Differential or Curvature Stress? Modus Vivendi

Abstract

It has been known for decades that mammalian cells actively maintain an asymmetric membrane composition. Very recently, advanced lipidomics of human red blood cells has revealed nearly the full lipidome on a per-leaflet basis (1Lorent J.H. Ganesan L. Levental I. et al.The molecular and structural asymmetry of the plasma membrane.bioRxiv. 2019; https://doi.org/10.1101/698837Crossref Google Scholar). What are the biophysical consequences of this asymmetry? In this issue of the Biophysical Journal, Hossein and Deserno (2Hossein A. Deserno M. Spontaneous curvature, differential stress, and bending modulus of asymmetric lipid membranes.Biophys. J. 2020; 118: 624-642Abstract Full Text Full Text PDF PubMed Scopus (72) Google Scholar) consider the consequences of asymmetry for the curvature elastic properties of the membrane. They begin by considering two different kinds of asymmetry that might resolve into a curved membrane. The first is an asymmetry of lipid type. Consider a bilayer whose leaflets are allowed to adjust their area independently. Composing one leaflet of DOPC and the other of DOPE, in numbers such that each leaflet is under no area strain and so the tension of each leaflet is zero, yields a bilayer that relaxes to a curved surface with a radius of ∼20 nm, with the DOPC on the positive curvature leaflet and the DOPE on the inner leaflet. (Lipid flip-flop is prohibited.) This effect, first described by Helfrich (3Helfrich W. Elastic properties of lipid bilayers: theory and possible experiments.Z. Naturforsch. C. 1973; 28: 693-703Crossref PubMed Scopus (4961) Google Scholar), is sometimes called “bilayer spontaneous curvature.” One can also create a curved membrane from a symmetric composition, simply by moving lipids from one leaflet to the other and coupling the leaflets’ areas. Concave (inner) leaflets have less relative area than convex leaflets. The area difference results in area difference elasticity that favors bending (4Miao L. Seifert U. Döbereiner H.G. et al.Budding transitions of fluid-bilayer vesicles: the effect of area-difference elasticity.Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics. 1994; 49: 5389-5407Crossref PubMed Scopus (440) Google Scholar). Hossein and Deserno (2Hossein A. Deserno M. Spontaneous curvature, differential stress, and bending modulus of asymmetric lipid membranes.Biophys. J. 2020; 118: 624-642Abstract Full Text Full Text PDF PubMed Scopus (72) Google Scholar) observe that these two mechanisms describe different elastic contributions to the bilayer energy density (bending for spontaneous curvature, stretching for area difference elasticity), are not “either/or” as is sometimes presented in the literature, and therefore should be considered simultaneously when contemplating asymmetric bilayers. The simultaneous action of these two different mechanisms implies a whole family of physically permissible curvature states for asymmetric membranes. Imagine that you wish to run a molecular dynamics simulation of a bilayer that mimics the asymmetry of the outer leaflet of the plasma membrane. You know the mole fraction of each lipid type. How many of each type should you put into each leaflet? Recently, several answers have been floated: match the areas of the leaflets, ensure that the tensions in each leaflet are both zero, or conduct any one of a number of other increasingly complex suggestions (5Doktorova M. Weinstein H. Accurate in silico modeling of asymmetric bilayers based on biophysical principles.Biophys. J. 2018; 115: 1638-1643Abstract Full Text Full Text PDF PubMed Scopus (25) Google Scholar,6Miettinen M.S. Lipowsky R. Bilayer membranes with frequent flip-flops have tensionless leaflets.Nano Lett. 2019; 19: 5011-5016Crossref PubMed Scopus (36) Google Scholar). According to Hossein and Deserno (2Hossein A. Deserno M. Spontaneous curvature, differential stress, and bending modulus of asymmetric lipid membranes.Biophys. J. 2020; 118: 624-642Abstract Full Text Full Text PDF PubMed Scopus (72) Google Scholar), the nonequilibrium (but metastable) conditions that give rise to the target membrane may determine the best choice. To understand why this is the case, we consider in more detail the two different sources of spontaneous curvature. An asymmetric bilayer composed of independent leaflets with different spontaneous curvatures minimizes to a state with a spontaneous curvature that is the bending modulus weighted difference of the leaflet spontaneous curvatures, called K0b in Hossein and Deserno (2Hossein A. Deserno M. Spontaneous curvature, differential stress, and bending modulus of asymmetric lipid membranes.Biophys. J. 2020; 118: 624-642Abstract Full Text Full Text PDF PubMed Scopus (72) Google Scholar). Thus, if the bending moduli and the spontaneous curvatures of the lipids (or mixtures of lipids) that comprise the leaflets are known, K0b can be estimated. (This is how the figure of 20 nm is arrived at above.) It is also clearly a curvature elastic mechanism and a material property, composed