Direct Evaluation of Vapour-Liquid Equilibria of Mixtures by Molecular Dynamics Using Gibbs-Duhem Integration

Abstract

We present an extension of the Gibbs-Duhem integration method that permits direct evaluation of vapour-liquid equilibria of mixtures by molecular dynamics. The Gibbs-Duhem integration combines the best elements of the Gibbs ensemble Monte Carlo technique and thermodynamic integration. Given conditions of coexistence of pure substances, simultaneous but independent molecular dynamics simulations of each phase at constant number of particles, constant pressure, constant temperature and constant fugacity fraction of species 2 are carried out in succession along coexistence lines. In each simulation, the coexistence pressure is adjusted to satisfy the Clapeyron-type equation. The Clapeyron-type equation is a first-order nonlinear differential equation that prescribes how the pressure must change with the fugacity fraction of species 2 to maintain coexistence at constant temperature. The Clapeyron-type equation is solved by the predictor-corrector method. Running averages of mole fraction and compressibility factor for the two phases are used to evaluate the right-hand side of the Clapeyron-type equation. The Gibbs-Duhem integration method is applied to three prototypes of binary mixtures of the two-centre Lennard-Jones fluid having various elongations. The starting points on the coexistence curve were taken from published data.     

Lipid-cholesterol interactions. Monte Carlo simulations and theory.

Results of Monte Carlo calculations of order parameter profiles of lipid chains interacting with cholesterol are presented. Cholesterol concentrations in the simulations are sufficiently large that it is possible to analyze profiles for chains which are near neighbors of two or more cholesterol molecules, chains which are neighbors to a single cholesterol, and chains which are not near any cholesterol molecules. The profiles, show that cholesterol acts to significantly decrease the ability of neighboring chains to undergo trans-gauche isomeric rotations, although these chains are not all forced into all-trans conformations. The effect is significantly greater for chains which are neighbors to more than one cholesterol. The Monte Carlo results are next used as a guide to develop a theoretical model for lipid-cholesterol mixtures. The properties of this model and the phase diagram which it predicts are described. The phase diagram is then compared with experimentally determined phase diagrams. The model calculations and the computer simulations upon which they are based yield a molecular mechanism for several of the observed phases exhibited by lipid-cholesterol mixtures. The theoretical model predicts that at low temperatures the system should exhibit solid phase immiscibility.     

Hexagonal Substructure and Hydrogen Bonding in Liquid-Ordered Phases Containing Palmitoyl Sphingomyelin

All-atom simulation data are presented for ternary mixtures of palmitoyl sphingomyelin (PSM), cholesterol, and either palmitoyl oleoyl phosphatidyl choline or dioleoyl phosphatidyl choline (DOPC). For comparison, data for a mixture of dipalmitoyl phosphatidyl choline (DPPC), cholesterol, and DOPC are also presented. Compositions corresponding to the liquid-ordered phase, the liquid-disordered phase, and coexistence of the two phases are simulated for each mixture. Within the liquid-ordered phase, cholesterol is preferentially solvated by DOPC if it is available, but if DOPC is replaced by POPC, cholesterol is preferentially solvated by PSM. In the DPPC mixtures, cholesterol interacts preferentially with the saturated chains via its smooth face, whereas in the PSM mixtures, cholesterol interacts preferentially with PSM via its rough face. Interactions between cholesterol and PSM have a very particular character: hydrogen bonding between cholesterol and the amide of PSM rotates the tilt of the amide plane, which primes it for more robust hydrogen bonding with other PSM. Cholesterol-PSM hydrogen bonding also locally modifies the hexagonal packing of hydrocarbon chains in the liquid-ordered phase of PSM mixtures.     

Monte-Carlo Calculations for Methane and Argon over a Wide Range of Density and Temperature,Including the Two-Phase Vapor-Liquid Region

Abstract

The Gibbs-ensemble simulation technique provides a powerful method to calculate vapor-liquid phase behavior [1]. To evaluate the configurational energy of a system of molecules, commonly used experessions describe the interaction between two molecules. Contributions from higher-body forces are usually implicitly taken into account by adjusting two-body potential parameters to give agreement with experimental data. Explicit expressions for higher-body potentials are not commonly used in simulations [8]. The work by Smit et al. [9] gives the appropriate expressions to evaluate the pressure as well as the chemical potential from a density-dependent two-body potential in an NVT ensemble.

In the present work, contributions to the potential from two-body interactions are separated from those due to higher-body interactions; to take higher-body forces into account, a mean-field term, proportional to (density)^0.9, is added to the two-body potential. NPT-simulations over a wide range of temperature and density, as well as Gibbs-ensemble simulations, are used to evaluate phase behavior of argon and of methane. The results indicate that a simple mean-field correction to the “true” two-body Kihara potential provides good agreement between experiment and simulation.     

