Anisotropic surface tension of buckled fluid membranes

Computing the influence of lipids and lipid complexes on membrane mechanics

Methodology for extracting the spontaneous curvature, bending modulus, and neutral surface of a lipid bilayer is described. The "SPEX" method is a robust technique for computing the bilayer bending modulus while allowing for resolution of the spontaneous curvature of specific interacting lipids and complexes, and the dependence of spontaneous curvature on wavelength. The method is described referring to the publicly available MembraneAnalysis.jl software package.

Analyzing lipid distributions and curvature in molecular dynamics simulations of complex membranes

We describe methods to analyze lipid distributions and curvature in membranes with complex lipid mixtures and embedded membrane proteins. We discuss issues involved in these analyses, available tools to calculate curvature preferences of lipids and proteins, and focus on tools developed in our group for visual analysis of lipid-protein interactions and the analysis of membrane curvature.

Curvature sensing of curvature-inducing proteins with internal structure

Many types of peripheral and transmembrane proteins can sense and generate membrane curvature. Laterally isotropic proteins and crescent proteins with twofold rotational symmetry, such as Bin/Amphiphysin/Rvs superfamily proteins, have been studied theoretically. However, proteins often have an asymmetric structure or a higher rotational symmetry. We studied theoretically the curvature sensing of proteins with asymmetric structures and structural deformations. First, we examined proteins consisting of two rodlike segments. When proteins have mirror symmetry, their sensing ability is similar to that of single-rod proteins; hence, with increasing protein density on a cylindrical membrane tube, a second- or first-order transition occurs at a middle or small tube radius, respectively. As asymmetry is introduced, this transition becomes a continuous change and metastable states appear at high protein densities. Protein with threefold, fivefold, or higher rotational symmetry has laterally isotropic bending energy. However, when a structural deformation is allowed, the protein can have a preferred orientation and stronger curvature sensing.

Relationship between Protein-Induced Membrane Curvature and Membrane Thermal Undulation

This work studied the membrane curvature generated by anchored proteins lacking amphipathic helices and intrinsic morphologies, including the Epsin N-terminal homology domain, intrinsically disordered C-terminal domain, and truncated C-terminal fragments, by using coarse-grained molecular dynamics simulations. We found that anchored proteins can stabilize the thermal undulation of membranes at a wavelength five times the protein's binding size. This proportional connection is governed by the membrane bending rigidity and protein density. Extended intrinsically disordered proteins with relatively high hydrophobicity favor colliding with the membrane, leading to a much larger binding size, and show superiority in generating membrane curvature at low density over folded proteins.

Membrane buckling and the determination of Gaussian curvature modulus

Biological membranes can exhibit various morphology due to the fluidity of the lipid molecules within the monolayers. The shape transformation of membranes has been well described by the classical Helfrich theory, which consists only a few phenomenological parameters, including the mean and the Gaussian curvature modulus. Though various methods have been proposed to measure the mean curvature modulus, determining the Gaussian curvature modulus remains difficult both in experiments and in simulations. In this paper we study the buckling process of a rectangular membrane and a circular membrane subject to compressive stresses and under different boundary conditions. We find that the buckling of a rectangular membrane takes place continuously, while the buckling of a circular membrane can be discontinuous depending on the boundary conditions. Furthermore, our results show that the stress-strain relationship of a buckled circular membrane can be used to determine the Gaussian curvature modulus effectively.

Curvature Matters: Modeling Calcium Binding to Neutral and Anionic Phospholipid Bilayers

In this work, the influence of membrane curvature on the Ca²⁺ binding to phospholipid bilayers is investigated by means of molecular dynamics simulations. In particular, we compared Ca²⁺ binding to flat, elastically buckled, or uniformly bent zwitterionic and anionic phospholipid bilayers. We demonstrate that Ca²⁺ ions bind preferably to the concave membrane surfaces in both types of bilayers. We also show that the membrane curvature leads to pronounced changes in Ca²⁺ binding including differences in free ion concentrations, lipid coordination distributions, and the patterns of ion binding to different chemical groups of lipids. Moreover, these effects differ substantially for the concave and convex membrane monolayers. Comparison between force fields with either full or scaled charges indicates that charge scaling results in reduction of the Ca²⁺ binding to curved phosphatidylserine bilayers, while for phosphatidylcholine membranes, calcium binds only weakly for both force fields.

