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Single-cell electroporation using an electrolyte-filled capillary is an emerging technique for transient pore formation in adherent cells. Because adherent cells do not have a simple and consistent shape and because the electric field emanating from the tip of the capillary is inhomogeneous, the Schwan equation based on spherical cells in homogeneous electrical fields does not apply. We sought to determine experimental and cell parameters that influence the outcome of a single-cell electroporation experiment. A549 cells were exposed to the thiol-reactive dye Thioglo-1, leading to green fluorescence from intracellular thiol adducts. Electroporation causes a decrease with time of the intracellular fluorescence intensity of Thioglo-1-loaded cells from diffusive loss of thiol adducts. The transient curves thus obtained are well-described by a simple model originally developed by Puc et al. We find that the final fluorescence following electroporation is related to the capillary tip-to-cell distance and cell size (specifically, 2(A/π)1/2 where A is the area of the cell's image in pixels. This quantity is the diameter if the image is a circle). In separate experiments, the relationship obtained can be used to control the final fluorescence following electroporation by adjusting the tip-to-cell distance based on cell size. The relationship was applied successfully to A549 as well as DU 145 and PC-3 cells. Finally, F-tests show that the variability in the final fluorescence (following electroporation) is decreased when the tip-to-cell distance is controlled according to the derived relationship in comparison to experiments in which the tip–cell distance is a constant irrespective of cell size.
Electroporation1–9 is a technique that uses electric fields to create nanopores (~1–10 nm)10 in a cell's membrane, dramatically increasing its permeability. With dependence upon the conditions used for electroporation, the pores can either be transient or short-lived in reversible electroporation or long-lived in irreversible electroporation. Cell viability is maintained in reversible electroporation as the pores reseal in this case, whereas cell death occurs in irreversible electroporation as the pores do not reseal. This technique has been studied and applied for almost 3 decades. Over that time, it has been used to transport molecules, chiefly DNA, across cell membranes for a wide range of applications such as gene therapy,11,12 molecular biology,4,13 and clinical chemotherapy.14
There are two types of electroporation, namely, bulk electroporation and single-cell electroporation. In bulk electroporation, a batch of cells in suspension is exposed to a macroscopically uniform electric field of a few kilovolts per centimeter resulting in the permeabilization of many cells at the same time.7,8,15–17 Neumann et al. in the early 1980s demonstrated gene transfection by bulk electroporation of mammalian cells.4 Ever since, bulk electroporation has become a standard method for introduction of DNA into cells. However, bulk electroporation results in a distribution of outcomes among the cells in the suspension. It would be preferable, especially to study the membrane biophysics of cell permeabilization, to electroporate at the single-cell level.
Single-cell electroporation has shown promise, especially for application to natural, adherent cells, but is not as well studied as bulk electroporation. Single-cell electroporation should not be confused with bulk electroporation in which numerous cells are permeabilized but only individual cells are studied.18–21 In single-cell electroporation, a localized field is applied to a single adherent cell or a cell in suspension flowing through a microfabricated device, while the neighboring cells are exposed to very low or no field.22–28 Lundqvist et al. first demonstrated single-cell electroporation using carbon-fiber microelectrodes in 1998.29 Since then, numerous other single-cell electroporation techniques using electrolyte-filled capillaries,25,30 micropipets,31 microfabricated chips,32–34 and multiwalled carbon nanotubes35 have been developed.
In bulk electroporation experiments, several workers have determined how to increase transport of DNA, dyes, and drugs across the cell membrane.36–38 Normally, optimizing conditions for gene transfection in bulk electroporation has been the focus of studies.4,39–41 Although transfection is of practical importance, it does not give a direct measure of transport across the membrane because transcription and translation involve many steps in addition to the transport of DNA such as DNA/membrane interaction, the translocation and migration of the plasmid into the cytoplasm, its passage into the nucleus, and the activation of expression. Other bulk electroporation studies have measured dye uptake to determine the effect of conditions on the success of electroporation.37 A few bulk electroporation studies have quantified uptake of small fluorescent molecules using flow cytometry following bulk electroporation.9,36 Small molecule uptake in bulk electroporation depends on electroporation parameters such as field strength, pulse length, number of pulses,42–44 cell size, and shape19 and other experimental conditions such as temperature, buffer,45 etc. However, significant variability both in the observed result and in viability under constant conditions is generally observed in bulk electroporation.44,46 Prausnitz and Canatella47 used a statistical approach to predict the uptake of calcein and cell viability following bulk electroporation. Their prediction is based on a nonlinear regression approach to an empirical equation. On the basis of data for a single cell type, the experimental parameters important to permeabilization and viability were field strength, pulse length, and number of pulses. The empirical expression was useful for 60 different cell types. This rather impressive result solidifies the idea that it is the lipid bilayer's response to an electric field that controls bulk electroporation. We note that the model predicts the average behavior of a large population of cells. It is a telling measure of the inherent variability in the experiment that predictions within a factor of 2 of the observed result were considered satisfactory. To summarize, there are abundant quantitative studies of the electrical response of a cell in a homogeneous electric field and of the mass transport of molecules into an electroporated cell in bulk electroporation.
