Bending Lipid Bilayers: A Closed-Form Collective Variable for Effective Free-Energy Landscapes in Quantitative Biology

Abstract

Curvature-related processes are of major importance during protein-membrane interactions. The illusive simplicity of membrane reshaping masks a complex molecular process crucial for a wide range of biological functions like fusion, endo- and exocytosis, cell division, cytokinesis, and autophagy. To date, no functional expression of a reaction coordinate capable of biasing molecular dynamics simulations to produce membrane curvature has been reported. This represents a major drawback given that the adequate identification of proper collective variables to enhance sampling is fundamental for restrained dynamics techniques. In this work, we present a closed-form equation of a collective variable that induces bending in lipid bilayers in a controlled manner, allowing for straightforward calculation of free energy landscapes of important curvature-related events, using standard methods such as umbrella sampling and metadynamics. As a direct application of the collective variable, we calculate the bending free energies of a ternary lipid bilayer in the presence and the absence of a Bin/Amphiphysin/Rvs domain with an N-terminal amphipathic helix (N-BAR), a well-known peripheral membrane protein that induces curvature.

Curvature as a Collective Coordinate in Enhanced Sampling Membrane Simulations

“…Interestingly, the saddle-shape conformations observed at the onset of the bending mechanism were also reported in previous studies inducing curvature indirectly (for instance by controlling the lipid density 22 or along a predefined path). 43 However, the present study suggests that introducing secondary bending axes is cheaper in free energy than increasing the bend along the axis of the original saddle: the order of magnitude of the free energy penalty is more than twice lower compared to the 80 kT found by Bubnis et al for the bending of a DOPC patch along a single axis 43 , itself ∼ 5 times lower than the result obtained by Masone et al on a similar deformation of DPPC.…”

“…43 However, the present study suggests that introducing secondary bending axes is cheaper in free energy than increasing the bend along the axis of the original saddle: the order of magnitude of the free energy penalty is more than twice lower compared to the 80 kT found by Bubnis et al for the bending of a DOPC patch along a single axis 43 , itself ∼ 5 times lower than the result obtained by Masone et al on a similar deformation of DPPC. 22 Compared to methods de facto restricted to the formation of a single feature in the center of the patch, the ability of the method under consideration to gives access to 'rugged' membrane topologies with multiple peaks and throughs is a crucial advantage for the exploration of lowenergy pathways. A larger conformational volume can can also be considered a disadvantage if the Γ range it maps to is too small, especially when exhaustive sampling is required; however, such conformational degeneracies can Figure 2: Top: distributions of Γ values for conformations segmented into a given number of regions by the region-growing algorithm.…”

Cholesterol plays a decisive role in tetraspanin assemblies during bilayer deformations

“…The collective variable, defined Ψ, forces lipid molecules to increase their local density at the centre of the bilayer, inducing the membrane to make slight displacements along its normal axis (Z). This rearrangement of lipids bends the membrane in a saddle-like shape which emerges spontaneously 18,52 . Unlike other approaches 72 , Ψ does not allow for a direct control of the bilayer curvature but is rather an indirect mechanism analogous to the protein-membrane surface interplay, as previously described for α-synuclein 13,36 and the N-BAR domain 52 , both known for inducing curvature in lipid bilayers 57,74 .…”

“…This rearrangement of lipids bends the membrane in a saddle-like shape which emerges spontaneously 18,52 . Unlike other approaches 72 , Ψ does not allow for a direct control of the bilayer curvature but is rather an indirect mechanism analogous to the protein-membrane surface interplay, as previously described for α-synuclein 13,36 and the N-BAR domain 52 , both known for inducing curvature in lipid bilayers 57,74 . Besides, the complex saddle-shape mode in which the bilayer bends follows two principal curvatures defined by two fitted circles with different radii 3,4,22 and requires significantly more energy than other modes of simpler geometry.…”

Mechanical Activation of MscL Revealed by a Locally Distributed Tension Molecular Dynamics Approach

“…1a). Similarly defined CVs have been used by others to study pore formation (74,75), membrane bending (76), and hydrophobic peptide insertion (77).…”