Determination of elastic parameters of lipid membranes from simulation under varied external pressure

Many cellular processes such as endocytosis, exocytosis, and vesicle trafficking involve membrane deformations, which can be analyzed in the framework of the elastic theories of lipid membranes. These models operate with phenomenological elastic parameters. A connection between these parameters and the internal structure of lipid membranes can be provided by three-dimensional (3D) elastic theories. Considering a membrane as a 3D layer, Campelo et al. [F. Campelo et al., Adv. Colloid Interface Sci. 208, 25 (2014)10.1016/j.cis.2014.01.018] developed a theoretical basis for the calculation of elastic parameters. In this work we generalize and improve this approach by considering a more general condition of global incompressibility instead of local incompressibility. Crucially, we find an important correction to the theory of Campelo et al., which if not taken into account leads to a significant miscalculation of elastic parameters. With the total volume conservation taken into account, we derive an expression for the local Poisson's ratio, which determines how the local volume changes upon stretching and permits a more precise determination of elastic parameters. Also, we substantially simplify the procedure by calculating the derivatives of the moments of the local tension with respect to stretching instead of calculating the local stretching modulus. We obtain a relation between the Gaussian curvature modulus as a function of stretching and the bending modulus, showing that these two elastic parameters are not independent, as was previously assumed. The proposed algorithm is applied to membranes composed of pure dipalmitoylphosphatidylcholine (DPPC), dioleoylphosphatidylcholine (DOPC), and their mixture. The following elastic parameters of these systems are obtained: the monolayer bending and stretching moduli, spontaneous curvature, neutral surface position, and local Poisson's ratio. It is shown that the bending modulus of the DPPC/DOPC mixture follows a more complex trend than predicted by the classical Reuss averaging, which is often employed in theoretical frameworks.

Determination of Elastic Parameters of Lipid Membranes with Molecular Dynamics: A Review of Approaches and Theoretical Aspects

Lipid membranes are abundant in living organisms, where they constitute a surrounding shell for cells and their organelles. There are many circumstances in which the deformations of lipid membranes are involved in living cells: fusion and fission, membrane-mediated interaction between membrane inclusions, lipid-protein interaction, formation of pores, etc. In all of these cases, elastic parameters of lipid membranes are important for the description of membrane deformations, as these parameters determine energy barriers and characteristic times of membrane-involved phenomena. Since the development of molecular dynamics (MD), a variety of in silico methods have been proposed for the determination of elastic parameters of simulated lipid membranes. These MD methods allow for the consideration of details unattainable in experimental techniques and represent a distinct scientific field, which is rapidly developing. This work provides a review of these MD approaches with a focus on theoretical aspects. Two main challenges are identified: (i) the ambiguity in the transition from the continuum description of elastic theories to the discrete representation of MD simulations, and (ii) the determination of intrinsic elastic parameters of lipid mixtures, which is complicated due to the composition-curvature coupling effect.

Undulation of a moving fluid membrane pushed by filament growth

Biomembranes experience out-of-equilibrium conditions in living cells. Their undulation spectra are different from those in thermal equilibrium. Here, we report on the undulation of a fluid membrane pushed by the stepwise growth of filaments as in the leading edge of migrating cells, using three-dimensional Monte Carlo simulations. The undulations are largely modified from equilibrium behavior. When the tension is constrained, the low-wave-number modes are suppressed or enhanced at small or large growth step sizes, respectively, for high membrane surface tensions. In contrast, they are always suppressed for the tensionless membrane, wherein the wave-number range of the suppression depends on the step size. When the membrane area is constrained, in addition to these features, a specific mode is excited for zero and low surface tensions. The reduction of the undulation first induces membrane buckling at the lowest wave-number, and subsequently, other modes are excited, leading to a steady state.

