Curvature Elasticity Research Articles

Determining the Mean Curvature Modulus of a Lipid Membrane by Simulating Buckling

Membrane remodeling is central to a large number of cell biological processes, and the cost for elastically deforming the lipid bilayer features prominently in the energetic budget of such events. It is quantified by the mean curvature modulus, the determination of which is hence a prominent task for experimental and computational membrane scientists alike. The standard approach in simulations is to monitor the undulation spectrum [1], but in order to reach the continuum regime one needs large membranes that take disproportionately long to equilibrate. It has been suggested to instead simulate curved membrane tethers and measure their axial force [2], but this method has technical difficulties for models that are not strongly coarse grained. Here we consider an alternative strategy recently proposed by Noguchi [3], namely, measuring the response of a membrane to buckling. We provide highly accurate analytical expressions to analyze parallel and perpendicular stresses, valid far into the highly nonlinear regime, and we derive fluctuation corrections. Using a variety of membrane models, ranging from strongly coarse grained to atomistic, we show that highly accurate values of the mean curvature modulus can be obtained with remarkable computational ease. The technique also permits to check whether deviations from quadratic curvature elasticity are important, and it offers insights into the thermodynamics of the bending energetics itself.[1] Goetz R., Gompper G., and Lipowsky R., “Mobility and elasticity of self-assembled membranes”, Phys. Rev. Lett. 82, 221-224 (1999).[2] Harmandaris V.A. and Deserno M., “A novel method for measuring the bending rigidity of model lipid membranes by simulating tethers”, J. Chem. Phys. 125, 204905 (2006).[3] Noguchi H., “Anisotropic surface tension of buckled fluid membranes”, Phys. Rev. E 83, 061919 (2011).

Gaussian curvature elasticity determined from global shape transformations and local stress distributions: a comparative study using the MARTINI model

We calculate the Gaussian curvature modulus kappa of a systematically coarse-grained (CG) one-component lipid membrane by applying the method recently proposed by Hu et al. [Biophys. J., 2012, 102, 1403] to the MARTINI representation of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC). We find the value kappa/kappa = -1.04 +/- 0.03 for the elastic ratio between the Gaussian and the mean curvature modulus and deduce kappa(m)/kappa(m) = -0.98 +/- 0.09 for the monolayer elastic ratio, where the latter is based on plausible assumptions for the distance z0 of the monolayer neutral surface from the bilayer midplane and the spontaneous lipid curvature K(0m). By also analyzing the lateral stress profile sigma0(z) of our system, two other lipid types and pertinent data from the literature, we show that determining K(0m) and kappa through the first and second moment of sigma0(z) gives rise to physically implausible values for these observables. This discrepancy, which we previously observed for a much simpler CG model, suggests that the moment conditions derived from simple continuum assumptions miss the effect of physically important correlations in the lipid bilayer.

Shape dynamics, lipid hydrodynamics, and the complex viscoelasticity of bilayer membranes

Biological membranes are continuously brought out of equilibrium, as they shape organelles, package and transport cargo, or respond to external actions. Even the dynamics of plain lipid membranes in experimental model systems are very complex due to the tight interplay between the bilayer architecture, the shape dynamics, and the rearrangement of the lipid molecules. We formulate and numerically implement a continuum model of the shape dynamics and lipid hydrodynamics, which describes the bilayer by its midsurface and by a lipid density field for each monolayer. The viscoelastic response of bilayers is determined by the stretching and curvature elasticity, and by the inter-monolayer friction and the membrane interfacial shear viscosity. While the bilayer equilibria are well understood theoretically, dynamical calculations have relied on simplified continuum approaches of uncertain transferability, or on molecular simulations reaching very limited length and time scales. Our approach incorporates the main physics, is fully nonlinear, does not assume predefined shapes, and can access a wide range of time and length scales. We validate it with the well understood tether extension. We investigate the tubular lipid transport between cells, the dynamics of bud absorption by a planar membrane, and the fate of a localized lipid density asymmetry in vesicles. These axisymmetric examples bear biological relevance and highlight the diversity of dynamical regimes that bilayers can experience.

