Modeling lipid membranes
Modeling lipid membranes
Christophe Chipot,a Mounir Tarek,a Michael L. Klein b
a
Equipe de dynamique des assemblages membranaires,
Unité mixte de recherche C NRS /U HP 7565,
Institut nancéien de chimie moléculaire,
Université Henri Poincaré, BP 239,
54506 Vandœuvre–lès–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
2
Chipot, Tarek & Klein: Modeling lipid membranes
Abstract
Modeling of membranes has become over the past fifteen years an emerging field that has benefited
from developments on both the hardware and the software fronts. Large computer simulations, in
particular molecular dynamics simulations are now able to provide novel insights into the structure
and the dynamics of extremely complex, three–dimensional assemblies like lipid bilayers, thereby
serving as an additional, complementary source of information to the current arsenal of experimental tools. An overview of the simulations performed in recent years to improve our understanding
of the function of lipid membranes is presented. Successes, failures and limitations of the current
methodology are discussed through a series of illustrative case examples. Last, a glimpse into new
directions and future prospects is outlined, emphasizing on alternative approaches for modeling
the complexity of the lipid environment.
Chipot, Tarek & Klein: Modeling lipid membranes
3
1. Introduction
Membranes consist of an assembly of a wide variety of lipids,1 proteins and carbohydrates that
self–organize to assume a host of biological functions in the cell machinery, like the passive and
active 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 delimit
the interior of the cell from the outside environment, preventing the free translocation of small
molecules from one side to the other. At an atomic level, knowledge of both the structure and
the dynamics of membranes remains to a large extent fragmentary, on account of the remarkable
fluidity of these systems under physiological conditions. As a result, the amount of experimental
information 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 conventional experimental techniques would, therefore, be extremely valuable for improving our understanding of how membranes function. It would further constitute a bridge between observations at
the macroscopic and the microscopic levels, and possibly reconcile the two views. Atomic simulations,2 in general, and molecular dynamics (MD) simulations, in particular, have proven to be an
effective approach for investigating lipid aggregates, providing new insights into both the structure
and the dynamics of these systems.
Basic structural characteristics of the membrane are determined by the nature of the lipids, and how
the latter self–organize into complex three–dimensional arrangements, exposing their polar head
groups to the aqueous environment, while protecting the aliphatic domain to form the hydrophobic
core. Atomic simulations have developed over the past two decades to such an extent that it
possible to model with the desired accuracy these structural features.
Statistical simulations rely on models that have undeniably improved over the years, getting inexorably 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, necessary
to confront as systematically as possible the results of numerical simulations to the experimental
data available. Only when the models have proven to have reached the appropriate robustness and
reliability, can they serve as an explanatory, possibly predictive tool, capable of (i) rationalizing
experimental findings, (ii) providing additional insights into experimentally observed phenomena,
and (iii) suggesting new experiments. In the particular case of water–lipid assemblies, there is a
considerable wealth of experimental information that can potentially be used to support or contradict 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 satisfactory 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 molecular simulations, de facto rubs out significant ranges of the wavelength spectrum that corresponds,
Chipot, Tarek & Klein: Modeling lipid membranes
4
for instance, to bending and splay motions. Current limitations in the available computational
resources not only impose restrictions on the size of the system, but also on the time–scales explored. In silico experiments, like MD simulations, nonetheless, represent a powerful tool which
is 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 guidance for setting up and understanding statistical simulations of lipid–water assemblies, together
with key–references for further reading. Up–to–date comprehensive reviews on modeling membranes 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 to
model these systems are described. Next, examples of atomic simulations of lipid bilayers are
presented emphasizing how the results can be compared to experiment. Last, selected simulations
of more complex membrane assemblies are described and discussed critically, with a glimpse into
the 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 unsaturated. In the case of phospholipids, the head group usually consists of a phosphate group bonded to
a 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. dimyristoylphosphatidylglycerol (DMPG), or neutral — e.g. dimyristoylphosphatidylcholine (DMPC). At the so–called
sn–3 position, the phosphate group is attached to a glycerol hydroxyl moiety, the two remaining
hydroxyl moieties being connected to aliphatic chains by means of ester linkages at position sn–1
and sn–2.
At low concentrations, lipids or surfactants in an aqueous medium usually remain in a monomeric
state. Beyond the critical micelle concentration (CMC), they self–assemble into a wide variety of
unique three–dimensional structures, that encompass micelles, inverse micelles, bilayers, hexagonal tubular phases and more complicated bicontinuous labyrinths (see Figure 1). The nature of
the lipid determines the morphology of the three–dimensional arrangement.11,12 In water, lipids
aggregate in such a fashion that the polar head group be hydrated adequately, while protecting
the 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–water
assembly. For instance, lipids featuring a large head group and a single alkyl chain usually form
5
Chipot, Tarek & Klein: Modeling lipid membranes
direct micelles, whereas lipids characterized by a smaller head group and possibly two alkyl chains
tend to self–organize into inverse micelles.1
For lipids forming planar bilayer assemblies, the net charge borne by the head group plays a
noteworthy role in the self–organization process. Small, charged head groups show an interesting
tendency to associate by means of intermolecular hydrogen bonds, resulting in compact structures
with 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 intra
molecular charge pairs between the oppositely charged choline and phosphate groups.14
(a)
(c)
(b)
Figure 1: Polymorphism of lipid–water assemblies. (a) The cross–sectional area of the head group is
larger than that of the alkyl tail. In an aqueous environment, this specie forms direct micelles, which further
organize into hexagonal HI phases. (b) The cross–sectional area of the head group is smaller than that of
the 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 planar
bilayers, in the gel, Lβ 0 , phase (left) or in the liquid crystal, Lα , phase.
