New Continuum Approaches for Determining Protein-Induced Membrane Deformations

Abstract

The influence of the membrane on transmembrane proteins is central to a number of biological phenomena, notably the gating of stretch activated ion channels. Conversely, membrane proteins can influence the bilayer, leading to the stabilization of particular membrane shapes, topological changes that occur during vesicle fission and fusion, and shape-dependent protein aggregation. Continuum elastic models of the membrane have been widely used to study protein-membrane interactions. These mathematical approaches produce physically interpretable membrane shapes, energy estimates for the cost of deformation, and a snapshot of the equilibrium configuration. Moreover, elastic models are much less computationally demanding than fully atomistic and coarse-grained simulation methodologies; however, it has been argued that continuum models cannot reproduce the distortions observed in fully atomistic molecular dynamics simulations. We suggest that this failure can be overcome by using chemically and geometrically accurate representations of the protein. Here, we present a fast and reliable hybrid continuum-atomistic model that couples the protein to the membrane. We show that the model is in excellent agreement with fully atomistic simulations of the ion channel gramicidin embedded in a POPC membrane. Our continuum calculations not only reproduce the membrane distortions produced by the channel but also accurately determine the channel's orientation. Finally, we use our method to investigate the role of membrane bending around the charged voltage sensors of the transient receptor potential cation channel TRPV1. We find that membrane deformation significantly stabilizes the energy of insertion of TRPV1 by exposing charged residues on the S4 segment to solution.

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

“…In Sections 6 and 7, we highlighted our most recent advances to our continuum elasticity solver, in which we have developed new methods for defining the protein-membrane boundary and applying boundary conditions [153]. The model does a very good job at quantitatively predicting membrane deformations around proteins when benchmarked against fully-atomistic MD simulations, but at a tiny fraction of the computational cost.…”

“…We assume that the total bilayer elastic energy is given by the sum of the contributions from each monolayer, we employ a Monge gauge representation, and we ignore spontaneous curvature. Near the protein, where there is not always a one-to-one correspondence between a patch in the upper leaflet with a patch in the lower leaflet, it becomes difficult to define the compression, and we have recently developed a method for handling these complex boundaries (currently in preparation [153]). In the past, we used a finite difference approach to solve the underlying PDEs in Cartesian or radial coordinates, but more recently we have developed a finite volume approach [154] that is more appropriate for solving biharmonic equations on complex boundaries by using a level set function to describe the membrane-protein boundary curves ([153]).…”

Fibronectin-Based Nanomechanical Biosensors to Map 3D Strains in Live Cells and Tissues

“…The use of the NMBS allows for a discretization of these continuous surfaces, sampling the surface at many points. If a dense enough mesh was used, the complete continuum of the surface of the cell could be estimated using interpolation between adjacent nodes in the NMBS 48 . This would allow for not only a complete strain analysis of the surface in 3D but also a measurement of the energy in the surface 49 .…”

Modeling membrane nanotube morphology: the role of heterogeneity in composition and material properties

“…The interaction energy of transmembrane proteins with the lipid bilayer is written as a constitutive function of the protein density σ (number per unit area). While the exact form of this energy is yet to be experimentally verified, based on thermodynamic arguments, a quadratic dependence of the energy on the local protein density has been proposed as [48,60,63,[70][71][72],…”