Modeling cell membrane electrodeformation by alternating electric fields

With the aim of characterizing and gaining insight into the frequency response of cells suspended in a fluid medium and deformed with a controlled alternating electric field, a continuum-based analysis is presented for modeling electrodeformation (ED) via Maxwell stress tensor (MST) calculation. Our purpose here is to apply this approach to explain the fact that the electric field anisotropy and electrical conductivity ratio Λ of the cytoplasm and the extracellular medium significantly impact the MST exerted on the cytoplasm-membrane interface. One important finding is that the modulation of electrical cues and MST force by the frequency of the applied electric field provides an extremely rich tool kit for manipulating cells. We show the extreme sensitivity of proximity-induced capacitive coupling arising concomitantly when the magnitude of the MST increases as the distance between cells is decreased and the spatial anisotropy becomes important. Moreover, our model highlights the strongly localized character of the electrostatic field effect emanating from neighboring cells and suggests the possibility of exploiting cell distribution as a powerful tool to engineer the functional performance of cell assemblies by controlling ED and capacitive coupling. We furthermore show that frequency has a significant impact on the attenuation-amplification transition of MST, suggesting that shape anisotropy has a much weaker influence on ED of the cell membrane compared to the anisotropy induced by the orientation angle itself.

Figure 1

Schematic of a typical square computational domain filled with the ECM to study the amplitude of the MST at point $a^{\prime }$. The two cells are initially separated by distance $r$ (not to scale). The orientation angle denoted by θ refers to the angle of the line joining the cell centers of mass relative to the electric field direction applied along the $y$ direction. Right and left boundaries are insulated while the top and bottom walls are prescribed with an applied potential. Dirichlet boundary conditions at the top and bottom boundaries are justified as the electric field is applied through discrete electrodes; $a$ and $a^{\prime }$ define the points on the outer and inner layer of the cell membrane allowing us to calculate the TMP at the pole. Surface charge distributions of $\pm 6.2\times {10}^{\text{–}4}\mathrm{C}{\mathrm{m}}^{-2}$ were added to both surfaces $A_{\mathrm{in}}$ and $A_{\mathrm{out}}$ in order to account for a resting potential of –70 mV.

Figure 2

(a) TMP at the pole for the reference single cell as a function of frequency of the electric field. Solid (resp., dotted) lines correspond to Λ set to 15 (resp., 0.1); (b) as in (a) for the two-cell configuration (Fig. S1 [20]). Blue (resp., red) lines correspond to $\theta =0$ (resp., $\theta =\pi /2$).

Figure 3

(a) MST at the pole for the reference single cell as a function of frequency of the electric field. Solid (resp., dotted) lines correspond to Λ set to 15 (resp., 0.1); (b) as in (a) for the two-cell configuration (Fig. S1 [20]). Blue (resp., red) lines correspond to $\theta =0$ (resp., $\theta =\pi /2$).

Figure 4

(a) Effect of increasing the rotation angle θ on the local distribution of the cell displacement arising from ED forces for the single-cell and two-cell configurations. The electric field is vertically oriented and frequency is fixed to 10 MHz. White arrows represent the direction of the local displacement field and the color bar shows the total displacement. (b) Evolution of the deformation induced by the electric field in terms of the aspect ratio $b$/$a$ (defined as the ratio of semimajor axis $b$ to semiminor axis $a$ of the ellipsoid) as a function of frequency of the electric field. The solid and dotted lines correspond to $\mathrm{\Lambda }=15$ and $\mathrm{\Lambda }=0.1$, respectively, while the red and blue curves correspond to values of θ set to π/2 and 0, respectively. Black arrows are guides to the eye for the curves associated with different values of θ.

Figure 5

Characterizing the anisotropy of the attenuation-amplification transition of the excess MST taken at the pole of the reference cell as a function of frequency of the electric field for values of Λ ranging from 0.1 (bottom) to 15 (top). Τhe color bar represents the excess MST at the pole of the reference cell in percent. The blue and red regions correspond, respectively, to attenuating and amplifying proximity effect induced by the presence of the neighboring cell. A cubic spline interpolation algorithm in the python programming language was used to plot the diagram. The simulation data on the $y$ axis correspond to values of $\theta =i(\pi /10)$, with $i=0,...,5$ and data on the $x$ axis were logarithmically sampled at a rate of eight points per decade.