Dielectrophoretic forces and potentials induced on pairs of cells in an electric field

Computationally efficient simulation of electrical activity at cell membranes interacting with self-generated and externally imposed electric fields

OBJECTIVE

We present a computational method that implements a reduced set of Maxwell's equations to allow simulation of cells under realistic conditions: sub-micron cell morphology, a conductive non-homogeneous space and various ion channel properties and distributions.

APPROACH

While a reduced set of Maxwell's equations can be used to couple membrane currents to extra- and intracellular potentials, this approach is rarely taken, most likely because adequate computational tools are missing. By using these equations, and introducing an implicit solver, numerical stability is attained even with large time steps. The time steps are limited only by the time development of the membrane potentials.

MAIN RESULTS

This method allows simulation times of tens of minutes instead of weeks, even for complex problems. The extracellular fields are accurately represented, including secondary fields, which originate at inhomogeneities of the extracellular space and can reach several millivolts. We present a set of instructive examples that show how this method can be used to obtain reference solutions for problems, which might not be accurately captured by the traditional approaches. This includes the simulation of realistic magnitudes of extracellular action potential signals in restricted extracellular space.

SIGNIFICANCE

The electric activity of neurons creates extracellular potentials. Recent findings show that these endogenous fields act back onto the neurons, contributing to the synchronization of population activity. The influence of endogenous fields is also relevant for understanding therapeutic approaches such as transcranial direct current, transcranial magnetic and deep brain stimulation. The mutual interaction between fields and membrane currents is not captured by today's concepts of cellular electrophysiology, including the commonly used activation function, as those concepts are based on isolated membranes in an infinite, isopotential extracellular space. The presented tool makes simulations with detailed morphology and implicit interactions of currents and fields available to the electrophysiology community.

Hybrid finite element method for describing the electrical response of biological cells to applied fields

A novel hybrid finite element method (FEM) for modeling the response of passive and active biological membranes to external stimuli is presented. The method is based on the differential equations that describe the conservation of electric flux and membrane currents. By introducing the electric flux through the cell membrane as an additional variable, the algorithm decouples the linear partial differential equation part from the nonlinear ordinary differential equation part that defines the membrane dynamics of interest. This conveniently results in two subproblems: a linear interface problem and a nonlinear initial value problem. The linear interface problem is solved with a hybrid FEM. The initial value problem is integrated by a standard ordinary differential equation solver such as the Euler and Runge-Kutta methods. During time integration, these two subproblems are solved alternatively. The algorithm can be used to model the interaction of stimuli with multiple cells of almost arbitrary geometries and complex ion-channel gating at the plasma membrane. Numerical experiments are presented demonstrating the uses of the method for modeling field stimulation and action potential propagation.

Dielectric characterization of bacterial cells using dielectrophoresis

Measurements of dielectrophoretic collection spectra of Escherichia coli and Staphylococcus aureus suspensions are used for obtaining dielectric characteristics of both types of bacteria. The experiments are interpreted using a numerical method that models the cells as compartmented spherical or rod-like particles. We show the usefulness of this simple method to extract significant information about the electrical properties of Gram-negative and -positive bacteria.

Dielectrophoresis based-cell patterning for tissue engineering

Engineering functional tissues and organs in vitro is considered integral to regenerative medicine. Many recent cell patterning technique developments position cells at a pre-designated pattern to improve tissue engineering efficiency and quality and to facilitate 3-D cell-cell interaction exploration. Among these techniques, dielectrophoresis (DEP)-based cell patterning advantageously offers speed, ease of operation, low degree of cell damage, and precision. This article reviews recent advances in DEP-based patterning techniques, including electrode design, suitable buffer and hydrogel, effects of the electric current to cells, combination potential with other techniques, as well as efforts to generate 3-D tissues.

