Modeling lipid membranesphysics.iitm.ac.in/...notes/...chapter_membranes.pdfChipot, Tarek & Klein: Modeling lipid membranes 3 1. Introduction Membranes consist of an assembly of a

Modeling lipid membranes

Modeling lipid membranes

Christophe Chipot,a Mounir Tarek,a Michael L. Klein b

a Equipe de dynamique des assemblages membranaires,Unite mixte de rechercheCNRS/UHP 7565,

Institut nanceien de chimie moleculaire,Universite Henri Poincare, BP 239,

54506 Vandœuvre–les–Nancy cedex, France

b Center for molecular modeling, Chemistry department,University of Pennsylvania, 231 South 34th street,

Philadelphia, Pennsylvania 19104–6323, USA

Version: January 23, 2004

Keywords: atomic simulations; lipid bilayers; membranes; molecular dynamics

Chipot, Tarek & Klein: Modeling lipid membranes 2

Abstract

Modeling of membranes has become over the past fifteen years an emerging field that has benefitedfrom developments on both the hardware and the software fronts. Large computer simulations, inparticular molecular dynamics simulations are now able to provide novel insights into the structureand the dynamics of extremely complex, three–dimensional assemblies like lipid bilayers, therebyserving as an additional, complementary source of information to the current arsenal of experimen-tal tools. An overview of the simulations performed in recent years to improve our understandingof the function of lipid membranes is presented. Successes, failures and limitations of the currentmethodology are discussed through a series of illustrative case examples. Last, a glimpse into newdirections and future prospects is outlined, emphasizing on alternative approaches for modelingthe complexity of the lipid environment.


1. Introduction

Membranes consist of an assembly of a wide variety of lipids,1proteins and carbohydrates thatself–organize to assume a host of biological functions in the cell machinery, like the passive andactive transport of matter, the capture and storage of energy, the control of the ionic balance,or the intercellular recognition and signalling. In essence, membranes act as walls that delimitthe interior of the cell from the outside environment, preventing the free translocation of smallmolecules from one side to the other. At an atomic level, knowledge of both the structure andthe dynamics of membranes remains to a large extent fragmentary, on account of the remarkablefluidity of these systems under physiological conditions. As a result, the amount of experimentalinformation that can be interpreted directly in terms of positions and motions is still rather limited.

A method that could provide the atomic detail of lipid bilayers, that is often inaccessible to conven-tional experimental techniques would, therefore, be extremely valuable for improving our under-standing of how membranes function. It would further constitute a bridge between observations atthe macroscopic and the microscopic levels, and possibly reconcile the two views. Atomic simu-lations,2 in general, and molecular dynamics (MD) simulations, in particular, have proven to be aneffective approach for investigating lipid aggregates, providing new insights into both the structureand the dynamics of these systems.

Basic structural characteristics of the membrane are determined by the nature of the lipids, and howthe latter self–organize into complex three–dimensional arrangements, exposing their polar headgroups to the aqueous environment, while protecting the aliphatic domain to form the hydrophobiccore. Atomic simulations have developed over the past two decades to such an extent that itpossible to model with the desired accuracy these structural features.

Statistical simulations rely on models that have undeniably improved over the years, getting in-exorably closer to the chemical, physical and biological reality of the systems investigated. Yet,they remain models, subject to a host of underlying approximations. It is, therefore, necessaryto confront as systematically as possible the results of numerical simulations to the experimentaldata available. Only when the models have proven to have reached the appropriate robustness andreliability, can they serve as an explanatory, possibly predictive tool, capable of (i) rationalizingexperimental findings, (ii) providing additional insights into experimentally observed phenomena,and (iii) suggesting new experiments. In the particular case of water–lipid assemblies, there is aconsiderable wealth of experimental information that can potentially be used to support or contra-dict in silico studies, albeit immediate confrontation often turns out to be rather cumbersome.

Modeling biological membranes raises a number of difficulties, that still have not found a satisfac-tory solution. A lipid bilayer is, in essence, a disordered liquid crystal of virtually infinite extent.Truncation of this system into a finite–size patch, to comply with the current limitations of molec-ular simulations,de factorubs out significant ranges of the wavelength spectrum that corresponds,


for instance, to bending and splay motions. Current limitations in the available computationalresources not only impose restrictions on the size of the system, but also on the time–scales ex-plored. In silico experiments, likeMD simulations, nonetheless, represent a powerful tool whichis able to offer new insights into the structural and dynamical properties of lipid bilayers.

This chapter is aimed at introducing this field to non–specialists, yet providing the necessary guid-ance for setting up and understanding statistical simulations of lipid–water assemblies, togetherwith key–references for further reading. Up–to–date comprehensive reviews on modeling mem-branes can be found elsewhere.3–10 After outlining the properties that govern self–organization,and the type of structural information accessible from experiment, the methodologies utilized tomodel these systems are described. Next, examples of atomic simulations of lipid bilayers arepresented emphasizing how the results can be compared to experiment. Last, selected simulationsof more complex membrane assemblies are described and discussed critically, with a glimpse intothe future of this very promising research area.

2. lipid–water assemblies

2.1. What are the factors that determine the morphology ?

By and large, lipids and surfactants are amphipathic chemical species formed, roughly speaking,by a hydrophilic head group and a hydrophobic, alkyl tail. As a function of the chemical specie,this non–polar tail may be constituted by one or two aliphatic chains, either saturated or unsatu-rated. In the case of phospholipids, the head group usually consists of a phosphate group bonded toa variety of functional moieties, like a choline, an ethanolamine, a serine, or a glycerol fragment.Depending upon the type of fragment, the lipid is either charged —e.g.dimyristoylphosphatidyl-glycerol (DMPG), or neutral —e.g. dimyristoylphosphatidylcholine (DMPC). At the so–calledsn–3 position, the phosphate group is attached to a glycerol hydroxyl moiety, the two remaininghydroxyl moieties being connected to aliphatic chains by means of ester linkages at positionsn–1andsn–2.