Virtual bending method to calculate bending rigidity, saddle-splay modulus, and spontaneous curvature of thin fluid membranes

A method to calculate the bending rigidity κ, saddle-splay modulus κ[over ¯], and spontaneous curvature C_{0} of a fluid membrane is proposed. Virtual work for the bending deformations into cylindrical and spherical shapes is calculated for a flat membrane. This method does not require a force decomposition, unlike the existing stress-profile method. The first derivative of the deformation gives κC_{0} and is a discrete form of the first moment of the stress profile. The second derivatives give κ and κ[over ¯] and include the variance terms of the first derivatives, which are not accounted for in the stress-profile method. This method is examined for a solvent-free meshless membrane model and a dissipative-particle-dynamics two-bead amphiphilic molecular model. It is concluded that κ and κ[over ¯] of a thin membrane can be accurately calculated, whereas for a thick membrane or one with an explicit solvent, a further extension to include the volume-fluctuation effects is required for an accurate estimation. The amplitude of the volume-fluctuation effects can be evaluated using the parameter dependence in the present method.

Elastic moduli of lipid membranes: Reproducibility of AFM measures

Membrane elastic properties play a major role in membrane remodeling events, such as vesicle fusion and fission. They are also crucial in drug delivery by liposomes. Different experimental techniques are available to measure elastic properties. Among them, atomic force microscopy (AFM) presents the unique advantage of being directly applicable to nano-sized liposomes. Unfortunately, different AFM measures reported in the literature show little agreement among each other and are difficult to compare with measures of bending modulus obtained by other experimental techniques or by molecular simulations. In this work we determine the bending rigidity of Egg PC liposomes in terms of Young modulus via AFM measurements, using two different tip shapes and different cantilever force constants. We interpret the measures using the Hertz and Shell models, and observe a clear dependency of the Young modulus values on the tip properties and on the interpretative theory. The effect of the AFM tip shape is less important than the effect of the cantilever force constant, and the mathematical model has a major effect on the interpretation of the data. The Shell theory provides the closest agreement between AFM data and other experimental data for the membrane bending modulus. Finally, we compare the results to calculations of bending modulus from molecular dynamics simulations of membrane buckles. Simulations provide values of bending modulus consistent with literature data, but the agreement with AFM experiments is reasonable only for some specific experimental conditions.

Spontaneous Curvature, Differential Stress, and Bending Modulus of Asymmetric Lipid Membranes

Lipid 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. For instance, asymmetric flat bilayers, whose specific areas in each leaflet are matched to those of corresponding tensionless symmetric flat membranes, still exhibit a residual differential stress because the conditions of vanishing area strain and vanishing bending moment differ. We also measure the curvature rigidity of asymmetric bilayers and find that a sufficiently strong differential stress, but not compositional asymmetry alone, can increase the bending modulus. The likely cause is a stiffening of the compressed leaflet, which appears to be related to its gel transition but not identical with it. We finally show that the impact of cholesterol on differential stress depends on the relative strength of elastic and thermodynamic driving forces: if cholesterol solvates equally well in both leaflets, it will redistribute to cancel both leaflet tensions almost completely, but if its partitioning free energy prefers one leaflet over the other, the resulting distribution bias may even create differential stress. Because cells keep most of their lipid bilayers in an asymmetric nonequilibrium steady state, our findings suggest that biomembranes are elastically more complex than previously thought: besides a spontaneous curvature, they might also exhibit significant differential stress, which could strongly affect their curvature energetics.

Curvature induction and sensing of the F-BAR protein Pacsin1 on lipid membranes via molecular dynamics simulations

F-Bin/Amphiphysin/Rvs (F-BAR) domain proteins play essential roles in biological processes that involve membrane remodelling, such as endocytosis and exocytosis. It has been shown that such proteins transform the lipid membrane into tubes. Notably, Pacsin1 from the Pacsin/Syndapin subfamily has the ability to transform the membrane into various morphologies: striated tubes, featureless wide and thin tubes, and pearling vesicles. The molecular mechanism of this interesting ability remains elusive. In this study, we performed all-atom (AA) and coarse-grained (CG) molecular dynamics simulations to investigate the curvature induction and sensing mechanisms of Pacsin1 on a membrane. From AA simulations, we show that Pacsin1 has internal structural flexibility. In CG simulations with parameters tuned from the AA simulations, spontaneous assembly of two Pacsin1 dimers through lateral interaction is observed. Based on the complex structure, we show that the regularly assembled Pacsin1 dimers bend a tensionless membrane. We also show that a single Pacsin1 dimer senses the membrane curvature, binding to a buckled membrane with a preferred curvature. These results provide molecular insights into polymorphic membrane remodelling.

