Tilt Modulus Research Articles

Mechanical response of double-stranded DNA: Bend, twist, and overwind.

We employed all-atom molecular dynamics simulations to explore the mechanical response of bending, twisting, and overwinding for double-stranded DNA (dsDNA). We analyzed the bending and twisting deformations, as well as their stiffnesses, using the tilt, roll, and twist modes under stretching force. Findings indicate that the roll and twist angles vary linearly with the stretching force but show opposite trends. The tilt, roll, and twist elastic moduli are considered constants, while the coupling between roll and twist modes slightly decreases under stretching force. The effect of the stretching force on the roll and twist modes, including both their deformations and elasticities, exhibits sequence-dependence, with symmetry around the base pair step. Furthermore, we examined the overwinding path and mechanism of dsDNA from the perspective of the stiffness matrix, based on the tilt, roll, and twist modes. The correlations among tilt, roll, and twist angles imply an alternative overwinding pathway via twist-roll coupling when dsDNA is stretched, wherein entropic contribution prevails.

Membrane free-energy landscapes derived from atomistic dynamics explain nonuniversal cholesterol-induced stiffening.

All lipid membranes have inherent morphological preferences and resist deformation. Yet adaptations in membrane shape can and do occur at multiple length scales. While this plasticity is crucial for cellular physiology, the factors controlling the morphological energetics of lipid bilayers and the dominant mechanisms of membrane remodeling remain to be fully understood. An ongoing debate regarding the universality of the stiffening effect of cholesterol underscores the challenges facing this field, both experimentally and theoretically, even for simple lipid mixtures. On the computational side, we have argued that enhanced-sampling all-atom molecular dynamics simulations are uniquely suited for the quantification of membrane conformational energetics, as they minimize a priori assumptions and permit analysis of bilayers in deformed states. To showcase this approach, we examine reported inconsistencies between alternative experimental measurements of bending moduli for cholesterol-enriched membranes. Specifically, we analyze lipid bilayers with different chain saturation and compute free-energy landscapes for curvature deformations distributed over areas from ∼5 to ∼60 nm. These enhanced simulations, totaling over 100 μs of sampling time, enable us to directly quantify both bending and tilt moduli and to dissect the contributing factors and molecular mechanisms of curvature generation at each length scale. Our results show that the effects of cholesterol on bending rigidity are lipid-specific and suggest that this specificity arises from differences in the torsional dynamics of the acyl chains. In summary, we demonstrate that quantitative relationships can now be established between lipid structure and bending energetics, paving the way for addressing open fundamental questions in cell membrane mechanics.

Understanding the phase behavior of a protobiomembrane.

The rich thermotropic behavior of lipid bilayers is addressed using phenomenological theory informed by many experiments. The most recent experiment not yet addressed by theory has shown that the tilt modulus in DMPC lipid bilayers decreases dramatically as the temperature is lowered toward the main transition temperature T_{M}. It is shown that this behavior can be understood by introducing a simple free energy functional for tilt that couples to the area per molecule. This is combined with a chain melting free energy functional in which the area is the primary order parameter that is the driver of the main transition. Satisfactory agreement with experiment is achieved with values of the model parameters determined by experiments, but the transition is directly into the gel phase. The theory is then extended to include the enigmatic ripple phase by making contact with the most recent experimentally determined ripple structure.

Modelling natural cell membranes and how lipid asymmetry influences structure.

Cell membranes are integral to the structure and function of the cell for all organisms and have a complex mixture of lipids that vary between cellular organelles and between leaflets of a membrane. With improved lipidomic and selective labeling experimental studies, computational modeling of realistic cell membranes has been feasible . In this presentation, a summary of various models of cell membranes will be presented ranging from simple bacteria to those found in mammals. The outer membrane of gram-negative bacteria has extreme leaflet asymmetry with lipopolysaccharides in the outer leaflet. Even the inner membrane has charge asymmetry that yields varied membrane properties in thickness, lipid packing and elastic tilt modulus. Considering plants, the plasma membrane of Arabidopsis thaliana has an enriched outer leaflet with sterols and glycolipids. The outer leaflet tends to be more rigid with the formation of clusters induced by sterol-glycolipid interaction. Sterols play an important role to the structure and dynamics of mammalian membranes. Diabetic patients tend to have a lower concentration of cholesterol and unsaturated fatty acids in the peripheral myelin sheath that yield a more fluid and thinner membrane. For the ocular lens, the cholesterol concentration in these layered membranes of the lens is at least 50% and the levels depend on age and presence of cataracts. Lens of older patients and those with cataracts have higher acyl chain order, lower lipid diffusion and increased lateral structures of lipids. Finally, models for the phases present the lipid matrix of the stratum corneum were developed and based on the asymmetric partitioning of cholesterol in the multilayered structure of the long periodicity phase. Our simulations resulted in some ceramide lipids with extended acyl chains between layers and permeation estimates that matched that in natural skin.

