Determining the Lipid Tilt Modulus by Simulating Membrane Buckles

Predicting protein curvature sensing across membrane compositions with a bilayer continuum model

Cytoplasmic proteins must recruit to membranes to function in processes such as endocytosis and cell division. Many of these proteins recognize not only the chemical structure of the membrane lipids, but the curvature of the surface, binding more strongly to more highly curved surfaces, or 'curvature sensing'. Curvature sensing by amphipathic helices is known to vary with membrane bending rigidity, but changes to lipid composition can simultaneously alter membrane thickness, spontaneous curvature, and leaflet symmetry, thus far preventing a systematic characterization of lipid composition on such curvature sensing through either experiment or simulation. Here we develop and apply a bilayer continuum membrane model that can tractably address this gap, quantifying how controlled changes to each material property can favor or disfavor protein curvature sensing. We evaluate both energetic and structural changes to vesicles upon helix insertion, with strong agreement to new in vitro experiments and all-atom MD simulations, respectively. Our membrane model builds on previous work to include both monolayers of the bilayer via representation by continuous triangular meshes. We introduce a coupling energy that captures the incompressibility of the membrane and the established energetics of lipid tilt. In agreement with experiment, our model predicts stronger curvature sensing in membranes with distinct tail groups (POPC vs DOPC vs DLPC), despite having identical head-group chemistry; the model shows that the primary driving force for weaker curvature sensing in DLPC is that it is thinner, and more wedge shaped. Somewhat surprisingly, asymmetry in lipid shape composition between the two leaflets has a negligible contribution to membrane mechanics following insertion. Our multi-scale approach can be used to quantitatively and efficiently predict how changes to membrane composition in flat to highly curved surfaces alter membrane energetics driven by proteins, a mechanism that helps proteins target membranes at the correct time and place.

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.

Self-Assembly of Unusually Stable Thermotropic Network Phases by Cellobiose-Based Guerbet Glycolipids

Bicontinuous thermotropic liquid crystal (LC) materials, e.g., double gyroid (DG) phases, have garnered significant attention due to the potential utility of their 3D network structures in wide-ranging applications. However, the utility of these materials is significantly constrained by the lack of robust molecular design rules for shape-filling amphiphiles that spontaneously adopt the saddle curvatures required to access these useful supramolecular assemblies. Toward this aim, we synthesized anomerically pure Guerbet-type glycolipids bearing cellobiose head groups and branched alkyl tails and studied their thermotropic LC self-assembly. Using a combination of differential scanning calorimetry, polarized optical microscopy, and small-angle X-ray scattering, our studies demonstrate that Guerbet cellobiosides exhibit a strong propensity to self-assemble into DG morphologies over wide thermotropic phase windows. The stabilities of these assemblies sensitively depend on the branched alkyl tail structure and the anomeric configuration of the glycolipid in a previously unrecognized manner. Complementary molecular simulations furnish detailed insights into the observed self-assembly characteristics, thus unveiling molecular motifs that foster network phase self-assembly that will enable future designs and applications of network LC materials.

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.

Tilt Modulus of Bilayer Membranes Self-Assembled from Rod-Coil Diblock Copolymers

Quantitatively understanding membrane fission and fusion requires a mathematical model taking their underlying elastic degrees of freedom, such as the molecule's tilt, into account. Hamm-Kozlov's model is such a framework that includes a tilt modulus along with the bending modulus and Gaussian modulus. This paper investigates the tilt modulus of liquid-crystalline bilayer membranes by applying self-consistent field theory. Unlike the widely used method in molecular dynamics simulation which extracts the tilt modulus by simulating bilayer buckles with various single modes, we introduce a tilt constrain term in the free energy to stabilize bilayers with various tilt angles. Fitting the energy curve as a function of the tilt angle to Hamm-Kozlov's elastic energy allows us to extract the tilt modulus directly. Based on this novel scheme and focused on the bilayers self-assembled from rod-coil diblock copolymers, we carry out a systematic study of the dependence of the tensionless A-phase bilayer's tilt modulus on the microscopic parameters.

