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. 2008 Aug;95(4):1547-63.
doi: 10.1529/biophysj.107.121921. Epub 2008 Apr 11.

Active mechanisms are needed to describe cell responses to submicrosecond, megavolt-per-meter pulses: cell models for ultrashort pulses

Kyle C Smith  1 , James C Weaver
Affiliations

Active mechanisms are needed to describe cell responses to submicrosecond, megavolt-per-meter pulses: cell models for ultrashort pulses

Kyle C Smith et al. Biophys J. 2008 Aug.

Abstract

Intracellular effects of submicrosecond, megavolt-per-meter pulses imply changes in a cell's plasma membrane (PM) and organelle membranes. The maximum reported PM transmembrane voltage is only 1.6 V and phosphatidylserine is translocated to the outer membrane leaflet of the PM. Passive membrane models involve only displacement currents and predict excessive PM voltages (approximately 25 V). Here we use a cell system model with nonconcentric circular PM and organelle membranes to demonstrate fundamental differences between active (nonlinear) and passive (linear) models. We assign active or passive interactions to local membrane regions. The resulting cell system model involves a large number of interconnected local models that individually represent the 1), passive conductive and dielectric properties of aqueous electrolytes and membranes; 2), resting potential source; and 3), asymptotic membrane electroporation model. Systems with passive interactions cannot account for key experimental observations. Our active models exhibit supra-electroporation of the PM and organelle membranes, some key features of the transmembrane voltage, high densities of small pores in the PM and organelle membranes, and a global postpulse perturbation in which cell membranes are depolarized on the timescale of pore lifetimes.

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Figures

FIGURE 1
FIGURE 1
Cell system model geometry. (A) The isolated cell is centered in a large (80 μm × 80 μm) region. The upper (anode) and lower (cathode) boundaries are planar electrodes. (B) The radii of the plasma membrane (PM), large organelle membrane (LOM), and small organelle membrane (SOM) are rPM = 8 μm, rLOM = 3 μm, and rSOM = 0.5 μm. These idealized, single-layer membranes represent the plasma membrane, nuclear envelope, and mitochondrial membrane. The poles, or polar regions, discussed in the text refer to the regions of greatest |y| for each membrane. The polar regions of membrane are perpendicular to the applied electric field.
FIGURE 2
FIGURE 2
Cell system mesh and Voronoi cells. The cell system (A) mesh and (B) Voronoi cells are shown at four scales with black dots indicating membrane-surface nodes. The mesh has 19,061 nodes, 38,061 triangles, and 44,691 edges. For a sense of scale, the lengths of the subfigure sides are (left to right) 80, 24, 7, and 0.3 μm.
FIGURE 3
FIGURE 3
Two-dimensional transport system. (A) Triangular mesh and Voronoi cells (VCs). The two-dimensional system is discretized into a set of VCs (solid) associated with the nodes connected by triangulation (shaded). (B) Adjacent Voronoi cells. The VCs have depth d and an interface of length wj,k, and the distance between the VC nodes is lj,k. The VCs have electric potentials φj and φk and, at the VC interface, there is an electric field formula image and current density formula image which can be broken into components normal (formula image and formula image) and parallel (formula image and formula image) to the interface.
FIGURE 4
FIGURE 4
Local model equivalent subcircuits for cell system model (Figs. 1 and 2). An electrolyte or membrane subcircuit is placed between each pair of adjacent nodes in the cell system. The electrical transport is determined by the local mesh geometry, passive electrical properties of the electrolyte and membranes, and, at the membranes, by the instantaneous pore density and associated time-dependent conductance, which provide a rapidly changing (active) response mechanism. In the active model, each membrane subcircuit has an associated pore density (units: m−2) subcircuit (35) that is used to calculate the total current through pores. In the passive model, there are no pores, and the membrane conductance is constant.
FIGURE 5
FIGURE 5
Electric field pulse. The pulse applied to the cell system model was an idealized trapezoidal version of the pulse used experimentally by Frey et al. (37) (95 kV/cm; 6 ns rise-time, 55 ns plateau, and 10 ns fall-time). Solid dots indicate the times at which the results are plotted in Figs. 6 and 7.
FIGURE 6
FIGURE 6
Passive and active cell responses. The electric potential and pore density are shown for the cell models (A–D) during and (E–G) after the electric pulse. For the active model, pore density is indicated by the white line thickness (1014, 1015, 1016 m−2). Twenty-one contour lines are uniformly spaced between the extreme values of their associated grayscale bars. Note that the intracellular and extracellular electric field magnitude are not equal, even early in the pulse, because σi = σe/4 (Table 1). See Fig. 10 for the case σi = σe.
FIGURE 7
FIGURE 7
Angular distributions of responses. The (A–F) transmembrane voltage and (G–I) pore density are shown as a function of angle for the plasma membrane (PM) and each organelle membrane (LOM, SOM) for the passive and active cell models. Times shown (from dark to light gray) are 4, 6, 25, and 61 ns. Θ = 90° is the anodic pole and Θ = 270° is the cathodic pole. Note that there are few changes in Vm and N between 25 ns and 61 ns and, consequently, the 61 ns traces obscure the 25 ns traces in many of the plots.
FIGURE 8
FIGURE 8
Temporal responses. (A and B) Transmembrane voltage and (C) pore density at the anodic (solid lines) and cathodic (dashed lines) poles of the plasma membrane (PM) and organelle membranes (LOM, SOM) for the (A) passive and (B and C) active cell models. Each plot is shown for 10 ns (top with dotted-line indicating end of pulse rise-time) and 100 ns (bottom) timescales. Note the significant differences in the voltage scales for passive and active models. The initial, pre-electroporation Vm are the same for both models, but the post-electroporation Vm differ greatly. For the passive model, the Vm increase throughout the pulse and peak at the end of the applied pulse plateau (61 ns). For the active model, the maximum Vm occur at ∼4 ns because the accompanying burst in pore creation and REB cause a large increase the membrane conductance and concomitant decrease in Vm.
FIGURE 9
FIGURE 9
Membrane current. The ratio of the conduction current to the displacement current for the anodic (solid) and cathodic (dashed) PM sides for the active (solid) and passive (shaded) models. After ∼4 ns, the conduction current is ∼3 orders-of-magnitude larger than the displacement current for the active model. In contrast, the conduction current remains ∼5 orders-of-magnitude smaller than the displacement current for the passive model.
FIGURE 10
FIGURE 10
Passive and active cell responses for σi = σe = 1.2 S/m. The electric potential and pore density are shown for the passive and active models at (A) 6 ns and (B) 61 ns for σi = σe = 1.2 S/m. For the active model, pore density is indicated by the white line thickness (1014, 1015, 1016 m−2). Twenty-one contour lines are uniformly spaced between the extreme values of their associated grayscale bars. (A) Initially, the electric field magnitude in the intracellular and extracellular regions is approximately equal. (B) For the passive model, at the end of the pulse plateau the intracellular electric field magnitude is significantly smaller than the extracellular electric field magnitude. For the active model, the electric field magnitude in the intracellular and extracellular regions remains approximately equal because of the high conductance of the electroporated membranes.
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References

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