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.     

Effect of electric field induced transmembrane potential on spheroidal cells: theory and experiment

The transmembrane potential on a cell exposed to an electric field is a critical parameter for successful cell permeabilization. In this study, the effect of cell shape and orientation on the induced transmembrane potential was analyzed. The transmembrane potential was calculated on prolate and oblate spheroidal cells for various orientations with respect to the electric field direction, both numerically and analytically. Changing the orientation of the cells decreases the induced transmembrane potential from its maximum value when the longest axis of the cell is parallel to the electric field, to its minimum value when the longest axis of the cell is perpendicular to the electric field. The dependency on orientation is more pronounced for elongated cells while it is negligible for spherical cells. The part of the cell membrane where a threshold transmembrane potential is exceeded represents the area of electropermeabilization, i.e. the membrane area through which the transport of molecules is established. Therefore the surface exposed to the transmembrane potential above the threshold value was calculated. The biological relevance of these theoretical results was confirmed with experimental results of the electropermeabilization of plated Chinese hamster ovary cells, which are elongated. Theoretical and experimental results show that permeabilization is not only a function of electric field intensity and cell size but also of cell shape and orientation.     

Electric Field-Induced Transmembrane Potential Depends on Cell Density and Organizatio

Electrochemotherapy is a novel technique to enhance the delivery of chemotherapeutic drugs into tumor cells. In this procedure, electric pulses are delivered to cancerous cells, which induce membrane permeabilization, to facilitate the passage of cytotoxic drugs through the cell membrane. This study examines how electric fields interact with and polarize a system of cells. Specifically, we consider how cell density and organization impact on induced cell transmembrane potential due to an external electric field. First, in an infinite volume of spherical cells, we examined how cell packing density impacts on induced transmembrane potential. With high cell density, we found that maximum induced transmembrane potential is suppressed and that the transmembrane potential distribution is altered. Second, we considered how orientation of cell sheets and strands, relative to the applied field, affects induced transmembrane potential. Cells that are parallel to the field direction suppress induced transmembrane potential, and those that lie perpendicular to the applied field potentiate its effect. Generally, we found that both cell density and cell organization are very important in determining the induced transmembrane potential resulting from an applied electric field.     

Effects of oscillatory electric fields on internal membranes: an analytical model

We derive an analytical model of the potential differences induced across plasma and internal organelle membranes in suspended cells exposed to oscillatory electric fields. Multiple shells are modeled using iterative applications of the single-shell calculation with mobile charges. This work is motivated, in part, by recent results suggesting the ability to use alternating current (ac) fields to noninvasively monitor enzyme activity within internal membranes, particularly the mitochondrial electron transport chain. Previous work, on induced transmembrane voltages in cells subjected to ac fields, has mainly been limited to oscillatory potentials across the plasma membrane. Here we first develop a three-membrane model, consisting of a plasma membrane surrounding inner and outer membranes representing an internal organelle, such as a mitochondrion. Frequency-dependent transmembrane potentials are modeled for spherical, weakly conducting membrane shells enclosing a conductive cytoplasm surrounding an idealized internal organelle. We then use a two-shell model to simulate induced ac membrane potentials of a suspended isolated mitochondrion in which the outer membrane is usually much more permeable than the inner membrane.     

Current noise parameters derived from voltage noise and impedance in embryonic heart cell aggregates.

We have recorded membrane impedance and voltage noise in the pacemaker range of potentials (-70 to -59 mV) from spheroidal aggregates of 7-d embryonic chick ventricle cells made quiescent by exposure to tetrodotoxin in medium containing 4.5 mM K+. The input capacitance is proportional to aggregate volume and therefore to total membrane area. The specific membrane capacitance is 1.24 microF/cm2. The input resistance at constant potential is inversely proportional to aggregate volume and therefore to total membrane area. The specific membrane resistance in 18 k omega . cm2 at -70 mV and increases to 81 k omega . cm2 at -59 mV. The RC time constant is 22 ms at -70 mV and increases to 146 ms at -59 mV. The aggregate transmembrane small-signal impedance can be represented by a parallel RC circuit itself in parallel with an inductive branch consisting of a resistor (rL) and an inductor (L) in series. The time constant of the inductive branch (L/rL) is 340 ms, and is only weakly dependent on potential. Correlation functions of aggregate voltage noise and the impedance data were modeled by a population of channels with simple open-close kinetics. The time constant of a channel (tau s) derived from the noise analysis is 300 ms. The low frequency limit of the pacemaker current noise (SI[0]), derived from the voltage noise and impedance, increases from 10(-20) A2/Hz . cm2 at -67 mV to 10(-19) A2/Hz . cm2 at -61 mV.     

