matt_papers/canidacy_paper/canidacy_paper.tex

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\begin{document}


\title{A Mesoscale Model for Phospholipid Simulations}

\author{Matthew A. Meineke\\
Department of Chemistry and Biochemistry\\
University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}
\maketitle

\section{Background and Research Goals}

\section{Methodology}

\subsection{Length and Time Scale Simplifications}

The length scale simplifications are aimed at increaseing the number
of molecules simulated without drastically increasing the
computational cost of the system. This is done by a combination of
substituting less expensive interactions for expensive ones and
decreasing the number of interaction sites per molecule. Namely,
charge distributions are replaced with dipoles, and unified atoms are
used in place of water and phospholipid head groups.

The replacement of charge distributions with dipoles allows us to
replace an interaction that has a relatively long range, $\frac{1}{r}$
for the charge charge potential, with that of a relitively short
range, $\frac{1}{r^{3}}$ for dipole - dipole potentials
(Equations~\ref{eq:dipolePot} and \ref{eq:chargePot}). This allows us
to use computaional simplifications algorithms such as Verlet neighbor
lists,\cite{allen87:csl} which gives computaional scaling by $N$. This
is in comparison to the Ewald sum\cite{leach01:mm} needed to compute
the charge - charge interactions which scales at best by $N
\ln N$.

\begin{equation}
V^{\text{dp}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
        \boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[
        \frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}}
        -
        \frac{3(\boldsymbol{\mu} \cdot \mathbf{r}_{ij}) %
                (\boldsymbol{\mu} \cdot \mathbf{r}_{ij}) }{r^{5}_{ij}} \biggr]
\label{eq:dipolePot}
\end{equation}

\begin{equation}
V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) = \frac{q_{i}q_{j}}%
        {4\pi\epsilon_{0} r_{ij}}
\label{eq:chargePot}
\end{equation} 

The second step taken to simplify the number of calculationsis to
incorporate unified models for groups of atoms. In the case of water,
we use the soft sticky dipole (SSD) model developed by
Ichiye\cite{Liu96} (Section~\ref{sec:ssdModel}). For the phospholipids, a
unified head atom with a dipole will replace the atoms in the head
group, while unified $\text{CH}_2$ and $\text{CH}_3$ atoms will
replace the alkanes in the tails (Section~\ref{sec:lipidModel}).

The time scale simplifications are taken so that the simulation can
take long time steps. By incresing the time steps taken by the
simulation, we are able to integrate the simulation trajectory with
fewer calculations. However, care must be taken to conserve the energy
of the simulation. This is a constraint placed upon the system by
simulating in the microcanonical ensemble. In practice, this means
taking steps small enough to resolve all motion in the system without
accidently moving an object too far along a repulsive energy surface
before it feels the affect of the surface.

In our simulation we have chosen to constrain all bonds to be of fixed
length. This means the bonds are no longer allowed to vibrate about
their equilibrium positions, typically the fastest periodical motion
in a dynamics simulation. By taking this action, we are able to take
time steps of 3 fs while still maintaining constant energy. This is in
contrast to the 1 fs time step typically needed to conserve energy when
bonds lengths are allowed to oscillate.

\subsection{The Soft Sticky Water Model}
\label{sec:ssdModel}

\begin{floatingfigure}{55mm}
\includegraphics[width=45mm]{ssd.epsi}
\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference. \vspace{5mm}}
% The dipole magnitude is 2.35 D and the Lennard-Jones parameters are $\sigma = 3.051 \mbox{\AA}$ and $\epsilon = 0.152$ kcal/mol.}
\label{fig:ssdModel}
\end{floatingfigure} 

The water model used in our simulations is a modified soft Stockmayer
sphere model. Like the soft Stockmayer sphere, the SSD
model\cite{Liu96} consists of a Lennard-Jones interaction site and a
dipole both located at the water's center of mass (Figure
\ref{fig:ssdModel}). However, the SSD model extends this by adding a
tetrahedral potential to correct for hydrogen bonding.

