trunk/mattDisertation/lipid.tex



\chapter{\label{chapt:lipid}Phospholipid Simulations}

\section{\label{lipidSec:Intro}Introduction}

In the past 10 years, computer speeds have allowed for the atomistic
simulation of phospholipid bilayers.  These simulations have ranged
from simulation of the gel phase ($L_{\beta}$) of
dipalmitoylphosphatidylcholine (DPPC), \cite{Lindahl:2000} to the
spontaneous aggregation of DPPC molecules into fluid phase
($L_{\alpha}$ bilayers. \cite{Marrinck:2001} With the exception of a
few ambitious
simulations,\cite{Marrinch:2001b,Marrinck:2002,Lindahl:2000} most
investigations are limited to 64 to 256
phospholipids.\cite{Lindal:2000,Sum:2003,Venable:2000,Gomez:2003,Smondyrev:1999,Marrinck:2001a}
This is due to the expense of the computer calculations involved when
performing these simulations.  To properly hydrate a bilayer, one
typically needs 25 water molecules for every lipid, bringing the total
number of atoms simulated to roughly 8,000 for a system of 64 DPPC
molecules. Added to the difficluty is the electrostatic nature of the
phospholipid head groups and water, requiring the computationally
expensive Ewald sum or its slightly faster derivative particle mesh
Ewald sum.\cite{Nina:2002,Norberg:2000,Patra:2003} These factors all
limit the potential size and time lenghts of bilayer simulations.

Unfortunately, much of biological interest happens on time and length
scales unfeasible with current simulation. One such example is the
observance of a ripple phase ($P_{\beta'}$) between the $L_{\beta}$
and $L_{\alpha}$ phases of certain phospholipid
bilayers.\cite{Katsaras:2000,Sengupta:2000} These ripples are shown to
have periodicity on the order of 100-200~$\mbox{\AA}$. A simulation on
this length scale would have approximately 1,300 lipid molecules with
an additional 25 water molecules per lipid to fully solvate the
bilayer. A simulation of this size is impractical with current
atomistic models.

Another class of simulations to consider, are those dealing with the
diffusion of molecules through a bilayer.  Due to the fluid-like
properties of a lipid membrane, not all diffusion across the membrane
happens at pores.  Some molecules of interest may incorporate
themselves directly into the membrane.  Once here, they may possess an
appreciable waiting time (on the order of 10's to 100's of
nanoseconds) within the bilayer.  Such long simulation times are
difficulty to obtain when integrating the system with atomistic
detail.

Addressing these issues, several schemes have been proposed.  One
approach by Goetz and Liposky\cite{Goetz:1998} is to model the entire
system as Lennard-Jones spheres. Phospholipids are represented by
chains of beads with the top most beads identified as the head
atoms. Polar and non-polar interactions are mimicked through
attractive and soft-repulsive potentials respectively.  A similar
model proposed by Marrinck \emph{et. al.}\cite{Marrinck:2004}~ uses a
similar technique for modeling polar and non-polar interactions with
Lennard-Jones spheres. However, they also include charges on the head
group spheres to mimic the electrostatic interactions of the
bilayer. While the solvent spheres are kept charge-neutral and
interact with the bilayer solely through an attractive Lennard-Jones
potential.

The model used in this investigation adds more information to the
interactions than the previous two models,

\section{\label{lipidSec:Methods}Methods}

\subsection{\label{lipidSec:lipidMedel}The Lipid Model}

\begin{figure}

\caption{Schematic diagram of the single chain phospholipid model}

\label{lipidFig:lipidModel}

\end{figure}

The phospholipid model used in these simulations is based on the
design illustrated in Fig.~\ref{lipidFig:lipidModel}.  The head group
of the phospholipid is replaced by a single Lennard-Jones sphere of
diameter $fix$, with $fix$ scaling the well depth of its van der Walls
interaction.  This sphere also contains a single dipole of magnitude
$fix$, where $fix$ can be varied to mimic the charge separation of a
given phospholipid head group.  The atoms of the tail region are
modeled by unified atom beads.  They are free of partial charges or
dipoles, containing only Lennard-Jones interaction sites at their
centers of mass.  As with the head groups, their potentials can be
scaled by $fix$ and $fix$.

