The current-voltage relation for electropores with conductivity gradients

Long-term changes in transmembrane voltage after electroporation are governed by the interplay between nonselective leak current and ion channel activation

Electroporation causes a temporal increase in cell membrane permeability and leads to prolonged changes in transmembrane voltage (TMV) in both excitable and non-excitable cells. However, the mechanisms of these TMV changes remain to be fully elucidated. To this end, we monitored TMV over 30 min after exposing two different cell lines to a single 100 µs electroporation pulse using the FLIPR Membrane Potential dye. In CHO-K1 cells, which express very low levels of endogenous ion channels, membrane depolarization following pulse exposure could be explained by nonselective leak current, which persists until the membrane reseals, enabling the cells to recover their resting TMV. In U-87 MG cells, which express many different ion channels, we unexpectedly observed membrane hyperpolarization following the initial depolarization phase, but only at 33 °C and not at 25 °C. We developed a theoretical model, supported by experiments with ion channel inhibitors, which indicated that hyperpolarization could largely be attributed to the activation of calcium-activated potassium channels. Ion channel activation, coupled with changes in TMV and intracellular calcium, participates in various physiological processes, including cell proliferation, differentiation, migration, and apoptosis. Therefore, our study suggests that ion channels could present a potential target for influencing the biological response after electroporation.

Gene Electrotransfer into Mammalian Cells Using Commercial Cell Culture Inserts with Porous Substrate

Gene electrotransfer is one of the main non-viral methods for intracellular delivery of plasmid DNA, wherein pulsed electric fields are used to transiently permeabilize the cell membrane, allowing enhanced transmembrane transport. By localizing the electric field over small portions of the cell membrane using nanostructured substrates, it is possible to increase considerably the gene electrotransfer efficiency while preserving cell viability. In this study, we expand the frontier of localized electroporation by designing an electrotransfer approach based on commercially available cell culture inserts with polyethylene-terephthalate (PET) porous substrate. We first use multiscale numerical modeling to determine the pulse parameters, substrate pore size, and other factors that are expected to result in successful gene electrotransfer. Based on the numerical results, we design a simple device combining an insert with substrate containing pores with 0.4 µm or 1.0 µm diameter, a multiwell plate, and a pair of wire electrodes. We test the device in three mammalian cell lines and obtain transfection efficiencies similar to those achieved with conventional bulk electroporation, but at better cell viability and with low-voltage pulses that do not require the use of expensive electroporators. Our combined theoretical and experimental analysis calls for further systematic studies that will investigate the influence of substrate pore size and porosity on gene electrotransfer efficiency and cell viability.

Molecular dynamics simulations of the effects of lipid oxidation on the permeability of cell membranes

The formation of transient pores in their membranes is a well-known mechanism of permeabilization of cells exposed to high-intensity electric pulses. However, the formation of such pores is not able to explain all aspects of the so-called electroporation phenomenon. In particular, the reasons for sustained permeability of cell membranes, persisting long after the pulses' application, remain elusive. The complete resealing of cell membranes takes indeed orders of magnitude longer than the time for electropore closure as reported from molecular dynamics (MD) investigations. Lipid peroxidation has been suggested as a possible mechanism to explain the sustainable permeability of cell membranes. However, theoretical investigations of membrane lesions containing excess amounts of hydroperoxides have shown that the conductivities of such lesions were not high enough to account for the experimental measurements. Here, expanding on these studies, we investigate quantitatively the permeability of cell membrane lesions that underwent secondary oxidation. MD simulations and free energy calculations of lipid bilayers show that such lesions provide a better model of post-pulse permeable and conductive electropermeabilized cells. These results are further discussed in the context of sonoporation and ferroptosis, respectively a procedure and a phenomenon, among others, in which, alike electroporation, substantial lipid oxidation might be triggered.

