MOLECULAR MODELING OF BIOMEMBRANE DEFORMATIONS—THEROLE OF LIPIDS

By

ERIC R. MAY

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2006

Copyright 2006

by

Eric R. May

I dedicate this work to parents David and Linda May, who instilled in me the

importance of education and encouraged me to undertake science and engineering.

I also dedicate this work to my brother Todd May who has shown me what can be

achieved through persistence and hard work. Lastly, I dedicate this to my fiancee

Jill Todisco who has loved and supported me throughout my academic career and

made sacrifices so that I could achieve my goals, for which I will always be grateful.

ACKNOWLEDGMENTS

I take this opportunity thank my mentor, Dr. Atul Narang, for proposing this

problem to me and introducing me to the field of biological modeling. I greatly

appreciate the guidance he provided me in my research, and also his advice and

wisdom on life’s other matters. I also would like to thank Dr. Dmitry Kopelevich

for all the time he has spent working closely with me over the last several years.

Additionally, I would also like to thank the other members of my committee,

Dr. Ranganathan Narayanan and Dr. Gerry Shaw, for their advice and availability.

I would like to thank Dr. Karthik Subramanian, Dr. Shakti Gupta, Jason Noel,

Ved Sharma, Ashish Gupta, Gunjan Mohan and Valere Chen for their assistance,

suggestions and the enjoyable scientific discussions we’ve had. I would also like

to thank Shirley Kelly and Debbie Sandoval for their assistance throughout my

graduate studies.

iv

TABLE OF CONTENTSpage

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Specific Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Role of Lipids in Biomembrane Deformations . . . . . . . . 31.2.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Broader Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 PHASE TRANSITIONS IN MIXED LIPID SYSTEMS: A MD STUDY . 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Molecular Model . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 LPA Micelles . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Phase Behavior of Pure DOPA System . . . . . . . . . . . 192.3.3 Phase Behavior of Mixed Lipid Systems . . . . . . . . . . . 20

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 DETERMINATION OF ELASTIC PROPERTIES FOR BILAYERS . . 31

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Bending Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Curvature Elasticity . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Molecular Model . . . . . . . . . . . . . . . . . . . . . . . . 353.2.3 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . 363.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Line Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Pressure Calculation from Molecular Dynamics Simulations 403.3.2 Description of Potentials and Forces . . . . . . . . . . . . . 423.3.3 Calculation of Virial . . . . . . . . . . . . . . . . . . . . . . 453.3.4 Distribution of Virial . . . . . . . . . . . . . . . . . . . . . 463.3.5 Calculation of Line Tension . . . . . . . . . . . . . . . . . . 49

v

3.3.6 Simulations Details . . . . . . . . . . . . . . . . . . . . . . 513.3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Tilt Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 Energy Model for Tilt . . . . . . . . . . . . . . . . . . . . . 563.4.2 Nature of the HII Phase and Evidence of Tilt . . . . . . . . 59

4 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 Major Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 Additional Lipid Species . . . . . . . . . . . . . . . . . . . 644.2.2 Continuum Scale Modeling . . . . . . . . . . . . . . . . . . 65

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

vi

LIST OF FIGURESFigure page

1–1 Membrane diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1–2 Coat protein assembly and vesicle formation . . . . . . . . . . . . . . 4

1–3 Potential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1–4 Comparision of atomistic and CG structures . . . . . . . . . . . . . . 9

2–1 Detailed atomic structures and the corresponding CG models . . . . . 16

2–2 Dependence of the size and shape of LPA micelles on [Mg2+] . . . . . 18

2–3 Dependence of the micelle asphericity on the Mg2+ concentration. . . 19

2–4 Bilayer configuration of pure DOPA systems . . . . . . . . . . . . . . 20

2–5 Temperature induced phase transitions . . . . . . . . . . . . . . . . . 21

2–6 Simulations of temperature–induced phase transitions . . . . . . . . . 23

2–7 Molecular geometry characterization for pure LPA systems . . . . . . 25

2–8 Molecular geometry characterization in mixed lipid systems . . . . . . 27

2–9 Proposed transition states in vesicle fusion . . . . . . . . . . . . . . . 28

2–10 Stalk formation in DOPE/LPA system . . . . . . . . . . . . . . . . . 29

3–1 Comparision of theoretical and experimental vesicle shapes . . . . . . 33

3–2 Atomisitic and CG representation of phosphoinositides . . . . . . . . 35

3–3 Depictions of PI4P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3–4 Angle probability distributions . . . . . . . . . . . . . . . . . . . . . . 38

3–5 Spectral intensity for PI4P/DPPC systems . . . . . . . . . . . . . . . 39

3–6 Bending modulus for PI4P/DPPC bilayer . . . . . . . . . . . . . . . . 39

3–7 Definition of rij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3–8 Angle Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3–9 Slab orientation for calculation of bilayer surface tension . . . . . . . 47

vii

3–10 Phases within a bilayer . . . . . . . . . . . . . . . . . . . . . . . . . 50

3–11 Patch system with repulsive walls . . . . . . . . . . . . . . . . . . . . 52

3–12 Density of pressure of CG systems . . . . . . . . . . . . . . . . . . . . 54

3–13 Virial and pressure profile comparision . . . . . . . . . . . . . . . . . 55

3–14 Bond potential comparison . . . . . . . . . . . . . . . . . . . . . . . . 55

3–15 Density and pressure profiles for patch systems . . . . . . . . . . . . . 57

3–16 Tilt and bending deformations . . . . . . . . . . . . . . . . . . . . . . 58

3–17 Construction of HII phase . . . . . . . . . . . . . . . . . . . . . . . . 59

3–18 Examination of HII phase from MD . . . . . . . . . . . . . . . . . . . 61

3–19 Tilt in a HII phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4–1 Proposed CG models of additional lipids: a) PI(4,5)2. b) DAG. . . . . 65

4–2 Domains within a membrane, taken from ref [1] . . . . . . . . . . . . 66

viii

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

MOLECULAR MODELING OF BIOMEMBRANE DEFORMATIONS—THEROLE OF LIPIDS

By

Eric R. May

December 2006

Chair: Atul NarangMajor Department: Chemical Engineering

Cellular transport processes such as endocytosis and exocytosis involve the

budding of vesicles from the donor membrane (fission) and their integration into

the membrane of the acceptor compartment (fusion). Both fission and fusion are

energetically unfavorable, since they involve the formation of highly curved non-

bilayer intermediates. Consequently, these processes do not occur spontaneously;

they are under the strict control of specific proteins.

In the classical paradigm of fission, membrane deformation was thought to

be driven entirely by interactions between proteins. Evidence is now accumulat-

ing that these proteins do not act alone. They operate in concert with specific

membrane lipids, such as phosphoinositides (PI), which localize in the region

of deformation. It is of interest to understand the extent to which the localized

phosphoinositides change the mechanical properties of biomembranes.

In this work, a coarse grain molecular model is used to conduct molecular

dynamics (MD) simulations of mixed lipid systems. The initial part of the work

focuses on the verification of the molecular model by comparing simulation data

to experimental phase transition data for mixed lipid systems containing dioleoyl

ix

phosphatidyle ethanolamine (DOPE), dioleoyl phosphatidic acid (DOPA) and

lysophosphatidic acid (LPA). Also, the molecular geometries of the previously

mentioned lipids are characterized and related the the mechanisms driving the

phase transitions. Additionally, MD simulations and analyses are performed

on another mixed lipid system consisting of dipalmitoyl phosphatidyl choline

(DPPC) and phosphatidylinositol-4-phosphate (PI4P). In this study two elastic

parameters of the membrane are measured from the simulations, namely, the

bending modulus and the coefficient of line tension between membrane domains of

different composition.

x

CHAPTER 1INTRODUCTION

1.1 Specific Aims

The goal of this research is to better understand the mechanisms underlying

the deformations of biological membranes. Biological membranes are very complex

structures, consisting of a variety of phospholipid species, cholesterol, and proteins

as shown in Figure 1–1a [2]. The membrane is an essential component of the

cell. It serves to enclose the inner contents, while also performing a number of

other functions. The membrane selectively allows passage of certain ions into

and out of the cell to maintain chemical gradients, which are essential to the

cell’s survival. Also, the membrane is embedded with receptors which receive

chemical signals from the environment instructing the cell to perform specific

operations. Biomembranes, as well as being compositionally complex, are also

structurally dynamic, changing shape to perform a variety of tasks. Eukaryotic

cells are capable of endocytosis and exocytosis, the process by which material is

brought into or shipped out from the cell, respectively, or passed between different

compartments of the cell. These processes involve the formation of transport

vesicles, which are small packages with an outer lipid membrane to enclose the

contents which are transported. A schematic of the budding and fission process

is shown in Figure 1–1b. Inside of the cell there are several orangelles, some of

which are enclosed by a membrane as well, such as the Golgi body, mitochondria,

endoplasmic reticulum, and nucleus. These organelles also undergo membrane

shape transformations. Lipid molecules make up the majority of these membranes,

which contain many different lipid species. During these shape changing processes,

certain lipid species will localize in and around the region of deformation.

1

2

(a) (b)

Figure 1–1: Membrane diagrams: a) Example of a biological membrane [2]. b)Depiction of budding and fission of a transport vesicle.

The hypothesis of this study is that the alteration of the chemical composition

of a biomembrane will lead to a change in the mechanical properties of the mem-

brane. This change in the mechanical property could then lead to a spontaneous

deformation of the membrane. The specific questions this study will try to answer

are the following:

1. Are certain elastic properties affected by a change in the chemical composi-

tion of the membrane?

2. What is the molecular mechanism driving the alteration of an elastic prop-

erty?

3. Does a change in chemical composition of the membrane lead to a configura-

tional change of the membrane?

With these questions answered, it will be the ultimate goal to relate the changes

in mechanical properties to the biochemistry occurring inside the cell when these

deformations occur. In essence, it is desired to develop a complete model starting

from chemical reactions inside the biochemical pathways effecting compositional

3

changes inside the membrane, effecting mechanical changes of the membrane,

leading to a continuum level model which can predict global membrane shape.

1.2 Background

1.2.1 Role of Lipids in Biomembrane Deformations

Lipid molecules are the main component of biomembranes. These lipids are

amphiphilic in nature, having one part which has an affinity for hydrophilic envi-

ronments (head) and one part which has an affinity for hydrophobic environments

(tail). The tail(s) consists mainly of saturated hydrocarbons, but unsaturated

bonds do commonly exist as well. There can be either a single tail or two tails

connected to the head. The head and tail are connected through a glycerol group.

The head group often contains a phosphate group, as well as other polar groups,

which often carry a net charge. In biological systems the environment in which

the cells lives is aqueous, as is the interior of the cell. This naturally leads to the

bilayer configuration of the lipid molecules to separate the internal contents of

the cell from the environment, as shown in Figure 1–1a. In this configuration, the

heads are submerged in both the interior of the cell and the exterior environments

while the tails form a hydrophobic region at the interior of the membrane itself.

This interior region of the membrane is very important because it provides a region

for proteins to reside since they often contain hydrophobic regions.

The mechanism by which biomembranes change shape is the focus of this

work. It had originally been postulated that the driving force behind the formation

of vesicles from biomembranes was driven by protein interactions. It is believed

that interactions between membrane bound proteins and external proteins known

as coat proteins, especially the protein complex clathrin, act to increase curvature

of the membrane and lead to bud formation [3, 4]. This is an incremental process

in which the addition of more coat proteins continuously deform the membrane

until a bud is formed and it pinches off from the original membrane. The coat then

4

Figure 1–2: Depiction of coat protein assembly and the formation of a transportvesicle. [2].

will dissociate from the vesicle to allow for fission with the target destination, see

Figure 1–2.

The process of fusion and fission of vesicles from membranes is an energetically

unfavorable one. The intermediate states can be highly curved, requiring more

energy to form than available from typical thermal fluctuations of the system [5].