The long range interactions between lipids are given by the following:
\begin{equation}
EQ Here
\label{lipidEq:LJpot}
\end{equation}
and
\begin{equation}
EQ Here
\label{lipidEq:dipolePot}
\end{equation}
Where $V_{\text{LJ}}$ is the Lennard-Jones potential and
$V_{\text{dipole}}$ is the dipole-dipole potential.  As previously
stated, $\sigma_{ij}$ and $\epsilon_{ij}$ are the Lennard-Jones
parameters which scale the length and depth of the interaction
respectively, and $r_{ij}$ is the distance between beads $i$ and $j$.
In $V_{\text{dipole}}$, $\mathbf{r}_{ij}$ is the vector starting at
bead$i$ and pointing towards bead $j$.  Vectors $\mathbf{\Omega}_i$
and $\mathbf{\Omega}_j$ are the orientational degrees of freedom for
beads $i$ and $j$.  $|\mu_i|$ is the magnitude of the dipole moment of
$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation
vector of $\boldsymbol{\Omega}_i$.

The model also allows for the bonded interactions of bonds, bends, and
torsions.  The bonds between two beads on a chain are of fixed length,
and are maintained according to the {\sc rattle} algorithm. \cite{fix}
The bends are subject to a harmonic potential:
\begin{equation}
eq here
\label{lipidEq:bendPot}
\end{equation}
where $fix$ scales the strength of the harmonic well, and $fix$ is the
angle between bond vectors $fix$ and $fix$.  The torsion potential is
given by:
\begin{equation}
eq here
\label{lipidEq:torsionPot}
\end{equation}
Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine
power series to the desired torsion potential surface, and $\phi$ is
the angle between bondvectors $fix$ and $fix$ along the vector $fix$
(see Fig.:\ref{lipidFig:lipidModel}).  Long range interactions such as
the Lennard-Jones potential are excluded for bead pairs involved in
the same bond, bend, or torsion.  However, internal interactions not
directly involved in a bonded pair are calculated.


