Transport Lattice Approach to Describing Cell ... - Semantic Scholar

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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 4, AUGUST 2004

Transport Lattice Approach to Describing Cell Electroporation: Use of a Local Asymptotic Model Donald A. Stewart, Jr., T. R. Gowrishankar, and James C. Weaver

Abstract—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. Index Terms—Asymptotic model, electroporation, supra-electroporation, transport lattice.

I. INTRODUCTION

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HE biophysical mechanism by which electrical pulses alter cell membranes has focused on the hypothesis of hydrophilic pore formation due to electrically decreased pore formation energy and thermal fluctuations [1]–[8]. Although a wide variety of electrical pulsing conditions have been used in conventional electroporation protocols, significant membrane changes are generally known to occur if the transmembrane , exceeds about 0.2 V for pulses longer than ms voltage, and about 1 V for shorter pulses with durations between s ms. and The main features of electroporation include: 1) application of electrical pulses; 2) charging of lipid bilayer membranes; 3) rapid, localized structural rearrangements within the membrane; 4) transitions to water—filled membrane pores (hypothesized to be “hydrophilic pores”) which perforate the membrane; and 5) tremendous increase in ionic conductance and molecular transport when pores are significantly large. Both applications

Manuscript received September 30, 2003; revised March 23, 2004. This work was supported in part by the NIH under Grant RO1-GM63857 and in part by an AFOSR/DOD MURI Grant on Subcellular Responses to Narrowband and Wideband Radio Frequency Radiation, administered through Old Dominion University. The authors are with the HST Biomedical Engineering Center, Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPS.2004.832639

and mechanistic understanding have received growing attention over the past two decades. Recently, there has been significant interest in the finding that submicrosecond, megavolt-per-meter pulses can alter subcellular structures without overwhelming effects at a cell’s plasma membrane [7], [9]–[11]. Early experiments demonstrated that reversible electrical breakdown onset occurs quickly for submicrosecond pulses. The maximum transmembrane voltage achieved is only about 1.2 V [12], [13], consistent with “supra-electroporation,” a new concept discussed in this paper. We have developed a transport lattice approach to the simulation of electrical, chemical, and thermal transport in multi-cellular and subcellular systems [14]. Here we apply the transport lattice technique to simulate cell membrane electroporation for a variety of systems. We first address “conventional electroporation” conditions, with single skeletal muscle cell electroporation experiments [15] representing an exemplary case. These experiments involve real cell membranes (not artificial planar bilayer membranes), but avoid the complication of spaover the membrane. These experiments tial variation of can be modeled on a one–dimensional (1-D) lattice. We then consider two–dimensional (2-D) models; first a circular cell model and then a budding yeast cell pair model for exposure to electric fields of 10 s duration, and also faster pulses of 60 s duration. A transport lattice approach allows a variety of local models to be combined into a system model. Here, the system consists of cell membranes, electrolyte regions, and electrodes. The local models can range from simple (e.g., fixed resistances and capacitances) to nonlinear, hysteretic interactions that are described by differential equations (e.g., the asymptotic membrane electroporation model demonstrated here). This provides an approach to hypothesis testing in which multiple local models for transport and storage of charge can be combined to construct a system model of one or more cells, and the predictions of the model compared with experimental observations. II. BACKGROUND AND MODELING METHOD A. Muscle Cell Experiment Background A set of experiments on individual frog skeletal muscle cells was carried out previously by one of the present authors [15], [16]. A feature of these experiments is that single skeletal muscle cells were subjected to both space-clamp and voltage-clamp conditions, which allows the same applied transmembrane voltage across the entire cell membrane as a function of time. Fig. 1(a) shows a schematic view of the experiment. A cylindrical muscle cell of radius 50 m and length

0093-3813/04$20.00 © 2004 IEEE

STEWART et al.: TRANSPORT LATTICE APPROACH TO DESCRIBING CELL ELECTROPORATION

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Fig. 1. (a) Schematic of the space and voltage-clamped frog muscle cell experiment. The cell is a cylinder of length 300 m and radius 50 m. A voltage pulse is applied between the external electrolyte and the internal electrolyte. (b) The corresponding 1-D system with a square patch of membrane is suspended between the interior (gray) and exterior (white) electrolyte regions. (c) The equivalent circuit for the frog muscle cell membrane system. The membrane separates two macroscopic electrolyte regions which are connected to the voltage source. The current through the membrane is governed by the membrane capacitance, C , and the membrane ion conduction current I from (6). The resting potential of 90 mV is supplied by the voltage pulse source V = V (t). (d) Part of the transport lattice used in the 2-D system models. (e) Equivalent circuit diagrams for the local transport models linking the nodes in panel (d). The intracellular electrolyte (M ) and extracellular electrolyte (M ) models are passive RC elements and the membrane model (M ) is a nonlinear current source in parallel with the local membrane capacitance and the resting potential model. G and V together provide a simplified resting potential source. The membrane model is sandwiched between layers of conducting media. The nonlinear current source I (t) represents the asymptotic model of electroporation.