However, evidence has been accumulating that the proteins do not act alone, but

act in concert with the lipid molecules themselves [6]. Some of the key lipids are

ones which exist in relatively low concentrations inside the membrane, but play

a crucial role in these morphological changes. Specifically, phosphoinositides,

phosphatidic acid, lysophosphatidic acid and diacylglycerol have been implicated

as essential lipids in the processes of fusion and fission of transport vesicles [7–9].

To this point, it has been unclear as to what function these lipids play in altering

the configuration of biomembranes. It is the goal of this work to study the effect

different lipids have on bilayers in the absence of proteins.

5

1.2.2 Molecular Dynamics

Molecular dynamics (MD) is a computational tool used to study molecular

systems both in equilibrium and non-equilibrium regimes. In MD, all particles are

treated classically, where the particles interact through different bonded and non-

bonded potentials. The bonded interactions can consist of bond length, bond angle

and dihedral potentials, which are typically modelled by a harmonic potential.

The non-bonded interactions are due to van der Waals and electrostatic forces.

For computational purposes, only pair potentials are typically considered, taking

3-body and higher term interactions becomes very time consuming because they

involve summing over triplets (or greater) of molecules [10]. However, the 2-body

potentials have shown to produce very good results in comparison to experimental

measurements.

To determine the particles position and velocities, Newtons equation’s of

motion must be solved

mi

d2ri

dt2= Fi, i = 1...N (1.1)

where i refers to the particle number, running from 1 to N , r is the position vector,

m is the mass of the particle and F is the force vector acting on the particle. The

force is given by the negative derivative of the potential with respect to position,

Fi= - dVdri

. The algorithm typically used to solve these equations is the Verlet

Algorithm [10].

v(t +4t

2) = v(t − 4t

2) +

F(t)

m4t (1.2)

r(t + 4t) = r(t) + v(t +4t

2)4t (1.3)

It can be seen Eq. 1.2 updates the velocity at time t + 4t

2from the velocity at time

t − 4t

2and the force at time t. The position is then updated, as shown in Eq. 1.3,

using the velocity at time t + 4t

2and the position at time t to get the position at

time t + 4t.

6

(a) (b)

Figure 1–3: Potential functions: a) Lennard-Jones potential b) Electrostatic poten-tial

The non bonded forces are often specified by a Lennard-Jones potential of the

form VLJ(rij) = 4ε((σ/rij)12 − (σ/rij)

6) as shown in Figure 1–3a. The parameter

ε gives the depth of the potential well and σ is the hard sphere radius. Since

the potential flattens out at large r, a cut-off radius is specified to increase the

computational efficiency. If two particles are separated by a distance greater than

the cut-off radius, it is assumed they have no influence on each other. Special care

must be taken when using cut-off radii to avoid discontinuities in the potential

functions. To avoid this, a shifting function can be added to the potential in order

for it to smoothly approach a zero slope at the cut-off radius. In addition to the

Lennard-Jones potential, an electrostatic potential must be accounted for if there

are charged particles in the system. The form of the electrostatic potential is

VES(rij) =qiqj

4πε0rijwhere qi is the charge on molecule i and ε0 is the permittivity of

free space, and is shown in Figure 1–3b. Again, a cut-off radius can be chosen for

the electrostatic potential or a more complex method of accounting for the long

range interactions known as Ewald summation may be employed.

In molecular dynamic simulations periodic boundary conditions are often

used. This is to eliminate the edge effects which would be introduced by the

presence of walls. When periodic conditions are used, it allows molecules to exit

7

from the simulation box on one side and then reappear, entering on the opposite

side of the box. The molecules will interact with the nearest periodic image of

the other molecules; this is known as the minimum image convention. In order

for the periodic simulation to represent the macroscopic system the length of the

simulation box should be at least 6σ where σ is the Lennard-Jones parameter

representing the hard sphere radius of the particles [10].

It is often desired for the simulation system to maintain a constant tempera-

ture and pressure, so that the results are comparable to experiments. One method

to control temperature is through the Berendsen algorithm, which couples the

system to a heat bath using first-order kinetics. In this weak coupling scheme the

system responds exponentially when it deviates from the bath temperature, T0,

given by Eq. 1.4, where τ is the coupling time scale. This is achieved by rescaling

the velocities of all particles by a factor λ given in Eq. 1.5. τT is related to τ by

Eq. 1.6, where CV is the total heat capacity of the system, k is the Boltzmann

constant and Ndf is the total number of degrees of freedom [11].

dT

dt=

T0 − T

τ(1.4)

λ = 1 +4t

τT

(T0

T− 1) (1.5)

τ = 2CV τT /Ndfk (1.6)

The weak coupling method of temperature coupling is easily implemented,

but it does not represent a correct statistical mechanical ensemble. An alternative

method which does probe a correct canonical ensemble, is the extended system

approach, which introduces a “Temperature Piston” to control the temperature,

thereby introducing and additional degree of freedom to the system. The equations

of motion become modified as shown in Eq. 1.7, where ξ is the heat bath variable

or friction parameter. This parameter is dynamic and is governed by it’s own

equation of motion Eq. 1.8. Q is defined by Eq. 1.9 where τT is coupling parameter

8

which defines the timescale of oscillation of kinetic energy between the heat bath

and the system [12].

d2ri

dt2=

Fi

mi

− ξdri

dt(1.7)

dξ

dt=

1

Q(T − T0) (1.8)

Q =τT T0

4π2(1.9)

Similar coupling schemes exist for pressure coupling as well. In pressure

coupling, the box vectors are scaled to adjust the volume and therefore the pressure

of the system. This can been achieved through weak coupling where the pressure

obeys first-order kinetics or through an extend system approach, where the

equations of motion of the particles are modified and the box vectors must obey

their own equation of motion. Pressure coupling can be done either isotropically or

anisotropically, which allows for deformation of the simulation box from the initial

aspect ratios. When dealing with lipid bilayers, the choice of anisotropic coupling

to equal pressure in all directions leads to a zero surface tension of the bilayer,

which is often desirable. The study of lipid bilayers using MD, is well established,

where numerous studies have shown good agreement with experimental data

regarding the density of these systems, diffusion coefficients and permeability [13].

The molecular models used in MD can be atomistic, in which all molecules,

bonds, bond angles and partial charges are specified (possibly excluding hydrogens)

or coarse grained in which multiple atoms are defined by a single interaction site.

There are methods of deriving coarse grain potentials from, atomistic potentials

one of which is the Iterative Boltzmann Inversion [? ]. In this method an initial

potential is guessed using a known radial distribution function, g(r), such that

V0(r) = −kT ln(g(r)). Simulating the system with this potential will then produce a

different radial distribution, g0(r) and the potential is corrected by adding the term

9

(a) (b)

Figure 1–4: Comparison of atomistic and coarse grain structures a) DPPC b) Wa-ter

−kT ln( g0(r)g(r)

) to the initial potential guess. This process can be iterated several

times until reasonable convergence of the radial distribution function is achieved.

To date several coarse grain models have been published, including one by Marrink

et al. which includes models for several lipid species [? ]. In Marrink’s model,

a coarse grain particle typically represents 4-6 atoms; shown in Figure 1–4 are

depictions of the atomistic and coarse grain representations of DPPC, a common

biological lipid, and water. The labelling of the coarse grain particles corresponds

the the force field designation of the particles published by Marrink [? ]. The main

advantage of using a coarse grain model is that it reduces the degrees of freedom

in the system and also allows for a larger time step, which make this model much

more computationally efficient than an atomistic model.

1.3 Broader Impact

The foundation of the work is the advancement of scientific knowledge. The

work has been conducted without a direct application intended. However, it is

hopeful that the findings of this research could lead to more directed studies. Biol-

ogy is a field which often offers qualitative and sometimes speculative explanations

of the phenomena. Taking an engineering approach to biological problems offers

10

the possibility of quantitative and mechanistic explanations. The motivation be-

hind this research is understanding the mechanical implications of observed lipid

localization in biomembranes. Furthermore, the goal of this work is to understand

how changes in chemical composition of membranes affect elastic properties and

what are the underlying molecular mechanisms driving these changes in the mem-

brane properties. These effects can then be compared with the forces generated

inside the cell including pressure, osmotic pressure, polymerization of filaments and

protein interactions to see if chemical changes within the membrane are playing a

significant role in the deformation of membranes.

CHAPTER 2PHASE TRANSITIONS IN MIXED DOPE-DOPA AND LPA-DOPA LIPID

SYSTEMS: A MOLECULAR DYNAMICS STUDY

2.1 Introduction

Cellular transport processes such as endocytosis, exocytosis, and subcellular

trafficking involve the budding of vesicles from the donor membrane (fission) and

their integration into the membrane of the acceptor compartment (fusion). Both

fission and fusion are energetically unfavorable, since they involve the formation of

highly curved non-bilayer intermediates. Consequently, these processes do not occur

spontaneously—they are under the strict control of specific proteins.

In the classical paradigm of fission, membrane deformation was thought to be

driven entirely by interactions between proteins. According to this model, fission

is initiated by the recruitment of certain cytosolic proteins called coat proteins to

the membrane. The stepwise assembly of coat proteins then incrementally deforms

the membrane into a spherical shape, thus producing a vesicle which is ultimately

released along with its encased coat.

Evidence is now accumulating that these proteins do not act alone. They

operate in concert with particular membrane lipids, namely, phosphoinositides,

diacylglycerol, and phosphatidic acid. One function of the lipids is to recruit

proteins involved in membrane deformation. It has been shown, for instance, that

phospholipase D, an enzyme required for vesicle formation in the Golgi complex,

is recruited to the membrane by diacylglycerol. More importantly, however, lipids

appear to be directly involved in deformation of biomembranes. Early experiments

showed that enzymes that acylate lysophosphatidic acid (LPA) to phosphatidic

acid (PA) strongly activate vesicle formation. Since LPA and PA are one- and

11

12

two-tailed phospholipids, it seems plausible to hypothesize that LPA is cone-

shaped (4) and PA has the shape of an inverted cone (∇). Now, it is known

from condensed-matter physics that amphiphilic molecules self-assemble into

aggregates whose morphology is intimately linked to the shape of the individual

molecules [14]. Specifically, ∇- and 4-shaped molecules form structures with

positive and negative curvatures because these geometries reduce bending stresses

within the membrane. This has led to the hypothesis that the shape change

induced by LPA-acyltransferases is ultimately driven by the distinct shapes of LPA

and PA molecules.

Recently, Kooijman et al. performed experiments in a cell-free system to

ascertain the molecular shape of LPA and PA [15]. To this end, systems of LPA,

PA, and mixtures of these lipids with various other lipids were studied in a variety

of environments in which pH, temperature and divalent cation concentrations were

varied to mimic the conditions in the cytosol and Golgi complex. In the mixed lipid

experiments, LPA or PA was mixed with either dioleoyl phosphatidylethanolamine

(DOPE) or dioleoyl phosphatidylethanolamine (DEPE), and again the pH, tem-

perature, and ionic conditions were altered. Using 31P-NMR measurements the

system phase was distinguished for a given set of conditions. It was observed that

a system of a given lipid composition would transition to a different phase by

altering environmental conditions or temperature. These phase transitions involve

a change in curvature of the of the equilibrium structure, often from a relatively

flat lamellar phase to a high curved inverted hexagonal phase. LPA and PA are

of particular interest because in the Golgi body the acylation of LPA to form PA

has been shown to induce the formation of a vesicle from the Golgi membrane [7].

LPA and PA are very different from each other from a geometrical perspective.

The spontaneous curvature of LPA was determined to be highly positive, desiring

13

to form micelles, while PA has a negative spontaneous curvature, under physi-

ological conditions [16]. It is believed that this transformation from LPA to PA

induces stress in the membrane leading to the destabilization of the bilayer and the

formation of a bud.

