Tesi di Dottorato di Ricerca in Fisica - UniTrentosega/thesis/phd_sega.pdfTesi di Dottorato di...

Universita degli Studi di Trento

Facolta di Scienze Matematiche

Fisiche e Naturali

Tesi di Dottorato di Ricerca in FisicaPh.D. Thesis in Physics

Structural and dynamical

properties of a gm3 bilayer

studied by computer simulation

Marcello Sega

Dottorato di Ricerca in Fisica, XVII Ciclo

23 Febbraio 2005

To my parents,

Germana and Roberto

Contents

Introduction 1

1 Gangliosides and their aggregates 3

1.1 Structure and Function of Gangliosides . . . . . . . . . . . . . . . 4

1.2 Amphiphiles Supramolecular Aggregates . . . . . . . . . . . . . . 9

1.3 GM3 Vesicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Molecular Dynamics Simulation 17

2.1 Details on the Simulation Package . . . . . . . . . . . . . . . . . 18

2.2 Force Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Changes to the gromos87 Force Field . . . . . . . . . . . . . . 26

2.4 GM3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 System Preparation and Equilibration 35

3.1 Set-up of the starting configuration . . . . . . . . . . . . . . . . 36

3.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 The equilibration phase . . . . . . . . . . . . . . . . . . . . . . . 38

4 Structural Properties 45

4.1 Structure Determination by X-Ray Scattering . . . . . . . . . . . 45

4.2 Mass Density Profiles . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Orientational Order of the Lipid Chains . . . . . . . . . . . . . . 53

4.4 Orientational Order of the Headgroups . . . . . . . . . . . . . . 56

4.5 Headgroup Hydration . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6 Electric Field and Water Orientational Order . . . . . . . . . . . 60

5 Dynamical Properties 67

5.1 Ganglioside Headgroup Dynamics . . . . . . . . . . . . . . . . . . 67

5.2 Ceramide Rotational Dynamics . . . . . . . . . . . . . . . . . . . 71

v

vi CONTENTS

5.3 Water Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Summary and Conclusions 91

Appendix: GM3 topology 95

Acknowledgements 109

Nomenclature

Chemical compounds

aliphatic:

Compound that contains carbon atoms linked in open or closed

chains without the presence of aromatic structural units. chains.

amide:

Any organic compound containing the group –CON–. The amide

bond has the peculiarity of being planar and rotations around it

are not allowed at room temperature.

amphiphile:

A molecule that presents both hydrophobic and hydrophilic char-

acter.

carbohydrate:

Synonymous of sugar or saccharide; the hystorical name of carbo-

hydrate derives from the chemical formula, which for some of them

reads C6(H2O)6.

ceramide:

Sphingoid base linked to a fatty acid via an amide bond.

cerebrosides:

Historical name for a class of glycosphingolipids.

chair conformation:

One of the possible conformations, like the inverted chair and the

boat conformation, of a 6-ring system. See Section 2.3.2.

vii

viii Nomenclature

counterion:

Mobile ion of opposite charge with respect to a macroion.

enzyme:

Protein that acts as catalyst in specific biochemical reactions.

exo-anomeric effect :

Rotational preference of the gauche conformation of the exocyclic

oxygen in sugar rings, see also gauche effect.

gauche effect :

Lowering of the energetic rotational barrier between the anti and

gauche conformation.

glycocalyx :

Carbohydrate layer that covers the cell surface.

glycoconjugates:

Family of organic molecules composed by a carbohydrate moiety

covalently linked to a non sugar one.

glycolipids:

Subgroup of glycoconjugates containing one or more monosaccha-

ridic residue, linked via a glycosidic linkage to a lipid moiety.

glycosidic linkage:

A link between two sugar units, made of a C–O–C bond.

glycosphingolipids:

Class of glycolipids where the lipid moyety is a ceramide or a sph-

ingoid.

glycosyltransferase:

Enzyme that catalyzes the transfer of sugar residues during the

degradation and biosynthesis of some organic compounds, such as

glycolipids.

Golgi apparatus:

Center of production and storing of molecules synthesized in the

cell.

ix

hydrocarbon:

Compound made of carbon and hydrogen atoms.

hydroxymethyl :

A methyl group with hydroxide replacing an hydrogen atom.

lipid :

Any oily organic compound insoluble in water.

methyl :

The univalent radical –CH3 derived from methane.

neurolipidoses:

Set of neurodegenerative diseases presenting an anomalous concen-

tration of some lipid in the neural tissue is reported.

phospholipids:

Class of amphiphilic molecules composed of fatty acids, phosphoric

acid and a nitrogenous base.

sialic acid :

General name of neuraminic acid derivatives, including N-acetyl

neuraminic acid (NANA, or Neu5Ac).

sialo-glycosphingolipids :

Class of glycosphingolipids bearing a sialic acid residue, ganglio-

sides.

sphingoid :

Long-chain aliphatic amines, containing two or three hydroxyl groups,

and often a trans double bond. Sphingoid are structural units of

sphingolipids.

stearic acid :

The fully saturated fatty acid, CH3(CH2)16COOH.

stereoisomers:

Isomers having the same atoms bonded to each other but differing

in the spatial arrangement of them.

x Nomenclature

surfactant :

Surfact-active agent, a substance capable of reducing the surface

tension of a liquid in which is dissolved.

zwitterionic :

Neutral compound characterized by the presence of two or more

oppositely charged groups.

Abbreviations

DOPC dioleoyl-phosphatidylcholine.

msd Mean square displacement.

pme Particle Mesh Ewald.

saxs Small angle X-ray scattering.

waxs Wide angle X-ray scattering.

Cer Ceramide.

DPPC dipalmitoyl-phosphatidylcholine.

Gal Galactose.

GalNAc N-acetyl galactosamine.

Glc Glucose.

GSL glycosphingolipids.

Neu5Ac N-acetyl neuraminic acid.

spc simple point charge water model. See Section 2.2.2.

Introduction

Alice thought it would never do to have an argument at

the very beginning of their conversation, so she smiled

and said, ’If your Majesty will only tell me the right way

to begin, I’ll do it as well as I can.’

L. Carrol, Through the Looking–Glass.

In recent years large supramolecular aggregates — such as lipid bilayers — have

been the subject of numerous investigations by means of molecular dynamics

simulation at full atomistic detail. The interest in lipid membranes stems pri-

marily from their fundamental role for the conformation and activity of the living

cell. However, almost all the simulations performed until now, have dealt with

phospholipids, which are the main constituent of the cell membrane. Little atten-

tion has been devoted to the simulation of aggregates of other surfactants that

are likewise present in the cell membrane, such as gangliosides. Despite ganglio-

sides share with phospholipids the general features of surfactant molecules, the

saccharidic and acidic nature of the extended headgroup of gangliosides leads to

uncommon and interesting physical properties of their aggregates.

This thesis reports the results of a two-years work aimed at the realization

of the first molecular dynamics simulation at atomistic detail of a ganglioside

aggregate, namely a fully hydrated bilayer of GM3 molecules, as well at the

structural and dynamical characterization of the system. The thesis is organized

as follows: The first chapter presents a brief introduction to gangliosides, de-

scribing their general molecular composition and principal implications in the

biophysical processes of the cell. The process of supramolecular aggregation is

also described, with emphasis on the formation of GM3 vesicles. The second

chapter is devoted to the description of the molecular dynamics technique and

to the peculiarities of the force field employed to perform the simulation of the

1

2 Introduction

GM3 bilayer. The protocol applied to generate the starting configuration is the

subject of the third chapter, together with the description of the approach to

equilibrium. In chapters 4 and 5 we present the results regarding the structural

and dynamical properties of GM3 molecules, surrounding water and counterions.

Finally, conclusions are drawn in the sixth chapter.

Many of the results described in this thesis have been published in or sub-

mitted to interational journals:

M. Sega, R. Vallauri, P. Brocca and S. Melchionna, Molecular Dynamics Simu-

lation of a GM3 Ganglioside Bilayer, J. Phys. Chem., B108(52); 20322-20330

M. Sega, R. Vallauri, and S. Melchionna, Diffusion of water in confied geometry:

the case of a multilamellar bilayer, submitted

1.

Gangliosides and their aggregates

53‡‡†305))6*;4826)4‡.)4‡);806*;48†8¶60))85;1‡(;:‡8

†83(88)5*†;46(;88*96*?;8)*‡(;485);5*†2:*‡(;4956*2

(5*-4)8¶8*;4069285);)6†8)4‡‡;1(‡9;48081;8:8‡1;48†8

5;4)485†528806*81(‡9;48;(88;4(‡?34;48)4‡;161;:188;‡?;

E. A. Poe, The gold bug.

In the late 1930s the german chemist Ernst Klenk was investigating ganglia cells

of patients presenting the Tay–Sachs syndrome, a rare, inherited neurodegnera-

tive disease. Like in the case of other neurolipidoses, in Tay–Sachs disease Klenk

discovered an unusually large presence of lipids, namely a previously unknown

class of glycolipids, which presented an acidic nature [1]. These new compounds

were named by Klenk gangliosides after their origin tissue and in analogy to

the cerebrosides, and their acid moiety was called neuraminic acid. Later it

became clear that the neuraminic acid, which is shown in Figure 1.1, was the

same compound found by G. Blix et al. in 1952 and called sialic acid [2].

After the discovery of gangliosides, which can be dated for convenience 1942

[3], some years elapsed before reaching an understanding of the heterogeneity of

this lipid class and achieving an elucidation of the chemical structure. In 1956

Lars Svennerholm provided a first detailed analysis of a wide group of ganglio-

sides [4]. Subsequently, the first clarification of the structure of a ganglioside,

namely that of GM1, was attained by Kuhn and Wiegandt in 1963 [5].

Since the work of Klenk, several other natural gangliosides have been dis-

covered, showing one of the highest variability in the glycolipid subgroup, and

many implications of gangliosides in fundamental biological processes as well as

3

4 Gangliosides and their aggregates

NH

O

CH3

OH

OH

OH

HO

HO

O

COO−

Figure 1.1: Structure formula of N-acetyl-neuraminic acid, the sialic acid moietyof GM3, here shown in the dissociated form.

in serious diseases were found. Today research activity on gangliosides is in full

swing, mainly due to their contribution in formation of lipid rafts [6] and rele-

vance in important biochemical processes such as cell recognition, toxin binding

[7], signal transduction, as well as regulation of receptor function [8].

§ 1.1 Structure and Function of Gangliosides

Gangliosides are part of the glycoconjugates family, a vast set of organic molecules

constituted by carbohydrates covalently linked to a non-sugar moiety. To be able

to identify in a comfortable way the highly varied compounds belonging to gly-

coconjugates, a complex, branched classification scheme, partially reproduced

in Figure 1.2, has been established. The family of glycoconjugates is thus com-

posed, among the others, by the group of glycolipids, which are characterized by

having the sugar part of the molecule bound via a glycosidic linkage to the lipid

moiety. When the hydrophobic lipid moiety is either a ceramide or a sphingoid,

the glycolipids are called glycosphingolipids (GSLs).

Gangliosides are a kind of acidic — or charged — GSLs characterized by the

presence of one or more sialic acid residues in the sugar moiety and are therefore

sialo-GSLs. The functional roles of gangliosides are fulfilled mainly by the carbo-

hydrate moiety, which presents a very wide range of structural forms. Although

the structure of the sugar moiety can be very variable both in number and in

type of monosaccharidic units, the lipid tail of gangliosides tends to be gener-

ally simple. Sphingosine is the main sphingoid composing the ceramide, together

with a fatty acid, such as the stearic one (18:0)1 as fatty acid. Small percent-

age (' 10%) of palmitic (16:0), arachidic (20:0) or erucic (22:0) acid are also

found in place of stearic acid. To identify the structure of some gangliosides, the

1stearic acid is composed by 18 saturated carbons and zero unsaturated bonds, hence thenotation (18:0)

Structure and Function of Gangliosides 5

Glycoconjugates

��

))TTTTTTTTTTTTTTTTTTT

Glycolipids

uukkkkkkkkkkkkkk

�� ((RRRRRRRRRRRRRRRRR

. . .

Glycosphingolipids(gsl)

��

%%KK

KK

KK

KK

KK

KK

KK

Glycophosphatidylinositols Glycoglycerolipids

Neutral gsl

glycosyl–sphingoidsglycosyl–ceramides

Charged gsl

sialo-GSLs(gangliosides)

urono-GSLssulfo-GSLsphospho-GSLsphosphono-GSLs

Figure 1.2: Schematic representation of the family of glycolipids. Gangliosidesare composed by a lipid moiety — sphingosine — and a charged saccharidicheadgroup, thus belonging to the family of acidic (charged) GSLs. Being theacidic moiety of gangliosides a sialic acid, they are also named sialo-GSLs.


Structure Shorthand notation

Neu5Acα3Galβ4GlcCer GM3

GalNAcβ4(Neu5Acα3)Galβ4GlcCer GM2

Galβ3GalNAcβ4(Neu5Acα3)Galβ4GlcCer GM1a

Neu5Acα3Galβ3GalNAcβ4Galβ4GlcCer GM1b

Neu5Acα8Neu5Acα3Galβ4GlcCer GD3

GalNAcβ4(Neu5Acα8Neu5Acα3)Galβ4GlcCer GD2

Neu5Acα3Galβ3GalNAcβ4(Neu5Acα3)Galβ4GlcCer GD1a

Galβ3GalNAcβ4(Neu5Acα8Neu5Acα3)Galβ4GlcCer GD1b

Table 1.1: Formulae of various gangliosides as well as their shorthand notation.Note the variability in the headgroup structure, and the characteristic presenceof N-acetyl neuraminic acid (Neu5Ac) as sialic acid in all gangliosides.

shorthand nomenclature system introduced by Svennerholm [9] is still nowadays

widely employed. Gangliosides are identified by applying a three-character code,

composed by two letters and a number. The first letter is always G, indicating

that the designed molecule is a ganglioside, whereas the number of sialic acid

residues beared by the ganglioside is described by setting the second letter in

the code to M, D, T or Q for mono–, di–, tri– or tetra–sialo-glycosphingolipids,

respectively. The last element of the code is the number Ng, which represents the

migration order in a chromatographic system used by Svennerholm, and char-

acterizes the carbohydrate sequence. This number can be related to the number

of sugar units Ns present in the carbohydrate part of the ganglioside via the

relation Ns = 5 − Ng; for example, the abbreviation GM1 identifies a ganglio-

side bearing one sialic acid and other 4 sugar rings. Even though Svennerholm

notation is widely adopted, the actual large number – approximately 300 – of

known gangliosides prohibits to adopt this notation to uniquely identify all the

gangliosides, and other abbreviations schemes are recommended when dealing

with less common gangliosides [10]. In Table 1.1 the formulae of some relevant

gangliosides are indicated, as well as their shorthand name. Moreover, the struc-

ture formula of GM3, which is the subject of this thesis, is reported in Figure

1.3, having highlighted their constituent parts.

Gangliosides are exclusively present in eukaryotic cells and are not found out-

Structure and Function of Gangliosides 7

NH

O

CH3

OH

HO

HO

O

COO

O

OHOH

O

OH

O HO

HOHO

O

OH

O

HO H

NH

O

H

galneu−ac glcfatty acid

sphingosine

−

Figure 1.3: Upper panel: structure formula of GM3. From left to right it is pos-sible to observe the three saccharidic groups composing the headgroup, namelysialic acid (Neu5Ac), galactose (Gal) and glucose (Glc). The hydrophobic part(ceramide) follows, composed by a fatty acid attached to the sphingosine. Lowerpanel: snapshot of a single GM3 molecule, excerpt from a configuration of thefully hydrated bilayer molecular dynamics simulation. Having employed united-atoms, hydrogens in non-polar groups such as CH2 and CH3 are not explicitlyrepresented. See chapter 2 for details.


Figure 1.4: Atomic force microscopy image of DOPC monolayer (0.9 micron longedge) containing GM3 molecules. The brighter parts indicates taller regions,and therefore a domain rich in gangliosides, since they are more elongated thanDOPC molecules. The formation of microdomains (also called rafts) is clearlyevident. Reproduced from [12].

side the animal kingdom. They are mostly abundant in the brain, reaching 6% in

weigth of the lipids, but are present in all other tissues as well. Their biosynthesis

takes place by means of the activity of enzymes such as glycosyltransferase in

the Golgi apparatus. These enzymes are bound to the membranes of the Golgi

apparatus in a peculiar order, which corresponds to the addition order of the

different sugar units. After the synthesis, gangliosides are transported via a flow

of vesicles to the plasma membrane, where they form, together with other gly-

colipids and glycoproteins, the carbohydrate layer – called glycocalyx – that

covers the surface of the cell. It is indeed in the glycocalyx that gangliosides,

like other glycolipids, carry out most of the functional roles (described later in

this section), mainly due to the specificity of the carbohydrate moiety structure

and to the formation of glycolipid enriched zones. Moreover, stable gangliosides

patterns on individual cell surfaces have to be preserved by a precise control

of gangliosides biosynthesis, degradation and intracellurar transport [11]. It is

indeed because of defects in these processes that human genetic diseases (such

as the previously mentioned Tay–Sachs syndrome) may occur, leading to degen-

eration of the nervous system.

In the naive picture of the cellular membrane, the function of lipids is that of

solvent for membrane proteins. In the real membrane bilayer, however, different

specieses of lipids are present, such as phospholipids, cholesterol and glycolipids,

which are found to be inhomogeneously distributed. In particular, glycolipids are

Amphiphiles Supramolecular Aggregates 9

more present in the exoplasmic (outer) leaflet than in the cytoplasmic (inner)

leaflet of the membrane. Moreover, glycolipids are ordered also on the membrane

surface, presenting a lateral organization. It is thought that this lateral organiza-

tion results from a peculiar packing of sphingolipids, which organizes themselves

in so-called lipid rafts, or microdomains. Segregation of GSLs should be a conse-

quence of their strong amphiphilic character, with respect to the phospholipids.

Formation and structure of microdomains on the cell membrane cannot be stud-

ied with a high spatial accuracy, and therefore a number of experiments have been

carried out on model membranes such as micelles, liposomes and lipid monolay-

ers. As an example, Figure 1.4 shows an atomic force microscopy (afm) image of

a GM3 and dioleoyl-phosphatidylcholine (DOPC ) monolayer, reproduced from

[12], where the formation of GM3 domains can be appreciated.

As already pointed out, gangliosides play an important role in many biolog-

ical functions, even if their precise role is not always well understood. It is clear

that they fulfill essential functions for the living organism, because of the conser-

vation of GSLs structure during evolution. Among these functions, there is cell

growth, proliferation, adhesion, and differentiation by regulating the activities

of transmembrane receptors and signal transduction pathways. In fact specific

gangliosides interact with key transmembrane receptors or signal transducers,

involved in the processes just mentioned. Moreover, it has been observed that

ganglioside composition depends on changes in the morfology and function of

cells, in particular during cell proliferation, brain developement, differentiation

and neoplasia in various cell types [13].

§ 1.2 Amphiphiles Supramolecular Aggregates

Gangliosides have extensively been studied not only in their natural environ-

ment and in model membrane for the purpose of studying their physiochemical

properties, but also in various spontaneous supramolecular aggregates to inves-

tigate their physical properties [14]. The tendency of gangliosides to segregate

from solvent and form supramolecular aggregates when dissolved in water above

a certain critical concentration, comes from the amphiphilic character that, like

all other glycolipids, characterizes gangliosides. The word “amphiphile” – from

the greek words �� (of both kind) and � �� (love) – indicates the presence

in the same molecule of two parts, one with hydrophilic and the other with hy-

drophobic character. In general, the hydrophilic part, also called the head of the


S

V h

Figure 1.5: Schematic representation of an amphiphile molecule and the associ-ated geometrical quantities employed to define the packing parameter P , namelythe occupied surface S, the height h and the total volume V .

amphiphile, can bear a net charge, as in the case of anionic or cationic groups,

or be globally neutral, as in zwitterionic or polar compounds. The impressively

large number of phases and aggregation forms that amphiphiles can show, has

its roots in their dualistic character, since self-associating is a way to reduce the

energetically unfavorable contact of the hydrophobic moiety with water [15].

The kind of aggregate formed by the amphiphile depends generally on many

extrinsic parameters such as temperature, volume fraction of the amphiphile,

concentration of added salt or type of counterion, as well as on intrinsic parame-

ters, such as length or saturation degree of the hydrocarbon chains, or the steric

hindrance of the hydrophilic moiety with respect to the hydrophobic one. Geo-

metrical considerations on the hindrance of different parts of the molecule can

especially be of help in predicting the possible shape of the aggregate. Assuming

that no water can penetrate into the hydrophobic region of an aggregate, the

hydrophobic group of an amphiphilic molecule can be represented as a truncated

cone (see Figure 1.5). Three parameters, namely height h, volume V and area at

the interface S, uniquely identify the truncated cone. It is then possible to define

a dimensionless packing parameter P = V/(S ·h), which resumes the geometrical

characteristics of the molecule. Two peculiar cases are that of a cone (P = 1/3)

and of a cylinder (P = 1). The more the amphiphile packing parameters is close

to 1/3 or 1, the more the resulting aggregate will approximate a spherical micelle

or a bilayer, respectively. Intermediate values are known to give rise to elongated

micelles (1/3 < P < 1/2), whereas for P > 1/2 vesicles and bilayers are formed.

All aggregate forms present the hydrophilic part (headgroups) in direct contact

Amphiphiles Supramolecular Aggregates 11

with water, whereas the hydrophobic moieties (tails) interact only with each

others. In the case of micelles, the form of the aggregate is approximately that of

a sphere or ellipsoid, presenting the outer shell as a unique hydrophilic surface.

In the case of bilayers, the amphiphilic molecules segregate so that a structure

with two hydrophilic parallel planes is formed, with the tails in the inner part of

the bilayer. Vesicles show the local structure of a bilayer, but they are bended

over to form a closed structure, which encloses the solvent.

The concentration of the amphiphile will naturally play an important role in

determining the type of aggregate. At low concentration, a region of isotropic

solution is found in the phase diagram, usually denoted by L1, where the am-

phiphile is present in the form of isolated molecules. By increasing the concen-

tration up to the range of the critical micelle concentration (c.m.c.) a strongly

cooperative association of surfactant molecules into micelles takes place. A simple

model can explain the arising of a c.m.c. by applying the mass action law. Call-

ing Xn the concentration of surfactant organized in aggregates of n molecules, so

that the total concentration of solute is C =∑

n=1Xn, the equilibrium between

the isolated molecules M1 and the aggregate state with n components Mn

Mn−→←− nM1

can be stated asXn

n= Xn

1 e−βn(µn−µ1),

where µn is the free energy per molecule in aggregates of n molecules and β =

1/KT . By rewriting the previous equation as

Xn = n[X1e

−β(µn−µ1)]n,

it is easy to observe that for small amphiphilic concentration, namely when

X1e−βn(µn−µ1) � 1, the presence of large aggregates becomes unlikely (and the

likelihood decreases when the aggregation number grows). The concentration

of structures with aggregation number n becomes appreciable only when the

condition X1e−β(µn−µ1) ' 1 is met. The concentration of isolated molecules that

satisfy the previous condition represents the critical micelle concentration, and

can be thus estimated as

Xc.m.c. ' eβ(µn−µ1).

Despite the name, the c.m.c refers not only to the formation of micelles, intended


a) b) c)

d) e)

Figure 1.6: Pictorical representation of the amphiphile arrangement in variouskind of lamellar phases: gel Lβ (a), tilted gel Lβ′ (b), ripple Pβ′ (c), fluid (usuallyimproperly called liquid crystalline) Lα (d) and interdigitated gel LβI (e).

as globular aggregates, but also to the formation of, for example, disklike or

rodlike aggregates in the isotropic phase. An increase of the concentration over

the c.m.c. usually leads to the transition from the isotropic solution to a more

ordered liquid crystalline phase, where the solution retains the liquid structure

at microscopic scale, but presents also the long-range order typical of crystals.