Aside from the nature of the lipid, external conditions, like the concentration, the temperature, the
pressure or the ionic strength of the solvent, strongly influence self–organization into a particular
structure. Extensive variables, for instance, can be used to control the transition between phases.
At low temperatures, lipid bilayers remain in the gel, Lβ 0 , phase, wherein the alkyl chains, mostly
in an all–trans conformation, are well ordered and exhibit a reduced mobility. At higher temperatures, the gel phase transforms into a liquid crystal, Lα , phase characterized by an increase of the
surface 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 chemical
nature of the lipid. For instance, it increases with the length of the alkyl chains, but it decreases
with the number of unsaturations.
Most cell membranes in vivo exist in the fluid, liquid crystal phase, barring a few cases, e.g.
stratum corneum specialized membrane.15 It is, therefore, not too surprising that, at the exception
Chipot, Tarek & Klein: Modeling lipid membranes
6
of a handful of simulations of lipid bilayers in the gel phase, most investigations have focused on
the 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 tools
for determining structures of lipid bilayers at an atomic resolution.16–19 A particularly pertinent
information supplied by diffraction experiments are density distributions,20 that can be deconvoluted in terms of atomic positions in the direction normal to the water–membrane interface, for
different types of atoms.
High–resolution x–ray diffraction experiments may offer additional, valuable information, that
can directly serve as a reference for computational studies. Such is the case of the surface area
per 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 the
large uncertainty in the values supplied by the literature.20 .
Whereas x–ray and neutron diffraction on multilayered samples have historically been a source
of high–resolution structural information of model membranes, neutron reflectivity has provided
unique 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 a
function of the scattering wave–vector transfer (Q). Recently it has been shown that it is also
possible to invert directly the reflectivity spectra to obtain SLD profiles.21 It is important to note,
however, that only the total SLD profile is determined. For more complex systems, atomistic modeling can provide valuable insight into such structures, thereby complementing the experimental
studies.22–24
Nuclear magnetic resonance (NMR) techniques are also used extensively to probe the molecular
organization in lipid membranes. Earlier on, 2 H NMR experiments on oriented lipid matrices
supplied lipid order parameters, against which the average orientational order along the acyl chains
calculated from simulations could be confronted. Today, thanks to the introduction of magic angle
spinning (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 as NMR experiments may also be used as
a possible source of comparison of dynamical properties against MD simulations. As will be
seen in what follows, the significant computational effort involved in atomic simulations of large
lipid–water assemblies limits, from a biological perspective, their length to relatively short times.
Short time–scale dynamics is yet amenable to MD, and the data determined by this approach can
7
Chipot, Tarek & Klein: Modeling lipid membranes
be confronted directly to scattering experiments26,4 , and, for instance, to nuclear Overhauser enhancement spectroscopy (NOESY) cross–relaxation rates27 , which probe motions occurring over
comparable time–scales.
3. Modeling lipid bilayers
In order to eliminate edge effects and to mimic a macroscopic system, simulations of lipid bilayers
consist of considering a small patch of lipid and water molecules confined in a central simulation
cell, and replicating the latter using periodic boundary conditions (PBCs) in the three directions of
Cartesian 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 a
multilamellar oriented lipid stack, similar to those deposited on a substrate (see Figure 2). The size
of the simulated sample results in artefactual, symmetry–induced effects and the impossibility to
witness collective phenomena like bending or splay motions that occur over length–scales above
the size of the cell.28,29,20
If needed, one may render a more biologically or physically meaningful picture, consistent with experimentally 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–scales
exceeding 10 ns (see Figure 2).
10−8 s
(c)
10−11s
(a)
10+4 s
(b)
10−9 s
10−6 s
Figure 2: Left: Small patch of lipid bilayer replicated by PBCs. Right: Characteristic time–scales in lipid
bilayers. Overall, motions occur on times that range between a few ps for the separation of sn–1 and sn–2
alkyl chains, to a few hours for the so–called flip–flop, where in a lipid unit migrates from one leaflet to the
other.