Atomistic simulations of liquid–liquid coexistence in confinement: comparison of thermodynamics and kinetics with bulk

We review a few simulation methods and results related to the structure and non-equilibrium dynamics in the coexistence region of immiscible symmetric binary fluids, in bulk as well as under confinement, with special emphasis on the latter. Monte Carlo methods to estimate interfacial tensions for flat and curved interfaces have been discussed. The latter, combined with a thermodynamic integration technique, provides contact angles for coexisting fluids attached to the wall. For such three-phase coexistence, results for the line tension are also presented. For the kinetics of phase separation, various mechanisms and corresponding theoretical expectations have been discussed. A comparative picture between the domain growth in bulk and confinement (including thin-film and semi-infinite geometry) has been presented from molecular dynamics simulations. Applications of finite-size scaling technique have been discussed in both equilibrium and non-equilibrium contexts.     

Phospholipid Chain Interactions with Cholesterol Drive Domain Formation in Lipid Membranes

Cholesterol is a key component of eukaryotic membranes, but its role in cellular biology in general and in lipid rafts in particular remains controversial. Model membranes are used extensively to determine the phase behavior of ternary mixtures of cholesterol, a saturated lipid, and an unsaturated lipid with liquid-ordered and liquid-disordered phase coexistence. Despite many different experiments that determine lipid-phase diagrams, we lack an understanding of the molecular-level driving forces for liquid phase coexistence in bilayers with cholesterol. Here, we use atomistic molecular dynamics computer simulations to address the driving forces for phase coexistence in ternary lipid mixtures. Domain formation is directly observed in a long-timescale simulation of a mixture of 1,2-distearoyl-sn-glycero-3-phosphocholine, unsaturated 1,2-dilinoleoyl-sn-glycero-3-phosphocholine, and cholesterol. Free-energy calculations for the exchange of the saturated and unsaturated lipids between the ordered and disordered phases give insight into the mixing behavior. We show that a large energetic contribution to domain formation is favorable enthalpic interactions of the saturated lipid in the ordered phase. This favorable energy for forming an ordered, cholesterol-rich phase is opposed by a large unfavorable entropy. Martini coarse-grained simulations capture the unfavorable free energy of mixing but do not reproduce the entropic contribution because of the reduced representation of the phospholipid tails. Phospholipid tails and their degree of unsaturation are key energetic contributors to lipid phase separation.     

Hydration pressure and phase transitions of phospholipids: I. Piezotropic approach

Dehydration reduces the main phase transition pressure of phospholipids. An analysis based on the Gibbs-Duhem equation shows how the shift of the transition pressure is correlated to the hydration pressure.By using Fourier transform infrared (FT-IR) spectroscopy we determined the hydration-dependent phase transition pressure. The application of our new approach gives hydration pressure values which agree with the values obtained with the osmotic stress method.     

Hydration pressure and phase transitions of phospholipids. I. Piezotropic approach

Dehydration reduces the main phase transition pressure of phospholipids. An analysis based on the Gibbs-Duhem equation shows how the shift of the transition pressure is correlated to the hydration pressure. By using Fourier transform infrared (FT-IR) spectroscopy we determined the hydration-dependent phase transition pressure. The application of our new approach gives hydration pressure values which agree with the values obtained with the osmotic stress method.     

Liquid-ordered phases induced by cholesterol: A compendium of binary phase diagrams

Mixtures of phospholipids with cholesterol are able to form liquid-ordered phases that are characterised by short-range orientational order and long-range translational disorder. These L_o-phases are distinct from the liquid-disordered, fluid L_α-phases and the solid-ordered, gel L_β-phases that are assumed by the phospholipids alone. The liquid-ordered phase can produce spatially separated in-plane fluid domains, which, in the form of lipid rafts, are thought to act as platforms for signalling and membrane sorting in cells. The areas of domain formation are defined by the regions of phase coexistence in the phase diagrams for the binary mixtures of lipid with cholesterol. In this paper, the available binary phase diagrams of lipid-cholesterol mixtures are all collected together. It is found that there is not complete agreement between different determinations of the phase diagrams for the same binary mixture. This can be attributed to the indirect methods largely used to establish the phase boundaries. Intercomparison of the various data sets allows critical assessment of which phase boundaries are rigorously established from direct evidence for phase coexistence.     

Direct Determination of Fluid Phase Equilibria by Simulation in the Gibbs Ensemble: A Review

This paper provides an extensive review of the literature on the Gibbs ensemble Monte Carlo method for direct determination of phase coexistence in fluids. The Gibbs ensemble technique is based on performing a simulation in two distinct regions in a way that ensures that the conditions of phase coexistence are satisfied in a statistical sense. Contrary to most other available techniques for this purpose, such as thermodynamic integration, grand canonical Monte Carlo or Widom test particle insertions, the Gibbs ensemble technique involves only a single simulation per coexistence point. A significant body of literature now exists on the method, its theoretical foundations, and proposed modifications for efficient determination of equilibria involving dense fluids and complex intermolecular potentials. Some practical aspects of Gibbs ensemble simulation are also discussed in this review. Applications of the technique to date range from studies of simple model potentials (for example Lennard–Jones, square-well or Yukawa fluids) to calculations of equilibria in mixtures with components described by realistic potentials. We conclude by discussing the limitations of the technique and potential future applications.     