Micelles in solution at high densities can form, for example, cylindrical mi-

celles, going into an hexagonal phase. At even higher densities, a transition to

the lamellar phase will commonly occur, providing a suitable packing at high

fraction of solute. A large number of different phases have been observed in sur-

factant solutions, ranging from the lamellar phases, like Lα, Lβ, to cubic (Q) or

hexagonal (H) phases. Table 1.2 summarizes the most important phases, and

Figures 1.6 and 1.7 display the surfactant molecules arrangement in a pictorial

way.

§ 1.3 GM3 Vesicles

The peculiar variability in the head-group composition of gangliosides allows

them to aggregate into very different structures, namely micelles, vesicles and

GM3 Vesicles 13

Figure 1.7: Arrangement of amphiphilic molecules in the cubic bicontinuous (left)and hexagonal inverse micellar (right) phases (reproduced from [15]).

Phase Description

L1 isotropic micellar solutionLα fluid lamellar phaseLβ untilted lamellar gelLβ′ tilted lamellar gelLβI interdigitated lamellar gelPβ′ ripple gelH hexagonalQ cubic

Table 1.2: Some phases and their descriptions. For a representation of the corre-sponding aggregates see Figure 1.6 and 1.7. L1 identifies the solution of isotropi-cally distributed micelles, which occurs at moderately low surfactant concentra-tions.


Ganglioside Sn Wm Nagg S(A2) P

GM4 2 1015 18000 80 > 0.5GM3 3 1195 14000 80 > 0.5GM2 4 1389 451 92 0.445GM1 5 1560 301 95.4 0.428GD1a 6 1851 226 98.1 0.416GD1b 6 1851 170 100.8 0.405GT1b 7 2142 176 100.8 0.405

Table 1.3: Properties of some ganglioside aggregates: number of sugar rings inthe headgroup (Sn), molecular weight (Wm), typical aggregation number (Nagg),estimated occupied surface per head (S) and packing parameter (P ). Reproducedfrom [14]

bilayers. The tendency to form a specific aggregate is largely due to the sterical

hindrance of their bulky head-group that keeps the geometrical packing parame-

ter close to the value of 0.5, so that, for example, GM3 tends to form flat bilayer

or vesicles, whereas GM2, bearing one more sugar ring with respect to GM3,

forms micelles. Some properties of ganglioside aggregates, like packing parame-

ter are presented in Table 1.3.

To describe the formation of some aggregates, beside the geometrical con-

siderations previously remarked, the so-called curvature model (see e. g.[16]) can

be employed, for which the configurational energy of a vesicle can be written

via the principal curvatures c1 and c2, in terms of the mean ([c1 + c2]/2) and

gaussian (c1c2) curvature as

E =

∮−κc0(c1 + c2) +

κ

2(c1 + c2)

2 + κGc1c2 dS, (1.1)

where κ, κG and c0 are the bending rigidity, the gaussian curvature modulus

and the spontaneous curvature, respectively; the integration is performed over

the whole surface of the bilayer. In the case of single amphiphile bilayer, the

spontaneous curvature parameter c0 is zero for symmetry reasons, and the energy

required to form a vesicle can be estimated in a simple way in the spherical

case, where c1 = c2 = 1/R, being R the sphere radius. After a straightforward

integration one obtains, from Equation 1.1

E = 4π(2κ + κG). (1.2)

GM3 Vesicles 15

When the bilayer is not in a closed form, but flat, the curvature energy is zero,

but another term in the free energy has to be considered, namely the contribution

due to the entropically unfavored exposition to the solvent of the hydrophobic

portion of the edge. It is important to notice that this term grows like L, the

edge of the bilayer patch, being therefore not bound on the upper side. On the

contrary, the energy of a spherical vesicle (1.2) is independent on the vesicle

size, thus indicating that it always exists a critical size, over which the closed,

vesicle form, has a lower free energy with respect to the flat bilayer. Nevertheless,

before completing the formation of a vesicle, some energy has to be given to the

bilayer to acquire curvature, whereas the edge energy term remains constant.

Therefore vesicles are generally believed not to aggregate without a supply of

external energy. Spontaneous formation of vesicles has been although reported in

many cases of mixed bilayers (see [17] and references within), due to a softening

of the bending rigidity.

The case of GM3 is rather peculiar, since it has been found [18] to spon-

taneously form vesicles without addition of other surfactants, thus becoming a

very interesting subject of investigation for physicists. This behavior is most

probabily due to negative gaussian modulus and low bending rigidity, since both

these conditions contribute in lowering the vesicle formation energy. The nega-

tive gaussian modulus should be a consequence of the larger area occupied by

the headgroup with respect to that of the lipid tails, and the bending rigidity

is expected to be low, possibly due to high disorder in the lipid tail region and

interdigitation [19].

2.

Molecular Dynamics Simulation

El universo (que otros llaman la Biblioteca) se compone de un

numero indefinido, y tal vez infinito, de galerıas hexagonales

[ . . . ] Yo me atrevo a insinuar esta solucion del antiguo proble-

ma: La biblioteca es ilimitada y periodica. Si un eterno viajero

la atravesara en cualquier direccion, comprobara al cabo de

los siglos que los mismos volumenes se repiten en el mismo

desorden.

J. L. Borges, La biblioteca de Babel.

Molecular dynamics is a methodology the theoretical basis of which is deeply

rooted into the eighteenth century physics. Indeed, the fundamental objects that

molecular dynamics deals with are classical particles, and little more than New-

ton’s law is required to perform a molecular dynamics simulation. A complete

description of the microscopic world would require a quantum mechanical ap-

proach, and in fact the interaction potential function used to let the classical

particle interact, are “merely” effective potentials, which mimic the real interac-

tion energies between electrons and nuclei in the molecules. Molecular dynamics

is therefore not the appropriate tool to account for chemical reactions, or to

describe correctly interactions whose energy is comparable with KT . Despite

its intimate quantum-mechanical nature, many aspects of the condensed-state

matter can be actually investigated and understood in terms of classical physics,

and the methods of molecular dynamics play an important role especially in

understanding the liquid state.

17

18 Molecular Dynamics Simulation

§ 2.1 Details on the Simulation Package

Simulating an amphiphile aggregate is by no means a trivial task, and the case of

GM3 makes no exception. Indeed, GM3 is a more complex molecule with respect

to phospholipids – the usual amphiphiles employed to perform simulation of bi-

layer aggregates – the reasons being its higher molecular weight, ionic nature

and branched structure of the oligosaccharidic head group. The high number

of atoms composing GM3 implies that a high number of interactions has to be

computed, especially the Lennard-Jones and Coulomb ones, which take much of

the computation time. On the other hand, the presence of a bulky head group,

which is branched and rich in polar groups, suggests that the GM3–GM3 inter-

actions, both of sterical and electrostatical nature, will lead to slower relaxation

times, with respect to the simpler phospholipids, due to steric hindrance and in-

teraction via hydrogen bonds. Given these characteristics it is expected that the

simulation of the GM3 bilayer will require significant computational resources.

The gromacs [20, 21] molecular dynamics package is a highly optimized par-

allel code that has been chosen to perform the simulations. It is one of the fastest

general-purpose codes available, and therefore well-suited for the demanding task

of simulating the GM3 bilayer, despite it lacks some features that would have

been appreciated, like a correct implementation of the Parrinello–Rahman [22]

pressure coupling algorithm. On the other hand, a quite wide set of analysis and

trajectory/topology processing tools are provided within the package, as well as

a set of function libraries that permits to easily write analysis code when needed.

2.1.1 Algorithms

The gromacs package provides the possibility of choosing among different al-

gorithms to perform various tasks like, for example, computation of long range

forces, treatment of cut-off, temperature or pressure coupling. In the following,

some of the principal algorithms specifically employed in the simulation of the

GM3 bilayer will be shortly described.

Boundary conditions and equations of motion integration — Unless

the aim of a simulation is to explicitly study the behavior of atoms or molecules

in interaction with rigid walls, one would usually investigate unbounded systems.

To achieve this goal with a simulation of 103–104 atoms it is necessary to avoid

placing them in a simulation box with rigid boundaries, because the number of

Details on the Simulation Package 19

particles in interaction with the walls would be of the order N2/3, which turns

out to be a not negligible fraction of the total number. By employing periodic

boundary conditions, the space results to be filled by an infinite number of

replicas of the simulation box. In this way the system becomes unlimited and

periodic, and particles that leave the simulation box from one of the faces, reenter

through the opposite one. In the present case the simulation box has been chosen

to be a parallelepiped, with the three box vectors orthogonal to each other. This

choice was made both for ease and for symmetry considerations, having the

investigated system a planar structure.

The movement of particles within the simulation box is computed by gro-

macs using the algorithm known as leapfrog [23]. The starting point to obtain the

leapfrog algorithm is the Taylor expansion of position and velocity of a particle

as a function of time,

r(t+ ∆t) = r(t) + v(t)∆t+ a(t)/2∆t2 +O(∆t3

)(2.1)

v(t+ ∆t/2) = v(t) + a(t)∆t/2 + a(t)∆t2/8 +O(∆t3

)(2.2)

Equation 2.1 can be rewritten, substituting the expression for v(t) that appears

in (2.2), as

r(t+ ∆t) = r(t) + v(t+ ∆t/2)∆t+O(∆t3

). (2.3)

Therefore, to propagate the position at time t+ ∆t, one has to know in advance

the value of the velocity at time t+ ∆t/2. The computation of the velocity can

be actually done by taking the difference of Equation 2.2 and the same equation

evaluated at t−∆t/2, so that

v(t+ ∆t/2) = v(t−∆t/2) + a(t)∆t+O(∆t3

). (2.4)

As it can be noticed, in this algorithm positions and velocities are evaluated at

different times, therefrom the name leapfrog. To allow an increase of the integra-

tion timestep, which is limited on the basis of the frequency of the fastest degree

of freedom, bonds are kept rigid by means of two algorithms provided by the

gromacs package, namely settle [24] (a version of the well known shake [25]

algorithm for water molecules) and lincs [26] for gangliosides.

Computation of long-range forces — The computation of long-range con-

tribution has been found to be of fundamental importance to avoid large artifacts


in the simulation of non-ionic amphiphiles, which would appear by employing

a cut-off scheme for coulomb interaction [27]. The interaction between periodic

images of the system can be accounted for in many ways, for example with

reaction-field, multipole expansion, or Ewald-like methods. Ewald-like methods

are generally preferable than reaction-field, because the former ones take natu-

rally into account the periodic structure imposed by the boundary conditions.

The gromacs molecular dynamics package provides the implementation of

various method to compute the electrostatic interaction, such as simple cut-off,

reaction field or Ewald-like methods, like the Ewald summation itself, and the

particle-mesh Ewald [28] (pme) method. For big enough system, pme results to

be much faster than Ewald summation, because the former scales as N logN ,

and the latter as N3/2, being N the number of particles. Therefore the pme

method has been the obvious choice to compute electrostatic interactions.

Temperature and pressure coupling — The practice of molecular dynam-

ics has led to the use of many well-known thermodynamic ensembles, like the

microcanonical or grancanonical ones, as well as to the development of new ones,

like — in the case of interfaces — the NγT ensemble [29] at constant number

of particles N, temperature T and surface tension γ. The choice for the present

case was to simulate the NPT ensemble, therefore the need for the employment

of temperature and pressure coupling. For the equilibration phase, as well as

for the sampling phase, the Berendsen temperature and pressure coupling algo-

rithms [30] have been used.

The Berendsen approach consists in performing a rescaling of the velocities

every timestep, in the case of the temperature coupling, and of both particle

positions and simulation box vectors for the pressure coupling. Velocities are

scaled, before being used to propagate positions, by a factor

λ =

(1 +

∆t

τT

{T0

T (t−∆t/2)− 1

})1/2

, (2.5)

where ∆t is the timestep, τT an appropriate time constant, T0 the desired equi-

librium temperature and T (t−∆t/2) is the temperature computed using the

unscaled velocities. In this way, a corrections to the temperature is applied, ac-

cording todT

dt=T0 − Tτ

. (2.6)

Force Field 21

The relation between τ and τT reads

τ = 2CV τT/NdKB , (2.7)

where CV is the heat capacity of the system, and Nd is the number of degrees

of freedom.

Similarly to temperature, pressure is kept constant by rescaling box vectors

and particle coordinates every timestep by applying a scaling matrix

µij = δij −∆t

3τpβij [P0ij − Pij(t)] , (2.8)

where τp is the scaling time, P0ij and Pij(t) are the equilibrium and instanta-

neous pressure, respectively, and βij is the isothermal compressibility. It has been

also chosen not to scale the system in a whole anisotropical manner, but to keep

the box angles fixed at 90 degrees, letting the box vector lengths to scale inde-

pendently, with an isotropic compressibility βij = βδij . Notably, this procedure

is quite different from employing an isotropic coupling scheme, where instead of

the pressure tensor Pij a mean scalar pressure P = 1/3 Tr Pij is kept constant,

thus scaling the box vectors by the same factor.

§ 2.2 Force Field

When molecular dynamics became a tool for investigating “real systems” in

computer experiments1 during the 1960’s [32], the target systems were simple

fluids like liquid argon, simulated as Lennard-Jones fluids. With the increase of

available computational resources, the technique of molecular dynamics has been

applied to disparate, and bigger, systems. The description of complex molecular

liquids requires, however, not only ingent computational resources, but also the

knowledge of a substantial number of interaction potentials. Indeed, complex

molecules are usually modeled as a connected set of atoms, and the interactions

due to the formation of molecular orbitals are represented in an effective way

by means of relatively simple single- and many-body potentials. A force field is

precisely the set of potentials required to correctly describe the interactions that

characterize a system.

1The first molecular dynamics simulation can be actually dated 1955 with the investigationabout ergodicity in a model system of anharmonically coupled particles by Fermi, Pasta andUlam [31].


Table 2.1: Atom types, their atomic weight and a short description. See also thetopology listed in the Appendix

Name mass Description

C 12.011 Bare carbon, as in carbonyl or amide groupsCB 12.011 Bare carbon in ringsCH1 13.019 Aliphatic CH groupCH2 14.027 Aliphatic CH2 groupCH3 15.035 Aliphatic CH3 groupCS1 13.019 CH group in sugarsCS2 14.027 CH2 group in sugarsH 1.008 Hydrogen bonded to nitrogenHO 1.008 Hydrogen in hydroxyl groupLP2 14.027 CH2 group in aliphatic chains, Berger’s parametersLP3 15.035 CH3 group in aliphatic chains, Berger’s parametersO 15.999 Oxygen in carbonyl groupOA 15.999 Oxygen in hydroxyl groupOM 15.999 Oxygen in carboxyl groupOS 15.999 Oxygen in sugar ringOSE 15.999 Exocyclic oxygen (exo-anomeric effect)

Differently from the case of Lennard-Jones simple fluids, where only two

parameters – namely, the Lennard-Jones radius and energy – have to be ad-

justed to fit experimental results, in the case of more complex molecular liquids

the number of parameters can easily rise to many hundreds. Apart from the

difficulties involved in obtaining accurate estimates of these parameters from

experiments, the potentials used in simulations of complex molecular liquids suf-

fer from another drawback. It is indeed impossible to mimic the features of the

real interaction with sufficiently simple, analytical potential as in the case of

the monoatomic fluids. Notwithstanding these difficulties, molecular dynamics

simulations of complex fluids have been proved to satisfactorily reproduce many

properties of these systems, thus being a reliable tool for their investigation.

The functional form for the interaction potentials chosen for the simulation of

the GM3 molecule is the gromos87 one, implemented in the gromacs molec-

ular dynamics package, although the parameters used, which are described in

appendix 6, have been modified with respect to the original set (see Section 2.3).

For the sake of convenience the labels adopted to indicate different atom types

have been reported in Table 2.1 with a brief description.

Force Field 23

2.2.1 Potential Functions

The interactions employed to model interatomic forces can be subdivided into

two families, namely the non-bonded interactions and the bonded ones. Poten-

tials like Lennard-Jones or Coulomb ones are part of the non-bonded interactions,

whereas bonded ones account for stretching, bending and torsion of covalently

bonded set of atoms.

Non-bonded Interactions — The Lennard-Jones potential VLJ between

two particles i and j at distance rij is characterized by its repulsive part, used

to model the steric hindrance of the atom and its attractive part that describes

induced-dipole interaction with other atoms

VLJ(rij) =C12

ij

r12ij

−C6

ij

r6ij, (2.9)

where C6ij and C12

ij are the interaction strengths. The only long-ranged interaction

present in the gromos87 potential functions is coulomb one, which takes the

form

VC(rij) =1

4πε

qiqjrij

, (2.10)

where qi is the charge of the i−th atom and ε is the dielectric permittivity.

It has to be noticed that for the purpose of correctly computing the Coulomb

interaction, Ewald-like methods have to be employed, as described in Section

2.1.1, thus splitting the potential into a short-ranged and a long-ranged part,

computed separately in the real and momentum space, respectively.

Bonded interactions — Differently from the non-bonded interactions, the

bonded ones can be not only pair interaction, but also three- or four-body in-

teractions. The bond stretching between a pair of atoms is represented by an

harmonic potential

Vs(rij) =kb

ij

2(rij − r0ij)2, (2.11)

where kbij is the interaction strength and r0ij the equilibrium bond length.

The bending term involves a triplet of atoms and, as the stretching term, is

represented by a harmonic potential


i j

kj

i

l

i

j

k

Figure 2.1: Schematical representation of the stretching (left), bending (center),and torsional (right) interaction terms.

ji

j k

l ji

l

k

i

kl

Figure 2.2: Schematical representation of three cases that require the use ofimproper dihedrals: a cis double bond (left, ξ0ijkl = 0), a planar structure (center,

ξ0ijkl = 0), and a tetrahedral structure, where the missing apolar hydrogen atom

is represented in white (right, ξ0ijkl 6= 0).

Vb(θijk) =kθ

ijk

2(θijk − θ0

ijk)2, (2.12)

where

θijk =arccos(rij · rkj)

rijrkj

and where θ0ijk is the equilibrium angle. The atom at the center of the triplet is

designed in position rj.

Another important contribution to the total potential energy is the rotational

term that involves four atoms. The interaction potential is represented as a

truncated Fourier–like series, and their element are written as

V nt (φijkl) = kφ

ijkl

[1 + cos

(nφijkl − φ0

ijkl

)], (2.13)

where φijkl is the dihedral angle defined by the planes identified by (ri − rj) ×(rk − rj) and (rj − rk) × (rl − rk), so that φijkl = 0 in the case of an eclipsed

configuration. The integer number n is called multiplicity and determines the

number of minima of the potential.

In the case of saturated hydrocarbon chain, a different torsional term is used,

Force Field 25

namely the Ryckaert–Bellemans potential

Vrb(φijkl) =5∑

n=0

Cn cosn(φijkl − π), (2.14)

wich improves the representation of the angular distribution with respect to the

standard torsional term. To correctly reproduce the rotational energy barrier

profile it is however required to exclude any non-bonded interaction between

pairs connected by three bonds, the so-called 1–4 interactions.

An harmonic potential is also employed to model the so-called improper

dihedral interactions, namely

Vid(ξijkl) = kξijkl

(ξijkl − ξ0ijkl

), (2.15)

where the improper dihedral angle ξijkl is defined the same way as for the proper

dihedral. However, the improper dihedral interactions not necessarily act on

atoms in a linear sequence, and are used for multiple purposes, like keeping some

structures planar, in cis or trans configuration, or to preserve the chirality of

structures involving united-atoms (see Figs. 2.1 and 2.2). In the latter case, with-

out the use of improper dihedrals, the absence of one or more hydrogen atoms

around a chiral center would lead to the appearence of wrong stereoisomers.

Practical examples are those of peptide and double bonds, and of axial or

equatorial configuration of OH groups in the carbohydrate rings. It has to be

noticed that the harmonic potential doesn’t take care of the periodicity, and since

angles are defined in the interval between -180 and 180 degrees, a discontinuity

appears when a fluctuation brings ξ0ijkl to ±180 degrees. To avoid this drawback,

the equilibrium angle has been kept far away from ±180 degrees.

2.2.2 Water Model

The force fields provided with the gromacs molecular dynamics package are

consistent with some water models, such as the simple point charge (spc), spc

extended (spc/e), tip3p and tip4p ones. Because of the high number of water

molecules employed in the simulation of the GM3 bilayer (see Section 3.1), the

choice of a single point charge model is preferable, with respect to more complex

models like tip4p, for the sake of efficiency. Although the spc/e model gives

more accurate results regarding the bulk water energy, single point charge po-

tential (spc) has been found to be an accurate choice for reproducing interfacial


θ

q

Q

σ

d

Figure 2.3: Schematic representation of the SPC water model

properties, as pointed out in ref. [33]. spc water model, shown in Figure 2.3 con-

sists in three centers, representing the charges on oxygen and hydrogen atoms,

bearing a charge of Q = −0.82 and q = 0.41e, respectively, where e denotes the

charge of the electron. On the oxygen charge center a Lennard-Jones potential

with radius σ = 0.3166 nm and energy ε = 0.650 kJ/mol is set. The two positive

charges are at a distance of d = 0.1 nm from the central one, and disposed at an

angle θ = 109.47 degrees.

§ 2.3 Changes to the gromos87 Force Field

The gromos87 force field is a very widely employed and tested one, and it

has been used to simulate disparate systems of biological interest, ranging from

oligosaccharides like cyclodextrin [34], to DNA [35] and phospholipid bilayers

[36]. Of course, every force field has its own limitations, and in the case of

gromos some drawbacks emerged during the simulations performed by several

authors of both hydrocarbon chains and oligosaccharides.

2.3.1 Hydrocarbon Moiety

During one of the first simulations of a fully hydrated phospholipid membrane

[36] by Egberts and coworkers in 1993, it happened that at temperatures well

above the gel–fluid transition temperature, the system went into a gel-like Lβ

phase. The authors argued that the lack of screening of the electrostatic force

by spc water, resulted in an overestimated electrostatic interaction among the

Changes to the gromos87 Force Field 27

zwitterionic headgroups. Therefore, to avoid these unphysical effects, the authors

reduced the electrostatic interactions by a factor of 2, recovering the correct

Lα phase. Obviously this solution was not satisfactory, and indeed short after

van Buuren, Marrink and Berendsen [37] proposed a different method. The idea

was to change the interaction between CHn groups and water, since the value of

solubility of simulated decane had been found to be too low with respect to the

experimental values. Although the new choice of parameters solved the problem

of obtaining the right phase, by decreasing the water-mediated interaction among

the lipid tails, this solution did not get a footing.

In 1997 Berger et al. proposed a different interpretation of the wrong ap-

pearance of the Lβ phase [38]. They noticed that usually, comparison between

simulation results and experimental data regarded the occupied surface per head,

which is experimentally known with a rather low accuracy, ranging from 0.56 to

0.72 nm2 for the DPPC. However, the mean occupied volume per lipid has been

practically ignored, although it is a quantity that is known with an accuracy of

less than 1% [39]. Therefore they re-parametrized the gromos87 force field in its

hydrocarbon part, modifying the Lennard-Jones parameter of the CHn groups.

The system employed for the optimization of the Lennard-Jones parameter was

bulk pentadecane, since its saturated chain length is the same as in DPPC, and

the quantities used for comparison with experiment were density and heat of

vaporization. The value of volume per molecule obtained using the gromos87

parameters for the simulation of liquid pentadecane was indeed far too low (30%)

with respect to the experimental value. Surprisingly, the main changes needed to

reach the optimal agreement regarded the Lennard-Jones energy rather than the

radius. The decreasing of the interaction energy implies a lowering of the phase

transition temperature of the bilayer and, with the optimized parameters, Berger

et al. obtained indeed a fluid phase at a temperature of 50◦C. As the Berger re-

parametrization seems to be the more reliable for simulating bilayers, it has been

adopted in the present simulations of the GM3 molecule. Ryckaert–Bellemans

dihedrals have been employed, as usual. Table 2.2 shows the Lennard-Jones pa-

rameters of the gromos standard force field, the opls ones and those resulting

from the optimization.