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Chipot, Tarek & Klein: Modeling lipid membranes
3.1. Choice of the thermodynamic ensemble
From a technical perspective, the simplest thermodynamical ensemble for simulating lipid–water
assemblies 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 experimental value and leave an appropriate head space of air in contact with the water lamellae, above and
below the membrane. Whereas this protocol is ad hoc in the case of a simple, homogeneous lipid
bilayer, one may legitimately wonder how it will perform when additives — e.g. small solutes to
large proteins, are introduced into the membrane or in its vicinity. A better adapted thermodynamic
ensemble should then be employed to allow the participating lipid chains to relax in response to
the modification of the surface tension imposed by the additive. A very tempting solution consists
in turning to the isobaric–isothermal, (N, P, T ), ensemble, that makes use of rigorous barostats
and 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 monotonously with the surface area, as the system minimizes the contact area between the two liquids. In
the case of lipids interacting with water — viz. typically a hydrated lipid bilayer, the picture is
somewhat more intricate. Just like for a mixture of oil and water, by virtue of the hydrophobic
effect, the free energy increases with the surface area. This is evidently not the sole contribution
governing the behavior of lipid bilayers, the surface area of which would be minimized regardless
of the temperature, thereby forcing the system in the gel, Lβ 0 , 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 the
contrary, 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 lateral
pressure, Pk , be strictly equal to the pressure normal to the water–lipid interface, P⊥ :
γ=
Z h
i
P⊥ − Pk (z) dz = 0
(1)
This important result, which was expected for a self–organized system, prompted a host of authors
to 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, it
should be kept in mind that in atomic simulations one model patches of finite size. Feller and Pastor put forward that a finite surface tension should be introduced to compensate for such finite–size
effects that eliminate the possibility to observe collective phenomena like undulations over significant length–scales,32,33 as in ripple, P0β , phases, for instance. Tieleman and Berendsen argued
that in the systems they investigated, the dependence of the surface tension with the surface area
was marginal.34 Lindahl and Edholm later showed that an applied surface tension in the order of
10 mN/m would correct for large fluctuations in the surface area per lipid unit that are witnessed
Chipot, Tarek & Klein: Modeling lipid membranes
9
in simulations of lipid–water assemblies of limited size.35 One thing is certain: In atomic simulations of lipid bilayers, Pk and P⊥ are anticipated to vary differently on account of the anisotropy
of the environment. It is, therefore, strongly recommended to adopt an algorithm that generates
the (N, P, T ) distribution, so that the dimensions of the simulation cell are rescaled independently
in the x, y (in-plane) and in the z–directions.36,37,2
3.2. The potential energy function
In atomic statistical simulations of membranes, all atoms pertaining to the system are treated
classically as point masses, which, in the harmonic approximation, are connected to each other
by means of springs. In some instances, for the sake of computational effort, certain groups of
atoms, 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–water
assemblies made use of the available multi–purpose force fields, often aimed at the modeling of
solvated 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 a
fortuitous cancellation of errors due to the conjunction of inadequate parameters and excessively
short runs.
In the following years, it was realized that a specific potential energy function should be employed
to mimic accurately the properties of lipids, like the subtle trans–gauche equilibrium in the alkyl
chains. A dearth of efforts was and is still invested to improve the representation of lipids and
surfactants by means of an appropriate parameterization of the force–field contributions likely to
affect 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 ascribed
to trans–gauche defects, the standard low–order Fourier series that is often used in conventional
macromolecular force fields, was replaced by the more sophisticated Ryckaert–Bellemans torsional potential.44
In addition, correct packing of the alkyl chains depends to a large extent on the quality of the
van der Waals parameters utilized. One of the underlying assumptions made for the design of
force fields is the transferability of these parameters between molecules — e.g. the van der Waals
radius and well depth of an aliphatic sp3 carbon should be the same regardless of the chemical
environment. The interaction parameters of the united methylene and methyl groups were originally derived from statistical simulations of short hydrocarbons like n–butane, as is the case of
the O PLS force field.45 The transferability hypothesis has proven to be inadequate when handling
long alkyl chains, prompting a number of authors to reoptimize van der Waals interactions based
on simulations of large hydrocarbons like pentadecane.31
Determination of net atomic charges for lipids and surfactants from sophisticated quantum me-
Chipot, Tarek & Klein: Modeling lipid membranes
10
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 satisfactory solution among the arsenal of approaches available to the modeler.46 Yet, as has been
demonstrated, point charges are inherently conformation–dependent,47 thus making the derivation
of a unique set of charges representative of all possible conformations questionable. To circumvent the difficulties connected to the size of the molecules, it has been proposed to derive the net
atomic charges as independent fragments, that are ultimately pieced together. This scheme, although tempting, should be considered with great care if local charges are delocalized over large
spatial extents.
3.3. Intermolecular interactions
As has been mentioned previously, physically and biologically realistic simulations should involve
a sufficiently large number of lipid and water molecules to minimize finite–size effects. Much of
the computation effort involved in atomic simulations of lipid–water assemblies lies in the evaluation of pairwise interactions, the number of which increases dramatically with the number of
particles in the system. Based upon the assumption that intermolecular interactions decay with the
distance, earlier studies have employed a cut–off sphere, beyond which the interactions are truncated. This approximation is expected to be ad hoc for the short–range, van der Waals contribution.