A Monte Carlo scheme based on mid-density in a hysteresis loop to determine equilibrium phase transition

A new and simple method to determine equilibrium phase transition in adsorption systems exhibiting a hysteresis loop is presented as an alternative to methods such as multiple histogram reweighting, gauge cell method and thermodynamic integration. This method is based on the NVT-grand canonical Monte Carlo mid-density scheme to determine the coexistence chemical potential and coexistence densities of an adsorption system. We illustrate this new scheme with argon and methane adsorption in a number of model solids having slit and cylindrical pores. This method does not have a strong basis on thermodynamic ground, but it does provide a simple heuristic approach that is simpler to understand physically.     

Exclusion of alcohols from spermidine-DNA assemblies: probing the physical basis of preferential hydration

The interaction of the alcohols 2-methyl-2,4-pentanediol (MPD) and 2-propanol and of glycerol with condensed spermidine(3+)-DNA arrays are investigated with direct force measurements using osmotic stress coupled with X-ray scattering. Thermodynamic forces between DNA helices are measured from the dependence of helical interaxial spacings on the osmotic pressure applied by poly(ethylene glycol) solutions in equilibrium with the DNA phase. The sensitivity of these forces to solute concentration can be transformed into a change in the number of excess or deficit solutes or waters in the DNA phase by applying the Gibbs-Duhem equation. The alcohols examined are excluded from the condensed DNA array and strongly affect the osmotic stress force curves. DNA is preferentially hydrated. MPD is significantly more excluded than 2-propanol. The exclusion of these alcohols, however, is not due to a steric repulsion since glycerol that is intermediate in size between MPD and 2-propanol does not observably affect DNA force curves. As the distance between DNA helices varies, the change in the number of excess waters is independent of alcohol concentration for each alcohol. These solutes are acting osmotically on the condensed array. The distance dependence of exclusion indicates that repulsive water structuring forces dominate the interaction of alcohols with the DNA surface. The exclusion measured for these condensed arrays can quantitatively account for the effect of these alcohols on the precipitation of DNA from dilute solution by spermidine(3+).     

Fluctuation Simulations and the Calculation of Mechanical Partial Molar Properties

Abstract

A method for the direct calculation of partial molar volumes, energies, and enthalpies in multicomponent mixtures in which all species have finite concentrations is presented. The approach, which is based on fluctuation theory, allows the simultaneous determination of the properties of all components in the mixture. The advantages and limitations of the method are illustrated through the (N, U, V) molecular dynamics calculation of the mechanical partial molar properties of two binary Lennard-Jones mixtures.     

Complex Phase Behavior of GUVs Containing Different Sphingomyelins

Understanding the lateral organization of biological membranes plays a key role on the road to fully appreciate the physiological functions of this fundamental barrier between the inside and outside regions of a cell. Ternary lipid bilayers composed of a high and a low melting temperature lipid and cholesterol represent a model system that mimics some of the important thermodynamical features of much more complex lipid mixtures such as those found in mammal membranes. The phase diagram of these ternary mixtures can be studied exploiting fluorescence microscopy in giant unilamellar vesicles, and it is typically expected to give rise, for specific combinations of composition and temperature, to regions of two-phase coexistence and a region with three-phase coexistence, namely, the liquid-ordered, liquid-disordered, and solid phases. Whereas the observation of two-phase coexistence is routinely possible using fluorescence microscopy, the three-phase region is more elusive to study. In this article, we show that particular lipid mixtures containing diphytanoyl-phosphatidylcholine and cholesterol plus different types of sphingomyelin (SM) are prone to produce bilayer regions with more than two levels of fluorescence intensity. We found that these intensity levels occur at low temperature and are linked to the copresence of long and asymmetric chains in SMs and diphytanoyl-phosphatidylcholine in the lipid mixtures. We discuss the possible interpretations for this observation in terms of bilayer phase organization in the presence of sphingolipids. Additionally, we also show that in some cases, liposomes in the three-phase coexistence state exhibit extreme sensitivity to lateral tension. We hypothesize that the appearance of the different phases is related to the asymmetric structure of SMs and to interdigitation effects.     