Mechanics of the Formation, Interaction, and Evolution of Membrane Tubular Structures

Membrane nanotubes, also known as membrane tethers, play important functional roles in many cellular processes, such as trafficking and signaling. Although considerable progresses have been made in understanding the physics regulating the mechanical behaviors of individual membrane nanotubes, relatively little is known about the formation of multiple membrane nanotubes due to the rapid occurring process involving strong cooperative effects and complex configurational transitions. By exerting a pair of external extraction upon two separate membrane regions, here, we combine molecular dynamics simulations and theoretical analysis to investigate how the membrane nanotube formation and pulling behaviors are regulated by the separation between the pulling forces and how the membrane protrusions interact with each other. As the force separation increases, different membrane configurations are observed, including an individual tubular protrusion, a relatively less deformed protrusion with two nanotubes on its top forming a V shape, a Y-shaped configuration through nanotube coalescence via a zipper-like mechanism, and two weakly interacting tubular protrusions. The energy profile as a function of the separation is determined. Moreover, the directional flow of lipid molecules accompanying the membrane shape transition is analyzed. Our results provide new, to our knowledge, insights at a molecular level into the interaction between membrane protrusions and help in understanding the formation and evolution of intra- and intercellular membrane tubular networks involved in numerous cell activities.

Curvature sensing by cardiolipin in simulated buckled membranes

Cardiolipin is a non-bilayer phospholipid with a unique dimeric structure. It localizes to negative curvature regions in bacteria and is believed to stabilize respiratory chain complexes in the highly curved mitochondrial membrane. Cardiolipin's localization mechanism remains unresolved, because important aspects such as the structural basis and strength for lipid curvature preferences are difficult to determine, partly due to the lack of efficient simulation methods. Here, we report a computational approach to study curvature preferences of cardiolipin by simulated membrane buckling and quantitative modeling. We combine coarse-grained molecular dynamics with simulated buckling to determine the curvature preferences in three-component bilayer membranes with varying concentrations of cardiolipin, and extract curvature-dependent concentrations and lipid acyl chain order parameter profiles. Cardiolipin shows a strong preference for negative curvatures, with a highly asymmetric chain order parameter profile. The concentration profiles are consistent with an elastic model for lipid curvature sensing that relates lipid segregation to local curvature via the material constants of the bilayers. These computations constitute new steps to unravel the molecular mechanism by which cardiolipin senses curvature in lipid membranes, and the method can be generalized to other lipids and membrane components as well.

BUMPy: A Model-Independent Tool for Constructing Lipid Bilayers of Varying Curvature and Composition

Molecular dynamics is a powerful tool to investigate atomistic and mesoscopic phenomena in lipid bilayer systems. These studies have progressed with the advent of increased computational power, and efforts are now increasingly being directed toward investigating the role of curvature and bilayer morphology, as these are critical features of biological processes. Computational studies of lipid bilayers benefit from tools that can create starting configurations for molecular dynamics simulations, but the majority of such tools are restricted to generating flat bilayers. Generating curved bilayer configurations comes with practical complications and potential ramifications on physical properties in the simulated system if the bilayer is initiated in a high-strain state. We present a new tool for creating curved lipid bilayers that combines flexibility of shape, force field, model resolution, and bilayer composition. A key aspect of our approach is the use of the monolayer pivotal plane location to accurately estimate interleaflet area differences in a curved bilayer. Our tool is named BUMPy (Building Unique Membranes in Python), is written in Python, is fast, and has a simple command line interface.