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#	User	Rev	Content
1	mmeineke	971
2
3			\chapter{\label{chapt:lipid}Phospholipid Simulations}
4
5			\section{\label{lipidSec:Intro}Introduction}
6
7	mmeineke	1001	In the past 10 years, computer speeds have allowed for the atomistic
8			simulation of phospholipid bilayers. These simulations have ranged
9			from simulation of the gel phase ($L_{\beta}$) of
10			dipalmitoylphosphatidylcholine (DPPC), \cite{Lindahl:2000} to the
11			spontaneous aggregation of DPPC molecules into fluid phase
12			($L_{\alpha}$ bilayers. \cite{Marrinck:2001} With the exception of a
13			few ambitious
14			simulations,\cite{Marrinch:2001b,Marrinck:2002,Lindahl:2000} most
15			investigations are limited to 64 to 256
16			phospholipids.\cite{Lindal:2000,Sum:2003,Venable:2000,Gomez:2003,Smondyrev:1999,Marrinck:2001a}
17			This is due to the expense of the computer calculations involved when
18			performing these simulations. To properly hydrate a bilayer, one
19			typically needs 25 water molecules for every lipid, bringing the total
20			number of atoms simulated to roughly 8,000 for a system of 64 DPPC
21			molecules. Added to the difficluty is the electrostatic nature of the
22			phospholipid head groups and water, requiring the computationally
23			expensive Ewald sum or its slightly faster derivative particle mesh
24			Ewald sum.\cite{Nina:2002,Norberg:2000,Patra:2003} These factors all
25			limit the potential size and time lenghts of bilayer simulations.
26
27			Unfortunately, much of biological interest happens on time and length
28			scales unfeasible with current simulation. One such example is the
29			observance of a ripple phase ($P_{\beta'}$) between the $L_{\beta}$
30			and $L_{\alpha}$ phases of certain phospholipid
31			bilayers.\cite{Katsaras:2000,Sengupta:2000} These ripples are shown to
32			have periodicity on the order of 100-200~$\mbox{\AA}$. A simulation on
33			this length scale would have approximately 1,300 lipid molecules with
34			an additional 25 water molecules per lipid to fully solvate the
35			bilayer. A simulation of this size is impractical with current
36			atomistic models.
37
38			Another class of simulations to consider, are those dealing with the
39			diffusion of molecules through a bilayer. Due to the fluid-like
40			properties of a lipid membrane, not all diffusion across the membrane
41			happens at pores. Some molecules of interest may incorporate
42			themselves directly into the membrane. Once here, they may possess an
43			appreciable waiting time (on the order of 10's to 100's of
44			nanoseconds) within the bilayer. Such long simulation times are
45			difficulty to obtain when integrating the system with atomistic
46			detail.
47
48			Addressing these issues, several schemes have been proposed. One
49			approach by Goetz and Liposky\cite{Goetz:1998} is to model the entire
50			system as Lennard-Jones spheres. Phospholipids are represented by
51			chains of beads with the top most beads identified as the head
52			atoms. Polar and non-polar interactions are mimicked through
53			attractive and soft-repulsive potentials respectively. A similar
54			model proposed by Marrinck \emph{et. al.}\cite{Marrinck:2004}~ uses a
55			similar technique for modeling polar and non-polar interactions with
56			Lennard-Jones spheres. However, they also include charges on the head
57			group spheres to mimic the electrostatic interactions of the
58			bilayer. While the solvent spheres are kept charge-neutral and
59			interact with the bilayer solely through an attractive Lennard-Jones
60			potential.
61
62			The model used in this investigation adds more information to the
63			interactions than the previous two models,
64
65	mmeineke	971	\section{\label{lipidSec:Methods}Methods}
66
67			\subsection{\label{lipidSec:lipidMedel}The Lipid Model}
68
69			\begin{figure}
70
71			\caption{Schematic diagram of the single chain phospholipid model}
72
73			\label{lipidFig:lipidModel}
74
75			\end{figure}
76
77			The phospholipid model used in these simulations is based on the
78			design illustrated in Fig.~\ref{lipidFig:lipidModel}. The head group
79			of the phospholipid is replaced by a single Lennard-Jones sphere of
80			diameter $fix$, with $fix$ scaling the well depth of its van der Walls
81			interaction. This sphere also contains a single dipole of magnitude
82			$fix$, where $fix$ can be varied to mimic the charge separation of a
83			given phospholipid head group. The atoms of the tail region are
84			modeled by unified atom beads. They are free of partial charges or
85			dipoles, containing only Lennard-Jones interaction sites at their
86			centers of mass. As with the head groups, their potentials can be
87			scaled by $fix$ and $fix$.
88
89			The long range interactions between lipids are given by the following:
90			\begin{equation}
91			EQ Here
92			\label{lipidEq:LJpot}
93			\end{equation}
94			and
95			\begin{equation}
96			EQ Here
97			\label{lipidEq:dipolePot}
98			\end{equation}
99			Where $V_{\text{LJ}}$ is the Lennard-Jones potential and
100			$V_{\text{dipole}}$ is the dipole-dipole potential. As previously
101			stated, $\sigma_{ij}$ and $\epsilon_{ij}$ are the Lennard-Jones
102			parameters which scale the length and depth of the interaction
103			respectively, and $r_{ij}$ is the distance between beads $i$ and $j$.
104			In $V_{\text{dipole}}$, $\mathbf{r}_{ij}$ is the vector starting at
105			bead$i$ and pointing towards bead $j$. Vectors $\mathbf{\Omega}_i$
106			and $\mathbf{\Omega}_j$ are the orientational degrees of freedom for
107			beads $i$ and $j$. $\|\mu_i\|$ is the magnitude of the dipole moment of
108			$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation
109			vector of $\boldsymbol{\Omega}_i$.
110
111			The model also allows for the bonded interactions of bonds, bends, and
112			torsions. The bonds between two beads on a chain are of fixed length,
113			and are maintained according to the {\sc rattle} algorithm. \cite{fix}
114			The bends are subject to a harmonic potential:
115			\begin{equation}
116			eq here
117			\label{lipidEq:bendPot}
118			\end{equation}
119			where $fix$ scales the strength of the harmonic well, and $fix$ is the
120			angle between bond vectors $fix$ and $fix$. The torsion potential is
121			given by:
122			\begin{equation}
123			eq here
124			\label{lipidEq:torsionPot}
125			\end{equation}
126			Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine
127			power series to the desired torsion potential surface, and $\phi$ is
128			the angle between bondvectors $fix$ and $fix$ along the vector $fix$
129			(see Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as
130			the Lennard-Jones potential are excluded for bead pairs involved in
131			the same bond, bend, or torsion. However, internal interactions not
132			directly involved in a bonded pair are calculated.
133
134