Table 2.2: Parameters for Berger et al. are taken from [38], whereas the gromosones refer to the ifp37C4 set.

LP2 LP3

source σ (nm) ε (kJ/mol) σ (nm) ε (kJ/mol)

gromos 0.396 0.585 0.379 0.753opls 0.3905 0.4932 0.3905 0.7315Berger et al. 0.396 0.380 0.396 0.570

Figure 2.4: Schematic representation of two cyclohexane conformers in chair andboat conformation.

2.3.2 Carbohydrate Moiety

The limitations of the gromos force field regard not only the hydrocarbon

chains, but also the sugar rings. Due to their branched and cyclic structure, car-

bohydrates show a very rich set of accessible conformations. In a series of works,

Spieser, Klewinghaus, Kroon-Batenburg and coworkers [40, 41, 42] showed that

the gromos force field can be improved in three aspects. The first one regards

possible conformations — chair, inverted chair and boat — that the sugar rings

can adopt (Figure 2.4 shows as an example the cyclohexane molecule in chair and

boat conformation). In particular, the potential energy barrier between the nor-

mal chair and the other two conformations obtained with the gromos87 force

field is too low with respect to that estimated by ab initio calculations. This dif-

ference leads to frequent unphysical changes of conformation during molecular

Changes to the gromos87 Force Field 29

Table 2.3: Parameters for Berger et al. are taken from [38], whereas the gromosones refer to the ifp37C4 set.

Force constants Equilibrium values

(kJ mol−1rad−2) (deg)

gromos Spieser et al. gromos Spieser et al.

Bending termCS1–OS–CS1 334.720 460.24 109.5 109.5CS1–CS1–O 284.512 460.24 109.5 107.0CS1–CS1–CS1 251.040 460.24 109.5 109.5CS2–CS1–OS 284.512 460.24 109.5 107.0CS2–CS1–CS1 251.040 460.24 109.5 109.5CS1–CS21–OA 284.512 460.24 109.5 107.0OS–CS1–OS 284.512 460.24 109.5 107.5CS1–OSE–CS1 334.72 111.0

Force constants Multiplicity Phase anglekJ/mol (deg)

gromos Spieser et al.

Torsional termOS–CS1–OSE–CS1 4.184 2 0OA–CS2–CS1–OS 2.092 4.184 2 0

dynamics runs. The energy barrier between the chair and inverted chair con-

formations resulted to be 4.5 kJ/mol for the gromos force field, whereas from

various molecular mechanics models as well as from ab initio calculations the

energy barrier was found to be in the range from 16.0 to 30.0 kJ/mol. To rise

the energy barrier Spieser et al. increased the force constant of the bending term

for some atom types, bringing the energy barrier between chair and inverted

chair conformations to 13.2 kJ/mol.

Moreover, the oxygen lone pairs induce peculiar effects like the gauche and

exo-anomeric ones, which strongly influence the conformation of the sugar rings

and that have to be taken into account. The gauche effect is the lowering of the

energetic barrier between a gauche and the anti conformation in a X–C–C–Y

structure (being X an electronegative group), due to the energetically favored

interaction of the C–H and C–X orbitals. As shown in Figure 2.5, the interac-

tion via donation of electronic density from a lone pair of the oxygen to the

antibonding C–H orbital is geometrically favored in the gauche conformation.


��

��

C

H

OO

H

H

� ��

� ��

��

C

H

O

O

HH

Figure 2.5: Qualitative explanation of the gauche effect in the specific case of theO–C–C–O bond. Although the relative disposition of the dipole moments wouldhave privileged the anti conformation of the oxygens around the C–C bond (left),the total energy in the gauche conformation (right) is lower, thanks to the factthat in this conformation the oxygen can donate electrons to the C–H bond.

Figure 2.6: Representation of a butane molecule in anti (left), gauche (center)and eclipsed (right) conformation.

The conformational local minima (anti, gauche and eclipsed) of the rotational

potential energy of a butane molecule are shown in Figure 2.6.

The nature of the exo-anomeric effect is similar to that of the gauche effect. It

takes place due to the donation of electronic density by the oxygen lone pair, as

in the case of the gauche effect, and influences the rotational preference around

the glycosidic bond. In this case the donor is the exocyclic oxygen, and the

acceptor is the endocyclic C–O bond. The exo-anomeric effect has to be taken

into account especially in the case of oligo- or polysaccharides, since it regards

the glycosidic linkages, and therefore it can influence substantially the global

conformation of the molecule.

The particular case of glucose can be used to show explicitly the atoms

subjected to these two effects. With reference to Figure 2.7, the dihedral angle

that shows the gauche effect is that embracing the hydroxymethyl group, O5–

C5–C6–O6, whereas the exo-anomeric effect regard the O5–C1–O1–C7 dihedral

GM3 Topology 31

� � ��

� � ��

� � � ��

� � � ��

� � � ��

C1

C2

C3

C5

O5C4C6

O6

H6

H4

O4 O3

H3 O2

H2

O1

C7

Figure 2.7: Structure formula of glucose

angle.

The solution adopted by Spieser to reproduce the gauche effect for the hy-

droxymethyl groups of the sugar rings is to strengthen the torsional force of the

OA–CS2–CS1–OS dihedral angle with multiplicity 2, therefore privileging the

gauche configurations. In the case of the exo-anomeric torsion, the atom type

OSE is added, which is absent in the gromos force field. The parameters of the

force field for OSE are the same as in the case of atom type OS, except for the

force constant of the OS–CS1–OSE–CS1 torsion with multiplicity 2, which is

doubled with respect to the OS–CS1–OS–CS1 case. Again, this choice was made

to correct the tendency for the dihedral to be too often localized near the anti

conformation. The modifications introduced by Spieser et al., employed for the

simulation of GM3 ganglioside, are summarized in Table 2.3.

§ 2.4 GM3 Topology

The list of atom sets subjected to the different bonded and non-bonded interac-

tions, together with the force field parameters that have to be employed, is called

topology and is meant to be a comprehensive description of the molecule. This

section is devoted to a review of the main points of the topology that has been

written for the GM3 ganglioside; the complete topology is reported in appendix

6. Some excerpted topology lines will be presented as reference for the sake of

clarity. With reference to equations (2.9–2.15), the topology lines take the form

displayed in Table 2.4, namely the sequential numbers of the involved atoms, a

number specifying the kind of interaction and the force field parameters.


Table 2.4: Meaning of the interaction parameters appearing in the topology.

atoms interaction type parameters description

i, j 1 C6

ij C12

ij Lennard–Jones (2.9)i, j 1 r0ij kb

ij Stretching term (2.11)

i, j, k 1 θ0ijk kθijk Bending term (2.12)

i, j, k, l 1 φ0

ijkl kφijkl n Proper dihedral (2.13)

i, j, k, l 2 ξ0ijkl kξijkl Improper dihedral (2.15)

i, j, k, l 3 Cn n = 0, . . . 5 RB dihedral (2.14)

2.4.1 Ceramide

In designing the part of topology relative to the hydrocarbon chains, it has

been used the implementation of the Berger parameters for gromacs topologies,

published by the biocomputing group at the University of Calgary[43]. Thereby

new atom types, namely LP2 and LP3 have been introduced. Since Ryckaert–

Bellemans dihedrals have been employed, no 1–4 interaction has been added. To

model the peculiar trans double bond (see Figure 2.8) a shorter bond length has

been adopted, with respect to the single bonds, as well as different rest angles.

140 28 29 1 0.13900 418400. ; CH1 CH1 double bond

408 25 28 29 1 120.000 418.400 ; CH1 CH1 CH1 double bond

409 28 29 30 1 120.000 418.400 ; CH1 CH1 LP2 double bond

To guarantee the trans conformation around the double bond, an improper di-

hedral has been added.

712 25 28 30 29 2 0.000 167.360 ; trans double bond CH1 CH1 LP2 CH1

The order of the last two atoms has been inverted with respect to the standard

one, so that the dihedral angle is equal to zero in the trans conformation. This

ordering allows to set the equilibrium angle at zero degrees, avoiding problems

with the periodicity.

There are other improper dihedrals employed in the description of sphingo-

sine, namely that relative to the amide group, which has to be kept planar, and

a chiral center on the carbon atom next to the sphingosine nitrogen. The amide

group is kept planar by the use of two improper dihedrals

GM3 Topology 33

Figure 2.8: Schematic representation of two butene molecules in trans (left) andcis (right) configuration around the central double bond.

707 20 19 22 21 2 0.000 167.360 ; amide in sphingosine ; C LP2 N O

708 22 20 23 24 2 0.000 167.360 ; amide in sphingosine ; N C H CH1

and of a proper dihedral with multiplicity 2.

539 19 20 22 24 1 180.000 33.472 2 ; LP2 C N CH1

The high rotational barrier, about one order of magnitude higher than that of

other dihedrals, permits to avoid transitions to the cis conformation of the LP2–

C–N–CH1 bond at room temperature. Such transitions have indeed never been

found in any of the performed simulation runs of the GM3 molecule, both in

isolated form and in bilayer aggregate. The correct tetrahedral geometry around

the chiral carbon is obtained by introducing out of plane improper dihedrals,

735 59 60 90 58 2 -35.246 334.720 ; ; CS1 CS1 OS OSE

736 60 63 59 61 2 35.246 334.720 ; ; CS1 CS1 CS1 OA

therefore avoiding unphysical transitions to the stereoisomer of the ceramide.


2.4.2 Sugar Rings

The part of topology relative to sugar rings is rather complex with respect to

ceramide, which is almost composed by linear groups. Indeed, in the gromos

force field, to account for the rotation around a bond, a dihedral term has to be

introduced for every set of four connected atoms sharing that bond, and some

quadruplets are subjected to multiple dihedrals with different multiplicity. As an

example, in the case of the C2–C1 bond of the glucose ring shown in Figure 2.7,

dihedrals interactions have to be introduced for four quadruplets, namely C3–

C2–C1–O1, O2–C2–C1–O1, C3–C2–C1–O5 and O2–C2–C1–O5. Apart from the

modifications introduced by Spieser et al., the parameters for the sugar rings are

that employed by Koehler and coworkers [34] in the simulation of α-cyclodextrin

hexahydrate. The complexity in modeling carbohydrate rings resides not only in

the high number of employed proper dihedrals, but also in the need to model the

chiral centers of the sugar rings themselves, that is, the five sp3 carbon atoms

in the ring. In this way, as in the case of ceramide, unphysical transitions to the

relative stereoisomers are avoided. Groups like OH or CH2OH, are therefore kept

by the improper dihedrals in equatorial or in axial conformation, as needed. As

an example, in the case of galactose the following out of plane improper dihedrals

have been introduced

735 59 60 90 58 2 -35.246 334.720 ; ; CS1 CS1 OS OSE

736 60 63 59 61 2 35.246 334.720 ; ; CS1 CS1 CS1 OA

737 63 95 60 64 2 -35.246 334.720 ; ; CS1 CS1 CS1 OSE

738 95 91 63 96 2 -35.246 334.720 ; ; CS1 CS1 CS1 OA

739 91 90 95 92 2 -35.246 334.720 ; ; CS1 OS CS1 CS2

3.

System Preparation and Equilibration

Round about the cauldron go;

In the poison’d entrails throw.

Toad, that under cold stone

Days and nights has thirty-one

Swelter’d venom sleeping got,

Boil thou first i’ the charmed pot.

W. Shakespeare,

The Tragedy of MacBeth.

Undertaking a molecular dynamics simulation of a system as complex as an hy-

drated ganglioside bilayer requires a careful setup of the starting configuration,

which should be not too different from the expected equilibrium configuration.

Whereas from a theoretical point of view, the starting configuration should not

matter on the equilibrium properties, the equilibration time can vary largely, de-

pending on how far the starting configuration is from the equilibrium one. While

for simple liquids this time can be in general reasonably short, for a complex sys-

tem it can be unattainable. It is therefore crucial to start the simulation with a

configuration as close as possible to the expected equilibrium one, namely, in the

present case with surfactant molecules already arranged in a bilayer structure,

and water molecules already confined outside the lipid region.

35

36 System Preparation and Equilibration

§ 3.1 Set-up of the starting configuration

The starting configuration of the hydrated bilayer was constructed by applying

the following protocol. One single GM3 molecule was brought to a configuration

of minimum potential energy by firstly applying the conjugate gradients method,

and thereafter by performing repeatedly a simulated annealing in effective water,

letting temperature drop ten times from 6000 to 0 K over a period of 100 ps.

Effective water was implemented by the use of a stochastic temperature coupling

with friction, and setting the dielectric permittivity εr = 80. The GM3 molecule

was eventually rotated to have its principal axis aligned along the z direction.

Two monolayers were then assembled by putting together copies of the GM3

energetically minimized structure on a 8× 8 square lattice, with lattice spacing

of 1 nm. To add static disorder, molecules were then randomly rotated around

the z axis. These operations resulted in a distance larger than 0.1 nm between

atoms belonging to different GM3 molecules. The two monolayers were then

assembled to form a bilayer composed by 128 GM3 molecules.

It is worth noticing that the choice of the distance between the two monolay-

ers plays a critical role in the bilayer setup, because the starting interdigitation

between tails belonging to different monolayers happens to strongly regulate the

formation of the bilayer. As a matter of fact, the hydrophobic tails start com-

pletely stretched, being in the minimum energy conformation, but their end-

to-end distance decreases rapidly when coupled to the thermal bath, as it is

possible to observe in Figure 3.6. Therefore, if no tail interdigitation is present,

a void region in the middle of the bilayer begins to grow, leading to an unphys-

ical situation. On the contrary, an appropriate degree of interdigitation attains

to avoid, via the interaction between Lennard-Jones centers, the separation of

the two monolayers. To avoid overlaps between atoms in the tail region, intro-

duced by the operation of joining the two monolayers together, the system was

energy-minimized with the conjugate gradients method.

To neutralize the total charge, 128 Na+ counterions were placed within a

simulation box of z-edge of 11.2 nm at random positions, at distance larger than

0.25 nm from any atom of the GM3 molecules. A short run of 50 ps at constant

volume and constant temperature of 600 K was then performed, keeping the

GM3 molecules fixed in space, to allow the counterions to partially condensate

on the negative charges of GM3.

At this stage, the surface density of the membrane turned out to be very

Simulation details 37

low. Therefore, we have performed a simulation run to relax the x and y box

vectors, while keeping the z box vector fixed as well as the simulation box angles

at 90◦. In this way, the GM3 molecules rapidly packed without destroying the

membrane structure. The final box vectors were equal to 6.7, 6.6 and 11.2 nm

in the x, y and z directions respectively. Subsequently, water molecules were

added in the simulation box, taking their positions from a configuration of equi-

librated spc water, and deleting molecules with distance smaller than the sum

of Van der Waals radii of water atoms and any other atom belonging to GM3

or counterions. The final number of added water molecules amounts to 6724

units. The choice of the water layer size is a compromise between the need to

minimize spurious interactions between periodic images and the computational

time needed to perform the simulation. Since the procedure adopted to add water

molecules is based only on geometric considerations, the obtained system is ener-

getically unfavored. Therefore, to further relax the system, a short run of 50 ps

at the constant temperature of 333 K and at constant volume was performed

before starting the equilibration run.

§ 3.2 Simulation details

All simulations were performed at constant pressure, temperature and number

of atoms using the gromacs molecular dynamics package [20, 21]. In Section

2.1.1 it has already been pointed out that it is crucial to compute electrostatics

without truncating the interaction when dealing with non-ionic amphiphiles. The

case of GM3, which is an ionic amphiphile, is even more problematic because, as

it will be shown, the electric field is only partially screened by the counterions

and therefore is long ranged. Electrostatics was thus treated using the Ewald

summation method in the smooth particle mesh Ewald implementation [28],

with a mesh spacing of 0.12 nm and a spline order of 4.

A cutoff of 0.9 nm was applied for both the Lennard-Jones interaction and

short range contribution to the Ewald sum. As already mentioned, all bond

lengths were constrained, using the shake algorithm [25] in its implementation

named settle [24] for water and lincs [26] for GM3. The time step of integra-

tion was set to 2 fs exhibiting excellent energy conservation when tested in the

microcanonical ensemble.

Both temperature and pressure were kept constant at 333 K and 1 atm,

respectively, by means of the weak-coupling method of Berendsen et. al. [30].


10 20 30 40Time (ns)

-3.6×105

-3.6×105

-3.5×105

-3.5×105

U (

kJ m

ol-1

)

Figure 3.1: Time evolution of the total potential energy of the system.

At the chosen temperature and pressure the GM3 bilayer is known to be well

within the liquid-crystalline (Lα) region of the phase diagram. The time con-

stants for the thermostat and piston were set to 0.1 and 1.0 ps, respectively.

During both equilibration and production runs, we let the system volume to

fluctuate anisotropically, while fixing the simulation box angles at 90◦. In this

way the three box edges were allowed to scale independently from each other, re-

laxing the internal stress tensor accordingly. Periodic boundary conditions were

applied in all directions. During both equilibration and production runs, config-

urations were stored every 125 ps for subsequent analysis.

§ 3.3 The equilibration phase

As already pointed out in the introduction, one of the most peculiar character-

istics of the GM3 molecule is that the head-group has approximately the same

longitudinal extension of the ceramide tail (' 1.8 nm). For the sake of compari-

son, phospholipidic heads have approximately half the longitudinal extension of

the GM3 heads, whereas DPPC’s lipid tails have 15 carbon atoms, in contrast

to GM3, which has two chains, made of 18 and 20 carbon atoms respectively.

The equilibration phase 39

10 20 30 40Time (ns)

5

6

7

8

9

10

11

Box

vec

tor

leng

th (

nm)

Figure 3.2: Time evolution of the three box vectors. From top to bottom thesolid lines represent the value of the z, x and y box vector lengths, respectively.


0 10 20 30 40Time (ns)

-100

-80

-60

-40

-20

0D

ihed

ral a

ngle

(de

gree

s)

Figure 3.3: Time evolution of the three dihedral angles involving rotation aroundthe ceramide–glucose (solid line), glucose–galactose (dashed line) and galactose–sialic acid (dotted line) bonds, respectively. The dihedral angles are defined bythe four atoms C−O−C−OR, where OR is the oxygen belonging to the sugarring. Data have been averaged over the 128 GM3 molecules.

Moreover, apart from the flexibility due to the glycosidic linkages, the sugar rings

of GM3 present very rigid chemical structures. These characteristics, as well as

the fact that GM3 bears a net negative charge, have posed some non-trivial

problems in the equilibration stage of the simulation.

The rigidity of the components of the head-group resulted in very long re-

laxation times for the system and, consequently, a long equilibration time. As

illustrated in Figure 3.1, the total potential energy reached a stationary state

in about 30 ns. This period is about one order of magnitude longer than that

reported for phospholipids (see for example ref. [38]). On the other hand, the

dimensions of the simulation box reported in Figure 3.2 exhibit an apparent

stationary behavior for the x, y and z components. Importantly, however, the x

and y lengths of the simulation cell retain practically the same value on average.

We have identified two possible mechanisms that can contribute to the ob-

served slow equilibration time. The first one is due to the slow relaxation of the


sialic acid residue, which rotates around the glycosidic linkage with galactose.

The time evolution of the three dihedral angles around the glycosidic linkages,

averaged over the 128 GM3 molecules, is presented in Figure 3.3. It is clear that

the equilibration time of the dihedral angle of the sialic acid residue, the most

exposed to the solvent, is by far the longest one, reaching equilibrium over the

same timescale of energy, i.e. after 30 ns. The equilibrium histograms of the three

dihedral angles are reported in Figure 3.4. In the inset we report the histogram

for the innermost torsional angle averaged over the time intervals (0 , 10), (10 , 20)

and (20 , 30) ns, so as to observe whether 30 ns is a sufficient time to reach a

stationary state. The absence of a drift in the time evolution after 10 ns confirms

that the histograms of the inner dihedrals can be considered equilibrium ones.

It is worth noticing that the distribution of the torsional angle relative to the

sialic acid residue turns out to be bimodal and nearly equally populated. This

result can explain why the system spends a long time in reaching a well defined

conformational state. As usual, quasi-degenerate states often produce long equi-

libration times, even if the energetic barrier is thermally activated. Viceversa,

in the case of the other two dihedral angles, only a tiny percentage of molecules

shifts to larger angles even on the longest time scale explored by the simulation.

The absence of a drift in the time evolution both of the average (see Figure

3.3) and of the whole distribution (see the inset of Figure 3.4) confirms that a

definitive equilibrium has been reached after 30 ns.

The second identified contribution to the long equilibration time is the de-

hydration process involving the interior rings. In Figure 3.5 the radial distri-

bution function (see Section 4.3 for the definition) of water oxygen around the

center of mass of the glucose residue is shown. It appears that the number of

water molecules coordinated with glucose rapidly decreases in the first 2.5 ns.

As time proceeds, water is progressively expelled from the interior part of the

bilayer, reducing the first peak height by a factor 2.7 with respect to the ini-

tial value, and the radial distribution function converges toward its equilibrium

profile. From these results we conclude that the system has reached equilibrium

after 30 ns of the relaxation stage. In Figure 3.6 two snapshots of the system

are reported, the first one at the beginning of the equilibration run, and the

second one after the equilibration has taken place, exhibiting the high degree of

conformational disorder attained by the tails over 40 ns.


-100 0 100 200angle (degrees)

0

0.005

0.01

0.015

0.02

0.025

0.03

Prob

abili

ty d

ensi

ty

-100 -80 -600

0.01

0.02

Figure 3.4: Normalized histograms of the torsional angles involving rotationaround the ceramide–glucose (solid line), glucose–galactose (dashed line) andgalactose–sialic acid (dotted line) bonds, respectively, computed as describedin Figure 3.3. Inset: histograms for the torsional angle around ceramide–glucosebond sampled in the time intervals (0 , 10), (10 , 20) and (20 , 30) ns(solid, dashedand dotted line respectively).


0.5 1 1.5 2 2.5r (nm)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

g(r)

Figure 3.5: Radial distribution function of water oxygen around the glucoseresidue, averaged over 1 ns, at different times, namely at the beginning of thesimulation (solid line), at 3 (dotted line), 6 (dashed line), 12 (dot–dashed line)and 24 ns (dot–dot–dashed line).


Figure 3.6: Snapshots of a 3 nm thick slice of GM3 bilayer in solution. Oxygen,carbon, and hydrogen atoms have been drawn in red, gray and white respectively.Nitrogen atoms as well as sodium counterions have been drawn in blue. Two GM3molecules are highlighted using fat bonds. Bottom panel: starting configuration.Top panel: equilibrium configuration after 40 ns from the starting configuration.Drawing was made using Raster3D [44].

4.

Structural Properties

With her anchor at the bow and clothed in canvas to her

very trucks, my command seemed to stand as motionless

as a model ship set on the gleams and shadows of polished

marble. It was impossible to distinguish land from wather

in the enigmatical tranquillity of the immense forces of

the world.

J. Conrad, The Shadow Line.

Once the equilibrium has been attained, it is possible to begin the sampling of the

quantities of interest, taking care of validating the chosen interaction potentials

by comparing the results obtained from computer simulation with the available

experimental data.