Cell Membrane Disruption by Vertical Micro-/Nanopillars: Role of Membrane Bending and Traction Forces

Gaining access to the cell interior is fundamental for many applications, such as electrical recording and drug and biomolecular delivery. A very promising technique consists of culturing cells on micro-/nanopillars. The tight adhesion and high local deformation of cells in contact with nanostructures can promote the permeabilization of lipids at the plasma membrane, providing access to the internal compartment. However, there is still much experimental controversy regarding when and how the intracellular environment is targeted and the role of the geometry and interactions with surfaces. Consequently, we investigated, by coarse-grained molecular dynamics simulations of the cell membrane, the mechanical properties of the lipid bilayer under high strain and bending conditions. We found out that a high curvature of the lipid bilayer dramatically lowers the traction force necessary to achieve membrane rupture. Afterward, we experimentally studied the permeabilization rate of the cell membrane by pillars with comparable aspect ratios but different sharpness values at the edges. The experimental data support the simulation results: even pillars with diameters in the micron range may cause local membrane disruption when their edges are sufficiently sharp. Therefore, the permeabilization likelihood is connected to the local geometric features of the pillars rather than diameter or aspect ratio. The present study can also provide significant contributions to the design of three-dimensional biointerfaces for tissue engineering and cellular growth.

Particle-based membrane model for mesoscopic simulation of cellular dynamics

We present a simple and computationally efficient coarse-grained and solvent-free model for simulating lipid bilayer membranes. In order to be used in concert with particle-based reaction-diffusion simulations, the model is purely based on interacting and reacting particles, each representing a coarse patch of a lipid monolayer. Particle interactions include nearest-neighbor bond-stretching and angle-bending and are parameterized so as to reproduce the local membrane mechanics given by the Helfrich energy density over a range of relevant curvatures. In-plane fluidity is implemented with Monte Carlo bond-flipping moves. The physical accuracy of the model is verified by five tests: (i) Power spectrum analysis of equilibrium thermal undulations is used to verify that the particle-based representation correctly captures the dynamics predicted by the continuum model of fluid membranes. (ii) It is verified that the input bending stiffness, against which the potential parameters are optimized, is accurately recovered. (iii) Isothermal area compressibility modulus of the membrane is calculated and is shown to be tunable to reproduce available values for different lipid bilayers, independent of the bending rigidity. (iv) Simulation of two-dimensional shear flow under a gravity force is employed to measure the effective in-plane viscosity of the membrane model and show the possibility of modeling membranes with specified viscosities. (v) Interaction of the bilayer membrane with a spherical nanoparticle is modeled as a test case for large membrane deformations and budding involved in cellular processes such as endocytosis. The results are shown to coincide well with the predicted behavior of continuum models, and the membrane model successfully mimics the expected budding behavior. We expect our model to be of high practical usability for ultra coarse-grained molecular dynamics or particle-based reaction-diffusion simulations of biological systems.

Computing Curvature Sensitivity of Biomolecules in Membranes by Simulated Buckling

Membrane curvature sensing, where the binding free energies of membrane-associated molecules depend on the local membrane curvature, is a key factor to modulate and maintain the shape and organization of cell membranes. However, the microscopic mechanisms are not well understood, partly due to absence of efficient simulation methods. Here, we describe a method to compute the curvature dependence of the binding free energy of a membrane-associated probe molecule that interacts with a buckled membrane, which has been created by lateral compression of a flat bilayer patch. This buckling approach samples a wide range of curvatures in a single simulation, and anisotropic effects can be extracted from the orientation statistics. We develop an efficient and robust algorithm to extract the motion of the probe along the buckled membrane surface, and evaluate its numerical properties by extensive sampling of three coarse-grained model systems: local lipid density in a curved environment for single-component bilayers, curvature preferences of individual lipids in two-component membranes, and curvature sensing by a homotrimeric transmembrane protein. The method can be used to complement experimental data from curvature partition assays and provides additional insight into mesoscopic theories and molecular mechanisms for curvature sensing.

