Mesoscopic undulations and thickness fluctuations in lipid bilayers from molecular dynamics simulations

Molecular dynamics simulations of fully hydrated Dipalmitoylphosphatidylcholine bilayers, extending temporal and spatial scales by almost one order of magnitude, are presented. The present work reaches system sizes of 1024 lipids and times 10-60 ns. The simulations uncover significant dynamics and fluctuations on scales of several nanoseconds, and enable direct observation and spectral decomposition of both undulatory and thickness fluctuation modes. Although the former modes are strongly damped, the latter exhibit signs of oscillatory behavior. From this, it has been possible to calculate mesoscopic continuum properties in good agreement with experimental values. A bending modulus of 4 x 10(-20) J, bilayer area compressibility of 250-300 mN/m, and mode relaxation times in the nanosecond range are obtained. The theory of undulatory motions is revised and further extended to cover thickness fluctuations. Finally, it is proposed that thickness fluctuations is the explanation to the observed system-size dependence of equilibrium-projected area per lipid.     

A New Computational Method for Membrane Compressibility: Bilayer Mechanical Thickness Revisited

Because lipid bilayers can bend and stretch in ways similar to thin elastic sheets, physical models of bilayer deformation have utilized mechanical constants such as the moduli for bending rigidity $\left({\kappa }_C\right)$ and area compressibility (K_A). However, the use of these models to quantify the energetics of membrane deformation associated with protein-membrane interactions, and the membrane response to stress is often hampered by the shortage of experimental data suitable for the estimation of the mechanical constants of various lipid mixtures. Although computational tools such as molecular dynamics simulations can provide alternative means to estimate K_A values, current approaches suffer significant technical limitations. Here, we present a novel, to our knowledge, computational framework that allows for a direct estimation of K_A values for individual bilayer leaflets. The theory is based on the concept of elasticity and derives K_A from real-space analysis of local thickness fluctuations sampled in molecular dynamics simulations. We explore and validate the model on a large set of single and multicomponent bilayers of different lipid compositions and sizes, simulated at different temperatures. The calculated bilayer compressibility moduli agree with values estimated previously from experiments and those obtained from a standard computational method based on a series of constrained tension simulations. We further validate our framework in a comparison with an existing polymer brush model and confirm the polymer brush model’s predicted linear relationship with proportionality coefficient of 24, using elastic parameters calculated from the simulation trajectories. The robustness of the results that emerge from the method allows us to revisit the origins of the bilayer mechanical (compressible) thickness and in particular its dependence on acyl-chain unsaturation and the presence of cholesterol.     

Local compressibilities of proteins: comparison of optical experiments and simulations for horse heart cytochrome-c

Spectroscopy with probe molecules yields local information on the environment of the probe. In this article we compare local compressibilities of cytochrome-c as obtained from molecular dynamics simulations with experimental results as obtained from spectroscopic measurements. The simulations show that the protein-core around the heme is much less compressible in a glycerol/water solvent than in pure water. The pocket is also much less compressible than the protein as a whole, although the compressibility of the water inside the rather incompressible protein-core is almost liquidlike. We show that the local compressibility values capture the collective correlations of local volume fluctuations with volume fluctuations in the surrounding protein-solvent system. The decoupling of the volume fluctuations of the core from the solvent shell explains the reduction of the heme-core-compressibility in glycerol/water solvent. This decoupling could be traced back to the suppression of the exchange between pocket-water and hydration-shell-water upon addition of glycerol as co-solvent.     

Bridging microscopic and mesoscopic simulations of lipid bilayers

A lipid bilayer is modeled using a mesoscopic model designed to bridge atomistic bilayer simulations with macro-scale continuum-level simulation. Key material properties obtained from detailed atomistic-level simulations are used to parameterize the meso-scale model. The fundamental length and time scale of the meso-scale simulation are at least an order of magnitude beyond that used at the atomistic level. Dissipative particle dynamics cast in a new membrane formulation provides the simulation methodology. A meso-scale representation of a dimyristoylphosphatidylcholine membrane is examined in the high and low surface tension regimes. At high surface tensions, the calculated modulus is found to be slightly less than the atomistically determined value. This result agrees with the theoretical prediction that high-strain thermal undulations still persist, which have the effect of reducing the value of the atomistically determined modulus. Zero surface tension simulations indicate the presence of strong thermal undulatory modes, whereas the undulation spectrum and the calculated bending modulus are in excellent agreement with theoretical predictions and experiment.     