The goal of this work is to develop a model for the lipids LPA and PA to

be used in molecular dynamics studies. A coarse grain (CG) model developed by

Marrink et al. is capable of reproducing experimentally observed phase transitions

for a phospholipid system consisting of dioleoyl phosphatidylcholine (DOPC)

and dioleoyl phosphatidylethanolamine (DOPE) [? ]. Using Marrink’s force

field parameters, models for PA and LPA were constructed and simulations were

conducted and compared to experimental results of Kooijman et al. [15]. In

addition, it is desired to characterize the molecular geometry of these molecules

to gain insight into the mechanism driving the phase transitions. This work is

a preliminary study to verify CG models for the lipids LPA and PA. With the

assurance that the models are reliable, further study is planned to fully investigate

the mechanism driving membrane deformation in mixed lipid systems.

2.2 Methods

2.2.1 Molecular Model

In order to accurately evaluate the elastic (i.e., macroscopic) properties using

microscopic simulations, it is necessary to perform simulations of systems contain-

ing thousands of lipids for hundreds or thousands of nanoseconds. The application

of a detailed atomistic model to such length and time scales is extremely time-

consuming even with modern computing systems and therefore limits the scope of

the systems that can be investigated within a reasonable time frame. Therefore,

simulations of lipid systems often employ less detailed models, such as the Brown-

ian dynamics simulations [? ], dissipative particle dynamics [? ], and coarse-grained

molecular dynamics (CGMD) models [17? ? ? , 18]. In this work, we use a CGMD

14

model, which allows one to perform simulations on near atomic length scales, while

reducing the computational time by orders of magnitude and therefore allowing

exploration of the systems on sufficiently large time- and length-scales. The CGMD

models approximate small groups of atoms by a single united atom (bead). Several

such models have been introduced and applied to simulations of various complex

molecular systems [17, 19? ? ? ? , 20]. Development of coarse-grained molecular

models is a subject of active ongoing research [? ? ? ]. In this work, we will use

the coarse-grained model proposed by Marrink et al. [? ]. This model has been

shown to yield good agreement with experiments and atomistically detailed simu-

lations for such quantities as density and elasticity of pure lipid bilayers. Within

this model, for example, 4 methyl groups of lipid tails are represented by a single

hydrophobic coarse-grained bead and 4 water molecules are represented by a single

spherical polar bead. This model also provides several types of beads that can be

used to model various groups of atoms (such as glycerol and phosphate) within

lipid headgroups. The interactions between non-bonded beads are modeled by the

Lennard-Jones potential, whereas the interactions of bonded beads are modeled by

the harmonic bond and angle vibration potentials. Also, electrostatic interactions

between charged beads are taken into account.

The parameters for these potentials as well as models for several molecules

were previously published by Marrink et al. [? ] and the same parameters were

used in this work. For the nonbonded interactions a cut off distance of 1.2 nm was

used. To avoid discontinuities of the potentials at the cut off point the LJ potential

is smoothly shifted to zero between 0.9 nm and the cutoff point. The electrostatic

potential is also shifted to zero beginning at 0.0 nm, to represent the effect of ionic

screening.

In this work, we will use the coarse-grained model for water, metal ions, and

DOPE lipids proposed in by Marrink et al. [? ]. In addition, we have developed

15

coarse-grained models of the additive lipids using the methodology of Marrink

et al. [? ] for mapping different functional groups within a molecule to coarse-

grained beads with specific properties. The detailed atomistic structures and the

corresponding course-grained representations of some of these lipid molecules

are shown in Fig. 2–1. The structure and CG model for the lipid LPA shown in

Fig. 2–1a consists of 7 CG particles. Each bead is classified with respect to the

system published by Marrink et al. [? ]. The glycerol-ester linkage is modelled

by a single bead in LPA with the classification Nda, indicating a non-polar

particle capable of both hydrogen bonding and donating. Glycerol in double

tailed molecules is typically modelled with two beads, however for LPA the choice

to use only a single bead model for glycerol was based upon a closer representation

of the true mass of the molecule and the qualitative agreement of the simulations

with experiments using this model. An attempt to model the glycerol of LPA with

2 beads, as in DOPA, for a total of 8 beads in the model, produced a simulation

result which did not agree with the experiments. In the absence of ions the self-

assembled structure of the 8 bead LPA formed a worm–like micelle, where the

experiments predict spherical micelles.

The structure and model for dioleoyl phosphatidic acid (DOPA) is shown in

Fig. 2–1b. The CG model consists of 13 CG particles. The glycerol ester linkage is

modelled in the typical way using two beads of type Na, indicating it is non polar

and can act as a hydrogen acceptor. LPA has the ability to be a hydrogen donor

due to the hydroxyl group on the glycerol, where in DOPA the oxygen is bound to

the ester linkage. Both of these lipids are modelled with oleoyl fatty acid tails (18

carbons, 1 double bond), the double bond causing the kink in the chain. Lastly, a

model for a divalent cation is introduced using a single bead of type Qda, carrying

a charge of 1.4 e. The charge is reduced from 2.0 e to account for the hydration

shell surrounding the ion.

16

(a) (b)

Figure 2–1: Detailed atomic structures and the corresponding coarse-grained mod-els of considered lipids: (a) LPA and (b) DOPA. Mapping between different groupsof atoms and the coarse-grained beads is shown for some groups. The types of thelipids (C = hydrophobic, P = polar, etc.) and their potential parameters are thesame as in Marrink et al. [? ]

2.2.2 Simulation Details

Molecular dynamic (MD) simulations were conducted, using the GROMACS

MD simulations package [? ], version 3.2.1. In all simulations anisotropic Berendsen

pressure coupling was used with a reference pressure of 1.0 bar. All systems were

also coupled to a temperature bath using Berendsen coupling. The time step used

for the simulations was 40 fs. Periodic boundary conditions were imposed in all

three directions.

2.3 Results

In this study molecular dynamic simulations were used to investigate the

polymorphic phase behavior of lipid systems. Experimental results have indicated

that lipid systems are sensitive to pH, temperature, ion concentration and relative

water hydration levels [15, 21, 22]. We use the coarse-grained (CG) molecular

model developed by Marrink et al. and discussed above. Recall that in this model

four water molecules are represented by a single CG particle. Therefore, in what

follows when referring to the amount of water in the simulation system the amount

17

of “real” water molecules will be given. Also, it is known that the dynamics of

coarse-grained models is typically faster than that of a real system. This is due

to the smoothing of the potentials. For the specific model considered here, the

CG time is 4 times faster than the real time. Therefore, the times reported for

the simulations are given as the effective time, which is calculated by scaling the

observed simulation time by a factor of 4.

2.3.1 LPA Micelles

Kooijman et al. studied the shape change of LPA aggregates in response

to increasing concentrations of Mg2+ (Mg2+/LPA = 0, 0.2, and 1.0) [15]. The

experiments were performed at 37�

and a pH of 7.2. They observed that in the

absence of Mg2+, LPA forms micellar aggregates. However, as the Mg2+/LPA ratio

is increased, the shape of the LPA aggregates changes from a micellar configuration

to a system of bilayer disks.

To test the ability of the CG model to reproduce these observations, we

conducted self-assembly simulations of the LPA lipids (shown in Fig. 2–1a), water

and different concentrations of divalent ions (representing Mg2+). The simulations

were performed at 37�

. The simulation box was randomly populated with 500

molecules of LPA, 192 molecules of water per LPA molecule, and the appropriate

amount of Mg2+ (determined by the Mg2+/LPA ratio that was being studied).

Initial random dispersions of LPA and Mg2+ in water quickly self-assembled into

micelles and equilibrated within 800 ns.

Figures 2–2a–d show the equilibrium configurations achieved at the end of the

simulation for the three Mg2+/LPA concentration ratios used in the experiments.

The simulations show that in the absence of Mg2+, six small micelles are formed

(Fig. 2–2a). When the Mg2+/LPA ratio is increased to 0.2, five micelles are formed

(Fig. 2–2b). If the Mg2+/LPA ratio is increased even further to 1.0, the larger

micelles aggregate to form a single aggregate. Figures 2–2c,d show the top and

18

(a) (b)

(c) (d)

Figure 2–2: Dependence of the size and shape of LPA micelles on [Mg2+]: (a) Mo-lar ratio [Mg2+]/[LPA] = 0. (b) Molar Ratio [Mg2+]/[LPA] = 0.2. (c) Molar ratio[Mg2+]/[LPA] = 1.0, side view (d) Molar ratio [Mg2+]/[LPA] = 1.0, top view. Inthese plots the water molecules and ions were removed for clarity.

side views of this single aggregate. It is clear from these figures that the aggregate

has a disk-like configuration. Therefore, as the concentration of the ions increases,

the micellar size increases and micelles acquire disk-like shapes, which is consistent

with experimental 31P-NMR measurements of Kooijman et al. [15].

To quantify the shape change of the LPA aggregates shown in Fig. 2–2, we

calculated their asphericity, which is defined as

A =

∑3i<j(R

2i − R2

j)2

∑3i=1(R

2i )

2(2.1)

where Ri are the radii of gyration. The asphericity characterizes the extent to

which the shape of an aggregate deviates from that of a sphere. The asphericity of

the aggregates was calculated by averaging over the data corresponding to the final

19

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Mg+2/LPAA

sphe

ricity

Figure 2–3: Dependence of the micelle asphericity on the Mg2+ concentration.

80 ns of the simulation. Fig. 2–3 shows that as the concentration of Mg2+ increases,

so does the asphericity of the lipid aggregates.

It is of interest to understand the molecular basis for the shape changes of the

LPA aggregates shown in Fig. 2–2. According to the shape-structure concept of

lipid polymorphism, the shape of an aggregate is determined largely by the shape

of the component molecules. Therefore, we expect that the change from spherical

to disk-like micelles indicates a change in the molecular geometry of LPA lipids.

2.3.2 Phase Behavior of Pure DOPA System

Kooijman et al. also studied the phase behavior of pure DOPA systems. It

was observed that a DOPA system formed a lamellar phase at neutral pH and a

temperature of 310 K both in the absence and equimolar presence of Ca2+. Self-

assembly simulations were conducted to match this result. 500 DOPA molecules

and 17500 CG waters were randomly placed in a simulations box, for the system

with Ca2+, 500 of CG divalent cations were also added to the system. The systems

were simulated at 310 K for 800ns and in both cases a bilayer phase was observed.

This indicates the model for DOPA and divalent cation can match the experimental

observations.

20

(a) (b)

Figure 2–4: Bilayer configuration of pure DOPA system observed after 800ns ofsimulation. a) No divalent ions. b) Divalent ion concentration equal to lipid con-centration.

2.3.3 Phase Behavior of DOPE-LPA and DOPE-DOPA Mixed System

We also investigated phase transitions between the lamellar and inverted

hexagonal phases in the mixed DOPE-LPA and DOPE-DOPA systems. The exper-

imental data [15] summarized in Fig. 2–5 indicate phase transitions from lamellar

to inverted hexagonal phases within the bilayer systems as the temperature of the

system increases.

Kooijman et al. [15] observed that when the temperature is increased, pure

DOPE undergoes a transition at T≈ 275 K from the lamellar (Lα) phase to the

inverted hexagonal (HII) phase (curve labeled 4 in Fig. 2–5). Upon addition of

10 mol% LPA to DOPE, the phase transition temperature increases to T≈ 300 K

(© curve of Fig. 2–5). The addition of 10 mol% DOPA to DOPE results in slight

increase in the phase transition temperature to T≈ 280 K (� curve of Fig. 2–5).

Therefore, the experimental data show that the change of the minor mixture

component from LPA to DOPA has a destabilizing effect on the lamellar phase by

lowering the phase transition temperature by ≈ 20 K. We perform simulations to

check if this transition was mirrored by the model.