Single-cell mechanical analysis and tension quantification via electrodeformation relaxation

The mechanical behavior and cortical tension of single cells are analyzed using electrodeformation relaxation. Four types of cells, namely, MCF-10A, MCF-7, MDA-MB-231, and GBM, are studied, with pulse durations ranging from 0.01 to 10 s. Mechanical response in the long-pulse regime is characterized by a power-law behavior, consistent with soft glassy rheology resulting from unbinding events within the cortex network. In the subsecond short-pulse regime, a single timescale well describes the process and indicates the naive tensioned (prestressed) state of the cortex with minimal force-induced alteration. A mathematical model is employed and the simple ellipsoidal geometry allows for use of an analytical solution to extract the cortical tension. At the shortest pulse of 0.01 s, tensions for all four cell types are on the order of 10^{-2} N/m.

Membrane Electroporation and Electropermeabilization: Mechanisms and Models

Exposure of biological cells to high-voltage, short-duration electric pulses causes a transient increase in their plasma membrane permeability, allowing transmembrane transport of otherwise impermeant molecules. In recent years, large steps were made in the understanding of underlying events. Formation of aqueous pores in the lipid bilayer is now a widely recognized mechanism, but evidence is growing that changes to individual membrane lipids and proteins also contribute, substantiating the need for terminological distinction between electroporation and electropermeabilization. We first revisit experimental evidence for electrically induced membrane permeability, its correlation with transmembrane voltage, and continuum models of electropermeabilization that disregard the molecular-level structure and events. We then present insights from molecular-level modeling, particularly atomistic simulations that enhance understanding of pore formation, and evidence of chemical modifications of membrane lipids and functional modulation of membrane proteins affecting membrane permeability. Finally, we discuss the remaining challenges to our full understanding of electroporation and electropermeabilization.

Characterization of Cell Membrane Permeability In Vitro Part II: Computational Model of Electroporation-Mediated Membrane Transport

Electroporation is the process by which applied electric fields generate nanoscale defects in biological membranes to more efficiently deliver drugs and other small molecules into the cells. Due to the complexity of the process, computational models of cellular electroporation are difficult to validate against quantitative molecular uptake data. In part I of this two-part report, we describe a novel method for quantitatively determining cell membrane permeability and molecular membrane transport using fluorescence microscopy. Here, in part II, we use the data from part I to develop a two-stage ordinary differential equation model of cellular electroporation. We fit our model using experimental data from cells immersed in three buffer solutions and exposed to electric field strengths of 170 to 400 kV/m and pulse durations of 1 to 1000 μs. We report that a low-conductivity 4-(2-hydroxyethyl)-1 piperazineethanesulfonic acid buffer enables molecular transport into the cell to increase more rapidly than with phosphate-buffered saline or culture medium-based buffer. For multipulse schemes, our model suggests that the interpulse delay between two opposite polarity electric field pulses does not play an appreciable role in the resultant molecular uptake for delays up to 100 μs. Our model also predicts the per-pulse permeability enhancement decreases as a function of the pulse number. This is the first report of an ordinary differential equation model of electroporation to be validated with quantitative molecular uptake data and consider both membrane permeability and charging.

Numerical study of the effect of soft layer properties on bacterial electroporation

We present a numerical model of electroporation in a gram-positive bacterium, which accounts for the presence of a negatively charged soft polyelectrolyte layer (which may include a periplasmic space, peptidoglycan layer, cilia, flagella, and other surface appendages) surrounding its plasma membrane. We model the ion transport within and outside the soft layer using the soft layer electrokinetics-based Poisson-Nernst-Planck formalism. Additionally, we model the electroporation dynamics on the plasma membrane using the pore nucleation-based electroporation formalism developed by Krassowska and Filev. We find that ion transport within the soft layer (surface conduction), which depends on the relative importance of the soft layer charged group concentration compared to the buffer concentration, significantly alters the transmembrane voltage across the plasma membrane and hence the pore characteristics. Our numerical simulations suggest that surface conduction significantly lowers the pore number in the plasma membrane. This observation is consistent with experimental studies that show that gram-positive bacteria, in general, have lower transformation efficiencies compared to gram-negative bacteria. Our studies highlight a strong dependence of bacterial electroporation on cell envelope properties and buffer conditions, which need to be taken into consideration when designing electroporation protocols.