We are unaware of experimental reports that quantify mass transport out of cells following single-cell electroporation. An advantage of single-cell electroporation is the ability to relate a result to properties of the cell, not only to experimenter-controlled parameters. However, this dependence is hard to predict because single-cell electroporation lacks the symmetry of bulk electroporation in which suspended, nominally spherical cells are exposed to a nominally homogeneous electric field. In the present work, we investigate mass transport from single cells following electroporation quantitatively The experimental method is based on first labeling thiols inside the cell with a maleimide-based fluorogenic reagent, Thioglo-1. As the most abundant thiol in the cell is reduced glutathione, GSH, we can say that the decrease in fluorescence intensity inside the cell following application of the pulse is predominantly due to the efflux of labeled glutathione from the cell. As the GSH–ThioGlo-1 conjugate has a formula weight of 684.7 Da, we view it as an indicator of the flux of freely diffusing molecules. Loss of fluorescence from diffusion of Thioglo-1 conjugates out of the cell was measured as a function of time. In a first set of experiments, which we will call set 1, the final fluorescence following electroporation and pore closing was correlated with experimental variables and parameters of the cell. In a second set of experiments, which we will call set 2, information from the first set was used to control the final fluorescence, which corresponds to a particular material loss from the cell, following electroporation. We show that a simple model developed for bulk electroporation describing the uptake of small molecules by cells in suspension46 applies, following suitable modification, to the release of molecules from single, adherent cells on a nonconductive surface. We show that for constant experimental conditions (set 1), there is considerable variability in the extent of freely diffusing molecule release in single-cell electroporation. Finally, we show that, with statistical significance, we can control the magnitude of molecular flux from single cells and decrease the variability of the outcome if the size of the individual, adherent cell is taken into account in establishing the electroporation conditions (set 2).
The chemicals used for buffer preparation were all analytical grade and were purchased from Sigma (St. Louis, MO). Thioglo-1 was purchased from Covalent Associates (Woburn, MA). Propidium iodide and calcein AM were purchased from Invitrogen/Molecular Probes (Eugene, OR). The cell lines A549, DU 145, and PC-3 were obtained from ATCC (Manassas, VA). Basal medium Eagle (BME), RPMI-1640, trypsin-EDTA, fetal bovine serum, L-glutamine, penicillin, and streptomycin were all obtained from Gibco-BRL (Carlsbad, CA). Milli-Q (Millipore Synthesis A 10, Billerica, MA) water was used. Extracellular buffer consisted of (mM) NaCl, 140.0; KCl, 5.0; MgCl2, 2.0; CaCl2, 2.0; D-glucose, 10; HEPES, 10.0; pH adjusted to 7.40.
Human lung cancer A549 cells were cultured in BME supplemented with 10% fetal bovine serum, 100 U/mL penicillin, and 100 μg/mL streptomycin. DU 145 and PC-3 cells were grown in RPMI-1640 medium with 0.3 g/L L-glutamine, 10% fetal bovine serum, 100 U/mL penicillin, and 100 μg/mL streptomycin. Cells were grown as a monolayer in 75 mL cell culture flasks at 37 °C and 95% air/5% CO2 (HERA cell incubator, Newtown, CT). Before the experiments, cells were plated on 35 mm, glass-bottom cell culture dishes (MatTek Corp., Ashland, MA) and were grown for up to 3 days. Experiments were performed on the second and third days following the cell plating. Before the experiments, the cells were stained with cell-permeable dye Thioglo-1 (2 μM in extracellular buffer) for 30 s at room temperature, then washed with extracellular buffer twice. Thioglo-1 is a maleimide-based reagent that gives a highly fluorescent product upon its reaction with active SH groups.48,49 Cell dishes were mounted on the cell chamber (DH 35i culture dish incubator, Warner Instruments, Holliston, MA) and transferred to the stage of the microscope.