At low concentrations, lipids or surfactants in an aqueous medium usually remain in a monomericstate. Beyond the critical micelle concentration (CMC), they self–assemble into a wide variety ofunique three–dimensional structures, that encompass micelles, inverse micelles, bilayers, hexag-onal tubular phases and more complicated bicontinuous labyrinths (see Figure 1). The nature ofthe lipid determines the morphology of the three–dimensional arrangement.11,12 In water, lipidsaggregate in such a fashion that the polar head group be hydrated adequately, while protectingthe alkyl chains from exposure towards the aqueous environment. As a consequence, the cross–sections of both the head group and the chains dictate the morphology of the resulting lipid–waterassembly. For instance, lipids featuring a large head group and a single alkyl chain usually form


direct micelles, whereas lipids characterized by a smaller head group and possibly two alkyl chainstend to self–organize into inverse micelles.1

For lipids forming planar bilayer assemblies, the net charge borne by the head group plays anoteworthy role in the self–organization process. Small, charged head groups show an interestingtendency to associate by means of intermolecular hydrogen bonds, resulting in compact structureswith a small surface area per lipid —e.g.dilaureylphosphatidylethanolamine (DLPE).13 In larger,zwitterionic lipids , like phosphatidylcholine, lipid head groups are organized in inter and intramolecular charge pairs between the oppositely charged choline and phosphate groups.14

(c)(a)

(b)

Figure 1 : Polymorphism of lipid–water assemblies. (a) The cross–sectional area of the head group islarger than that of the alkyl tail. In an aqueous environment, this specie forms direct micelles, which furtherorganize into hexagonal HI phases. (b) The cross–sectional area of the head group is smaller than that ofthe alkyl tail. The lipids form inverted micelles in water, which may aggregate into hexagonal HII phases.(c) The cross–sectional areas of the head group and the tail are comparable. The lipids assemble into planarbilayers, in the gel, Lβ′ , phase (left) or in the liquid crystal, Lα, phase.

Aside from the nature of the lipid, external conditions, like the concentration, the temperature, thepressure or the ionic strength of the solvent, strongly influence self–organization into a particularstructure. Extensive variables, for instance, can be used to control the transition between phases.At low temperatures, lipid bilayers remain in the gel, Lβ′, phase, wherein the alkyl chains, mostlyin an all–transconformation, are well ordered and exhibit a reduced mobility. At higher tempera-tures, the gel phase transforms into a liquid crystal, Lα, phase characterized by an increase of thesurface area per lipid and a decrease of the thickness of the bilayer, as a direct consequence of the“melting” of the participating alkyl chains. The transition temperature, depends on the chemicalnature of the lipid. For instance, it increases with the length of the alkyl chains, but it decreaseswith the number of unsaturations.

Most cell membranesin vivo exist in the fluid, liquid crystal phase, barring a few cases,e.g.stratum corneumspecialized membrane.15 It is, therefore, not too surprising that, at the exception


of a handful of simulations of lipid bilayers in the gel phase, most investigations have focused onthe so–called, biologically relevant Lα phase.

2.2. Experimental available information

To this date, neutron and x–ray diffraction experiments probably remain the most powerful toolsfor determining structures of lipid bilayers at an atomic resolution.16–19 A particularly pertinentinformation supplied by diffraction experiments are density distributions,20 that can be deconvo-luted in terms of atomic positions in the direction normal to the water–membrane interface, fordifferent types of atoms.

High–resolution x–ray diffraction experiments may offer additional, valuable information, thatcan directly serve as a reference for computational studies. Such is the case of the surface areaper lipid, that may be derived from gravimetric x–ray methods or from electron density profiles.It should be underlined, however, that the highly disordered nature of liquid crystal, Lα, phases,and their fluctuations makes the observation of such systems particularly difficult, and explains thelarge uncertainty in the values supplied by the literature.20.

Whereas x–ray and neutron diffraction on multilayered samples have historically been a sourceof high–resolution structural information of model membranes, neutron reflectivity has providedunique data on single lipid bilayers in contact with bulk water. Scattering length density (SLD)profiles along the normal to a layered system are deduced from the information collected as afunction of the scattering wave–vector transfer (Q). Recently it has been shown that it is alsopossible to invert directly the reflectivity spectra to obtainSLD profiles.21 It is important to note,however, that only the totalSLD profile is determined. For more complex systems, atomistic mod-eling can provide valuable insight into such structures, thereby complementing the experimentalstudies.22–24

Nuclear magnetic resonance (NMR) techniques are also used extensively to probe the molecularorganization in lipid membranes. Earlier on,2H NMR experiments on oriented lipid matricessupplied lipid order parameters, against which the average orientational order along the acyl chainscalculated from simulations could be confronted. Today, thanks to the introduction of magic anglespinning (MAS) techniques, a very large number of parameters from lipid bilayers are available,providing a wealth of information on the conformation of all lipid segments.25

X–ray and neutron scattering measurements as well asNMR experiments may also be used asa possible source of comparison of dynamical properties againstMD simulations. As will beseen in what follows, the significant computational effort involved in atomic simulations of largelipid–water assemblies limits, from a biological perspective, their length to relatively short times.Short time–scale dynamics is yet amenable toMD, and the data determined by this approach can


be confronted directly to scattering experiments26,4, and, for instance, to nuclear Overhauser en-hancement spectroscopy (NOESY) cross–relaxation rates27, which probe motions occurring overcomparable time–scales.