Membrane response to current pulses in spheroidal aggregates of embryonic heart cells

Hearts from chick embryos aged 4,7, or 14 days were dissociated into their component cells, and the cells allowed to reassociate in the form of smooth-surfaced spheroidal aggregates on a gyratory shaker. Records from intracellular electrodes inserted into two widely spaced cells in a spontaneously beating aggregate indicated that the action potentials occurred virtually simultaneously. In aggregates made quiescent with tetrodotoxin, the voltage response to a current pulse injected in one cell could be noted by recording with a second microelectrode at various distance from the current source. The magnitude of the response was found not to vary with distance. It is concluded that the component cells in an aggregate are normally tightly coupled electrically; the cell boundaries do not constitute an appreciable resistive barrier. Such ag-regates behave as virtually isopential systems, with properties similar to those of single spherical cells, as modeled by Eisenberg and Engel (1970. J. Gen. Physiol. 55:736-757). Passive membrane time constant ranged from 11 to 31 ms, with a mean value of 17 ms; this value did not vary with aggregate size. Input resistance (V/I) varied inversely with aggregate size, as predicted, but with much scatter in the measured values. Specific membrane resistance was calculated as either 13,000 or 800 ohm-cm2 depending on whether input resistance was attributed to the total cell surface membrane area or to the outer surface of the sphere alone. No systematic difference in passive electrical properties of aggregates composed of 4-, 7-, and 14-day cells was seen. It is concluded that these aggregates may be suitable for voltage clamp analysis of their excitable membrane properties.     

Nonlinear Dispersive Model of Electroporation for Irregular Nucleated Cells

In this work, the electroporation phenomenon induced by pulsed electric field on different nucleated biological cells is studied. A nonlinear, non‐local, dispersive, and space–time multiphysics model based on Maxwell’s and asymptotic Smoluchowski’s equations has been developed to calculate the transmembrane voltage and pore density on both plasma and nuclear membrane perimeters. The irregular cell shape has been modeled by incorporating in the numerical algorithm the analytical functions pertaining to Gielis curves. The dielectric dispersion of the cell media has been modeled considering the multi‐relaxation Debye‐based relationship. Two different irregular nucleated cells have been investigated and their response has been studied applying both the dispersive and non‐dispersive models. By a comparison of the obtained results, differences can be highlighted confirming the need to make use of the dispersive model to effectively investigate the cell response in terms of transmembrane voltages, pore densities, and electroporation opening angle, especially when irregular cell shapes and short electric pulses are considered. Bioelectromagnetics. 2019;40:331–342. © 2019 Wiley Periodicals, Inc.     

Non-uniform Distribution of Outer Hair Cell Transmembrane Potential Induced by Extracellular Electric Field

Intracochlear electric fields arising out of sound-induced receptor currents, silent currents, or electrical current injected into the cochlea induce transmembrane potential along the outer hair cell (OHC) but its distribution along the cells is unknown. In this study, we investigated the distribution of OHC transmembrane potential induced along the cell perimeter and its sensitivity to the direction of the extracellular electric field (EEF) on isolated OHCs at a low frequency using the fast voltage-sensitive dye ANNINE-6plus. We calibrated the potentiometric sensitivity of the dye by applying known voltage steps to cells by simultaneous whole-cell voltage clamp. The OHC transmembrane potential induced by the EEF is shown to be highly nonuniform along the cell perimeter and strongly dependent on the direction of the electrical field. Unlike in many other cells, the EEF induces a field-direction-dependent intracellular potential in the cylindrical OHC. We predict that without this induced intracellular potential, EEF would not generate somatic electromotility in OHCs. In conjunction with the known heterogeneity of OHC membrane microdomains, voltage-gated ion channels, charge, and capacitance, the EEF-induced nonuniform transmembrane potential measured in this study suggests that the EEF would impact the cochlear amplification and electropermeability of molecules across the cell.     