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300 m is clamped in a chamber. Electrodes provide a voltage difference across the membrane. The equivalent 1-D geometry is shown in Fig. 1(b) with a square patch of cell membrane suspended between two electrolyte regions. The area of the patch corresponds to the area of the cylindrical cell membrane, 0.094 mm . Although this is a 1-D simulation, the electrolyte compartments and the membrane are treated as volumes with cross sectional area 0.094 mm , and depths of 0.1 mm, 5 nm, and 5 mm for the internal electrolyte, membrane, and external electrolyte compartments respectively. The experiment applied a 4-ms square voltage pulse across the two aqueous electrolyte regions and the frog skeletal muscle cell membrane in between. The equivalent circuit of this setup is shown in Fig. 1(c). The , was set to a baseline of mV (the applied voltage, cell’s plasma membrane resting potential) and then pulsed with amplitudes ranging from 220 to 440 mV. The resulting currents through the membrane were recorded. We compare the membrane current of the experiment with the system model’s solutions over the range of applied voltage amplitudes. The parameters of the asymptotic model were adjusted to improve the agreement between the model prediction and the experiment.

B. Cylindrical Cell We next apply the transport lattice method to a 2-D cylindrical cell electroporation system model, with the geometry shown in Fig. 2. The direction of the electric field, the cell radius, the membrane thickness, and the respective conductivities and permittivities are labeled in Fig. 2(a). Fig. 2(b) shows the mapping of the cylindrical cell geometry of the system model to the 101 101 node lattice. The cylindrical cell has a radius of m. For compu10 m in a square area with side length tational purposes, all the 2-D simulations have a depth dimension equal to the 2-D lattice spacing, , but this choice makes no difference in the simulation results since there is no transport in or out of the 2-D plane. The applied electric field is supplied by electrodes along the outer edge of the system model that provide an electric field at 45 to the horizontal. The asymptotic model parameters are taken from [17] and are listed in Table I. The internal electrolyte conductivity, 1.2 Sm , is set to be the same as the external electrolyte. Often, the internal conductivity is set to a smaller value (e.g., 0.3 S m [18]) in cell simulations, which is presumably done to take into account the effective increase in resistance due to subcellular structures. In the present didactic model, we keep the electrolyte

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duration and 60 kV cm amplitude [9]. We extract the transmembrane voltage response as a function of time and position around the cell from the simulation. C. Budding Yeast Cells Finally, we simulate the response of a 2-D model of budding yeast cells. Budding yeast (here Saachromyces cerevisae) represents an asymmetric cell pair in which two different size cells (mother and daughter) can have a significant area of their membranes in close proximity. Yeast are of great importance to biotechnology and biological research, and have been used in some of the few electroporation studies which make quantitative determinations of molecular uptake by electroporation [19]–[21]. Realistic models of yeast cells should be valuable in comparing molecular uptake measurements with predictions based on models of membrane electroporation. The geometry [Fig. 3(a)] was taken from a published image of budding yeast cells [22]. The larger cell is roughly elliptical with major and m and m, respectively. minor axis dimensions of The smaller cell has major and minor axis dimensions of m and m. The system model volume has a side length of m and a depth of m. Just as in the cylindrical cell simulation, the applied electric field is at 45 to the horizontal. The asymptotic model parameters are again from [17] and listed in Table I. The same standard pulse and ultrashort pulse durations and amplitudes are used for the budding yeast simulations. D. Asymptotic Electroporation Model The transient aqueous pore hypothesis of electroporation is based on continuum models of membrane pores, electrostatic energy differences and thermal fluctuations, usually in the form of the Smoluchowski equation [23]–[30]. The Smoluchowski Equation (SE) describes the evolution of the pore population (pores per unit membrane area per unit pore radensity, dius) that describes the number and sizes of pores in a membrane

(1)

Fig. 2. (a) Schematic showing the 2-D cylindrical cell model. An electric ~ , is applied at 45 to the horizontal axis. The cell has a radius field, E r 1 = 10 m, and a membrane thickness d = 5 nm. Each region has a characteristic conductivity , and permittivity . The dielectric properties of aqueous electrolytes become important for rapidly ( 1 ns) changing local electric fields. The position along the membrane is determined by the angle ~ measured relative to E . The membrane thickness scale in this figure is exaggerated relative to the radius to show that the membrane region is not infinitely thin. The actual values of r and d are used in the local membrane model [Fig. 1(e)]. (b) Diagram showing the position of the cylindrical cell within the system model, which is a Cartesian lattice of 101 101 nodes with a side length of 35 m.

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is the hydrophilic pore forwhere is the pore radius and depends on and as shown in Fig. 4, mation energy. which shows the pore formation energy curves as a function of for both hydrophobic and pore radius for several values of is the diffusion coefficient in pore radius hydrophilic pores. is the thermal energy, and is an assumed pore space, source term. The source term describes transitions between hyand hydrophilic pores, governed by drophobic which pore type is energetically favored at a given radius. The expression for the source term in the asymptotic model [31] is (2)

conductivities uniform with the intention of addressing electroporation of subcellular structures in a subsequent publication. For the 2-D cell simulation, we use a standard pulse of 10- s duration and 1 kV cm amplitude and a shorter pulse of 60-ns

where and are pore creation and destruction rates, respecis the membrane thickness, is the hydrophobic tively, pore formation energy as a function of , and is the value of at which the and curves intersect (at ).