Theoretical Study of Molecular Transport Through a Permeabilized Cell Membrane in a Microchannel

A two-dimensional model is developed to study the molecular transport into an immersed cell in a microchannel and to investigate the effects of finite boundary (a cell is suspended in a microchannel), amplitude of electric pulse, and geometrical parameter (microchannel height and size of electrodes) on cell uptake. Embedded electrodes on the walls of the microchannel generate the required electric pulse to permeabilize the cell membrane, pass the ions through the membrane, and transport them into the cell. The shape of electric pulses is square with the time span of 6 ms; their intensities are in the range of 2.2, 2.4, 2.6, 3 V. Numerical simulations have been performed to comprehensively investigate the molecular uptake into the cell. The obtained results of the current study demonstrate that calcium ions enter the cell from the anodic side (the side near positive electrode); after a while, the cell faces depletion of the calcium ions on a positive electrode-facing side within the microchannel; the duration of depletion depends on the amplitude of electric pulse and geometry that lasts from microseconds to milliseconds. By keeping geometrical parameters and time span constant, increment of a pulse intensity enhances molecular uptake and rate of propagation inside the cell. If a ratio of electrode size to cell diameter is larger than 1, the transported amount of Ca ²⁺ into the cell, as well as the rate of propagation, will be significantly increased. By increasing the height of the microchannel, the rate of uptake is decreased. In an infinite domain, the peak concentration becomes constant after reaching the maximum value; this value depends on the intra-extracellular conductivity and diffusion coefficient of interior and exterior domains of the cell. In comparison, the maximum concentration is changed by geometrical parameters in the microchannel. After reaching the maximum value, the peak concentration reduces due to the depletion of Ca ²⁺ ions within the microchannel. Electrophoretic velocity has a significant effect on the cell uptake.

The Influence of Vesicle Shape and Medium Conductivity on Possible Electrofusion under a Pulsed Electric Field

The effects of electric field on lipid membrane and cells have been extensively studied in the last decades. The phenomena of electroporation and electrofusion are of particular interest due to their wide use in cell biology and biotechnology. However, numerical studies on the electrofusion of cells (or vesicles) with different deformed shapes are still rare. Vesicle, being of cell size, can be treated as a simple model of cell to investigate the behaviors of cell in electric field. Based on the finite element method, we investigate the effect of vesicle shape on electrofusion of contact vesicles in various medium conditions. The transmembrane voltage (TMV) and pore density induced by a pulsed field are examined to analyze the possibility of vesicle fusion. In two different medium conditions, the prolate shape is observed to have selective electroporation at the contact area of vesicles when the exterior conductivity is smaller than the interior one; selective electroporation is more inclined to be found at the poles of the oblate vesicles when the exterior conductivity is larger than the interior one. Furthermore, we find that when the exterior conductivity is lower than the internal conductivity, the pulse can induce a selective electroporation at the contact area between two vesicles regardless of the vesicle shape. Both of these two findings have important practical applications in guiding electrofusion experiments.

Dependence of Electroporation Detection Threshold on Cell Radius: An Explanation to Observations Non Compatible with Schwan's Equation Model

It is widely accepted that electroporation occurs when the cell transmembrane voltage induced by an external applied electric field reaches a threshold. Under this assumption, in order to trigger electroporation in a spherical cell, Schwan's equation leads to an inversely proportional relationship between the cell radius and the minimum magnitude of the applied electric field. And, indeed, several publications report experimental evidences of an inverse relationship between the cell size and the field required to achieve electroporation. However, this dependence is not always observed or is not as steep as predicted by Schwan's equation. The present numerical study attempts to explain these observations that do not fit Schwan's equation on the basis of the interplay between cell membrane conductivity, permeability, and transmembrane voltage. For that, a single cell in suspension was modeled and the electric field necessary to achieve electroporation with a single pulse was determined according to two effectiveness criteria: a specific permeabilization level, understood as the relative area occupied by the pores during the pulse, and a final intracellular concentration of a molecule due to uptake by diffusion after the pulse, during membrane resealing. The results indicate that plausible model parameters can lead to divergent dependencies of the electric field threshold on the cell radius. These divergent dependencies were obtained through both criteria and using two different permeabilization models. This suggests that the interplay between cell membrane conductivity, permeability, and transmembrane voltage might be the cause of results which are noncompatible with the Schwan's equation model.