Fused-silica capillaries from Polymicro Technology (Phoenix, AZ) were used as a probe to perform electroporation experiments. Capillaries were pulled with a CO2 laser puller (Sutter Instruments Co. P-2000, Novato, CA) to create reproducible capillaries with a short pulled tip having an inner diameter of ~5 μm. The final length of the capillaries was 15 cm.
To electroporate a cell (Figure 1), the pulled tip of the capillary was submerged in the cell dish containing extracellular buffer. The other end of the capillary was placed in a vial filled with extracellular buffer. A platinum electrode placed in this vial was connected to the electroporator (ECM 830, BTX Instruments, San Diego, CA), and the electrical circuit was completed with a grounded platinum counter electrode placed in the cell dish. The counter electrode is a Pt wire situated at the periphery of the 35 mm diameter cell dish, typically about 15 mm from any electroporated cell. The capillary resistance was measured immediately prior to electroporation. The resistance determining circuit consisted of a synthesized function generator (SRS model DS 340, Stanford Research Systems, Inc., Sunnyvale, CA) and a lock-in amplifier (SRS model SR 830 DSP) which has an internal current-to-voltage converter. The function generator applied a continuous 2 V sine wave ac signal at 100 Hz across the aforementioned Pt electrodes with the electrolyte-filled capillary in place. The current resulting from the 2 V across the resistance of the capillary was measured by the lock-in amplifier. The resistance informed us about the status of the tip. Tips with an opening of 5 μm that are not clogged have a resistance of about 1.4 × 107 Ω. The test circuit is switched off during electroporation.
Cells were observed through a 40× 1.3 NA oil immersion objective using an inverted microscope (Olympus, IX 71, Melville, NY) equipped with an HBO 100 W mercury lamp as the excitation source. For Thioglo-1, an Omega fluorescence cube (specially built, Omega, Brattleboro, VT) was used with filters for excitation at 378 nm and emission at 480 nm. For live/dead imaging, a triple band “Pinkel” filter set from Semrock (Rochester, NY) was used (exciter 1 387 nm, exciter 2 494 nm; exciter 3 575 nm; dichroic mirror, 394–414, 484–504, 566–586 nm; emitter, 457, 530, 628 nm). A CCD camera (Hamamatsu, ORCA-285, Bridgewater, NJ) imaged cells. The image collection frequency was 1 frame/s. Image processing was performed by the image acquisition software Simple PCI (Compix, Inc., Sewickley, PA).
There were two sets of experiments. In “set 1”, A549 cells were exposed to single pulses of duration 300–500 ms and magnitude 500 V. This will be referred as “pulsing” of the cells. The tip–cell distance, dm (see Figure 1) was set at 2.0, 3.5, or 5.0 μm. At a time of 25 s from the initiation of the acquisition of images, the pulse was applied. Diffusive loss of fluorescent thiol (chiefly reduced glutathione) adducts from the cell was determined quantitatively with Simple PCI software. Further details of single-cell electroporation can be found elsewhere.30,50
In the second set (“set 2”), A549, PC-3, and DU 145 cells were used. Information from set 1 experiments was used to determine what value of dm to use for a given cell, based on its size (the average diameters of the cells were A549, 25 μm; DU 145, 23.5 μm; and PC-3, 23 μm), to achieve a particular final fluorescence.
Cell parameters representing size, shape, and intensity were obtained by Simple PCI software. The size factors used were area, diameter, perimeter, and breadth. The area, A, is the total number of pixels in the image of a cell, the diameter is computed as 2(A/π)1/2, the perimeter is the distance traveled around the boundary of the cell, and the breadth is the average width of the cell's image. The shape factor is the aspect ratio (maximum length/maximum breadth). Intensity is the average gray level of the pixels in the object.
