Dielectric permittivity of water confined in stacks of charged lipid layers: extracting profiles from molecular dynamics simulations using a modified Poisson-Boltzmann equation

TL;DR

This work tackles how confinement between charged lipid stacks alters water dielectric response and interlayer forces. It introduces a spatially varying dielectric modified Poisson-Boltzmann framework with explicit Born solvation terms, calibrated against MD data to extract ε_perp(z) and ion distributions. The results show a sharp dehydration-driven drop in permittivity with a plateau attributable to lipid headgroups, and a local maximum at intermediate hydration, supporting a link between lowered permittivity and intermembrane attraction. Overall, the mPB approach provides a generic, tractable method to estimate dielectric profiles and related forces in confined electrolyte systems with free charges, extendable to other biological and solid-wall interfaces.

Abstract

Most organic and inorganic surfaces (e.g., glass, nucleic acids or lipid membranes) become charged in aqueous solutions. The resulting ionic distribution induces effective interactions between the charged surfaces. Stacks of like-charged lipid bilayers immersed in multivalent ion solutions exhibit strong coupling (SC) effects, where ion correlations cause counter-intuitive membrane attraction. A similar attraction observed with monovalent ions is explained by SC theory through reduced dielectric permittivity under confinement. To explore this phenomenon, we propose a modified Poisson-Boltzmann (mPB) model with spatially varying dielectric permittivity and explicit Born solvation energy for ions. We use the model to investigate the dielectric permittivity profile of confined water in molecular dynamics simulations of charged lipid layers stacks at varying hydration levels, and compare the results with alternative computational methods. The model captures a sharp decrease in permittivity upon dehydration, converging to a plateau value that we attribute to lipid headgroups. The generic nature of the mPB framework allows application to other systems, such as other biological interfaces or solid walls, provided ions follow Boltzmann statistics. Finally, the increase of the area per lipid in our tension-free simulations of the fluid membranes hints that the permittivity decrease upon dehydration is concomitant with an intermembrane attraction.

Figures (18)

• Figure 1: (a, b) Snapshots of systems at HN = 35, respectively at $T$ = 353.15 K (bilayer in the fluid phase) and $T$ = 333.15 K (bilayer in the gel phase). The blue boxes represent the simulation boxes, replicated in all directions of space. For clarity, the lipids' hydrogen atoms are not drawn in the snapshots, nor are water molecules in the simulation boxes. The $z$ coordinate is set at 0 at the centre of the left bilayer. It reaches $z_{\mathrm{max}}$ = $L_z / 2$ at the middle of the interlayer water. (c) Detail of the DPPS lipid molecule. The gold and silver colours respectively correspond to the heads and the tails of the lipids as drawn in the snapshots. (d) Interlayer species.
• Figure 2: Results from the modified Poisson Boltzmann model at 353.15 K, at HN = 35 (left) and HN = 10 (right). (a, d) Relative dielectric permittivity profile. The lipid in the background represents the approximative $z$ position of the tails and the heads in the lipid layer. (b, e) Reduced electric potential $\phi(z)$ and electric potential $V(z)$. (c, f) Charge densities. Markers show charge densities extracted from the MD simulation (for clarity, only one data point out of seven is plotted). Lines show charge densities obtained with the mPB model, i.e. with Eq. \ref{['eq:modifiedIonsBoltzmannDistribution']}. According to Eq. \ref{['eq:modifiedPoisson1D']}, both blue plots include the lipid charge density extracted from the MD simulation.
• Figure 3: Maximum dielectric permittivity versus hydration number at different temperatures, for bilayers in the fluid phase (top) and bilayers in the gel phase (bottom). The colours are the same as in Fig. \ref{['fig:ApL_vs_HN']} and the open/closed markers have the same meaning. The purple region shows where we identify an attractive regime between the membranes (see Sect. \ref{['Subsec:Discussion:forces']}). The mPB model cannot be used in the hatched region because the ions no longer follow a Boltzmann distribution.
• Figure 4: Evolution of the plasma parameter at different HN. Above the dashed line at $\Gamma$ = 1, ion correlations start to become non negligible.
• Figure 5: Comparison between $\varepsilon_{\perp}^{\mathrm{mPB}}$, $\varepsilon_{\perp}^{\mathrm{Born}}$ and $\varepsilon_{\perp}^{\mathrm{water~density}}$ for two systems at $T$ = 353.15 K. (a) HN = 35. (b) HN = 10.