Temperature and configurational effects on the Young’s modulus of poly (methyl methacrylate): a molecular dynamics study comparing the DREIDING,AMBER and OPLS force fields

The effects of the configuration and temperature on the Young’s modulus of poly (methyl methacrylate) (PMMA) have been studied using molecular dynamics simulations. For the DREIDING force field under ambient temperatures, increasing the number of monomers significantly increases the modulus of isotactic and syndiotactic PMMA while the isotactic form has a greater modulus. The effects of temperature on the modulus of isotactic PMMA have been simulated using the DREIDING, AMBER, and OPLS force fields. All these force fields predict the effects of temperature on the modulus from 200 to 350 K that are in close agreement with experimental values, while at higher temperatures the moduli are greater than those measured. The glass transition temperature determined by the force fields, based on the variation of the modulus with temperature, is greater than the experimental values, but when obtained from a plot of the volume as a function of the temperature, there is closer agreement. The Young’s moduli calculated in this study are in closer agreement to the experimental data than those reported by previous simulations.     

Structure and interactions of fully hydrated dioleoylphosphatidylcholine bilayers.

This study focuses on dioleoylphosphatidylcholine (DOPC) bilayers near full hydration. Volumetric data and high-resolution synchrotron x-ray data are used in a method that compares DOPC with well determined gel phase dipalmitoylphosphatidylcholine (DPPC). The key structural quantity obtained is fully hydrated area/lipid A0 = 72.2 +/- 1.1 A2 at 30 degrees C, from which other quantities such as thickness of the bilayer are obtained. Data for samples over osmotic pressures from 0 to 56 atmospheres give an estimate for the area compressibility of KA = 188 dyn/cm. Obtaining the continuous scattering transform and electron density profiles requires correction for liquid crystal fluctuations. Quantitation of these fluctuations opens an experimental window on the fluctuation pressure, the primary repulsive interaction near full hydration. The fluctuation pressure decays exponentially with water spacing, in agreement with analytical results for soft confinement. However, the ratio of decay length lambda(fl) = 5.8 A to hydration pressure decay length lambda = 2.2 A is significantly larger than the value of 2 predicted by analytical theory and close to the ratio obtained in recent simulations. We also obtain the traditional osmotic pressure versus water spacing data. Our analysis of these data shows that estimates of the Hamaker parameter H and the bending modulus Kc are strongly coupled.     

Elastic deformation and failure of lipid bilayer membranes containing cholesterol.