3. Modeling lipid bilayers

In order to eliminate edge effects and to mimic a macroscopic system, simulations of lipid bilayersconsist of considering a small patch of lipid and water molecules confined in a central simulationcell, and replicating the latter using periodic boundary conditions (PBCs) in the three directions ofCartesian space, as is being done in the simulations of molecular liquids and crystals. In doing so,the simulated system corresponds to a small fragment of either a multi–lamellar liposome or of amultilamellar oriented lipid stack, similar to those deposited on a substrate (see Figure 2). The sizeof the simulated sample results in artefactual, symmetry–induced effects and the impossibility towitness collective phenomena like bending or splay motions that occur over length–scales abovethe size of the cell.28,29,20

If needed, one may render a more biologically or physically meaningful picture, consistent with ex-perimentally observed phenomena, by incorporating a large number of lipid and water molecules.30

Even then, the length of the simulation constitutes another critical aspect in the modeling of lipid–water assemblies, essentially because a number motions in lipid bilayers, occur over time–scalesexceeding 10 ns (see Figure 2).

(b)

(a)

(c)

−11

10 s

10 s

−6

10 s

10 s−8

+4

10 s−9

Figure 2 : Left: Small patch of lipid bilayer replicated byPBCs. Right: Characteristic time–scales in lipidbilayers. Overall, motions occur on times that range between a few ps for the separation ofsn–1 andsn–2alkyl chains, to a few hours for the so–called flip–flop, where in a lipid unit migrates from one leaflet to theother.


3.1. Choice of the thermodynamic ensemble

From a technical perspective, the simplest thermodynamical ensemble for simulating lipid–waterassemblies is undeniably the microcanonical,(N, V, E ), ensemble, or possibly the canonical,(N, V, T ), ensemble, wherein the temperature is controlled rigorously by means of a thermostat.In this event, the modeler may choose to fix the cross–sectional area per lipid unit to its experimen-tal value and leave an appropriate head space of air in contact with the water lamellae, above andbelow the membrane. Whereas this protocol isad hocin the case of a simple, homogeneous lipidbilayer, one may legitimately wonder how it will perform when additives —e.g. small solutes tolarge proteins, are introduced into the membrane or in its vicinity. A better adapted thermodynamicensemble should then be employed to allow the participating lipid chains to relax in response tothe modification of the surface tension imposed by the additive. A very tempting solution consistsin turning to the isobaric–isothermal,(N, P, T ), ensemble, that makes use of rigorous barostatsand thermostats to maintain, respectively, the pressure and the temperature at the desired values.This raises, however, difficulties of its own.

In a mixture of oil and water with a positive surface tension, the free energy increases monotonous-ly with the surface area, as the system minimizes the contact area between the two liquids. Inthe case of lipids interacting with water —viz. typically a hydrated lipid bilayer, the picture issomewhat more intricate. Just like for a mixture of oil and water, by virtue of the hydrophobiceffect, the free energy increases with the surface area. This is evidently not the sole contributiongoverning the behavior of lipid bilayers, the surface area of which would be minimized regardlessof the temperature, thereby forcing the system in the gel, Lβ′, phase. Small surface areas, indeed,restrain the alkyl chains in an ordered state, consequently decreasing the entropy of the lipid–water assembly. As a result, the free energy no longer increases with the surface area, but, on thecontrary, exhibits a minimum that corresponds to an optimum surface area for a given temperature.This also implies that the surface tension,γ, should be strictly zero, and, therefore, that the lateralpressure,P‖, be strictly equal to the pressure normal to the water–lipid interface,P⊥:

γ =∫ [

P⊥ − P‖(z)]

dz = 0 (1)

This important result, which was expected for a self–organized system, prompted a host of authorsto simulate lipid bilayers in the isotropic isobaric–isothermal ensemble,(N, P, T ).31 Whereas,strictly speaking, equation (1) is true for a lipid–water assembly of virtually infinite extent, itshould be kept in mind that in atomic simulations one model patches of finite size. Feller and Pas-tor put forward that a finite surface tension should be introduced to compensate for such finite–sizeeffects that eliminate the possibility to observe collective phenomena like undulations over signif-icant length–scales,32,33 as in ripple, P′β, phases, for instance. Tieleman and Berendsen arguedthat in the systems they investigated, the dependence of the surface tension with the surface areawas marginal.34 Lindahl and Edholm later showed that an applied surface tension in the order of10 mN/m would correct for large fluctuations in the surface area per lipid unit that are witnessed


in simulations of lipid–water assemblies of limited size.35 One thing is certain: In atomic simula-tions of lipid bilayers,P‖ andP⊥ are anticipated to vary differently on account of the anisotropyof the environment. It is, therefore, strongly recommended to adopt an algorithm that generatesthe(N, P, T ) distribution, so that the dimensions of the simulation cell are rescaled independentlyin thex, y (in-plane) and in thez–directions.36,37,2

3.2. The potential energy function

In atomic statistical simulations of membranes, all atoms pertaining to the system are treatedclassically as point masses, which, in the harmonic approximation, are connected to each otherby means of springs. In some instances, for the sake of computational effort, certain groups ofatoms, like methylene, –CH2–, or methyl, –CH3, moieties, are represented as a single, “united”atom of appropriate van der Waals radius and well depth.38 Seminal simulations of lipid–waterassemblies made use of the available multi–purpose force fields, often aimed at the modeling ofsolvated proteins and nucleic acids. It is, therefore, not too surprising that in early investigations,the agreement with experiment was either far from optimal, or clearly too good to not suspect afortuitous cancellation of errors due to the conjunction of inadequate parameters and excessivelyshort runs.