Effective Parameters Controlling Sterol Transfer: A Time-Resolved Small-Angle Neutron Scattering Study.

Though cholesterol is the most prevalent and essential sterol in mammalian cellular membranes, its precursors, post-synthesis cholesterol products, as well as its oxidized derivatives play many other important physiological roles. Using a non-invasive in situ technique, time-resolved small angle neutron scattering, we report on the rate of membrane desorption and corresponding activation energy for this process for a series of sterol precursors and post-synthesis cholesterol products that vary from cholesterol by the number and position of double bonds in B ring of cholesterol's steroid core. In addition, we report on sterols that have oxidation modifications in ring A and ring B of the steroid core. We find that sterols that differ in position or the number of double bonds in ring B have similar time and energy characteristics, while oxysterols have faster transfer rates and lower activation energies than cholesterol in a manner generally consistent with known sterol characteristics, like Log P, the n-octanol/water partitioning coefficient. We find, however, that membrane/water partitioning which is dependent on lipid-sterol interactions is a better predictor, shown by the correlation of the sterols' tilt modulus with both the desorption rates and activation energy.

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.

Bending Modulus of Lipid Membranes from Density Correlation Functions.

The bending modulus κ quantifies the elasticity of biological membranes in terms of the free energy cost of increasing the membrane corrugation. Molecular dynamics (MD) simulations provide a powerful approach to quantify κ by analyzing the thermal fluctuations of the lipid bilayer. However, existing methods require the identification and filtering of non-mesoscopic fluctuation modes. State of the art methods rely on identifying a smooth surface to describe the membrane shape. These methods introduce uncertainties in calculating κ since they rely on different criteria to select the relevant fluctuation modes. Here, we present a method to compute κ using molecular simulations. Our approach circumvents the need to define a mesoscopic surface or an orientation field for the lipid tails explicitly. The bending and tilt moduli can be extracted from the analysis of the density correlation function (DCF). The method introduced here builds on the Bedeaux and Weeks (BW) theory for the DCF of fluctuating interfaces and on the coupled undulatory (CU) mode introduced by us in previous work. We test the BW-DCF method by computing the elastic properties of lipid membranes with different system sizes (from 500 to 6000 lipid molecules) and using coarse-grained (for POPC and DPPC lipids) and fully atomistic models (for DPPC). Further, we quantify the impact of cholesterol on the bending modulus of DPPC bilayers. We compare our results with bending moduli obtained with X-ray diffraction data and different computer simulation methods.

Leaflet Asymmetry Modeling in the Lipid Composition of Escherichia coli Cytoplasmic Membranes.

Lipid composition asymmetry between leaflets is important to cell function and plays a key role in the "positive inside" rule in transmembrane proteins. In this work, Escherichia coli inner plasma membrane models reflecting this asymmetry have been investigated at the early-log and stationary stages during the bacterial lifecycle using all-atom molecular dynamics simulations. The CHARMM36 lipid force field is used, and selected membrane properties are tested for variations between two leaflets and whole membranes. Our models include bacterial lipids with a cyclopropane moiety on the sn-2 acyl chain in the stationary membrane model. The PE/PG ratio for two leaflets reflects the "positive inside" rule of membrane proteins, set to 6.8 and 2.8 for the inner and outer leaflets of the two models, respectively. We are the first to model leaflet asymmetry in the lipid composition of E. coli cytoplasmic membranes and observe the effect on membrane properties in leaflets and whole membranes. Specifically, our results show that for the stationary phase bilayer, the surface area per lipid (SA/lipid) is larger, the thickness (2DC and DB) is smaller, the tilt angle is larger, the tilt modulus is smaller, and the deuterium order parameters (SCD) of sn-1 and sn-2 tails are lower, compared to the early-log stage. Moreover, the stationary stage bilayer has a positive spontaneous curvature, while the early-log stage has a near flat spontaneous curvature. For leaflet asymmetry, the inner leaflet has a larger SA/lipid, a smaller thickness, a smaller elastic tilt modulus (a larger tilt angle), and lower SCD, compared to the outer leaflet in both stages. Moreover, an asymmetric membrane involves a lipid tilt and a lateral extension, varying from a reference state of a pre-equilibrium membrane. This work encourages a more profound exploration of leaflet asymmetry in various other membrane models and how this might affect the structure and function of membrane-associated peptides and proteins.