Recently, Marrink et al. [? ] studied the phase behavior of the CG model for

pure DOPE. They found, for a hydration level in the range of 9-12 waters per lipid,

21

240 260 280 300 320 340

0

10

20

30

40

50

60

70

80

90

100

% B

ilaye

r

Temperatuer(K)

Pure DOPE (Exp)

DOPE/LPA (Exp)

DOPE/DOPA (Exp)

Pure DOPE (Sim)

DOPE/LPA (Sim)

DOPE/DOPA (Sim)

Figure 2–5: Temperature-induced phase transition between lamellar and invertedhexagonal phases in the mixed lipid systems: Comparison of experimental data [15]and simulations for DOPE/DOPA (�) , DOPE/LPA systems (©) and pure DOPE(4). The experimental data are shown by open symbols and the simulations resultsare shown by closed symbols. The lamellar phase corresponds to 100% bilayer andthe inverted hexagonal phase corresponds to 0%. The composition of the mixedsystems is 90% DOPE and 10% of a minor lipid (LPA or DOPA). The simulationdata for mixed systems are from the current work, where as the simulation data ofthe pure DOPE system was taken from Marrink et al. [? ].

the phase transition temperature to be above 287 K (N curve of Fig. 2–5). This

is quite close to the experimentally observed phase transition temperature. To

determine the phase transition, temperature simulations were performed at several

temperatures in which the initial configuration of the system consisted of four

pre-simulated bilayers stacked on top of each other. In some of the simulations,

the bilayers would form stalks between the layers, which would then grow and the

(HII) phase would spontaneously form.

We performed similar simulations to check if the CG model could capture

the phase behavior of the DOPE-LPA system. The initial conditions for our

simulations were taken to be lamellar phases prepared following the simulation

protocol of Marrink et al. [? ]. First, we used self-assembly of initially randomly

22

dispersed lipids in water to prepare a lipid bilayer. The self-assembly simulations

were performed in a system with relatively low lipid concentration which leads to

bilayer formation within 200 ns, the system was allowed to run for and additional

600ns to allow for equilibration. The self assembly simulation was conducted at

a water to lipid ratio of 80 and a temperature of 280 K. We then dehydrate the

bilayer down to a water lipid ratio of 15 and stack 2 identical dehydrated bilayers

on top of each other. In an attempt to promote formation of the HII phase, the

bilayers were given a perturbation so they would resemble a sine wave in one

direction. The direction of the perturbation was alternated between the bilayers

so that the peak of one bilayer would coincide with the valley of the neighboring

bilayer. This configuration was then energy minimized using the steepest descent

method in GROMACS for 50000 steps. This energy minimized configuration was

then copied in the normal direction to obtain a system consisting of 4 bilayers.

This initial configuration is shown in Fig. 2–6a. Each bilayer consists of 450 DOPE

molecules and 50 LPA molecules. The simulations were performed at various

temperatures between 250 K and 300 K.

Examples of final configuration of the simulations for the DOPE-LPA system

are shown in Fig. 2–6b and c. At temperatures at or below 265 K the lamellar

phase remains stable on the timescale of simulations (4.0µs), see Fig. 2–6b. On

the other hand, at T = 275 K and above, the same system exhibits an instability

which leads to formation of a stalk and then ultimately the inverted hexagonal

phase is reached. The final configuration for the simulation at 300 K is shown in

Fig. 2–6c. All of the configurations shown in Fig. 2–6 were generated by copying

the simulation cell 3 times in the lateral direction to better illustrate the geometry

of the systems.

23

(a) (b) (c)

Figure 2–6: Simulations of temperature-induced phase transition between lamellarand inverted hexagonal phases in the mixed DOPE-LPA lipid systems: (a) Initialperturbed configuration for DOPE-LPA simulations. (b) Final configuration ofDOPE-LPA system at 250 K (lamellar phase). (c) Final configuration of DOPE-LPA system at 300 K . Lipid headgroup beads are shown by black spheres, tailbeads are shown by gray spheres, and the water beads are shown by white spheres.

Figure 2–5 shows a comparison between the experimental system and the

simulation results. The model system displays a phase transition temperature

above 265 K whereas the experimental system transitioned around 300 K.

For the DOPE-DOPA simulations, a bilayer consisting of 120 lipid molecules

(10% DOPA, 90% DOPE) was simulated by self-assembly at water to lipid ratio

of 120 and at 240 K for 80ns. These small bilayers were then copied laterally to

form a bilayer consisting of 480 lipids. A perturbed bilayer system was then created

in the same manner as for the DOPE-LPA system, again at a reduced water to

lipid ratio of 15. This perturbed system was then simulated for up 4.0µs at varying

temperatures between 240 K and 310 K. At a temperature of 265 K and greater the

systems spontaneously formed the HII phase. At temperatures of 240 K and 250

K the system remained stable in the bilayer phase, but at temperature at or above

265 K the systems converted into the HII phase.

Figure 2–5 shows that the simulations are in qualitative agreement with

the experimental data. The model DOPE-DOPA system undergoes a phase

transition at a temperature approximately 15 K lower than the phase-transition

24

temperature for DOPE-LPA system, compared to an approximate 20 K shift

for the experimental system. However, the precise values of the phase transition

temperature predicted by the simulations are different from those observed in the

experiments. The experiments indicate phase transition temperatures of 280 K

and 300 K for the DOPE-DOPA and DOPE-LPA systems, but the corresponding

simulated values transitions temperatures are above 250 K and 265 K, respectively.

As can be seen, the current coarse-grained model does not capture the exact

phase transition temperature, it does capture important qualitative features of the

systems, such as the destabilizing effect on the lamellar phase due to the change

from LPA to DOPA.

2.4 Discussion

We also studied the molecular mechanism involved in temperature-dependent

phase transitions. It is evident that as the temperature increases, so do the bilayer

fluctuations, thus facilitating the possibility of a phase change. However, for the

hexagonal phase to be stable, it is necessary that the molecular geometry also

change. To investigate the molecular shape changes in temperature-dependent

phase transitions, the simulations of the DOPE-LPA system were analyzed.

Molecular packing theory states the microstructure of the lipid aggregate should

be representative of the mean molecular geometry of the system [14]. In order to

verify this hypothesis, the packing parameter was measured, P = v/lca0, where

v is the volume occupied by the hydrocarbon tails, lc is the length of the tails in

the direction normal to the water lipid interface and a0 is the area occupied by the

head group.

For the pure LPA system, a transition between several spherical micelles to

a single bilayer disk is observed upon the addition of an equimolar concentration

of divalent ions. In Fig. 2–7a, the packing parameter is shown for the LPA system

at the varying divalent ion concentration. A significant increase in the packing

25

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

Divalent Ion/Lipid Molar Ratio

Pac

king

Par

amet

er

(a)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

Divalent Ion/Lipid Molar Ratio

Mol

ecul

ar P

rope

rty

Mea

sure

men

ts

a0 (nm2)Tail Length (nm)

Tail Volume (nm3)

(b)

Figure 2–7: Molecular Geometry Characterization of LPA in a pure LPA system:a) LPA Packing Parameters. b) LPA measurements of tail volume, tail length andhead spacing.

parameter occurs upon the increase in ion /lipid ratio from 0.2 to 1.0, which

coincides with the transition from micelle to disk. To understand why the packing

parameter is changing, the individual measurements of a0, lc, and v are shown

in Fig. 2–7b. Both a0 and lc decrease upon addition of the ions, while v remains

relatively constant. The volume of the tails should remain constant since the

temperature is fixed; the change in a0 can be understood because the divalent ions

act to screen the negatively charged LPA headgroups from each other, causing an

effective decrease in head group size.

At 250 K and 265 K the DOPE-LPA system remained in the lamellar phase,

and at 285 K and 300 K the system transitions into the HII phase; these state

points were used for the packing parameter analysis. Data was analysed from the

final 80ns of the simulation after which the system had formed its final phase and

equilibrated. The packing parameter for DOPE, LPA and the mean value for the

system are shown in Fig. 2–8a. Discontinuous jump is observed between the Lα

points and the HII points, indicating that the shape of the molecules becomes

more like an inverted cone (5). In Fig. 2–8b the measurements of a0, lc and v are

26

shown. It is observed that lc is the parameter which is changing the most. Since lc

is decreasing with increasing temperature, the tails sample configurations farther

away from the equilibrium configurations, essentially reducing their extension

normal to the water lipid interface. The theory states that values of the packing

parameter above 1 lead to inverted phases, however it can been seen that even the

lamellar phase points are above 1. This is due to the assumption in the calculation

of the head group area, a0, that DOPE and LPA have equal size head group

areas. The head group area was estimated by calculating the area of the water

lipid interface and dividing through by the number of lipid molecules at that

interface. There was no accurate way to individually measure each of the species

a0’s separately, therefore, for the purpose of the packing parameter calculation it

was assumed both species had the same ao. From this analysis it is evident that the

molecular mechanism driving the phase transition is the configuration of the tails.

The increased temperature imparts a greater energy on the tails to deviate away

from relatively straight configurations to a more bent and splayed structure.

The transition states which a lipid systems samples as it transitions from a Lα

to a HII has been studied both experimentally and theoretically[23–27]. Biologi-

cally this transition is of particular interest for understanding the mechanism of

vesicle fusion and understanding the how antimicrobial peptides destroy bacterial

membranes [28]. A mechanism proposed by Siegel, proposes the bilayers first form

a stalk phase and then the outer monolayers contract and form a contact know as

the trans monolayer contact (TMC), Fig. 2–9a. The TMC phase involves formation

of hydrophobic void spaces which Kozlovsky et al. states are energetically unfavor-

able, and a different intermediate stalk structure is proposed [26]. This alternative

stalk model is shown in Fig. 2–9b; in order to reach this configuration the lipid tails

must tilt away from normal to water–lipid interface to prevent the formation of

void space.

27

250 255 260 265 270 275 280 285 290 295 3001.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Temperature (K)

Pac

king

Par

amet

er

(a)

250 260 270 280 290 3000.4

0.6

0.8

1

1.2

1.4

1.6

Temperature (K)M

olec

ular

Pro

pert

y

a0 (nm2)Lc (nm)

Single Tail Volume (nm3)

(b)

235 240 245 250 255 260 265 270 275 2800.6

0.8

1

1.2

1.4

1.6

1.8

2

Temperature (K)

Mea

n P

acki

ng P

aram

eter

(c)

235 240 245 250 255 260 265 270 275 2800.4

0.6

0.8

1

1.2

1.4

1.6

Temperature (K)

Mol

ecul

ar P

rope

rty

a0 (nm2)Lc (nm)

Single Tail Volume (nm3)

(d)

Figure 2–8: Molecular Geometry Characterization of DOPE, LPA and DOPA inmixed system: a) DOPE-LPA system packing parameters. b) DOPE-LPA systemmeasurements of tail volume, tail length and head spacing. c) DOPE-DOPA systempacking parameters d) DOPE-DOPA measurements of tail volume, tail length andhead spacing.

28

(a) (b)

Figure 2–9: Proposed transitions in vesicle fusion a) Siegal stalk and TMCmodel [23]. b) Kozlovsky and Kozlov stalk model [26].

To verify which of these mechanisms our simulations underwent the a system

of 90% DOPE and 10% LPA was analyzed at 300 K. The formation of a stalk is

depicted in Fig. 2–10a–d. It is clear form examining the simulation images that the

stalk structure is very similar to that proposed by Kozlovsky et al.. There is no

evidence of TMC of void region, furthermore the water pores are long narrow ovals

are shown by Kozlovsky et al. [26].

Using a coarse grain model for molecular dynamics has proved to be a qual-

itatively reliable tool for studying lipid systems. In real system there are many

parameters and a high level of complexity. However, in our model system the

system is greatly simplified, but the simulations are capable of matching the trends

shown experimentally. The key result from these simulations is the shift in the

transition temperature, not the temperature itself. The the transition temperature

is lowered by changing the minor component from LPA to DOPA both experi-

mentally and in our system. Therefore there is a range of temperature, in between

the phase transition temperature, which the conversion of LPA to DOPA or vice

29

(a) (b)

(c) (d)

Figure 2–10: Stalk formation in a DOPE/LPA system: a) 96 ns. b) 112 ns. c) 128ns. d) 144 ns. The images were zoomed in on just 2 of the bilayers to better il-lustrate the stalk formation. Water was removed for clarity, the head groups arecolored black and the tails are grey.

versa can induce a transformation. This model can also be used to quantify the

molecular geometry of the molecules.