Giant bilayer vesicles were reconstituted from several lipids and lipid/cholesterol (CHOL) mixtures: stearolyloleoylphosphatidylcholine (SOPC), bovine sphingomyelin (BSM), diarachidonylphosphatidylcholine (DAPC), SOPC/CHOL, BSM/CHOL, DAPC/CHOL, and extracted red blood cell (RBC) lipids with native cholesterol. Single-walled vesicles were manipulated by micropipette suction and several membrane material properties were determined. The properties measured were the elastic area compressibility modulus K, the critical areal strain alpha c, and the tensile strength tau lys, from which the failure energy or membrane toughness Tf was calculated. The elastic area expansion moduli for these lipid and lipid/cholesterol bilayers ranged from 57 dyn/cm for DAPC to 1,734 dyn/cm for BSM/CHOL. The SOPC/CHOL series and RBC lipids had intermediate values. The results indicated that the presence of cholesterol is the single most influential factor in increasing bilayer cohesion, but only for lipids where both chains are saturated, or mono- or diunsaturated. Multiple unsaturation in both lipid chains inhibits the condensing effect of cholesterol in bilayers. The SOPC/CHOL system was studied in more detail. The area expansion modulus showed a nonlinear increase with increasing cholesterol concentration up to a constant plateau, indicating a saturation limit for cholesterol in the bilayer phase of approximately 55 mol% CHOL. The membrane compressibility was modeled by a property-averaging composite theory involving two bilayer components, namely, uncomplexed lipid and a lipid/cholesterol complex of stoichiometry 1/1.22. The area expansion modulus of this molecular composite membrane was evaluated by a combination of the expansion moduli of each component scaled by their area fractions in the bilayer. Bilayer toughness, which is the energy stored in the bilayer at failure, showed a maximum value at approximately 40 mol% CHOL. This breakdown energy was found to be only a fraction of the available thermal energy, implying that many molecules (approximately 50-100) may be involved in forming the defect structure that leads to failure. The area expansion modulus of extracted RBC lipids with native cholesterol was compared with recent measurements of intact RBC membrane compressibility. The natural membrane was also modeled as a simple composite made up to a compressible lipid/cholesterol matrix containing relatively incompressible transmembrane proteins. It appears that the interaction of incompressible proteins with surrounding lipid confers enhanced compressibility on the composite structure.     

Determination of electron density profiles and area from simulations of undulating membranes

The traditional method for extracting electron density and other transmembrane profiles from molecular dynamics simulations of lipid bilayers fails for large bilayer systems, because it assumes a flat reference surface that does not take into account long wavelength undulations. We have developed what we believe to be a novel set of methods to characterize these undulations and extract the underlying profiles in the large systems. Our approach first obtains an undulation reference surface for each frame in the simulation and subsequently isolates the long-wavelength undulations by filtering out the intrinsic short wavelength modes. We then describe two methods to obtain the appropriate profiles from the undulating reference surface. Most combinations of methods give similar results for the electron density profiles of our simulations of 1024 DMPC lipids. From simulations of smaller systems, we also characterize the finite size effect related to the boundary conditions of the simulation box. In addition, we have developed a set of methods that use the undulation reference surface to determine the true area per lipid which, due to undulations, is larger than the projected area commonly reported from simulations.     

The influence of short-chain alcohols on interfacial tension, mechanical properties, area/molecule, and permeability of fluid lipid bilayers

We used micropipette aspiration to directly measure the area compressibility modulus, bending modulus, lysis tension, lysis strain, and area expansion of fluid phase 1-stearoyl, 2-oleoyl phosphatidylcholine (SOPC) lipid bilayers exposed to aqueous solutions of short-chain alcohols at alcohol concentrations ranging from 0.1 to 9.8 M. The order of effectiveness in decreasing mechanical properties and increasing area per molecule was butanol>propanol>ethanol>methanol, although the lysis strain was invariant to alcohol chain-length. Quantitatively, the trend in area compressibility modulus follows Traube's rule of interfacial tension reduction, i.e., for each additional alcohol CH(2) group, the concentration required to reach the same area compressibility modulus was reduced roughly by a factor of 3. We convert our area compressibility data into interfacial tension values to: confirm that Traube's rule is followed for bilayers; show that alcohols decrease the interfacial tension of bilayer-water interfaces less effectively than oil-water interfaces; determine the partition coefficients and standard Gibbs adsorption energy per CH(2) group for adsorption of alcohol into the lipid headgroup region; and predict the increase in area per headgroup as well as the critical radius and line tension of a membrane pore for each concentration and chain-length of alcohol. The area expansion predictions were confirmed by direct measurements of the area expansion of vesicles exposed to flowing alcohol solutions. These measurements were fitted to a membrane kinetic model to find membrane permeability coefficients of short-chain alcohols. Taken together, the evidence presented here supports a view that alcohol partitioning into the bilayer headgroup region, with enhanced partitioning as the chain-length of the alcohol increases, results in chain-length-dependent interfacial tension reduction with concomitant chain-length-dependent reduction in mechanical moduli and membrane thickness.     

