Hydrodynamic liquid crystal models for lipid bilayers

TL;DR

The paper addresses the limitation of purely curvature-based Helfrich models by introducing a scalar order parameter $β$ that encodes molecular alignment normal to the membrane surface. It develops two thermodynamically consistent, fully hydrodynamic models on moving surfaces: a surface Beris–Edwards (BE) model for symmetric lipid bilayers and a surface Landau–Helfrich (LH) model for asymmetric bilayers, with curvature–order coupling governed by $β$ and the mean curvature $\mathcal{H}$. In the fully ordered limit $β=\frac{2}{3}$, both models reduce to the conventional surface (Navier–)Stokes–Helfrich system, providing an alternative derivation for these classical dynamics. The LH formulation further enables curvature-order interactions that break up–down symmetry, enabling spontaneous curvature effects tied to molecular ordering and richer dynamic behavior for asymmetric bilayers.

Abstract

Coarse-grained continuous descriptions for lipid bilayers are typically based on minimizing the Helfrich energy. Such models consider the fluid properties of these structures only implicitly and have been shown to nicely reproduce equilibrium properties. Model extensions that also address the dynamics of these structures are surface (Navier--)Stokes--Helfrich models. They explicitly account for membrane viscosity. However, these models also usually treat the lipid bilayer as a homogeneous continuum, neglecting the molecular degrees of freedom of the lipids. Here, we derive refined models which consider in addition a scalar order parameter representing the molecular alignment of the lipids along the surface normal. Starting from hydrodynamic surface liquid crystal models, we obtain a hydrodynamic surface Landau--Helfrich model for asymmetric lipid bilayers and a surface Beris--Edwards model for symmetric lipid bilayers. The fully ordered case for both models leads to the known surface (Navier--)Stokes--Helfrich models. Besides more detailed continuous models for lipid bilayers, we therefore also provide an alternative derivation of surface (Navier--)Stokes--Helfrich models.

Figures (4)

• Figure 1: Symmetric Lipid Bilayer. Left: lipid molecules are in a fully ordered state ($\beta=\frac{2}{3}$), i. e. they are perfectly aligned perpendicular to the surface $\mathcal{S}$. Right: the degree of orientational order $\beta$ decreases from left to right, while the mean molecular alignment remains normal to the surface. The bilayer is represented by a surface $\mathcal{S}$ (green line), and the molecular orientation by a Q-tensor field $\boldsymbol{Q}$ fulfilling ansatz \ref{['eq:QAnsatz']}, i. e. $\boldsymbol{Q}$ is depicting an apolar normal field (gray rods) with order parameter $\beta$ (grayscale). For $\beta \neq 0$, the lipid molecules are not in an isotropic state, the geometric minimal configuration is obtained by minimizing the Helfrich energy with zero spontaneous curvature, thus leading to a flat surface.
• Figure 2: Asymmetric Lipid Bilayer. The asymmetry may arise through various mechanisms. We provide some examples. Left: differing molecular compositions. Center & Bottom: differing molecular densities. Right: scaffold protein. The molecular orientation is represented by a Q-tensor field $\boldsymbol{Q}$ fulfilling ansatz \ref{['eq:QAnsatz']}, i. e. $\boldsymbol{Q}$ is depicting an apolar normal field (gray rods) with order parameter $\beta$ (grayscale). Top: In the ordered state ($\beta=\frac{2}{3}$), with all lipid molecules aligned normal to the surface $\mathcal{S}$ (green curve), the minimal geometric configuration is achieved when the mean curvature takes its spontaneous curvature value ($\mathcal{H}_{0}$). Bottom: A less ordered non-uniform state ($\beta < \frac{2}{3}$) may counteract this effect, since another spontaneous curvature ($\hat{\mathcal{H}}_0$) related to the isotropic state ($\beta = 0$) can be imposed additionally. The geometric minimal configuration is achieved for a curved surface with the mean curvature depending on $\beta$, $\mathcal{H}_{0}$ and $\hat{\mathcal{H}}_0$.
• Figure 3: Energy density plots of the double-well potential \ref{['eq:DW']}. In the left plot we stipulate $\hat{a}=0$ and choose some values for $\varpi$. Their additive inverse equal the minimum values at the fully ordered state $\beta=\frac{2}{3}$. In the right plot we use $\varpi=1$ and specify some values for $\hat{a}$. Only $\hat{a}=0$ yields an inflection point at the isotropic state $\beta=0$ instead of a minimum for $\hat{a} > 0$.
• Figure 4: Energy density plots of the energy \ref{['eq:LdGReduced__hatH0_equals_0']} ($\hat{\mathcal{H}}_0=\hat{a}=0$) with $L=\bar{\kappa}=0$. We stipulate $\kappa=5$ and $\mathcal{H}_{0}=\varpi=1$. In the left plot we fix the mean curvature $\mathcal{H}$ for some fixed values around $\mathcal{H}_{0}=1$. Additional we add the $\kappa=0$ case, which does not depend on $\mathcal{H}$ and leads $\beta$ to the fully ordered state $\beta=\frac{2}{3}$. The right plot shows the dependency w. r. t. $\mathcal{H}$ for some fixed $\beta$'s.