Buckling Under Pressure: Curvature-Based Lipid Segregation and Stability Modulation in Cardiolipin-Containing Bilayers

Mitochondrial metabolic function is affected by the morphology and protein organization of the mitochondrial inner membrane. Cardiolipin (CL) is a unique tetra-acyl lipid that is involved in the maintenance of the highly curved shape of the mitochondrial inner membrane as well as spatial organization of the proteins necessary for respiration and oxidative phosphorylation. Cardiolipin has been suggested to self-organize into lipid domains due to its inverted conical molecular geometry, though the driving forces for this organization are not fully understood. In this work, we use coarse-grained molecular dynamics simulations to study the mechanical properties and lipid dynamics in heterogeneous bilayers both with and without CL, as a function of membrane curvature. We find that incorporation of CL increases bilayer deformability and that CL becomes highly enriched in regions of high negative curvature. We further show that another mitochondrial inverted conical lipid, phosphatidylethanolamine (PE), does not partition or increase the deformability of the membrane in a significant manner. Therefore, CL appears to possess some unique characteristics that cannot be inferred simply from molecular geometry considerations.

Theory and algorithms to compute Helfrich bending forces: a review

Cell membranes are vital to shield a cell's interior from the environment. At the same time they determine to a large extent the cell's mechanical resistance to external forces. In recent years there has been considerable interest in the accurate computational modeling of such membranes, driven mainly by the amazing variety of shapes that red blood cells and model systems such as vesicles can assume in external flows. Given that the typical height of a membrane is only a few nanometers while the surface of the cell extends over many micrometers, physical modeling approaches mostly consider the interface as a two-dimensional elastic continuum. Here we review recent modeling efforts focusing on one of the computationally most intricate components, namely the membrane's bending resistance. We start with a short background on the most widely used bending model due to Helfrich. While the Helfrich bending energy by itself is an extremely simple model equation, the computation of the resulting forces is far from trivial. At the heart of these difficulties lies the fact that the forces involve second order derivatives of the local surface curvature which by itself is the second derivative of the membrane geometry. We systematically derive and compare the different routes to obtain bending forces from the Helfrich energy, namely the variational approach and the thin-shell theory. While both routes lead to mathematically identical expressions, so-called linear bending models are shown to reproduce only the leading order term while higher orders differ. The main part of the review contains a description of various computational strategies which we classify into three categories: the force, the strong and the weak formulation. We finally give some examples for the application of these strategies in actual simulations.

Anisotropic Membrane Curvature Sensing by Amphipathic Peptides

Many proteins and peptides have an intrinsic capacity to sense and induce membrane curvature, and play crucial roles for organizing and remodeling cell membranes. However, the molecular driving forces behind these processes are not well understood. Here, we describe an approach to study curvature sensing by simulating the interactions of single molecules with a buckled lipid bilayer. We analyze three amphipathic antimicrobial peptides, a class of membrane-associated molecules that specifically target and destabilize bacterial membranes, and find qualitatively different sensing characteristics that would be difficult to resolve with other methods. Our findings provide evidence for direction-dependent curvature sensing mechanisms in amphipathic peptides and challenge existing theories of hydrophobic insertion. The buckling approach is generally applicable to a wide range of curvature-sensing molecules, and our results provide strong motivation to develop new experimental methods to track position and orientation of membrane proteins.