Stabilizing Leaflet Asymmetry under Differential Stress in a Highly Coarse-Grained Lipid Membrane Model

We present a version of the coarse-grained Cooke lipid model, modified to simulate asymmetric lipid membranes. It is inspired by a method employed by Wang et al. [ Commun. Comput. Phys. 2013, 13, 1093-1106] for artificially penalizing lipid flip-flop but copes more robustly with differential stress, at the cost of one additional bead per lipid and the concomitant increase in computational overhead. Bilayer asymmetry ultimately breaks down beyond a system size dependent critical differential stress, which can be predicted from a simple analytical model. We remeasure many important material parameters for the new model and find it to be consistent with typical fluid lipid membranes. Maintaining a stable stress asymmetry has many applications, and we give two examples: (i) connecting monolayer stress to lipid number asymmetry in order to directly measure the monolayer area modulus and (ii) finding its strain-dependent higher-order correction by monitoring the equilibrium bilayer area.

A combined molecular/continuum-modeling approach to predict the small-angle neutron scattering of curved membranes

This paper develops a framework to compute the small-angle neutron scattering (SANS) from highly curved, dynamically fluctuating, and potentially inhomogeneous membranes. This method is needed to compute the scattering from nanometer-scale membrane domains that couple to curvature, as predicted by molecular modeling. The detailed neutron scattering length density of a small planar bilayer patch is readily available via molecular dynamics simulation. A mathematical, mechanical transformation of the planar scattering length density is developed to predict the scattering from curved bilayers. By simulating a fluctuating, curved, surface-continuum model, long time- and length-scales can be reached while, with the aid of the planar-to-curved transformation, the molecular features of the scattering length density can be retained. A test case for the method is developed by constructing a coarse-grained lipid vesicle following a protocol designed to relieve both the osmotic stress inside the vesicle and the lipid-number stress between the leaflets. A question was whether the hybrid model would be able to replicate the scattering from the highly deformed inner and outer leaflets of the small vesicle. Matching the scattering of the full (molecular vesicle) and hybrid (continuum vesicle) models indicated that the inner and outer leaflets of the full vesicle were expanded laterally, consistent with previous simulations of the Martini forcefield that showed thinning in small vesicles. The vesicle structure is inconsistent with a zero-tension leaflet deformed by a single set of elastic parameters, and the results show that this is evident in the scattering. The method can be applied to translate observations of any molecular model's neutron scattering length densities from small patches to large length and timescales.

Mechanical properties of lipid bilayers: a note on the Poisson ratio

We investigate the Poisson ratio ν of fluid lipid bilayers, i.e., the question how area strains compare to the changes in membrane thickness (or, equivalently, volume) that accompany them. We first examine existing experimental results on the area- and volume compressibility of lipid membranes. Analyzing them within the framework of linear elasticity theory for homogeneous thin fluid sheets leads us to conclude that lipid membrane deformations are to a very good approximation volume-preserving, with a Poisson ratio that is likely about 3% smaller than the common soft matter limit . These results are fully consistent with atomistic simulations of a DOPC membrane at varying amount of applied lateral stress, for which we instead deduce ν by directly comparing area- and volume strains. To assess the problematic assumption of transverse homogeneity, we also define a depth-resolved Poisson ratio ν(z) and determine it through a refined analysis of the same set of simulations. We find that throughout the membrane's thickness, ν(z) is close to the value derived assuming homogeneity, with only minor variations of borderline statistical significance.

A consistent quadratic curvature-tilt theory for fluid lipid membranes

The tilt of a lipid molecule describes the deviation of its orientation away from the local normal of its embedding membrane. Tilt is the subleading degree of freedom after a membrane's geometry, and it becomes relevant at scales comparable to lipid bilayer thickness. Building on earlier work by Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)], who envisioned lipid membranes as thin prestressed fluid elastic films, and Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)], who discovered a new coupling term between splay and tilt divergence, we construct a theory of membrane elasticity that is quadratic in geometry and tilt and complete at order 1/length². We show that a general and consistent treatment of both lateral and transverse depth-dependent shear stresses creates several contributions to the elastic energy density, of which only a subset had previously been identified. Apart from the well-known penalty of lipid twist (the curl of tilt), these terms generate no qualitatively new phenomenology, but they quantitatively revise the connections between the moduli of a tilt-curvature theory and its underlying microscopic foundation. In particular, we argue that the monolayer Gaussian curvature modulus κ¯_m, widely believed to be equal to the second moment of the transmonolayer stress profile, acquires a second contribution from lipid twist, which is always negative. This could resolve the long-standing conundrum that many measured values of κ¯_m appeared to have a sign that violates basic stability considerations. We also show that the previously discovered novel coupling between splay and tilt divergence is not simply proportional to κ¯_m but acquires its own splay-tilt coupling modulus, κ_st,m. We explore the predictions of our theory for various elastic moduli and their mutual interrelations and use an extensive set of existing atomistic molecular dynamics simulations for 12 different lipid types to collectively reason about such predictions. We find that bending rigidities are captured fairly well by existing theories, while reliable predictions for local moduli, especially the splay-tilt coupling modulus, remain challenging.