§ 4.1 Structure Determination by X-Ray Scattering

Direct contact with experiments on lipid bilayers can be made by comparing the

small angle (saxs) and wide angle (waxs) X-ray scattering intensity data. By

looking at the radiation scattered in different angular ranges, these two tech-

niques investigate the structural properties of the target sample with different

spatial resolution, that is, that of the bilayer thickness in the case of saxs and

that of the interatomic distances in the case of waxs, corresponding to a scat-

tering vector modulus qsaxs ∼ 1 and qwaxs ∼ 10 nm−1, respectively.

45

46 Structural Properties

4.1.1 Comparison with saxs Data

While for the computation of the waxs spectrum no problem arises due to the

size of the simulated system, in the case of the small angle range the lower

limit and resolution of the accessible scattering vectors are strongly limited. To

overcome this limitation the average electron density profile has been extended

up to 20 nm both in positive and negative direction, by keeping the value of bulk

water electron density.

In a general framework, employing the kinematic approximation, the inelastic

scattering intensity of a continuous sample can be written as

〈I(q)〉 ∝⟨∣∣∣∣∫F(q, r) ei q · r d3r

∣∣∣∣2⟩, (4.1)

where hq = 2 hk sin(θ/2) is the transferred momentum of a radiation with wave

number k that scatters off the sample at an angle θ, and F(q, r) is the scat-

tering amplitude of the target portion located at r. At small scattering angles

the distribution of the scattering centers can be considered continuous and the

electron density ρe(r) can be taken as the scattering amplitude. It is convenient

to express the electron density as ρe(r) = 〈ρwe 〉+ρ∗e(z), where 〈ρw

e 〉 is the average

electron density of bulk water. In the approximation of a spherically symmetri-

cal vesicle, ρ∗e(z) depends only on the coordinate z normal to the vesicle surface.

The average scattering intensity takes then the form

〈I(q)〉 ∝⟨∣∣∣∣∫ ∞

0

∫ π

0(〈ρw

e 〉+ ρ∗e(z)) ei qz cos(α) sin(α) dz dα

∣∣∣∣2⟩

=

∣∣∣∣ 2∫ ∞

0〈ρ∗e(z)〉

sin(qz)

q zdz

∣∣∣∣2

, (4.2)

where the contribution at q = 0 coming from the constant water density back-

ground has been subtracted. Moreover it has been assumed that the electron den-

sity is not first-order correlated, that is 〈ρ∗e(z) ρ∗e(z′)〉 = 〈ρ∗e(z)〉〈ρ∗e(z′)〉, which

is a valid approximation at the spatial resolution of saxs. By shifting the z

coordinate origin at the middle of the bilayer one can write

〈I(q)〉 ∝∣∣∣∣∣

∫ `/2

−`/2〈ρe(z)〉

sin(qz) cos(qR) + cos(qz) sin(qR)

q (z +R)dz

∣∣∣∣∣

2

, (4.3)

where ρe(z) = ρ∗e(z − R) and R denotes the vesicle radius. Considering a sym-

Structure Determination by X-Ray Scattering 47

metric electron density profile of the bilayer, and approximating R+ z ' R, the

scattering intensity takes the simple form

〈I(q)〉 ∝ sin2(qR)

(qR)2

∣∣∣∣∣

∫ `/2

−`/2〈ρe(z)〉 ei qz dz

∣∣∣∣∣

2

(4.4)

In case of polydispersity of the vesicle size, the rapidly oscillating term sin2(qR)/R2

that appears in the previous expression has to be weighted accordingly to the

probability distribution of the size, R. Employing a uniform distribution in the

interval

I = [R0 −∆ , R0 + ∆]

leads to the first order approximation

〈I(q)〉 ∝ 1

q2

∣∣∣∣∣

∫ `/2

−`/2〈ρe(z)〉 ei qz dz

∣∣∣∣∣

2

, (4.5)

where the rapidly oscillating terms disappeared.

This approximation is valid only at moderately low scattering angles: it fails

both at high angles – due to inter-atomic correlations – and at very low angles,

where the global shape of the aggregate begins to be detected by the scattered

radiation. saxs data represent a valid benchmark to test the simulated GM3

against the real system, allowing to characterize the aggregate structure, partic-

ularly with regard to the bilayer width.

In Figure 4.1 the experimental saxs [45] and simulated spectra are reported.

Data refer to experiments conducted at the European Scattering Radiation Fa-

cility (esrf) on a sample of GM3 solution at ' 333K. The agreement is very

good in the whole q range starting from q = 0.3 nm−1. The minimum in the sim-

ulated intensity found at q ' 0.5 nm−1 is a direct manifestation of the bilayer

width, which turns out to be in good agreement with that of the real system. The

most apparent discrepancies between simulated and experimental data are the

pronounced minima in the simulated spectrum at about 2.1 and 3.0 nm−1. Since

the higher smoothness of experimental data can be thought as a consequence

of a polydispersity in the bilayer thickness, a possible explanation for this dis-

agreement can be the relatively small size of the simulation box in the xy plane.

Indeed, the use of a finite box size can reduce the amplitude of the peristaltic

modes of the membrane, thus reducing the accessible range of thicknesses.

Once assessed that the comparison between simulation and experiment re-


0 1 2 3 4

q (nm-1

)

0.0001

0.001

0.01

0.1

1

10

Scat

teri

ng in

tens

ity (

arb.

u.)

Figure 4.1: Scattering intensity (logarithmic scale) measured at esrf for a sampleof GM3 vesicle (solid line) and calculated from simulation (open squares) versusthe modulus of the scattering vector q. The calculated data have been shiftedby an arbitrary factor.

Mass Density Profiles 49

sults is satisfactory, the saved equilibrium configurations can be employed to

evaluated various quantities, in order to characterize the static and dynamical

properties of the GM3 bilayer.

§ 4.2 Mass Density Profiles

One of the more commonly evaluated properties in the investigation of a bilayer

structure is the mass density profile of species α, defined as

ραxy(z) =

⟨∑

i

mi δ(z − zi)⟩, (4.6)

where mi is the mass of the i–th atom at position zi along the z direction, Lx

and Ly are the length of the box vectors perpendicular to z, and the sum is

extended to the atoms of species α.

In Figures 4.2 and 4.4 the mass density profile of the GM3 bilayer is reported,

where the contributions of the main molecular components have been separated

out. The overall features are similar to those found for other lipid membranes (see

Figure 4.3 for comparison with DPPC), i.e. a well defined segregation between

solvent and lipid molecules, the former being in contact with the high density

region of the hydrophilic headgroups, and the latter enclosing the less dense,

hydrophobic lipid tails, in the central region.

Even though the general features of the density profile are similar to that

of bilayers of phospholipids like DPPC or DMPC, the peculiarities of the GM3

bilayer are already manifest, and regard both the hydrophobic and hydrophilic

regions. Indeed, the mass density inside the lipid region is slightly larger for GM3

than for DMPC or DPPC, even if the dry inner layer has an extension of about

2 nm, similar to what found in DPPC and DMPC bilayers (see e.g. [38], Figure

4 and [29], Figure 1).

Turning our attention to the region of the headgroups, a clear difference

between DPPC and GM3 membranes appears when the units composing the

head are separately observed. In Figure 4.4 the mass density profiles of the three

sugar units, as well as that of water and Na+, are reported. The three sugar

rings of the headgroup (i.e. sialic acid, galactose and glucose residues) are found

to have peaks at distinct positions. The same is not true for DPPC (see e.g.

[33]), where the peaks corresponding to the two charged groups are on the top of

each other. This particular distribution takes place because the charged groups


-4 -2 0 2 4z (nm)

0

200

400

600

800

1000

1200

1400

Den

sity

(kg

/m3 )

Figure 4.2: Mass density profile of the whole system (solid line), water (dottedline), ceramide tails (dashed line) and saccharidic headgroups (dot–dashed line).The profiles have been symmetrized with respect to the origin, placed at thecenter of the membrane.

Mass Density Profiles 51

0 2 4 6 8z (nm)

0

10

20

30

40

50

60

Num

eric

al d

ensi

ty (

atom

s/nm

3 )

Figure 4.3: Mass density profile of a DPPC bilayer from a molecular dynamicssimulation [46] at 325 K. The lines refer to the whole system (solid line), DPPC(dashed line), CH2 (dotted line), water (dot-dashed line), CH3 (dot dot-dashedline), N(CH3)3 (dashed thin line) and PO4 (solid thin line).


-4 -2 0 2 4z (nm)

100

200

300

400

500

600

Den

sity

(kg

/m3 )

Figure 4.4: Mass density profile of glucose (solid line), galactose (dashed line),sialic acid (dot–dashed line) residues, water (dotted line) and Na+ counterions(thin solid line). The profiles have been symmetrized with respect to the origin.

Orientational Order of the Lipid Chains 53

composing the phospholipid head lay preferentially on the bilayer surface so to

minimize the dipole–dipole interaction energy [36].

Both the differences in the hydrophobic and hydrophilic region, which emerge

from the analysis of the mass density profile, can be interpreted in terms of

orientational order as it will be shown in the following sections.

§ 4.3 Orientational Order of the Lipid Chains

The above mentioned characteristics of the hydrophobic region, suggest that

the tails of GM3 are more disordered than that of DMPC or DPPC. From a

qualitative point of view, the two tails, far from being aligned with the normal to

the membrane, are quite bended over and disordered, as it is possible to see from

the snapshot of the bilayer taken at equilibrium, and presented in Figure 3.6.

This observation can be validated by looking at the radial distribution functions

gα,β(r) of the terminal CH3 groups around the center of mass of the three sugar

rings, defined as

gα,β(r) =V

NαNβ

⟨∑

i,j

δ (r − |ri,α − rj,β|)⟩. (4.7)

Here V denotes the box volume, and the subscripts α and β identify distinct

species.

As shown in Figure 4.5, the terminal groups of the hydrocarbon chains have

quite high probability to be found close to the glucose residue. By integrating

the distribution function up to a cutoff, one obtains the coordination number

Nα,β(Rc) =

∫ Rc

04πρβgα,β(r)r2dr, (4.8)

where gα,β(r) is the radial distribution function of group β with respect to group

α, ρβ is the number density of group β, and the cutoff Rc is chosen to coincide

with the first minimum of the radial distribution function. The coordination

number Nglc,CH3of the CH3 group with respect to the center of mass of glucose

shows that every glucose residue has in its proximity about 1.5 terminal groups

of the hydrocarbon chains.

Even though ceramide tails are bended over, the degree of chain disorder can


0.5 1 1.5 2 2.5 3r (nm)

0

0.5

1

1.5

g (r

)

Figure 4.5: Radial distribution function of CH3 with respect to the center ofmass of glucose (solid line), galactose (dashed line) and sialic acid (dotted line)residues. The CH3 coordination number is equal to 1.5, 0.9 and 0.6 for glucose,galactose and sialic residue respectively.

Orientational Order of the Lipid Chains 55

5 10 15Atom number

0

0.02

0.04

0.06

0.08

0.1

0.12

-SC

D

Figure 4.6: Deuterium order parameter of the hydrocarbon chains of the ce-ramide, namely the fatty acid (squares) and the sphingosine (circles). The right-most points in the figure are referring to the end-terminal CH2 group. The linesare guide to the eye.

be quantified by examining the deuterium order parameter, SCD(n). defined as

SCD(n) =1

2Nc

⟨∑

i

(3 cos2(ηn,i)− 1

)⟩, (4.9)

where ηn,i is the angle encompassed by the C–H bond vector of the n–th carbon

on the i–th chain with the z axis, and Nc denotes the number of chains in the

present simulation, namely 128.

This order parameter is very important, because it is directly measurable by

nuclear magnetic resonance experiments on selectively deuterated chains. Com-

parison between simulation and experimental data sets regarding phospholipid

bilayers have shown an excellent agreement [47], thus indicating that this or-

der paramenter is one of the most reliable quantities accessible via molecular

dynamics techniques.

In the present case the computation of the deuterium order parameter cannot

be performed in a direct way, because of the use of united atoms. Therefore the


O

O

O

v

w

w

v

Figure 4.7: Schematic drawing showing the two vectors v and w used to definethe plane of a sugar ring (left) and the whole headgroup (right). See the text fordetails.

direction of the C–H bonds has been reconstructed by using the position of the

Cn, Cn−1 and Cn+1 atoms [48]. The order parameter of the hydrocarbon chains

in the GM3 bilayer, which is presented in Figure 4.6, is generally lower than in

phospholipid membranes (see e.g. [38], Figure 5) by at least a factor of 2, thus

confirming a high degree of disorder. It has to be noticed that the sphingosine’s

order parameter is lower than that of the fatty acid, and even negative in the

case of the first carbon atom. This feature can be ascribed to the presence of the

double bond, since unsaturated carbon atoms are known to be responsible for a

substantial reduction in the overall deuterium order parameter [49].

Nevertheless, in the present case, even the saturated hydrocarbon tail – the

fatty acid – presents a disorder higher than for phospholipids [38], which re-

flects out the relatively large steric occupancy of the carbohydrate headgroup.

Actually, the mean occupied surface per head results to be 0.67 nm2, about 10%

larger than for DPPC [38]. The steric hindrance of the hydrophilic moiety affects

the underlying hydrophobic region, where the tails tend to occupy the available

lateral space, thus becoming more disordered. Furthermore, the high degree of

disorder found in the tail region confirms that, at the chosen thermodynamic

conditions, the simulated system reproduces the Lα liquid crystalline phase [36].

§ 4.4 Orientational Order of the Headgroups

Even if the presented characteristics of the GM3 lipid region differ from that of

some phospholipids, it has been shown that the chemical nature of the hydrocar-

bon chains influences only in part the examined properties. An important role is

played also by the shape and overall dimension of the headgroup. Indeed, it can

be argued easily that the complexity of the GM3 headgroup can lead to major

effects in both the arrangement and dynamics of the whole GM3 molecules. For

Orientational Order of the Headgroups 57

30 60 90

Angle (degrees)

0.02

0.04

0.06

0.08

Prob

abili

ty d

ensi

ty30 60 90

0.05

0.1

0.15

30 60

0.01

0.02

0.03

0.04

0.05

30 60

0.01

0.02

0.03

0.04

0.05A

B C

D

Figure 4.8: Normalized distributions of the orientation of the vectors vin (solidline) and vp (dashed line), with respect to the bilayer’s normal. Panels A, B, Cand D refer to the entire headgroup, glucose, galactose and sialic acid residues,respectively.

example, the distinct position of the peaks in GM3, discussed in Section 4.2, is a

consequence of the orientational order of the three sugar rings, which are found

to be mainly aligned along the normal to the bilayer.

The orientational order of the head can be studied by looking at the dis-

tribution of the angles of the vectors that characterize the planes of the head

rings and the direction of the rings themselves, called θp and θin, respectively.

By choosing two vectors, v and w, that characterize the sugar rings plane, as

illustrated in Figure 4.7, we define the vectors

vin = (v + w) /|v + w|vp = (v ×w) / |v ×w|

to identify the longitudinal direction and the direction perpendicular to the

group’s plane, respectively. The directions are projected onto the unit vector


perpendicular to the layer plane n, so that

θin = arccos(vin · n)

θp = arccos(|vp · n|).

Given the angular histogram, 〈N(θ)〉, which represents the average number

of configurations forming an angle θ, the associated probability distribution ψ(θ)

can be computed as

ψ(θ) = 〈N(θ)〉 / sin(θ).

The normalized probability distributions for θp and θin relative to the whole

headgroup, glucose, galactose and sialic acid residues, are reported in Figure

4.8. For each group the plane is found to be highly aligned with the membrane

perpendicular axis, since the distribution of θp is peaked at 90 degrees. Moreover,

the distribution for θin has a maximum around zero, but is more spread out

than for θp, thus pointing out that the headgroups are tilted with some degree

of random orientation.

The alignment of the headgroup with the membrane perpendicular axis sug-

gests that GM3 molecules show good packing properties, and explains why phos-

pholipids and GM3 were found to have a comparable occupied area per head.

Given the large difference in the number of atoms, a naive picture would have

attributed a much larger area per head to GM3. On the other hand, the dipolar

interactions drive the phospholipid heads to lay on the bilayer surface, occupy-

ing a wide surface portion. The GM3 heads do not present a strong net dipole

moment, and consequently they align parallel to the membrane perpendicular

axis, thus reducing conspicuously the occupied surface.

§ 4.5 Headgroup Hydration

Let us now turn the attention to the study of the arrangement of both water and

Na+ ions in proximity of the GM3 headgroups. Water hydrates substantially the

sugar rings composing the headgroup, as monitored by the radial distribution

functions of water oxygen with respect to the center of mass of the three sugar

rings, reported in Figure 4.9.

As already mentioned in Section 4.3, one can obtain the coordination number

for water from the integration of the radial distribution function. This number

results to be equal to 19.7, 4.5 and 2.6 for the sialic acid, galactose and glucose

Headgroup Hydration 59

0.5 1 1.5 2 2.5 3r (nm)

0.2

0.4

0.6

0.8

1

1.2

g(r)

Figure 4.9: Radial distribution functions of water oxygen gw,α(r) with respect tothe center of mass of glucose (solid line), galactose (dashed line) and sialic acid(dotted line) residues.


residues, respectively. For the sake of comparison, we remind that in DPPC about

5.9 molecules of water are found around the phosphate group and 15.3 around

the N(CH3)3 group [33], and this fact points out the high packing achieved by the

GM3 headgroups. It is evident both from Figure 4.9, and from the computed co-

ordination numbers, that in the GM3 bilayer water cannot penetrate at the level

of the most buried groups of the head (glucose), despite the highly hydrophilic

nature of carbohydrate rings. Nevertheless, this behavior can be interpreted in

terms of a significant overlap of glucose residue with the two hydrocarbon tails,

which appears from the density profiles shown in Figures 4.2 and 4.4.

Like in the case of water, Na+ counterions structure is influenced by the

presence of GM3 molecules. Counterions are broadly distributed, as it is possible

to observe in Figure 4.4, presenting nevertheless a peak connected with the sialic

acid residues, that is, close to the negative charge, as expected. A non negligible

percentage of ions is found to be dissolved in bulk water. The total number

of dissolved Na+ ions can be obtained by integrating the corresponding density

profile in the range where their density profile itself is higher than that of GM3. It

is found that approximately 10% of the total number of counterions are dissolved

in solution.

In Figure 4.10 we report the radial distribution function of Na+ ions and

water around each of the oxygens of the COO− group. The first shell coordination

number obtained by integrating gO−,Na+(r) up to the first minimum is 0.3, and

therefore about 30% of Na+ ions are tightly bound to a COO− group. The strong

coordination of Na+ ions with COO− groups induces a Na+−Na+ ordering. The

well defined second peak of the Na+ radial distribution function indicates the

presence of this order, and turns out to be at the same distance of the first peak

in the O−−O− distribution function. Moreover, the radial distribution function

of water around O− shows that Na+ ions induce a marked coordination with the

solvent.

§ 4.6 Electric Field and Water Orientational Order

Due to the ionic character of the GM3 heads, the extent of the electrostatic

interaction acting between bilayers belonging to different images of the simulated

system can be, in principle, significant.

Therefore it is worth investigating whether finite size effects, related to the

electrostatic interaction, are important in the simulation of GM3 bilayers. One

Electric Field and Water Orientational Order 61

0.5 1 1.5 2r (nm)

1

2

3

4

5

6

7

g(r)

Figure 4.10: Radial distribution functions of the COO− oxygen atoms with re-spect to the water oxygen (solid line), Na+ ion (dashed line) and COO− oxygenatoms belonging to distinct GM3 molecules (dotted line).


-4 -2 0 2 4z (nm)

-2e+10

-1e+10

0

1e+10

2e+10

3e+10

Fiel

d (V

/nm

)

Figure 4.11: Contributions to the electric field arising from GM3 (solid line),water (dashed line) and Na+ ions (dot–dashed line), as well as the total electricfield (dotted line).


of the main quantities that can be looked at, is the profile of the electric field,

generated by the different molecular species, along the z direction. This quantity

can be estimated by integrating the charge density of the species of interest across

the simulation box according to the expression

E(z) =1

ε0

∫ z

−Lρq(z

′)dz′ + C, (4.10)

where ρq(z′) is the average charge density profile, −L corresponds to the mid-

point between periodic replicas of the membrane, and the constant C is chosen

so that the field at the center of the bilayer is zero, as it should be for an ideal

system with no fluctuations in the x–y plane and composed by two symmetric

monolayers [33].

Figure 4.11 illustrates the profile of the electric field across the simulation

box, having separated the contributions arising from different species. As it is

apparent, the charge unbalance arises mostly from the GM3 heads and sodium,

even if the contribution due to water is not negligible. From the figure it is also

clear that most of the counterions condensate in proximity of the membrane

surface, while a slow decay develops away from the membrane. Consistently,

water shows a similar slow decaying tail when approaching the bulk region. The

behavior of the electric field within the water layer suggests that the electrostatic

field induces a strong orientational polarization of the solvent, which persists over

a large region.

To quantify such an effect, we looked at the z dependence of the water dipole

unit vector projection D(z) onto the normal to the bilayer surface, defined as

D(z) =

⟨∑i cos(θi)δ(z − zi)∑

i δ(z − zi)

⟩, (4.11)

where θi is the angle encompassed by the dipole vector of the i–th water molecule

with the z axis. Results are shown in Figure 4.12, where the profile has been

refolded so that the origin coincides with the midpoint between periodic images

of the membrane.

The most important result is that the orientational order parameter decays

to zero rather slowly when going toward the bulk region. It has to be noted

that for every simulation box size the parameter must actually drop to zero at

the box edge, as imposed by the periodic boundary conditions. Reaching zero

is not a clear signature of the absence of finite size effects; rather, a plateau of


-3 -2 -1 0 1 2 3z (nm)

-0.05

0

0.05

0.1

0.15

0.2

D(z

)

Figure 4.12: Orientational order parameter of water. Data have been refolded sothat the the origin coincides with the midpoint between periodic images of thebilayer. The figure has been symmetrized with respect to the origin.


depolarization would be a better one. This slow decay seems to be a peculiarity

of the ionic nature of GM3, because in the case of non-ionic amphiphiles like

phospholipids, the polarization of water vanishes over an extension of about 1 nm

(see e. g.[50], Figure 3 and [29], Figure 1). In conclusion, even if the behavior of

the orientational order parameter suggests that the chosen size of the system is

large enough to allow water to depolarize, the lack of a clear plateau cannot leave

out the presence of finite size effects. This problem can probably be addressed

only by performing a molecular dynamics simulation with bigger interlamellar

spacing.

5.

Dynamical Properties

There the Loves a circle go,

The flaming circle of our days,

Gyring, spiring to and fro

In thos great ignorant leafy ways;

W.B. Yeats, The Two Trees.

§ 5.1 Ganglioside Headgroup Dynamics

It has already been shown in the section dedicated to the equilibration, that

the relaxation of the glycosidic linkage torsions takes place on a very long time

scale. Slow dynamics is also evident at the equilibrium, as we will see during

the presentation of the results regarding some dynamical quantities. The most

distinctive character is found to be in the dynamics of the hydrophilic region,

whereas for the hydrophobic one the features look very similar to that of other

amphiphilic molecules.