In the following years, it was realized that a specific potential energy function should be employedto mimic accurately the properties of lipids, like the subtletrans–gaucheequilibrium in the alkylchains. A dearth of efforts was and is still invested to improve the representation of lipids andsurfactants by means of an appropriate parameterization of the force–field contributions likely toaffect the structural and dynamical features of these systems.39–43 In some of these force-fields,to obtain a better description of the ordering in the fatty aliphatic chains, that can be ascribedto trans–gauchedefects, the standard low–order Fourier series that is often used in conventionalmacromolecular force fields, was replaced by the more sophisticated Ryckaert–Bellemans tor-sional potential.44

In addition, correct packing of the alkyl chains depends to a large extent on the quality of thevan der Waals parameters utilized. One of the underlying assumptions made for the design offorce fields is the transferability of these parameters between molecules —e.g. the van der Waalsradius and well depth of an aliphaticsp3 carbon should be the same regardless of the chemicalenvironment. The interaction parameters of the united methylene and methyl groups were orig-inally derived from statistical simulations of short hydrocarbons liken–butane, as is the case ofthe OPLS force field.45 The transferability hypothesis has proven to be inadequate when handlinglong alkyl chains, prompting a number of authors to reoptimize van der Waals interactions basedon simulations of large hydrocarbons like pentadecane.31

Determination of net atomic charges for lipids and surfactants from sophisticated quantum me-


chanical calculations may turn out to be a difficult task, on account of the size of the molecules.Unquestionably, partial charges derived from the electrostatic potential constitute the most sat-isfactory solution among the arsenal of approaches available to the modeler.46 Yet, as has beendemonstrated, point charges are inherently conformation–dependent,47 thus making the derivationof a unique set of charges representative of all possible conformations questionable. To circum-vent the difficulties connected to the size of the molecules, it has been proposed to derive the netatomic charges as independent fragments, that are ultimately pieced together. This scheme, al-though tempting, should be considered with great care if local charges are delocalized over largespatial extents.

3.3. Intermolecular interactions

As has been mentioned previously, physically and biologically realistic simulations should involvea sufficiently large number of lipid and water molecules to minimize finite–size effects. Much ofthe computation effort involved in atomic simulations of lipid–water assemblies lies in the eval-uation of pairwise interactions, the number of which increases dramatically with the number ofparticles in the system. Based upon the assumption that intermolecular interactions decay with thedistance, earlier studies have employed a cut–off sphere, beyond which the interactions are trun-cated. This approximation is expected to bead hocfor the short–range, van der Waals contribution.The use of a brute, finite spherical cut–off for truncating the short–range van der Waals interac-tions may, however, modulate the forces responsible for the cohesion of lipid–water assemblies.Accurate use of a cut–off requires to take into account the appropriate long–range corrections forboth the energy and the pressure,48 based on the classical formulae utilized for Lennard–Jonesfluids.49

For Coulomb interactions, the range of which varies inrn, wheren ≤ 3,49,2 truncation becomesparticularly arguable. In this event, the long–range character of the participating charge–charge(n = 1) and charge–dipole (n = 2) interactions makes the use of a spherical truncation unsuit-able. Probably the most accurate approach for handling the long–range nature of electrostaticinteractions in spatially replicated simulation cells is solving the Poisson equation. The Ewaldapproach,50 that decomposes the conditionnally convergent Coulombic sum over periodic boxesinto two rapidly decaying contributions evaluated respectively in the direct and reciprocal spacesis the most used method. Formally, the computational effort involved in this method scales as(N2), thus making statistical simulations of large ensembles of atoms particularly costly. Thiseffort can be reduced, scaling down the calculation to(N ln N), by solving the Poisson equationnumerically on a grid of points, over which the position of the particles are interpolated. Sucha scheme constitutes the central idea of algorithms like particle–mesh Ewald (PME) or particle–particle–particle–mesh (P3M).51 For completeness, while to our knowledge, it has not been yetapplied in membrane simulations, it is worth mentioning the fast multipole approach, a method al-ternative to Ewald summation, that treats long–range interactions in a rigorous fashion, and scales


linearly with N for very large systems —viz. on the order of 100,000 atoms.52

The substantial computational investment required to attain a physically consistent description ofthe simulated molecular assembly may be further reduced by taking advantage of recent advancesin theMD methodology. Considering that the different degrees of freedom involved in the systemrelax over distinct time–scales, it is not necessary that the corresponding force contributions beevaluated concurrently. This is, in essence, the basic principle of the so–called multiple time–step methods,53,54 in which intramolecular, van der Waals and Coulomb forces can be updatedwith different frequencies.55 In conjunction with constraint algorithms likeSHAKE or RATTLE,56

that virtually eliminate the vibrations due to hard degrees of freedom it is possible to explorelarge regions of the phase space for a lesser computational effort, thus making long simulations oflarge lipid–water assemblies somewhat more affordable — the reader is referred to the chapter ofTuckerman and Martyna dedicated to integrator and ensembles in statistical simulations.

4. Atomic simulations of lipid membranes

Traditionally, phospholipids have served as models for investigatingin silico the structural anddynamical properties of membranes. From both a theoretical and an experimental perspective,zwitterionic phosphatidylcholine (PC) lipids constitute the best characterized systems. HydratedDMPC13,57 and dipalmitoylphosphatidylcholine58,34,31,59–61 (DPPC) bilayers have been so far prob-ably the most extensively surveyed lipid membranes. Yet, on account of their intrinsic limita-tions —viz. the short alkyl chains inDMPC and the temperature of Lβ′ to Lα phase transition inDPPC, above physiological conditions — several authors have turned to biologically more rele-vant lipids like palmitoyloleylphosphatidylcholine62,63 (POPC), in particular for examining mem-brane proteins in a realistic environment, and lipids based on mixtures of saturated/polyunsaturatedalkyl chains (SDPC, 18:0/22:6PC).64,43 A variety of alternative lipids, featuring different, possi-bly charged, head groups, have also been explored —e.g. DLPE65,13 (DLPE), dipalmitoylphos-phatidylserine66,67 (DMPS) or glycerolmonoolein68,30 (GMO). In several cases, however, the mod-eler is faced with an absence of experimental data to which the results of atomic simulations canbe confronted.

Bilayers built fromPC lipids, nonetheless, represent remarkable test systems not only to probe themethodology, but also to gain additional insight into the physical properties of membranes. In thissection, the derivation of these properties fromMD trajectories and how a bridge with experimentcan be established will be detailed.