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TABLE I PARAMETERS USED FOR EACH SIMULATION

is the Heaviside step function which goes from 1 to . 0 at The basic assumption of the asymptotic model [17], [31], [32] is that source term dominates the diffusion and drift terms of the Smoluchowski equation that governs the pore population, . A more recent publication introduced modifications to the model to account for pore growth [33], but here we use the original asymptotic model. The resulting ordinary differential equation for the rate of change of the number of hydrophilic pores per local membrane area is (3) with the normalization relation (4) sets the scale of the electroporation voltage critical Here, for breakdown, is the steady-state pore density at is determined by the time constant of pore resealing when returns to 0, and is a parameter that governs the sensitivity of the hydrophilic pore creation rate to . All model parameter values are listed in Table I. An in-depth description of the asymptotic model can be found in [17] and [31]. The asymptotic model is an approximation to the Smoluchowski equation-based models [23]–[30], and since it is computationally simpler, the asymptotic model can readily be adapted to local membrane regions of a transport lattice system model. Computational simplicity is important to creating and

solving realistic cell models, as a large number of interacting local models is required [14]. E. System Model of One or Two Cells The 1-D transport lattice of Fig. 1(c) is the system model for the space-clamped, voltage-clamped cell membrane. Only six local models are needed because the space-clamp condition is a good approximation to a 1-D system (there is no voltage dependence on the position on the membrane). The membrane is described by the capacitance, , and the variable membrane resistance, which contains the asymptotic electroporation model, here represented by a nonlinear hysteretic current . On either side of the membrane are the internal and external electrolytes, described by parallel RC elements. The transport lattice of Fig. 1(d) shows a section of a system model for the 2-D model of a cylindrical cell. A Cartesian lattice of nodes are connected by local charge transport models, identified as hashed ( or ) or black rectangles. The basic features of a transport lattice system model with spatially assigned local models for charge transport and storage are shown in Fig. 1(e). The membrane model, , is the same in the 1-D and 2-D models except for parameter values and the resting potential. The 2-D cylindrical cell system model of Fig. 2 has local models distributed spatially on a 101 101 node lattice, with 228 local membrane models. The 2-D budding yeast system model features a mother and daughter cell pair (Fig. 3).

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Fig. 4. Pore formation energies as a function of pore radius, r , and transmembrane voltage, following Neu and Krassowska [31]. The curves for hydrophobic pores (W : dot-dashed) are steeply ascending, beginning close to r = 0 and rising to 100 kT at r 0:8 nm. The family of solid-line curves represents the hydrophilic pore formation energy, W , for an edge energy

= 1:8 10 J m , and a membrane surface tension of 0 = 10 J m . The W and W curves for U = 0 intersect at r 0:5 nm and the local 0:76 nm. minimum in the hydrophilic pore curve occurs at r

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Fig. 3. (a) Image of budding yeast cells (mother/daughter cells) from Niemistö et al. [22]. (b) Processed image [22] showing cell edges derived from the image in panel (a). (c) Projection of yeast cells onto a 2-D 101 101 node lattice with a side length of 40 m. Local models were constructed and assigned as for the circular cell [Fig. 1(e)]. Note that a thin aqueous layer separates the mother and daughter cells.

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The geometry for this model was obtained from a published image [22, Fig. 6(d)]. The budding yeast model has 290 discrete local membrane models along the surface of both cells.

Fig. 5. Asymptotic model ODE (3), is solved by this equivalent subcircuit in which the current source depends on the transmembrane voltage and a capacitance to ground. The voltage on the capacitor represents the pore density N (t) (not a physical voltage in the system model). = 1= and the equivalent current source is The capacitance is C I =e (1 N (t)=(N e )). The values of ; V ; N , and q are given in Table I for each system model.

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F. Local Model of Electrolytes The two resistances, and [Fig. 1(c)], represent the aqueous electrolyte regions that connect the membrane to the voltage source in the 1-D muscle cell system model. The and , represent both the dielectric two capacitances, properties of these electrolytes and the stray capacitance of the experimental arrangement. In the 2-D circular cell model of Fig. 1(e), the electrolyte regions are represented by a lattice of RC subcircuits that together represent the electrolyte’s conduction and dielectric properties. Following [14], the resistors and and capacitors are set according to the local conductivity , the node spacing and lattice voxel cross permittivity sectional area

(5)

For the 1-D system model, mm mm, and mm (node spacings for the intracellular and extra, and cellular regions). For the 2-D system models, is given in Table I. G. Local Model of Resting Potential A simple model for the contribution of a local membrane area to the cell’s resting potential is based on a single type of ion pump that represents the total pump activity within the and , particular local area. This is shown in Fig. 1(e) as the voltage source and the resistance in series, which are in parallel with the membrane capacitance and electroporation model [34]. In the 1-D model, the resting potential of mV is provided by the applied voltage source, , to