Effects of farnesol on the physical properties of DMPC membranes

Farnesol interacts with membranes in a wide variety of biological contexts, yet our understanding of how it affects lipid bilayers is not yet complete. This study investigates how the 15-carbon isoprenoid, farnesol, influences the phase behaviour, lateral organization, and mechanical stability of dimyristol phosphatidylcholine (DMPC) model membranes. Differential scanning calorimetry (DSC) of multilamellar DMPC-farnesol mixtures (up to 26 mol% farnesol) demonstrates how this isoprenoid lowers and broadens the gel-fluid phase transition. A gel-fluid coexistence region becomes progressively more dominant with increasing farnesol concentration and at concentrations of and greater than 10.8 mol%, an upper transition emerges at about 35 °C. Atomic force microscopy images of supported farnesol-DMPC bilayers containing 10 and 20 mol% farnesol provide structural evidence of gel-fluid coexistence around the main transition. Above this coexistence region, membranes exhibit homogeneous lateral organization but at temperatures below the main gel-fluid coexistence region, another form of phase coexistence is observed. The solid nature of the gel phase is confirmed using micropipette aspiration. The combined thermodynamic, structural, and mechanical data allow us to construct a phase diagram. Our results show that farnesol preferentially partitions into the fluid phase and induces phase coexistence in membranes below the main transition of the pure lipid.     

Effects of farnesol on the physical properties of DMPC membranes

Farnesol interacts with membranes in a wide variety of biological contexts, yet our understanding of how it affects lipid bilayers is not yet complete. This study investigates how the 15-carbon isoprenoid, farnesol, influences the phase behaviour, lateral organization, and mechanical stability of dimyristol phosphatidylcholine (DMPC) model membranes. Differential scanning calorimetry (DSC) of multilamellar DMPC-farnesol mixtures (up to 26 mol% farnesol) demonstrates how this isoprenoid lowers and broadens the gel-fluid phase transition. A gel-fluid coexistence region becomes progressively more dominant with increasing farnesol concentration and at concentrations of and greater than 10.8 mol%, an upper transition emerges at about 35 degrees C. Atomic force microscopy images of supported farnesol-DMPC bilayers containing 10 and 20 mol% farnesol provide structural evidence of gel-fluid coexistence around the main transition. Above this coexistence region, membranes exhibit homogeneous lateral organization but at temperatures below the main gel-fluid coexistence region, another form of phase coexistence is observed. The solid nature of the gel phase is confirmed using micropipette aspiration. The combined thermodynamic, structural, and mechanical data allow us to construct a phase diagram. Our results show that farnesol preferentially partitions into the fluid phase and induces phase coexistence in membranes below the main transition of the pure lipid.     

Direct measurement of phase coexistence in DPPC/cholesterol vesicles using Raman spectroscopy

The phase behavior of bilayers of binary mixtures of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) and cholesterol has been studied using Raman spectroscopy. It is observed that the shape of the cholesterol vibrational spectrum in lipid-cholesterol binary mixtures does not vary significantly with either the cholesterol concentration or the temperature. This permits determination of the lipid vibrational signatures of the liquid-disordered (l(d)), solid-ordered (s(o)) and liquid-ordered (l(o)) phases. Within the phase coexistence region, the measured spectra are described very well by a linear combination of the different spectral components, which permits a quantitative analysis of the phase diagram. In contrast to earlier findings, our experiments provide no indication of a phase boundary at low cholesterol concentration. The upper boundary of the phase coexistence region is found at approximately 27 and approximately 22 mol% for l(d)-l(o) and s(o)-l(o) coexistence region, respectively. Within these phase coexistence regions, the partitioning of cholesterol between the cholesterol-poor and the cholesterol-rich phases is in close agreement with the lever rule.     

Phase fluctuation in phospholipid membranes revealed by Laurdan fluorescence.

The organization of lipids surrounding membrane proteins can influence their properties. We have used 6-dodecanoyl-2-dimethylaminonaphthalene (Laurdan) to study phase coexistence and phase interconversion in membrane model systems. The fluorescence properties of Laurdan provide a unique possibility to study lipid domains because of the different excitation and emission spectra of this probe in the gel and in the liquid-crystalline phase. The difference in excitation spectra allows photoselection of Laurdan molecules in one of the two phases. Using the difference in emission spectra it is then possible to observe interconversion between the two phases. We have performed experiments in dipalmitoyl-phosphatidylcholine (DPPC) vesicles at different temperatures, in particular in the region of the phase transition, where phase coexistence and interconversion between phases is likely to be maximal. We have also studied vesicles of different lipids and mixtures dilauroyl-phosphatidylcholine (DLPC), DPPC, and 50% DLPC in DPPC. Both steady-state fluorescence intensity and polarization data have been collected. To quantitate phase coexistence and interconversion we have introduced the concept of "generalized polarization." We have also performed time-resolved experiments to directly prove the interconversion process. We have found that in DLPC-DPPC mixtures, at 20 degrees C, phase interconversion occurs in approximately 30-40 ns.