4.1. Bilayer structure

4.1. 1. Density distributions

As can be seen in Figure 3, the spatial extent encompassed by the head–group region of theDMPC

units in a bilayer arrangement is remarkably broad. This is clearly seen in the number densityprofiles computed from theMD trajectory — an analysis along the direction normal to water–membrane interface of the in–plane densities of lipid and water atoms. A striking feature emergingfrom these distributions is the penetration of water far in the head–group region. The farthestextent of water molecules roughly coincides with the ester moieties of the lipids. The width ofthe interfacial region, on the order of 8 to 10A for a fully hydratedDMPC bilayer highlightsthe significant static and dynamic roughness of the membrane surface68,69, therefore, refining thetraditional textbook picture, like that of Figures 1 and 2.

Figure 3 : Left: Snapshot taken from anMD simulation of a fully hydratedDMPC bilayer. Methylene andmethyl carbon atoms of the alkyl chains are shown in grey and pink, respectively. Ester carbon and oxygenatoms are shown in blue and purple, respectively. Phosphate phosphorus and choline nitrogen atoms areshown in orange and green, respectively. Note the protrusion of head groups that results locally in a roughwater–membrane interface. Right: Density distributions for selected groups of atoms in a fully hydratedDMPC bilayer examined at 303 K (from reference [70]).

Interestingly enough, phosphate and choline groups lie approximately at the same depth in thebilayer, indicating that head groups are rather oriented in the plane of the bilayer. The average ori-entation of the head–groupP—N bond dipoles with respect to the normal of the water–membraneinterface arises around 700, pointing towards the aqueous medium, albeit the orientational dis-tribution is remarkably wide, and depends upon the temperature and, as expected, the potentialenergy function utilized.14 In addition, the level of lipid hydration has been shown to play a note-worthy role in the orientation of the head groups.9 Under any circumstances, it is crucial that the


slow reorientation of the lipid head groups be considered when interpreting results from shortMD

trajectories. Estimates from single–molecule anisotropy imaging for fluorophore–taggedPOPC

molecules71 indicate a rotational diffusion coefficient ofca. 0.7 rad2/ns, slightly below estimatesfrom MD simulations,72 suggesting that sampling of the whole rotational space for each moleculewould necessitate over few tens of a nanosecond.

The information provided by the density distributions can be confronted directly to x–ray and neu-tron diffraction measurements17 (considering respectively the atomic scattering length densities,or the electron densities). TheMD trajectories can further be used in conjunction with the scatter-ing experiments in order to refine the data by, for instance including fraction volumes extractedfrom the simulation.73

4.1. 2. Lipid tail conformation

Deuterium quadrupole splitting measured by2H NMR on non–oriented samples of membranepreparations, is mainly determined by the average conformation of the phospholipid molecules,and, as such, supplies valuable structural information about the system. Order parameters can bederived fromMD trajectory, and can be expressed as a tensor, the elements of which write:74

Sαβ =1

2〈3 cos ϕα cos ϕβ − δαβ〉 (2)

Here, ϕα is the angle formed by theα–th molecular axis and the normal to the water–bilayerinterface, and〈· · ·〉 is an ensemble average over all lipid chains. In most circumstances, based onsymmetry relationships, it is assumed that the order parameters,SCD, for an alkyl chain bearingdeuterium labels can be expressed as:

SCD =1

2

⟨3 cos2 θ − 1

⟩(3)

whereθ is simply the angle between the C—D chemical bond and the normal to the bilayer.When C—D is uniformly distributed,SCD = 0, and when the chain is all–trans, |SCD| = 0.5. Ingeneral for saturated lipids,|SCD| exhibits a plateau value atca. 0.2 for the upper chain segments.Force fields of the new generation reproduce quite well these order parameters, barring smalldiscrepancies for the second carbon atom of the alkyl chains.

Further analysis ofMD trajectories may be aimed at extracting additional information from theNMR experiments. For instance, one may refine those methods targeted at obtaining such quantitiesas the average chain length or the surface area per molecule.76 Another study exemplifies thesuccessful combination ofMD simulation with experiment to probe alkyl chain packing in lipidmembranes. Such is the case of infrared (IR) data, that have been reinterpreted to estimate theconcentrations ofgauche–gauche, trans–gaucheand trans–transconformational sequences in aDPPCbilayer.77.


2 3 4 5 6 7 8 9 10 11 12 13 14carbon atom number

0.05

0.10

0.15

0.20

0.25

-SC

D

Figure 4 : Order parameters of a fully hydratedDMPC bilayer at 323 K. red: derived from the simulation(conditions explicited in [75]) and black: experimental values.74

4.1. 3. Hydration of the head–group region

In atomic simulations, solvation properties are often measured by means of radial distribution orpair correlation functions (RDFs):

gij(r) =〈Nj(r; r + δr)〉

4π%j

∫r2dr

(4)

whereNj is the number of particlesj at a distance fromi comprised betweenr andr+δr and%j isthe density of particlesj. In essence, this definition is targeted at isotropic fluids, and, in principle,should not be applied, as is, to anisotropic lipid–water assemblies.78 To estimate the coordinationnumber of sitei — e.g.PChead groups, it seems far more appropriate to merely evaluate〈Nj(r; r+

δr)〉 as a function of the separationr and determine its value at the first minimum of a qualitativeRDF computed using equation (4).

