Three-slab model for the dielectric permittivity of a lipid bilayer

Abstract

A model for the tensorial dielectric permittivity of phospholipid membranes is presented here. The four-nanometer-thick membrane is treated as a composite made up of three dielectric slabs: one for each of the two phospholipid head-group regions, and one for the entire domain spanned by the lipid tails. Equal and opposite bound surface charge densities surround each head-group slab, and account for the membrane dipole potential. Three-slab model parameters are obtained from molecular dynamics simulations, and capture both the zero-field electric potential and the membrane response to applied electric fields. The tail region is well-approximated as having vacuum permittivity, while the head-group region is highly anisotropic due to the configurations of molecular dipoles. For the bilayers studied, the out-of-plane permittivity of the head-group region is 10--15 times that of the vacuum, while the in-plane permittivity is an order of magnitude larger. Membrane responses to applied electric fields up to 30 millivolts per nanometer are found to be in the linear regime. The model overcomes a fundamental limitation of microscopic theories -- where the out-of-plane permittivity is ill-posed in the head-group region due to large gradients in the local electric field -- by averaging over slab widths, thereby introducing new length scales. Our approach can be extended to characterize general interfacial systems with similarly ill-defined permittivities.

Figures (4)

• Figure 1: DPPC phospholipid bilayer under zero applied field, from MD simulations. (a) Representative structure of phospholipid molecules in the upper leaflet, along with nearby water molecules. Each molecule has an electric dipole moment, which is separated into in-plane and out-of-plane components as $\bm{m} = \bm{m}^{\mkern3mu\newline\mkern2mu\mkern3mu} + m^\perp \mkern1mu \bm{e}_z$. (b) Probability distribution of $\lvert \bm{m} \rvert$ and $\lvert \bm{m}^{\mkern3mu\newline\mkern2mu\mkern3mu} \rvert$ for phospholipid molecules. (c) Average out-of-plane polarization density $\langle P_{\mkern-1mu\mkern-1mu z} \rangle_0$ of water, lipids, and the combined system. While phospholipid dipoles in the top (resp. bottom) leaflet point up (resp. down), intercalated water molecules are oriented in the opposite direction and dominate the overall polarization density. Inset: $\langle P_{\mkern-1mu\mkern-1mu z} \rangle_0$ for the system and three-slab model.
• Figure 2: Three-slab model parametrized with results from molecular simulations. (a) Model parameters and geometry, as described in the text. (b) Zero-field electric potential of a DPPC membrane as a function of distance $z$ from the bilayer midplane. Three-slab model parameters are chosen to match the MD dipole potential, $\langle \phi (z = 0) \rangle^{}_0 - \langle \phi (z = \ell/2) \rangle^{}_0$, where $\ell$ is the height of the MD unit cell.
• Figure 3: Permittivity of a DPPC membrane. (a) The in-plane permittivity $\epsilon^{\mkern3mu\newline\mkern2mu\mkern3mu} (z)$ is calculated from MD simulations via the linear response and fluctuation formulas in Eq. \ref{['eq_epspar']}. The two approaches agree for electric fields up to 30 mV$/$nm, beyond which the system response is nonlinear (see the SM supplemental). The in-plane permittivity of the three-slab model (thin dashed line) captures essential features of the MD result according to Eq. \ref{['eq_in_plane']}. (b) Reciprocal of the out-of-plane permittivity $\epsilon^\perp (z)$, as calculated from Eq. \ref{['eq_epsperp_fluct']}. In the head-group region, $\epsilon^\perp$ takes unphysical values (grey shading), for which $1 / \epsilon^\perp$ crosses zero.
• Figure 4: Change in electric potential (a) and out-of-plane polarization density (b) of a DPPC bilayer due to an electric field $\bm{\hat{E}} = \hat{E}_z \bm{e}_z$. The three-slab parameters (see Table \ref{['tab_params']}) are obtained with $\hat{E}_z = 20$ mV$/$nm, and well-approximate the results here, for which $\hat{E}_z = 70$ mV$/$nm. (a) The model captures the MD response, with $\langle \Delta E_z \rangle := - \langle \Delta \phi' (z) \rangle$ largely confined to the tail region. (b) The change in polarization is predominantly in the head-group regions and surrounding water, and is reasonably described by the model.