Determining the pivotal plane of fluid lipid membranes in simulations

Each leaflet of a curved lipid membrane contains a surface at which the area strain vanishes, the so-called pivotal plane. Its distance z0 from the bilayer's midplane arises in numerous contexts, for instance the connection between monolayer and bilayer moduli, stress-profile moments, or area-difference elasticity theories. Here, we propose two precise methods for determining the location of the pivotal plane in computer simulations, both of which rely on monitoring the lipid imbalance across a curved bilayer. The first method considers the ratio of lipid number between the two leaflets of cylindrical or spherical vesicles; it hence requires lipid flip-flop for equilibration. The second method looks at the leaflet difference across local sections cut out from a buckled membrane; this observable equilibrates even in the absence of flip-flop. We apply our methods to two different coarse-grained lipid models, the generic three-bead solvent-free Cooke model and a ten-bead representation of dimyristoylphosphocholine with the explicit solvent MARTINI model. The Cooke model is amenable to both methods and gives results that agree at the percent level. Using it, we also show that the pivotal plane moves outward as lipid curvature becomes more positive. The MARTINI model can only be analyzed with the buckling method; the obtained value z0 = 0.850(11) nm lies about 0.4 nm inwards of the glycerol backbone and is hence unexpectedly small. We attribute this to limitations of the coarse-grained description, suggesting that the location of the pivotal plane might be a good indicator for how well lipid models capture the microscopic origins of curvature elasticity. Finally, we also show that the pivotal plane position itself moves as the membrane is bent. The leading correction is linear in curvature, dependent on the Poisson ratio, and can matter when analyzing experimental results obtained from highly curved inverse hexagonal phases.

Calculation of membrane bending rigidity using field-theoretic umbrella sampling

The free-energy change of membrane shape transformations can be small, e.g., as in the case of membrane bending. Therefore, the calculation of the free-energy difference between different membrane morphologies is a challenge. Here, we discuss a computational method - field-theoretic umbrella sampling - to compute the local chemical potential of a non-equilibrium configuration and illustrate how one can apply it to study free-energy changes of membrane transformations using simulations. Specifically, the chemical potential profile of the bent membrane and the bending rigidity of membrane are calculated for a soft, coarse-grained amphiphile model and the MARTINI model of a dioleoylphosphatidylcholine (DOPC) membrane.

Monte Carlo study of the frame, fluctuation and internal tensions of fluctuating membranes with fixed area

Three types of surface tensions can be defined for lipid membranes: the internal tension, σ, conjugated to the real membrane area in the Hamiltonian, the mechanical frame tension, τ, conjugated to the projected area, and the "fluctuation tension", r, obtained from the fluctuation spectrum of the membrane height. We investigate these surface tensions by means of a Monge gauge lattice Monte Carlo simulation involving the exact, nonlinear, Helfrich Hamiltonian and a measure correction for the excess entropy of the Monge gauge. Our results for the relation between σ and τ agrees well with the theoretical prediction of [J.-B. Fournier and C. Barbetta, Phys. Rev. Lett., 2008, 100, 078103] based on a Gaussian approximation. This provides a valuable knowledge of τ in the standard Gaussian models where the tension is controlled by σ. However, contrary to the conjecture in the above paper, we find that r exhibits no significant difference from τ over more than five decades of tension. Our results appear to be valid in the thermodynamic limit and are robust to changing the ensemble in which the membrane area is controlled.

Curvature Softening and Negative Compressibility of Gel-Phase Lipid Membranes

We show that gel-phase lipid membranes soften upon bending, leading to curvature localization and a negative compressibility. Using simulations of two very different lipid models to quantify shape and stress-strain relation of buckled membranes, we demonstrate that gel phase bilayers do not behave like Euler elastica and hence are not well described by a quadratic Helfrich Hamiltonian, much unlike their fluid-phase counterparts. We propose a theoretical framework which accounts for the observed softening through an energy density that smoothly crosses over from a quadratic to a linear curvature dependence beyond a critical new scale [Formula: see text](-1). This model captures both the shape and the stress-strain relation for our two sets of simulations and permits the extraction of bending moduli, which are found to be about an order of magnitude larger than the corresponding fluid phase values. We also find surprisingly large crossover lengths [Formula: see text], several times bigger than the bilayer thickness, rendering the exotic elasticity of gel-phase membranes more strongly pronounced than that of homogeneous compressible sheets and artificial metamaterials. We suggest that such membranes have unexpected potential as nanoscale systems with striking materials characteristics.

Formation of adhesion domains in stressed and confined membranes

The adhesion bonds connecting a lipid bilayer to an underlying surface may undergo a condensation transition resulting from an interplay between a short range attractive potential between them, and a long range fluctuation-induced potential of mean force. Here, we use computer simulations of a coarse-grained molecular model of supported lipid bilayers to study this transition in confined membranes, and in membranes subjected to a non-vanishing surface tension. Our results show that confinement may alter significantly the condensation transition of the adhesion bonds, whereas the application of surface tension has a very minor effect on it. We also investigate domain formation in membranes under negative tension which, in free membranes, causes the enhancement of the amplitude of membrane thermal undulations. Our results indicate that in supported membranes, this effect of a negative surface tension on the fluctuation spectrum is largely eliminated by the pressure resulting from the mixing entropy of the adhesion bonds.