Calcium Ions Promote Membrane Fusion by Forming Negative-Curvature Inducing Clusters on Specific Anionic Lipids.

Vesicles enriched in certain negatively charged lipids, such as phosphatidylserine and PIP2, are known to undergo fusion in the presence of calcium ions without assistance from protein assemblies. Other lipids do not exhibit this propensity, even if they are negatively charged. Using our recently developed methodology, we extract elastic properties of a representative set of lipids. This allows us to trace the origin of lipid-calcium selectivity in membrane fusion to the formation of lipid clusters with long-range correlations that induce negative curvature on the membrane surface. Furthermore, the clusters generate lateral tension in the headgroup region at the membrane surface, concomitantly also stabilizing negative Gaussian curvature. Finally, calcium binding also reduces the orientational polarization of water around the membrane head groups, potentially reducing the hydration force acting between membranes. Binding calcium only weakly increases membrane bending rigidity and tilt moduli, in agreement with experiments. We show how the combined effects of calcium binding to membranes lower the barriers along the fusion pathway that lead to the formation of the fusion stalk as well as the fusion pore.

Confinement in Nanodiscs Anisotropically Modifies Lipid Bilayer Elastic Properties.

Lipid nanodiscs are small synthetic lipid bilayer structures that are stabilized in solution by special circumscribing (or scaffolding) proteins or polymers. Because they create native-like environments for transmembrane proteins, lipid nanodiscs have become a powerful tool for structural determination of this class of systems when combined with cryo-electron microscopy or nuclear magnetic resonance. The elastic properties of lipid bilayers determine how the lipid environment responds to membrane protein perturbations, and how the lipid in turn modifies the conformational state of the embedded protein. However, despite the abundant use of nanodiscs in determining membrane protein structure, the elastic material properties of even pure lipid nanodiscs (i.e., without embedded proteins) have not yet been quantitatively investigated. A major hurdle is due to the inherently nonlocal treatment of the elastic properties of lipid systems implemented by most existing methods, both experimental and computational. In addition, these methods are best suited for very large “infinite” size lipidic assemblies, or ones that contain periodicity, in the case of simulations. We have previously described a computational analysis of molecular dynamics simulations designed to overcome these limitations, so it allows quantification of the bending rigidity (KC) and tilt modulus (κt) on a local scale even for finite, nonperiodic systems, such as lipid nanodiscs. Here we use this computational approach to extract values of KC and κt for a set of lipid nanodisc systems that vary in size and lipid composition. We find that the material properties of lipid nanodiscs are different from those of infinite bilayers of corresponding lipid composition, highlighting the effect of nanodisc confinement. Nanodiscs tend to show higher stiffness than their corresponding macroscopic bilayers, and moreover, their material properties vary spatially within them. For small-size MSP1 nanodiscs, the stiffness decreases radially, from a value that is larger in their center than the moduli of the corresponding bilayers by a factor of ∼2–3. The larger nanodiscs (MSP1E3D1 and MSP2N2) show milder spatial changes of moduli that are composition dependent and can be maximal in the center or at some distance from it. These trends in moduli correlate with spatially varying structural properties, including the area per lipid and the nanodisc thickness. Finally, as has previously been reported, nanodiscs tend to show deformations from perfectly flat circular geometries to varying degrees, depending on size and lipid composition. The modulations of lipid elastic properties that we find should be carefully considered when making structural and functional inferences concerning embedded proteins.

A local instantaneous surface method for extracting membrane elastic moduli from simulation: Comparison with other strategies

Advances over the past decade have made it possible to extract elastic constants of lipid assemblies from molecular dynamics simulations. We summarize existing strategies for obtaining membrane elastic moduli and clarify the differences in the underlying approaches. We analyze these strategies in depth, including several important advantages and limitations. By addressing these limitations, we obtain a newly formulated spatially local methodology for extracting bending and tilt moduli: The Real Space Instantaneous Surface Method (ReSIS). With its freely available implementation, this method is designed for highly dynamic systems with arbitrary interface geometries. We demonstrate how the method provides consistent results for membranes of arbitrary size. In addition, we describe alternative implementations of various Fourier-space methods, and use these to compare the results from the different available methods and from published computational data. We specifically focus on the tilt modulus, where very large differences between Fourier and real-space based methods are observed, including those derived using ReSIS. These discrepancies are likely due to the known difference between model moduli and thermodynamic moduli that are derived from the corresponding response functions. In addition, we reexamine the issue of angular degeneracy and its effect on conformational ensembles. Finally, a van ’t Hoff analysis of the tilt and bending moduli reveals that both modes act as entropic springs and that enthalpy favors nonzero tilt, perhaps heralding the spontaneous lipid chain tilt of the gel phase at lower temperatures.