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Fig. 6. Results of the 1-D simulation with applied voltage V = 440 mV. (a) The transmembrane voltage follows the applied voltage across the system model region with a voltage drop arising from the resistance of the electrolyte regions. (b) The total current through the system model region including the displacement current charging the membrane (the spikes at 1 and 5 ms). (c) The number of pores in the membrane as a function of time. (d) The membrane conductance, which depends on the number of pores as described by (6).

mimic the experimental setup. Although asymmetric spatial distributions of membrane proteins are widely known [35]–[38], for simplicity at this stage of model development, we assume a spatially uniform distribution of the representative pump that . accounts for H. Local Model of Electroporation As illustrated in Fig. 5, the local electroporation model is based on simple subcircuit that solves the asymptotic model ordinary differential equation (ODE). In the asymptotic electroporation model [31], the total ionic current through a flat membrane is governed by the pore conductance, , the number , and the transmembrane of pores in the local area, voltage,

, where is the electron voltage is defined as is the thermal energy. All quantities are defined charge, and in Tables I and II. In our local membrane electroporation model, we use the subcircuit of Fig. 5 to solve (3) by exploiting a mathematical analogy. The use of circuits to solve differential equations involving nonelectrical quantities is well established [40]. in this subcircuit is not a physSpecifically, the voltage on ical voltage in the lattice, but represents the local pore popu. Similarly, the current source, , represents the lation, rate of change of for the local area. The local transmembrane voltage, , is coupled to the subcircuit by (8). The capacitance and the rate of change in the local pore population, , are

(6)

(8)

Here, to maintain consistency with Neu and Krassowska, we of Glaser et al. [39], which use the pore conductance, considers the Born energy change upon moving a monovalent ion through a trapezoidal edged cylindrical pore (7)

, and are given in Table I for each The values of system model. In principle, the membrane subcircuit could include the full Smoluchowski Equation (1), the Goldman Equation for the resting potential [41], [42], or other candidate mechanisms that . Here, we use only the asymptotic model due to influence its relative simplicity to demonstrate the ability of the transport lattice method to include nonlinear local models. For purposes of comparison, we also model the membrane response without electroporation. This “passive membrane” , indemodel has a constant, small membrane conductivity, , set according to Table I. pendent of

(7) The parameter is a barrier energy threshold (in units of ) and is a geometrical factor that smooths the sharp edge of a simple cylindrical pore. The dimensionless transmembrane

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Fig. 7. Experimental current measurements of Gowrishakar et al. for the space- and voltage-clamped skeletal muscle cell (dashed lines) compared with the model transmembrane currents (solid lines). Each panel in the figure shows the temporal change in transmembrane current at different applied voltages (inset V ).

TABLE II NOTATION REFERRED TO IN TEXT AND FIGURES

I. Transport Lattice Model Solution As described previously [14], transport lattices are solved by Kirchhoff’s Laws, initially using robust circuit simulation software. Here, we used Berkeley SPICE [43] version 3f5. We created the SPICE input files using MATLAB (Version 6.5.1, MathWorks, Natick, MA). We also used MATLAB to compute equipotentials, transmembrane voltages, and to display results. III. RESULTS AND DISCUSSION A. Space-Clamped Skeletal Cell Model (1-D) In the muscle cell analysis (Figs. 6 and 7), some of the model parameters are constrained by the geometry, and others by the results of the experiment. While exploring parameter space, we found that the asymptotic electroporation parameters with the

largest discrepancy from the nominal values given by DeBruin and Krassowska [17] were the pore creation rate parameter , and the average pore radius . DeBruin and Krassowska considered experiments with membrane resealing exponential decay time constants of 1.5 s. This is four orders of magnitude longer than the membrane conduction recovery time (0.16 ms) observed by Gowrishankar et al. for the clamped skeletal cell membranes [15]. If the membrane conductance recovery is attributed to pore destruction, rather than pore shrinkage, then the parameter had to be changed by two orders of magnitude. The other critical parameter was the average pore radius. The membrane conductivity depends on the pore radius squared, so we increased the average pore radius from 0.76 nm to 20 nm to improve agreement with the data. The Gowrishankar et al. experiment employed a long (4 ms) pulse and also noted a recovery time component much longer than 0.16 ms, but this was not adequately measured in their 10-ms time window. In contrast with the full SE model, the asymptotic model has no clear mechanism for generating two widely different components for recovery time constants. Our general strategy was to first explore parameter space to determine whether a set of parameters exists that yields reasonable agreement with the experimentally measured membrane currents (Fig. 7). Fig. 6 shows the muscle cell membrane model response mV. The transmembrane to an applied voltage, voltage shown in Fig. 6(a) follows the applied voltage but with a voltage drop in the electrolyte regions. The total current through the system model, which includes the displacement current charging the membrane, is plotted in Fig. 6(b). The


