--- trunk/matt_papers/canidacy_paper/canidacy_paper.tex	2002/08/24 15:55:38	97
+++ trunk/matt_papers/canidacy_paper/canidacy_paper.tex	2003/04/28 16:03:10	507
@@ -1,8 +1,12 @@
 \documentclass[11pt]{article}
 
 \usepackage{graphicx}
+\usepackage{color}
+\usepackage{floatflt}
 \usepackage{amsmath}
 \usepackage{amssymb}
+\usepackage{subfigure}
+\usepackage{palatino}
 \usepackage[ref]{overcite}
 
 
@@ -19,6 +23,7 @@
 
 \begin{document}
 
+
 \title{A Mesoscale Model for Phospholipid Simulations}
 
 \author{Matthew A. Meineke\\
@@ -31,86 +36,164 @@ Notre Dame, Indiana 46556}
 
 \section{Background and Research Goals}
 
-\section{Methodology}
+Simulations of phospholipid bilayers are, by necessity, quite
+complex. The lipid molecules are large molecules containing many
+atoms, and the head group of the lipid will typically contain charge
+separated ions which set up a large dipole within the molecule. Adding
+to the complexity are the number of water molecules needed to properly
+solvate the lipid bilayer, typically 25 water molecules for every
+lipid molecule. Because of these factors, many current simulations are
+limited in both length and time scale due to to the sheer number of
+calculations performed at every time step and the lifetime of the
+researcher. A typical
+simulation\cite{saiz02,lindahl00,venable00,Marrink01} will have around
+64 phospholipids forming a bilayer approximately 40~$\mbox{\AA}$ by
+50~$\mbox{\AA}$ with roughly 25 waters for every lipid. This means
+there are on the order of 8,000 atoms needed to simulate these systems
+and the trajectories are integrated for times up to 10 ns. 
 
-\subsection{Length and Time Scale Simplifications}
+These limitations make it difficult to study certain biologically
+interesting phenomena that don't fit within the short time and length
+scale requirements. One such phenomenon is the existence of the ripple
+phase ($P_{\beta'}$) of the bilayer between the gel phase
+($L_{\beta'}$) and the fluid phase ($L_{\alpha}$). The $P_{\beta'}$
+phase has been shown to have a ripple period of
+100-200~$\mbox{\AA}$.\cite{katsaras00,sengupta00} A simulation of this
+length scale would require approximately 1,300 lipid molecules and
+roughly 25 waters for every lipid to fully solvate the bilayer. With
+the large number of atoms involved in a simulation of this magnitude,
+steps \emph{must} be taken to simplify the system to the point where
+the numbers of atoms becomes reasonable.
 
-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.
+Another system of interest would be drug molecule diffusion through
+the membrane. Due to the fluid-like properties of a lipid membrane,
+not all diffusion takes place at membrane channels. It is of interest
+to study certain molecules that may incorporate themselves directly
+into the membrane. These molecules may then have an appreciable
+waiting time (on the order of nanoseconds) within the
+bilayer. Simulation of such a long time scale again requires
+simplification of the system in order to lower the number of
+calculations needed at each time step or to increase the length of
+each time step.
 
-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}
+\section{Methodology}
 
-\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} 
+\subsection{Length and Time Scale Simplifications}
 
-The second step taken to simplify the number of calculationsis to
+The length scale simplifications are aimed at increasing the number of
+molecules that can be simulated without drastically increasing the
+computational cost of the simulation. This is done through a
+combination of substituting less expensive interactions for expensive
+ones and decreasing the number of interaction sites per
+molecule. Namely, groups of point charges are replaced with single
+point-dipoles, and unified atoms are used in place of water,
+phospholipid head groups, and alkyl groups.
+
+The replacement of charges with dipoles allows us to replace an
+interaction that has a relatively long range ($\frac{1}{r}$ for the
+coulomb potential) with that of a relatively short range
+($\frac{1}{r^{3}}$ for dipole - dipole potentials). Combined with
+Verlet neighbor lists,\cite{allen87:csl} this should result in an
+algorithm which scales linearly with increasing system size. This is
+in comparison to the Ewald sum\cite{leach01:mm} needed to compute
+periodic replicas of the coulomb interactions, which scales at
+best\cite{darden93:pme} by $N \ln N$.
+
+The second step taken to simplify the number of calculations is 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}).
+Ichiye\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md}
+(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 alkyl
+units in the tails (Section~\ref{sec:lipidModel}). This model is
+similar in practice to that of Lipowsky and Goetz\cite{goetz98}, where
+the whole system is reduced to attractive and repulsive Lennard-Jones
+spheres. However, our model gives a greater level of detail to each
+unified molecule, namely through dipole, bend, and torsion
+interactions.
 