The role of scaffold reshaping and disassembly in dynamin driven membrane fission

The large GTPase dynamin catalyzes membrane fission in eukaryotic cells, but despite three decades of experimental work, competing and partially conflicting models persist regarding some of its most basic actions. Here we investigate the mechanical and functional consequences of dynamin scaffold shape changes and disassembly with the help of a geometrically and elastically realistic simulation model of helical dynamin-membrane complexes. Beyond changes of radius and pitch, we emphasize the crucial role of a third functional motion: an effective rotation of the filament around its longitudinal axis, which reflects alternate tilting of dynamin’s PH binding domains and creates a membrane torque. We also show that helix elongation impedes fission, hemifission is reached via a small transient pore, and coat disassembly assists fission. Our results have several testable structural consequences and help to reconcile mutual conflicting aspects between the two main present models of dynamin fission—the two-stage and the constrictase model.

When cells take up material from their surroundings, they must first transport this cargo across their outer membrane, a flexible sheet of tightly organized fat molecules that act as a barrier to the environment. Cells can achieve this by letting their membrane surround the object, pulling it inwards until it is contained in a pouch that bulges into the cell. This bag is then corded up so it splits off from the outer membrane. The ‘cord’ is a protein called dynamin, which is thought to form a tight spiral around the bag’s neck, closing it over and pinching it away. The structure of dynamin is fairly well known, and yet several theories compete to explain how it may snap the bag off the outer membrane.

Here, Pannuzzo et al. have created a computer simulation that faithfully replicates the geometry and the elasticity of the membrane and of dynamin, and used it to test different ways the protein could work. The first test featured simple constriction, where the dynamin spiral contracts around the membrane to pinch it; this only separated the bag from the membrane after implausibly tight constriction. The second test added elongation, with the spiral lengthening as well as reducing its diameter, but this further reduced the ability for the protein to snap off the membrane. The final test combined constriction and rotation, whereby dynamin ‘twirls’ as it presses on the neck of the bag: this succeeded in efficiently severing the membrane once the dynamin spiral disassembled. Indeed, the simulations suggested that dynamin might start to dismantle while it constricts, without compromising its role. In fact, getting rid of excess length as the protein contracts helps to dissolve any remnants of a membrane connection.

Defects in dynamin are associated with conditions such as centronuclear myopathy and Charcot‐Marie‐Tooth peripheral neuropathy. Recent research also indicates that the protein is involved in a much wider range of neurological disorders that include Alzheimer's, Parkinson's, Huntington's, and amyotrophic lateral sclerosis. The models created by Pannuzzo et al. are useful tools to understand how dynamin and similar proteins work and sometimes fail.

Novel tilt-curvature coupling in lipid membranes

On mesoscopic scales, lipid membranes are well described by continuum theories whose main ingredients are the curvature of a membrane's reference surface and the tilt of its lipid constituents. In particular, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)] have shown how to systematically derive such a tilt-curvature Hamiltonian based on the elementary assumption of a thin fluid elastic sheet experiencing internal lateral pre-stress. Performing a dimensional reduction, they not only derive the basic form of the effective surface Hamiltonian but also express its emergent elastic couplings as trans-membrane moments of lower-level material parameters. In the present paper, we argue, though, that their derivation unfortunately missed a coupling term between curvature and tilt. This term arises because, as one moves along the membrane, the curvature-induced change of transverse distances contributes to the area strain-an effect that was believed to be small but nevertheless ends up contributing at the same (quadratic) order as all other terms in their Hamiltonian. We illustrate the consequences of this amendment by deriving the monolayer and bilayer Euler-Lagrange equations for the tilt, as well as the power spectra of shape, tilt, and director fluctuations. A particularly curious aspect of our new term is that its associated coupling constant is the second moment of the lipid monolayer's lateral stress profile-which within this framework is equal to the monolayer Gaussian curvature modulus, κ¯_m. On the one hand, this implies that many theoretical predictions now contain a parameter that is poorly known (because the Gauss-Bonnet theorem limits access to the integrated Gaussian curvature); on the other hand, the appearance of κ¯_m outside of its Gaussian curvature provenance opens opportunities for measuring it by more conventional means, for instance by monitoring a membrane's undulation spectrum at short scales.