The use of a brute, finite spherical cut–off for truncating the short–range van der Waals interactions 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 for
both the energy and the pressure,48 based on the classical formulae utilized for Lennard–Jones
fluids.49
For Coulomb interactions, the range of which varies in rn , where n ≤ 3,49,2 truncation becomes
particularly 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 unsuitable. Probably the most accurate approach for handling the long–range nature of electrostatic
interactions in spatially replicated simulation cells is solving the Poisson equation. The Ewald
approach,50 that decomposes the conditionnally convergent Coulombic sum over periodic boxes
into two rapidly decaying contributions evaluated respectively in the direct and reciprocal spaces
is the most used method. Formally, the computational effort involved in this method scales as
(N 2 ), thus making statistical simulations of large ensembles of atoms particularly costly. This
effort can be reduced, scaling down the calculation to (N ln N ), by solving the Poisson equation
numerically on a grid of points, over which the position of the particles are interpolated. Such
a scheme constitutes the central idea of algorithms like particle–mesh Ewald (PME) or particle–
particle–particle–mesh (P3 M).51 For completeness, while to our knowledge, it has not been yet
applied in membrane simulations, it is worth mentioning the fast multipole approach, a method alternative to Ewald summation, that treats long–range interactions in a rigorous fashion, and scales
Chipot, Tarek & Klein: Modeling lipid membranes
11
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 of
the simulated molecular assembly may be further reduced by taking advantage of recent advances
in the MD methodology. Considering that the different degrees of freedom involved in the system
relax over distinct time–scales, it is not necessary that the corresponding force contributions be
evaluated 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 updated
with different frequencies.55 In conjunction with constraint algorithms like SHAKE or RATTLE,56
that virtually eliminate the vibrations due to hard degrees of freedom it is possible to explore
large regions of the phase space for a lesser computational effort, thus making long simulations of
large lipid–water assemblies somewhat more affordable — the reader is referred to the chapter of
Tuckerman and Martyna dedicated to integrator and ensembles in statistical simulations.
4. Atomic simulations of lipid membranes
Traditionally, phospholipids have served as models for investigating in silico the structural and
dynamical properties of membranes. From both a theoretical and an experimental perspective,
zwitterionic phosphatidylcholine (PC) lipids constitute the best characterized systems. Hydrated
DMPC 13,57 and dipalmitoylphosphatidylcholine 58,34,31,59–61 ( DPPC ) bilayers have been so far probably the most extensively surveyed lipid membranes. Yet, on account of their intrinsic limitations — viz. the short alkyl chains in DMPC and the temperature of Lβ 0 to Lα phase transition in
DPPC , above physiological conditions — several authors have turned to biologically more relevant lipids like palmitoyloleylphosphatidylcholine62,63 (POPC), in particular for examining membrane proteins in a realistic environment, and lipids based on mixtures of saturated/polyunsaturated
alkyl chains (SDPC, 18:0/22:6 PC).64,43 A variety of alternative lipids, featuring different, possibly charged, head groups, have also been explored — e.g. DLPE65,13 (DLPE), dipalmitoylphosphatidylserine66,67 (DMPS) or glycerolmonoolein68,30 (GMO). In several cases, however, the modeler is faced with an absence of experimental data to which the results of atomic simulations can
be confronted.
Bilayers built from PC lipids, nonetheless, represent remarkable test systems not only to probe the
methodology, but also to gain additional insight into the physical properties of membranes. In this
section, the derivation of these properties from MD trajectories and how a bridge with experiment
can be established will be detailed.
Chipot, Tarek & Klein: Modeling lipid membranes
12
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 the DMPC
units in a bilayer arrangement is remarkably broad. This is clearly seen in the number density
profiles computed from the MD trajectory — an analysis along the direction normal to water–
membrane interface of the in–plane densities of lipid and water atoms. A striking feature emerging
from these distributions is the penetration of water far in the head–group region. The farthest
extent of water molecules roughly coincides with the ester moieties of the lipids. The width of
the interfacial region, on the order of 8 to 10 Å for a fully hydrated DMPC bilayer highlights
the significant static and dynamic roughness of the membrane surface68,69 , therefore, refining the
traditional textbook picture, like that of Figures 1 and 2.
Figure 3: Left: Snapshot taken from an MD simulation of a fully hydrated DMPC bilayer. Methylene and
methyl carbon atoms of the alkyl chains are shown in grey and pink, respectively. Ester carbon and oxygen
atoms are shown in blue and purple, respectively. Phosphate phosphorus and choline nitrogen atoms are
shown in orange and green, respectively. Note the protrusion of head groups that results locally in a rough
water–membrane interface. Right: Density distributions for selected groups of atoms in a fully hydrated
DMPC bilayer examined at 303 K (from reference [70]).