Impact of external medium conductivity on cell membrane electropermeabilization by microsecond and nanosecond electric pulses

The impact of external medium conductivity on the efficiency of the reversible permeabilisation caused by pulsed electric fields was investigated. Pulses of 12 ns, 102 ns or 100 μs were investigated. Whenever permeabilisation could be detected after the delivery of one single pulse, media of lower conductivity induced more efficient reversible permeabilisation and thus independently of the medium composition. Effect of medium conductivity can however be hidden by some saturation effects, for example when pulses are cumulated (use of trains of 8 pulses) or when the detection method is not sensitive enough. This explains the contradicting results that can be found in the literature. The new data are complementary to those of one of our previous study in which an opposite effect of the conductivity was highlighted. It stresses that the conductivity of the medium influences the reversible permeabilization by several ways. Moreover, these results clearly indicate that electropermeabilisation does not linearly depend on the energy delivered to the cells.

Properties of lipid electropores I: Molecular dynamics simulations of stabilized pores by constant charge imbalance

Molecular dynamics (MD) simulations have become a powerful tool to study electroporation (EP) in atomic detail. In the last decade, numerous MD studies have been conducted to model the effect of pulsed electric fields on membranes, providing molecular models of the EP process of lipid bilayers. Here we extend these investigations by modeling for the first time conditions comparable to experiments using long (μs-ms) low intensity (~kV/cm) pulses, by studying the characteristics of pores formed in lipid bilayers maintained at a constant surface tension and subject to constant charge imbalance. This enables the evaluation of structural (size) and electrical (conductance) properties of the pores formed, providing information hardly accessible directly by experiments. Extensive simulations of EP of simple phosphatidylcholine bilayers in 1M NaCl show that hydrophilic pores with stable radii (1-2.5 nm) form under transmembrane voltages between 420 and 630 mV, allowing for ionic conductance in the range of 6.4-29.5 nS. We discuss in particular these findings and characterize both convergence and size effects in the MD simulations. We further extend these studies in a follow-up paper (Rems et al., Bioelectrochemistry, Submitted), by proposing an improved continuum model of pore conductance consistent with the results from the MD simulations.

Modelling the Bioelectronic Interface in Engineered Tethered Membranes: From Biosensing to Electroporation

This paper studies the construction and predictive models of three novel measurement platforms: (i) a Pore Formation Measurement Platform (PFMP) for detecting the presence of pore forming proteins and peptides, (ii) the Ion Channel Switch (ICS) biosensor for detecting the presence of analyte molecules in a fluid chamber, and (iii) an Electroporation Measurement Platform (EMP) that provides reliable measurements of the electroporation phenomenon. Common to all three measurement platforms is that they are comprised of an engineered tethered membrane that is formed via a rapid solvent exchange technique allowing the platform to have a lifetime of several months. The membrane is tethered to a gold electrode bioelectronic interface that includes an ionic reservoir separating the membrane and gold surface, allowing the membrane to mimic the physiological response of natural cell membranes. The electrical response of the PFMP, ICS, and EMP are predicted using continuum theories for electrodiffusive flow coupled with boundary conditions for modelling chemical reactions and electrical double layers present at the bioelectronic interface. Experimental measurements are used to validate the predictive accuracy of the dynamic models. These include using the PFMP for measuring the pore formation dynamics of the antimicrobial peptide PGLa and the protein toxin Staphylococcal α-Hemolysin; the ICS biosensor for measuring nano-molar concentrations of streptavidin, ferritin, thyroid stimulating hormone (TSH), and human chorionic gonadotropin (pregnancy hormone hCG); and the EMP for measuring electroporation of membranes with different tethering densities, and membrane compositions.

Cell scale modeling of electropermeabilization by periodic pulses

In this paper, we focus on the behaviour of periodic solutions to a cell-scale electropermeabilization model previously proposed by Kavian et al. [6]. Since clinical permeabilization protocols mostly submit cancer cells to trains of periodic pulses, we investigate on parameters that modify significantly the resulting permeabilization. Theoretical results of existence and uniqueness of periodic solutions are presented, for two different models of membrane electric conductivity. Numerical simulations were performed to corroborate these results and illustrate the asymptotic convergence to periodic solutions, as well as the dependency on biological parameters such as the cell size and the extracellular conductivity.