Calculating the bulk modulus for a lipid bilayer with nonequilibrium molecular dynamics simulation

Nonequilibrium molecular dynamics (NEMD) computer simulations are used to calculated the bulk modulus for a dimyristoylphosphatidylcholine bilayer. A methodology is developed whereby NEMD can be effectively used to calculate material properties for complex systems that undergo long time-scale conformational changes. It is found that the bulk modulus upon expansion from a zero stress state agrees well with experimental estimates. However, it is also found that the modulus upon contraction from a zero stress state is larger. From a molecular perspective, it is possible to explain this phenomena by examining the molecular origins of the pressure response. The finding that the two moduli are not equal upon compression and expansion is in apparent contradiction to osmotic stress experiments where the area modulus was found to be the same upon expansion and contraction. This issue is addressed.     

Comparing experimental and simulated pressure-area isotherms for DPPC

Although pressure-area isotherms are commonly measured for lipid monolayers, it is not always appreciated how much they can vary depending on experimental factors. Here, we compare experimental and simulated pressure-area isotherms for dipalmitoylphosphatidylcholine (DPPC) at temperatures ranging between 293.15 K and 323.15 K, and explore possible factors influencing the shape and position of the isotherms. Molecular dynamics simulations of DPPC monolayers using both coarse-grained (CG) and atomistic models yield results that are in rough agreement with some of the experimental isotherms, but with a steeper slope in the liquid-condensed region than seen experimentally and shifted to larger areas. The CG lipid model gives predictions that are very close to those of atomistic simulations, while greatly improving computational efficiency. There is much more variation among experimental isotherms than between isotherms obtained from CG simulations and from the most refined simulation available. Both atomistic and CG simulations yield liquid-condensed and liquid-expanded phase area compressibility moduli that are significantly larger than those typically measured experimentally, but compare well with some experimental values obtained under rapid compression.     

The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus

The elastic moduli of human subchondral, trabecular, and cortical bone tissue from a proximal tibia were experimentally determined using three-point bending tests on a microstructural level. The mean modulus of subchondral specimens was 1.15 GPa, and those of trabecular and cortical specimens was 4.59 GPa and 5.44 GPa respectively. Significant differences were found in the modulus values between bone tissues, which may have mainly resulted from the differences in the microstructures of each bone tissue rather than in the mineral density. Furthermore, the size-dependency of the modulus was examined using eight different sizes of cortical specimens (heights h = 100-1000 microns). While the modulus values for relatively large specimens (h greater than 500 microns) remained fairly constant (approximately 15 GPa), the values decreased as the specimens became smaller. A significant correlation was found between the modulus and specimen size. The surface area to volume ratio proved to be a key variable to explain the size-dependency.     

Structure of lipid bilayers

The quantitative experimental uncertainty in the structure of fully hydrated, biologically relevant, fluid (L(alpha)) phase lipid bilayers has been too large to provide a firm base for applications or for comparison with simulations. Many structural methods are reviewed including modern liquid crystallography of lipid bilayers that deals with the fully developed undulation fluctuations that occur in the L(alpha) phase. These fluctuations degrade the higher order diffraction data in a way that, if unrecognized, leads to erroneous conclusions regarding bilayer structure. Diffraction measurements at high instrumental resolution provide a measure of these fluctuations. In addition to providing better structural determination, this opens a new window on interactions between bilayers, so the experimental determination of interbilayer interaction parameters is reviewed briefly. We introduce a new structural correction based on fluctuations that has not been included in any previous studies. Updated measurements, such as for the area compressibility modulus, are used to provide adjustments to many of the literature values of structural quantities. Since the gel (L(beta)') phase is valuable as a stepping stone for obtaining fluid phase results, a brief review is given of the lower temperature phases. The uncertainty in structural results for lipid bilayers is being reduced and best current values are provided for bilayers of five lipids.     