The simulation results give good qualitative results, but there is a discrepancy

between the reported experimental transition temperature and the observed values

in the simulations. There is potential to produce better quantitative agreement by

changing some parameters in the model. Increasing the hydration of the system

will stabilize the bilayer by increasing the separation of the stacked bilayers,

making a transition more difficult, and should shift the transition temperature to a

higher value. Also, the Lennard-Jones parameters can be altered as Marrink et al.

did in their study of DOPE and DOPC to make the bilayer phase more stable [? ].

However in this study we chose not to modify the force field to remain consistent

with previous simulations.

30

One limitation of this coarse grain approach is that the pH of the system

cannot be explicitly modelled. The charges on the molecules can be modified to

reflect the desired pH, but H+ ions are too small to be modeled in this approach.

Kooijman et al. do observe an effect of changing the pH on the phase behavior of

these lipid systems, without changing the net charge on the lipids.

CHAPTER 3MOLECULAR MODELING OF KEY ELASTIC PROPERTIES FOR

INHOMOGENEOUS LIPID BILAYERS

3.1 Introduction

In biology, membranes can be very non-uniform in their structure and com-

position. The major component of the biological membrane are lipid molecules,

while they also consist of cholesterol and proteins [2]. One type of molecule

which is implicated in many biological shape changing process are phosphoinosi-

tides [9, 29–31]. In this study the effect of introducing a phosphoinositide molecule,

phosphatidylinositol-4-phosphate (PI4P) to a homogeneous dipalmitoyl phos-

phatidyl choline membrane is studied. Two important elastic constants will be

measured.

The bending modulus is a measure of the amount of energy required to change

the curvature of a membrane. The bending modulus has previously been measured

both experimentally and from simulations for homogeneous systems [32, 33]. In

this study, molecular dynamics simulations of mixed PI4P and DPPC bilayers

are simulated and analyzed to determine the effect of PI4P concentration on

the bending modulus. Understanding the effect of concentration on the bending

modulus is important since in biological membranes phosphoinositides will localize

in one region of the membrane. This region of localization is often the site a

membrane deformation in the form of a protrusion, budding or endocytosis.

A second key elastic parameter is the line tension existing between lipid

phases. If a tension exists between phases this can act to cause budding or en-

docytosis as the tension force acts to minimize the size of the interface, resulting

in an out of plane deformation [1, 34, 35]. The line tension existing between a

31

32

homogeneous DPPC section and an inhomogeneous PI4P/DPPC membrane section

will be determined, again with the use of molecular dynamics simulations.

A third elastic parameter will also be investigated. When lipid molecules in a

monolayer tilt away from the lipid-water interface normal direction and energy is

associated with this type of deformation. The tilt modulus characterized the energy

associated with this type of deformation. In this work, it will be shown that tilt

deformations in a hexagonal phase observed through MD simulations are consistent

with theoretical and experimental predictions [36–38].

3.2 Bending Modulus

3.2.1 Curvature Elasticity

A continuum model for describing the shape of bilayer vesicles was first

proposed by Helfrich [39]. The Helfrich model predicts a quadratic relationship

between the mean curvature, C, of a homogeneous bilayer and the free energy of

the bilayer per unit area, f , in Eq. 3.1,

f =1

2κ(C − C0)

2 + κ̄K (3.1)

where κ is the bending modulus, C0 is the spontaneous curvature of the bilayer, κ̄

is the Gaussian modulus and K is the Gaussian curvature. The mean and Gaussian

curvatures are defined below:

C = κ1 + κ2

K = κ1κ2

where κ1 and κ2 are the principal curvatures of the bilayer surface.

The minimum energy configuration of the bilayer can be solved by limiting

the solutions to spherically symmetric shapes for a given volume, V , and surface

area, A, of the enclosed bilayer. This is a constrained minimization problem, where

the object is find the curve which minimizes the total energy of the bilayer given

33

Figure 3–1: Comparison of theoretical and experimental vesicle shapes. [40].

the constant volume and surface area restrictions. The total functional is given by

Eq. 3.2 where, Σ, and P are Lagrange multipliers.

Fb =

∮

fdA + ΣA + PV (3.2)

All functions can be parameterized by the arc length of the curve defining the

axisymmetric surface. A set of Euler-Lagrange shape equations can then be

derived and numerically solved to yield the minimum energy shape given a set

of constraints. The phase diagram has been mapped out by Seifert et al. and

several biologically relevant shapes were determined to be minimum energy

configurations [40, 41]. Figure 3–1 shows a comparison of experimental and

theoretical vesicles shapes as the temperature of the system is altered thereby

altering the surface area and volume of the vesicles. Budding shapes, as well as, the

biconcave shape of the red blood cell are observed.

In solving the shape equations, the integral of the Gaussian curvature, over

the surface area for closed vesicles remains constant and therefore need not be

considered in variational problem. This leaves only the mean curvature term, and

it is desired to know the value of the bending modulus. This has been determined

34

experimentally [42] and also from molecular dynamic simulations [33, 43] and good

agreement has been shown between them. In order to determine κ from simulations

the thermal fluctuations of an equilibrium bilayer can be analyzed. If we assume

C0 = 0 and neglect the Gaussian bending term the free energy per unit area is

given by

f =1

2κ(hxx + hyy)

2 =1

2κ(∇2(h(r))2

where h is the height of the bilayer and the subscripts denote derivatives with

respect to a direction, where the bilayer normal is the z direction. Taking the

Fourier transform of the surface, Eq. 3.3, and the inverse transform, Eq. 3.4 yields

and expression for h(r) in terms of the wave vector q.

h(q) =1√A

∫

h(r)e−iq·rdr (3.3)

h(r) =1√A

∑

q

h(q)eiq·r (3.4)

By taking the Fourier transform of the surface, the bending modes become decou-

pled and the total energy of the surface is given by Eq. 3.5.

F =

∫

fdA =1

2κ∑

q

q4|h(q)|2 (3.5)

From the equipartition theorem, the energy of each mode can be can be related to

the Fourier coefficients shown in Eq. 3.6

< |h(q)|2 >=kBT

κq4(3.6)

where kB is Boltzmann’s constant and T is the temperature of the system. The

bending modulus can then be determined from the simulation data by fitting

< |h(q)|2 > versus q4.

35

(a) (b)

Figure 3–2: Atomistic and coarse grain representation of phosphoinositides: a)PI4P. b) DPPC.

3.2.2 Molecular Model

DPPC is one of the major components of most biological membranes. PI4P is

an important molecule in membrane systems because it has been implicated to play

an essential role in Golgi membrane functions and it is a precursor to PI(4, 5)P2 [9].

From a geometric perspective, simulating systems which contained PI molecules

was intriguing because of the large head group due to the inositol ring and attached

phosphate groups. The phosphate group on PI4P carries a net -2e charge which

can cause the effective head size to be even larger if it is in the presence of other

negatively charged lipids. The CG model and atomistic structure of PI4P and

DPPC are shown in Figure 3–2, it can be seen that the inositol ring is modeled as a

single polar bead due to it’s ability to form hydrogen bonds and its high solubility

in water.

Experimental evidence has shown that the orientation of the inositol ring of

PI4P, when in the presence of DPPC, is not normal to the bilayer, but becomes

bent [44]. Figure 3–3a shows how the inositol ring gets pulled down towards the

36

(a) (b)

Figure 3–3: Depictions of PI4P a) Atomistic representation in the presence ofDPPC. b) Head group of the CG model shown with the imposed equilibrium an-gles, no equilibrium value is imposed on α1.

plane of the bilayer, rather than extend out into the aqueous phase. It has been

speculated that this structural feature of PI4P is caused by the electrostatic

interaction between the negatively charge phosphate group of PI4P and the

positively charged choline group of DPPC. Figure 3–3b shows the CG model of

the head group of PI4P and the equilibrium angles. To try to mimic the structural

features of PI4P no potential has been imposed on the angle α1, where the angles

α2 and α3 have equilibrium angles of 180oand 120o, respectively. Since no angle

potential is acting on the α1 angle this will allow for the top phosphate group to

move and adopt a conformation due solely to the intermolecular interactions and of

course the bond length potential.

3.2.3 Simulation Details

Self-assembly simulations of DPPC and PI4P were conducted at relative PI4P

concentrations of 0%, 5%, 10% and 20%. This was done by populating a simula-

tions box with total of 200 lipid molecules at the proper relative concentrations

and 28 waters per lipid (7 CG W’s), which was the same concentration used in the

37

experimental determination of the head group orientation by Bradshaw et al. [44].

The systems were then simulated for 800ns at a temperature of 298 K, in all sys-

tems a bilayer was formed. The bilayer system was then copied laterally four times

to create a bilayer system consisting of 800 lipid molecules. The 800 lipid system

was then simulated for 1.6µs. The first 400ns were considered an equilibration,

and the final 1.2µs were used for in the analysis of the system. In both the self-

assembly and bilayer simulations Berendsen temperature coupling and anisotropic

pressure coupling (1 bar) were used as well as periodic boundary conditions in all 3

directions.

3.2.4 Results

To determine the structure of the PI4P head group the α1 angle was calculated

for all systems. The probability distribution of α1 and α2 angles in the PI4P head

group for each of the systems containing PI4P is shown in Figure 3–4a and b,

respectively. All of the systems display the same trend with a peak at θ ≈2.1 rads

(120o) for α1 and at θ = π for α2. Since α2 is normal to the bilayer water interface,

it can be determined that the upper phosphate group is being pulled down towards

the bilayer-water interface as indicated by the α1 angle data. Since the phosphate

group is partially lying in the interface, it acts to increase the head group size of

PI4P. This data indicates that the model is qualitatively reflecting the true nature

of the PI4P head group in a PI4P/DPPC system.

As described in Section 3.2.1 the relationship between the fluctuations of the

bilayer surface and the bending modulus can be determined, Eq. 3.6. To obtain

h(q) for a surface, a well defined surface is required. An imaginary lateral grid

was imposed and each bead was assigned to a bin based upon it’s lateral position

(x, y). For each bin the average normal (z) position of all molecules in the bin

was determined. The surface was defined by assigning each bin center the average

z position of all molecules in the bin giving a point-wise function z = f(x, y),

38

0 0.5 1 1.5 2 2.5 3 3.50

0.005

0.01

0.015

0.02

0.025

0.03

Angle (radians)

Pro

babi

lity

5% PI4P

10% PI4P

20% PI4P

(a)

0 0.5 1 1.5 2 2.5 3 3.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Angle (radians)

Pro

babi

lity

5% PI4P

10% PI4P

20% PI4P

(b)

Figure 3–4: Probability distribution for systems of PI4P/DPPC at concentrationsof 5%, 10% and 20% a) α1angle. b) α2angle.

representing the surface. At each point in time, during the analysis, a 2D Fourier

transform of the surface was taken as described by Eq. 3.3. Each (h(q))2 was

averaged and then log(< |h(q)|2 >) was plotted versus log(|q|), these spectral

intensity plots for each system are shown in Figure 3–5a-d. The longer wavelength

modes, |q| > 1.0 nm, were used to fit a line by the least-squares method. A function

can be fit to the data of the form

log(< |h(q)|2 >) = −4log(|q|) + C1

where the intercept value, C1, is equal to log(kBT/κ)(Eq. 3.6) allowing for κto be

determined. The uncertainty in C1, σC1was propagated to give the error in κ as

shown in Eq. 3.7.

σκ =dκ

dC1σC1

= (d(kbT

eC1)

dC1)σC1

= −κσC1(3.7)

The bending modulus and error at varying concentrations of PI4P is shown

in Figure 3–6. At the time of this report there was only 200ns of data for the pure

DPPC system, accounting for the larger error in that system. The values reported

here are in the same range as previously reported for simulated and experimental

systems[33, 42].