Translational and Rotational Diffusion, and Conformational Dynamics of fd Virus on Freestanding Cationic Lipid Membranes

Understanding of the mechanisms of interaction of macromolecules and colloidal particles with lipid membranes is far from complete, and the questions related to the effects of membrane local curvature and bending elasticity of the interacting partners are still largely unsolved. Recently, using single-molecule fluorescence video microscopy we discovered unexpected effects in interaction of single DNA molecules with freestanding cationic lipid membranes [1]. In the present contribution, using the same experimental approach, we investigate how much more rigid negatively charged particles - fd viruses [2] - behave upon electrostatic binding to freestanding cationic lipid membranes. We study the translational and rotational diffusion of fd viruses with the lengths in the range of (1-11)×0.88 μm bound to freestanding membranes and observe the crossover from the 2D membrane-controlled to 3D bulk fluid-controlled diffusion dynamics. For virus particles with lengths >4.4 μm we additionally study the effects of the membrane on their conformations and conformational dynamics.[1] C. Herold, P. Schwille, and E. P. Petrov, Phys. Rev. Lett. 104 (2010) 148102.[2] S. Fraden and Z. Dogic, in: Soft Matter, Vol. 2: Complex Colloidal Suspensions (eds. G. Gompper and M. Schick) Wiley-VCH, Weinheim, 2006, pp. 1–86.

Curvature elasticity of a weak polyelectrolyte brush and shape transitions in assemblies of amphiphilic diblock copolymers

For a weak polyelectrolyte brush swollen in electrolyte solution, we obtain an analytical description of the change produced by curvature deformation. Expressions are given for elasticity constants and for the change in the thickness and concentration profiles of a box-like brush in equilibrium with an external reservoir. We show that curvature expansion of the free energy leads to reliable predictions of shape transitions for crew-cut diblock copolymer aggregates. We study the dependence of the curvature elasticity of a diblock copolymer membrane on the salinity and the acidity of the environment and predict a nontrivial mechanical behavior of the amphiphilic membrane at pH around pKa in the physiological salinity interval.

The Role of the Membrane Confinement for Cell Morphology and Surface Area Regulation

Commonly, the membranes of mammalian and plant cells are not isolated, but rather they are confined to an actin cortex on the inner side and to an extracellular matrix, cell wall or a substrate on the outer side. The confinement restricts the modes of the membrane deformation and so influences the mechanisms for shape remodeling and surface area regulation. To study the mechanics of confined membranes we developed an in vitro system, which couples a lipid bilayer to the strain-controlled deformation of an elastic sheet (Staykova et al., PNAS 108:9084-9088, 2011). We demonstrate that upon contracting the elastic support, membrane protrusions grow out of the membrane plane, thus reducing its area; upon stretching, the bilayer absorbs the protrusions and expands its area without the lost of integrity. Most of our observations can be understood in terms of free energy minimization in a theoretical model we develop to account for stretch and curvature elasticity of each monolayer, the adhesion with the substrate, and the limited amount of area and volume available to the protrusions. However, some of the observed morphologies are dynamical in nature, and simulations suggest that the friction between the lower monolayer and the substrate plays a key role. Our experimental and theoretical results closely reproduce membrane processes found in, for example, shrinking neurons, and plant and muscle cells, and thus offer a mechanistic approach towards understanding the area regulation in cells.

A quantitative model and experimental investigations of wall shear stress between solid and gaseous fluid using liquid crystal coating

Measurement of wall shear stress (WSS) has important significance in many fields, such as spacecraft surface flow visualization and drag reduction of solid-liquid interface. Various measuring methods have been developed with the rapid development of micro and nano technology. This paper focuses on the measurement of WSS by using nematic liquid crystal coating. A quantitative model is built to describe the relationship between the variations of director orientation and the WSS based on the curvature elasticity theory of liquid crystal. Then an experiment setup is built to verify this model using 5 CB and 7 CB through the optical method. There is good consistence between the theoretical model and experiment results. Finally, the WSS measuring limits and influencing factors are discussed such as the thickness of the liquid crystal layer, the elastic coefficient of the molecule and measurement resolution.

Ring formation mechanism of single-walled carbon nanotubes: Energy conservation between curvature elasticity and inter-tube adhesion

Rings of single-walled carbon nanotubes (SWNTs) exist widely during water evaporation from their dispersions at low concentration on such substrates as silica wafer. We examine the phenomenon in terms of energy conservation between the increased significant curvature energy and the inherent inter-tube van der Waals (vdW) attraction potentials. And thereby, the observed multi ring structures for coarse and long SWNT bundles have also gained detailed interpretation. We conclude that carbon nanotubes (CNTs) coil into rings by their own elastic mechanism. The formed rings with different width and diameters originate from appropriate sizes of SWNTs or the bundles. Specially, the associated elasticity may have prospective potentials to reveal other fascinating self-assembling phenomenon on CNTs, for instance, the known liquid crystallinity of them. Besides, we have also analyzed the external factors to the ring formation, both statistically and dynamically.