4.1. 4. Transmembrane electrostatic potentials

Orientation of water molecules near the head–group region of the lipid bilayer is clearly aniso-tropic, compared to the bulk aqueous medium. This can be shown by measuring the average cosineof the angle formed by the dipole moment of the water molecules and the normal to the bilayer,as a function of the distance from its geometrical center. A marked peak emerges at a distancecharacteristic of the phosphate groups, emphasizing the orienting power exerted by this moiety onthe surrounding aqueous environment. The preferential orientation of the dipole moment borne bythe water molecules is at the origin of the vocabulary “dipole potential”, that has been employed


extensively to denote the electrostatic potential across the water–membrane interface.79,80 Thisconspicuous ordering of water molecules was recently directly evidenced using coherent anti–Stokes Raman scattering microscopy.81

In a number ofin silico investigations, the electrostatic potential has been estimated from theknowledge of the charge density. In the spirit of atomic density distributions, charges are accu-mulated as a function of their position along the direction normal to water–bilayer interface. Thenegative of the first integral of the charge density yields the electric field. In turn, integral of thefield provides the electrostatic potential. Not too surprisingly, the resulting “dipole potential” in-herently depends upon the choice of the potential energy function and should, thus, be interpretedcautiously.4,31,57

4.2. Dynamics

The increasing level of interaction between experimental studies and numerical simulations oflipid bilayers evidenced in the previous section also holds for the dynamics of lipid bilayers.

Felleret al.27 have usedMD simulations to analyzeNOESY cross relaxation rates in lipid bilayers.Magnetic dipole–dipole correlation in such systems occurs over a variety of time scales and de-pends upon the probability of close approach for proton–proton interactions. The relaxation rateshave been calculated directly from a 10 nsMD simulation ofDPPC. Fitting the autocorrelationfunctions yields characteristic correlation times and weight factors that determine the relative con-tributions of the individual type of motions. Combining simulations and experiments, relaxationrates may, therefore, be assigned to various motions —viz. less than 1 ps for chemical bond vibra-tions, 50–100 ps fortrans–gaucheisomerization, 1–2 ns for molecular rotation and wobble, andbeyond 100 ns for lateral diffusion.

A model for the dynamics of individual lipid molecules has also been proposed based on a thor-ough comparison of simulation data and experimental measurements of the13C NMR T1 relaxationin DPPCalkyl chains.82 Employing Brownian dynamics andMD simulations associated to fits ofexperimental data, it was found that lipid molecules confine themselves into a cylinder within the100 ps time scale, and wobble in a cone–like potential on the nanosecond time scale.

A similar model for lipid dynamics has emerged from anMD study aimed at interpreting inelasticneutron scattering (INS) data. One particular aspect of such experiments, probing the motion ofindividual hydrogen nuclei —i.e. self correlation of single particle, is that they are space– and timeresolved. In the case ofDPPCbilayers, a good agreement between simulations and experimentsprobing the 100 ps time scale is attained.83 The analysis corroborates the fact that the motion of thecenter of mass and the internal motions of lipid molecules are decoupled. Moreover, the former iswell described as a diffusion in a confined space,i.e. a cylinder. A refined picture of the internal


dynamics arising from the simulation shows that protons of the alkyl chains move according to achain defect model, where kinks or chain defects form and disappear randomly —i.e. stochasticmodel — along the lipid tail, rather than diffuse along the chain.

Collective dynamics of lipid bilayers have also been examined carefully as simulations over in-creasingly significant time scales and length scaled are feasible. Large systems involving 1,024lipid molecules studied over 10 ns led to the direct observation of bilayer undulations and thick-ness fluctuations of mesoscopic nature.35 Continuum properties such as bending modulus, surfacecompressibility and mode relaxation times were calculated and agreed nicely with experiment.Several processes occurring at different length scales were identified. The undulatory motionscould be separated in two regimes — one involving more than 50 lipids, that can be ascribed tomesoscopic undulations, and the other, involving less than 25 lipids, that is attributed to collec-tive lipid protrusion. Peristaltic modes —i.e. anti–correlated modes between the two layers —could also be distinguished in two types: bending modes involving 50 to 400 lipids, and protrusionmodes over shorter length scales.

Shorter wavelength collective dynamics may be probed using coherent inelastic,viz. neutron orx–ray, scattering. Density fluctuations of length scales comparable to the interlipid distance arebelieved to play a pivotal role in the transport of small molecules across the bilayer. Recently,MD simulations have been used to complement inelastic x–ray data of lipid bilayers, both in thegel, Lβ′, and the liquid crystal, Lα, phases.84 The results support the applicability of generalizedhydrodynamics to describe the motion of carbon atoms in the hydrophobic core, thus allowing themodeler to extract key–parameters, such as sound mode propagation velocity, thermal diffusivityand kinematic longitudinal viscosity.

4.3. Modeling transport phenomena

Models of lipid bilayers have been employed widely to investigate diffusion properties acrossmembranes through assisted and non–assisted mechanisms. Simple ions,e.g. Na+, K+, Ca2+ orCl−, have been shown to play a significant role in the cell machinery, in particular at the level ofintercellular communication. In order to enter the cell, the ion must preliminarily permeate thelipid bilayer that acts as a rampart towards the cytoplasm. Wilson and Pohorille have investigatedthe passive transport of Na+ and Cl− ions across a lipid bilayer formed by glycerolmonooleinunits, which undergoes severe deformations as the ions translocate across the water–membraneinterface. This process is accompanied by thinning defects and the formation of water fingers thatensure an appropriate hydration of the ion as it penetrates in the non–polar environment.85

Ideally, atomic simulations could also serve as a predictive tool for estimating water–membranepartition coefficients of small drugs, in strong connection with the so–called blood–brain barrier— the ultimate step in thede novodesign of pharmacologically active molecules. Diffusion of


small, organic solutes in lipid bilayers was examined for a variety of molecular species rangingfrom benzene86,87 to more complex anesthetics.88–90 Yet, access to partition coefficients by meansof statitistical simulations implies the determination of the underlying free energy behavior alongthe direction normal to the interface.91 In the specific instance of inhaled anesthetics, an analysisof the variations of the free energy for translocating the solute from the aqueous medium intothe interior of the bilayer suggests that potent anesthetics reside preferentially near the water–membrane interface. Contrary to the dogmatic Meyer–Overton hypothesis,92 potency is shown tocorrelate with the interfacial concentration of the anesthetic, rather than its sole lipophilicity.93

The considerable free energy associated to the transfer of ions from the aqueous medium to theinterior of the membrane rationalizes the use in cells of specific transmembrane channels, pumpsor carriers that facilitate while controlling selectively the passage of ionic species across the lipidbilayer.94 Recent complete reviews of the theoretical developments and simulation capabilities inion channels modelling can be found in references [95] and [96]. Here, we briefly describe someof the complex systems examined hitherto.