39

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−4

−3

−2

−1

0

1

2

3

4

Log q(1/nm)

Log(

fc2 )

(nm

4 )

(a)

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−4

−3

−2

−1

0

1

2

3

4

Log q(1/nm)

Log(

fc2 )

(nm

4 )

(b)

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−4

−3

−2

−1

0

1

2

3

4

Log q(1/nm)

Log(

fc2 )

(nm

4 )

(c)

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−4

−3

−2

−1

0

1

2

3

4

Log q(1/nm)

Log(

fc2 )

(nm

4 )

(d)

Figure 3–5: Spectral intensity for PI4P/DPPC systems a) Pure DPPC. b) 5%PI4P. c) 10% PI4P. d) 20% PI4P.

0 5 10 15 20 258

8.5

9

9.5

x 10−20

% PI4P

Ben

ding

Mod

ulus

(J)

Figure 3–6: Bending modulus of PI4P/DPPC bilayer at varying concentrations ofPI4P

40

3.3 Line Tension

The existence of a tension force existing between lipid phases can be a driving

force for shape change of biomembranes. From a molecular prospective a tension

force between lipid phases can be generated due to the differing interaction energies

between the molecules of the respective phases. The lipid phases maybe a pure

raft or can be a region of the membrane where a specific lipid has localized. A key

membrane lipid in many biological processes are phosphoinositides which are known

to play a role both inter and intracellular trafficking [7–9, 31]. At the outer cellular

membrane there is a cytoskeleton which plays a role of structural stability for the

cell as well as effecting shape change. At the outer membrane the mechanism of

shape change is quite complex involving the interaction between lipids, proteins

and the cytoskeleton. However, intracellular compartments, such as the Golgi body

and the endoplasmic reticulum do not have a cytoskeleton on their interior, but

are still able to mediate shape changes. Therefore the mechanism of shape change

at the intracellular membrane is less complex and is location where this work is

focused.

3.3.1 Pressure Calculation from Molecular Dynamics Simulations

Pressure in a fluid is made up of two-components: a kinetic part due to the

molecular motions and a static part due the the intermolecular potentials [45].

Pressure is a force acting on the unit volume, so the kinetic part can be viewed

as the momentum transfer across an imaginary surface. The static part is due to

particles on opposites sides of the imaginary surface exerting a force on each other

due to the intermolecular potentials. The kinetic part of pressure can be calculated

from the temperature of the system or from the molecular velocities as shown in

Eq. 3.9 and Eq. 3.10.

41

P = PK + PS (3.8)

PK =NkBT

V(3.9)

PK =2

VEK =

1

V

N∑

i

mivi ⊗ vi (3.10)

Where PK is the scalar or hydrostatic pressure, it is related to the pressure

tensor, PK, by PK = trace(PK)/3. N is the number of particles in the system,

V is the volume of the system, EK is the kinetic energy tensor, mi is the mass of

each particles and vi is the velocity vector of each particle. The calculation of the

kinetic pressure from molecular dynamics is trivial. If the velocities of each particle

are stored to an output trajectory file the kinetic pressure can be calculated easily

from Eq. 3.10.

The static pressure calculation can be more complex, it is related to the virial

of the system [46].

PS = − 2

VVir (3.11)

Vir = −1

2

N∑

i

ri ⊗ Fi (3.12)

The virial is a 2nd order tensor shown in Eq. 3.12, where ri is the vector describing

the particle position and Fi is the force vector acting on particle i. The complexity

of the virial calculation comes in determining the forces acting on the particles.

In theory these forces can be due to external and internal potentials, but in this

work only internal forces were present. The potentials are an input to the model

and the forces on each bead are calculated from the negative derivative of the

potential with respect to particle position, F(i)= - dVdri

. The potentials present in the

coarse-grain model simulations are described in Section 3.3.2.

42

3.3.2 Description of Potentials and Forces

Lennard-Jones Potential

The Lennard-Jones(LJ) potential is a non-bonded potential which represents

the the long range attractive van der Waals force and a short range repulsive part

due to overlap of the electron clouds. For the simulations conducted in this work,

the LJ potential is shifted beyond a certain radius so the forces go smoothly to zero

at the cut-off radius. In the simulation described in this work, the cut off distance

was rLJc= 1.2 nm and the shift distance was rLJs

= 0.9 nm. The LJ interactions

between nearest bonded neighbors are excluded. The potentials, VLJ , and forces,

FLJ(i) acting on bead i , for the unshifted and shifted parts of the potential are

given by

VLJ =

C12

r12ij

− C6

r6ij

rij < rLJs

12C12

[

112r12

ij

− A12

3(rij − rLJs

)3 − B12

4(rij − rLJs

)4 − D12

]

−6C6

[

16r6

ij

− A6

3(rij − rLJs

) − B6

4(rij − rLJs

) − D6

]

rLJs≤ rij < rLJc

0 rij ≥ rLJc

and

F (i)LJ =

−12C12

r13ij

+ 6C6

r7ij

rij

rijrij < rLJs

12C12

[

1r13ij

+ A12(rij − rLJs)2 + B12(rij − rLJs

)3]

−6C6

[

1r7ij

+ A6(rij − rLJs)2 + B6(rij − rLJs

)3]

rij

rijrLJs

≤ rij < rLJc

0 rij ≥ rLjc

43

Figure 3–7: Definition of rij

where rij is the vector pointing from bead i to bead j as described in Figure 3–7,

rij is the magnitude of rij, and

Aα = −(α + 4)rLJc− (α + 1)rLJs

rα+2LJc

(rLJc− rLJs

)2

Bα =(α + 3)rLJc

− (α + 1)rLJs

rα+2LJC

(rLJc− rLJs

)3

Dα =1

αrαLJc

− Aα

3(rLJc

− rLJs)3 − Bα

4(rLJc

− rLJs)4.

The force on bead j is equal and opposite to the force on bead i, F(j)LJ =

−F(i)LJ . This is also true of the 2-body potentials for electrostatic and bonded

interactions described below.

Electrostatic Potential

The electrostatic potential (ES) is a non-bonded potential which acts between

beads with a net charge. The potential is shifted smoothly to zero at the cut off

radius, rESc= 1.2 nm starting from rESs

= 0.0 nm. The cut off and the shifting

and of the electrostatic potential is done to mimic the distance dependant screening

phenomenon [? ]. Electrostatic interactions between nearest bonded neighbors are

excluded. The electrostatic potential and resultant force are described below.

44

VES

=qiqj

4πεoεr

[

1rij

− A1

3(rij − rESs

)3 − B1

4(rij − rESs

)4 − D1

]

rij < rESC

= 0 rij ≥ rESc

F(i)ES

= − qiqj

4πε0εr

[

1r2ij

+ A1(rij − rESS)2 + B1(rij − rESS

)3]

rij

rijrij < rESC

= 0 rij ≥ rESc

VES is the electrostatic potential, F(i)ES is the force due to the electrostatic

potential acting on bead i, qi is the charge on bead i, εo the permittivity of free

space and εr is the relative dielectric constant, which was taken to be εr = 20. The

constants A1, B1and D1 are describe by the same equations for the constants in the

LJ potential.

Bonded Potential

The bonded potential, VB, is a harmonic potential which corrects for devia-

tions from equilibrium bond lengths. The bond potential and resultant force, FB,

are described below.

VB =1

2kbond(rij − r0)

2

F(i)B = kbond(rij − r0)rij

rij

The bonding force constant for a single bond is kbond = 1250 kJmol∗nm2 , and the

equilibrium bond length is taken to be r0 = 0.47 nm.

Angle Potential

The angle potential can be used to constrain three consecutively bonded

beads. The potential is a harmonic function of the cosine of the angle made about

the central bead. The angle orientation and resultant forces are described in

Figure 3–8. The angle potential, Vθ, and forces, Fθ , are described below,

45

Figure 3–8: Angle Orientation

Vθ =1

2kθ(cos θ − cos θ0)

2

F(i)θ = kθ(cos θ − cos θ0)

[

(rji · rjk)rji

r3jirjk

− rjk

rjirjk

]

F(j)θ =kθ(cos θ − cos θ0)

rjirjk

[

rji + rjk − (rji · rjk)

(

rji

r2jh

+rjk

r2ji

)]

F(k)θ = kθ(cos θ − cos θ0)

[

(rji · rjk)rjk

r3jkrji

− rji

rjirjk

]

where the angle force constant for single bonds is kθ = 25 kJmol

and the equilibrium

angle isθ0 = π for. The cosine of the angle can be determined by taking the dot

product of rji and rjk, cos(θ) =rji·rjk

rjirjk[47].

3.3.3 Calculation of Virial

To calculate the virial of the system, the particle positions and total force on

each particle must be known as described in Eq. 3.12. For the 2-body potentials

the force is a function of separation distance between two interacting particles. It

is computationally efficient to calculate the virial due to each interaction Vir(i, j),

Eq. 3.13 and then sum over all interactions Eq. 3.14.

46

Vir(i, j) = −1

2[ri ⊗ Fij + rj ⊗ Fji] =

1

2[rij ⊗ Fij] (3.13)

Vir2−Body =1

2

∑

i<j

[rij ⊗ Fij] (3.14)

Fij is the force on i due to j, and Fji is equal and opposite to Fij from the

conservation of linear momentum. The calculation of the virial due to the angle

potential can be achieved by summing over all angles in the system, Eq. 3.15.

Virθ = −1

2

∑

Angles

[ri ⊗ Fi + rj ⊗ Fj + rk ⊗ Fk] (3.15)

3.3.4 Distribution of Virial

In this work the pressure profile, or the spatial variation in the pressure

tensor, is of interest. This variation in components of the pressure tensor is the

source of surface tension and line tension. To determine pressure as a function of

space the pressure must be determined locally within the system. To do this the

system is section off into slabs and the pressure is calculated in each slab. The

slab is taken to have the same dimensions in as the simulation box in two of the

dimensions, but in the direction normal to the interface of interest the dimension

will be much smaller. The slab orientation for the calculation of a bilayer surface

tension is described by Figure 3–9. The slab is moved through the z-direction and

the pressure is calculated at each slab location so P(z) is obtained. To calculate

P(z)K Eq. 3.10 is used in which the sum is taken over only those molecules which

lie within the slab.

The calculation of P(z)S is not well defined since the location of the force

acting between particles is ambiguous [48]. The static pressure at a point R can

be written in a general form by Eq 3.16, where C0i is any contour connecting the

particle position, ri, with and arbitrary point, R0 [49].

47

Figure 3–9: Slab orientation for calculation of bilayer surface tension

P αβS (R, t) =

∑

i

[∇αi V ({ri})]

∮

C0i

dlβδ(R− l) (3.16)

The most natural choice for the contour is the Irving-Kirkwood contour which is a

straight line between interacting particles [50]. Other contours have been proposed,

most notably the Harasima contour, but in this work only the Irving-Kirkwood

contour is used [51]. A generalized expression for the static pressure for an m-body

potential can be derived from Eq. 3.16 is described by 3.17.

P αβSm−Body

(R, t) =1

m

∑

<j>

∑

<k,l>

{

∇αjk

Vm −∇αjlVm

}

∮

Cjljk

δ(R − l)dlβ (3.17)

where < j > represents all “clusters”, which means a group of interacting particles,

and k and l are the indices of particles within the cluster. The average pressure in

each slab is what is desired to be known, so Eq. 3.17 must be integrated over the

dimensions of the slab,

P αβSm−body

(Zi, t) =1

V ol∆Z

∫ Lx

0

dx

∫ Ly

0

∫ Zi+∆Z

Zi

dzP αβSm−body

(R, t) (3.18)

48

where Zi is the ith slab, ∆Z is the slab thickness, and V ol∆Z is the volume of the

slab. The choice of contour must be defined to evaluated the integral, here we use a

straight line contour (Irving-Kirkwood) connecting interacting particles

lβ = rβjk

+ λ(rβjl− rβ

jk), λ ε [0, 1] (3.19)

dlβ = (rβjl− rβ

jk)dλ = rβ

jkjldλ (3.20)

∮

Cjljk

δ(R− l)dlβ = −rβjkjl

∫ 1

0

δ(R − (rjk+ λrjkjl

)dλ (3.21)

where rjkjl= rjl

− rjk. Combining Eqs. 3.17, 3.18 and 3.21, an expression which

can be evaluated for the static pressure in a slab is derived

P αβS (Zi,t) = − 1

mV ol∆Z

∑

<j>

∑

<k,l>

{

∇αjk

Vm −∇αjlVm

}

rβjkjl

f(zjk, zjl

, Zi)

where zjkis the z position coordinate of particle k in j cluster, and

f(zjk, zjl

, Zi) =

∫ Lx

0

∫ Ly

0

∫ Zi+∆Z

Zi

dxdydz

∫ 1

0

δ(R− (rjk+ rjl

))dλ (3.22)

∫ Lx

0

∫ Ly

0

∫ Zi+∆Z

Zi

δ(R − (rjk+ rjl

))dxdydz =

1 R = rjk+ λrjkjl

R ε V ol∆Z

0 R 6= rjk+ λrjkjl

(3.23)

so R must lie on the rjkjland be within slab i for the spatial integration to be non-

zero. Since the slab spans the system in the x and y directions only the particle z

positions are critical in evaluating the pressure in a slab. Eq. 3.23 can be re-written

49

with the use of the Heaviside step function.