Interestingly enough, phosphate and choline groups lie approximately at the same depth in the
bilayer, indicating that head groups are rather oriented in the plane of the bilayer. The average orientation of the head–group P — N bond dipoles with respect to the normal of the water–membrane
interface arises around 700 , pointing towards the aqueous medium, albeit the orientational distribution is remarkably wide, and depends upon the temperature and, as expected, the potential
energy function utilized.14 In addition, the level of lipid hydration has been shown to play a noteworthy role in the orientation of the head groups.9 Under any circumstances, it is crucial that the
13
Chipot, Tarek & Klein: Modeling lipid membranes
slow reorientation of the lipid head groups be considered when interpreting results from short MD
trajectories. Estimates from single–molecule anisotropy imaging for fluorophore–tagged POPC
molecules71 indicate a rotational diffusion coefficient of ca. 0.7 rad2 /ns, slightly below estimates
from MD simulations,72 suggesting that sampling of the whole rotational space for each molecule
would necessitate over few tens of a nanosecond.
The information provided by the density distributions can be confronted directly to x–ray and neutron diffraction measurements17 (considering respectively the atomic scattering length densities,
or the electron densities). The MD trajectories can further be used in conjunction with the scattering experiments in order to refine the data by, for instance including fraction volumes extracted
from the simulation.73
4.1. 2. Lipid tail conformation
Deuterium quadrupole splitting measured by 2 H NMR on non–oriented samples of membrane
preparations, is mainly determined by the average conformation of the phospholipid molecules,
and, as such, supplies valuable structural information about the system. Order parameters can be
derived from MD trajectory, and can be expressed as a tensor, the elements of which write:74
Sαβ =
1
h3 cos ϕα cos ϕβ − δαβ i
2
(2)
Here, ϕα is the angle formed by the α–th molecular axis and the normal to the water–bilayer
interface, and h· · ·i is an ensemble average over all lipid chains. In most circumstances, based on
symmetry relationships, it is assumed that the order parameters, SCD , for an alkyl chain bearing
deuterium labels can be expressed as:
SCD =
E
1 D
3 cos2 θ − 1
2
(3)
where θ 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. In
general for saturated lipids, |SCD | exhibits a plateau value at ca. 0.2 for the upper chain segments.
Force fields of the new generation reproduce quite well these order parameters, barring small
discrepancies for the second carbon atom of the alkyl chains.
Further analysis of MD trajectories may be aimed at extracting additional information from the
NMR experiments. For instance, one may refine those methods targeted at obtaining such quantities
as the average chain length or the surface area per molecule.76 Another study exemplifies the
successful combination of MD simulation with experiment to probe alkyl chain packing in lipid
membranes. Such is the case of infrared (IR) data, that have been reinterpreted to estimate the
concentrations of gauche–gauche, trans–gauche and trans–trans conformational sequences in a
DPPC bilayer.77 .
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Chipot, Tarek & Klein: Modeling lipid membranes
0.25
-SCD
0.20
0.15
0.10
0.05
2
3
4
5
6
7
8
9
10
11
12
13
14
carbon atom number
Figure 4: Order parameters of a fully hydrated DMPC 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 or
pair correlation functions (RDFs):
gij (r) =
hNj (r; r + δr)i
4π%j
Z
r2 dr
(4)
where Nj is the number of particles j at a distance from i comprised between r and r +δr and %j is
the density of particles j. 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 coordination
number of site i — e.g. PC head groups, it seems far more appropriate to merely evaluate hNj (r; r+
δr)i as a function of the separation r and determine its value at the first minimum of a qualitative
RDF 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 anisotropic, compared to the bulk aqueous medium. This can be shown by measuring the average cosine
of 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 distance
characteristic of the phosphate groups, emphasizing the orienting power exerted by this moiety on
the surrounding aqueous environment. The preferential orientation of the dipole moment borne by
the water molecules is at the origin of the vocabulary “dipole potential”, that has been employed
Chipot, Tarek & Klein: Modeling lipid membranes
15
extensively to denote the electrostatic potential across the water–membrane interface.79,80 This
conspicuous ordering of water molecules was recently directly evidenced using coherent anti–
Stokes Raman scattering microscopy. 81
In a number of in silico investigations, the electrostatic potential has been estimated from the
knowledge of the charge density. In the spirit of atomic density distributions, charges are accumulated as a function of their position along the direction normal to water–bilayer interface. The
negative of the first integral of the charge density yields the electric field. In turn, integral of the
field provides the electrostatic potential. Not too surprisingly, the resulting “dipole potential” inherently depends upon the choice of the potential energy function and should, thus, be interpreted
cautiously.4,31,57
4.2. Dynamics
The increasing level of interaction between experimental studies and numerical simulations of
lipid bilayers evidenced in the previous section also holds for the dynamics of lipid bilayers.
Feller et al.27 have used MD simulations to analyze NOESY cross relaxation rates in lipid bilayers.
