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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