-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.
+The time scale simplifications are introduced so that we can take
+longer time steps. By increasing the size of the time steps taken by
+the simulation, we are able to integrate a given length of time using
+fewer calculations. However, care must be taken that any
+simplifications used still conserve the total energy of the
+simulation. 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 effect 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.
+their equilibrium positions. Bond vibrations are typically the fastest
+periodic 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}
+\subsection{The Soft Sticky Dipole Water Model}
 \label{sec:ssdModel}
 
-The water model used in our simulations is 
+\begin{figure}
+\centering
+\includegraphics[width=50mm]{ssd.epsi}
+\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference.}
+\label{fig:ssdModel}
+\end{figure} 
 
+The water model used in our simulations is a modified soft
+Stockmayer-sphere model.\cite{stevens95} Like the Stockmayer-sphere, the SSD
+model 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.
+
+The SSD water potential for a pair of water molecules is then given by
+the following equation:
 \begin{equation}
-\label{eq:ssdTotPot}
-V_{\text{ssd}} = V_{\text{LJ}}(r_{i\!j}) + V_{\text{dp}}(\mathbf{r}_{i\!j},
+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}
+where $\mathbf{r}_{ij}$ is the vector between molecules $i$ and $j$,
+and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the
+Euler angles of molecule $i$ or $j$ respectively. $V_{\text{LJ}}$ is
+the Lennard-Jones potential:
+\begin{equation}
+V_{\text{LJ}} = 
+	4\epsilon_{ij} \biggl[
+	\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12}
+	- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6}
+	\biggr]
+\label{eq:lennardJonesPot}
+\end{equation}
+where $\sigma_{ij}$ scales the length of the interaction, and
+$\epsilon_{ij}$ scales the energy of the potential. For SSD,
+$\sigma_{\text{SSD}} = 3.051 \mbox{
+\AA}$ and $\epsilon_{\text{SSD}} = 0.152\text{ kcal/mol}$.
+$V_{\text{dp}}$ is the dipole potential:
+\begin{equation}
+V_{\text{dp}}(\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}_i \cdot \mathbf{r}_{ij}) %
+		(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) }
+		{r^{5}_{ij}} \biggr]
+\label{eq:dipolePot}
+\end{equation}
+where $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$,
+$\boldsymbol{\mu}_i$ points along the z-axis in the body fixed
+frame. This frame is then oriented in space by the Euler angles,
+$\boldsymbol{\Omega}_i$. The SSD model specifies a dipole magnitude of
+2.35~D for water.
 
+The hydrogen bonding is modeled by the $V_{\text{sp}}$
+term of the potential. Its form is as follows:
 \begin{equation}
-\label{eq:spPot}
 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},
@@ -118,26 +201,42 @@ V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i
 	+
 	s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
 		\boldsymbol{\Omega}_{j})]
+\label{eq:spPot}
 \end{equation}
-
+Where $v^\circ$ scales 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}
-\label{eq:apPot2}
 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:spPot2}
 \end{equation}
-
+and
 \begin{equation}
-\label{eq:spCorrection}
 \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 \\
-	&\phantom{=} + (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ}
+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 angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical
+coordinates of the position of molecule $j$ in the reference frame
+fixed on molecule $i$ with the z-axis aligned with the dipole moment.
+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 $180^\circ$.
 
+Finally, the sticky potential is scaled by a switching function,
+$s(r_{ij})$, that scales smoothly between 0 and 1. It is represented
+by:
 \begin{equation}
-\label{eq:spCutoff}
 s(r_{ij}) = 
 	\begin{cases}
 	1&	\text{if $r_{ij} < r_{L}$}, \\
@@ -146,12 +245,300 @@ s(r_{ij}) = 
 		\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}
 