Mechanics of membrane–membrane adhesion

Curvature elasticity is used to derive the equilibrium conditions that govern the mechanics of membrane–membrane adhesion. These include the Euler–Lagrange equations and the interface conditions which are derived here for the most general class of strain energies permissible for fluid surfaces. The theory is specialized for homogeneous membranes with quadratic ‘Helfrich’-type energies with non-uniform spontaneous curvatures. The results are employed to solve four-point boundary value problems that simulate the equilibrium shapes of lipid vesicles that adhere to each other. Numerical studies are conducted to investigate the effect of relative sizes, osmotic pressures, and adhesion-induced spontaneous curvature on the morphology of adhered vesicles.

SHAPE TRANSFORMATION OF HALLOYSITE PARTICLES

A mechanism of shape transformation from spheres to tubes in halloysite is proposed from the standpoint of curvature elasticity theory, using the size and morphological data of samples separated from chronological series of tephras in Hachijo Island. Transmission electron microscopic observations indicated that when the diameters of halloysite particles exceeded about 200 nm, the shape showed prominent deviation from spherical to tubular and/or partially curled platy ones. This fact leads to that the spherical halloysite with 200 nm in diameter is subjected to the curvature elastic energy of about 9 kJ/m 2 due to the anti-bending of layers, which is an estimate using approximately the elastic constants of kaolinite and taking into account a preferable curvature for a single layer of halloysite. The curvature elastic energy acts as a driving force for the shape transformation. In reality irreversible dehydration process plays a trigger to transform such flustrated spherical halloysites to more stable shapes such as tubular ones.

Systems biology and physical biology of clathrin-mediated endocytosis.

In this review, we describe the application of experimental data and modeling of intracellular endocytic trafficking mechanisms with a focus on the process of clathrin-mediated endocytosis. A detailed parts-list for the protein-protein interactions in clathrin-mediated endocytosis has been available for some time. However, recent experimental, theoretical, and computational tools have proved to be critical in establishing a sequence of events, cooperative dynamics, and energetics of the intracellular process. On the experimental front, total internal reflection fluorescence microscopy, photo-activated localization microscopy, and spinning-disk confocal microscopy have focused on assembly and patterning of endocytic proteins at the membrane, while on the theory front, minimal theoretical models for clathrin nucleation, biophysical models for membrane curvature and bending elasticity, as well as methods from computational structural and systems biology, have proved insightful in describing membrane topologies, curvature mechanisms, and energetics.

Does maltose influence on the elasticity of SOPC membrane?

Thermally induced shape fluctuations of giant quasi-spherical lipid vesicles are used to study the influence of the disaccharide maltose, dissolved in the aqueous solution, on the curvature elasticity kc of a lipid membrane. The influence of the carbohydrate solute is investigated throughout a considerably wide interval of concentrations. The values of the bending elastic modulus for 200 mM and 400 mM of maltose in the water solution are obtained. The data for kc in presence of maltose is compared with previously obtained results for this constant for the most popular hydrocarbons: monosaccharides glucose and fructose and disaccharides sucrose and trehalose. It is shown that the presence of maltose, dissolved in the aqueous phase surrounding the membrane does not influence on the bending elasticity with the increase of its concentration in the aqueous solution. Up to our knowledge this is the first sugar that does not show decrease of the bending elastic modulus of the lipid membrane, when present in the water surrounding it in concentration up to 400mM.

Ordered micellar and inverse micellar lyotropic phases

In this article we review the ordered micellar and inverse micellar lyotropic liquid-crystalline phases that can be formed by amphiphilic molecules such as lipids and surfactants. We focus first on the self-assembly of amphiphiles into aggregates, and then consider the interfacial curvature and the role of curvature elasticity and packing constraints in determining the allowed structures. We then review the range of ordered micellar and inverse micellar phases that have so far been observed in a variety of surfactant and lipid systems. Finally, we describe certain characteristic properties, such as the epitaxy between phases, and the self-diffusion and electrical conductivity within such ordered micellar phases.

Functional Consequences of Lipid-Mediated Interaction Between Rhodopsin Molecules