Gramicidin A, a prototypical channel for assisted ion transport, has been the object of thoroughanalyses from both experimental and theoretical perspectives. Dimerization of individual pro-tein units results in membrane–spanning channels suitable for ion conduction.MD simulations ofgramicidin A embedded in hydrated lipid bilayers,e.g. DMPC, were able to reproduce the struc-tural features observed experimentally.97 Such studies have clearly shown that important questionsrelated to ion selectivity, ion binding, gating and proton transfer mechanisms may be addressedwith some confidence.

Internal arrangement of water molecules in single–file chain of water molecules, characteristic incomplex transporters,98 was also witnessed in a somewhat more rudimentary, synthetic channelformed by stacked cyclic peptides of alternatedD– andL–chiralities (see Figure 6).70 Such nan-otubes have been recognized to modify in a selective fashion the permeability of cell membranesand are envisioned to act as potent therapeutic agents in response to bacterial resistance.99 Aqua-porins, membrane channels ubiquitous to most living species controlling the water contents of thecells, have also focused much attention lately. They are formed of tetramers that organize to facil-itate the transport of water, and possibly other small solutes, across the lipid bilayer. The resultingwater pores remain, however, impervious to the passage of small ions to ensure a proper conser-vation of the electrochemical potential.100 As a final note, it is worth mentioning that, as expected,the determination of the high resolution structure of KscA, a bacterial K+ channel, has motivateda large number of realistic simulations taking into account the lipidic environment studies aimedat deciphering the underlying complex conduction mechanism.


Figure 5 : Snapshot taken from anMD simulation of a synthetic channel formed of cyclic peptides ofalternatedD– andL–chiralities, embedded in a fully hydratedDMPC bilayer. Color coding of the atoms isidentical to that in Figure 3. Note the antiparallelβ–sheet like conformation of the nanotube spanning themembrane. Within a few hundreds of ps, a single–file chain of water molecules is established in the hollowtubular structure.

4.4. Interaction of small molecules, peptides and proteins with membranes

In most circumstances, the biological membrane is described at the theoretical level as a simple,homogeneous bilayer formed by a single type of lipid — usually the well–studied, zwitterionicPC lipids. Membranes, however, are infinitely more complex and consist of an heterogeneous as-sembly of different lipids, either charged or not, carbohydrates and proteins. Approaching the finedetail of the biological picture by incorporating in atomic simulations chemical species of differ-ent natures is evidently the direction towards which the modeler is evolving. From a modelingperspective, the influence of cholesterol,101–105 and more generally sterols,106 on the structure anddynamics of lipid bilayers has attracted a lot of attention in recent years. Although the limitedsampling in some simulations calls into question the conclusions reached by the authors, choles-terol is shown to increase the order parameters of the alkyl chains while decreasing their tilt anglewith respect to the normal to the water–membrane interface, in qualitative agreement with experi-ment.107

Aside from transporters and channels that assist the transport of chemical species across lipidbilayers, a vast array of key–cellular functions are accomplished by proteins that interact with themembrane, either spanning the latter, or bound to its surface.108 Yet, interfacial and transmembraneproteins generally play distinct roles in the cell machinery, albeit the frontier between these twoclasses of proteins remains somewhat fuzzy. A number of proteins, for instance, are only partiallyburied in the membrane —e.g. melittin or alamethicin, the insertion of which is conditioned bythe transmembrane electric field.109–112


The strength ofin silico experiments is to provide glimpses into the atomic detail of biologicalmembranes that conventional experimental techniques cannot capture. Of particular interest isthe molecular interplay that govern membrane–protein association, accessible through large–scaleatomic simulations.MD simulations illuminated, for instance, how the presence of a protein per-turbs the structure of the lipid membrane. For example, the helices of theInfluenza AM2 channeltilt in a DMPC bilayer to maximize membrane–protein hydrophobic contacts.113,114 In the case ofgramicidin A, key–residues located in the head–group region have been shown to stabilize thechannel in the membrane.97

The influence of the protein on the lipid bilayer can be viewed as the subtle balance betweenhydrophobic and hydrophilic contributions that, in principle, can be captured byMD simulations.Differences in the order parameters of lipid units adjacent to the protein and far from it have led tothe concept of “boundary lipids”. In a vast number of instances, among which theMycobacteriumtuberculosisMscL channel,115 theInfluenzaA virus M2 protein,116 and theEscherichia coliOmpFtrimer,117 it was observed that the membrane protein induces an increasing disorder of the lipidalkyl chains in its neighborhood. In sharp contrast, alkyl chains close to the transporter gramicidinA tend to be more ordered, compared to those pertaining to the bulk lipid environment.118,97

In the light of these computational investigations, it would, therefore, appear thattrans–gaucheequilibria in lipid chains are dictated by the very nature of the membrane protein. Yet, as wasshown recently,70 drawing definitive conclusions based on limited simulation lengths may turn outto give a distorted vision of the actual behavior of the lipid bilayer. In principle, exceedingly shortsimulations do not permit the complete relaxation of lipid chains in the vicinity of the protein, andshould, thus, be interpreted cautiously.