An engineered membrane to measure electroporation: effect of tethers and bioelectronic interface

This article reports on the construction and predictive models for a platform comprised of an engineered tethered membrane. The platform provides a controllable and physiologically relevant environment for the study of the electroporation process. The mixed self-assembled membrane is formed via a rapid solvent exchange technique. The membrane is tethered to the gold electrode and includes an ionic reservoir separating the membrane and gold surface. Above the membrane, there is an electrolyte solution, and a gold counterelectrode. A voltage is applied between the gold electrodes and the current measured. The current is dependent on the energy required to form aqueous pores and the conductance of each pore. A two-level predictive model, consisting of a macroscopic and a continuum model, is developed to relate the pore dynamics to the measured current. The macroscopic model consists of an equivalent circuit model of the tethered membrane, and asymptotic approximations to the Smoluchowski-Einstein equation of electroporation that is dependent on the pore conductance and the energy required to form aqueous pores. The continuum model is a generalized Poisson-Nernst-Planck (GPNP) system where an activity coefficient to account for steric effects of ions is added to the standard PNP system. The GPNP is used to evaluate the conductance of aqueous pores, and the electrical energy required to form the pores. As an outcome of the setup of the device and the two-level model, biologically important variables can be estimated from experimental measurements. To validate the accuracy of the two-level model, the predicted current is compared with experimentally measured current for different tethering densities.

Scaling relationship and optimization of double-pulse electroporation

The efficacy of electroporation is known to vary significantly across a wide variety of biological research and clinical applications, but as of this writing, a generalized approach to simultaneously improve efficiency and maintain viability has not been available in the literature. To address that discrepancy, we here outline an approach that is based on the mapping of the scaling relationships among electroporation-mediated molecular delivery, cellular viability, and electric pulse parameters. The delivery of Fluorescein-Dextran into 3T3 mouse fibroblast cells was used as a model system. The pulse was rationally split into two sequential phases: a first precursor for permeabilization, followed by a second one for molecular delivery. Extensive data in the parameter space of the second pulse strength and duration were collected and analyzed with flow cytometry. The fluorescence intensity correlated linearly with the second pulse duration, confirming the dominant role of electrophoresis in delivery. The delivery efficiency exhibited a characteristic sigmoidal dependence on the field strength. An examination of short-term cell death using 7-Aminoactinomycin D demonstrated a convincing linear correlation with respect to the electrical energy. Based on these scaling relationships, an optimal field strength becomes identifiable. A model study was also performed, and the results were compared with the experimental data to elucidate underlying mechanisms. The comparison reveals the existence of a critical transmembrane potential above which delivery with the second pulse becomes effective. Together, these efforts establish a general route to enhance the functionality of electroporation.

Cell electrofusion using nanosecond electric pulses

Electrofusion is an efficient method for fusing cells using short-duration high-voltage electric pulses. However, electrofusion yields are very low when fusion partner cells differ considerably in their size, since the extent of electroporation (consequently membrane fusogenic state) with conventionally used microsecond pulses depends proportionally on the cell radius. We here propose a new and innovative approach to fuse cells with shorter, nanosecond (ns) pulses. Using numerical calculations we demonstrate that ns pulses can induce selective electroporation of the contact areas between cells (i.e. the target areas), regardless of the cell size. We then confirm experimentally on B16-F1 and CHO cell lines that electrofusion of cells with either equal or different size by using ns pulses is indeed feasible. Based on our results we expect that ns pulses can improve fusion yields in electrofusion of cells with different size, such as myeloma cells and B lymphocytes in hybridoma technology.

Preface to Special Topic: Papers from the 13th International Conference on Surface and Colloid Science (ICSCS) and the 83rd ACS Colloid and Surface Science Symposium, Columbia University, New York, 2009

This Special Topic section is a compilation of several original contributions covering both fundamental and practical aspects of electrokinetic microfluidic phenomena that were presented during the Electrokinetics and Microfluidics sessions held at the conference.