In the retina, rhodopsin is densely packed in a rod outer segment disk membrane rich in phospholipids with PC and PE headgroups. Increasing rhodopsin packing density in model PC membranes has been shown to alter metarhodopsin-II (MII) formation. This observation is deemed to be the consequence of rhodopsin association promoted by non-specific properties of the membrane. Here, we studied the effect of rhodopsin packing density on MII formation in membranes characterized by different intrinsic curvature and different interfacial hydrogen bonding propensity. Rhodopsin was reconstituted into a series of POPC bilayers doped with DOPC, di-and mono-methylated DOPE or DOPE at rhodopsin/lipid ratios ranging from 1:250 to 1:70. The level of rhodopsin activation and rate of MII formation were determined by steady-state and time-resolved UV/vis spectroscopy. In PC membranes, lower rhodopsin concentrations shift the MI/MII equilibrium towards MII and result in a faster rate of MII formation, in agreement with previous findings. On the contrary, in membranes rich in lipids with PE headgroups, the MII concentration is independent of rhodopsin packing density and rates of MII formation, while reduced, show only a slight dependence on rhodopsin crowding. In addition, at low or high protein density, the amount of MII formed depends to a larger extent on the ability of the annular PE headgroups to establish hydrogen bonds with the MII state than on changes in membrane curvature elasticity. These results show clearly that MII formation and interaction between rhodopsin molecules depend strongly on interactions between annular lipids and rhodopsin, highlighting the fundamental role of the first layer of lipids surrounding the protein.

Global Bilayer Properties can Modulate Membrane Protein Oligomerization

While sequence dependent oligomerization of individual transmembrane α-helices has been studied to some extent in the recent years, the influence of the lipid bilayer properties on defined helix-helix interactions remains largely uncharacterized. To study the potential impact of changing bilayer properties on a defined transmembrane helix-helix interaction we have followed association of fluorescently labeled glycophorin A transmembrane peptides in model membranes by fluorescence spectroscopy. Changes in Forster resonance energy transfer strongly suggest that the lipid bilayer thickness does significantly influence the monomer-dimer equilibrium of the transmembrane domain. Furthermore, the presence of cholesterol in model membranes promotes self-association of transmembrane helices by modulating the bilayer thickness and -more importantly- by affecting lipid acyl chain ordering. In addition, changes in the lipid composition, which modulate lipid bilayer curvature elasticity and the lateral pressure profile, affect GpA dimerization. In conclusion, the findings show that the physical state of a membrane can be critically involved in controlling specific and promiscuous interactions of α-helical transmembrane domains, as e. g. involved in membrane protein folding and assembly as well as in transmembrane signaling.

Molecular field theory with atomistic modeling for the curvature elasticity of nematic liquid crystals

Liquid crystals oppose a restoring force to distortions of the main alignment axis, the so-called director. For nematics this behavior is characterized by the three elastic moduli associated with the splay (K(11)), twist (K(22)), and bend (K(33)) modes; in addition, two moduli for mixed splay-bend (k(13)) and saddle-splay (k(24)) can be defined. The elastic constants are material properties which depend on the mesogen structure, but the relation between molecular features and deformations on a much longer scale has not been fully elucidated. The prediction of elastic properties is a challenge for theoretical and computational methods: atomistic simulations require large samples and must be integrated by statistical thermodynamics models to connect intermolecular correlations and elastic response. Here we present a molecular field theory, wherein expressions for the elastic constants of nematics are derived starting from a simple form of the single molecule orientational distribution function; this is parametrized according to the amount of molecular surface aligned to the nematic director. Such a model allows a detailed account of the chemical structure; moreover the conformational freedom, which is a common feature of mesogens, can be easily included. Given the atomic coordinates, the elastic constants can be calculated without any adjustable parameter at a low computational cost. The example of 4-n-pentyl,4(')-cyanobiphenyl (5CB) is used to illustrate the capability of the developed methodology; even for this mesogen, which is usually taken as a prototypal rodlike system, we predict a significant dependence of the elastic moduli on the molecular conformation. We show that good estimates of magnitude and temperature dependence of the elastic constants are obtained, provided that the molecular geometry is correctly taken into account.

Relaxation dynamics of fluid membranes

We study the effect of membrane viscosity in the dynamics of liquid membranes-possibly with free or internal boundaries-driven by conservative forces (curvature elasticity and line tension) and dragged by the bulk dissipation of the ambient fluid and the friction occurring when the amphiphilic molecules move relative to each other. To this end, we formulate a continuum model which includes a form of the governing equations for a two-dimensional viscous fluid moving on a curved, time-evolving surface. The effect of membrane viscosity has received very limited attention in previous continuum studies of the dynamics of fluid membranes, although recent coarse-grained discrete simulations suggest its importance. By applying our model to the study of vesiculation and membrane fusion in a simplified geometry, we conclude that membrane viscosity plays a dominant role in the relaxation dynamics of fluid membranes of sizes comparable to those found in eukaryotic cells, and is not negligible in many large synthetic systems of current interest.