The close adequation between the thickness of the lipid bilayer and the length of the hydrophobicsegment of the protein spanning the latter constitutes yet another important facet of the protein–membrane interplay. By providing the microscopic detail of the interactions of integral proteinswith the lipid environment, atomic statistical simulations may contribute to advance our under-standing of the underlying physical principles that govern the function and structure of mem-branes.119 In the light of a series of experimental investigations on model peptides embedded inPC membranes with alkyl chains of increasing length, it was found that if the hydrophobic thick-ness of the peptide is greater than that of the bilayer, the latter becomes thinner, andvice versa.120

A similar phenomenon was observed recently in theMD simulation of a single peptide nanotubeinserted in a hydratedDMPC bilayer.70 The hydrophobic thickness of the membrane adjusts itselfas the synthetic channel tilts concurrently to adapt to its host lipid environment. Whereas theso–called hydrophobic mismatch121 does not appear to induce perturbations in peptide nanotubes,it can, however, modulate strongly the function of more complex proteins. As was observedrecently for gramicidin A, minute changes in the length of the lipid alkyl chains —viz. fromthe 18–carbon oleyl– to the 20–carbon eicosenoylphosphatidylcholine, switch the protein froma stretch–activated to a stretch–inactivated channel. Symmetrically, the hydrophobic mismatch


may alter the phase behavior of the membrane, as demonstrated in the case ofWALP peptidesthat promote the formation of non–lamellar phases.122 These remarkable results should, therefore,incline the modeler to be cautious when solvating membrane proteins in lipid surroundings. Thechoice of the lipid unit for a given protein may turn out as a genuine leap of faith if attention hasnot been paid to the possible imbalance in the hydrophobic thicknesses of the membrane and theprotein, likely to render a physically unrealistic picture of the assembly. When devised appropri-ately, atomic simulations can, nonetheless, shed new light on the nature of the protein–membraneinterplay, by allowing the modeler to not only visualize, but also possibly quantify the strengthof the participating interactions. Of particular interest, the non–covalent chemical bonds formedby L–Trp residues and acceptor moieties of the head–group region have been recognized to actas anchoring points of the protein into the lipid bilayer.123,124 As has been shown in the case ofgramicidin A, the presence of severalL–Trp amino acids at the level of the lipid head groups isexpected to mediate the overall stabilization of the channel in the membrane.97

5. Discussion, outlook and future prospects

Retrospectively, with about fifteen years of hindsight, it has become clear that atomic simulations,in particular MD simulations of lipid–water assemblies have contributed in a large measure toimprove our knowledge of these very complex systems from both a structural and a dynamicalpoint of view. It is also obvious that the successes of pioneering, tantalizing investigations, whichnot only ignited the field of lipid simulations, but were also rapidly fueled by many studies onlarger assemblies, often reflected as much good fortune as they did science. Yet, major advanceson both the hardware and the software, algorithmic fronts progressively allowed the modeler totackle systems of increasing complexity over time–scales compatible with the physical, chemicaland biological reality. Among these advances, the development of specific methods for performingthe simulation in apt thermodynamic ensembles, the improvement of potential energy functionstargeted at the specific modeling of lipid–water assemblies, and the continuous decrease of theprice/performance ratio of modern computers have helped pushing back the intrinsic limitationsof MD simulations. More recent studies have demonstrated that simulations at least an order ofmagnitude longer than those reported when the field was only in its infancy, were required toobtain reliable and reproducible results.125

Simulation of lipid–water systems still constitutes a research area seething with excitement. Thedevelopment of all the ingredients to investigatein silico lipid bilayers with full confidence opensnew perspectives, in particular on the biological front, and should rapidly allow the modeler to uselipids in a routine fashion, just like any other solvent. In this spirit, theoretical studies of membraneproteins in a realistic environment should continue to flourish in the near future. Unfortunately, asthe level of sophistication of atomic simulations increases, together with the available computa-tional power, so does the ambition of the modeler, attempting to deal with molecular systems yet


even more complex, both in terms of size– and time–scales. This explains the current teeming ac-tivity in the development of approximate schemes that could serve as alternatives to a full–atomicdescription for the modeling of large lipid–water assemblies over long times.

Among these alternatives, a dearth of effort has been invested in recent years in the field of implicitsolvation.8 Since the seminal work of Onsager on continuum electrostatics,126 the temptation torepresent explicit surroundings by a simple dielectric medium for a myriad of chemical systemshas been the object of tremendous interest. Modeling the complexity of lipid bilayers by meansof a continuum description has been used, for example, to investigate the insertion ofα–helicalpeptides in a membrane,127 or the interaction of a small toxin with the latter.111 Results of con-tinuum electrostatics simulations, which are based on solving the Poisson–Boltzmann equationnumerically, are, in general, in qualitative agreement with atomic simulations. Yet, not too unex-pectedly, this approximate description cannot capture subtle, specific interactions that govern thestability of the solute —e.g.a short peptide, at the water–membrane interface. As was underlinedrecently by Linet al., the reproduction of membrane dipole potentials based on a sole continuumelectrostatics representation is usually erroneous, but can be significantly improved by inclusionof explicit layers of water molecules near the head–group region.128

Aside from implicit solvation approaches, the use of coarse–grained representations, wherein eachlipid unit is described by a limited number of interacting sites, is probably the most promising. Theunderlying assumption that the formation of a lipid vesicle is a sufficiently robust process to besimulated by simplified models of lipids was ascertained recently by Marrink and Mark througha study of the aggregation ofDPPC units into small unilamellar vesicles.129 By and large, thestrength of coarse–grained models resides in their ability to make simulations self–assembly pro-cesses substantially more affordable than conventional all– or even united–atom models.130 Thelevel of representation offered by this alternative is, in sharp contrast, incompatible with the finedescription of specific interactions of the participating lipid units with small solutes, like anesthet-ics. It is obvious that much attention should still be paid in the optimization of the potentials ofthese rudimentary models to warrant compatibility with a full–atomic descriptions.


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