+Despite the apparent complexity of Equation \ref{eq:spPot}, the SSD
+model is still computationally inexpensive. This is due to Equation
+\ref{eq:spCutoff}. With $r_{L}$ being 2.75~$\mbox{\AA}$ and $r_{U}$
+being equal to either 3.35~$\mbox{\AA}$ for $s(r_{ij})$ or
+4.0~$\mbox{\AA}$ for $s'(r_{ij})$, the sticky potential is only active
+over an extremely short range, and then only with other SSD
+molecules. Therefore, it's predominant interaction is through the
+point dipole and the Lennard-Jones sphere. Of these, only the dipole
+interaction can be considered ``long-range''.
+
 \subsection{The Phospholipid Model}
 \label{sec:lipidModel}
 
+\begin{figure}
+\centering
+\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.}
+\label{fig:lipidModel}
+\end{figure}
 
-\bibliographystyle{achemso}
-\bibliography{canidacy_paper} 
-\end{document}
\ No newline at end of file
+The lipid molecules in our simulations are unified atom models. Figure
+\ref{fig:lipidModel} shows a schematic 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 the dipole moment is 20.6 D, chosen to match
+that of DPPC\cite{Cevc87}. The tail atoms are unified $\text{CH}_2$
+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_{\text{lipid}} =
+	\sum_{i}V_{i}^{\text{internal}}
+	+ \sum_i \sum_{j>i} \sum_{\alpha_i} 
+	\sum_{\beta_j}
+	V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}})
+	+\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j})
+\label{eq:lipidModelPot}
+\end{equation}
+where,
+\begin{equation}
+V_{i}^{\text{internal}} = 
+	\sum_{\text{bends}}V_{\text{bend}}(\theta_{\alpha\beta\gamma})
+	+ \sum_{\text{torsions}}V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta})
+	+ \sum_{\alpha_i} \sum_{\beta_i > (\alpha_i + 4)}V_{\text{LJ}}
+		(r_{\alpha_i \beta_i})
+\label{eq:lipidModelPotInternal}
+\end{equation}
+
+The non-bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are
+the Lennard-Jones and dipole-dipole interactions respectively. For the
+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_{\alpha\beta\gamma}) 
+	= k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \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_{\alpha\beta\gamma\zeta})
+	= c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}]
+	+ c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] 
+	+ c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})]
+\label{eq:torsPot}
+\end{equation}
+All parameters for bonded and non-bonded potentials in the tail atoms
+were taken from TraPPE.\cite{Siepmann1998} The bonded interactions for
+the head atom were also taken from TraPPE, however it's dipole moment
+and mass were based on the properties of the phosphatidylcholine head
+group. The Lennard-Jones parameter for the head group was chosen such
+that it was roughly twice the size of a $\text{CH}_3$ atom, and it's
+well depth was set to be approximately equal to that of $\text{CH}_3$.
+
+\section{Initial Simulation: 25 lipids in water}
+\label{sec:5x5}
+
+\subsection{Starting Configuration and Parameters}
+\label{sec:5x5Start}
+
+\begin{figure}
+\centering
+\mbox{
+	\subfigure[The starting configuration of the 25 lipid system.]{%
+		\label{fig:5x5Start}%
+		\includegraphics[width=70mm]{5x5-initial.eps}}\quad
+	\subfigure[The 25 lipid system at 6.27~ns.]{%
+		\label{fig:5x5Final}%
+		\includegraphics[width=70mm]{5x5-6.27ns.eps}}
+}
+\caption{Snapshots of the 25 lipid system. Carbon tail atoms are drawn in gray, the phospholipid head groups are colored blue, and the waters are scaled down for visibility. A box has been drawn around the periodic image.}
+\end{figure}
+
+Our first simulation is an array of 25 single chain lipids in a sea
+of water (Figure \ref{fig:5x5Start}). The total number of water
+molecules is 1386, giving a final of water concentration of 70\%
+wt. The simulation box measures 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$
+x 39.4~$\mbox{\AA}$ with periodic boundary conditions imposed. The
+system is simulated in the micro-canonical (NVE) ensemble with an
+average temperature of 300~K.
+
+\subsection{Results}
+\label{sec:5x5Results}
+
+\begin{figure}
+\centering
+	\subfigure[The self correlation of the phospholipid head groups. $g(r)$ is on the top, the bottom chart is the $g_\gamma(r)$.]{%
+		\label{fig:5x5HHCorr}%
+		\includegraphics[angle=-90,width=80mm]{all5x5-HEAD-HEAD.epsi}%