Magnetic dipole–dipole correlation in such systems occurs over a variety of time scales and depends upon the probability of close approach for proton–proton interactions. The relaxation rates
have been calculated directly from a 10 ns MD simulation of DPPC. Fitting the autocorrelation
functions yields characteristic correlation times and weight factors that determine the relative contributions of the individual type of motions. Combining simulations and experiments, relaxation
rates may, therefore, be assigned to various motions — viz. less than 1 ps for chemical bond vibrations, 50–100 ps for trans–gauche isomerization, 1–2 ns for molecular rotation and wobble, and
beyond 100 ns for lateral diffusion.
A model for the dynamics of individual lipid molecules has also been proposed based on a thorough comparison of simulation data and experimental measurements of the 13 C NMR T1 relaxation
in DPPC alkyl chains.82 Employing Brownian dynamics and MD simulations associated to fits of
experimental data, it was found that lipid molecules confine themselves into a cylinder within the
100 ps time scale, and wobble in a cone–like potential on the nanosecond time scale.
A similar model for lipid dynamics has emerged from an MD study aimed at interpreting inelastic
neutron scattering (INS) data. One particular aspect of such experiments, probing the motion of
individual hydrogen nuclei — i.e. self correlation of single particle, is that they are space– and time
resolved. In the case of DPPC bilayers, a good agreement between simulations and experiments
probing the 100 ps time scale is attained.83 The analysis corroborates the fact that the motion of the
center of mass and the internal motions of lipid molecules are decoupled. Moreover, the former is
well described as a diffusion in a confined space, i.e. a cylinder. A refined picture of the internal
Chipot, Tarek & Klein: Modeling lipid membranes
16
dynamics arising from the simulation shows that protons of the alkyl chains move according to a
chain defect model, where kinks or chain defects form and disappear randomly — i.e. stochastic
model — along the lipid tail, rather than diffuse along the chain.
Collective dynamics of lipid bilayers have also been examined carefully as simulations over increasingly significant time scales and length scaled are feasible. Large systems involving 1,024
lipid molecules studied over 10 ns led to the direct observation of bilayer undulations and thickness fluctuations of mesoscopic nature.35 Continuum properties such as bending modulus, surface
compressibility and mode relaxation times were calculated and agreed nicely with experiment.
Several processes occurring at different length scales were identified. The undulatory motions
could be separated in two regimes — one involving more than 50 lipids, that can be ascribed to
mesoscopic undulations, and the other, involving less than 25 lipids, that is attributed to collective 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 protrusion
modes over shorter length scales.
Shorter wavelength collective dynamics may be probed using coherent inelastic, viz. neutron or
x–ray, scattering. Density fluctuations of length scales comparable to the interlipid distance are
believed 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 the
gel, Lβ 0 , and the liquid crystal, Lα , phases.84 The results support the applicability of generalized
hydrodynamics to describe the motion of carbon atoms in the hydrophobic core, thus allowing the
modeler to extract key–parameters, such as sound mode propagation velocity, thermal diffusivity
and kinematic longitudinal viscosity.
4.3. Modeling transport phenomena
Models of lipid bilayers have been employed widely to investigate diffusion properties across
membranes through assisted and non–assisted mechanisms. Simple ions, e.g. Na+ , K+ , Ca2+ or
Cl− , have been shown to play a significant role in the cell machinery, in particular at the level of
intercellular communication. In order to enter the cell, the ion must preliminarily permeate the
lipid bilayer that acts as a rampart towards the cytoplasm. Wilson and Pohorille have investigated
the passive transport of Na+ and Cl− ions across a lipid bilayer formed by glycerolmonoolein
units, which undergoes severe deformations as the ions translocate across the water–membrane
interface. This process is accompanied by thinning defects and the formation of water fingers that
ensure 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–membrane
partition coefficients of small drugs, in strong connection with the so–called blood–brain barrier
— the ultimate step in the de novo design of pharmacologically active molecules. Diffusion of
Chipot, Tarek & Klein: Modeling lipid membranes
17
small, organic solutes in lipid bilayers was examined for a variety of molecular species ranging
from benzene86,87 to more complex anesthetics.88–90 Yet, access to partition coefficients by means
of statitistical simulations implies the determination of the underlying free energy behavior along
the direction normal to the interface.91 In the specific instance of inhaled anesthetics, an analysis
of the variations of the free energy for translocating the solute from the aqueous medium into
the 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 to
correlate 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 the
interior of the membrane rationalizes the use in cells of specific transmembrane channels, pumps
or carriers that facilitate while controlling selectively the passage of ionic species across the lipid
bilayer.94 Recent complete reviews of the theoretical developments and simulation capabilities in
ion channels modelling can be found in references [95] and [96]. Here, we briefly describe some
of the complex systems examined hitherto.