A 15 cm-long pulled capillary with a tip of ~5 μm in diameter was positioned near a cell using an MP-285 motorized micromanipulator from Sutter (Novato, CA). The capillary was first positioned at about a 45° angle with respect to the cell dish normal 5 μm above the surface (Figure 1). The distance between the capillary tip and the cell (dm) was taken to be from the closest approach of the cell and the projection of the capillary image in the horizontal imaging plane.
Fluorescence vs time data from 130 cells comprise the set 1 data. Stepwise forward and backward linear regression analyses were performed using the program STATA (Intercooled 9.0). The dependent variable was K, the ratio of mass transport rate constant to pore resealing rate constant (see below, eq 2). K is also related to the long-time (final or limiting) fluorescence intensity as shown in eq 1.
where F0 is the fluorescence intensity at the time of the application of the pulse before any decrease caused by mass transport out of the cell and Ff is the final fluorescence at long times. The independent variables used in the regression were the cell parameters area, diameter, perimeter, breadth, aspect ratio, and intensity and the experimental conditions, dm and pulse duration. The cell parameters were transformed to get a normal distribution. The transformations are shown in Table 1. Regression analysis was based on the transformed variables.
In order to validate the regression results, set 2 electroporation experiments were performed again on the A549 cell line. Further, to validate our results for different cell lines, experiments were performed on two more cell lines: prostate adenocarcinoma cell line PC-3 and prostate cancer cell line DU 145. For this set of experiments we fixed the percent fluorescence intensity loss 100(1 − Ff/F0), which we will call ΔF%. For example, in the case of the PC-3 cell line, cells were pulsed to achieve a 50% loss. In order to do so, the diameter of the cell to be pulsed was first measured using the Simple PCI software, then according to eq 4 the dm at which the cell should be pulsed was determined. Similarly, experiments were performed on A549 cells to obtain ΔF% equal to 20%, and 40%, and on cell line DU 145 to obtain ΔF% equal to 40%.
Following electroporation experiments, live/dead analysis was performed on the cells. The buffer in the cell dish was replaced by 2 mL of fresh growth medium. The cells were allowed 5–6 h of recovery in the 37 °C incubator with 95% air/5% CO2, followed by a 30 min staining with 2 μM (each) calcein AM and propidium iodide. Cell survival percentage was calculated for each experimental condition based on this live/dead assay. It is important to note that a measurement of cell viability was made on each cell that was electroporated.
In the present work, extracellular buffer (140 mM Na+, 5 mM K+) was used to perform the experiments as opposed to intracellular buffer (140 mM K+, 5 mM Na+) used in our previous work.30,50 The range of pulse durations used in the previous experiments30,50 performed with intracellular buffer was 50–300 ms at 500 V. The range of pulse durations used in the current work with extracellular buffer was 300–500 ms at 500 V. When extracellular buffer at pulse durations below 300 ms was used, the electroporation success for A549 cells was very low (<20%). Moreover, we found that the survival rate was higher with extracellular buffer as compared to intracellular buffer under the same set of experimental conditions. For example, at pulsing conditions of 300 ms and dm of 2 μm, the survival rate with intracellular buffer was only ~0–5%, whereas in the case of extracellular buffer, the survival rate was ~65%. At longer distance, e.g., at pulse duration of 300 ms and dm of 5 μm, the survival rate with intracellular buffer was 60%, whereas in the case of extracellular buffer, it was 95%. Thus, we have used only extracellular buffer in this work.
In both set 1 and set 2 experiments, the function given in eq 2 was fitted to fluorescence intensity vs time curves.
Here F is the fluorescence intensity at time t, F0 is the fluorescence intensity at time zero when the pulse is applied, k is the photobleaching rate constant, K = M/α where M is the first order rate constant for transport of Thioglo-1 labeled molecules out of the cell, and α is the first order pore resealing rate constant. This equation results from incorporating a photobleaching term into the equation from Puc et al.46
Equation 2 incorporates the assumption that concentration polarization within the cell is minimal. We think that this is justified. The time scale over which the fluorescence intensity changes, and thus over which the intracellular concentration of thiol adducts changes, is on the order of 10 s. The relaxation time for a concentration perturbation in a sphere is approximately a2/15D (a is the sphere radius and D is the solute diffusion coefficient).51 We estimate the diffusion coefficient of the Thioglo adduct of GSH in water to be 3 × 10−10 m2 s−1. Diffusion coefficients in the cytoplasm for noninteracting species (dextrans, ficolls) of a similar molecular size as the Thioglo adduct of GSH are about ¼ of the magnitude of the diffusion coefficients in free solution.52,53 We thus estimate the relaxtion time in the cell to be on the order of 10−1 s, which is significantly smaller than the time over which concentrations change due to electroporation-induced flux. Thus, concentration polarization within the cell is not likely to influence the mass transport of low-molecular weight solutes out of the cell.