5.1.1 Headgroup Rotational Dynamics

Gangliosides dynamics can be usefully investigated by means of rotational cor-

relation functions

CR(t) = 〈n(t) ·n(0)〉 , (5.1)

where n is a unitary vector defined on the basis of atomic positions, which iden-

tifies some direction of interest. In the case of the heagroups, it is convenient to

67

68 Dynamical Properties

0 2000 4000 6000time (ps)

0.9

0.92

0.94

0.96

0.98

1

CR

T(t

)

Figure 5.1: Rotational autocorrelation function of the vectors vin associated toglucose (solid line), galactose (dotted line), and sialic acid (dashed line) moieties,as well as that of the whole headgroup (dot-dashed line).

use the vectors vin and vp — already introduced to investigate the headgroups

arrangement in Section 4.4 — that identify the longitudinal and perpendicu-

lar directions to the group’s plane, respectively. Therefore, for every sugar ring

composing the headgroup, two correlation functions can be computed, namely

CRT(t) and CRP(t), defined as

CRT(t) = 〈vin(t) ·vin(0)〉 (5.2)

CRP(t) = 〈vp(t) ·vp(0)〉 . (5.3)

The results are presented in Figures 5.1 and 5.2, where the autocorrela-

tion functions CRT and CRP for the three sugar rings as well as for the whole

headgroup are shown. The most striking feature of all the presented correlation

functions is their significantly slow decay. Notably, more than 6 ns are necessary

to let the correlation functions to decay by roughly 1–10%. For comparison, the

Ganglioside Headgroup Dynamics 69

0 2000 4000 6000time (ps)

0.9

0.92

0.94

0.96

0.98

1

CR

P(t)

Figure 5.2: Rotational autocorrelation function of the vectors vp associated toglucose (solid line), galactose (dotted line), and sialic acid (dashed line) moieties,as well as that of the whole headgroup (dot-dashed line).


typical time of 90% decorrelation is of the order of the picosecond for water and,

as we will see in Section 5.2, about 100–1000 ps for the lipid tails. On the other

hand, the differences among various sugar units are not big indeed, both for the

in-plane and for the out-of-plane rotations, and only the magnification of the

ordinate axis in the range from 0.9 to 1 makes them appreciable.

Going into detail of the decay patterns, the vin vector presents a faster re-

orientation time for the sialic acid residue (shown in Figure 5.1), with respect

to the other two rings. This fact should be primarily the consequence of the

sialic acid being connected to only one residue, as well as being the ring more in

contact with the solvent.

In the case of vector vp (shown in Figure 5.2) the three sugar rings follow

almost the same pattern, but for an initial slightly different decay. Even the

sialic residue does not show significant differences with respect to glucose and

galactose. On the contrary, the reorientation of the out-of-plane vector associ-

ated to the whole headgroup seems to relax significantly faster, with respect to

the in-plane one. This difference is likely to be caused by the GM3 headgroups

orientation: they are forced to remain aligned, pointing toward the water layer,

and on the other hand they have no limitation regarding the rotation around

the z direction, thus presenting a faster decorrelation of CRP.

As it could be expected from the results presentedin Section 3.3, the time

scale involved in the headgroup relaxation is so long that little information can

be extracted, even from a multi-nanosecond simulation such as the present one:

With respect to the time scales of the processes involving water molecules or

even hydrocarbon chains (as we will see in Sections 5.2 and 5.3) the headgroups

can be considered almost static.

5.1.2 Headgroup Mean Square Displacement

In addition to the rotational dynamics of ganglioside headgroups, it is interesting

to observe the translational behavior by means of the mean square displacement

(msd) µ(t), defined as

µ(t) =1

N

⟨N∑

i

[ri(t)− ri(0)]2

⟩, (5.4)

where the sum is extended to the N atoms or centers of mass of the molec-

ular groups. Moreover, because of the planar arrangement of the ganglioside

Ceramide Rotational Dynamics 71

molecules, it is natural to compute separately the msd in the bilayer plane, µxy,

and in the orthogonal direction, µz. Figure 5.3 shows the results of the molec-

ular dynamics simulation regarding the msd in the xy plane and along the z

direction. As it will be addressed more specifically in Section 5.3, a linear msd

time dependence identifies a diffusion process for molecules in an homogeneous,

unbounded medium. The data regarding the motion in the xy plane reveal that

the msd never attains a linear behavior in the whole investigated time span,

thus suggesting that gangliosides do not reach a diffusive regime. Moreover, the

region explored by GM3 headgroup after 1 ns is very small, having a radius of

about 0.3 nm, comparable to a carbohydate ring diameter. Notably, this pat-

tern is very close to that observed for phospholipids, such as, for example, that

of DPPC (Essmann and Berkowitz [51], Figures 3 and 4). An explanation for

this behavior can not be formulated easily: the displacement achieved by GM3

headgroup within the investigated time span is too small with respect to the

molecular size to appreciate motions other than the wobbling in the cage of

neighboring molecules. However, increasing the investigated time span, and at

the same time keeping an appropriate accuracy, would require simulation runs

that are by far longer than that available at present.

The case of the z direction, instead, is markedly different from the previous

one, because it does not seem to grow any longer after 200 ps. This behavior is

strongly related to the planar nature of the GM3 bilayer: as it will be shown in

Section 5.3, the presence of a plateau in the msd is consistent with the confined

condition along the z axis of the investigated system.

§ 5.2 Ceramide Rotational Dynamics

In the case of hydrocarbon chains, to define the vector employed in the calcu-

lation of the rotational correlation function, it is natural to choose triplets of

consecutive CH2 group (i.e., two bonds sharing one CH2 group), at position r0,

r0 + r1, and r0 + r2. Vector n can therefore be defined as

n =r1 × r2

|r1 × r2|. (5.5)

Figure 5.4 reports the correlation functions CR(t; k) for selected triplets along

the fatty acid chain, where k indicates the carbon atom number, following the

labeling scheme of Figure 5.5.


0 200 400 600 800 1000time (ps)

0

0.02

0.04

0.06

0.08

0.1

MSD

(nm

2 )

Figure 5.3: msd of the heagroup center of mass in the xy plane (solid line) andalong the z direction (dashed line). The contribution given by the system centerof mass motion has been subtracted.


10 100 1000time (ps)

0.01

0.1

1

CR

T(t

)

Figure 5.4: Rotational autocorrelation functions CR for selected atoms. The solid,dotted, dashed, dot-dashed and dot dot-dashed lines refer to the central atomsof the triplets labeled in Figure 5.5 as 1,3,5,7, and 9, respectively.

HO

NH

O

CH3

OH

HO

HO

O

COO

O

OHOH

O

OH

O HO

HO

O

OH

O

HO H

NH

O

H

Galneu−ac Glcfatty acid

sphingosine

1 3

2

5

4

7

6

−

Figure 5.5: Structure formula of GM3 along with the numbering scheme adoptedto identify the carbon atoms employed in the computation of the rotationalautocorrelation functions


All the correlation functions are quite different in magnitude, although from

a qualitative point of view they seem to share the general shape. The decrease

in overall magnitude observed in concomitance with the increased distance from

the headgroup region can indeed be justified by the following simplified model.

Instead of representing the hydrocarbon chain in a detailed fashion, the vec-

tors nk employed in the calculation of the autocorrelation functions can be mod-

eled as rotating around one common axis. The conformation of the chain, which

— in the real system — is determined by N dihedral angles acting as Lagrange

generalized coordinates, is now identified by N angles that describe the rota-

tion around the single axis. Therefore, care has to be taken in distinguishing

between the rotations around the axis regarded as Lagrange coordinates (αk)

and regarded as angles in the fixed reference frame of the simulation box (θk).

In the real system case, the orientation of nk+1 can be written as a more or

less complicated function of the previous vector nk and of the torsional angle

αk+1. In the model system this dependence can be introduced, for example, by

imposing that

θk+1 = θk + αk+1. (5.6)

This relation is obviously not the only possible choice, nevertheless it is useful to

catch the general properties of the real chain. The system configuration can be

therefore described using the distribution probability Ψ(α, t) for the generalized

coordinate α and Φk(θ, t) for the angle formed by vector nk with respect to the

fixed frame coordinates. The distribution Ψ has been written explicitly without

a subscript, emphasizing that different segments behave the same way, therefore

ignoring environmental inhomogeneities. Having observed that the rotational

autocorrelation function of vectors belonging to the headgroup decays on a longer

timescale with respect to that of tail vectors (compare Figures 5.1 and 5.4 ), it is

a fairly good approximation to consider the headgroup to be static, and therefore

a fixed reference frame for the tails. The presence of this reference implies that

in the model system the distribution for the first angle θ1, which is next to the

headgroup, coincides with that of angle α, so that

θ1 = α1,

thereby imposing that the first angle θ1 is distributed the same way as the

torsional angles, namely

Φ1(θ, t) = Ψ(α, t). (5.7)


It appears now feasible to express the probability distribution function for the

generic angle θk, and therefore the autocorrelation function CR(t; k), in terms of

the probability distribution of the first angle θ1. The distribution for the generic

angle θk can be written by requiring that

θk = θk−1 + αk,

in the following way

Φk(θ) =

∫Φk−1(θ1)Ψ(α) δ(θ − θ1 − α) dθ1 dα

=

∫Φk−1(θ1)Ψ(θ − θ1) dθ1

= [Φk−1 ?Ψ1] (θ) = [Φk−1 ? Φ1] (θ), (5.8)

where in Equation 5.8 the operation of convolution has been denoted by the

symbol ?. In order to obtain the last expression, equivalence (5.7) has been

employed. It is straightforward to recursively apply the previous equation to

obtain

Φk = Φ1 ? Φ1 . . . ? Φ1︸︷︷︸k times

(5.9)

Equation 5.9 is the starting point for the computation of the reorientational

correlation function of the k–th vector, which reads

CR(t; k) =

∫cos(θ)Φk(θ) dθ (5.10)

=

∫eiqθΦk(θ) dθ

∣∣∣∣q=1

(5.11)

=

∫eiqθΦ1 ?Φ1 . . . ? Φ1(θ) dθ

∣∣∣∣q=1

(5.12)

=[Φ1(1)

]k(5.13)

where Φ1(q) denotes the Fourier transform of Φ1(θ), apart from a constant factor,

and the symbol CR is employed to distinguish the correlation function given by

the model from the sampled one, CR. As an auxiliary hypothesis, the distribution

function Φk(θ) is considered to be symmetric around θ = 0, thus allowing to write

Equation 5.11. The case k = 1 states that

Φ1(1) = CR(t; 1), (5.14)


0 500 1000time (ps)

0

0.2

0.4

0.6

0.8

1

CR(t

)

0 500 1000

Figure 5.6: Autocorrelation functions for different carbon atoms along the fattyacid chain (left panel) and the rescaled functions (right panel). The differentcurves, from the upper to the lower one, refer to the carbon atoms 1–7, followingthe convention presented in Figure 5.5

and therefore it is possible to write in simple form the rotational autocorrelation

function of the k−th vector as a function of the correlation of the first one.

Namely, from (5.13) and (5.14) follows that

CR(t; k) =[CR(t; 1)

]k. (5.15)

This relation can be written in a form that highlights the value of the ratio

CR(t; k)/CR(t; 1):

CR(t; k)

CR(t; 1)=[CR(t; 1)

]k−1. (5.16)

Within this model the correlation functions for different carbon centers can

be related via a power law. It can be therefore expected that even in the real

— and more complex — case, some kind of power-law relation can be found

between different correlation functions. The simplest generalization of Equation

Water Dynamics 77

5.16 isCR(t; k)

CR(t; 1)= [CR(t; 1)]α(k−1) , (5.17)

where α can be considered as an adjustable parameter. Indeed, this relation

appears to reproduce well the patterns of the reorientational correlation func-

tions, by setting the free parameter α to 1/6. Figure 5.6 reports the comparison

between the correlation functions CR(t; k) (k ranging from 1 to 7) and those

obtained by the scaling procedure (5.17), namely

[CR(t; 1)]−(k−1)/6 CR(t; k). (5.18)

In the ideal case, the scaled functions (5.18) would be coincident with CR(t, 1),

as follows from Equation 5.17. From the right panel of Figure 5.6 one can ob-

serve that the scaled curves differ from each other less than 10%, thus showing

that the simple model, here introduced, accounts for an important amount of

the dynamical correlation. Nevertheless, differences cannot be ascribed to statis-

tical error, and reflect the influence of the system inhomogeneities, as well as, of

course, the approximations inherent to the model.

§ 5.3 Water Dynamics

To study the dynamics of water in interaction with the ganglioside bilayer sur-

face, it is natural to look at the dependence of water mobility on the distance

from the surface itself. The two quantities more commonly employed to look at

molecular mobility are the velocity autocorrelation function and the mean square

displacement. Generally, a parameter employed to characterize the dynamics of

liquids, which can be derived from the msd or from the velocity autocorrelation

function is the diffusion coefficient D, defined for continuous media by Fick’s law

ρ(r, t)u(r, t) = −D∇ρ(r, t), (5.19)

where ρ and u are the fluid density and local velocity, respectively. In case of

isotropic, unbounded systems, the second Fick’s law can be derived,

∂ρ

∂ t= D∇2ρ, (5.20)


which permits to relate the diffusion coefficient D to the msd via the Einstein

relation

D = limt→∞

µ(t)

6 t(5.21)

The previous expression is obviously no more true in the case of inhomoge-

neous or confined fluids, and other methods have to be developed to relate the

diffusion coefficient appearing in equation 5.19 with quantities that are directly

accessible via the molecular dynamics simulation technique.

In recent years the diffusion of liquids in confined geometries has attracted

the attention of many investigators [52, 50, 53, 54, 55, 56, 57, 58, 59]. As a matter

of fact it has been observed that in many physical situations the msd of a single

particle does not grow linearly in time, as predicted by the brownian approxima-

tion in the long time regime. The onset of spatial and temporal correlations has

in general been invoked to explain this phenomenon. From a phenomenological

point of view, the presence of such correlations can be introduced by assuming

that anomalous diffusion and non-exponential relaxation manifest themselves in

presence of a corrugated surface, so that the msd can be written as

µ(t) = 6Dtα,

as pointed out by different authors [59, 58, 60]. In particular, this approach has

been used to explain the diffusion of water in proximity of a macromolecular

surface or a multilamellar bilayer, as revealed by experimental and computer

simulation works. However it has also been argued that the effect of confinement

alone, i.e. independent on the microscopic details of the confining medium, can

explain the observed non linear time behavior of the msd. Lindahl and Edholm

[53] suggested that diffusion of water close to the surface of a protein (myoglobin),

modeled as an infinite and sharp reflecting plane, leads to a non linear time

dependence of the out-of-plane diffusion. However, their statement has not been

clearly substantiated by numerical experiments. In the following, besides the

presentation of simulation results, an attempt to clarify the interpretation of the

data on the basis of the diffusion equation with appropriate boundaries will be

presented.

In a hydrated bilayer system, like the GM3 one, it is natural to study sep-

arately the motion of water in the direction perpendicular (z direction) and

parallel (xy plane) to the bilayer surface, as well as the dependence of the msd

from the initial position along the z axis.

Water Dynamics 79

-4 -2 0 2 4Box (nm)

0

200

400

600

800

1000

1200

1400

Den

sity

(kg

/m3 )

(a)(b)(c)

Figure 5.7: Mass density profile of the simulated system (solid line) decomposedfor different species, namely water (dotted line), lipid tails (dashed line) andsaccharidic headgroups (dot–dashed line). The vertical dotted double lines (a),(b) and (c) indicate the initial position of water molecules whose msd is detailedin Figs.(5.8, 5.9, 5.11). The solid vertical line indicates the resulting position ofthe outer edge of the effective water layer.


0.1

1

10

MSD

(nm

2 )

10 100 1000time (ps)

0.1

1

10

Figure 5.8: msd along the z direction (upper panel) and in the xy plane (lowerpanel) in the whole time range. Squares and diamonds refer to molecules startingfrom slabs (a) and (c), respectively. For comparison the linear time dependenceis shown as a solid line.

Water Dynamics 81

It has been evaluated the in-plane and out-of-plane msd of molecules that at

t = 0 were located in slabs of thickness ∆z = 0.1 nm and at distance z0 between

0 and 3 nm from the midpoint of the water layer. An overview of the full diffusive

behavior is presented in fig 5.8 for two selected slabs (shown in Figure 5.7). In all

cases three distinct timescales can be identified; a short time regime (t < 20 ps),

an intermediate regime (20 < t < 500 ps) and a long time span above 500 ps. The

short time scale is characterized, for both the in-plane and out-of-plane msd, by

a linear behavior, which ends at the intermediate time region. In the third region

the in-plane msds of molecules starting from different slabs converge to the same

linear behavior, whereas the perpendicular msd reaches plateau values dependent

on the starting slab.

In view of these results it has been used a simplified theoretical model to

give an interpretation of the behavior of the msd. At first one has to write the

diffusion equation for the probability density distribution G(r, t; z0) and impose

appropriate boundary conditions to model the presence of the bilayer. These

conditions can be included by considering the bilayer surfaces as rigid reflecting

planes, i.e. by imposing the Neumann boundary conditions,

∂

∂zG(r, t; z0) = 0,

at the bilayer surface. Moreover, because the solvent is definitely inhomogeneous,

it is reasonable to introduce a position dependent diffusion coefficient D(z). The

complete diffusion equation then reads

G(r, t; z0) = ∇ · (D(z)∇G(r, t; z0)) . (5.22)

Since the quantities of interest are computed by performing an average either

on the xy plane or on the z direction, it is useful to introduce two reduced

probability distributions defined as

P‖(x, y, t; z0) =

∫G(r, t; z0)dz

P⊥(z, t; z0) =

∫G(r, t; z0) dxdy,

so that the perpendicular msd can be written as

|z(t)− z(0)|2 =

∫ `

0|z(t)− z(0)|2 P⊥(z, t; z0)dz. (5.23)


Here ` represents the effective width of the water layer. Integrating the diffusion

equation (5.22) on the z or xy domains, one derives the evolution equations for

P‖ and P⊥,

P‖(x, y, t; z0) =

[∂2

∂x2+

∂2

∂y2

] ∫ `

0D(z)G(r, t; z0)dz (5.24)

P⊥(z, t; z0) =∂

∂z

[D(z)

∂

∂zP⊥(z, t; z0)

], (5.25)

respectively. It is worth noticing that the equation for P⊥ is completely decoupled

from the dynamics in the xy plane, whereas the time evolution for P‖ does indeed

depend on the full probability density distribution G(r, t; z0). However, in the

two time regimes of short and long times, it is possible to decouple the evolution

of P‖ from the dynamics on z, as we will soon see.

5.3.1 Short Time msd

Let us start by examining the short time behavior. A linear fit performed in the

range 5-20 ps shows a good agreement with the simulation data (see Figure 5.9).

Thus, in this time interval the molecules undergo a standard brownian motion

with a diffusion coefficient dependent on the initial condition, as illustrated in

Figure 5.10. The simple diffusion behavior that characterizes the short time

regime can be easily derived: equations (5.24) and (5.25) can be trivially solved

at times short enough to consider D(z) ' D(z0) (i.e. constant along the spatial

scale explored by the molecules, an approximation valid for particles traveling

distances shorter than D/|∇D|) because under this condition they reduce to

simple diffusion equations.

From the previous discussion it follows that the presence and extension of

this linear regime strongly depends on the steepness of D(z) and consequently on

the degree of inhomogeneity. In the time interval 0-20 ps water molecules have

traveled ∼ 0.3 nm along the z direction, corresponding to about 3 slabs. Con-

sequently one can safely neglect the effect of boundary conditions for molecules

not too close to the surface.

5.3.2 Long Time msd

Equations (5.24) and (5.25), which correctly describe the msd in the short time

span, are even very informative in the long time regime, when the effects of

Water Dynamics 83

0 10 20 300

0.5

1

0 10 20 30time (ps)

0

0.2

0.4

MSD

(nm

2 )

Figure 5.9: Short time behavior of the out-of-plane (left panel) and in-plane msd(right panel). Diamonds, circles and squares refer to molecules starting from slabs(a), (b) and (c), shown in Fig.5.7, respectively. The continuous lines representthe result of a linear fit in the interval 5-20 ps.


2 3 4 5bin position (nm)

0

2

4

6

D (

10-3

nm2 ps

-1)

Figure 5.10: Profile of the diffusion coefficient computed from the short timedependence of the msd along the z direction (squares) and in the xy plane(circles) with an estimate of an error bar. The solid horizontal line indicatesthe value of the diffusion coefficient of spc water at the same thermodynamicconditions.

Water Dynamics 85

confinement and the spatial inhomogeneity produce the rich phenomenology seen

in Figure 5.8. At sufficiently long times the motion in the z and xy plane becomes

uncorrelated, so that the distribution can be factorized as

G(r, t; z0) ' P‖(x, y, t; z0)P⊥(z, t; z0), (5.26)

thus allowing to compute the msd separately for the z direction and xy plane.

In-plane msd — By substituting the factorized probability density distribu-

tion (5.26) into the evolution Equation 5.24 one obtains

P‖(x, y, t; z0) = D(z0, t)

(∂2

∂x2+

∂2

∂y2

)P‖(x, y, t; z0), (5.27)

where

D(z0, t) =

∫ `

0D(z)P⊥(z, t; z0) dz.

In the long time regime P⊥ becomes stationary and, if particles are not

trapped by the bilayer, the distribution does not depend any more on the starting

point z0 and converges to the normalized water density profile along the z axis,

n(z). The asymptotical independence of D(z0, t) from the initial condition is

confirmed by the observation that all in-plane msds converge to the same slope,

as evident from Figure 5.8, upper panel. Moreover, it can be verified that at

enough long times, the distribution attains the stationary state by computing

separately the two terms of the expression

∫ `

0D(z)n(z)dz = lim

t→∞D(z0, t) (5.28)

where the l.h.s. is the weighted integral of the diffusion coefficient computed at

short times and the r.h.s. is taken as a linear fit of the in-plane msd at long

times. The values obtained from the simulation data for the two members of

equation 5.28 show an excellent agreement, being 5.7 and 5.6± 0.1 10−3nm2ps−1,

respectively. Therefore, even in the long time regime, the in-plane diffusion is well

described by the presented theoretichal scheme.

Out-of-plane msd — Differently from the previous case, here the evolution

equation does not produce any longer the standard Einstein diffusion, because

the diffusion coefficient depends on the z variable. Nevertheless one can derive


the limiting solution at long times by ignoring this dependence, thus substituting

D(z) = D, but retaining the same reflecting boundary conditions. In this case

the diffusion equation can be integrated, for example, by variable separation and

integration in terms of the laplacian eigenfunctions [61]. Variable separation is

obtained by factorizing the distribution function P⊥(z, t; z0) as

P⊥(z, t; z0) = T (t)Z(z; z0),

thus allowing to rewrite the diffusion equation for P⊥ in the form

T (t)Z(z; z0) = D T (t)d2

dz2Z(z; z0). (5.29)

Dividing both members of equation 5.29 by P⊥(z, t; z0), after simple algebra one

obtainsT (t)

T (t)=

DZ(z; z0)

d2

dz2Z(z; z0). (5.30)

Since the l.h.s and r.h.s members of previous equation depend solely on the t

and z variable, respectively, the solution can be found by solving the following

two ordinary differential equations

T (t) = −λDT (t) (5.31)

d2

dz2Z(z; z0) = −λZ(z; z0). (5.32)

Equation 5.32 can be solved, taking care of the boundary conditions, by a

linear superposition of the following normalized eigenfunctions un:

{un(x) = An cos(

√λnz)

λn = (πn/`)2,

where the normalization is assured by the proper setting of the coefficients

An =

√2` n 6= 0√1` n = 0

.

With this choice for the coefficients An, the set of eigenfunctions un becomes an

Water Dynamics 87

orthonormal basis, that is

∫ `

0un(x)um(x) = δnm,

and the general solution to the diffusion in the [0, `] interval can thus be written

P⊥(z, t; z0) =∑

n

cnun(z)e−λnDt, (5.33)

where cn are generic constant coefficients. It is possible to determine the value

of the coefficients cn by computing the scalar product

∫ `

0P⊥(z, 0; z0) um(z) dz =

∫ `

0

∑

n

cnun(z)um(z) dz (5.34)

=∑

n

cnδnm = cm, (5.35)

and at the same time by imposing as initial condition

P⊥(z, 0; z0) = δ(z − z0).

Therefore, the l.h.s. of Equation 5.34 can be evaluated explicitly, leading to the

determination of the coefficients cn, which read

cn = un(z0).