Intramolecular structural parameters are key modulators of the gel-liquid transition in coarse grained simulations of DPPC and DOPC lipid bilayers

The capability of coarse-grained models based on the MARTINI mapping to reproduce the gel-liquid phase transition in saturated and unsaturated model lipids was investigated. We found that the model is able to reproduce a lower critical temperature for 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) with respect to 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC). Nonetheless, the appearance of a gel phase for DOPC is strictly dependent on the intramolecular parameters chosen to model its molecular structure. In particular, we show that the bending angle at the coarse-grained bead corresponding to the unsaturated carbon-carbon bond acts as an order parameter determining the temperature of the phase transition. Structural analysis of the molecular dynamics simulations runs evidences that in the gel phase, the packing of the lipophilic tails of DOPC assume a different conformation than in the liquid phase. In the latter phase, the DOPC geometry resembles that of the relaxed free molecule. DPPC:DOPC mixtures show a single phase transition temperature, indicating that the observation of a phase separation between the two lipids requires the simulation of systems with sizes much larger than the ones used here.

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.

Determination of bending rigidity and tilt modulus of lipid membranes from real-space fluctuation analysis of molecular dynamics simulations

We have recently developed a novel computational methodology (termed RSF for Real-Space Fluctuations) to quantify the bending rigidity and tilt modulus of lipid membranes from real-space analysis of fluctuations in the tilt and splay degrees of freedom as sampled in molecular dynamics (MD) simulations. In this article, we present a comprehensive study that combines results from the application of the RSF method to a wide range of lipid bilayer systems that encompass membranes of different fluidities and sizes, including lipids with saturated and unsaturated lipid tails, single and multi-component lipid systems, as well as non-standard lipids such as the four-tailed cardiolipin. By comparing the material properties calculated with the RSF method to those obtained from experimental data and from other computational methodologies, we rigorously demonstrate the validity of our approach and show its robustness. This should allow for future applications of even more complex lipidic assemblies, whose material properties are not tractable by other computational techniques. In addition, we discuss the relationship between different definitions of the tilt modulus appearing in current literature to address some important unresolved discrepancies in the field.

X-ray scattering reveals molecular tilt is an order parameter for the main phase transition in a model biomembrane

Synchrotron diffuse x-ray scattering data reveal a dramatic softening of the molecular tilt modulus K_{θ} of the model biomembrane composed of DMPC lipids as the temperature is lowered towards the main phase transition temperature at T_{M}=24^{∘}C. Spontaneous tilt occurs below T_{M}, suggesting that tilt is a symmetry breaking order parameter. Consistent with this hypothesis, it is also found that a different lipid POPS has no spontaneous tilt below its T_{M} at 14^{∘}C and correspondingly its tilt modulus did not soften as T_{M} was approached from above. As previously known, the bending modulus K_{C} of DMPC also softens close to T_{M}, but unlike the tilt modulus, K_{C} has a maximum 3^{∘} above T_{M}, which also marks the limit of the well-known anomalous swelling regime. Tilt adds a different perspective to our previous understanding of the main phase transition in lipid bilayers.

Toward Chemically Resolved Computer Simulations of Dynamics and Remodeling of Biological Membranes

Cellular membranes are fundamental constituents of living organisms. Apart from defining the boundaries of the cells, they are involved in a wide range of biological functions, associated with both their structural and the dynamical properties. Biomembranes can undergo large-scale transformations when subject to specific environmental changes, including gel-liquid phase transitions, change of aggregation structure, formation of microtubules, or rupture into vesicles. All of these processes are dependent on a delicate interplay between intermolecular forces, molecular crowding, and entropy, and their understanding requires approaches that are able to capture and rationalize the details of all of the involved interactions. Molecular dynamics-based computational models at atom-level resolution are, in principle, the best way to perform such investigations. Unfortunately, the relevant spatial and time dimensionalities involved in membrane remodeling phenomena would require computational costs that are today unaffordable on a routinely basis. Such hurdles can be removed by coarse-graining the representations of the individual molecular components of the systems. This procedure anyway reduces the possibility of describing the chemical variations in the lipid mixtures composing biological membranes. New hybrid particle field multiscale approaches offer today a promising alternative to the more traditional particle-based simulations methods. By combining chemically distinguishable molecular representations with mesoscale-based computationally affordable potentials, they appear as one of the most promising ways to keep an accurate description of the chemical complexity of biological membranes and, at the same time, cover the required scales to describe remodeling events.