Effect of chain length and unsaturation on elasticity of lipid bilayers

Micropipette pressurization of giant bilayer vesicles was used to measure both elastic bending k(c) and area stretch K(A) moduli of fluid-phase phosphatidylcholine (PC) membranes. Twelve diacyl PCs were chosen: eight with two 18 carbon chains and degrees of unsaturation from one double bond (C18:1/0, C18:0/1) to six double bonds per lipid (diC18:3), two with short saturated carbon chains (diC13:0, diC14:0), and two with long unsaturated carbon chains (diC20:4, diC22:1). Bending moduli were derived from measurements of apparent expansion in vesicle surface area under very low tensions (0.001-0.5 mN/m), which is dominated by smoothing of thermal bending undulations. Area stretch moduli were obtained from measurements of vesicle surface expansion under high tensions (>0.5 mN/m), which involve an increase in area per molecule and a small-but important-contribution from smoothing of residual thermal undulations. The direct stretch moduli varied little (< +/-10%) with either chain unsaturation or length about a mean of 243 mN/m. On the other hand, the bending moduli of saturated/monounsaturated chain PCs increased progressively with chain length from 0.56 x 10(-19) J for diC13:0 to 1.2 x 10(-19) J for diC22:1. However, quite unexpectedly for longer chains, the bending moduli dropped precipitously to approximately 0.4 x 10(-19) J when two or more cis double bonds were present in a chain (C18:0/2, diC18:2, diC18:3, diC20:4). Given nearly constant area stretch moduli, the variations in bending rigidity with chain length and polyunsaturation implied significant variations in thickness. To test this hypothesis, peak-to-peak headgroup thicknesses h(pp) of bilayers were obtained from x-ray diffraction of multibilayer arrays at controlled relative humidities. For saturated/monounsaturated chain bilayers, the distances h(pp) increased smoothly from diC13:0 to diC22:1 as expected. Moreover, the distances and elastic properties correlated well with a polymer brush model of the bilayer that specifies that the elastic ratio (k(c)/K(A))(1/2) = (h(pp) - h(o))/24, where h(o) approximately 1 nm accounts for separation of the headgroup peaks from the deformable hydrocarbon region. However, the elastic ratios and thicknesses for diC18:2, diC18:3, and diC20:4 fell into a distinct group below the correlation, which showed that poly-cis unsaturated chain bilayers are thinner and more flexible than saturated/monounsaturated chain bilayers.     

Membrane lateral compressibility determined by NMR and x-ray diffraction: effect of acyl chain polyunsaturation.

The elastic area compressibility modulus, Ka, of lamellar liquid crystalline bilayers was determined by a new experimental approach using 2H-NMR order parameters of lipid hydrocarbon chains together with lamellar repeat spacings measured by x-ray diffraction. The combination of NMR and x-ray techniques yields accurate determination of lateral area per lipid molecule. Samples of saturated, monounsaturated, and polyunsaturated phospholipids were equilibrated with polyethylene glycol (PEG) 20,000 solutions in water at concentrations from 0 to 55 wt % PEG at 30 degrees C. This procedure is equivalent to applying 0 to 8 dyn/cm lateral pressure to the bilayers. The resulting reductions in area per lipid were measured with a resolution of +/-0.2 A2 and the fractional area decrease was proportional to applied lateral pressure. For 1,2-dimyristoyl(d54)-sn-glycero-3-phosphocholine, 1-stearoyl(d35)-2-oleoyl-sn-glycero-3-phosphocholine (SOPC-d35), and 1-stearoyl(d35)-2-docosahexaenoyl-sn-glycero-3-phosphocholine (SDPC-d35) cross-sectional areas per molecule in excess water of 59.5, 61.4, and 69.2 A2 and bilayer elastic area compressibility moduli of 141, 221, and 121 dyn/cm were determined, respectively. Combining NMR and x-ray results enables the determination of compressibility differences between saturated and unsaturated hydrocarbon chains. In mixed-chain SOPC-d35 both chains have similar compressibility moduli; however, in mixed-chain polyunsaturated SDPC-d35, the saturated stearic acid chain appears to be far less compressible than the polyunsaturated docosahexaenoic acid chain.     