as it is from bending moduli and spontaneous curvatures of the lipids. On the other hand, the curvature arising from an area difference in area-coupled leaflets has its origin in entirely different physics. By packing more lipids into one leaflet than the other, the two leaflets have different preferred areas and therefore would like to relax to a surface with nonzero curvature, denoted K0s, with the subscript “s” as a reminder that the origin lies in a differential (across the leaflets) area strain. Penalizing deviations from K0s introduces a new term to the curvature elastic energy density, which (at the quadratic level) has the same form as the usual material terms, with a bending modulus times the squared deviation of the average curvature from K0s. The bending modulus associated with K0s has its physical origin in stretching/compression of the two leaflets and is therefore related to the area expansion modulus. Because both mechanisms may be at work in the same membrane, it is sensible to ask how they conspire to yield a curved membrane (or not) and what one expects for the curvature elastic properties of such a membrane. For surfaces of constant curvature, K0s and K0b are averaged (weighted by their associated bending moduli) to yield the spontaneous curvature of an asymmetric bilayer, K0∗. If it happens that K0s = K0b, then when the membrane assumes its energy-minimized state with curvature equal to K0∗, there will be no differential tension across the leaflets; both leaflets will be in a tensionless state. But note that this is a special case, and in general, an energy-minimized asymmetric bilayer will have a nonzero differential stress. At the other extreme, it could be the case that K0s and K0b cancel one another, such that the energy-minimized state of the asymmetric bilayer is flat. Caution is advised especially in the context of simulations that are constrained to be flat by the boundary conditions of a simulation. It would be quite possible, for example, to set up a simulation with a significant differential curvature asymmetry (such as the DOPC/DOPE example described above) but without a compensating differential tension. Such a membrane under different boundary conditions would be unstable against tubulation, but this would not be observed in a small all-atom simulation. The same reasoning is used by Hossein and Deserno (2Hossein A. Deserno M. Spontaneous curvature, differential stress, and bending modulus of asymmetric lipid membranes.Biophys. J. 2020; 118: 624-642Abstract Full Text Full Text PDF PubMed Scopus (72) Google Scholar) to argue that the asymmetric vesicles in recent experiments reported by Elani et al. (7Elani Y. Purushothaman S. Ces O. et al.Measurements of the effect of membrane asymmetry on the mechanical properties of lipid bilayers.Chem. Commun. (Camb.). 2015; 51: 6976-6979Crossref PubMed Google Scholar) are under significant differential stress, precisely because they do not tubulate, and instead interpret the stiffening observed by Elani et al. (7Elani Y. Purushothaman S. Ces O. et al.Measurements of the effect of membrane asymmetry on the mechanical properties of lipid bilayers.Chem. Commun. (Camb.). 2015; 51: 6976-6979Crossref PubMed Google Scholar) as arising from differential stress. (Compressed leaflets are stiffer than their zero-tension counterparts.) A control simulation tests that compositional asymmetry alone (designed to be at the state in which there is zero tension per leaflet) is insufficient to generate stiffening. All of the preceding discussion concerns metastable states of the bilayer. Given enough time, an asymmetric bilayer will relax via exchange of lipids between the leaflets until the chemical potentials of each species are equal in the two leaflets. However, the timescale for (uncatalyzed) lipid flip-flop is slow compared to relevant biological timescales and very slow compared to accessible simulation timescales; a vesicle prepared by any number of asymmetric protocols takes between 24 and 150 h to lose asymmetry (8Marquardt D. Heberle F.A. Pabst G. et al.1H NMR shows slow phospholipid flip-flop in gel and fluid bilayers.Langmuir. 2017; 33: 3731-3741Crossref PubMed Scopus (75) Google Scholar). Therefore, the spontaneous bilayer curvature arising from asymmetry is “only” metastable but is certainly relevant. It also suggests that different vesicle preparation protocols may end at different metastable equilibria because the kinetics of cyclodextrin-mediated exchange (9Chiantia S. Schwille P. London E. et al.Asymmetric GUVs prepared by MβCD-mediated lipid exchange: an FCS study.Biophys. J. 2011; 100: L1-L3Abstract Full Text Full Text PDF PubMed Scopus (99) Google Scholar), phase transfer (10Pautot S. Frisken B.J. Weitz D.A. Engineering asymmetric vesicles.Proc. Natl. Acad. Sci. USA. 2003; 100: 10718-10721Crossref PubMed Scopus (363) Google Scholar), and asymmetric supported bilayers produced by vesicle rupture (11Kiessling V. Crane J.M. Tamm L.K. Transbilayer effects of raft-like lipid domains in asymmetric planar bilayers measured by single