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Lipid rafts are nanoscopic assemblies of sphingolipids, cholesterol, and specific membrane proteins. They are believed to underlie the experimentally observed lateral heterogeneity of eukaryotic plasma membranes and implicated in many cellular processes, such as signaling and trafficking. Ternary model membranes consisting of saturated lipids, unsaturated lipids, and cholesterol are common proxies because they exhibit phase coexistence between a liquid-ordered (lo) and liquid-disordered (ld) phase and an associated critical point. However, plasma membranes are also asymmetric in terms of lipid type, lipid abundance, leaflet tension, and corresponding cholesterol distribution, suggesting that rafts cannot be examined separately from questions about elasticity, curvature torques, and internal mechanical stresses. Unfortunately, it is challenging to capture this wide range of physical phenomenology in a single model that can access sufficiently long length- and time scales. Here we extend the highly coarse-grained Cooke model for lipids, which has been extensively characterized on the curvature-elastic front, to also represent raft-like lo/ld mixing thermodynamics. In particular, we capture the shape and tie lines of a coexistence region that narrows upon cholesterol addition, terminates at a critical point, and has coexisting phases that reflect key differences in membrane order and lipid packing. We furthermore examine elasticity and lipid diffusion for both phase separated and pure systems and how they change upon the addition of cholesterol. We anticipate that this model will enable significant insight into lo/ld phase separation and the associated question of lipid rafts for membranes that have compositionally distinct leaflets that are likely under differential stress—like the plasma membrane. 

Many biological membranes host different lipid species in their two leaflets. Since their spontaneous curvatures are typically not the same, this compositional asymmetry generally entails bending torques, which can be counteracted by differential stress—the difference between the two leaflet tensions. This stress, in turn, can affect elastic parameters or phase behavior of the membrane or each individual leaflet, or push easily flippable species, especially cholesterol, from the compressed leaflet into the tense leaflet. In short, breaking the symmetry of a single observable (to wit: composition), essentially breaks all other symmetries as well, with many potentially interesting consequences. This brief report examines the elastic aspects of this interplay, focusing on some elementary conditions of mechanical and thermodynamic equilibrium, but also shows how this poses novel questions that we are only beginning to appreciate. 

Biological lipid membranes are generally asymmetric, not only with respect to the composition of the two membrane leaflets but also with respect to the state of mechanical stress on the two sides. Computer simulations of such asymmetric membranes pose unique challenges with respect to the choice of boundary conditions and ensemble in which such simulations are to be carried out. Here, we demonstrate an alternative to the usual choice of fully periodic boundary conditions: The membrane is only periodic in one direction, with free edges running parallel to the single direction of periodicity. In order to maintain bilayer asymmetry under these conditions, nanoscale “sticky tapes” are adhered to the membrane edges in order to prevent lipid flip-flop across the otherwise open edge. In such semi-periodic simulations, the bilayer is free to choose both its area and mean curvature, allowing for minimization of the bilayer elastic free energy. We implement these principles in a highly coarse-grained model and show how even the simplest examples of such simulations can reveal useful membrane elastic properties, such as the location of the monolayer neutral surface. 

The quantum free energy of a system governed by a standard Hamiltonian is larger than its classical counterpart. The lowest-order correction, first calculated by Wigner, is proportional to ℏ2 and involves the sum of the mean squared forces. We present an elementary derivation of this result by drawing upon the Zassenhaus formula, an operator-generalization for the main functional relation of the exponential map. Our approach highlights the central role of non-commutativity between kinetic and potential energy and is more direct than Wigner's original calculation, or even streamlined variations thereof found in modern textbooks. We illustrate the quality of the correction for the simple harmonic oscillator (analytically) and the purely quartic oscillator (numerically) in the limit of high temperature. We also demonstrate that the Wigner correction fails in situations with sufficiently rapidly changing potentials, for instance, the particle in a box. 

Many cellular lipid bilayers consist of leaflets that differ in their lipid composition — a non-equilibrium state actively maintained by cellular sorting processes that counter passive lipid flip-flop. While this lipidomic aspect of membrane asymmetry has been known for half a century, its elastic and thermodynamic ramifications have garnered attention only fairly recently. Notably, the torque arising when lipids of different spontaneous curvature reside in the two leaflets can be counterbalanced by a difference in lateral mechanical stress between them. Such membranes can be essentially flat in their relaxed state, despite being compositionally strongly asymmetric, but they harbor a surprisingly large but macroscopically invisible differential stress. This hidden stress can affect a wide range of other membrane properties, such as the resistance to bending, the nature of phase transitions in its leaflets, and the distribution of flippable species, most notably sterols. In this short note we offer a concise overview of our recently proposed basic framework for capturing the interplay between curvature, lateral stress, leaflet phase behavior, and cholesterol distribution in generally asymmetric membranes, and how its implied signatures might be used to learn more about the hidden but physically consequential differential stress.