The complete solution for the diffusion equation is, eventually,

P⊥(z, t; z0) =1

`+

∞∑

n=1

2

`exp

[−D(πn)2t

`2

]×

cos

(πnz

`

)cos

(πnz0`

). (5.36)

It is straightforward to compute the corresponding msd from Equation 5.36,

which turns out to be

|z(t)− z0|2 =1

3`

[(`− z0)3 + z3

0

]+

∞∑

n=1

4`

(nπ)2×

exp

[−D(πn)2t

`2

]cos

(πnz0`

)[z0 + (`− z0)(−1)n] . (5.37)


0 500 1000 1500 2000time (ps)

0

1

2

3

4

5

6

7

8

9

MSD

(nm

2 )

50 100 150

1

Figure 5.11: msd along the z direction for molecules starting from slab (a) (dia-monds), (b) (circles), and (c) (squares). The lines represent the result of Equation5.37 with an effective parameter D = 6.5 10−3nm2ps−1. In the inset moleculardynamics data are compared with the numerical solution in the intermediatetime range.

Water Dynamics 89

Before being able to compare this result with simulation data, one has to note

that, because the water density profile does not go abruptly to zero at the bilayer

surface, it is not obvious how to define the thickness ` of this region. However,

an effective thickness can be estimated by noticing that at very long times the

msd for molecules starting from the center of the region (z0 = `/2) reaches the

asymptotic limit `2/12. A thickness of 5.1 nm can therefore be deduced from

the simulation data. This value turns out to be compatible with the thickness of

the water slab obtained from the density profile, corresponding in particular to

water molecules that are found slightly beyond the position of the peaks of the

bilayer hydrophilic part (see Figure 5.7).

The coefficient D introduced in eqs. (5.36–5.37) plays the role of an effective

parameter that fixes the time scale over which P⊥(z, t; z0) becomes uniform.

By taking D = 6.5 10−3nm2ps−1 as an educated guess, a very good agreement

between the simulation results and Equation 5.37 is found, as shown in Figure

5.11, and in particular the agreement for water molecules starting from slab (c)

is almost perfect in the whole explored range. The chosen value for parameter

D is well within the range of estimated values for the diffusion coefficient and

comparable with the average value.

Having set D = const, effects introduced by spatial inhomogeneities are cer-

tainly neglected. However, the good agreement reinforces the idea that geomet-

rical constraints alone correctly describe the time evolution of msd in a wide

temporal range. Interestingly, the solutions show both a superlinear (Figure 5.11,

inset) and a sublinear behavior as often reported in the literature [59, 62] thus

accounting qualitatively for the observed time dependence of msd.

6.

Summary and Conclusions

“Forty-two,” said Deep Thought, with infinite majesty

and calm.

D. Adams, The Hitchhiker’s Guide to the Galaxy.

This thesis reports the results of the first molecular dynamics simulation of a

GM3 ganglioside bilayer at full atomistic detail. The complete topology of the

GM3 molecule has been written from scratch, and the gromos force field has

been chosen to model the system. Some modifications to the standard force field

have been introduced to achieve a more realistic description of certain groups.

Namely, the potential model for the hydrocarbon moiety has been improved

by using the parameters suggested by Berger, whereas the potentials describing

the sugar moiety have been modified by following the procedure suggested by

Spieser, thus accounting for the rigidity of the sugar rings as well as for the

gauche and exo-anomeric effects.

A considerable amount of time was spent in finding a suitable starting con-

figuration and bringing the system to the equilibrium. Very long relaxation times

have been observed during this phase, which resulted to be characterized by a

slow hydration process, as well as by a relaxation of specific glycosidic torsional

angles. These two processes appear to be suitable indicators for the progress

toward equilibrium.

The quality of the simulated system was tested against the results of angle

X-ray scattering experiments. The comparison turns out to be particularly good,

indicating that the employed modelization of the GM3 bilayer provides a good

91

92 Summary and Conclusions

description of the real system.

Several properties have been studied in order to characterize the GM3 bilayer.

Moreover, when feasible, a comparison with similar properties of phospholipid

bilayers has been made. The tails are found to be more disordered in GM3

than in phospholipids at similar thermodynamic conditions. In particular this

finding supports the hypothesis of a significantly flexible structure, introduced

to explain the experimental evidence of spontaneous vesicles formation. On the

contrary, the headgroups are found to exhibit a much more ordered orientational

structure. In particular, the sugar rings are found to lay preferentially on a plane,

which is orthogonal to the bilayer surface, whereas the longitudinal axis of the

headgroups presents some degree of tilting.

Looking at the membrane–water interface, the hydration of the whole head-

group resulted to be more pronounced in GM3 than in phospholipids, and this

fact is possibly due to the strong hydrophilic character of the GM3 headgroups.

About 90% of the whole Na+ ions are found to be condensed in the headgroup

region. However, the little percentage of uncondensed counterions is enough to

introduce long-range effects on the solvent, resulting namely in a slowly decaying

polarization of water away from the membrane, in sharp contrast to the case of

phospholipids.

The analisys of the dynamical properties revealed that three main timescales

can be identified in the system, namely that of the solvent, of the lipid tails

and of the headgroups. The slowest one turned out to be that of the headgroup

rotations, for which a complete relaxation has not been observed within the

time span of the whole simulation. Similarly, the headgroup center of mass is

constrained to very small displacements on the bilayer plane. The dynamics of

the headgroup can therefore considered to be almost decoupled from that of the

lipid chains and, a fortiori, from that of the solvent.

The orientational dynamics of GM3 molecules shows a dramatic change when

looking at the motion of the chains. Due to their much lower degree of orienta-

tional order, the relative relaxation times are much faster than the corresponding

ones for the headgroups. For example, the time scale is found to be ' 100 and

1000 ps for the middle of the chain and for the carbon atoms close to the hea-

group, respectively. A simple model has been suggested, which shows how this

difference can be explained in terms of topological and statistical considerations.

The presence of long relaxation processes required significantly long simula-

tion runs. Having at disposal a large number of water molecules and long trajec-

93

tories, we were allowed to investigate with high statistical accuracy the dynamical

features of water molecules.

It has been demonstrated that spatial inhomogeneities and boundary con-

ditions play a fundamental role for the interpretation of diffusive processes in

presence of non-permeable surfaces, like the GM3 bilayer. Moreover, it has been

found that the dynamics of water can be well described, at different time scales,

by a simple brownian model, supplemented by appropriate boundary conditions.

Ordinary Fick’s law correctly describes diffusion in the plane parallel to the bi-

layer surface, provided that a judicious choice of the diffusion coefficient is made.

In the direction perpendicular to the bilayer surface the msd has been found to

be again linear for enough short times, whereas at long times the effect of con-

finement becomes crucial in determining the msd pattern. However, even in the

latter case, the msd can be quantitatively reproduced by means of a diffusion

equation — once the correct boundary conditions are imposed — and, notably,

both superlinear and sublinear behaviors can be accounted for by this simple

model.

In conclusion, we have proved that a realistic simulation of a biologically im-

portant molecule — namely GM3 ganglioside — in the bilayer aggregation form

is feasible on the time scale accessed by simulation. This achievement paves

the way to more extensive studies concerning both the structural and dynami-

cal properties of gangliosides aggregates at different thermodynamic conditions.

There are several key points that have to be addressed:

– The contribution to the bilayer stability coming from the hydrogen bonds

has to be estabilished, by clarifying to which extent water plays a role in

building a hydrogen bonds network together with sugar rings.

– A finite size study has to be completed, in order to investigate long range

effects on the polarization of water and the influence of the interaction

between periodic images of the system on the bilayer structure.

– A comparison between simulation and experimental data given by wide

angle X-ray scattering has to be performed, by investigating especialy the

contribution to the scattering intensity coming from the spatial ordering

of the headgroups.

– The vibrational and peristaltic modes of the membrane have to be stud-

ied, to account quantitatively for the value of the GM3 bilayer bending

94 Summary and Conclusions

modulus.

– Collective phenomena interesting the headgroups have to be investigated.

This point is of particular interest, in view of the simulation of other gan-

glioside aggregates, such as the GM1 micelles, whose aggregation number

is thought to be strongly dependent on cooperative phenomena at the

headgroup level.

Appendix: GM3 topology

In this appendix the complete topology employed to describe the GM3 molecule

in the format required by the gromacs molecular dynamics package is reported.

The topology file is divided in many sections: the atoms name, type, charge

and mass, as well as the atom sequential number are listed in the [atoms]

section, while the bond, angle, torsion and 1–4 interaction parameters are listed

in the [bonds], [angles], [dihedrals], and [pairs] sections, respectively.

The parameters, as well as the atom names, are those presented in Section 2.2.

At the end of the topology a list of the Lennard-Jones parameters is presented.

At the beginning of every section a comment line is present, which explains the

content of subsequent fields, following the notation:

nr sequential atom number

resnr residue number

resid residue name

cgnr charge group number

funct interaction type (see Table 2.4)

i, j, k, l atom numbers involved in interaction

r0, theta0 equilibrium lenght and angle

kr, tkheta stretching and bending interaction strengths

C6, C12 Lennard-Jones coefficients

phase, f dihedral term phase and interaction strength

mult dihedral multiplicity

95

96 Appendix: GM3 topology

6.0.3 GM3 topology

[ moleculetype ]2 ; Name nrexcl

GM3 34 ; all non-bonded interactions between atoms that dist

; nrexcl or less consecutive bonds are excluded.6 ;

[ atoms ]8 ; nr type resnr resid atom cgnr charge mass

;10 1 LP3 1 GM3 CDP 1 0.000 15.035 ; qtot 0

2 LP2 1 GM3 CDD 1 0.000 14.027 ; qtot 012 3 LP2 1 GM3 CCQ 1 0.000 14.027 ; qtot 0

4 LP2 1 GM3 CBU 1 0.000 14.027 ; qtot 014 5 LP2 1 GM3 CBD 2 0.000 14.027 ; qtot 0

6 LP2 1 GM3 CAP 2 0.000 14.027 ; qtot 016 7 LP2 1 GM3 CAO 3 0.000 14.027 ; qtot 0

8 LP2 1 GM3 CAN 3 0.000 14.027 ; qtot 018 9 LP2 1 GM3 CAM 4 0.000 14.027 ; qtot 0

10 LP2 1 GM3 CAL 4 0.000 14.027 ; qtot 020 11 LP2 1 GM3 CAK 5 0.000 14.027 ; qtot 0

12 LP2 1 GM3 CAJ 5 0.000 14.027 ; qtot 022 13 LP2 1 GM3 CAI 6 0.000 14.027 ; qtot 0

14 LP2 1 GM3 CAH 6 0.000 14.027 ; qtot 024 15 LP2 1 GM3 CAG 7 0.000 14.027 ; qtot 0

16 LP2 1 GM3 CAF 7 0.000 14.027 ; qtot 026 17 LP2 1 GM3 CAE 8 0.000 14.027 ; qtot 0

18 LP2 1 GM3 CAD 8 0.000 14.027 ; qtot 028 19 LP2 1 GM3 CAC 9 0.000 14.027 ; qtot 0

20 C 1 GM3 CBB 10 0.395 12.011 ; qtot 0.39530 21 O 1 GM3 OBC 10 -0.625 15.9994 ; qtot -0.23

22 N 1 GM3 NBA 10 0.129 14.0067 ; qtot -0.10132 23 H 1 GM3 HAQ 10 -0.019 1.008 ; qtot -0.12

24 CH1 1 GM3 CBT 10 0.120 13.019 ; qtot 034 25 CH1 1 GM3 CCC 11 0.133 13.019 ; qtot 0.133

26 OA 1 GM3 ODB 11 -0.165 15.9994 ; qtot -0.03236 27 HO 1 GM3 HAI 11 0.032 1.008 ; qtot 0

28 CH1 1 GM3 CCD 12 -0.034 13.019 ; qtot -0.03438 29 CH1 1 GM3 CCE 12 -0.034 13.019 ; qtot -0.068

30 LP2 1 GM3 CCF 12 0.034 14.027 ; qtot -0.03440 31 LP2 1 GM3 CCG 12 0.034 14.027 ; qtot 0

32 LP2 1 GM3 CCH 13 0.000 14.027 ; qtot 042 33 LP2 1 GM3 CCI 13 0.000 14.027 ; qtot 0

34 LP2 1 GM3 CCJ 14 0.000 14.027 ; qtot 044 35 LP2 1 GM3 CCK 14 0.000 14.027 ; qtot 0

36 LP2 1 GM3 CCL 15 0.000 14.027 ; qtot 046 37 LP2 1 GM3 CCM 15 0.000 14.027 ; qtot 0

38 LP2 1 GM3 CCN 16 0.000 14.027 ; qtot 048 39 LP2 1 GM3 CCO 16 0.000 14.027 ; qtot 0

40 LP2 1 GM3 CCP 16 0.000 14.027 ; qtot 050 41 LP2 1 GM3 CDC 16 0.000 14.027 ; qtot 0

42 LP3 1 GM3 CDO 16 0.000 15.035 ; qtot 052 43 CH2 1 GM3 CBS 17 0.141 14.027 ; qtot 0.141

44 OSE 1 GM3 OBR 17 -0.279 15.9994 ; qtot -0.13854 45 CS1 1 GM3 CBQ 17 0.138 13.019 ; qtot 0

46 CS1 1 GM3 CAY 18 0.133 13.019 ; qtot 0.13356 47 OA 1 GM3 OAZ 18 -0.165 15.9994 ; qtot -0.032

48 HO 1 GM3 HAC 18 0.032 1.008 ; qtot 058 49 CS1 1 GM3 CAX 19 0.133 13.019 ; qtot 0.133

50 OA 1 GM3 OAB 19 -0.165 15.9994 ; qtot -0.03260 51 HO 1 GM3 HAA 19 0.032 1.008 ; qtot 0

52 OS 1 GM3 OBP 20 -0.197 15.9994 ; qtot -0.19762 53 CS1 1 GM3 CBO 20 0.197 13.019 ; qtot 0

54 CS2 1 GM3 CCA 21 0.072 14.027 ; qtot 0.072

97

64 55 OA 1 GM3 OCB 21 -0.118 15.9994 ; qtot -0.04656 HO 1 GM3 HAG 21 0.046 1.008 ; qtot 0

66 57 CS1 1 GM3 CAW 22 0.139 13.019 ; qtot 0.13958 OSE 1 GM3 OAV 22 -0.278 15.9994 ; qtot -0.139

68 59 CS1 1 GM3 CAU 22 0.139 13.019 ; qtot 060 CS1 1 GM3 CBM 23 0.133 13.019 ; qtot 0.133

70 61 OA 1 GM3 OBN 23 -0.165 15.9994 ; qtot -0.03262 HO 1 GM3 HAE 23 0.032 1.008 ; qtot 0

72 63 CS1 1 GM3 CBL 24 0.150 13.019 ; qtot 0.1564 OSE 1 GM3 OBZ 24 -0.258 15.9994 ; qtot -0.108

74 65 CB 1 GM3 CCZ 24 0.108 12.011 ; qtot 066 C 1 GM3 CDM 25 0.410 12.011 ; qtot 0.41

76 67 OM 1 GM3 ODN 25 -0.645 15.9994 ; qtot -0.23568 OM 1 GM3 ODL 25 -0.645 15.9994 ; qtot -0.88

78 69 OS 1 GM3 OCY 25 -0.168 15.9994 ; qtot -1.04870 CS2 1 GM3 CDA 25 0.048 14.027 ; qtot -1

80 71 CS1 1 GM3 CDX 26 0.099 13.019 ; qtot -0.90172 OA 1 GM3 OEB 26 -0.222 15.9994 ; qtot -1.123

82 73 HO 1 GM3 HAP 26 0.024 1.008 ; qtot -1.09974 CS1 1 GM3 CDW 27 0.099 13.019 ; qtot -1

84 75 N 1 GM3 NEC 37 0.048 14.0067 ; qtot -0.95276 H 1 GM3 HAS 37 -0.048 1.008 ; qtot -1

86 77 C 1 GM3 CED 28 0.397 12.011 ; qtot -0.60378 CH3 1 GM3 CEF 28 0.004 15.035 ; qtot -0.599

88 79 O 1 GM3 OEE 28 -0.622 15.9994 ; qtot -1.22180 CS1 1 GM3 CDV 28 0.221 13.019 ; qtot -1

90 81 CH1 1 GM3 CDU 29 0.133 13.019 ; qtot -0.86782 OA 1 GM3 OEA 29 -0.165 15.9994 ; qtot -1.032

92 83 HO 1 GM3 HAO 29 0.032 1.008 ; qtot -184 CH1 1 GM3 CDT 30 0.133 13.019 ; qtot -0.867

94 85 OA 1 GM3 ODZ 30 -0.165 15.9994 ; qtot -1.03286 HO 1 GM3 HAN 30 0.032 1.008 ; qtot -1

96 87 CH2 1 GM3 CDS 34 0.072 14.027 ; qtot -0.92888 OA 1 GM3 ODY 34 -0.118 15.9994 ; qtot -1.046

98 89 HO 1 GM3 HAM 34 0.046 1.008 ; qtot -190 OS 1 GM3 OAT 24 -0.197 15.9994 ; qtot -1.197

100 91 CS1 1 GM3 CAS 24 0.197 13.019 ; qtot -192 CS2 1 GM3 CAR 25 0.072 14.027 ; qtot -0.928

102 93 OA 1 GM3 OAQ 25 -0.118 15.9994 ; qtot -1.04694 HO 1 GM3 HAB 25 0.046 1.008 ; qtot -1

104 95 CS1 1 GM3 CBK 26 0.195 13.019 ; qtot -0.80596 OA 1 GM3 OBJ 26 -0.198 15.9994 ; qtot -1.003

106 97 HO 1 GM3 HBI 26 0.003 1.008 ; qtot -1.000;

108 [ bonds ];

110 ; i j funct r0 kr;

112 1 2 1 0.15300E+00 0.33470E+06 ; LP3 LP22 3 1 0.15300E+00 0.33470E+06 ; LP2 LP2

114 3 4 1 0.15300E+00 0.33470E+06 ; LP2 LP24 5 1 0.15300E+00 0.33470E+06 ; LP2 LP2

116 5 6 1 0.15300E+00 0.33470E+06 ; LP2 LP26 7 1 0.15300E+00 0.33470E+06 ; LP2 LP2

118 7 8 1 0.15300E+00 0.33470E+06 ; LP2 LP28 9 1 0.15300E+00 0.33470E+06 ; LP2 LP2

120 9 10 1 0.15300E+00 0.33470E+06 ; LP2 LP210 11 1 0.15300E+00 0.33470E+06 ; LP2 LP2

122 11 12 1 0.15300E+00 0.33470E+06 ; LP2 LP212 13 1 0.15300E+00 0.33470E+06 ; LP2 LP2

124 13 14 1 0.15300E+00 0.33470E+06 ; LP2 LP214 15 1 0.15300E+00 0.33470E+06 ; LP2 LP2

126 15 16 1 0.15300E+00 0.33470E+06 ; LP2 LP216 17 1 0.15300E+00 0.33470E+06 ; LP2 LP2

128 17 18 1 0.15300E+00 0.33470E+06 ; LP2 LP218 19 1 0.15300E+00 0.33470E+06 ; LP2 LP2

130 19 20 1 0.15300 334720. ; LP2 C


20 21 1 0.12300 502080. ; C O132 20 22 1 0.13300 418400. ; C N

22 23 1 0.10000 374468. ; N H134 22 24 1 0.14700 376560. ; N CH1

24 25 1 0.15300 334720. ; CH1 CH1136 24 43 1 0.15300 334720. ; CH1 CH2

25 26 1 0.14300 334720. ; CH1 OA138 25 28 1 0.15300 334720. ; CH1 CH1

26 27 1 0.10000 313800. ; OA HO140 28 29 1 0.13900 418400. ; CH1 CH1 double bond

29 30 1 0.15300 334720. ; CH1 LP2142 30 31 1 0.15300E+00 0.33470E+06 ; LP2 LP2

31 32 1 0.15300E+00 0.33470E+06 ; LP2 LP2144 32 33 1 0.15300E+00 0.33470E+06 ; LP2 LP2

33 34 1 0.15300E+00 0.33470E+06 ; LP2 LP2146 34 35 1 0.15300E+00 0.33470E+06 ; LP2 LP2

35 36 1 0.15300E+00 0.33470E+06 ; LP2 LP2148 36 37 1 0.15300E+00 0.33470E+06 ; LP2 LP2

37 38 1 0.15300E+00 0.33470E+06 ; LP2 LP2150 38 39 1 0.15300E+00 0.33470E+06 ; LP2 LP2

39 40 1 0.15300E+00 0.33470E+06 ; LP2 LP2152 40 41 1 0.15300E+00 0.33470E+06 ; LP2 LP2

41 42 1 0.15300E+00 0.33470E+06 ; LP2 LP3154 43 44 1 0.14300 251040. ; CH2 OSE

44 45 1 0.14350 251040. ; OSE CS1156 45 46 1 0.15200 251040. ; CS1 CS1

45 52 1 0.14350 251040. ; CS1 OS158 46 47 1 0.14300 251040. ; CS1 OA

46 49 1 0.15200 251040. ; CS1 CS1160 47 48 1 0.10000 313800. ; OA HO

49 50 1 0.14300 251040. ; CS1 OA162 49 57 1 0.15200 251040. ; CS1 CS1

50 51 1 0.10000 313800. ; OA HO164 52 53 1 0.14350 251040. ; OS CS1

53 54 1 0.15200 251040. ; CS1 CS2166 53 57 1 0.15200 251040. ; CS1 CS1

54 55 1 0.14300 251040. ; CS2 OA168 55 56 1 0.10000 313800. ; OA HO

57 58 1 0.14350 251040. ; CS1 OSE170 58 59 1 0.14350 251040. ; OSE CS1

59 60 1 0.15200 251040. ; CS1 CS1172 59 90 1 0.14350 251040. ; CS1 OS

60 61 1 0.14300 251040. ; CS1 OA174 60 63 1 0.15200 251040. ; CS1 CS1

61 62 1 0.10000 313800. ; OA HO176 63 64 1 0.14350 251040. ; CS1 OSE

63 95 1 0.15200 251040. ; CS1 CS1178 64 65 1 0.14350 251040. ; OSE CB

65 66 1 0.15300 334720. ; CB C180 65 69 1 0.14350 251040. ; CB OS

65 70 1 0.13900 334720. ; CB CS2182 66 67 1 0.12500 418400. ; C OM

66 68 1 0.12500 418400. ; C OM184 69 80 1 0.14350 251040. ; OS CS1

70 71 1 0.15200 251040. ; CS2 CS1186 71 72 1 0.14300 251040. ; CS1 OA

71 74 1 0.15200 251040. ; CS1 CS1188 72 73 1 0.10000 313800. ; OA HO

74 75 1 0.14700 376560. ; CS1 N190 74 80 1 0.15200 251040. ; CS1 CS1

75 76 1 0.10000 374468. ; N H192 75 77 1 0.13300 418400. ; N C

77 78 1 0.15300 334720. ; C CH3194 77 79 1 0.12300 502080. ; C O

80 81 1 0.15300 334720. ; CS1 CH1196 81 82 1 0.14300 334720. ; CH1 OA

81 84 1 0.15300 334720. ; CH1 CH1

99

198 82 83 1 0.10000 313800. ; OA HO84 85 1 0.14300 334720. ; CH1 OA

200 84 87 1 0.15300 334720. ; CH1 CH285 86 1 0.10000 313800. ; OA HO

202 87 88 1 0.14300 334720. ; CH2 OA88 89 1 0.10000 313800. ; OA HO

204 90 91 1 0.14350 251040. ; OS CS191 92 1 0.15200 251040. ; CS1 CS2

206 91 95 1 0.15200 251040. ; CS1 CS192 93 1 0.14300 251040. ; CS2 OA

208 93 94 1 0.10000 313800. ; OA HO95 96 1 0.14300 251040. ; CS1 OA

210 96 97 1 0.10000 313800. ; OA HO;