A program written in Mathcad (Mathsoft, Cambridge, MA) was used to fit decay curves, to estimate the parameters k, K, and α, and provide a goodness of fit R2. Figure 2 gives an example of eq 2 fitted to a set of data. Here, the circles represent the data from a cell that was pulsed for 300 ms with a cell-to-capillary tip distance of 3.5 μm. The solid line is the best fit that was obtained using the Mathcad program. Table 2 gives a list of parameter definitions.
In initial explorations of the curve-fitting procedure, we used eq 3.
which, except for the photobleaching term, directly reflects the derivation of Puc et al.46 It was found that M and α were somewhat correlated (the correlation coefficient between M and α arising from the nonlinear fitting, ρMα, is 0.48). We repeated the nonlinear curve fitting with the ratio M/α replaced by the new parameter K. We then found no correlation between the fit parameters K and α (ρKα = −0.06). By minimization of the correlation between these two fit parameters, the error in the result is also minimized. We should mention that K and α are not “adjusted” to compensate for photobleaching. Rather, the photobleaching rate constant is one of the parameters determined along with K and α. It turns out that the parameter K is useful in its own right. It is related to the fraction of the cell's final fluorescent contents, as shown above in eq 1. The value of K is important because it is a time-independent result of electroporation, the variability of which we seek to control. Our goal is to explain the variability (set 1 experiments) and then use that understanding to predict the result (set 2 experiments).
Recall that set 1 experiments represent a survey of ΔF% for a set of cells of various sizes and shapes and of various tip–cell distances and pulse times, where ΔF%isthe percent fluorescence intensity loss 100 (1 − Ff/F0). In this set of experiments, the tip–cell distance was set arbitrarily for any given cell. A histogram of ΔF% for all the cells pulsed at 2 μm dm is shown in Figure 3. The average ΔF% for these cells was 53%, with a standard deviation of percent loss in intensity (ΔF%) of 24%. The wide distribution of ΔF% shows that there is a large variability in the outcome even when the tip–cell distance is constant.
In order to understand what variables affect K, we did multiple linear regression of K on the transformed independent variables pulse duration, tip–cell distance dm, and the cell parameters in Table 1. The “size” (parameters area, diameter, perimeter, and breadth) are all highly correlated, but we do not know which is a better predictor (if any) of K. Thus, we used stepwise regression. The results from stepwise forward and backward linear regression of K on the transformed variables (Table 1) are shown in Table 3 and eq 4. There is significant relationship (R2 = 0.30, p < 0.001 for the coefficients of dm and dc−1, = p 0.694 for the constant) between dependent variable K and independent variables dm and cell diameter dc (μm). The fact that cell-size parameters other than diameter do not appear in the regression should not be taken to mean that they are not important.
Numerical simulations50 show that the transmembrane potential induced in an electroporation experiment depends on the tip–cell distance, the polar angle from the long axis of the capillary, and the cell size. The cell-size dependence of the fraction of the cell's surface area that is electroporated is smaller in single-cell electroporation (∞ dc0.33) than in the bulk experiment (∞ dc1.0).50 In the single-cell experiment, the most critical variable is dm, as it governs the electric field at the cell for a given applied voltage and capillary length. Thus, it is gratifying to find that both dc and dm are significant predictors.
The regression results were used to make a contour plot of ΔF% as a function of dm and dc (Figure 4). The contour plot shows that at a given dm, a smaller cell has a larger ΔF% than a larger cell. It is instructive to consider the likely spread of values of ΔF% for a fixed dm given the cell size distribution shown in the Box plot in Figure 4. The Box plot in Figure 4 represents the distribution of cell sizes in percentiles: 5th, 25th, 50th, 75th, and 95th (12, 19, 22, 27, and 36 μm, respectively). It is apparent that the median cell is predicted to lose 50% of its fluorescence at a dm of about 3.9 μm. The spread of expected values (5th to 95th percentile) of fluorescence loss is fairly high at any value of dm. It is clear that fixing the capillary tip–cell distance, dm, is not sufficient to tightly control the field strength at the cell and thus the outcome of electroporation.