This SSD water's motion is then governed by the following potential:
\begin{equation}
V_{\text{ssd}} = V_{\text{LJ}}(r_{i\!j}) + V_{\text{dp}}(\mathbf{r}_{i\!j},
        \boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
        + V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i},
        \boldsymbol{\Omega}_{j})
\label{eq:ssdTotPot}
\end{equation}
$V_{\text{LJ}}$ is the Lennard-Jones potential with $\sigma_{\text{w}}
= 3.051 \mbox{ \AA}$ and $\epsilon_{\text{w}} = 0.152\text{
kcal/mol}$. $V_{\text{dp}}$ is the dipole potential with
$|\mu_{\text{w}}| = 2.35\text{ D}$.

The hydrogen bonding of the model is governed by the $V_{\text{sp}}$ term of the potentail. Its form is as follows:
\begin{equation}
V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i},
        \boldsymbol{\Omega}_{j}) =
        v^{\circ}[s(r_{ij})w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
                \boldsymbol{\Omega}_{j})
        +
        s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
                \boldsymbol{\Omega}_{j})]
\label{eq:spPot}
\end{equation}
Where $v^\circ$ is responsible for scaling the strength of the
interaction.
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
and
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
are responsible for the tetrahedral potential and a correction to the
tetrahedral potential respectively. They are,
\begin{equation}
w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) =
        \sin\theta_{ij} \sin 2\theta_{ij} \cos 2\phi_{ij}
        + \sin \theta_{ji} \sin 2\theta_{ji} \cos 2\phi_{ji}
\label{eq:apPot2}
\end{equation}
and
\begin{equation}
\begin{split}
w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) =
        &(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij} + 0.8)^2 \\
        &+ (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ}
\end{split}
\label{eq:spCorrection}
\end{equation}
The correction 
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
is needed because
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical polar coordinates of the position of sphere $j$ in the reference frame fixed on sphere $i$ with the z-axis alligned with the dipole moment. 

Finaly, the  sticky potentail is scaled by a cutoff function, $s(r_{ij})$ that scales smoothly between 0 and 1. It is represented by:
\begin{equation}
s(r_{ij}) = 
        \begin{cases}
        1&      \text{if $r_{ij} < r_{L}$}, \\
        \frac{(r_{U} - r_{ij})^2 (r_{U} + 2r_{ij} 
                - 3r_{L})}{(r_{U}-r_{L})^3}&
                \text{if $r_{L} \leq r_{ij} \leq r_{U}$},\\
        0&      \text{if $r_{ij} \geq r_{U}$}.
        \end{cases}
\label{eq:spCutoff}
\end{equation}


\subsection{The Phospholipid Model}
\label{sec:lipidModel}

\begin{floatingfigure}{90mm}
\includegraphics[angle=-90,width=80mm]{lipidModel.epsi}
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length. \vspace{5mm}}
\label{fig:lipidModel}
\end{floatingfigure}

The lipid molecules in our simulations are unified atom models. Figure
\ref{fig:lipidModel} shows a template drawing for one of our
lipids. The Head group of the phospholipid is replaced by a single
Lennard-Jones sphere with a freely oriented dipole placed at it's
center. The magnitude of it's dipole moment is 20.6 D. The tail atoms
are unifed $\text{CH}_2$ and $\text{CH}_3$ atoms and are also modeled
as Lennard-Jones spheres. The total potential for the lipid is
represented by Equation \ref{eq:lipidModelPot}.

\begin{equation}
V_{\mbox{lipid}} = \overbrace{%
        V_{\text{bend}}(\theta_{ijk}) +% 
        V_{\text{tors.}}(\phi_{ijkl})}^{bonded}
        + \overbrace{%
        V_{\text{LJ}}(r_{i\!j}) + 
        V_{\text{dp}}(r_{i\!j},\Omega_{i},\Omega_{j})%
        }^{non-bonded}
\label{eq:lipidModelPot}
\end{equation}

