Thickness effects in the electromechanical stability of charged biological membranes

Abstract

Understanding how electric fields destabilize biological membranes is important for electroporation-based technologies and bioelectronic interfaces. However, theoretical descriptions of this phenomenon remain fragmented. Existing theories treat either electrostatics in membranes of finite thickness or electrohydrodynamic flows at idealized zero-thickness interfaces, leaving unresolved a unified description that simultaneously incorporates finite membrane thickness, surface charge, and bulk electrohydrodynamics. Here, we apply a recently-developed, dimension-reduction framework that captures the coupled electrohydrodynamic and mechanical effects governing height fluctuations of a charged lipid bilayer of thickness $δ$ in an electrolyte characterized by Debye screening length $λ$. We derive voltage- and charge-dependent renormalizations of the effective surface tension and bending rigidity, along with a dispersion relation governing undulatory instabilities. A wide range of prior theoretical results arise as limiting cases of our more general theory when finite-thickness effects are neglected or screening is asymptotically strong. The key new contribution arises from traction moments generated across the finite membrane thickness, which are absent in zero-thickness descriptions. Under physiological screening ($δ/λ\sim 4$), these contributions account for more than $>70\%$ of the total electrostatic correction to both surface tension and bending rigidity. The theory further reveals that surface charges can stabilize the membrane at physiological ionic strengths, increasing the effective tension and shifting electroporation thresholds in a manner that depends on charge asymmetry between the leaflets.

Figures (6)

• Figure 1: A lipid membrane of thickness $\delta$ bearing intrinsic surface charges (large purple spheres on upper/lower leaflets) is subjected to a transverse voltage ($+V$ at the upper electrode, $-V$ at the lower electrode). Each electrode is placed at a distance ${L \gg \lambda}$ from the nearest membrane surface, when the membrane is flat. The upper and lower membrane leaflets carry surface charge densities $\breve{\sigma}^+$ and $\breve{\sigma}^-$, respectively. The applied field induces electrostatic interactions with the surrounding electrolyte ions (blue and orange spheres), modifying the membrane's intrinsic surface tension $\Lambda$ and bending rigidity $k_\text{b}$ to their effective values $\Lambda^{\mathrm{eff}}$ and $k_{\text{b}}^{\mathrm{eff}}$. These field-renormalized mechanical properties regulate membrane undulations $h(x,y,t)$ and stability.
• Figure 2: Schematic of the membrane surface showing the local tangent basis vectors $\boldsymbol{g}^\pm_{\alpha}$ and the unit normal vector $\boldsymbol{n}$. The total traction vectors acting on the upper and lower membrane leaflets are denoted by $\boldsymbol{t}^{+}$ and $\boldsymbol{t}^{-}$, respectively. Their projections along the tangent directions $\boldsymbol{t}^{\pm}\!\cdot \boldsymbol{g}^\pm_{\alpha}$ and the normal direction $\boldsymbol{t}^{\pm}\!\cdot \boldsymbol{n}$ are indicated.
• Figure 3: Bar charts decomposing the effective surface tension (left) and bending rigidity (right) into their voltage-driven ($V$), surface-charge-driven ($\sigma$), and mixed (${V\sigma}$) contributions for four representative membrane conditions (Cases 1--4). The dimensionless voltage is fixed at $\bar{V}=3$, and the mean dimensionless surface charge density is ${\langle \bar{\sigma}\rangle=-3.6}$. The corresponding charge asymmetry $\alpha$ and dimensionless thickness $\bar{\delta}$ for each case are indicated above the bars. For the surface tension, the voltage contribution is destabilizing in all cases. The surface-charge contribution is stabilizing when ${\bar{\delta}>1}$ and destabilizing when ${\bar{\delta}<1}$. Charge asymmetry further modulates stability: ${\alpha>0}$ (outer leaflet more negatively charged) stabilizes the membrane, whereas ${\alpha<0}$ (inner leaflet more negatively charged) promotes destabilization for ${\bar{V}>0}$. The bending rigidity panel is shown on a symmetric logarithmic scale to resolve contributions spanning multiple orders of magnitude.
• Figure 4: (a,b) Dispersion relations $\omega(q)$ for four representative cases: Bare (no applied voltage and no surface charge), $\text{V}$ (nonzero voltage but no surface charge), $\breve{\sigma}$ (nonzero surface charge but no voltage), and $V \,\& \, \breve{\sigma}$ (both present). Parameter values are $\bar{V}\in \{0,3\}$, $\langle \bar{\sigma} \rangle\in \{0,-3.6\}$, and $\alpha\in \{0,0.5\}$. Panel (a) corresponds to strong screening ($\bar{\delta}=4$, $\lambda=1\,\mathrm{nm}$), and panel (b) to weak screening ($\bar{\delta}=0.4$, $\lambda=10\,\mathrm{nm}$). Regions with ${\omega(q)<0}$ indicate stable decay of fluctuations, while ${\omega(q)>0}$ signals unstable growth. (c,d) Stability diagrams in the $(\bar{V},\langle \bar{\sigma} \rangle,\alpha)$ parameter space for the same two screening regimes. Red regions denote instability ($\Lambda^{\mathrm{eff}}<0$), and blue regions correspond to stable configurations ($\Lambda^{\mathrm{eff}}>0$). Increasing the Debye length enlarges the unstable region, indicating the destabilizing influence of longer-ranged electrostatic interactions.
• Figure 5: Bar charts showing the percentage contribution of Term 1 to the total electrostatic renormalization (Term 1 + Term 2) of the surface tension $\Lambda$ (left) and bending rigidity $k_{\rm b}$ (right) for the voltage-driven, charge-driven, and mixed ($V\sigma$) mechanisms. Results are shown for the same four membrane conditions (Cases 1–4) as in Fig. \ref{['fig:barplot']}. The dimensionless voltage is fixed at $\bar{V}=3$, and the mean dimensionless surface charge density at $\langle\bar{\sigma}\rangle=-3.6$. The corresponding charge asymmetry $\alpha$ and dimensionless thickness $\bar{\delta}$ for each case are indicated in the figure annotations.