In these experiments, we sought to predict ΔF%. Equation 4 can be used to estimate the right tip-to-cell distance for a desired concentration decrease (or increase if diffusive flux is inward rather than outward). If, for example, a fluorescence decrease of 50% is desired and the cell diameter is 22 μm, then the equation directs us to use a dm of 3.9 μm. In Set 2, the tip–cell distance was adjusted for each cell to achieve a particular ΔF%. We performed additional electroporation experiments on A549 cells and also on PC-3 and DU 145 cells. Results are shown in Table 4 for three cell types and for several predicted values of ΔF%. For example, the top row shows results from our attempt to obtain a ΔF% of 20% from A549 cells. The 95% confidence interval for ΔF% for these cells was [15, 22]. In fact, for all cell lines (A549, DU 145, and PC-3), the predicted ΔF% lies within the 95% confidence interval of the observed ΔF%.
Figure 5 is a histogram of ΔF% for A549 cells electroporated in set 1 (dm = 2 μm) and set 2 experiments. The large spread in set 1 experiments shows that there is a large variance in the results (histogram A). Histograms B and C in Figure 5 show that we were able to reduce this variance by taking dc into account (eq 4) when setting dm. To further verify this reduction in the variance between set 1 and set 2 experiments, an F-test was performed between set 1 and each of the set 2 intensity loss observations. The results are shown in Table 5. The F-test results show that the variances in set 2 experiments were reduced significantly (p < 0.05) for all cell types. This is strong evidence that we can predict ΔF% by controlling dm in accordance with eq 4.
In our previous research (using high K+ buffer), we investigated conditions leading to successful electroporation and cell viability. The observations were binary in nature, i.e., either a cell was electroporated or it was not. In those experiments, we observed a large variability (e.g., the average ΔF% for all cells electroporated at 2 μm dm was 55% with a standard deviation of 30%) in success and survivability at a given set of conditions.30,50 We found that cell size and shape were significant factors in controlling success and survivability. Thus, at constant experimental conditions, the natural variability of the cells led to variability in the outcome. Nonetheless, we found conditions (experimental and cell parameters) at which we could maximize cell permeabilization and survivability. The present work goes one step beyond that. We have quantified the degree of electroporation. Also, we have not only found one of the factors that induces variability in the experiments but have also been able to reduce this variability by controlling the dm according to eq 4. We note also, see Table 4, that the survival rate for the set 2 experiments is quite acceptable.
The statistical results in Tables 4 and and55 show two things. Table 4 demonstrates that taking into account the cell size in setting the tip–cell distance is critical to achieving a particular ΔF% on average. Table 5 demonstrates that taking into account the cell size in setting the tip–cell distance significantly reduces the variability of the outcome of electroporation (in terms of ΔF%). The experimental significance of the reduction in variance is a decrease in the number of experiments required to achieve a particular standard error of the mean given a particular experimental variability. Thus, with statistical rigor, we have demonstrated the advantage of distance control based on cell size in single-cell electroporation. However, it is also true that the variability is still rather large. At this point, we do not know the cause of the variability, although it is by now well-known that cell-to-cell variability is to be expected in single-cell experiments of any sort. It is very likely that the parameter K depends on aspects of cell physiology that we do not now control. For example, we do not control for the number of passages of the cells or where the cells are in the cell cycle.
It is interesting to pursue the origin of eq 4 from a causal rather than an observational (regression) perspective. Transient permeabilization in an electric field results when the transmembrane potential exceeds a critical value. Estimates of the critical value vary, but we have found that the value established by Teissie and Rols54 of 250 mV is consistent with our observations.55 We have also previously determined the fraction of a hemispherical cell's exposed area (πdc2/2) that has a transmembrane potential greater than the critical value in single-cell electroporation with an electrolyte-filled capillary.30 We define fp as the fraction of the cell's surface area that has a transmembrane potential greater than or equal to the critical value at t = 0 (i.e., before the electrical conductance of the membrane increases because of the increased porosity). This permeabilized fraction, fp is a function of tip–cell distance and of cell diameter.50 The area of a cell that is permeabilized contributes to the efflux of a chemical species through the membrane.43 A simple first order rate constant for the loss of material from a cell can be given as
where A is the permeabilized area of the cell, V is the cell's volume, and P is the permeability of the solute through the (porated) membrane. The permeabilized area, A, is equal to the product of the cell's surface area, πdc2/2, and the fraction of the area that is permeabilized. The permeabilized area is thus π/2fpdc2. The cell's volume is πdc3/12.