The non bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are
the Lennard-Jones and dipole-dipole interactions respectively. For the
non-bonded potentials, only the bend and the torsional potentials are
calculated. The bond potential is not calculated, and the bond lengths
are constrained via RATTLE.\cite{leach01:mm} The bend potential is of
the form: 
\begin{equation}
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}\frac{(\theta_{ijk} - \theta_0)^2}{2}
\label{eq:bendPot}
\end{equation}
Where $k_{\theta}$ sets the stiffness of the bend potential, and $\theta_0$
sets the equilibrium bend angle. The torsion potential was given by:
\begin{equation}
V_{\text{tors.}}(\phi_{ijkl}= \cos(\phi_{ijkl})
\label{eq:torsPot}
\end{equation}

\pagebreak
\bibliographystyle{achemso}
\bibliography{canidacy_paper} \end{document}
Revision:	100
Committed:	Tue Aug 27 01:28:22 2002 UTC (23 years, 3 months ago) by mmeineke
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#	User	Rev	Content
1	mmeineke	95	\documentclass[11pt]{article}
2
3			\usepackage{graphicx}
4	mmeineke	98	\usepackage{floatflt}
5	mmeineke	95	\usepackage{amsmath}
6			\usepackage{amssymb}
7			\usepackage[ref]{overcite}
8
9
10
11			\pagestyle{plain}
12			\pagenumbering{arabic}
13			\oddsidemargin 0.0cm \evensidemargin 0.0cm
14			\topmargin -21pt \headsep 10pt
15			\textheight 9.0in \textwidth 6.5in
16			\brokenpenalty=10000
17			\renewcommand{\baselinestretch}{1.2}
18			\renewcommand\citemid{\ } % no comma in optional reference note
19
20
21			\begin{document}
22
23	mmeineke	98
24	mmeineke	95	\title{A Mesoscale Model for Phospholipid Simulations}
25
26			\author{Matthew A. Meineke\\
27			Department of Chemistry and Biochemistry\\
28			University of Notre Dame\\
29			Notre Dame, Indiana 46556}
30
31			\date{\today}
32			\maketitle
33
34			\section{Background and Research Goals}
35
36			\section{Methodology}
37
38	mmeineke	96	\subsection{Length and Time Scale Simplifications}
39	mmeineke	95
40			The length scale simplifications are aimed at increaseing the number
41			of molecules simulated without drastically increasing the
42			computational cost of the system. This is done by a combination of
43			substituting less expensive interactions for expensive ones and
44			decreasing the number of interaction sites per molecule. Namely,
45			charge distributions are replaced with dipoles, and unified atoms are
46			used in place of water and phospholipid head groups.
47
48			The replacement of charge distributions with dipoles allows us to
49			replace an interaction that has a relatively long range, $\frac{1}{r}$
50			for the charge charge potential, with that of a relitively short
51			range, $\frac{1}{r^{3}}$ for dipole - dipole potentials
52			(Equations~\ref{eq:dipolePot} and \ref{eq:chargePot}). This allows us
53			to use computaional simplifications algorithms such as Verlet neighbor
54			lists,\cite{allen87:csl} which gives computaional scaling by $N$. This
55			is in comparison to the Ewald sum\cite{leach01:mm} needed to compute
56			the charge - charge interactions which scales at best by $N
57			\ln N$.
58
59			\begin{equation}
60			V^{\text{dp}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
61			\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[
62			\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}}
63			-
64			\frac{3(\boldsymbol{\mu} \cdot \mathbf{r}_{ij}) %
65			(\boldsymbol{\mu} \cdot \mathbf{r}_{ij}) }{r^{5}_{ij}} \biggr]
66			\label{eq:dipolePot}
67			\end{equation}
68
69			\begin{equation}
70			V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) = \frac{q_{i}q_{j}}%
71			{4\pi\epsilon_{0} r_{ij}}
72			\label{eq:chargePot}
73			\end{equation}
74
75			The second step taken to simplify the number of calculationsis to
76			incorporate unified models for groups of atoms. In the case of water,
77			we use the soft sticky dipole (SSD) model developed by