212 [ pairs ];

214 ; i j funct C6 C12;

216 ; 1 4 1 ; commented pairs are that of ; LP3 LP2; 2 5 1 ; Ryckaert-Bellemans dihedrals ; LP2 LP2

218 ; 3 6 1 ; (don’t need 1-4 interaction) ; LP2 LP2; 4 7 1 ; LP2 LP2

220 ; 5 8 1 ; LP2 LP2; 6 9 1 ; LP2 LP2

222 ; 7 10 1 ; LP2 LP2; 8 11 1 ; LP2 LP2

224 ; 9 12 1 ; LP2 LP2; 10 13 1 ; LP2 LP2

226 ; 11 14 1 ; LP2 LP2; 12 15 1 ; LP2 LP2

228 ; 13 16 1 ; LP2 LP2; 14 17 1 ; LP2 LP2

230 ; 15 18 1 ; LP2 LP2; 16 19 1 ; LP2 LP2

232 ; 17 20 1 ; LP2 C18 21 1 0.32637E-02 0.30910E-05 ; LP2 O

234 18 22 1 0.33872E-02 0.46696E-05 ; LP2 N19 23 1 0.00000E+00 0.00000E+00 ; LP2 H

236 19 24 1 0.37030E-02 0.69383E-05 ; LP2 CH120 25 1 0.26103E-02 0.35506E-05 ; C CH1

238 20 43 1 0.33248E-02 0.48994E-05 ; C CH221 23 1 0.00000E+00 0.00000E+00 ; O H

240 21 24 1 0.25662E-02 0.16646E-05 ; O CH122 26 1 0.23473E-02 0.11203E-05 ; N OA

242 22 28 1 0.26633E-02 0.25147E-05 ; N CH122 44 1 0.23473E-02 0.11203E-05 ; N OSE

244 23 25 1 0.00000E+00 0.00000E+00 ; H CH123 43 1 0.00000E+00 0.00000E+00 ; H CH2

246 24 27 1 0.00000E+00 0.00000E+00 ; CH1 HO24 29 1 0.29117E-02 0.37364E-05 ; CH1 CH1

248 24 45 1 0.29117E-02 0.37364E-05 ; CH1 CS125 30 1 0.37030E-02 0.69383E-05 ; CH1 LP2

250 25 44 1 0.25662E-02 0.16646E-05 ; CH1 OSE26 29 1 0.25662E-02 0.16646E-05 ; OA CH1

252 26 43 1 0.32685E-02 0.22969E-05 ; OA CH227 28 1 0.00000E+00 0.00000E+00 ; HO CH1

254 28 31 1 0.37030E-02 0.69383E-05 ; CH1 LP228 43 1 0.37086E-02 0.51559E-05 ; CH1 CH2

256 29 32 1 0.37030E-02 0.69383E-05 ; CH1 LP2; 30 33 1 ; LP2 LP2

258 ; 31 34 1 ; LP2 LP2; 32 35 1 ; LP2 LP2

260 ; 33 36 1 ; LP2 LP2; 34 37 1 ; LP2 LP2

262 ; 35 38 1 ; LP2 LP2; 36 39 1 ; LP2 LP2

264 ; 37 40 1 ; LP2 LP2


; 38 41 1 ; LP2 LP2266 ; 39 42 1 ; LP2 LP3

43 46 1 0.37086E-02 0.51559E-05 ; CH2 CS1268 43 52 1 0.32685E-02 0.22969E-05 ; CH2 OS

44 47 1 0.22617E-02 0.74158E-06 ; OSE OA270 44 49 1 0.25662E-02 0.16646E-05 ; OSE CS1

44 53 1 0.25662E-02 0.16646E-05 ; OSE CS1272 45 48 1 0.00000E+00 0.00000E+00 ; CS1 HO

45 50 1 0.25662E-02 0.16646E-05 ; CS1 OA274 45 54 1 0.37086E-02 0.51559E-05 ; CS1 CS2

45 57 1 0.29117E-02 0.37364E-05 ; CS1 CS1276 46 51 1 0.00000E+00 0.00000E+00 ; CS1 HO

46 53 1 0.29117E-02 0.37364E-05 ; CS1 CS1278 46 58 1 0.25662E-02 0.16646E-05 ; CS1 OSE

47 50 1 0.22617E-02 0.74158E-06 ; OA OA280 47 52 1 0.22617E-02 0.74158E-06 ; OA OS

47 57 1 0.25662E-02 0.16646E-05 ; OA CS1282 48 49 1 0.00000E+00 0.00000E+00 ; HO CS1

49 52 1 0.25662E-02 0.16646E-05 ; CS1 OS284 49 54 1 0.37086E-02 0.51559E-05 ; CS1 CS2

49 59 1 0.29117E-02 0.37364E-05 ; CS1 CS1286 50 53 1 0.25662E-02 0.16646E-05 ; OA CS1

50 58 1 0.22617E-02 0.74158E-06 ; OA OSE288 51 57 1 0.00000E+00 0.00000E+00 ; HO CS1

52 55 1 0.22617E-02 0.74158E-06 ; OS OA290 52 58 1 0.22617E-02 0.74158E-06 ; OS OSE

53 56 1 0.00000E+00 0.00000E+00 ; CS1 HO292 53 59 1 0.29117E-02 0.37364E-05 ; CS1 CS1

54 58 1 0.32685E-02 0.22969E-05 ; CS2 OSE294 55 57 1 0.25662E-02 0.16646E-05 ; OA CS1

57 60 1 0.29117E-02 0.37364E-05 ; CS1 CS1296 57 90 1 0.25662E-02 0.16646E-05 ; CS1 OS

58 61 1 0.22617E-02 0.74158E-06 ; OSE OA298 58 63 1 0.25662E-02 0.16646E-05 ; OSE CS1

58 91 1 0.25662E-02 0.16646E-05 ; OSE CS1300 59 62 1 0.00000E+00 0.00000E+00 ; CS1 HO

59 64 1 0.25662E-02 0.16646E-05 ; CS1 OSE302 59 92 1 0.37086E-02 0.51559E-05 ; CS1 CS2

59 95 1 0.29117E-02 0.37364E-05 ; CS1 CS1304 60 65 1 0.26103E-02 0.35506E-05 ; CS1 CB

60 91 1 0.29117E-02 0.37364E-05 ; CS1 CS1306 60 96 1 0.25662E-02 0.16646E-05 ; CS1 OA

61 64 1 0.22617E-02 0.74158E-06 ; OA OSE308 61 90 1 0.22617E-02 0.74158E-06 ; OA OS

61 95 1 0.25662E-02 0.16646E-05 ; OA CS1310 62 63 1 0.00000E+00 0.00000E+00 ; HO CS1

63 66 1 0.26103E-02 0.35506E-05 ; CS1 C312 63 69 1 0.25662E-02 0.16646E-05 ; CS1 OS

63 70 1 0.37086E-02 0.51559E-05 ; CS1 CS2314 63 90 1 0.25662E-02 0.16646E-05 ; CS1 OS

63 92 1 0.37086E-02 0.51559E-05 ; CS1 CS2316 63 97 1 0.00000E+00 0.00000E+00 ; CS1 HO

64 67 1 0.22617E-02 0.74158E-06 ; OSE OM318 64 68 1 0.22617E-02 0.74158E-06 ; OSE OM

64 71 1 0.25662E-02 0.16646E-05 ; OSE CS1320 64 80 1 0.25662E-02 0.16646E-05 ; OSE CS1

64 91 1 0.25662E-02 0.16646E-05 ; OSE CS1322 64 96 1 0.22617E-02 0.74158E-06 ; OSE OA

65 72 1 0.23006E-02 0.15818E-05 ; CB OA324 65 74 1 0.26103E-02 0.35506E-05 ; CB CS1

65 81 1 0.26103E-02 0.35506E-05 ; CB CH1326 65 95 1 0.26103E-02 0.35506E-05 ; CB CS1

66 71 1 0.26103E-02 0.35506E-05 ; C CS1328 66 80 1 0.26103E-02 0.35506E-05 ; C CS1

67 69 1 0.22617E-02 0.74158E-06 ; OM OS330 67 70 1 0.32685E-02 0.22969E-05 ; OM CS2

68 69 1 0.22617E-02 0.74158E-06 ; OM OS

101

332 68 70 1 0.32685E-02 0.22969E-05 ; OM CS269 71 1 0.25662E-02 0.16646E-05 ; OS CS1

334 69 75 1 0.23473E-02 0.11203E-05 ; OS N69 82 1 0.22617E-02 0.74158E-06 ; OS OA

336 69 84 1 0.25662E-02 0.16646E-05 ; OS CH170 73 1 0.00000E+00 0.00000E+00 ; CS2 HO

338 70 75 1 0.33923E-02 0.34700E-05 ; CS2 N70 80 1 0.37086E-02 0.51559E-05 ; CS2 CS1

340 71 76 1 0.00000E+00 0.00000E+00 ; CS1 H71 77 1 0.26103E-02 0.35506E-05 ; CS1 C

342 71 81 1 0.29117E-02 0.37364E-05 ; CS1 CH172 75 1 0.23473E-02 0.11203E-05 ; OA N

344 72 80 1 0.25662E-02 0.16646E-05 ; OA CS173 74 1 0.00000E+00 0.00000E+00 ; HO CS1

346 74 78 1 0.44668E-02 0.67137E-05 ; CS1 CH374 79 1 0.25662E-02 0.16646E-05 ; CS1 O

348 74 82 1 0.25662E-02 0.16646E-05 ; CS1 OA74 84 1 0.29117E-02 0.37364E-05 ; CS1 CH1

350 75 81 1 0.26633E-02 0.25147E-05 ; N CH176 78 1 0.00000E+00 0.00000E+00 ; H CH3

352 76 79 1 0.00000E+00 0.00000E+00 ; H O76 80 1 0.00000E+00 0.00000E+00 ; H CS1

354 77 80 1 0.26103E-02 0.35506E-05 ; C CS180 83 1 0.00000E+00 0.00000E+00 ; CS1 HO

356 80 85 1 0.25662E-02 0.16646E-05 ; CS1 OA80 87 1 0.37086E-02 0.51559E-05 ; CS1 CH2

358 81 86 1 0.00000E+00 0.00000E+00 ; CH1 HO81 88 1 0.25662E-02 0.16646E-05 ; CH1 OA

360 82 85 1 0.22617E-02 0.74158E-06 ; OA OA82 87 1 0.32685E-02 0.22969E-05 ; OA CH2

362 83 84 1 0.00000E+00 0.00000E+00 ; HO CH184 89 1 0.00000E+00 0.00000E+00 ; CH1 HO

364 85 88 1 0.22617E-02 0.74158E-06 ; OA OA86 87 1 0.00000E+00 0.00000E+00 ; HO CH2

366 90 93 1 0.22617E-02 0.74158E-06 ; OS OA90 96 1 0.22617E-02 0.74158E-06 ; OS OA

368 91 94 1 0.00000E+00 0.00000E+00 ; CS1 HO91 97 1 0.00000E+00 0.00000E+00 ; CS1 HO

370 92 96 1 0.32685E-02 0.22969E-05 ; CS2 OA93 95 1 0.25662E-02 0.16646E-05 ; OA CS1

372 ;[ angles ]

374 ;; i j k funct theta0 ktheta

376 ;1 2 3 1 111.000 460.240 ; LP3 LP2 LP2

378 2 3 4 1 111.000 460.240 ; LP2 LP2 LP23 4 5 1 111.000 460.240 ; LP2 LP2 LP2

380 4 5 6 1 111.000 460.240 ; LP2 LP2 LP25 6 7 1 111.000 460.240 ; LP2 LP2 LP2

382 6 7 8 1 111.000 460.240 ; LP2 LP2 LP27 8 9 1 111.000 460.240 ; LP2 LP2 LP2

384 8 9 10 1 111.000 460.240 ; LP2 LP2 LP29 10 11 1 111.000 460.240 ; LP2 LP2 LP2

386 10 11 12 1 111.000 460.240 ; LP2 LP2 LP211 12 13 1 111.000 460.240 ; LP2 LP2 LP2

388 12 13 14 1 111.000 460.240 ; LP2 LP2 LP213 14 15 1 111.000 460.240 ; LP2 LP2 LP2

390 14 15 16 1 111.000 460.240 ; LP2 LP2 LP215 16 17 1 111.000 460.240 ; LP2 LP2 LP2

392 16 17 18 1 111.000 460.240 ; LP2 LP2 LP217 18 19 1 111.000 460.240 ; LP2 LP2 LP2

394 18 19 20 1 111.000 460.240 ; LP2 LP2 C19 20 21 1 121.000 502.080 ; LP2 C O

396 19 20 22 1 115.000 502.080 ; LP2 C N21 20 22 1 124.000 502.080 ; O C N

398 20 22 23 1 115.000 376.560 ; C N H


20 22 24 1 122.000 502.080 ; C N CH1400 23 22 24 1 115.000 376.560 ; H N CH1

22 24 25 1 109.500 460.240 ; N CH1 CH1402 22 24 43 1 109.500 460.240 ; N CH1 CH2

25 24 43 1 111.000 460.240 ; CH1 CH1 CH2404 24 25 26 1 109.500 460.240 ; CH1 CH1 OA

24 25 28 1 111.000 460.240 ; CH1 CH1 CH1406 26 25 28 1 109.500 460.240 ; OA CH1 CH1

25 26 27 1 109.500 397.480 ; CH1 OA HO408 25 28 29 1 120.000 418.400 ; CH1 CH1 CH1 double bond

28 29 30 1 120.000 418.400 ; CH1 CH1 LP2 double bond410 29 30 31 1 111.000 460.240 ; CH1 LP2 LP2

30 31 32 1 111.000 460.240 ; LP2 LP2 LP2412 31 32 33 1 111.000 460.240 ; LP2 LP2 LP2

32 33 34 1 111.000 460.240 ; LP2 LP2 LP2414 33 34 35 1 111.000 460.240 ; LP2 LP2 LP2

34 35 36 1 111.000 460.240 ; LP2 LP2 LP2416 35 36 37 1 111.000 460.240 ; LP2 LP2 LP2

36 37 38 1 111.000 460.240 ; LP2 LP2 LP2418 37 38 39 1 111.000 460.240 ; LP2 LP2 LP2

38 39 40 1 111.000 460.240 ; LP2 LP2 LP2420 39 40 41 1 111.000 460.240 ; LP2 LP2 LP2

40 41 42 1 111.000 460.240 ; LP2 LP2 LP3422 24 43 44 1 111.000 460.240 ; CH1 CH2 OSE

43 44 45 1 112.900 397.480 ; CH2 OSE CS1424 44 45 46 1 107.000 460.240 ; OSE CS1 CS1

44 45 52 1 107.500 460.240 ; OSE CS1 OS426 46 45 52 1 107.000 460.240 ; CS1 CS1 OS

45 46 47 1 109.500 284.512 ; CS1 CS1 OA428 45 46 49 1 109.500 460.240 ; CS1 CS1 CS1

47 46 49 1 109.500 284.512 ; OA CS1 CS1430 46 47 48 1 109.500 397.480 ; CS1 OA HO

46 49 50 1 109.500 284.512 ; CS1 CS1 OA432 46 49 57 1 109.500 460.240 ; CS1 CS1 CS1

50 49 57 1 109.500 284.512 ; OA CS1 CS1434 49 50 51 1 109.500 397.480 ; CS1 OA HO

45 52 53 1 109.500 460.240 ; CS1 OS CS1436 52 53 54 1 107.000 460.240 ; OS CS1 CS2

52 53 57 1 107.000 460.240 ; OS CS1 CS1438 54 53 57 1 109.500 460.240 ; CS2 CS1 CS1

53 54 55 1 107.000 460.240 ; CS1 CS2 OA440 54 55 56 1 109.500 397.480 ; CS2 OA HO

49 57 53 1 109.500 460.240 ; CS1 CS1 CS1442 49 57 58 1 107.000 460.240 ; CS1 CS1 OSE

53 57 58 1 107.000 460.240 ; CS1 CS1 OSE444 57 58 59 1 109.500 334.720 ; CS1 OSE CS1

58 59 60 1 107.000 460.240 ; OSE CS1 CS1446 58 59 90 1 107.500 460.240 ; OSE CS1 OS

60 59 90 1 107.000 460.240 ; CS1 CS1 OS448 59 60 61 1 109.500 284.512 ; CS1 CS1 OA

59 60 63 1 109.500 460.240 ; CS1 CS1 CS1450 61 60 63 1 109.500 284.512 ; OA CS1 CS1

60 61 62 1 109.500 397.480 ; CS1 OA HO452 60 63 64 1 107.000 460.240 ; CS1 CS1 OSE

60 63 95 1 109.500 460.240 ; CS1 CS1 CS1454 64 63 95 1 107.000 460.240 ; OSE CS1 CS1

63 64 65 1 109.500 334.720 ; CS1 OSE CB456 64 65 66 1 109.500 284.512 ; OSE CB C

64 65 69 1 109.500 284.512 ; OSE CB OS458 64 65 70 1 109.500 284.512 ; OSE CB CS2

66 65 69 1 109.500 284.512 ; C CB OS460 66 65 70 1 120.000 418.400 ; C CB CS2

69 65 70 1 109.500 284.512 ; OS CB CS2462 65 66 67 1 117.000 502.080 ; CB C OM

65 66 68 1 117.000 502.080 ; CB C OM464 67 66 68 1 126.000 502.080 ; OM C OM

65 69 80 1 109.500 334.720 ; CB OS CS1

103

466 65 70 71 1 111.000 460.240 ; CB CS2 CS170 71 72 1 109.500 284.512 ; CS2 CS1 OA

468 70 71 74 1 109.500 460.240 ; CS2 CS1 CS172 71 74 1 109.500 284.512 ; OA CS1 CS1

470 71 72 73 1 109.500 397.480 ; CS1 OA HO71 74 75 1 109.500 460.240 ; CS1 CS1 N

472 71 74 80 1 109.500 460.240 ; CS1 CS1 CS175 74 80 1 109.500 460.240 ; N CS1 CS1

474 74 75 76 1 115.000 376.560 ; CS1 N H74 75 77 1 122.000 502.080 ; CS1 N C

476 76 75 77 1 115.000 376.560 ; H N C75 77 78 1 115.000 502.080 ; N C CH3

478 75 77 79 1 124.000 502.080 ; N C O78 77 79 1 121.000 502.080 ; CH3 C O

480 69 80 74 1 107.000 460.240 ; OS CS1 CS169 80 81 1 109.500 284.512 ; OS CS1 CH1

482 74 80 81 1 109.500 251.040 ; CS1 CS1 CH180 81 82 1 109.500 284.512 ; CS1 CH1 OA

484 80 81 84 1 109.500 251.040 ; CS1 CH1 CH182 81 84 1 109.500 460.240 ; OA CH1 CH1

486 81 82 83 1 109.500 397.480 ; CH1 OA HO81 84 85 1 109.500 460.240 ; CH1 CH1 OA

488 81 84 87 1 111.000 460.240 ; CH1 CH1 CH285 84 87 1 109.500 460.240 ; OA CH1 CH2

490 84 85 86 1 109.500 397.480 ; CH1 OA HO84 87 88 1 109.500 460.240 ; CH1 CH2 OA

492 87 88 89 1 109.500 397.480 ; CH2 OA HO59 90 91 1 109.500 460.240 ; CS1 OS CS1

494 90 91 92 1 107.000 460.240 ; OS CS1 CS290 91 95 1 107.000 460.240 ; OS CS1 CS1

496 92 91 95 1 109.500 460.240 ; CS2 CS1 CS191 92 93 1 107.000 460.240 ; CS1 CS2 OA

498 92 93 94 1 109.500 397.480 ; CS2 OA HO63 95 91 1 109.500 460.240 ; CS1 CS1 CS1

500 63 95 96 1 109.500 284.512 ; CS1 CS1 OA91 95 96 1 109.500 284.512 ; CS1 CS1 OA

502 95 96 97 1 109.500 397.480 ; CS1 OA HO;

504 [ dihedrals ];

506 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

508 ; ceramide;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

510 ;; RB Dihedrals

512 ; i j k l funct;

514 ; (RB force field constants for rotations around LP2-LP2 bonds are; 9.2789 12.156 -13.120 -3.0597 26.240 -31.495 )

516 ;1 2 3 4 3 ; LP3 LP2 LP2 LP2

518 2 3 4 5 3 ; LP2 LP2 LP2 LP23 4 5 6 3 ; LP2 LP2 LP2 LP2







532 16 17 18 19 3 ; LP2 LP2 LP2 LP2


17 18 19 20 3 ; LP2 LP2 LP2 C534 ;

; Standard dihedrals536 ; i j k l funct phase f mult

;538 18 19 20 22 1 0.000 0.418 6 ; LP2 LP2 C N

19 20 22 24 1 180.000 33.472 2 ; LP2 C N CH1540 20 22 24 25 1 180.000 0.418 6 ; C N CH1 CH1

20 22 24 43 1 180.000 0.418 6 ; C N CH1 CH2542 22 24 25 28 1 0.000 5.858 3 ; N CH1 CH1 CH1

43 24 25 28 1 0.000 5.858 3 ; CH2 CH1 CH1 CH1544 22 24 43 44 1 0.000 5.858 3 ; N CH1 CH2 OSE

25 24 43 44 1 0.000 5.858 3 ; CH1 CH1 CH2 OSE546 24 25 26 27 1 0.000 1.255 3 ; CH1 CH1 OA HO

24 25 28 29 1 0.000 0.418 6 ; CH1 CH1 CH1 CH1 double bond548 28 29 30 31 1 0.000 0.418 6 ; CH1 CH1 LP2 LP2 double bond

;550 ; RB Dihedrals

;552 29 30 31 32 3 ; CH1 LP2 LP2 LP2

30 31 32 33 3 ; LP2 LP2 LP2 LP2554 31 32 33 34 3 ; LP2 LP2 LP2 LP2





;564 ; Standard dihedrals

; i j k l funct phase f mult566 ;

24 43 44 45 1 0.000 3.766 3 ; CH1 CH2 OSE CS1568 ;

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;570 ; glucose

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;572 ;

44 45 46 47 1 0.000 2.092 2 ; OSE CS1 CS1 OA574 44 45 46 49 1 0.000 5.860 3 ; OSE CS1 CS1 CS1

44 45 46 49 1 0.000 0.418 2 ; OSE CS1 CS1 CS1576 52 45 46 47 1 0.000 2.092 2 ; OS CS1 CS1 OA

52 45 46 49 1 0.000 0.418 2 ; OS CS1 CS1 CS1578 ;

46 45 52 53 1 0.000 3.770 3 ; CS1 CS1 OS CS1580 ;

45 52 53 57 1 0.000 3.770 3 ; CS1 OS CS1 CS1582 ;

52 53 57 49 1 0.000 0.418 2 ; OS CS1 CS1 CS1584 52 53 57 58 1 0.000 0.418 2 ; OS CS1 CS1 OSE

54 53 57 49 1 0.000 5.860 3 ; CS2 CS1 CS1 CS1586 54 53 57 58 1 0.000 0.418 2 ; CS2 CS1 CS1 OSE

;588 45 46 49 50 1 0.000 0.418 2 ; CS1 CS1 CS1 OA

45 46 49 57 1 0.000 5.860 3 ; CS1 CS1 CS1 CS1590 47 46 49 50 1 0.000 2.092 2 ; OA CS1 CS1 OA

47 46 49 57 1 0.000 0.418 2 ; OA CS1 CS1 CS1592 ;