We can thus write
This mass transport coefficient is the same as that in Puc et al. mentioned above. It is the mass transport coefficient in the presence of the field, assumed constant. Recall from eq 3 that the time dependent part of the dynamics (pore resealing) is embodied in the time variable, the parameters M, α, and K are time-independent. We also assume here that the mass transport occurs solely by diffusion (e.g., not by electrophoresis) because we do not see experimentally an instantaneous drop in fluorescence at the time the pulse is applied (called “g” in Puc et al.). The computed values of fp (called FEA in Figure 6 of Agarwal50) fit well to second order polynomials in dm (see Supporting Information, Table S1). Using these polynomials, we calculated values of the term 6fp/dc. This term is well-described by eq 7 (R2 = 0.87, p values for coefficients of dm and dc−1 are <0.0001, for the constant, p = 0.24, see Supporting Information, Table S2).
This dependence is remarkably similar to the dependence seen for K (eq 4 and Table 3). The magnitudes of the parameters cannot be compared directly for the two equations; however, the relative parameter values can be. We note that in both equations the coefficient of dm is negative, the coefficient of dc−1 is positive, and the constant term is statistically indistinguishable from zero (see Supporting Information, Table S2 for regression statistics for eq 7). Further, the ratios of the coefficients of dc−1 to the coefficients of dm are similar, on the order of 102.
It is possible to make an inference about the electroporation process from this similarity. The functional similarity of eqs 4 and 7 means that the ratio of the two equations is approximately constant (not dependent on the independent variables). Explicit computation of the ratio shows that the ratio is approximately constant except for large cells and long distances, cases in which there is little or no fluorescence loss. Algebraically, the ratio of the two quantities on the left-hand sides of eqs 4 and 7 is equal to P/α. We infer that this ratio is approximately constant.
Krassowska's group has developed a pore formation and closure model based on energies and measured or estimated parameters.56 In this model, after the electric pulse is removed all pores rapidly shrink to same pore area, i.e., the minimum-energy radius, rm. Hence, after removal of the electric field, cells reseal from the same minimum-energy radius rm.56,57 Thus, the pore resealing rate is considered to be a constant. Indeed, while our measured values vary over a range of values (~0.1–1 s−1), values of α were not correlated with any experimental or cell parameter. If the pore resealing rate is approximately constant and the ratio P/α is approximately constant for all cells, then we can conclude that, under the conditions of our experiments, the permeability of the cell immediately after the potential is returned to zero (P) is approximately independent of electroporation and cell-based parameters. Further, as we know the ratio P/α, and α, we can estimate P. This is necessarily highly approximate, so only an order of magnitude estimate is justified. We define P as the ratio of diffusion coefficient to membrane thickness, D/l, multiplied by a porosity, ϕ. The porosity ϕ refers to the effective fraction of the electroporated area (itself a fraction of the cell's surface area) through which molecules pass. Using D ~ 10−10 m2 s−1, l ~ 4 nm, α ~ 0.1 s−1, we obtain ϕ ~ 10−4, a reasonable value.21,37
One important application of our present work of controlled release of small molecules from single cells is in single-cell analysis. The analysis of cytoplasmic contents of single cells using microfluidics and separations has become an important research area.58–63 This is because highly efficient and sensitive detection of the components in a single cell will explain important physiological processes and therefore improve our understanding of basic cellular processes. Generally, although not always,64,65 in the process of single-cell analysis, the cell is destroyed. It would be helpful to be able to sample a cell's cytoplasm over time. Our present work is a major step in this direction as the ability to predict/control small molecular release will provide a better handle at reproducibility in single-cell small molecular release experiments.
This work was financially supported by the National Institutes of Health (Grant R01 GM066018), the Foundation for Strategic Research (SSF), Vetenskapsrådet (VR), and the Göran Gustafsson Foundation to whom we are grateful.
SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.