Curvature and Specific Lipid-Protein Interactions Modulate Activity of Rhodopsin

Membrane composition strongly modulates the ability of photo-activated rhodopsin to achieve the G-protein binding competent metarhodopsin II conformation (MII). In particular, MII concentration increases linearly with the bilayer concentration of non-bilayer forming PE lipids. This observation has prompted the membrane curvature hypothesis, which states that the continuum elastic properties of the lipid matrix play a dominant role on MII formation. Here, we aimed to separate the effect of membrane curvature elasticity from specific interactions between rhodopsin and PE headgroups. In a series of rhodopsin-containing proteoliposomes of different intrinsic curvature the level of rhodopsin activation was determined by steady-state and time-resolved UV/vis spectroscopy, membrane order and dynamics parameters were probed by 2H NMR and 13C MAS nuclear relaxation, and the structural response of rhodopsin to changes in membrane composition was followed by circular dichroism (CD). MII formation was increased by 18:0-22:6 PC and 18:0-22:6 PE, agents promoting negative curvature and decreased by lysophosphatidylcholine which promotes positive curvature, in agreement with the membrane curvature hypothesis. However, MII formation was also augmented by curvature-neutral lysophosphatidylethanolamine. In parallel, significant changes in helical content were observed by CD. Our results suggest that the structure and function of rhodopsin are modulated not only by membrane curvature elasticity, but also by specific interactions between rhodopsin and PE headgroups. The role of headgroup hydration, cation-pi interactions or salt bridge formation between rhodopsin and the annular lipids will be discussed on the basis of NMR experiments.

Effects of Ion and Water Channels Blockers and Uncouplers on the Dionaea Muscipula Ellis Trap Closure

The Venus flytrap (Dionaea muscipula Ellis) captures insects with one of the most rapid movements in the plant kingdom. Here we present detailed experiments for comparative study of effects of inhibitors of ion channels, aquaporins, and uncouplers on kinetics of the trap closing stimulated by mechanical or electrical triggering of the trap. Two method of inhibitors phytoextraction were used: (1) two 10 μL drops of channels blockers or uncouplers were placed on the midrib of the trap or (2) addition of 50 mL of inhibitors to the soil. Both methods of inhibitors phytoextraction give the same effects on the kinetics of the trap closing. Ion and water channels blockers such as HgCl2, TEACl, ZnCl2, BaCl2, as well as uncouplers CCCP, FCCP, 2,4-dinitrophenol, and pentachlorophenol decrease speed and increase time of the trap closing [1]. We applied for the evaluation of the mechanism of trap closing our new hydroelastic curvature mechanism, which is based on the assumption that the lobes possess curvature elasticity and are composed of outer and inner hydraulic layers with different hydrostatic pressure. The open state of the trap contains high elastic energy accumulated due to the hydrostatic pressure difference between the hydraulic layers of the lobe. Stimuli open pores connecting the two layers, water rushes from one hydraulic layer to another, and the trap relaxes to the equilibrium configuration corresponding to the closed state. The detailed mechanism of the trap closing is discussed.[1] A.G. Volkov, K.J. Coopwood, V.S. Markin, Plant Science 175 (2008) 642-649.

The Gaussian Curvature Elastic Energy of Membrane Fusion Intermediates, and a Possible Mode of Action of Fusion-Mediating Proteins

Calculating the Gaussian curvature elastic energy of fusion intermediates requires knowledge of the Gaussian curvature elastic modulus, κ. κ can be measured for lipids that form QII phases. It is shown that one can estimate κ for non-QII phase lipids by studying phase behavior of lipid mixtures. κ is between -5 and -10 kBT for biological lipid compositions. The Gaussian curvature elastic energy of fusion intermediates is positive and ≥ the total bending energy calculated previously: it increases the total energy of fusion intermediates by 100 kBT or more. This large contribution makes the predicted intermediate energies compatible with observed lipid phase behavior in excess water. An order-of-magnitude fusion rate equation is used to show that a current theory now predicts stalk energies that are slightly too large, by about 30 kBT, to rationalize the observed rates of stalk-mediated processes in pure lipids. Despite this discrepancy, when the effect of κ is included, current models of fusion intermediate energy can make semi-quantitative predictions about how proteins mediate biomembrane fusion. Fusion-mediating proteins must lower the stalk energy by several tens of kBT relative to lipidic stalks. One way proteins could do this is by altering the elastic constants of the patches of monolayer that fuse, by inserting peptides. Preliminary data using fusion peptides and membrane-spanning peptides is compatible with this role for peptides. The energies of stalks, fusion pores, rhombohedral (R) phase and QII phase relative to Lα phase all depend on approximately the same assembly of monolayer elastic constants; (2kmδ Js-κ). Thus the influence of peptides on stalk and fusion pore energy can be studied by measuring the effects of peptides on R and QII phase stability (Siegel, Biophys. J.; Dec. 2008).