biophys. J. 1993, 65, 138−148. (10) Guarrotxena, N.; Braun, G. Ag-Nanoparticle Fractionation by Low Melting Point Agarose Gel Electrophoresis. J. Nanopart. Res. 2012, 14, 1199. (11) Hanauer, M.; Pierrat, S.; Zins, I.; Lotz, A.; Sonnichsen, C. Separation of Nanoparticles by Gel Electrophoresis According to Size and Shape. Nano Lett. 2007, 7, 2881−2885. (12) Hasenoehrl, C.; Alexander, C. M.; Azzarelli, N. N.; Dabrowiak, J. C. Enhanced Detection of Gold Nanoparticles in Agarose Gel Electrophoresis. Electrophoresis 2012, 33, 1251−1254. (13) Kowalczyk, B.; Lagzi, I. N.; Grzybowski, B. A. Nanoseparations: Strategies for Size and/or Shape-Selective Purification of Nanoparticles. Curr. Opin. Colloid Interface Sci. 2011, 16, 135−148. (14) Li, F.; Hill, R. J. Nanoparticle Gel Electrophoresis: Bare Charged Spheres in Polyelectrolyte Hydrogels. J. Colloid Interface Sci. 2013, 394, 1−12. (15) Mesgari, S.; Sundramoorthy, A. K.; Loo, L. S.; Chan-Park, M. B. Gel Electrophoresis Using a Selective Radical for the Separation of Single-Walled Carbon Nanotubes. Faraday Discuss. 2014, 173, 351− 363. (16) Pellegrino, T.; Sperling, R. A.; Alivisatos, A. P.; Parak, W. J. Gel Electrophoresis of Gold-DNA Nanoconjugates. J. Biomed. Biotechnol. 2007, 2007, 26796. (17) Surugau, N.; Urban, P. L. Electrophoretic Methods for Separation of Nanoparticles. J. Sep. Sci. 2009, 32, 1889−1906. (18) Wu, T. C.; Dutta, M.; Stroscio, M. A. Agarose Gel Investigation of Quantum Dots Conjugated with Short ssDNA. IEEE Trans. NanoBiosci. 2013, 12, 282−288. (19) Zhu, X. M.; Mason, T. G. Nanoparticle Size Distributions Measured by Optical Adaptive-Deconvolution Passivated-Gel Electrophoresis. J. Colloid Interface Sci. 2014, 435, 67−74. (20) Zhu, X. M.; Mason, T. G. Passivated Gel Electrophoresis of Charged Nanospheres by Light-Scattering Video Tracking. J. Colloid Interface Sci. 2014, 428, 199−207. (21) Gorg, A.; Obermaier, C.; Boguth, G.; Harder, A.; Scheibe, B.; Wildgruber, R.; Weiss, W. The Current State of Two-Dimensional Electrophoresis with Immobilized pH Gradients. Electrophoresis 2000, 21, 1037−1053. (22) Wildgruber, R.; Harder, A.; Obermaier, C.; Boguth, G.; Weiss, W.; Fey, S. J.; Larsen, P. M.; Gorg, A. Towards Higher Resolution: Two-Dimensional Electrophoresis of Saccharomyces Cerevisiae Proteins Using Overlapping Narrow Immobilized pH Gradients. Electrophoresis 2000, 21, 2610−2616. (23) Ventzki, R.; Ruggeberg, S.; Leicht, S.; Franz, T.; Stegemann, J. Comparative 2-DE Protein Analysis in a 3-D Geometry Gel. BioTechniques 2007, 42, 271−279. (24) Millioni, R.; Miuzzo, M.; Antonioli, P.; Sbrignadello, S.; Iori, E.; Dosselli, R.; Puricelli, L.; Kolbe, M.; Tessari, P.; Righetti, P. G. SDSPage and Two-Dimensional Maps in a Radial Gel Format. Electrophoresis 2010, 31, 465−470. (25) Purcell, E. M.; Morin, D. J. Electricity and Magnetism, 3rd ed.; Cambridge, New York, 2013.

cylindrical geometry is unlikely to be universally advantageous over a simple Cartesian geometry, in certain situations, the cylindrical geometry can be advantageous, especially if there is a wide distribution of sizes of colloids or molecules to be separated. Smaller objects advance less rapidly as they move further away from their initial position, even in a homogeneous gel, so the duration of the electrophoresis experiment can be extended to enable better separation of larger objects that are closer to the inner electrode. Because of this potential advantage, the cylindrical geometry could be a useful choice when considering the separation of aqueous mixtures of biological origin that contain both smaller biomolecules as well as larger objects, such as organelles and viruses. Cutting out concentric rings of the gel after performing electrophoresis and eluting would enable recovery of separated biological objects having the same electrophoretic mobilities. Thus, future experiments could examine the bounds on the upper and lower limits of sizes of objects that can be practically separated using this approach. In addition, we anticipate that this non-Cartesian approach could potentially be extended to a spherical shell geometry in a fully three-dimensional electrophoresis apparatus, yielding an electric field that decreases as the inverse of the square of the radius away from the central point, representing an even more dramatic reduction in propagation velocity than that which has been achieved using a cylindrical geometry.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: 310-206-0828. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank UCLA for providing financial support for this work. We also thank and congratulate Bill Gelbart who has been such a wonderful friend, colleague, and mentor over many years.

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REFERENCES

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DOI: 10.1021/acs.jpcb.6b02165 J. Phys. Chem. B XXXX, XXX, XXX−XXX