Field stimulation of cells in suspension: use of a hybrid finite element method

Electric fields are used in a range of applications, including gene transfection, electrochemotherapy of tumors and cardiac defibrillation. Despite the widespread use of electric fields, most of the theoretical and computational studies on discrete cellular tissue have focused on a single cell. In this work, we propose a hybrid finite element method to simulate the effects of external electric fields on clusters of excitable cells. The method can be used to model cells of arbitrary cell geometries and non-linear membrane dynamics. The results show that the response of multiple cell, like a single cell, is a two-stage process consisting of the initial polarization that proceeds with cellular time constant (less than one microsecond) and the actual excitation of the cell membrane that proceeds with the membrane time constant (on the order of milliseconds). The results also show that the stimulation of a given cell depends in part on the arrangement of cells within the field and not simply the location within the field, suggesting that classical approaches that ignores the effect of the cells on the field do not adequately predict the cellular response.

Erythrocyte rouleau formation under polarized electromagnetic fields

We study the influence of an external electromagnetic field of 1.8 GHz in the formation or disaggregation of long rouleau of identical erythrocyte cells. In particular we calculate the variation of the transmembrane potential of an individual erythrocyte illuminated by the external field due to the presence of the neighboring erythrocytes in the rouleau, and compare the total electric energy of isolated cells with the total electric energy of the rouleau. We show that the polarization of the external electromagnetic field plays a fundamental role in the total energy variation of the cell system, and consequently in the formation or disaggregation of rouleau.

Cell hybridization by electrofusion on filters

Electric field pulses induce permeabilization and associated fusogenicity in cell membranes. Electrofusion of cells is usually performed in two steps: the first is the creation of close intercellular contacts; the second is an application of electric pulses that induces membrane fusion. Very large cell contacts can be obtained by a filter aspiration method. A cell monolayer is created by controlled suction on biocompatible filter. No spontaneous fusion results. Just after filtration, electrofusion is obtained by field pulses applied parallel to the filter. Cell viability is not strongly affected and cells recover their spherical shape in the minute time range after filtration. The electrical parameters, the cell density, and the flow rate control fusion. Fusion is obtained with cells of different origins with very different adhesion properties. Hybrid cells are easily formed. This approach appears to be a very efficient method for cell hybridization with an easy-to-use protocol.

Effective conductivity of cell suspensions

Using finite-element method (FEM) effective conductivity of cell suspension was calculated for different cell volume fractions and membrane conductivities. Cells were modeled as spheres having equivalent conductivity and were organized in cubic lattices, layers and clusters. The results were compared to different analytical expressions for effective conductivity and they showed that Maxwell theory is valid also for higher volume fractions.

Theoretical aspects of single pulse induced modulation of cell-cell interactions favoring iso-electrofusion

A single pulse-mediated electrofusion of freely suspending widely separated cells could reduce steps in the protocol. Prediction of such electrofusion is still unsolved. In pre-pulse condition, quantitative estimation with three-layered model cell surface reveals electrostatic repulsion (due to negative surface charge) is the major hindrance to membrane-to-membrane contact. On theoretical grounds we predict that designed single high voltage pulsing on widely separated model cell surface can counteract the non-specific repulsive interaction and favor approach of two apposed membranes. But hydrodynamic modulation of pulse exposed cell surface interaction can hamper approach of membranes contact when cells are held at a gap of more than approximately 450 A.