Gramicidin A, a prototypical channel for assisted ion transport, has been the object of thorough
analyses from both experimental and theoretical perspectives. Dimerization of individual protein units results in membrane–spanning channels suitable for ion conduction. MD simulations of
gramicidin A embedded in hydrated lipid bilayers, e.g. DMPC, were able to reproduce the structural features observed experimentally.97 Such studies have clearly shown that important questions
related to ion selectivity, ion binding, gating and proton transfer mechanisms may be addressed
with some confidence.
Internal arrangement of water molecules in single–file chain of water molecules, characteristic in
complex transporters,98 was also witnessed in a somewhat more rudimentary, synthetic channel
formed by stacked cyclic peptides of alternated D– and L–chiralities (see Figure 6).70 Such nanotubes have been recognized to modify in a selective fashion the permeability of cell membranes
and are envisioned to act as potent therapeutic agents in response to bacterial resistance.99 Aquaporins, membrane channels ubiquitous to most living species controlling the water contents of the
cells, have also focused much attention lately. They are formed of tetramers that organize to facilitate the transport of water, and possibly other small solutes, across the lipid bilayer. The resulting
water pores remain, however, impervious to the passage of small ions to ensure a proper conservation 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 motivated
a large number of realistic simulations taking into account the lipidic environment studies aimed
at deciphering the underlying complex conduction mechanism.
Chipot, Tarek & Klein: Modeling lipid membranes
18
Figure 5: Snapshot taken from an MD simulation of a synthetic channel formed of cyclic peptides of
alternated D– and L–chiralities, embedded in a fully hydrated DMPC bilayer. Color coding of the atoms is
identical to that in Figure 3. Note the antiparallel β–sheet like conformation of the nanotube spanning the
membrane. Within a few hundreds of ps, a single–file chain of water molecules is established in the hollow
tubular 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, zwitterionic
PC lipids. Membranes, however, are infinitely more complex and consist of an heterogeneous assembly of different lipids, either charged or not, carbohydrates and proteins. Approaching the fine
detail of the biological picture by incorporating in atomic simulations chemical species of different natures is evidently the direction towards which the modeler is evolving. From a modeling
perspective, the influence of cholesterol,101–105 and more generally sterols,106 on the structure and
dynamics of lipid bilayers has attracted a lot of attention in recent years. Although the limited
sampling in some simulations calls into question the conclusions reached by the authors, cholesterol is shown to increase the order parameters of the alkyl chains while decreasing their tilt angle
with respect to the normal to the water–membrane interface, in qualitative agreement with experiment.107
Aside from transporters and channels that assist the transport of chemical species across lipid
bilayers, a vast array of key–cellular functions are accomplished by proteins that interact with the
membrane, either spanning the latter, or bound to its surface.108 Yet, interfacial and transmembrane
proteins generally play distinct roles in the cell machinery, albeit the frontier between these two
classes of proteins remains somewhat fuzzy. A number of proteins, for instance, are only partially
buried in the membrane — e.g. melittin or alamethicin, the insertion of which is conditioned by
the transmembrane electric field.109–112
Chipot, Tarek & Klein: Modeling lipid membranes
19
The strength of in silico experiments is to provide glimpses into the atomic detail of biological
membranes that conventional experimental techniques cannot capture. Of particular interest is
the molecular interplay that govern membrane–protein association, accessible through large–scale
atomic simulations. MD simulations illuminated, for instance, how the presence of a protein perturbs the structure of the lipid membrane. For example, the helices of the Influenza A M2 channel
tilt in a DMPC bilayer to maximize membrane–protein hydrophobic contacts.113,114 In the case of
gramicidin A, key–residues located in the head–group region have been shown to stabilize the
channel in the membrane.97
The influence of the protein on the lipid bilayer can be viewed as the subtle balance between
hydrophobic and hydrophilic contributions that, in principle, can be captured by MD simulations.
Differences in the order parameters of lipid units adjacent to the protein and far from it have led to
the concept of “boundary lipids”. In a vast number of instances, among which the Mycobacterium
tuberculosis MscL channel,115 the Influenza A virus M2 protein,116 and the Escherichia coli OmpF
trimer,117 it was observed that the membrane protein induces an increasing disorder of the lipid
alkyl chains in its neighborhood. In sharp contrast, alkyl chains close to the transporter gramicidin
A 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 that trans–gauche
equilibria in lipid chains are dictated by the very nature of the membrane protein. Yet, as was
shown recently,70 drawing definitive conclusions based on limited simulation lengths may turn out
to give a distorted vision of the actual behavior of the lipid bilayer. In principle, exceedingly short
simulations do not permit the complete relaxation of lipid chains in the vicinity of the protein, and
should, thus, be interpreted cautiously.