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Fig. 8. Equipotentials and locations of significantly electroporated membrane in response to the 10 s 1 kV cm pulse. (a) and (b) passive membrane equipotentials at times 0.1 s and 10 s, respectively. The 24 equipotentials run from 2:5 V to +2:5 V. (c) Passive membrane model geometry showing the membrane with fixed conductivity = 9:5 nS m . “X” marks the spot where the U was extracted for Fig. 9. The “+” at the center marks the spot where the internal electric field (E ) is measured for Fig. 9. (d) and (e) Asymptotic model equipotentials at times 0.1 s and 10 s, with appreciable current flowing through the cell at 10 s because of electroporation. (f) Asymptotic model geometry identifying locations of increased membrane conductance (the absence of the black line near the cell poles) where the conductance increases by at least 3 orders of magnitude ( > 10 Sm ) by the end of the pulse.

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current through the membrane is only a fraction of the total current, and is easily extracted from the simulation output. The transient current spikes at the beginning and end of the pulse were roughly an order of magnitude larger than the component of the current passing though the membrane for mV. The increasing number of pores shown in Fig. 6(c), and the membrane conductance shown in Fig. 6(d), are extracted from the electroporation subcircuit (Fig. 5) according to (3), (6), and (7). The time-dependent membrane currents predicted by the system model of Fig. 1 are presented in Fig. 7. The system model’s results are in reasonable agreement with experimental measurements, but there are some notable discrepancies. One discrepancy is that although the peak currents agree well for mV, the model shows more electroporation current (membrane conduction) at lower . Another discrepancy is that the membrane recovery is not quite complete by 10 ms in the measurements, but is complete in the model’s predictions. This is most likely due to pore radius evolution that is not included in the asymptotic model. The asymptotic model may be better suited to describe short pulse behavior because pore creation and destruction is expected to be more important than pore radius evolution. B. Circular Cell Model (2-D) As described in Fig. 2, the applied field is created by treating the nodes along the boundaries of the square region of the system model as linear arrays of ideal (zero over-voltage) microelectrodes with evenly spaced ascending (or descending) potentials, which are varied synchronously in time. This repreis related to a previous 2-D transport lattice sentation of

of a multicellular system [14], but with the difference that in the previous case the top and bottom boundaries of a system model were a line of microelectrodes with a common applied potential, and the vertical boundaries were treated as insulators (applied field directed from top to bottom; here it is directed at a 45 angle). By selecting the distribution of applied potentials can be varied, e.g., a uniform along the microelectrodes, applied field can easily be rotated. As shown previously for a discretized spherical cell confined to a small three dimensional volume [14], the finite size of the system model forces currents to flow with a slightly different spatial distribution than would be the case for the cell(s) within an electrolyte of large volume. The equipotentials in Fig. 8 are thus, slightly different than for cell in a dilute cell suspension. However, in some cases, particularly within solid tissues, microbial colonies, biofilms, or centrifuged pellets of cells, the cells are close together. Fig. 8 shows equipotentials and the locations of electroporated membrane in response to the 10 s pulse. The applied electric field was kV cm for this simulation at 45 from the vertical. Fig. 8(a) and (b) shows passive membrane (no electroporation model) equipotentials at times 0.1 s and 10 s. The equipotentials at the earlier time slice show that the electric field inside the cell is almost as large as the external field, and that by 10 s, the cell interior roughly reaches an equipotential. Fig. 8(d) and (e) shows the corresponding equipotentials with the asymptotic model rather than the passive membrane. The cell interior has a significant electric field at 10 s due to the increased conductance of the membrane. Fig. 8(f) identifies the locations of increased membrane conductance (the missing black line near the cell poles), where the local conductivity increases

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Fig. 9. Cylindrical cell U (top) and internal electric field (bottom) time dependence for 10 s 1 kV cm pulse (left) and 60 ns 60 kV cm pulse (right). (a) Time dependence of the U response at the pole to the 10 s pulse. U is evaluated at the location marked with an “X” in Fig. 8. The solid line shows the response with a asymptotic electroporation model (“EP”) and the dotted line shows the response with the passive membrane model (“P”). (b) Time dependence of the U response at the pole to the 60 ns pulse. The solid line shows the response with a asymptotic electroporation model and the dotted line shows the response with the passive membrane model. (c) Time dependence of the electric field (E ) at the center of the cell with the 10-s pulse. The dot-dashed line shows the applied electric field. (d) Time dependence of the electric field (E ) at the center of the cell with the 60-ns pulse. The “E EP” and the E lines are almost identical (the dot-dashed line overlaps with the solid line).

