Thiruvallur R. Gowrishankar

An approach to electrical modeling of single and multiple cells

Previous theoretical approaches to understanding effects of electric fields on cells have used partial differential equations such as Laplaces equation and cell models with simple shapes. Here we describe a transport lattice method illustrated by a didactic multicellular system model with irregular shapes. Each elementary membrane region includes local models for passive membrane resistance and capacitance, nonlinear active sources of the resting potential, and a hysteretic model of electroporation. Field amplification through current or voltage concentration changes with frequency, exhibiting significant spatial heterogeneity until the microwave range is reached, where cellular structure becomes almost “electrically invisible.” In the time domain, membrane electroporation exhibits significant heterogeneity but occurs mostly at invaginations and cell layers with tight junctions. Such results involve emergent behavior and emphasize the importance of using multicellular models for understanding tissue-level electric field effects in higher organisms.

A brief overview of electroporation pulse strength-duration space: a region where additional intracellular effects are expected.

Electroporation (EP) of outer cell membranes is widely used in research, biotechnology and medicine. Now intracellular effects by organelle EP are of growing interest, mainly due to nanosecond pulsed electric fields (nsPEF). For perspective, here we provide an approximate overview of EP pulse strength-duration space. This overview locates approximately some known effects and applications in strength-duration space, and includes a region where additional intracellular EP effects are expected. A feature of intracellular EP is direct, electrical redistribution of endogenous biochemicals among cellular compartments. For example, intracellular EP may initiate a multistep process for apoptosis. In this hypothesis, initial EP pulses release calcium from the endoplasmic reticulum, followed by calcium redistribution within the cytoplasm. With further EP pulses calcium penetrates mitochondrial membranes and causes changes that trigger release of cytochrome c and other death molecules. Apoptosis may therefore occur even in the presence of apoptotic inhibitors, using pulses that are smaller, but longer, than nsPEF.

Transport lattice approach to describing cell electroporation: use of a local asymptotic model

Electroporation has been widely used to manipulate cells and tissues, but quantitative understanding of electrical behavior in cell membranes has not been achieved. According to the transient aqueous pore hypothesis, pore creation and expansion is a nonlinear, hysteretic process. Different membrane sites respond locally to their own transmembrane voltage history, so that a self-consistent description should involve the interaction of many different regions of a cell membrane model and its aqueous electrolytes. A transport lattice system model of a cell allows active and passive interaction models for local transport and storage of charge to be combined, yielding approximate solutions for this highly interacting system. Here, we use an asymptotic model for local membrane electroporation, which involves solving an ordinary differential equation for each local membrane area of the system model, subject to constraints imposed by self-consistency throughout the system model of the cell. To illustrate this approach, we first treat a model for a space- and voltage-clamped skeletal muscle cell. We then create and analyze models of a circular cell and of a budding yeast cell pair, both of which exhibit electroporation when exposed to pulsed electric fields.

Towards solid tumor treatment by irreversible electroporation: intrinsic redistribution of fields and currents in tissue.

Local and drug-free tissue treatment by irreversible electroporation (IRE) involves the creation of aqueous pores in a cells plasma membrane (PM) and leads to non-thermal cell death by necrosis. To investigate explicit pore-based effects we use two-dimensional system models with different spatial scales. The first is a multicellular system model (spatial scale 100 μm) that has irregularly shaped cells, and quantitatively describes dynamic (creation and destruction, evolution in pore size) pore behavior at the PM. The second is a tissue model (spatial scale 200 mm) that is constructed from a unit cell and uses the asymptotic (fixed pore size) electroporation model. Both system models show that significant redistribution of fields and currents occurs through transient PM pores. Pore histograms for the multicellular model demonstrate the simultaneous presence of small and large pores during IRE pulses. The associated significant increase of PM permeability may prove to be essential to understanding how cell death by necrosis occurs. The averaged tissue conductivity in both models increases during IRE pulses because of electroporation. This leads to greater electrical dissipation (heating) and, thus, to larger temperature increases than suggested by tissue models with passive and static electrical properties.

The Spatially Distributed Dynamic Transmembrane Voltage of Cells and Organelles due to 10 ns Pulses: Meshed Transport Networks

The authors describe two versions of a two-dimensional (2-D) cell model that contains three circular membranes representing the plasma membrane (PM) and single bilayer approximations to both the nuclear envelope and the mitochondrial membrane. The first version uses a Cartesian transport network, which respects topology but approximates geometry. The second version uses a meshed transport network, which respects both. The asymptotic electroporation model is assigned to all local membrane sites in order to assess the electrical response of the membranes. The predictions of these two models are presented for 10-ns trapezoidal pulses with 1.5 ns rise and fall times. The applied field magnitudes range from 1 to 100 kV/cm, corresponding to recent experiments. The spatially distributed electroporation models exhibit a supraelectroporation for the larger pulses with a maximum transmembrane voltage of Um~1.5 V for both the PM and organelle membranes. For the larger fields, the PM and organelle membranes electroporate essentially simultaneously. The meshed version of the transport network eliminates numerical artifacts and is more computationally efficient

The 1996 Lindberg Award : Calcium antagonists alter cell shape and induce procollagenase synthesis in keloid and normal human dermal fibroblasts