∫ Lx

0

∫ Ly

0

∫ Zi+∆Z

Zi

δ(R − (rjk+ rjl

))dxdydz =Θ(zjk+ λ(zjl

− zjk) − Zi)×

Θ(Zi + ∆Z + zjk+ λ(zjl

− zjk))

Θ(x) =

1 x > 0

0 x < 0

f(zjk, zjl

, Zi) =

∫ 1

0

Θ(zjk+ λ(zjl

− zjk) − Zi)×

Θ(Zi + ∆Z + zjk+ λ(zjl

− zjk))dλ

So depending on the location of the interacting particles with respect to slab i

location, f(zjk, zjl

, Zi) will take on different values. If both particles lie within slab

i, f = 1; if both particles lie outside of slab i and no part of rjkjllies within slab

i, f = 0; if both particles do not lie with slab i, but part of rjkjldoes pass through

slab i, then

f =rZi

jkjl

rjkjl

where rZi

jkjlis the portion of rjkjl

within slab i.

3.3.5 Calculation of Line Tension

Surface tension is calculated from the difference between lateral and normal

components of the pressure tensor [52].

γ =

∫ +∞

−∞

(PN(z) − PL(z))dz

The z-direction is normal to the interface, PN and PL are the normal and lateral

pressure, respectively. This deviation in the isotropy of the pressure tensor can be

due to both density deviations and changes in configuration energy. The integration

is theoretically taken from −∞ to +∞ since the pressure becomes isotropic away

from the interface and does not contribute to the integration.

50

Figure 3–10: Phases within a bilayer

To calculate line tension, the pressure must be localized to the surface defined

by the membrane and then the differences between the components of this 2-

dimensional pressure tensor will result in line tension. Figure 3–10 describes the

geometry of phases embedded within a bilayer.

In this situation pressure would be calculated as a function of the z-direction,

with slabs spanning the box in the x and y directions.

σ(t) =1

Nint

∫ LZ

0

(P ZZ2D (z, t) − P XX

2D (z, t))dz (3.24)

P2D(z, t) =

∫ Ly

0

P(z, y, t)dy

σ is line tension, P2Dis the 2-dimensional pressure tensor and Nint is the number of

interfaces in the system, since the line tension per interface is desired. Since P(z, t)

is what will be calculated and it is already averaged over the y-direction we can see

that P2D(z, t) = P(z, t)Ly and by combining this with Eq. 3.24 an expression for

line tension as a function of P(z, t) is derived.

σ(t) =Ly

Nint

∫ LZ

0

(P ZZ(z, t) − P XX(z, t))dz (3.25)

51

3.3.6 Simulations Details

The same CG models for PI4P and DPPC are used in simulations for the

calculation of line tension. The 800 lipid simulations generated for the bending

modulus calculations as described in Section 3.2.3 were used to assemble the line

tension systems, referred to from here forward as a patch system. Figure 3–10

describes the desired configuration for the line tension calculation in which the

α-phase represents a homogeneous pure DPPC section and the β-phase represents

an inhomogeneous section of PI4P and DPPC. The initial bilayer sections were

taken from the output of 1.6 µs for the 800 lipid systems. A new topology was

then generated by placing the inhomogeneous section in between to homogeneous

sections. To do this, the inhomogeneous section had to be modified so the median

bilayer height was the same as the homogeneous section. Also, it was required that

both systems have identical x and y dimensions, so which ever of the two systems

was larger was trimmed to match the dimensions of the smaller section. The patch

system was generated with both a 5% PI4P and 20% PI4P inhomogeneous section.

These patch systems were then energy minimized for 50000 steps using the steep

energy minimization method. Additional water was then added to the system

to bring the water level to 22 CG W per lipid and again the system was energy

minimized for 50000 steps. In order the keep the sections separated and to prevent

the PI4P molecules from diffusing out into the homogeneous section a repulsive

wall was constructed, which acted only on the PI4P molecules. The wall spanned

the simulation box in the x and y directions. The “dummy” atoms which made up

the wall were separated with 0.5 nm spacing in both the x and y directions. The z

coordinate location of the wall at high z value was determined by adding 0.4 nm to

the maximum z coordinate position of all PI4P beads. Likewise the wall at low z

value was determined by subtracting 0.4 nm from the minimal zcoordinate position

of all PI4P beads. The dummy atoms were kept at a fixed position throughout the

52

Figure 3–11: Patch system with repulsive walls

simulations. The interaction potential between the dummy atoms and the PI4P

beads was equivalent to the repulsive part of the LJ potential for polar - nonpolar

interactions

VDummy =

C12

r12ij

if interacting with PI4P

0 if interacting with bead other than PI4P

where C12 = 8.3658 × 10−2. This potential is shifted and cut-off with the same radii

and in the same manner as all other LJ potentials as described in Section 3.3.2.

Figure 3–11 is a simulation snapshot of the patch system configuration, water

was removed for clarity. The black beads extending through the bilayer are the

dummy wall atoms. The section between the walls is a mixture of PI4P and DPPC,

while the outer sections of the bilayer are pure DPPC. These systems were then

simulated for 400ns using Berendsen pressure and temperature coupling, set to

1 bar anisotropically and 323 K respectively. The pressure coupling was turned

off after the initial 400ns simulation and was run for an additional 400ns with no

pressure coupling.

3.3.7 Results

Comparison to Atomistic Results

Several studies have been conducted on the calculation of pressure profiles

and surface tension of homogeneous bilayers from atomistic molecular dynamics

53

simulations [48, 53, 54]. The potentials used in atomistic simulations differ from

CG simulations in many ways. Atomistic simulations typically have more poten-

tials, often including a dihedral potential which is not included in the CG model.

In atomistic simulations the electrostatics can be very different because of the

presence of partial charges on the majority of atoms, where in the CG model most

beads are modeled as neutral beads. Additionally, the electrostatics may be not

be cut off in atomistic simulations to allow for long range interactions. In the CG

model the LJ potential partially accounts for the electrostatic. Lastly the coeffi-

cients of all of the potentials will vary from CG to atomistic simulations making

the comparison between virial contributions difficult. However, if the CG model is

a reliable model the pressure profiles between CG and atomistic model should be

comparable.

The pressure profile for a pure DPPC bilayer was calculated for the same

system used for the bending modulus calculation (800 lipids, 7 CG W/lipid, 298 K,

anisotropic pressure coupling). The analysis was done on 1.2µs of data after 400ns

of equilibration. This data was compared to results obtained for a similar atomistic

system study conducted by Sonne et al. [48]. The system used by Sonne consisted

of 72 DPPC molecules simulated at 300 K with a ratio of 28 waters per lipid (same

concentration as our system). The pressure profile was calculated as a function of

the bilayer normal direction, the z direction for convenience. In the analysis done

here the system was divided into 60 slabs, in the work by Sonne the system was

divided into 70 slabs [48]. In Figure 3–12a the density profile is plotted, where the

units are number of beads per slab. In Figure 3–12 the diagonal components of the

static pressure tensor are plotted as a function of z position. It can be seen that

the normal component remains constant through the bilayer.

54

−4 −3 −2 −1 0 1 2 3 40

50

100

150

200

250

300

Z position (nm)

Num

ber

Bea

ds

WDPPC

(a)

−4 −3 −2 −1 0 1 2 3 4−700

−600

−500

−400

−300

−200

−100

Z position (nm)

Pre

ssur

e (b

ar)

Px(z)Py(z)Pz(z)

(b)

Figure 3–12: Density and pressure of CG systems and a function of bilayer posi-tion: a) Density profile. b) Pressure profile.

In Figure 3–13 the virial contributions and pressure profiles are compared for a

CG and atomistic models. The CG virial contributions are quite different from the

atomistic virial contributions, especially in the magnitude of the numbers.

This difference in magnitude can be explained by the difference in the po-

tentials and resultant forces. To illustrate the difference, the bonded potential is

examined for both the CG and atomistic models. Figure 3–14 plots the bonded

potential for both models. For a deviation from the equilibrium bond length on the

order of kBT the forces for each model were calculated and an order of magnitude

difference is observed. However, the Pressure profiles in Figures 3–13c and d show

good agreement between the atomistic and CG data.

FB ≈

1.3 × 103 Atomisitic

7.9 × 101 CG

55

−4 −3 −2 −1 0 1 2 3 4−600

−500

−400

−300

−200

−100

0

100

200

300

400

Z position (nm)

PL −

PN

(ba

r)

LJBondAngleElectrostaic

(a) (b)

−4 −3 −2 −1 0 1 2 3 4−300

−200

−100

0

100

200

300

Z position (nm)

PL −

PN

(ba

r)

(c) (d)

Figure 3–13: Virial and pressure profile comparison: a) CG virial profile. b) Atom-istic virial profile [48]. c) CG static pressure profile. d) Atomistic pressure pro-file [48].

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

0

5000

10000

15000

Bond Displacement (nm)

Ene

rgy

(kJ/

mol

)

Bond Energy

AtomisticCG

Figure 3–14: Bond potential comparison

56

Line Tension for Patch Bilayer Systems

Calculation of line tension was conducted for patch systems with inhomoge-

neous sections containing 5% and 20% PI4P. The analysis was done on the final

80ns of data for the simulations done without pressure coupling. In the analysis

the pressure was calculated as a function of the z direction, normal to the phase

interface, as described in Figure 3–10. The system was divided into 100 slabs to

calculate the pressure profiles. The pressure difference was calculated between the z

and x directions (in plane components) so that the line tension could be calculated

using Eq. 3.25. The number density of PI4P beads and the pressure differences for

the 5% and 20% patch systems are shown in Figure 3–15.

It can be observed that the 5% system shows no distinct deviation in the

pressures near the phase boundary. However, in the 20% patch system there is a

sharp change in the pressure difference profile at the phase boundary indicating the

presence of a line tension. The line tension was calculated for both systems, for the

5% system σ = −2.3 × 10−11N and for the 20% system σ = 2.0 × 10−10N. The value

for the 5% can be attributed to noise and poor statistics, however the 20% systems

shows a sharp deviation from isotropic behavior near the boundary. The value

obtained here is higher than reported for an experimental system (9 × 10−13N) and

theoretical estimate (10−11)N [1, 55].