78			Ichiye\cite{Liu96} (Section~\ref{sec:ssdModel}). For the phospholipids, a
79			unified head atom with a dipole will replace the atoms in the head
80			group, while unified $\text{CH}_2$ and $\text{CH}_3$ atoms will
81			replace the alkanes in the tails (Section~\ref{sec:lipidModel}).
82
83	mmeineke	96	The time scale simplifications are taken so that the simulation can
84			take long time steps. By incresing the time steps taken by the
85			simulation, we are able to integrate the simulation trajectory with
86			fewer calculations. However, care must be taken to conserve the energy
87			of the simulation. This is a constraint placed upon the system by
88			simulating in the microcanonical ensemble. In practice, this means
89			taking steps small enough to resolve all motion in the system without
90			accidently moving an object too far along a repulsive energy surface
91			before it feels the affect of the surface.
92	mmeineke	95
93	mmeineke	96	In our simulation we have chosen to constrain all bonds to be of fixed
94			length. This means the bonds are no longer allowed to vibrate about
95			their equilibrium positions, typically the fastest periodical motion
96			in a dynamics simulation. By taking this action, we are able to take
97			time steps of 3 fs while still maintaining constant energy. This is in
98			contrast to the 1 fs time step typically needed to conserve energy when
99	mmeineke	97	bonds lengths are allowed to oscillate.
100	mmeineke	95
101			\subsection{The Soft Sticky Water Model}
102			\label{sec:ssdModel}
103
104	mmeineke	98	\begin{floatingfigure}{55mm}
105			\includegraphics[width=45mm]{ssd.epsi}
106			\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference. \vspace{5mm}}
107			% The dipole magnitude is 2.35 D and the Lennard-Jones parameters are $\sigma = 3.051 \mbox{\AA}$ and $\epsilon = 0.152$ kcal/mol.}
108			\label{fig:ssdModel}
109			\end{floatingfigure}
110	mmeineke	96
111	mmeineke	98	The water model used in our simulations is a modified soft Stockmayer
112			sphere model. Like the soft Stockmayer sphere, the SSD
113			model\cite{Liu96} consists of a Lennard-Jones interaction site and a
114			dipole both located at the water's center of mass (Figure
115			\ref{fig:ssdModel}). However, the SSD model extends this by adding a
116			tetrahedral potential to correct for hydrogen bonding.
117
118			This SSD water's motion is then governed by the following potential:
119	mmeineke	96	\begin{equation}
120			V_{\text{ssd}} = V_{\text{LJ}}(r_{i\!j}) + V_{\text{dp}}(\mathbf{r}_{i\!j},
121			\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
122			+ V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i},
123			\boldsymbol{\Omega}_{j})
124	mmeineke	98	\label{eq:ssdTotPot}
125	mmeineke	96	\end{equation}
126	mmeineke	98	$V_{\text{LJ}}$ is the Lennard-Jones potential with $\sigma_{\text{w}}
127			= 3.051 \mbox{ \AA}$ and $\epsilon_{\text{w}} = 0.152\text{
128			kcal/mol}$. $V_{\text{dp}}$ is the dipole potential with
129			$\|\mu_{\text{w}}\| = 2.35\text{ D}$.
130	mmeineke	96
131	mmeineke	98	The hydrogen bonding of the model is governed by the $V_{\text{sp}}$ term of the potentail. Its form is as follows:
132	mmeineke	96	\begin{equation}
133			V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i},
134			\boldsymbol{\Omega}_{j}) =
135			v^{\circ}[s(r_{ij})w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
136			\boldsymbol{\Omega}_{j})
137			+
138			s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
139			\boldsymbol{\Omega}_{j})]
140	mmeineke	98	\label{eq:spPot}
141	mmeineke	96	\end{equation}
142	mmeineke	98	Where $v^\circ$ is responsible for scaling the strength of the
143			interaction.
144			$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
145			and
146			$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
147			are responsible for the tetrahedral potential and a correction to the