by at least 3 orders of magnitude ( Sm ) at the end of the pulse. and electric field time dependence for a Fig. 9 shows the 10- s pulse (left) and a 60 ns pulse (right) for the cylindrical cell. The 10 s pulse is typical for cell electroporation and the 60 ns pulse involves what we describe as “supra-electroporesponse ration.” Fig. 9(a) shows the time dependence of the at the pole to the 10- s 1 kV cm pulse. In the conventional plateaus at about 1.2 V during electroporation model, returns to the resting potential the pulse. After the pulse, mV in the passive model, but for the electroporated of membrane, is clamped at 0 V well beyond 25 s. The . Fig. 9(b) resting potential appears as a negative offset in response at the cell’s pole shows the time dependence of to the 60 ns 60 kV cm pulse. The 60 ns pulse shape is taken from [9]. If there were no electroporation (“passive membrane” peaks at 28 V, but with the asymptotic or “P” in the figure), peaks at about 1.4 V and plateaus at 1.2 V during model, the pulse. Fig. 9(c) shows the time dependence of the electric at the center of the cylindrical cell with the 10 s field 1 kV cm pulse. The solid line shows the response with a asymptotic electroporation model, and the dotted line shows the response with the passive membrane model. The dot-dashed

line shows the applied electric field. With the passive memhas a transient response with time constant brane model, of 0.16 s and settles at 0. The electroporation model’s settles at 0.18 kV cm during the pulse. After the pulse, returns to 0. Part (d) shows the time dependence of for the reaches about 50 kV cm with 60 ns 60 kV cm pulse. reaches nearly the full the passive membrane model, and applied field with the electroporation model. The amplitude of the internal electric field amplitude is sensitive to our choice of cytoplasm conductivity. For fast pulses, can be estimated by simple linear voltage division, so that when the cytoplasm conductivity is smaller than the external electrolyte conductance, the internal electric field can exceed the applied field amplitude. Fig. 10 shows the cylindrical cell polar angle dependence of for the [Fig. 10(a)] 10 s 1 kV cm pulse at time s ns [Fig. 10(b)]. and the 60 ns 60 kV cm pulse at time The responses for both passive membrane (dashed) and asymptotic electroporation model (solid) are shown on each plot. The electroporated regions in the vicinity of the poles show a plateau [corresponding to Fig. 9(d)]. The bumps in 10(b) around in are an artifact arising from the high-frequency response of the local membrane models in a Cartesian lattice.


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Fig. 10. Cylindrical cell polar angle dependence of U for the (a) 10 s 1 kV cm pulse at t = 10 s and the (b) 60 ns 60 kV cm pulse at t = 60 ns. The responses for both passive membrane (dashed) and electroporation (solid) models are shown on each plot. The electroporated regions at the pole show a plateau in U for both pulses, but the 10 s pulse electroporated regions extend 20 from the poles while for the 60-ns pulse the electroporated region extends over nearly all of the membrane. The bumps in (b) solid line around 90 are an artifact arising from the high frequency response of the local membrane models in a Cartesian lattice.

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The results in Figs. 9 and 10 are roughly consistent with the results of the Sea Urchin egg model in [17], obtained for m), larger electrolyte a much larger cell radius ( Sm ), and different applied pulses conductivities ( kV cm , pulse width 1 ms). ( C. Budding Yeast Model (2-D) Fig. 11 shows equipotentials and locations of electroporated membrane for the budding yeast cells in response to the 10- s kV cm at 45 pulse. The applied electric field was from the vertical. Fig. 11(a) and (b) show the passive membrane equipotentials at times 0.1 s and 10 s. The equipotentials run V to V. The cell interior roughly reaches an from equipotential by 10 s. Fig. 11(c) shows the passive membrane nS m fully surmodel geometry with conductivity rounding both cells. The location of the spot on the membrane is plotted in Fig. 12 is marked with an “X.” Fig. 11(d) where and (e) show the asymptotic model membrane equipotentials at times 0.1 s and 10 s. The cell has a significant interior electric field at 10 s. Fig. 11(f) identifies locations where the membrane conductance increases by at least 3 orders of magnitude Sm ) by the end of the pulse. The daughter cell’s ( membrane did not electroporate due to its smaller size. Fig. 12 shows the time dependence of the budding yeast cells and internal electric field for the 10 s pulse (left) and the response at the 60 ns pulse (right). The time dependence of cell’s pole to the 10- s 1 kV cm pulse is shown in Fig. 12(a).

response at Fig. 12(b) shows the time dependence of the the pole to the 60 ns 60 kV cm pulse. Without electroporapeaks at 27 V, but with electroporation, peaks at tion, about 1.4 V and plateaus at 1.2 V during the pulse. Fig. 12(c) inside the shows the time dependence of the electric field larger cell with the 10 s 1 kV cm pulse. Fig. 12(d) shows with the 60-ns pulse. The 60-ns the time dependence of pulse shape is taken from [9], an experimental pulse, not an idereaches alized pulse shape. The passive membrane model about 47 kV cm and the electroporation model reaches nearly the full applied field strength due to the increased conductivity of the membrane. D. “Supra-Electroporation” Both two dimensional models were considered for exposure to very large electric fields. For example, for the budding yeast with model, it is of interest to compare the magnitude of and without electroporation in Fig. 12 at a membrane site that corresponds to the “X” at the pole of the larger cell in Fig. 11. As in the circular cell model, the very large applied field results in electroporation over the entire membrane, so that the maximum transmembrane voltage is only about 1.5 V (without would be more than an order of magnitude electroporation larger). The approximate saturation of is a result of extensive membrane electroporation, far more than in conventional is protective, in electroporation. The prevention of a large the sense that large and possibly damaging electroconformation