Elastic constants of polymer-grafted lipid membranes

The surface expansion that is induced by the lateral pressure in the brush region of lipid membranes containing grafted polymers is deduced from the scaling and mean-field theories for the polymer brush, together with the equation of state for a lipid monolayer at the equivalence pressure with fluid lipid bilayers. Depending on the length and mole fraction of the polymer lipid, the membrane expansion can be appreciable. Direct experimental evidence for this lateral expansion comes from recent spin-label measurements with lipid membranes containing poly(ethylene glycol)-grafted lipids. The expansion in lipid area modifies the elastic constants of the polymer-grafted membranes in a way that opposes the direct elastic response of the polymer itself. Calculations as a function of polymer lipid content indicate that the net change in isothermal area expansion modulus of the membrane is negative but small, in contrast to previous predictions. A similar situation applies to the curvature elastic moduli of membranes containing short polymer lipids. For longer polymer lipids, however, the direct contribution of the polymer brush to the bending elastic constants dominates, and the increase in bending moduli with increasing polymer lipid content rapidly exceeds the basal values of the bare lipid membrane. The spontaneous (or intrinsic) curvature of the component monolayer of polymer lipid-containing membranes is calculated for the first time. The polymer brush contribution to spontaneous curvature scales quadratically with the polymer length, and at least quadratically with the mole fraction of polymer lipid.     

Ab initio predictions of structural and elastic properties of struvite: contribution to urinary stone research

In the present work, we carried out density functional calculations of struvite--the main component of the so-called infectious urinary stones--to study its structural and elastic properties. Using a local density approximation and a generalised gradient approximation, we calculated the equilibrium structural parameters and elastic constants C(ijkl). At present, there is no experimental data for these elastic constants C (ijkl) for comparison. Besides the elastic constants, we also present the calculated macroscopic mechanical parameters, namely the bulk modulus (K), the shear modulus (G) and Young's modulus (E). The values of these moduli are found to be in good agreement with available experimental data. Our results imply that the mechanical stability of struvite is limited by the shear modulus, G. The study also explores the energy-band structure to understand the obtained values of the elastic constants.     

Structure of Sphingomyelin Bilayers: A Simulation Study

We have carried out a molecular dynamics simulation of a hydrated 18:0 sphingomyelin lipid bilayer. The bilayer contained 1600 sphingomyelin (SM) molecules, and 50,592 water molecules. After construction and initial equilibration, the simulation was run for 3.8 ns at a constant temperature of 50°C and a constant pressure of 1 atm. We present properties of the bilayer calculated from the simulation, and compare with experimental data and with properties of dipalmitoyl phosphatidylcholine (DPPC) bilayers. The SM bilayers are significantly more ordered and compact than DPPC bilayers at the same temperature. SM bilayers also exhibit significant intramolecular hydrogen bonding between phosphate ester oxygen and hydroxyl hydrogen atoms. This results in a decreased hydration in the polar region of the SM bilayer compared with DPPC. Since our simulation system is very large we have calculated the power spectrum of bilayer undulation and peristaltic modes, and we compare these data with similar calculations for DPPC bilayers. We find that the SM bilayer has significantly larger bending modulus and area compressibility compared to DPPC.     

Molecular dynamics simulation shows large volume fluctuations of proteins

In this paper we present a new approach to study the volume fluctuations of proteins. From a 1 ns molecular dynamics simulation, the volume fluctuation of human lysozyme has been calculated. We used two different ways for the calculation. In the first one, the volume fluctuation is extracted directly from the trajectory. For the second one, a newly developed formalism based on principal component analysis is used. The r.m.s. volume fluctuations obtained from the two analyses agree well with each other. The isothermal intrinsic compressibility was found to be larger than the one reported by experiment. The difference is discussed and suggested to exist in the assumed uncertainty of the compressibility of hydrated water to deduce the isothermal intrinsic compressibility from the experimental value. Spectral analysis shows that low-frequency dynamics dominate the total volume fluctuation. The same aspect is found in the study using principal component analysis. This low-frequency region is related to large and slow motions of proteins. Therefore a long time dynamics simulation is necessary to describe the volume fluctuations of proteins.