The close adequation between the thickness of the lipid bilayer and the length of the hydrophobic
segment 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 proteins
with the lipid environment, atomic statistical simulations may contribute to advance our understanding of the underlying physical principles that govern the function and structure of membranes.119 In the light of a series of experimental investigations on model peptides embedded in
PC membranes with alkyl chains of increasing length, it was found that if the hydrophobic thickness of the peptide is greater than that of the bilayer, the latter becomes thinner, and vice versa.120
A similar phenomenon was observed recently in the MD simulation of a single peptide nanotube
inserted in a hydrated DMPC bilayer.70 The hydrophobic thickness of the membrane adjusts itself
as the synthetic channel tilts concurrently to adapt to its host lipid environment. Whereas the
so–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 observed
recently for gramicidin A, minute changes in the length of the lipid alkyl chains — viz. from
the 18–carbon oleyl– to the 20–carbon eicosenoylphosphatidylcholine, switch the protein from
a stretch–activated to a stretch–inactivated channel. Symmetrically, the hydrophobic mismatch
Chipot, Tarek & Klein: Modeling lipid membranes
20
may alter the phase behavior of the membrane, as demonstrated in the case of WALP peptides
that 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. The
choice of the lipid unit for a given protein may turn out as a genuine leap of faith if attention has
not been paid to the possible imbalance in the hydrophobic thicknesses of the membrane and the
protein, likely to render a physically unrealistic picture of the assembly. When devised appropriately, atomic simulations can, nonetheless, shed new light on the nature of the protein–membrane
interplay, by allowing the modeler to not only visualize, but also possibly quantify the strength
of the participating interactions. Of particular interest, the non–covalent chemical bonds formed
by L–Trp residues and acceptor moieties of the head–group region have been recognized to act
as anchoring points of the protein into the lipid bilayer.123,124 As has been shown in the case of
gramicidin A, the presence of several L–Trp amino acids at the level of the lipid head groups is
expected 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 to
improve our knowledge of these very complex systems from both a structural and a dynamical
point of view. It is also obvious that the successes of pioneering, tantalizing investigations, which
not only ignited the field of lipid simulations, but were also rapidly fueled by many studies on
larger assemblies, often reflected as much good fortune as they did science. Yet, major advances
on both the hardware and the software, algorithmic fronts progressively allowed the modeler to
tackle systems of increasing complexity over time–scales compatible with the physical, chemical
and biological reality. Among these advances, the development of specific methods for performing
the simulation in apt thermodynamic ensembles, the improvement of potential energy functions
targeted at the specific modeling of lipid–water assemblies, and the continuous decrease of the
price/performance ratio of modern computers have helped pushing back the intrinsic limitations
of MD simulations. More recent studies have demonstrated that simulations at least an order of
magnitude longer than those reported when the field was only in its infancy, were required to
obtain reliable and reproducible results.125
Simulation of lipid–water systems still constitutes a research area seething with excitement. The
development of all the ingredients to investigate in silico lipid bilayers with full confidence opens
new perspectives, in particular on the biological front, and should rapidly allow the modeler to use
lipids in a routine fashion, just like any other solvent. In this spirit, theoretical studies of membrane
proteins in a realistic environment should continue to flourish in the near future. Unfortunately, as
the level of sophistication of atomic simulations increases, together with the available computational power, so does the ambition of the modeler, attempting to deal with molecular systems yet
Chipot, Tarek & Klein: Modeling lipid membranes
21
even more complex, both in terms of size– and time–scales. This explains the current teeming activity in the development of approximate schemes that could serve as alternatives to a full–atomic
description 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 implicit
solvation.8 Since the seminal work of Onsager on continuum electrostatics,126 the temptation to
represent explicit surroundings by a simple dielectric medium for a myriad of chemical systems
has been the object of tremendous interest. Modeling the complexity of lipid bilayers by means
of a continuum description has been used, for example, to investigate the insertion of α–helical
peptides in a membrane,127 or the interaction of a small toxin with the latter.111 Results of continuum electrostatics simulations, which are based on solving the Poisson–Boltzmann equation
numerically, are, in general, in qualitative agreement with atomic simulations. Yet, not too unexpectedly, this approximate description cannot capture subtle, specific interactions that govern the
stability of the solute — e.g. a short peptide, at the water–membrane interface. As was underlined
recently by Lin et al., the reproduction of membrane dipole potentials based on a sole continuum
electrostatics representation is usually erroneous, but can be significantly improved by inclusion
of explicit layers of water molecules near the head–group region.128
Aside from implicit solvation approaches, the use of coarse–grained representations, wherein each
lipid unit is described by a limited number of interacting sites, is probably the most promising. The
underlying assumption that the formation of a lipid vesicle is a sufficiently robust process to be
simulated by simplified models of lipids was ascertained recently by Marrink and Mark through
a study of the aggregation of DPPC units into small unilamellar vesicles.129 By and large, the
strength of coarse–grained models resides in their ability to make simulations self–assembly processes substantially more affordable than conventional all– or even united–atom models.130 The
level of representation offered by this alternative is, in sharp contrast, incompatible with the fine
description of specific interactions of the participating lipid units with small solutes, like anesthetics. It is obvious that much attention should still be paid in the optimization of the potentials of
these rudimentary models to warrant compatibility with a full–atomic descriptions.
Chipot, Tarek & Klein: Modeling lipid membranes
22
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