Modeling assemblies of biological cells exposed to electric fields

Gap junctions are channels through the cell membrane that electrically connect the interiors of neighboring cells. Most cells are connected by gap junctions, and gaps play an important role in local intercellular communication by allowing for the exchange of certain substances between cells. Gap communication has been observed to change when cells are exposed to electromagnetic (EM) fields. In this work, we examine the behavior of cells connected by gap junctions when exposed to electric fields, in order to better understand the influence of the presence of gap junctions on cell behavior. This may provide insights into the interactions between biological cells and weak, low-frequency EM fields. Specifically, we model gaps in greater detail than is usually the case, and use the finite element method (FEM) to solve the resulting geometrically complex cell models. The responses of gap-connected cell configurations to both dc and time harmonic fields are investigated and compared with those of similarly shaped (equivalent) cells. To further assess the influence of the gap junctions, properties such as gap size, shape, and conductivity are varied. Our findings indicate that simple models, such as equivalent cells, are sufficient for describing the behavior of small gap-connected cell configurations exposed to dc electric fields. With larger configurations, some adjustments to the simple models are necessary to account for the presence of the gaps. The gap junctions complicate the frequency behavior of gap-connected cell assemblies. An equivalent cell exhibits low-pass behavior. Gaps effectively add a bandstop filter in series with the low-pass behavior, thus lowering the relaxation frequency. The characteristics of this bandstop filter change with changes to gap properties. Comparison of the FEM results to those obtained with simple models indicates that more complex models are required to represent gap-connected cells.

Biological cells with gap junctions in low-frequency electric fields

Biological effects have been observed from weak, low-frequency magnetic fields. It has been suggested that the observed effects are due to the induced currents and electric fields. The behavior of cells exposed to an electric field is investigated in this paper. The induced transmembrane potential (TMP) is examined in geometrically complex models of various cell configurations. The TMP is evaluated using the finite element method (FEM), a numerical technique that is well suited to complicated geometries. Because displacement currents can be neglected at very low frequencies, a FEM solver that considers only material conductivity is used. Therefore, our results apply only well below the relaxation frequency. Chains and clusters of gap-connected cells of various sizes are modeled. The conductivity and size of the gap junctions in the cell configurations are also varied. The results for small configurations are compared to models of ellipsoidal cells with shapes similar to those of the configurations. FEM estimates of TMP's in long, cylindrical cell chains are compared to the predictions of the leaky cable model. The FEM approach confirms that gap-junction-connected cells can be treated as a single similarly shaped cell. Gaps influence the potential in the interior of cell configurations, and these effects increase with gap size and conductivity. For configurations to which approximations such as the leaky cable model do not apply, the FEM approach can be used to estimate the TMP, if the model is adapted to fit within computational memory limits.

Membrane skeleton restraint of surface shape change during fusion of erythrocyte membranes: evidence from use of osmotic and dielectrophoretic microforces as probes

The role of the spectrin-based membrane skeleton in cell fusion was studied by following the condition-dependent diameter versus time expansion signature of the fusion zone in electrofused pairs of erythrocyte ghost membranes. Previous work showed that the presence of the dielectrophoresis-inducing alternating electric field, which is used to bring membranes into contact through pearl chain formation, had a detectable promoting effect on fusion zone expansion. Two new dielectrophoresis protocols were used in the present work to utilize this externally generated and controllable microforce field to probe the forces intrinsic to the system that drives the expansion of the fusion zone. First, fusion zones expanded to a greater diameter in a strong AC field compared to a weak AC field, and they expanded to a greater diameter if erythrocyte ghosts received a prior heat treatment (42 degrees C, 20 min). Furthermore, flat diaphragm fusion zones broke down into open lumen fusion zones sooner (i.e., had shorter lifetimes) when they were expanding more quickly. Second, changing the AC field strength at specific times during the fusion zone expansion led to an immediate visco-elastic response. However, shifting the AC field strength to zero after 5 s of fusion zone expansion resulted in a subsequent decrease in the average fusion zone diameter. This suggests not only that the spectrin-based membrane skeleton actually tends to prevent the rounding up process but that it may be capable of generating an antirounding force, which has broad implications for the role of the membrane skeleton in cell fusion. These results are consistent with the hypothesis that flat diaphragm fusion zones induced in heat-treated membranes were very easily stretched and that membrane-based forces that control or drive the expansion process must originate from membrane area that is outside rather than inside the fusion zone. Lastly, when an outward-directed osmotic pressure-based microforce was present at the time that erythrocyte ghosts were fused, the fusion zone diameter underwent a greater expansion in the 0-1 s interval after fusion. This suggests that an osmotic pressure-based microforce can be used to experimentally calibrate the dielectrophoretic force.