Small scale membrane mechanics

Abstract

Large scale changes to lipid bilayer shapes are well represented by the Helfrich model. However, there are membrane processes that take place at smaller length scales that this model cannot address. In this work, we present a one-dimensional continuum model that captures the mechanics of the lipid bilayer membrane at the length scale of the lipids themselves. The model is developed using the Cosserat theory of surfaces with lipid orientation, or ‘tilt’, as the fundamental degree of freedom. The Helfrich model can be recovered as a special case when the curvatures are small and the lipid tilt is everywhere zero. We use the tilt model to study local membrane deformations in response to a protein inclusion. Parameter estimates and boundary conditions are obtained from a coarse-grained molecular model using dissipative particle dynamics (DPD) to capture the same phenomenon. The continuum model is able to reproduce the membrane bending, stretch and lipid tilt as seen in the DPD model. The lipid tilt angle relaxes to the bulk tilt angle within 5–6 nm from the protein inclusion. Importantly, for large tilt gradients induced by the proteins, the tilt energy contribution is larger than the bending energy contribution. Thus, the continuum model of tilt accurately captures behaviors at length scales shorter than the membrane thickness.

Continuum descriptions of membranes and their interaction with proteins: Towards chemically accurate models

“…As discussed in Section 3, the membrane exhibits internal lipid orientation degrees of freedom that are independent of the curvatures and compression, and many studies have described how to couple membrane proteins to the lipid tilt [71, 106–111]. Fournier proposed one of the earliest phenomenological models based on symmetry expansions of a Helfrich-type Hamiltonian in two structural variables for each monolayer: one for lipid orientation and one for shape [106].…”

Bilayer-thickness-mediated interactions between integral membrane proteins

“…(3) and the corresponding "zeroth-order" models [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68] capturing curvature-and fluctuation-mediated protein interactions absorb the molecular details of lipids and membrane proteins into effective material parameters. To provide a more detailed description of bilayer-protein interactions, a number of extensions and refinements of these models have been developed [44,45,[69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85][86]. For instance, the effect of bilayer-protein interactions on the elastic properties of lipid bilayers can be captured [44,81,82,86] by allowing for spatial variations in the values of the elastic bilayer parameters.…”

“…The minimal model in Eq. (3) can be extended in a variety of ways to account for more detailed properties of lipid bilayers including lipid tilt [48,[75][76][77][78][79], lipid intrinsic curvature [39-42, 45, 70, 71], inhomogeneous deformation of lipid volume and effects of Gaussian curvature on protein-induced bilayer deformations [45,70,71], asymmetric bilayer thickness deformations [45,70,71,80], and protein-induced local modulation of bilayer elastic properties [44,[81][82][83][84][85][86].…”

“…The classic elasticity theory of protein-induced lipid bilayer deformations can be extended in various ways [44,45,[69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85][86] to account for detailed molecular properties of lipids such as lipid tilt and lipid intrinsic curvature [39][40][41][42]48], yielding additional modes of bilayermediated protein interactions. Theoretical studies of bilayer-mediated protein interactions have largely focused on idealized (often cylindrical or conical) protein shapes which do not correspond to any particular membrane protein, and proteins at large d. In contrast, bilayer-mediated protein interactions at small d are most relevant for the crowded environment provided by cell membranes [3,[21][22][23], while modern structural biology suggests a rich picture of membrane protein shape with experimental surveys of the protein content in various cell membranes [21,23,87] indicating great diversity in the oligomeric states and symmetries of membrane proteins.…”

Variable tilt on lipid membranes

“…In the first alternative, the tilt perturbation is localized in boundary layers near the edges y = ±l; in the second, this field is oscillatory and permeates the entire strip. Both modes have been simulated using coarsegrained molecular dynamics [28][29][30]. The oscillatory mode corresponds to ripple phases that have been detected experimentally [31].…”