Non-uniform Distribution of Outer Hair Cell Transmembrane Potential Induced by Extracellular Electric Field

Intracochlear electric fields arising out of sound-induced receptor currents, silent currents, or electrical current injected into the cochlea induce transmembrane potential along the outer hair cell (OHC) but its distribution along the cells is unknown. In this study, we investigated the distribution of OHC transmembrane potential induced along the cell perimeter and its sensitivity to the direction of the extracellular electric field (EEF) on isolated OHCs at a low frequency using the fast voltage-sensitive dye ANNINE-6plus. We calibrated the potentiometric sensitivity of the dye by applying known voltage steps to cells by simultaneous whole-cell voltage clamp. The OHC transmembrane potential induced by the EEF is shown to be highly nonuniform along the cell perimeter and strongly dependent on the direction of the electrical field. Unlike in many other cells, the EEF induces a field-direction-dependent intracellular potential in the cylindrical OHC. We predict that without this induced intracellular potential, EEF would not generate somatic electromotility in OHCs. In conjunction with the known heterogeneity of OHC membrane microdomains, voltage-gated ion channels, charge, and capacitance, the EEF-induced nonuniform transmembrane potential measured in this study suggests that the EEF would impact the cochlear amplification and electropermeability of molecules across the cell.     

Measuring the Induced Membrane Voltage with Di-8-ANEPPS

Placement of a cell into an external electric field causes a local charge redistribution inside and outside of the cell in the vicinity of the cell membrane, resulting in a voltage across the membrane. This voltage, termed the induced membrane voltage (also induced transmembrane voltage, or induced transmembrane potential difference) and denoted by ΔΦ, exists only as long as the external field is present. If the resting voltage is present on the membrane, the induced voltage superimposes (adds) onto it. By using one of the potentiometric fluorescent dyes, such as di-8-ANEPPS, it is possible to observe the variations of ΔΦ on the cell membrane and to measure its value noninvasively. di-8-ANEPPS becomes strongly fluorescent when bound to the lipid bilayer of the cell membrane, with the change of the fluorescence intensity proportional to the change of ΔΦ. This video shows the protocol for measuring ΔΦ using di-8-ANEPPS and also demonstrates the influence of cell shape on the amplitude and spatial distribution of ΔΦ.     

A polarization model overcoming the geometric restrictions of the laplace solution for spheroidal cells: obtaining new equations for field-induced forces and transmembrane potential.

We present a new model for a variety of electric polarization effects on oblate and prolate homogeneous and single-shell spheroids. For homogeneous spheroids the model is identical to the Laplace model. For single-shell spheres of cell-like geometry the calculated difference of the induced dipole moments is in the thousandths range. To solve Laplace's equation for nonspherical single-shell objects it is necessary to assume a confocal shell, which results in different cell membrane properties in the pole and equator regions, respectively. Our alternative model addresses this drawback. It assumes that the disturbance of the external field due to polarization may project into the medium to a characteristic distance, the influential radius. This parameter is related to the axis ratio of the spheroid over the depolarizing factors and allows us to determine the geometry for a finite resistor-capacitor model. From this model the potential at the spheroid's surface is obtained and, consequently, the local field inside a homogeneous spheroid is determined. In the single-shell case, this is the effective local field of an equivalent homogeneous spheroid. Finally, integration over the volume yields the frequency-dependent induced dipole moment. The resistor-capacitor approach allowed us to find simple equations for the critical and characteristic frequencies, force plateaus and peak heights of deformation, dielectrophoresis and electrorotation for homogeneous and single-shell spheroids, and a more generalized equation for the induced transmembrane potential of spheroidal cells.     

Transmembrane voltage induced on altered erythrocyte shapes exposed to RF fields

In this article, the transmembrane voltage induced on erythrocyte, codocyte, ovalocyte and spherocyte cell models exposed to a linearly polarised electromagnetic plane wave of frequency 1800 MHz is calculated. For this purpose, a finite element (FE) numerical technique with adaptive meshing is used. The results show that the value of the induced voltage on the original erythrocyte shape is higher than the one observed on the rest of the altered cell geometries studied. The erythrocyte shape and the membrane electric permittivity are shown to play a fundamental role on the values of the induced transmembrane voltage.     