148			tetrahedral potential respectively. They are,
149	mmeineke	96	\begin{equation}
150			w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) =
151			\sin\theta_{ij} \sin 2\theta_{ij} \cos 2\phi_{ij}
152			+ \sin \theta_{ji} \sin 2\theta_{ji} \cos 2\phi_{ji}
153	mmeineke	98	\label{eq:apPot2}
154	mmeineke	96	\end{equation}
155	mmeineke	98	and
156	mmeineke	96	\begin{equation}
157			\begin{split}
158	mmeineke	98	w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) =
159			&(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij} + 0.8)^2 \\
160			&+ (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ}
161	mmeineke	96	\end{split}
162	mmeineke	98	\label{eq:spCorrection}
163	mmeineke	96	\end{equation}
164	mmeineke	98	The correction
165			$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
166			is needed because
167			$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
168			vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical polar coordinates of the position of sphere $j$ in the reference frame fixed on sphere $i$ with the z-axis alligned with the dipole moment.
169	mmeineke	96
170	mmeineke	98	Finaly, the sticky potentail is scaled by a cutoff function, $s(r_{ij})$ that scales smoothly between 0 and 1. It is represented by:
171	mmeineke	96	\begin{equation}
172			s(r_{ij}) =
173			\begin{cases}
174			1& \text{if $r_{ij} < r_{L}$}, \\
175			\frac{(r_{U} - r_{ij})^2 (r_{U} + 2r_{ij}
176			- 3r_{L})}{(r_{U}-r_{L})^3}&
177			\text{if $r_{L} \leq r_{ij} \leq r_{U}$},\\
178			0& \text{if $r_{ij} \geq r_{U}$}.
179			\end{cases}
180	mmeineke	98	\label{eq:spCutoff}
181	mmeineke	96	\end{equation}
182
183	mmeineke	98
184	mmeineke	95	\subsection{The Phospholipid Model}
185			\label{sec:lipidModel}
186
187	mmeineke	99	\begin{floatingfigure}{90mm}
188			\includegraphics[angle=-90,width=80mm]{lipidModel.epsi}
189			\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length. \vspace{5mm}}
190			\label{fig:lipidModel}
191			\end{floatingfigure}
192	mmeineke	95
193	mmeineke	99	The lipid molecules in our simulations are unified atom models. Figure
194			\ref{fig:lipidModel} shows a template drawing for one of our
195			lipids. The Head group of the phospholipid is replaced by a single
196			Lennard-Jones sphere with a freely oriented dipole placed at it's
197			center. The magnitude of it's dipole moment is 20.6 D. The tail atoms
198			are unifed $\text{CH}_2$ and $\text{CH}_3$ atoms and are also modeled
199			as Lennard-Jones spheres. The total potential for the lipid is
200			represented by Equation \ref{eq:lipidModelPot}.
201
202			\begin{equation}
203			V_{\mbox{lipid}} = \overbrace{%
204			V_{\text{bend}}(\theta_{ijk}) +%
205			V_{\text{tors.}}(\phi_{ijkl})}^{bonded}
206			+ \overbrace{%
207			V_{\text{LJ}}(r_{i\!j}) +
208			V_{\text{dp}}(r_{i\!j},\Omega_{i},\Omega_{j})%
209			}^{non-bonded}
210			\label{eq:lipidModelPot}
211			\end{equation}
212
213			The non bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are
214			the Lennard-Jones and dipole-dipole interactions respectively. For the
215			non-bonded potentials, only the bend and the torsional potentials are
216			calculated. The bond potential is not calculated, and the bond lengths
217			are constrained via RATTLE.\cite{leach01:mm} The bend potential is of
218			the form:
219			\begin{equation}
220			V_{\text{bend}}(\theta_{ijk}) = k_{\theta}\frac{(\theta_{ijk} - \theta_0)^2}{2}
221			\label{eq:bendPot}
222			\end{equation}
223	mmeineke	100	Where $k_{\theta}$ sets the stiffness of the bend potential, and $\theta_0$
224			sets the equilibrium bend angle. The torsion potential was given by:
225			\begin{equation}
226			V_{\text{tors.}}(\phi_{ijkl}= \cos(\phi_{ijkl})
227			\label{eq:torsPot}
228			\end{equation}
229	mmeineke	99
230	mmeineke	100	\pagebreak
231			\bibliographystyle{achemso}
232	mmeineke	99	\bibliography{canidacy_paper} \end{document}