+		}
+	\subfigure[The $g(r)$ for the $\text{CH}_2$ molecules in the chain tails]{%
+		\label{fig:5x5CCg}%
+		\includegraphics[angle=-90,width=80mm]{all5x5-CH2-CH2.epsi}}
+	\subfigure%
+	[The pair correlations between the head groups and the water]{%
+		\label{fig:5x5HXCorr}%
+		\includegraphics[angle=-90,width=80mm]{all5x5-HEAD-X.epsi}}
+\caption{The pair correlation functions for the 25 lipid system}
+\end{figure}
+		
+
+Figure \ref{fig:5x5Final} shows a snapshot of the system at
+6.27~ns. Note that the system has spontaneously self assembled into a
+bilayer. Discussion of the length scales of the bilayer will follow in
+this section. However, it is interesting to note a key qualitative
+property of the system revealed by this snapshot, the tail chains are
+tilted to the bilayer normal. This is usually indicative of the gel
+($L_{\beta'}$) phase. In this system, the box size is probably too
+small for the bilayer to relax to the fluid ($P_{\alpha}$) phase. This
+demonstrates a need for an isobaric-isothermal ensemble where the box
+size may relax or expand to keep the system at 1~atm.
+
+The simulation was analyzed using the radial distribution function,
+$g(r)$, which has the form:
+\begin{equation}
+g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} 
+	\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle
+\label{eq:gofr}
+\end{equation}
+Equation \ref{eq:gofr} gives us information about the spacing of two
+species as a function of radius. Essentially, if the observer were
+located at atom $i$ and were looking out in all directions, $g(r)$
+shows the relative density of atom $j$ at any given radius, $r$,
+normalized by the expected density of atom $j$ at $r$. In a
+homogeneously mixed fluid, $g(r)$ will approach 1 at large $r$, as a
+fluid contains no long range structure to contribute peaks in the
+number density.
+
+For the species containing dipoles, a second pair-wise distribution
+function was used, $g_{\gamma}(r)$. It is of the form:
+\begin{equation}
+g_{\gamma}(r) = \langle \sum_i \sum_{j>i}
+	(\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle
+\label{eq:gammaofr}
+\end{equation}
+Where $\gamma_{ij}$ is the angle between the dipole of atom $j$ with
+respect to the dipole of atom $i$. This correlation will vary between
+$+1$ and $-1$ when the two dipoles are perfectly aligned and
+anti-aligned respectively. This then gives us information about how
+directional species are aligned with each other as a function of
+distance.
+
+Figure \ref{fig:5x5HHCorr} shows the two self correlation functions
+for the Head groups of the lipids. The first peak in the $g(r)$ at
+4.03~$\mbox{\AA}$ is the nearest neighbor separation of the heads of
+two lipids. This corresponds to a maximum in the $g_{\gamma}(r)$ which
+means that the two neighbors on the same leaf have their dipoles
+aligned. The broad peak at 6.5~$\mbox{\AA}$ is the inter-surface
+spacing. Here, there is a corresponding anti-alignment in the angular
+correlation. This means that although the dipoles are aligned on the
+same monolayer, the dipoles will orient themselves to be anti-aligned
+on a opposite facing monolayer. With this information, the two peaks
+at 7.0~$\mbox{\AA}$ and 7.4~$\mbox{\AA}$ are head groups on the same
+monolayer, and they are the second nearest neighbors to the head
+group. The peak is likely a split peak because of the small statistics
+of this system. Finally, the peak at 8.0~$\mbox{\AA}$ is likely the
+second nearest neighbor on the opposite monolayer because of the
+anti-alignment evident in the angular correlation.
+
+Figure \ref{fig:5x5CCg} shows the radial distribution function for the
+$\text{CH}_2$ unified atoms. The spacing of the atoms along the tail
+chains accounts for the regularly spaced sharp peaks, but the broad
+underlying peak with its maximum at 4.6~$\mbox{\AA}$ is the
+distribution of chain-chain distances between two lipids. The final
+Figure, Figure \ref{fig:5x5HXCorr}, includes the correlation functions
+between the Head group and the SSD atoms. The peak in $g(r)$ at
+3.6~$\mbox{\AA}$ is the distance of closest approach between the two,
+and $g_{\gamma}(r)$ shows that the SSD atoms will align their dipoles
+with the head groups at close distance. However, as one increases the
+distance, the SSD atoms are no longer aligned.
+