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Fig. 11. Budding yeast cell Equipotentials and locations of electroporated membrane in response to the 10 s E = 1 kV cm pulse. (a) and (b) Passive membrane equipotentials at times 0:1 s and 10 s. The 24 equipotentials range from 2:8 V to +2:8 V. (c) Passive membrane model geometry showing the membrane with fixed conductivity = 9:5 nS m . “X” marks the spot where the U was determined for Fig. 12. The “+” at the center marks the spot where the internal electric field (E ) is determined for for Fig. 12. (d) and (e) Asymptotic model membrane equipotentials at times 0.1 s and 10 s. The equipotentials range from 2:8 V to +2:8 V. (f) Asymptotic model geometry identifying locations of increased membrane conductance (segments of black line absent at the larger cell’s poles) where the conductance increases by at least 3 orders of magnitude ( > 10 Sm ) by the end of the pulse.

0

0

Fig. 12. Budding yeast cells U and internal electric field time dependence for the 10-s 1 kV cm pulse (left) and the 60 ns 60 kV cm pulse (right). (a) Time dependence of the U response at the pole to the 10-s pulse. The solid line shows the response with the electroporation model and the dotted line shows the response with the passive membrane model. (b) Time dependence of the U response at the pole to the 60-ns pulse. (c) Time dependence of the internal electric field (E ) for the larger cell with the 10 s pulse. The dot-dashed line shows E . (d) Time dependence of the internal electric field (E ) of the larger cell with the 60-ns pulse. The 60-ns pulse shape is taken from [9].


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changes [44]–[46] to membrane macromolecules are limited to V. This can be regarded as an imthose experienced at portant protective membrane response. We suggest the term “supra-electroporation” for the much more extensive local membrane electroporation by megavoltper-meter fields than is needed to achieve significant molecular uptake by conventional electroporation. Conventional electroporation pulses cause regions of a cell membrane to reach about 1 V or less, but the pulses are much longer, typically 0.1 to 1 ms. In this sense, the extensive local membrane electroporation predicted for megavolt-per-meter electric field pulses can be regarded as “supra-electroporation,” and may be expected to play a role in the induction of apoptosis by field pulses with magnitudes of order 10 to 100 kV cm [10], [11], [47]–[49], and in other experiments that involve subcellular structures [9]. In the case of apoptosis, electroporation-induced voltage-gating of the MPTP (mitochondrial permeability transition pore) or swelling of the mitochrondria by osmotic pressure may be involved [8]. Supra-electroporation is pervasive, occurring over the entire plasma membrane, and even in subcellular membranes. For a recylindrical cell with the passive membrane model, the sponse to the leading edge of a fast pulse is approximately

(e.g., membrane capacitance) to nonlinear, hysteretic local models (e.g., membrane voltage-gated channels and membrane electroporation). Presently, a disadvantage is that although topology of transport barriers is respected, the geometry is represented only approximately. This can be addressed by replacing the Cartesian lattice with an optimized mesh. Unlike precision engineered man-made devices, cells generally have irregular shapes (at least “roughness” at the plasma membrane level), and change with time. Precise geometry is therefore usually not an attribute of cellular systems. We assume that explicit inclusion of the topology of membrane barriers and their interaction mechanisms, which contribute to their barrier function, the several regions of aqueous electrolytes (e.g., extracellular vs. intra-cellular), and the subcellular compartments such as nuclei and mitochondria, are more important than precise geometrical representation.