Electroinsertion and activation of the C-terminal domain of colicin A, a voltage gated bacterial toxin, into mammalian cell membranes

The C-terminal fragment of colicin, a protein that is highly soluble in aqueous solution, is spontaneously and irreversibly inserted into the membranes of mammalian cells, which are locally permeabilized by a transmembrane voltage increase. Insertion is detected by immunodetection. This is obtained by mixing the protein with electropermeabilized cells. The same result is observed by pulsing the colicin/cell mixture. Electroinsertion is therefore obtained for the first time with a multi-fragment spanning protein. The cell viability is not affected beyond the effect of electropermeabilization. A train of low voltage repetitive transmembrane modulation, which cannot trigger membrane permeabilization, is applied a day after the electroinsertion. This induces no effect on unmodified cells but triggers the lysis of cells in which colicin has been inserted by the first electropulsation. The low-level electrical treatment is high enough to trigger the voltage gated opening of colicin and to induce the associated toxicity. A transmembrane configuration of colicin is therefore obtained by electroinsertion. The toxic effect of their voltage gating is only obtained when a critical number of voltage gated channels are activated.     

Passive Membrane Potentials: A Generalization of the Theory of Electrotonus

The theory of electrotonus, which has been well developed for small cylinders, is extended: the fundamental potential equations for a membrane of arbitrary shape are derived, and solutions are found for cylindrical and spherical geometries. If two purely conductive media are separated by a resistance-capacitance membrane, then Laplace's equation describes the potential in either medium, and two boundary equations relate the transmembrane potential to applied currents and to currents flowing into the membrane from each medium. The core conductor model, on which most previous work on cylindrical electrotonus has been based, gives rise to a one dimensional diffusion equation, the cable equation, for the transmembrane potential in a small cylinder. Under the assumptions of the core conductor model the more general equations developed here are shown to reduce to the cable equation. The two theories agree well in predicting the transmembrane potential in a small cylinder owing to an applied current step, and the extracellular potential for this cylinder is estimated numerically from the general theory. A detailed proof is given for the isopotentiality of a spherical soma membrane.     

Force versus axial deflection of pipette-aspirated closed membranes

The axial deformation of a pipette-pressurized fluid membrane bag produces minuscule yet well-defined, reproducible forces. The stiffness of this ultrasensitive force transducer is tunable and largely independent of the constitutive membrane behavior. Based on a rigorous variational treatment, we present both numerical as well as approximate analytical solutions for the force-deflection relation of this unique biophysical force probe. Our numerical results predict a measurably nonlinear force-deflection behavior at moderate-to-large deformations, which we confirm experimentally using red blood cells. Furthermore, considering nearly spherical membrane shapes and enforcing proper boundary conditions, we derive an analytical solution valid at small deformations. In this linear regime the pressurized membrane bag behaves like a Hookean spring, with a spring constant that is significantly larger than previously published for the biomembrane force probe.     

Induced Transmembrane Voltage and Its Correlation with Electroporation-Mediated Molecular Transport

Exposure of a cell to an electric field results in inducement of a voltage across its membrane (induced transmembrane voltage, ΔΨ _m) and, for sufficiently strong fields, in a transient increase of membrane permeability (electroporation). We review the analytical, numerical and experimental methods for determination of ΔΨ _m and a method for monitoring of transmembrane transport. We then combine these methods to investigate the correlation between ΔΨ _m and molecular transport through an electroporated membrane for isolated cells of regular and irregular shapes, for cells in dense suspensions as well as for cells in monolayer clusters. Our experiments on isolated cells of both regular and irregular shapes confirm the theoretical prediction that the highest absolute values of ΔΨ _m are found in the membrane regions facing the electrodes and that electroporation-mediated transport is confined to these same regions. For cells in clusters, the location of transport regions implies that, at the field strengths sufficient for electroporation, the cells behave as electrically insulated (i.e., as individual) cells. In contrast, with substantially weaker, nonelectroporating fields, potentiometric measurements show that the cells in these same clusters behave as electrically interconnected cells (i.e., as one large cell). These results suggest that sufficiently high electric fields affect the intercellular pathways and thus alter the electric behavior of the cells with respect to their normal physiological state.