molecule tracking.Biophys. J. 2006; 91: 3313-3326Abstract Full Text Full Text PDF PubMed Scopus (186) Google Scholar) are likely all different from one another. Indeed, this is likely true of different vesiculation pathways in vivo; there is no reason to expect that the asymmetry of a vesicle generated by clathrin-mediated endocytosis is identical to those generated by the ESCRT machinery or those that begin as caveolae. Cholesterol, however, is another matter. Although there are exceptions (12Garg S. Porcar L. Perez-Salas U. et al.Noninvasive neutron scattering measurements reveal slower cholesterol transport in model lipid membranes.Biophys. J. 2011; 101: 370-377Abstract Full Text Full Text PDF PubMed Scopus (69) Google Scholar), it seems that cholesterol flip-flop can be quite fast (13Steck T.L. Ye J. Lange Y. Probing red cell membrane cholesterol movement with cyclodextrin.Biophys. J. 2002; 83: 2118-2125Abstract Full Text Full Text PDF PubMed Scopus (210) Google Scholar), especially in membranes containing polyunsaturated chains. Thus, it is reasonable to consider a state in which the chemical potential of cholesterol has equilibrated across the leaflets, whereas the other lipids have not. What curvature stress states are possible in such a case? Using a simple and analytically tractable model, the authors consider to what state a membrane will relax via fast cholesterol flip-flop if it has an asymmetric composition of (much) more slowly exchanging lipids. If chemical interactions between the lipids and the mixing entropy of an uneven distribution of cholesterol across the leaflets are ignored, then cholesterol will flip until the leaflet areas are balanced and the bilayer is under zero differential stress. There is, however, an entropic cost to an uneven distribution of cholesterol that counteracts this process, which means that the metastable equilibrium (in which only the chemical potential of cholesterol is equilibrated) is not a zero-tension state. Mixing entropy therefore drives the system away from the zero-tension state. There are also chemical interactions that will drive the redistribution of cholesterol. Consider a bilayer with polyunsaturated chains on the inner leaflet and sphingolipids on the outer leaflet. Favorable interactions with the sphingolipids and unfavorable interactions with the PUFA will drive cholesterol to enrich in the outer leaflet. Indeed, using the coarse-grained Martini model and a similarly asymmetric membrane, the authors show that the cholesterol-favoring (DPPC) membrane recruits ∼80% of the total cholesterol, resulting in a differential tension of 3.7 mN/m, with the DPPC leaflet under compressive stress. This result is roughly consistent with recent estimates based on a theoretical approach to cholesterol redistribution reported by Allender et al. (14Allender D.W. Sodt A.J. Schick M. Cholesterol-dependent bending energy is important in cholesterol distribution of the plasma membrane.Biophys. J. 2019; 116: 2356-2366Abstract Full Text Full Text PDF PubMed Scopus (26) Google Scholar). Still, there are many other facets that deserve attention from simulations, such as precisely how the membrane resolves the competition between chemical interactions, (nonadditive) effects of cholesterol on curvature (15Pan J. Mills T.T. Nagle J.F. et al.Cholesterol perturbs lipid bilayers nonuniversally.Phys. Rev. Lett. 2008; 100: 198103Crossref PubMed Scopus (227) Google Scholar,16Sodt A.J. Venable R.M. Pastor R.W. et al.Nonadditive compositional curvature energetics of lipid bilayers.Phys. Rev. Lett. 2016; 117: 138104Crossref PubMed Scopus (52) Google Scholar) and area per molecule (17Edholm O. Nagle J.F. Areas of molecules in membranes consisting of mixtures.Biophys. J. 2005; 89: 1827-1832Abstract Full Text Full Text PDF PubMed Scopus (165) Google Scholar), and the balance between differential curvature and differential stress. Molecular simulations guided by the theory presented by Hossein and Deserno (2Hossein A. Deserno M. Spontaneous curvature, differential stress, and bending modulus of asymmetric lipid membranes.Biophys. J. 2020; 118: 624-642Abstract Full Text Full Text PDF PubMed Scopus (72) Google Scholar) are well suited to address this gap. Spontaneous Curvature, Differential Stress, and Bending Modulus of Asymmetric Lipid MembranesHossein et al.Biophysical JournalDecember 18, 2019In BriefLipid bilayers can exhibit asymmetric states, in which the physical characteristics of one leaflet differ from those of the other. This most visibly manifests in a different lipid composition, but it can also involve opposing lateral stresses in each leaflet that combine to an overall vanishing membrane tension. Here, we use theoretical modeling and coarse-grained simulation to explore the interplay between a compositional asymmetry and a nonvanishing differential stress. Minimizing the total elastic energy leads to a preferred spontaneous curvature that balances torques due to both bending moments and differential stress, with sometimes unexpected consequences. Full-Text PDF Open Archive