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[1] E. Neumann, A. E. Sowers, and C. A. Jordan, Eds., Electroporation and Electrofusion in Cell Biology. New York, NY: Plenum, 1989. [2] D. C. Chang, B. M. Chassy, J. A. Saunders, and A. E. Sowers, Eds., Guide to Electroporation and Electrofusion. New York: Academic, 1992. [3] J. C. Weaver, “Electroporation: A general phenomenon for manipulating cells and tissue,” J. Cellular Biochem., vol. 51, pp. 426–435, 1993. [4] U. Zimmermann, Ed., The Effects of High Intensity Electric Filed Pulses on Eukaryotic Cell Membranes: Fundamentals and Applications. Boca Raton, FL: CRC, 1996. [5] J. C. Weaver and Y. A. Chizmadzhev, “Theory of electroporation: A review,” Bioelectrochem. Bioenerget., vol. 41, pp. 135–160, 1996. [6] J. C. Weaver, “Electroporation of cells and tissues,” IEEE Trans. Plasma Sci., vol. 28, pp. 24–33, Feb. 2000. [7] K. H. Schoenbach, S. Katsuki, R. H. Stark, E. S. Buescher, and S. J. Beebe, “Bioelectrics-new applications for pulsed power technology,” IEEE Trans. Plasma Sci., vol. 30, pp. 293–300, Feb. 2002. [8] J. C. Weaver, “Electroporation of biological membranes from multicellular to nano scales,” IEEE Trans. Dielect. Elect. Insulation, vol. 10, pp. 754–768, Oct. 2003. [9] K. H. Schoenbach, S. J. Beebe, and E. S. Buescher, “Intracellular effect of ultrashort pulses,” Bioelectromagn., vol. 22, pp. 440–448, 2001. [10] S. J. Beebe, P. M. Fox, L. J. Rec, K. Somers, R. H. Stark, and K. H. Schoenbach, “Nanosecond pulsed electric field (nsPEF) effects on cells and tissues: Apoptosis induction and tumor growth inhibition,” IEEE Trans. Plasma Sci., vol. 30, pp. 286–292, Feb. 2002. [11] S. J. Beebe, P. M. Fox, L. J. Rec, L. K. Willis, and K. H. Schoenbach, “Nanosecond, high intensity pulsed electric fields induce apoptosis in human cells,” in Fed. Amer. Soc. Exp. Biol. J., vol. 17, 2003, pp. 1493–1495. [12] R. Benz and U. Zimmermann, “Relaxation studies on cell membranes and lipid bilayers in the high electric field range,” Bioelectrochem. Bioenerg., vol. 7, pp. 723–739, 1980. , “Pulse-length dependence of the electrical breakdown in lipid bi[13] layer membranes,” Biochim. Biophys. Acta, vol. 597, pp. 637–642, 1980. [14] T. R. Gowrishankar and J. C. Weaver, “An approach to electrical modeling of single and multiple cells,” Proc. Nat. Acad. Sci., vol. 100, pp. 3203–3208, 2003. [15] T. R. Gowrishankar, W. Chen, and R. C. Lee, “Non-linear microscale alterations in membrane transport by electropermeabilization,” Ann. NY Acad. Sci., vol. 11, pp. 205–216, 1998. [16] M. Bier, W. Chen, T. R. Gowrishankar, R. D. Astumian, and R. C. Lee, “Resealing dynamics of a cell membrane after electroporation,” Phys. Rev. E, vol. 66, pp. 062 905-1–062 905-4, 2002. [17] K. A. DeBruin and W. Krassowska, “Modeling electroporation in a single cell: I. Effects of field strength and rest potential,” Biophys. J., vol. 77, pp. 1213–1224, 1999.

where . For times much smaller than , this reduces to (10) which is independent of . Although this was derived for a cylindrical cell, this size independence applies generally to all cell shapes and subcellular membranes. For the “conventional electroporation” pulse, the daughter yeast cell was not electroporated. However, for the “supra-electroporation” pulse, both cells were electroporated even far from the “poles.” IV. SUMMARY The asymptotic model significantly simplifies creation of models, in that the asymptotic ODE (3) is used instead of a second order partial differential equation (1). Experimental conditions that involve expanded pores (e.g., transport of large molecules) may not be adequately described by the asymptotic model [31]. The present models demonstrate the flexibility of transport lattice method to predict nonlinear membrane response to a variety of cell membrane geometries. The motivation for considering a transport lattice approach to modeling of single and multiple cells includes: 1) description of both field and transmembrane voltage-dependent interactions within a single model; 2) provision for changes of local electrical properties (e.g., local membrane conductance increase due to electroporation); 3) irregular shaped cell membranes, sometimes in close proximity; and 4) inclusion of both “weak” and “strong” field effects in a single model, so that the model would appropriately respond to both small and large electric fields [14]. The advantage of a transport lattice model is the ability to include different local models, ranging from linear passive local models

ACKNOWLEDGMENT The authors thank Z. Vasilkoski, K. C. Smith, and A. Esser for comments, and K. G. Weaver for computer support. REFERENCES

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Donald A. Stewart, Jr. received the B.S. degree from the University of Illinois, Champaign, in 1981 and the Ph.D. degree in high energy physics from Indiana University, Bloomington, in 1988. He is a Research Scientist at the Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge. He worked as a Senior Research Scientist at Bio-logic Systems Corp. Mundelein, IL, from 1995 to 2001. He was a Research Fellow at University of Michigan, Ann Arbor, from 1992 to 1995. Prior to that, he was a postdoc at the Fermi National Accelerator Laboratory, Batavia, IL. His reserach interests include physics, modeling of biological systems, artificial neural networks, and biological signal processing.

T. R. Gowrishankar received the B.E. degree from Bangalore University, Bangalore, India, in 1986, the M.S. degree from George Mason University, Fairfax, VA, in 1989, and the Ph.D. degree in medical physics from the University of Chicago, Chicago, IL, in 1996. He is currently a Postdoctoral Associate at the Harvard-MIT Division of Health Sciences and Technology at the Massachusetts Institute of Technology, Cambridge, MA. His research interests include experimental and simulation studies of bioelectric phenomena, drug delivery, and electrical injury.

James C. Weaver, photograph and biography not available at the time of publication.