+\section{Second Simulation: 50 randomly oriented lipids in water}
+\label{sec:r50}
+
+\subsection{Starting Configuration and Parameters}
+\label{sec:r50Start}
+
+\begin{figure}
+\centering
+\mbox{
+	\subfigure[The starting configuration of the 50 lipid system.]{%
+		\label{fig:r50Start}%
+		\includegraphics[width=70mm]{r50-initial.eps}}\quad	
+	\subfigure[The 50 lipid system at 2.21~ns]{%
+		\label{fig:r50Final}%	
+		\includegraphics[width=70mm]{r50-2.21ns.eps}}
+}
+\caption{Snapshots of the 50 lipid system}
+\end{figure}
+
+The second simulation consists of 50 single chained lipid molecules
+embedded in a sea of 1384 SSD waters (54\% wt.). The lipids in this
+simulation were started with random orientation and location (Figure
+\ref{fig:r50Start} ) The simulation box measured 26.6~$\mbox{\AA}$ x
+26.6~$\mbox{\AA}$ x 108.4~$\mbox{\AA}$ with periodic boundary conditions
+imposed. The simulation was run in the NVE ensemble with an average
+temperature of 300~K.
+
+\subsection{Results}
+\label{sec:r50Results}
+
+\begin{figure}
+\centering
+	\subfigure[The self correlation of the phospholipid head groups.]{%
+		\label{fig:r50HHCorr}%
+		\includegraphics[angle=-90,width=80mm]{r50-HEAD-HEAD.epsi}%
+		}
+	\subfigure%
+	[The pair correlations between the head groups and the water]{%
+		\label{fig:r50HXCorr}%
+		\includegraphics[angle=-90,width=80mm]{r50-HEAD-X.epsi}%
+		}
+	\subfigure[The $g(r)$ for the $\text{CH}_2$ molecules in the chain tails]{%
+		\label{fig:r50CCg}%
+		\includegraphics[angle=-90,width=80mm]{r50-CH2-CH2.epsi}}
+
+\caption{The pair correlation functions for the 50 lipid system}
+\end{figure}
+
+Figure \ref{fig:r50Final} is a snapshot of the system at 2.21~ns. Here
+we see that the system has already aggregated into several micelles
+and two are even starting to merge. It will be interesting to watch as
+this simulation continues what the total time scale for the micelle
+aggregation and bilayer formation will be, in Marrink's\cite{Marrink01}
+simulation, bilayer aggregation is predicted to occur around 10~ns.
+
+Figures \ref{fig:r50HHCorr}, \ref{fig:r50HXCorr}, and \ref{fig:r50CCg} are
+the same correlation functions for the random 50 simulation as for the
+previous simulation of 25 lipids. What is most interesting to note, is
+the high degree of similarity between the correlation functions
+between systems. Even though the 25 lipid simulation formed a bilayer
+and the random 50 simulation is still in the micelle stage, both have
+an inter-surface spacing of 6.5~$\mbox{\AA}$ with the same
+characteristic anti-alignment between surfaces. Not as surprising, is
+the consistency of the closest packing statistics between
+systems. Namely, a head-head closest approach distance of
+4~$\mbox{\AA}$, and similar findings for the chain-chain and
+head-water distributions as in the 25 lipid system.
+
+\section{Future Directions}
+
+Current simulations indicate that our model is a feasible one, however
+improvements will need to be made to allow the system to be simulated
+in the isobaric-isothermal ensemble. This will relax the system to an
+equilibrium configuration at room temperature and pressure allowing us
+to compare our model to experimental results. Also, we are in the
+process of parallelizing the code for an even greater speedup. This
+will allow us to simulate large enough systems to examine phenomena
+such as the ripple phase and drug molecule diffusion
+
+Once the work has been completed on the simulation engine, we will
+then use it to explore the phase diagram for our model. By
+characterizing how our model parameters affect the bilayer properties,
+we will tailor our model to more closely match real biological
+molecules. With this information, we will then incorporate
+biologically relevant molecules into the system and observe their
+transport properties across the membrane.
+
+\section{Acknowledgments}
+
+I would like to thank Dr.~J.~Daniel Gezelter for his guidance on this
+project. I would also like to acknowledge the following for their help
+and discussions during this project: Christopher Fennell, Charles
+Vardeman, Teng Lin, Megan Sprague, Patrick Conforti, and Dan
+Combest. Funding for this project came from the National Science
+Foundation.
+
+\pagebreak 
+\bibliographystyle{achemso} 
+\bibliography{canidacy_paper}
+\end{document}