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.

Gramicidin A Channel Formation Induces Local Lipid Redistribution II: A 3D Continuum Elastic Model

To change conformation, a protein must deform the surrounding bilayer. In this work, a three-dimensional continuum elastic model for gramicidin A in a lipid bilayer is shown to describe the sensitivity to thickness, curvature stress, and the mechanical properties of the lipid bilayer. A method is demonstrated to extract the gramicidin-lipid boundary condition from all-atom simulations that can be used in the three-dimensional continuum model. The boundary condition affects the deformation dramatically, potentially much more than typical variations in the material stiffness do as lipid composition is changed. Moreover, it directly controls the sensitivity to curvature stress. The curvature stress and hydrophobic surfaces of the all-atom and continuum models are found to be in excellent agreement. The continuum model is applied to estimate the enrichment of hydrophobically matched lipids near the channel in a mixture, and the results agree with single-channel experiments and extended molecular dynamics simulations from the companion article by Beaven et al. in this issue of Biophysical Journal.

Experimentally determined tilt and bending moduli of single-component lipid bilayers

Values of the bending modulus K_C and the tilt modulus K_θ are reported for single component lipid bilayers. The lipids studied have the common names DOPC, DMPC, diC22:1PC, SOPC, POPC, diPhyPC, DLPC, DPPC, DHPC and DEPC, listed in the order of number of samples examined. The experimental method, thus far the only one that measures the tilt modulus of lipid bilayers, first obtains diffuse X-ray scattering data from oriented stacks of bilayers. The values of the moduli emerge from fitting the data to the accepted tilt-dependent continuum model for the free energy of a single bilayer, further enhanced by interactions between bilayers in the stack. The results indicate the broad trend that the tilt modulus for these PC lipids is smaller the closer the temperature is to the main transition temperature. Another trend is that inclusion of tilt raises the value of the bending modulus more for lipids with smaller values of the tilt modulus. Values of both moduli are compared to recent literature values obtained from simulations and values of the bending modulus are compared to the literature values obtained by other experimental methods.

Analytical calculation of the lipid bilayer bending modulus

Bending and Gaussian moduli of a homogenious single-component lipid bilayer are calculated analytically using microscopic model of the lipid hydrocarbon chains. The approach allows for thermodynamic averaging over different chains conformations. Each chain is modeled as a flexible string with finite bending rigidity and an incompressible cross-section area. The interchain steric repulsion is accounted for self-consistently determined single-chain confining parabolic potential. The model provides a simple analytical expression for the membrane bending modulus, which falls within a range of experimental values. An observed dependence of the modulus on hydrocarbon chain length is also reproduced. Correspondence between our microscopic model and the membrane theory of elasticity is established.

Coarse-grained molecular dynamics simulation of binary charged lipid membranes: Phase separation and morphological dynamics

Biomembranes, which are mainly composed of neutral and charged lipids, exhibit a large variety of functional structures and dynamics. Here, we report a coarse-grained molecular dynamics (MD) simulation of the phase separation and morphological dynamics in charged lipid bilayer vesicles. The screened long-range electrostatic repulsion among charged head groups delays or inhibits the lateral phase separation in charged vesicles compared with neutral vesicles, suggesting the transition of the phase-separation mechanism from spinodal decomposition to nucleation or homogeneous dispersion. Moreover, the electrostatic repulsion causes morphological changes, such as pore formation, and further transformations into disk, string, and bicelle structures, which are spatiotemporally coupled to the lateral segregation of charged lipids. Based on our coarse-grained MD simulation, we propose a plausible mechanism of pore formation at the molecular level. The pore formation in a charged-lipid-rich domain is initiated by the prior disturbance of the local molecular orientation in the domain.