Theoretical Modeling of Experimentally Determined Tilt Modulus of Lipid Bilayers

Values of the tilt moduli of single-component PC lipid bilayers recently determined by analysis of diffuse x-ray scattering (Chem. Phys. Lipids 205, 18-24 (2017)) show a strong correlation with how far the temperature was from the temperature TM of the main transition for each PC lipid. Consistent with this, the tilt modulus of DMPC decreased dramatically as temperature was lowered toward TM (Phys. Rev. E 96, 030401 (2017)). It is hypothesized that tilt is a symmetry breaking order parameter for the main transition of PC lipids that have spontaneous tilt below TM. Consistently, POPS does not have spontaneous tilt below TM and its tilt modulus is not temperature sensitive. Theoretical modeling will be presented that combines a tilt free energy functional with a chain melting free energy functional based on an older exact statistical mechanical theory that did not have tilt (Faraday Discussion 81, 151 (1986)). Coupling between the two terms obtains the experimental temperature dependence of the tilt modulus of DMPC.

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.

Molecular Insight into the Line Tension of Bilayer Membranes Containing Hybrid Polyunsaturated Lipids.

Line tension (γ) is a key parameter for the structure and dynamics of membrane domains. It was proposed that hybrid lipids, with mixed saturated and unsaturated acyl chains, participate in the relaxation of γ through different mechanisms. In this work, we used molecular dynamics simulations of the coarse-grained MARTINI model to measure γ in liquid-ordered-liquid-disordered (Lo-Ld) membranes, with increasingly larger relative proportion of the hybrid polyunsaturated lipid PAPC (4:0-5:4PC) to DAPC (di5:4PC) (i.e., XH). We also calculated an elastic contribution to γ by the Lo-Ld thickness mismatch, tilt moduli, and bending moduli, as predicted by theory. We found that an increase in XH decreased the overall γ value and the elastic contribution to line tension. The effect on the elastic line tension is driven by a reduced hydrophobic mismatch. Changes in the elastic constants of the phases due to an increase in XH produced a slightly larger elastic γ term. In addition to this elastic energy, other major contributions to γ are found in these model membranes. Increasing XH decreases both elastic and nonelastic contributions to γ. Finally, PAPC also behaves as a linactant, relaxing γ through an interfacial effect, as predicted by theoretical results. This study gives insight into the actual contribution of distinct energy terms to γ in bilayers containing polyunsaturated hybrid lipids.

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.

Erratum to: Implementation of a methodology for determining elastic properties of lipid assemblies from molecular dynamics simulations

The importance of the material properties of membranes for diverse cellular processes is well established. Notably, the elastic properties of the membrane, which depend on its composition, can directly influence membrane reshaping and fusion processes as well as the organisation and function of membrane proteins. Determining these properties is therefore key for a mechanistic understanding of how the cell functions. We have developed a method to determine the bending rigidity and tilt modulus, for lipidic assemblies of arbitrary lipid composition and shape, from molecular dynamics simulations. The method extracts the elastic moduli from the distributions of microscopic tilts and splays of the lipid components. We present here an open source implementation of the method as a set of Python modules using the computational framework OpenStructure. These modules offer diverse algorithms typically used in the calculatation the elastic moduli, including routines to align MD trajectories of complex lipidic systems, to determine the water/lipid interface, to calculate lipid tilts and splays, as well as to fit the corresponding distributions to extract the elastic properties. We detail the implementation of the method and give several examples of how to use the modules in specific cases. The method presented here is, to our knowledge, the only available computational approach allowing to quantify the elastic properties of lipidic assemblies of arbitrary shape and composition (including lipid mixtures). The implementation as python modules offers flexibility, which has already allowed the method to be applied to diverse lipid assembly types, ranging from bilayers in the liquid ordered and disordered phases to a study of the inverted-hexagonal phase, and with different force-fields (both all-atom and coarse grained representations). The modules are freely available through GitHub at https://github.com/njohner/ost_pymodules/ while OpenStructure can be obtained at http://www.openstructure.org .

Determining the Lipid Tilt Modulus by Simulating Membrane Buckles.

