Exploiting Lipid Permutation Symmetry to Compute Membrane Remodeling Free Energies

Abstract

A complete physical description of membrane remodeling processes, such as fusion or fission, requires knowledge of the underlying free energy landscapes, particularly in barrier regions involving collective shape changes, topological transitions, and high curvature, where Canham-Helfrich (CH) continuum descriptions may fail. To calculate these free energies using atomistic simulations, one must address not only the sampling problem due to high free energy barriers, but also an orthogonal sampling problem of combinatorial complexity stemming from the permutation symmetry of identical lipids. Here, we solve the combinatorial problem with a permutation reduction scheme to map a structural ensemble into a compact, nondegenerate subregion of configuration space, thereby permitting straightforward free energy calculations via umbrella sampling. We applied this approach, using a coarse-grained lipid model, to test the CH description of bending and found sharp increases in the bending modulus for curvature radii below 10 nm. These deviations suggest that an anharmonic bending term may be required for CH models to give quantitative energetics of highly curved states. DOI: 10.1103/PhysRevLett.117.188102 Membrane remodeling is essential for many cellular transport functions, notably fusion [1] and fission [2]. However, despite continued interest, the underlying lipidic mechanisms and energetics are not fully understood. Atomistic and near-atomistic molecular dynamics (MD) simulations provide sufficient temporal and spatial resolution to probe lipidic mechanisms. However, standard MD fails to reach the high energy barriers (≫k B T) which often dictate mechanisms and kinetics. Biasing methods are thus required to address the sampling problem for high energy barriers.An additional and ubiquitous obstacle, in both describing and biasing lipidic transitions, is the N! degeneracy of lipid configuration space arising from the permutation symmetry of N identical lipids. The same combinatorial symmetry lies at the heart of the Gibbs paradox [3][4][5], which is resolved by a 1=N! scaling of the partition function, to account for physically indistinguishable (degenerate) states.In contrast to this straightforward analytical correction, this permutation symmetry poses a severe challenge to atomistic simulations and free energy calculations involving lipids: sampling the space of degenerate states (via self-diffusion) shrouds the collective structural changes of interest and precludes the use of lipidic collective coordinate biasing schemes. Accordingly, special purpose methods (that circumvent this hindrance) have been developed to direct lipidic transitions using boundary conditions [6,7], external guiding potentials [8,9], probe particles [10], density biasing [11,12], and single-lipid restraints [13,14].Here, we describe and apply a rigorous method to directly address this N! degeneracy, using permutation reduction (PR) [15,16]. The method exploits permutation symmetry by remapping structures from the full N! degenerate configur...

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“…Instead, sampling can be alternatively enhanced via statistical biasing by starting simulations (trial runs) from states characterized by a forward progress along an defined reaction coordinate. These methods may have an advantage in collective, diffusive processes such as, for example, membrane fusion where the relationship between a defined reaction coordinate and the concomitant umbrella forces are not trivial (Bubnis and Risselada 2016). In addition, they gain additional insight into the actual rate of the reaction (kinetic prefactor).…”

“…Its application to membrane transformations is, however, complicated by the diffusion of the lipids within each leaflet. This problem has been solved recently by the permutation reduction method (Reinhard and Grubmüller 2007;Heinz and Grubmüller 2020), which un-does this diffusion without changing the physics of the system and, hence, allows for the application of Umbrella sampling (Bubnis and Risselada 2016). (Fig.…”

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