3.4 Tilt Deformations

3.4.1 Energy Model for Tilt

In the pioneering work by Helfrich, on the description of equilibrium vesicle

shapes, he choose to neglect the energy associated with tilt of lipid molecules away

from the bilayer normal direction [39]. In the case of flat bilayers the lipids on

average do not tilt away from the normal direction, however other lipid phases

including the HII and intermediate stalk phase (as described in Chapter 2) do

require the molecules to tilt and this energy must be accounted for when predicting

57

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

Z position (nm)

Num

ber

PI4

P B

eads

(a)

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

40

45

50

Z position (nm)N

umbe

r P

I4P

Bea

ds

(b)

0 5 10 15 20 25 30 35 40 45−400

−300

−200

−100

0

100

200

300

400

Z position (nm)

PX

X −

PZ

Z (

bar)

(c)

0 5 10 15 20 25 30 35−400

−300

−200

−100

0

100

200

300

400

Z position (nm)

PX

X −

PZ

Z (

bar)

(d)

Figure 3–15: Density and pressure profiles for patch system:a) 5% PI4P densityprofile. b) 20% PI4P density profile. c) 5% PI4P pressure difference profile. d) 20%PI4P pressure difference profile

58

(a) (b) (c)

Figure 3–16: Tilt and bending: a)Definition of tilt. b) Non-uniform tilt deforma-tion on a volume element of monolayer [36]. c) Bending deformation on a volumeelement of monolayer [36].

stability of phases. Hamm and Kozlov introduced a energy model for tilt and

used this to predict the stability of the lamellar and inverted lipid phases as well

as geometric properties of the inverted phases [37]. The definition of tilt and the

elastic energy per unit area per monolayer are described by Eq. 3.26 and Eq. 4.2,

respectively and graphical representation of tilt is shown in Figure 3–16a.

t =n

n · N − N (3.26)

f =1

2κ(tξξ + Js)

2 + κ̄det(tξζ) +1

2κθt

2 (3.27)

The tilt vector, t, is perpendicular to N, so if we take N to point in the

z direction, t will have components in the x and y directions. The tilt tensor

is defined by the derivatives of tilt tξζ = ∇t, where ∇ = ∂

∂ξwhere ξ = x, y.

tξξ = trace(tξζ)=∇ · t, Jsis the spontaneous curvature of the monolayer, andκθis the

tilt modulus. It is interesting to note that the same elastic parameters, κ and κ̄,

describe the energy of both tilt and bending deformations. This is explained in a

later paper by Hamm and Kozlov and can be visualized by the Figures 3–16b and

59

(a) (b) (c)

Figure 3–17: Construction of HII phase: a) Original construction of hexagonalphase unit cell [37]. b) Multiple original construction cells shown to illustrate thehydrophobic void region [38]. c) Recent hexagonal phase unit cell construction [37].

c [36]. Figure [36]b describes a non-uniform tilt deformation, where tilt changes

across the element. Figure 3–16c describes a bending deformation, it can be seen

that the shear and stretch of this element a essentaillyessentially the same in both

deformations, which is allows for both to be characterized by the same parameters.

3.4.2 Nature of the HII Phase and Evidence of Tilt

It had long been assumed that the structure of the HII phase consisted of

a circular hydrophilic interface and a hexagonal hydrophobic interface between

unit cells as shown in Figure 3–17. This construction allows for a mismatch

between the interior hydrophilic boundary and the outer hydrophobic boundary.

In this construction, the lipid tails must either stretch to fill the corners of outer

boundary or a hydrophobic void region will exist as illustrated in the center of

the three cells in Figure 3–17b; both situation are energetically unfavorable. To

alleviate this “frustration energy”, as it is know, the construction of the unit

cell was re-examined experimentally and theoretically to construct the current

description [37, 38, 56]. This current description allows for the corners of the cell to

be filled without stretching of the hydrocarbon chains, but comes at the expense of

tilt deformations along the faces of the hexagon.

60

From the MD simulations described in Chapter 2 the hexagonal phase was

examined. For a system of 10% LPA and 90% DOPE simulated at T = 300 K, as

described in Section 2.3.3, beginning from a multi-lamellar configuration formed

a stable HII phase. This HII was examined to evaluate if the current hexagonal

construction was observed. Figure 3–18a shows a simulation snapshot from the

above described system after 1.2µs of simulation, it water is removed to show

that the water pores have a hexagonal geometry. Figure 3–18b illustrates the

hexagonal nature of the hydrophobic by plotting the location of all tail beads in

the lipids. Figure 3–18c plots the location of the lipid-water interface, defined as

the mid-point of the bond connecting the phosphate bead with the glycerol bead

and fitted to a hexagon by least-squares. It is evident from the Figure 3–18 the

HII phase observed in our MD simulations do reflect the description of Hamm and

Kozlov [37].

In order for the lipids to accommodate the structure proposed in Figure 3–17c

and verified by Figures 3–18a-c, the lipids must tilt away from the normal direction

to the lipid-water interface. At the center of each face of the hexagonal phase the

lipid will be aligned with the normal, but away from the center, the lipids must

tilt. Hamm and Kozlov predict the tilt should be linear, that it will reach it’s

maximum values that the outer edge of each face, and along the pore direction tilt

should be zero [37]. This has been verified through analysis of the MD simulation

of the above described 10% LPA/DOPE system. To calculate tilt for each lipid

the director, n, must be suitably defined. In this work, n, is defined by the vector

pointing from the lipid-water interface location to the center of mass of the tail

beads. The tilt along a face of the hexagonal lipid-water interface, tx, and along

the pore direction ty averaged over 50 data points in time beginning after 800ns of

simulation, with a time step of 1.6ns. Figure a shows tx as a function of position

along the facet of the hexagonal pore. It can be seen tx is a linear function with

61

(a) (b)

−1 0 1 2 3 4 5 6

27

28

29

30

31

32

33

(c)

Figure 3–18: Examination of HII from MD simulations for a 10% LPA/DOPE sys-tem at T = 300 Kafter 1.2µs. a) Simulation snapshot of HIIphase. b) Tail beadlocations plotted to illustrate the hydrophobic boundary. c) Lipid-water interfacelocation (defined as the middle of the phosphate-glycerol bond) plotted for eachlipid to illustrate the hydrophilic boundary.

62

−1.5 −1 −0.5 0 0.5 1 1.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

X position (nm)

Tx

(a)

0 5 10 15−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Y position (nm)

Ty

(b)

Figure 3–19: Tilt in a HII phase :a) tx, tilt along the one face of the hexagon. b)ty, tilt along the direction of the water pore.

extremal values the edge of the face. Tilt along the pore direction is shown in

Figure b, it is nearly zero and has no sustained gradient. Both of these results in

Figure a and b correspond to the theoretical predictions of Hamm and Kozlov[37].

CHAPTER 4CONCLUSIONS

4.1 Major Conclusions

One of the advantages of coarse grained models in molecular dynamics is

that the systems can be visualized and measured on near atomic length scales,

while relatively large systems and long simulations can be simulated without being

extremely computationally expensive. This ability allows for insight to be gained

on the molecular scale as well as the ability to compute bulk properties for these

systems. Here we were able to verify the changing molecular geometries in lipid

systems as a function of the environment parameters.

1. The CG model developed for the lipids LPA and DOPA and the ion

Mg+2exhibit qualitative agreement with experimental temperature and

charge based phase transition data.

2. For pure LPA, a negatively charged lipid, the head size reduced with in-

creasing cationic concentration, making the molecule more cylindrical and

reducing the curvature of the system. For mixed DOPA-DOPE the increase

in temperature of the system caused the tails to splay further apart, while the

head group separation also increased with the increased vibrational energy

but to lesser degree. Thereby it was observed the increase in temperature for

this system causes the molecules to become more conical leading to a system

of higher curvature.This study provided a basis for further investigation in

biological systems with a coarse grained model for the lipids LPA and DOPA.

The ability the qualitatively reproduce experimental phases in both pure and

mixed lipid systems is justification for extending the work in this area.

63

64

3. The intermediate stalk phase observed in phase transition simulations

of mixed lipid systems verifies the structure proposed by Kozlovsky and

Kozlov [26].

4. Increasing the relative concentration of PI4P in a PI4P/DPPC bilayer causes

an increase in the bending modulus of the membrane compared to a pure

DPPC bilayer. This indicates that the bilayer becomes more difficult to

deform with the addition of more PI4P up to 20%.

5. A system in which PI4P localizes in one region of a DPPC bilayer while

neighboring regions remain homogeneous, exhibits a positive line tension on

the order of 10−10N when the concentration of PI4P in the inhomogeneous

section reaches 20%. If the inhomogeneous section only contains 5% PI4P no

anisotropy in the 2-dimension pressure tensor is observed.

6. The hexagonal phase in these MD simulations display hexagonal phases at

the both the hydrophilic and hydrophobic boundaries. The tilt along a side

of the hexagon hydrophilic interface is a linear increasing function. Both of

these results support the theoretical predictions of Hamm and Kozlov [37].

4.2 Future Directions

In this work molecular modeling and analysis techniques were developed which

produced results warranting further investigation. In this work only a few lipid

systems were studied, while it is believed that other lipid species could be key

molecules in eliciting changes in elastic parameters of membranes.

4.2.1 Additional Lipid Species

The lipid phosphatidylinositol-4,5-bisphosphate (PI(4,5)P2) has been impli-

cated to play a role in stimulating vesicle formation from liposomes as well as from

the outer cellular membrane [29, 30, 35]. PI(4,5)P2is structurally similar to PI4P,

but has an additional phosphate group extending from the inositol ring and carries

a net −4 charge compared to −3 of PI4P at physiological conditions [57]. It is

65

Figure 4–1: Proposed CG models of additional lipids: a) PI(4,5)2. b) DAG.

believe that the increased head group size and increased effective head group size

due to charge would potentially increase the anisotropy in the 2 dimension pressure

tensor in the line tension study. Additionally the lipid diacyl glycerol (DAG) is of

interest, due to it role in Golgi budding and bud fission from the Golgi network.

DAG is similar to the lipid DOPA except that it does not have the phosphate

group extending from the glycerol group. It is desired to conduct similar studies

done in this work, but using PI(4,5)P2 and DAG as the minor component in the

lipid systems. Proposed CG models of the lipids PI(4,5)P2and DAG as shown in

Figure 4–1.

4.2.2 Continuum Scale Modeling

Ultimately, it is desired to use the parameters determined from molecular mod-

eling and incorporated these findings into a continuum scale model for predicting

equilibrium membrane configurations, based upon the membrane chemical com-

position. Work has been done in this area starting with a theoretical description

of the bending energy by Helfrich and extending by Lipowsky for homogeneous

membranes [39, 41]. This work was further extended to incorporate the existence

of different homogeneous lipid phases with the same lipid by modifying the Helfrich

expression to incorporate line tension between phases [1, 34, 35]. The expression of

the bending and interface energy is for a system with distinct lipid phases is given

66

Figure 4–2: Domains within a membrane, taken from ref [1]

by Eq. 4.1

F =

∫

α

[κ(α)

2(C1 + C2 + C

(α)0 )2 + κ̄(α)C1C2]dA+

∫

β

[κ(β)

2(C1 + C2 + C

(β)0 )2 + κ̄(β)C1C2]dA+

∫

dα

σdl (4.1)

where κ(α), κ̄(α), C(α)o are the bending modulus, Gaussian curvature modulus

and spontaneous curvature of the α phase, dα is the length of the interface

separating the phases. The geometry of the system is described by Figure 4–2.

A model describing the energy of associated with tilt and bending of lipid

structures has been introduced by Hamm and Kozlov shown in Eq. 4.2 and

previously described in Chapter 3 of this work [36].

f =1

2κ(tξξ + Js)

2 + κ̄det(tξζ) +1

2κθt

2 (4.2)

It is desired to incorporate both the Julicher and Hamm models to develop a

model which accounts for bending, tilt and line tension. Further work must be done

to develop a method to extract the tilt modulus from the MD simulations [35, 36].

Using these energy models it is desired to compute the equilibrium shapes for

varying concentrations of mixed lipid systems using parameters extracted from

molecular simulations.

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BIOGRAPHICAL SKETCH

Eric May was born in New Haven, Connecticut, on August 29, 1978. He was

raised in Cheshire, Connecticut and attended Cheshire High School, graduating

in 1996. He received a Bachelor of Science from Bucknell University in chemical

engineering in 2001. During his tenure at Bucknell he completed summer intern-

ships with Neurogen Corporation (Branford, CT) and the Connecticut Department

of Environmental Protection (Hartford, CT). In 2001 he joined the Department

of Chemical Engineering at the University of Florida and Atul Narang’s research

group. His research interests are focused on mathematical modeling of biological

systems.

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