Computer investigations of influences of molar fraction and acyl chain length of lipids on the nanoparticle–biomembrane interactions

Structural variations of the heterogeneous membrane: (a) a water defect, (b) the membrane buckling.

The Effective Field Theory approach towards membrane-mediated interactions between particles

Fluid lipid membranes can mediate forces between particles bound to them: A local deformation of the surface geometry created by some object spreads to distant regions, where other objects can respond to it. The physical characteristics of these geometric interactions, and how they are affected by thermal fluctuations, are well described by the simple continuum curvature-elastic Hamiltonian proposed 40 years ago by Wolfgang Helfrich. Unfortunately, while the underlying principles are conceptually straightforward, the corresponding calculations are not-largely because one must enforce boundary conditions for finite-sized objects. This challenge has inspired several heuristic approaches for expressing the problem in a point particle language. While streamlining the calculations of leading order results and enabling predictions for higher order corrections, the ad hoc nature of the reformulation leaves its domain of validity unclear. In contrast, the framework of Effective Field Theory (EFT) provides a systematic way to construct a completely equivalent point particle description. In this review we present a detailed account for how this is accomplished. In particular, we use a familiar example from electrostatics as an analogy to motivate the key steps needed to construct an EFT, most notably capturing finite size information in point-like "polarizabilities," and determining their value through a suitable "matching procedure." The interaction (free) energy then emerges as a systematic cumulant expansion, for which powerful diagrammatic techniques exist, which we also briefly revisit. We then apply this formalism to derive series expansions for interactions between flat and curved particle pairs, multibody interactions, as well as corrections to all these interactions due to thermal fluctuations.

Structure formation of surfactant membranes under shear flow

Shear-flow-induced structure formation in surfactant-water mixtures is investigated numerically using a meshless-membrane model in combination with a particle-based hydrodynamics simulation approach for the solvent. At low shear rates, uni-lamellar vesicles and planar lamellae structures are formed at small and large membrane volume fractions, respectively. At high shear rates, lamellar states exhibit an undulation instability, leading to rolled or cylindrical membrane shapes oriented in the flow direction. The spatial symmetry and structure factor of this rolled state agree with those of intermediate states during lamellar-to-onion transition measured by time-resolved scatting experiments. Structural evolution in time exhibits a moderate dependence on the initial condition.

A guiding potential method for evaluating the bending rigidity of tensionless lipid membranes from molecular simulation

A new method is proposed to estimate the bending rigidity of lipid membranes from molecular dynamics simulations. An external cylindrical guiding potential is used to impose a sinusoidal deformation to a planar membrane. The bending rigidity is obtained from the mean force acting on the cylinder by calibrating against a discretized Helfrich model that accounts for thermal fluctuations of the membrane surface. The method has been successfully applied to a dimyristoyl phosphatidylcholine bilayer simulated with a coarse-grained model. A well-converged bending rigidity was obtained for the tension-free membrane and showed reasonable agreement with that obtained from the height fluctuation spectrum.

Estimation of the bending rigidity and spontaneous curvature of fluid membranes in simulations

Several numerical methods for measuring the bending rigidity and the spontaneous curvature of fluid membranes are studied using two types of meshless membrane models. The bending rigidity is estimated from the thermal undulations of planar and tubular membranes and the axial force of tubular membranes. We found a large dependence of its estimate value from the thermal undulation analysis on the upper-cutoff frequency q(cut) of the least-squares fit. The inverse power-spectrum fit with an extrapolation to q(cut)→0 yields the smallest estimation error among the investigated methods. The spontaneous curvature is estimated from the axial force of tubular membranes and the average curvature of bent membrane strips. The results of these methods show good agreement with each other.