Lipid tilt affects the energy of membrane deformations on scales comparable to the membrane's thickness. The surface divergence of the tilt field creates a local spontaneous curvature, while tilt itself is quadratically penalized with a strength given by the tilt modulus. Traditionally, this modulus is determined by measuring the power spectrum of lipid orientation fluctuations. Here we present a novel approach which does not rely on fluctuations but instead exploits the fact that curvature gradients induce a tilt field. Its implementation extends a technique previously developed by us for localizing the position of the pivotal plane in buckling simulations, which quantifies the lipid imbalance across segments cut out from a complete buckle. Lipid tilt affects this count in a predictable way, and the signal can be quantified well enough to back out the tilt modulus-at no additional cost and with about 5% precision for not too coarse models. We apply our technique to three lipid models of very different resolution: the highly coarse grained Cooke model, and two versions of DMPC, using both the (less highly coarse grained) MARTINI and the (united atom) Berger force field. For Cooke, we find an effective bilayer tilt modulus of 29 ± 9 pN/nm, and for the less generic DMPC lipid, we find 115 ± 6 pN/nm for MARTINI and 39 ± 2 pN/nm for Berger, both in reasonable agreement with existing studies for these models. We also show that the position of the pivotal plane for Berger DMPC lies just below the glycerol backbone-unlike for MARTINI DMPC, where this plane lies closer to the middle of the lipid.

Implementation of a methodology for determining elastic properties of lipid assemblies from molecular dynamics simulations.

BackgroundThe importance of the material properties of membranes for diverse cellular processes is well established. Notably, the elastic properties of the membrane, which depend on its composition, can directly influence membrane reshaping and fusion processes as well as the organisation and function of membrane proteins. Determining these properties is therefore key for a mechanistic understanding of how the cell functions.ResultsWe have developed a method to determine the bending rigidity and tilt modulus, for lipidic assemblies of arbitrary lipid composition and shape, from molecular dynamics simulations. The method extracts the elastic moduli from the distributions of microscopic tilts and splays of the lipid components. We present here an open source implementation of the method as a set of Python modules using the computational framework OpenStructure. These modules offer diverse algorithms typically used in the calculatation the elastic moduli, including routines to align MD trajectories of complex lipidic systems, to determine the water/lipid interface, to calculate lipid tilts and splays, as well as to fit the corresponding distributions to extract the elastic properties. We detail the implementation of the method and give several examples of how to use the modules in specific cases.ConclusionsThe method presented here is, to our knowledge, the only available computational approach allowing to quantify the elastic properties of lipidic assemblies of arbitrary shape and composition (including lipid mixtures). The implementation as python modules offers flexibility, which has already allowed the method to be applied to diverse lipid assembly types, ranging from bilayers in the liquid ordered and disordered phases to a study of the inverted-hexagonal phase, and with different force-fields (both all-atom and coarse grained representations). The modules are freely available through GitHub at https://github.com/njohner/ost_pymodules/ while OpenStructure can be obtained at http://www.openstructure.org.

Sugar does not affect the bending and tilt moduli of simple lipid bilayers

The diffuse X-ray scattering method has been applied to samples composed of SOPC, DOPC, DMPC, and POPC with added sugar, either sucrose, glucose, fructose, maltose, or trehalose. Several sugar concentrations in the range 200–500mM were investigated for each of the lipid/sugar samples. We observed no systematic change in the bending modulus KC or in the tilt modulus Kθ with increasing sugar concentration. The average values of both these moduli were the same as those of the respective pure lipid controls within statistical uncertainty of 2%. These results are inconsistent with previous reports of sugar concentration dependent values of KC.

Revisiting Tilt in Classical Curvature Elastic Theories for Membranes

The classical continuum description of a fluid lipid membrane is the curvature elastic Hamiltonian due to Helfrich, in which surface geometry is the only degree of freedom. This model does not consider the deviation of a lipid's orientation away from the local membrane normal, which is another degree of freedom, called lipid tilt. Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)] have shown how this variable emerges naturally during the dimensional reduction of a thin but still three-dimensional fluid sheet described by a functional quadratic in the permissible strains. They show that the tilt divergence acts as a locally varying spontaneous curvature, the tilt's magnitude is penalized by a quadratic term proportional to a new tilt modulus, and the Gaussian curvature gets entangled with derivatives of the tilt. We revisit and extend the Hamm-Kozlov theory and show that disentangling Gaussian curvature and tilt leads to a curvature-dependent modification of tilt modulus, which matters for strongly curved membranes. Moreover, by consistently collecting all curvature-tilt couplings, we also find that the effective tilt modulus becomes anisotropic, implying that in general the price for tilt depends on the tilt direction. If we assume that this quadratic continuum theory remains valid at a membrane edge, the corrections are large and significantly modify other observables, such as the edge tension.