Fibroblast cytomorphology is tightly coupled to phenotypic expression, particularly as it relates to extracellular matrix protein synthesis and degradation. We have observed that calcium antagonists, such as verapamil and trifluoperazine, depolymerize actin filaments and alter fibroblast cell shape from bipolar to spherical. Characteristically, the depolymerization of actin filaments, which mediates the cell shape change, turns on procollagenase gene expression in normal human skin fibroblasts. We have found the same effects of calcium antagonists on cell shape, cytoskeletal components, and induction of procollagenase in the keloid fibroblasts of three cell lines, CB792, CW792, and WT949. Rounded cells were seen in 74.8% of verapamil-treated and 86.7% of trifluoperazine-treated cells, whereas only 1.1% of the control cells were spherical. The percentage of cells that synthesized collagenase in the control, verapamil-treated, and trifluoperazine-treated groups was 3.8%, 42.8%, and 53.4%, respectively. Approximately 60% of rounded cells exhibited increased collagenase synthesis when the cells were treated with a calcium antagonist. These results indicate considerable heterogeneity in the phenotypic response to morphologic change. The amount of procollagenase synthesized in a cell was estimated by the fluorescence intensity of the fluorescein-labeled antibody. The normalized fluorescence intensity of procollagenase in the control cells was about 2 to 2.6 times that of background. In contrast, the normalized fluorescence intensity of procollagenase in the calcium antagonist-treated cells was about 2.4 to 12 times that of background. This high intensity level indicates an increase in procollagenase production in the calcium antagonist-treated cells. Calcium green dye used to study cytosolic calcium revealed that after cells were treated with verapamil, the cytosolic calcium ion concentration first increased and then decreased. The change of cytosolic calcium ion concentration may be related to the depolymerization of actin filaments and the alteration of cell shape.

Towards solid tumor treatment by nanosecond pulsed electric fields.

Local and drug-free solid tumor ablation by large nanosecond pulsed electric fields leads to supra-electroporation of all cellular membranes and has been observed to trigger nonthermal cell death by apoptosis. To establish pore-based effects as the underlying mechanism inducing apoptosis, we use a multicellular system model (spatial scale 100 μm) that has irregularly shaped liver cells and a multiscale liver tissue model (spatial scale 200 mm). Pore histograms for the multicellular model demonstrate the presence of only nanometer-sized pores due to nanosecond electric field pulses. The number of pores in the plasma membrane is such that the average tissue conductance during nanosecond electric field pulses is even higher than for longer irreversible electroporation pulses. It is shown, however, that these nanometer-sized pores, although numerous, only significantly change the permeability of the cellular membranes to small ions, but not to larger molecules. Tumor ablation by nanosecond pulsed electric fields causes small to moderate temperature increases. Thus, the underlying mechanism(s) that trigger cell death by apoptosis must be non-thermal electrical interactions, presumably leading to different ionic and molecular transport than for much longer irreversible electroporation pulses.

Cylindrical cell membranes in uniform applied electric fields: validation of a transport lattice method

The frequency and time domain transmembrane voltage responses of a cylindrical cell in an external electric field are calculated using a transport lattice, which allows solution of a variety of biologically relevant transport problems with complex cell geometry and field interactions. Here we demonstrate the method for a cylindrical membrane geometry and compare results with known analytical solutions. Results of transport lattice simulations on a Cartesian lattice are found to have discrepancies with the analytical solutions due to the limited volume of the system model and approximations for the local membrane model on the Cartesian lattice. Better agreement is attained when using a triangular mesh to represent the geometry rather than a Cartesian lattice. The transport lattice method can be readily extended to more sophisticated cell, organelle, and tissue configurations. Local membrane models within a system lattice can also include nonlinear responses such as electroporation and ion-channel gating.

Spatially constrained skin electroporation with sodium thiosulfate and urea creates transdermal microconduits

Controlled transport of molecules through the skins main barrier, the stratum corneum (SC), is a long standing goal of transdermal drug delivery. Traditional, needle-based injection provides delivery of almost any water soluble compound, by creating a single large aqueous pathway in the form of the hollow core of a needle, through which drug is delivered by pressure-driven flow. We extend previous work to show that SC-spanning microconduits (here with diameters of about 200 microm) can be created in vivo by skin electroporation and low-toxicity, keratolytic molecules (here sodium thiosulfate and urea). A single microconduit in isolated SC can support volumetric flow of the order of 0.01 ml s(-1) by a pressure difference of only 0.01 atm (about 10(2) Pa), demonstrating that the SC barrier has been essentially eliminated within this microscopic area.

Emergence of a large pore subpopulation during electroporating pulses.

Electroporation increases ionic and molecular transport through cell membranes by creating transient aqueous pores. These pores cannot be directly observed experimentally, but cell system modeling with dynamic electroporation predicts pore populations that produce cellular responses consistent with experiments. We show a cell system models response that illustrates the life cycle of a pore population in response to a widely used 1 kV/cm, 100 μs trapezoidal pulse. Rapid pore creation occurs early in the pulse, followed by the gradual emergence of a subpopulation of large pores reaching ~30 nm radius. After the pulse, pores rapidly contract to form a single thermally broadened distribution of small pores (~1 nm radius) that slowly decays. We also show the response of the same model to pulses of 100 ns to 1 ms duration, each with an applied field strength adjusted such that a total of 10,000±100 pores are created. As pulse duration is increased, the pore size distributions vary dramatically and a distinct subpopulation of large pores emerges for pulses of microsecond and longer duration. This subpopulation of transient large pores is relevant to understanding rapid transport of macromolecules into and out of cells during a pulse.