46 49 57 58 1 0.000 0.418 3 ; CS1 CS1 CS1 OSE594 46 49 57 53 1 0.000 5.860 3 ; CS1 CS1 CS1 CS1

50 49 57 58 1 0.000 2.092 3 ; OA CS1 CS1 OSE596 50 49 57 53 1 0.000 0.418 3 ; OA CS1 CS1 CS1

;598 52 53 54 55 1 0.000 2.090 2 ; gauche eff. ; OS CS1 CS2 OA

57 53 54 55 1 0.000 5.860 3 ; CS1 CS1 CS2 OA600 57 53 54 55 1 0.000 0.418 2 ; CS1 CS1 CS2 OA

;

105

602 43 44 45 46 1 0.000 3.770 3 ; CH2 OSE CS1 CS145 46 47 48 1 0.000 1.260 3 ; CS1 CS1 OA HO

604 46 49 50 51 1 0.000 1.260 3 ; CS1 CS1 OA HO49 57 58 59 1 0.000 3.770 3 ; CS1 CS1 OSE CS1

606 53 54 55 56 1 0.000 1.260 3 ; CS1 CS2 OA HO43 44 45 52 1 0.000 4.180 2 ; exo-anom. eff; CH2 OSE CS1 OS

608 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

610 ; galactose;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

612 ;58 59 60 61 1 0.000 2.090 2 ; OSE CS1 CS1 OA

614 58 59 60 63 1 0.000 5.860 3 ; OSE CS1 CS1 CS158 59 60 63 1 0.000 0.418 2 ; OSE CS1 CS1 CS1

616 90 59 60 91 1 0.000 2.090 2 ; OS CS1 CS1 CS190 59 60 63 1 0.000 0.418 2 ; OS CS1 CS1 CS1

618 ;59 60 63 64 1 0.000 0.418 2 ; CS1 CS1 CS1 OSE

620 59 60 63 95 1 0.000 5.860 3 ; CS1 CS1 CS1 CS161 60 63 64 1 0.000 2.090 2 ; OA CS1 CS1 OSE

622 61 60 63 95 1 0.000 0.418 2 ; OA CS1 CS1 CS1;

624 60 63 95 91 1 0.000 5.858 3 ; CS1 CS1 CS1 CS160 63 95 96 1 0.000 0.418 3 ; CS1 CS1 CS1 OA

626 64 63 95 91 1 0.000 0.418 3 ; OSE CS1 CS1 CS164 63 95 96 1 0.000 2.090 3 ; OSE CS1 CS1 OA

628 ;90 91 95 63 1 0.000 0.418 2 ; OS CS1 CS1 CS1

630 90 91 95 96 1 0.000 2.090 2 ; OS CS1 CS1 OA92 91 95 63 1 0.000 5.186 3 ; CS2 CS1 CS1 CS1

632 92 91 95 96 1 0.000 0.418 2 ; CS2 CS1 CS1 OA;

634 90 91 92 93 1 0.000 2.090 2 ; gauche eff. ; OS CS1 CS2 OA95 91 92 93 1 0.000 3.860 3 ; CS1 CS1 CS2 OA

636 95 91 92 93 1 0.000 0.418 2 ; CS1 CS1 CS2 OA;

638 59 90 91 92 1 0.000 3.770 3 ; CS1 OS CS1 CS2;

640 60 59 90 91 1 0.000 3.770 3 ; CS1 CS1 OS CS1;

642 57 58 59 60 1 0.000 3.770 3 ; CS1 OSE CS1 CS157 58 59 90 1 0.000 4.180 2 ; exo-anom. eff; CS1 OSE CS1 OS

644 ;59 60 61 62 1 0.000 1.260 3 ; CS1 CS1 OA HO

646 60 63 64 65 1 0.000 3.770 3 ; CS1 CS1 OSE CB63 95 96 97 1 0.000 1.260 3 ; CS1 CS1 OA HO

648 91 92 93 94 1 0.000 1.260 3 ; CS1 CS2 OA HO;

650 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; sialic acid

652 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

654 ;64 65 70 71 1 0.000 0.418 2 ; OSE CB CS2 CS1

656 64 65 70 71 1 0.000 5.860 3 ; OSE CB CS2 CS1;

658 65 70 71 72 1 0.000 0.418 2 ; CB CS2 CS1 OA65 70 71 74 1 0.000 5.860 3 ; CB CS2 CS1 CS1

660 ;70 71 74 75 1 0.000 0.418 3 ; CS2 CS1 CS1 N

662 70 71 74 80 1 0.000 5.860 3 ; CS2 CS1 CS1 CS172 71 74 75 1 0.000 2.090 3 ; OA CS1 CS1 N

664 72 71 74 80 1 0.000 0.418 3 ; OA CS1 CS1 CS1;

666 81 80 74 71 1 0.000 5.860 3 ; CH1 CS1 CS1 CS181 80 74 75 1 0.000 0.418 2 ; CH1 CS1 CS1 N

668 69 80 74 71 1 0.000 0.418 2 ; OS CS1 CS1 CS169 80 74 75 1 0.000 2.090 2 ; OS CS1 CS1 N

670 ;


69 80 81 82 1 0.000 2.092 2 ; OS CS1 CH1 OA672 69 80 81 84 1 0.000 0.418 2 ; OS CS1 CH1 CH1

74 80 81 82 1 0.000 0.418 2 ; CS1 CS1 CH1 OA674 74 80 81 84 1 0.000 5.860 3 ; CS1 CS1 CH1 CH1

;676 80 81 82 83 1 0.000 1.260 3 ; CS1 CH1 OA HO

;678 80 81 84 87 1 0.000 5.860 3 ; CS1 CH1 CH1 CH2

80 81 84 85 1 0.000 0.418 2 ; CS1 CH1 CH1 OA680 ;

81 84 87 88 1 0.000 5.858 3 ; CH1 CH1 CH2 OA682 81 84 87 88 1 0.000 0.418 2 ; CH1 CH1 CH2 OA

;684 84 87 88 89 1 0.000 1.260 3 ; CH1 CH2 OA HO

;686 81 84 85 86 1 0.000 1.260 3 ; CH1 CH1 OA HO

;688 65 69 80 81 1 0.000 3.770 3 ; CB OS CS1 CH1

;690 70 65 69 80 1 0.000 3.766 3 ; CS2 CB OS CS1

;692 63 64 65 70 1 0.000 3.770 3 ; CS1 OSE CB CS2

63 64 65 66 1 0.000 3.770 3 ; CS1 OSE CB C694 63 64 65 69 1 0.000 4.180 2 ; exo-anom. eff; CS1 OSE CB OS

;696 70 71 72 73 1 0.000 1.260 3 ; CS2 CS1 OA HO

71 74 75 77 1 180.000 0.418 6 ; CS1 CS1 N C698 64 65 66 67 1 180.000 5.858 2 ; OSE CB C OM

74 75 77 78 1 180.000 33.472 2 ; CS1 N C CH3700 ;

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;702 ; improper dihedrals

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;704 ;

;;;;; in plane ;;;;706 ;

20 19 22 21 2 0.000 167.360 ; amide in sphingosine ; C LP2 N O708 22 20 23 24 2 0.000 167.360 ; amide in sphingosine ; N C H CH1

66 65 68 67 2 0.000 167.360 ; O2- in sialic acid ; C CB OM OM710 75 74 76 77 2 0.000 167.360 ; amide in NeuAc ; N CS1 H C

77 75 78 79 2 0.000 167.360 ; amide in NeuAc ; C N CH3 O712 25 28 30 29 2 0.000 167.360 ; trans double bond CH1 CH1 LP2 CH1

; in sphingosine.714 ; (the order of last two atoms -30 and

; 29- is inverted to reproduce gauche716 ; conformation and avoid problems with

; the periodicity of the potential)718 ;

;;;;; out of plane ;;;;;720 ;

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;722 ; glucose

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;724 ;

45 52 46 44 2 35.246 334.720 ; ; CS1 OS CS1 OSE726 53 57 52 54 2 35.246 334.720 ; ; CS1 CS1 OS CS2

57 49 53 58 2 -35.246 334.720 ; ; CS1 CS1 CS1 OSE728 49 46 57 50 2 35.246 334.720 ; ; CS1 CS1 CS1 OA

46 45 49 47 2 -35.246 334.720 ; ; CS1 CS1 CS1 OA730 ;

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;732 ; galactose

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;734 ;

59 60 90 58 2 -35.246 334.720 ; ; CS1 CS1 OS OSE736 60 63 59 61 2 35.246 334.720 ; ; CS1 CS1 CS1 OA

63 95 60 64 2 -35.246 334.720 ; ; CS1 CS1 CS1 OSE738 95 91 63 96 2 -35.246 334.720 ; ; CS1 CS1 CS1 OA

91 90 95 92 2 -35.246 334.720 ; ; CS1 OS CS1 CS2

107

740 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

742 ; sialic acid;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

744 ;65 70 69 66 2 -35.246 334.720 ; ; CB CS2 OS C

746 65 70 69 64 2 35.246 334.720 ; ; CB CS2 OS OSE71 74 70 72 2 35.246 334.720 ; ; CS1 CS1 CS2 OA

748 74 80 71 75 2 -35.246 334.720 ; ; CS1 CS1 CS1 N80 69 74 81 2 35.246 334.720 ; ; CS1 OS CS1 CH1

750 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

752 ; ceramide;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

754 ;24 25 43 22 2 35.246 334.720 ; ; CH1 CH1 CH2 N

756 25 28 24 26 2 35.246 334.720 ; ; CH1 CH1 CH1 OA;

758 ;;; END

6.0.4 LJ parameters

[ nonbond_params ]2

4

; i j funct C6 C126 O O 1 0.22617E-02 0.74158E-06

O OA 1 0.22617E-02 0.13807E-058 O OW 1 0.24329E-02 0.18255E-05

O N 1 0.23473E-02 0.21861E-0510 OM OM 1 0.22617E-02 0.74158E-06

OM OA 1 0.22617E-02 0.22594E-0512 OM OW 1 0.24329E-02 0.29873E-05

OM N 1 0.23473E-02 0.35773E-0514 OA OA 1 0.22617E-02 0.15062E-05

OA OW 1 0.24329E-02 0.19915E-0516 OA N 1 0.23473E-02 0.23849E-05

OA OS 1 0.22617E-02 0.15062E-0518 OW OW 1 0.26171E-02 0.26331E-05

OW N 1 0.25250E-02 0.31532E-0520 OW OS 1 0.24329E-02 0.19915E-05

N N 1 0.24362E-02 0.16924E-0522 N OS 1 0.23473E-02 0.23849E-05

C C 1 0.23402E-02 0.33740E-0524 CH1 CH1 1 0.12496E-01 0.71747E-04

CH2 CH2 1 0.90975E-02 0.35333E-0426 CH3 CH3 1 0.88765E-02 0.26150E-04

CB CB 1 0.23402E-02 0.33740E-0528 OS OS 1 0.22617E-02 0.74158E-06

CS1 CS1 1 0.12496E-01 0.71747E-0430 CS2 CS2 1 0.90975E-02 0.35333E-04

LP LP 1 9.16000e-03 2.50700e-0532 LP LP2 1 7.33500e-03 2.38300e-05

LP LP3 1 8.96600e-03 2.91300e-0534 LP2 LP2 1 5.87400e-03 2.26500e-05

LP2 LP3 1 7.18000e-03 2.76900e-0536 LP3 LP3 1 8.77700e-03 3.38500e-05

LP C 1 5.872807e-03 8.652886e-0638 LP CB 1 5.872807e-03 8.652886e-06

LP CH1 1 1.357078e-02 3.990160e-0540 LP CH2 1 1.157925e-02 2.800133e-05

LP CH3 1 1.143774e-02 2.408931e-0542 LP CS1 1 1.357078e-02 3.990160e-05

LP CS2 1 1.157925e-02 2.800133e-0544 LP H 1 0.000000e+00 0.000000e+00


LP HO 1 0.000000e+00 0.000000e+0046 LP N 1 5.992054e-03 6.128299e-06

LP O 1 5.773468e-03 4.056649e-0648 LP OA 1 5.773468e-03 5.781357e-06

LP OM 1 5.773468e-03 4.056649e-0650 LP OS 1 5.773468e-03 0.57815E-05

LP2 C 1 4.614105e-03 1.091850e-0552 LP2 CB 1 4.614105e-03 1.091850e-05

LP2 CH1 1 1.066219e-02 5.034915e-0554 LP2 CH2 1 9.097500e-03 3.533300e-05

LP2 CH3 1 8.986321e-03 3.039668e-0556 LP2 CS1 1 1.066219e-02 5.034915e-05

LP2 CS2 1 9.097500e-03 3.533300e-0558 LP2 H 1 0.000000e+00 0.000000e+00

LP2 HO 1 0.000000e+00 0.000000e+0060 LP2 N 1 4.707795e-03 7.732889e-06

LP2 O 1 4.536057e-03 5.118813e-0662 LP2 OA 1 4.536057e-03 7.295106e-06

LP2 OM 1 4.536057e-03 5.118813e-0664 LP2 OS 1 4.536057e-03 5.118813e-06

LP3 C 1 4.557717e-03 9.393088e-0666 LP3 CB 1 4.557717e-03 9.393088e-06

LP3 CH1 1 1.053189e-02 4.331494e-0568 LP3 CH2 1 8.986321e-03 3.039668e-05

LP3 CH3 1 8.876500e-03 2.615000e-0570 LP3 CS1 1 1.053189e-02 4.331494e-05

LP3 CS2 1 8.986321e-03 3.039668e-0572 LP3 H 1 0.000000e+00 0.000000e+00

LP3 HO 1 0.000000e+00 0.000000e+0074 LP3 N 1 4.650261e-03 6.652538e-06

LP3 O 1 4.480623e-03 4.403671e-0676 LP3 OA 1 4.480623e-03 6.275917e-06

LP3 OM 1 4.480623e-03 4.403671e-0678 LP3 OS 1 4.480623e-03 4.403671e-06

LP OW 1 4.92600e-03 8.21000e-0680 LP2 OW 1 3.94400e-03 7.80300e-06

LP3 OW 1 4.82100e-03 9.53900e-0682 LP HW 1 0.000000e+00 0.000000e+00

LP2 HW 1 0.000000e+00 0.000000e+0084 LP3 HW 1 0.000000e+00 0.000000e+00

OA OSE 1 0.22617E-02 0.15062E-0586 OW OSE 1 0.24329E-02 0.19915E-05

N OSE 1 0.23473E-02 0.23849E-0588 OS OSE 1 0.22617E-02 0.74158E-06

OSE OSE 1 0.22617E-02 0.74158E-06

Acknowledgements

About two years have passed since this work began — without really knowing

it had to turn into a Ph.D. thesis —, pleasantely discussing at lita about the

strange world of gangliosides. Among all the tasks that had to be completed

before ending up in the hundred and more pages that you, dear reader, (may)

have read, the most enjoyable one is to thank the people that accompanied me,

and — directly or indirectly — helped me.

Firstly, I want to thank my advisor, prof. Renzo Vallauri, who patiently fol-

lowed, and carefully guided me step by step, while this work came into being. His

manners and considerateness made the collaboration of these years a significant

human, besides scientific, experience.

The realization of a meaningful simulation would have not been possible

without the aid of Simone Melchionna, to whom I am grateful, and whose both

sincerity and patience I greatly esteem.

Also, it would be unconceivable not to mention the whole research group led

by Mario Corti, and especially Laura and Paola, for the opportunity of working

on this enthralling subject, and for all the support they all have given me.

A special thanks is reserved to many people with whom I interacted during

these years: Pal Jedlovszky with his happy family, whose company I really ap-

preciated during the wonderful sojourn in Budapest; Graziano Guella, for his

thoughtful teaching of chemistry, and for the many useful discussions. My grat-

itude also goes to Luisa Rossi-Doria, for the valuable work she steadily carries

on, and for her great kindness. I would like to thank all of my friends: Herr

und Frau Doria–Bacca, with whom, although missing in deep Germany, I am

happy to keep an epistolary exchange; Vecchio, now back in Perugia with his

family, for neverending discussions about the latest kernel feature and happy

hours passed sealing vacuum chambers; Gianpaolo, aka Jester, with his irre-

sistible, italian-english hybrid tongue (horses, cats, owls, and the moon being his

favorite subjects of discussion); Giovanni ”Bu’a-Bu’a”, for his genuine friend-

ship, his always keen remarks about physics, and for having introduced me to

the joy of reading ”Il Vernacoliere”; Laura and Alessandro, for uncountable nice

suppers in Cimirlo; Sofia, for her politeness and for her pleasant New Year’s Eve

parties; Valerio, for the black magic that resurrected my hard disk, and many

other valuable computer tricks; Francesco, Stefano, Cesare, Marco and all other

fellows of the “pollaio”, every day bravely facing my terrible jokes.

A special thanks — which counts twice! — to my girlfriend Stefania: her

steady presence and support has been fundamental during both the easy and

troublesome situations of everyday life.

Last, but not least, a sincere thanks to my parents, Germana and Roberto

— to whom this thesis is dedicated — for all the care, love and trust they have

always given me.

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List of Tables

1.1 Formulae of various gangliosides as well as their shorthand no-

tation. Note the variability in the headgroup structure, and the

characteristic presence of N-acetyl neuraminic acid (Neu5Ac) as

sialic acid in all gangliosides. . . . . . . . . . . . . . . . . . . . . 6

1.2 Some phases and their descriptions. For a representation of the

corresponding aggregates see Figure 1.6 and 1.7. L1 identifies the

solution of isotropically distributed micelles, which occurs at mod-

erately low surfactant concentrations. . . . . . . . . . . . . . . . 13

1.3 Properties of some ganglioside aggregates: number of sugar rings

in the headgroup (Sn), molecular weight (Wm), typical aggrega-

tion number (Nagg), estimated occupied surface per head (S) and

packing parameter (P ). Reproduced from [14] . . . . . . . . . . 14

2.1 Atom types, their atomic weight and a short description. See also

the topology listed in the Appendix . . . . . . . . . . . . . . . . 22

2.2 Parameters for Berger et al. are taken from [38], whereas the gro-

mos ones refer to the ifp37C4 set. . . . . . . . . . . . . . . . . . 28

2.3 Parameters for Berger et al. are taken from [38], whereas the gro-

mos ones refer to the ifp37C4 set. . . . . . . . . . . . . . . . . . 29

2.4 Meaning of the interaction parameters appearing in the topology. 32

117

118 LIST OF FIGURES

List of Figures

1.1 Structure formula of N-acetyl-neuraminic acid, the sialic acid moi-

ety of GM3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Schematic representation of the family of glycolipids . . . . . . . 5

1.3 Upper panel: structure formula of GM3. Lower panel: snapshot of

a single GM3 molecule, excerpt from a configuration of the fully

hydrated bilayer molecular dynamics simulation . . . . . . . . . . 7

1.4 Atomic force microscopy of GM3 rafts in a DOPC monolayer . . 8

1.5 Schematic representation of an amphiphile molecule and the as-

sociated geometrical quantities employed to define the packing

parameter P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Pictorical representation of the amphiphile arrangement in various

kind of lamellar phases . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Arrangement of amphiphilic molecules in the cubic bicontinuous

and hexagonal inverse micellar phases . . . . . . . . . . . . . . . 13

2.1 Schematical representation of the stretching, bending, and tor-

sional interaction terms . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Schematical representation of three cases that require the use of

improper dihedrals: a cis double bond, a planar structure, and a

tetrahedral structure, where the missing apolar hydrogen atom is

represented in white . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Schematic representation of the SPC water model . . . . . . . . . 26

2.4 Schematic representation of two cyclohexane conformers in chair

and boat conformation . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Qualitative explanation of the gauche effect in the specific case of

the O–C–C–O bond . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Representation of a butane molecule in anti, gauche and eclipsed

conformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Structure formula of glucose . . . . . . . . . . . . . . . . . . . . . 31

LIST OF FIGURES 119

2.8 Schematic representation of two butene molecules in trans and cis

configuration around the central double bond . . . . . . . . . . . 33

3.1 Time evolution of the total potential energy of the system . . . . 38

3.2 Time evolution of the three box vectors . . . . . . . . . . . . . . 39

3.3 Time evolution of the three dihedral angles involving rotation

around the ceramide–glucose, glucose–galactose and galactose–

sialic acid bonds, respectively . . . . . . . . . . . . . . . . . . . . 40

3.4 Normalized histograms of the torsional angles involving rotation

around the ceramide–glucose, glucose–galactose and galactose–

sialic acid bonds, respectively . . . . . . . . . . . . . . . . . . . . 42

3.5 Radial distribution function of water oxygen around the glucose

residue, averaged over 1 ns, at different times . . . . . . . . . . . 43

3.6 Snapshots of a 3 nm thick slice of GM3 bilayer in solution. Bottom

panel: starting configuration. Top panel: equilibrium configuration

after 40 ns from the starting configuration . . . . . . . . . . . . . 44

4.1 Scattering intensity (logarithmic scale) measured at the European

Scattering Radiation Facility (esrf) for a sample of GM3 vesicle

and calculated from simulation versus the modulus of the scatter-

ing vector q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Mass density profile of the whole system, water, ceramide tail and

saccharidic headgroups . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 Mass density profile of a DPPC bilayer from a molecular dynamics

simulation [46] at 325 K . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Mass density profile of glucose, galactose, sialic acid residues, wa-

ter and Na+ counterions . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Radial distribution function of CH3 with respect to the center of

mass of glucose, galactose and sialic acid residues . . . . . . . . 54

4.6 Deuterium order parameter of the hydrocarbon chains of the ce-

ramide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.7 Schematic drawing showing the two vectors v and w used to define

the plane of a sugar ring and the whole headgroup . . . . . . . . 56

4.8 Normalized distributions of the orientation of the vectors vin and

vp, with respect to the bilayer’s normal . . . . . . . . . . . . . . 57

4.9 Radial distribution functions of water oxygen gw,α(r) with respect

to the center of mass of glucose, galactose and sialic acid residues 59

120 LIST OF FIGURES

4.10 Radial distribution functions of the COO− oxygen atoms with

respect to the water oxygen, Na+ ion and COO− oxygen atoms

belonging to distinct GM3 molecules . . . . . . . . . . . . . . . . 61

4.11 Contributions to the electric field arising from GM3, water and

Na+ ions, as well as the total electric field . . . . . . . . . . . . . 62

4.12 Orientational order parameter of water . . . . . . . . . . . . . . . 64

5.1 Rotational autocorrelation function of the vectors vin associated

to glucose, galactose, and sialic acid moieties, as well as that of

the whole headgroup . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Rotational autocorrelation function of the vectors vp associated

to glucose, galactose, and sialic acid moieties, as well as that of

the whole headgroup . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 msd of the heagroup center of mass in the xy plane and along the

z direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4 Rotational autocorrelation functions CR for selected atoms . . . 73

5.5 Structure formula of GM3 along with the numbering scheme adopted

to identify the carbon atoms employed in the computation of the

rotational autocorrelation functions . . . . . . . . . . . . . . . . . 73

5.6 Autocorrelation functions for different carbon atoms along the

fatty acid chain (left panel) and the rescaled functions (right panel) 76

5.7 Mass density profile of the simulated system and vertical lines

showing the initial position of water molecules choosen for the

computation of the z dependent msd . . . . . . . . . . . . . . . . 79

5.8 msd along the z direction (upper panel) and in the xy plane (lower

panel) in the whole time range . . . . . . . . . . . . . . . . . . . 80

5.9 Short time behavior of the out-of-plane (left panel) and in-plane

msd (right panel) . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.10 Profile of the diffusion coefficient computed from the short time

dependence of the msd along the z direction and in the xy plane

with an estimate of an error bar . . . . . . . . . . . . . . . . . . 84

5.11 msd along the z direction for molecules starting from different

slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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