--- trunk/tengDissertation/Lipid.tex	2006/06/06 20:49:05	2806
+++ trunk/tengDissertation/Lipid.tex	2006/06/23 20:21:54	2881
@@ -28,36 +28,34 @@ rapidly in the eukaryotic ER and the bacterial cytopla
 hydrophobic interior of the membrane, and for the hydrophobic tails
 to be exposed to the aqueous environment\cite{Sasaki2004}. A number
 of studies indicate that the flipping of phospholipids occurs
-rapidly in the eukaryotic ER and the bacterial cytoplasmic membrane
-via a bi-directional, facilitated diffusion process requiring no
-metabolic energy input. Another system of interest would be the
-distribution of sites occupied by inhaled anesthetics in membrane.
-Although the physiological effects of anesthetics have been
-extensively studied, the controversy over their effects on lipid
-bilayers still continues. Recent deuterium NMR measurements on
-halothane in POPC lipid bilayers suggest the anesthetics are
-primarily located at the hydrocarbon chain region\cite{Baber1995}.
-Infrared spectroscopy experiments suggest that halothane in DMPC
-lipid bilayers lives near the membrane/water
-interface\cite{Lieb1982}.
+rapidly in the eukaryotic endoplasmic reticulum and the bacterial
+cytoplasmic membrane via a bi-directional, facilitated diffusion
+process requiring no metabolic energy input. Another system of
+interest is the distribution of sites occupied by inhaled
+anesthetics in membrane. Although the physiological effects of
+anesthetics have been extensively studied, the controversy over
+their effects on lipid bilayers still continues. Recent deuterium
+NMR measurements on halothane on POPC lipid bilayers suggest the
+anesthetics are primarily located at the hydrocarbon chain
+region\cite{Baber1995}. However, infrared spectroscopy experiments
+suggest that halothane in DMPC lipid bilayers lives near the
+membrane/water interface\cite{Lieb1982}.
 
 Molecular dynamics simulations have proven to be a powerful tool for
 studying the functions of biological systems, providing structural,
 thermodynamic and dynamical information. Unfortunately, much of
 biological interest happens on time and length scales well beyond
-the range of current simulation technologies.
-%review of coarse-grained modeling
-Several schemes are proposed in this chapter to overcome these
-difficulties.
+the range of current simulation technologies. Several schemes are
+proposed in this chapter to overcome these difficulties.
 
-\section{\label{lipidSection:model}Model}
+\section{\label{lipidSection:model}Model and Methodology}
 
 \subsection{\label{lipidSection:SSD}The Soft Sticky Dipole Water Model}
 
 In a typical bilayer simulation, the dominant portion of the
 computation time will be spent calculating water-water interactions.
 As an efficient solvent model, the Soft Sticky Dipole (SSD) water
-model\cite{Chandra1999,Fennel2004} is used as the explicit solvent
+model\cite{Chandra1999,Fennell2004} is used as the explicit solvent
 in this project. Unlike other water models which have partial
 charges distributed throughout the whole molecule, the SSD water
 model consists of a single site which is a Lennard-Jones interaction
@@ -97,11 +95,11 @@ Although dipole-dipole and sticky interactions are mor
  w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij}  - 0.6)^2 (\cos \theta _{ij}  + 0.8)^2  - w_0  \\
  \end{array}
 \]
-Although dipole-dipole and sticky interactions are more
+Although the dipole-dipole and sticky interactions are more
 mathematically complicated than Coulomb interactions, the number of
 pair interactions is reduced dramatically both because the model
 only contains a single-point as well as "short range" nature of the
-higher order interaction.
+more expensive interaction.
 
 \subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model}
 
@@ -116,8 +114,14 @@ $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
 \begin{figure}
 \centering
 \includegraphics[width=3in]{coarse_grained.eps}
-\caption[A representation of coarse-grained phospholipid model]{}
-\label{lipidFigure:coarseGrained}
+\caption[A representation of coarse-grained phospholipid model]{A
+representation of coarse-grained phospholipid model. The lipid
+headgroup consists of $\text{{\sc PO}}_4$ group (yellow),
+$\text{{\sc NC}}_4$ group (blue) and a united $\text{{\sc C}}$ atom
+(gray) $ with a dipole, while the glycerol backbone includes dipolar
+$\text{{\sc CE}}$ (red) and $\text{{\sc CK}}$ (pink) groups. Alkyl
+groups in hydrocarbon chains are simply represented by gray united
+atoms.} \label{lipidFigure:coarseGrained}
 \end{figure}
 
 Accurate and efficient computation of electrostatics is one of the
@@ -131,13 +135,12 @@ Ewald summation method mathematically transforms this
 \]
 where $N_A$ and $N_B$ are the number of point charges in the two
 molecular species. Originally developed to study ionic crystals, the
-Ewald summation method mathematically transforms this
-straightforward but conditionally convergent summation into two more
-complicated but rapidly convergent sums. One summation is carried
-out in reciprocal space while the other is carried out in real
-space. An alternative approach is a multipole expansion, which is
-based on electrostatic moments, such as charge (monopole), dipole,
-quadruple etc.
+Ewald sum method mathematically transforms this straightforward but
+conditionally convergent summation into two more complicated but
+rapidly convergent sums. One summation is carried out in reciprocal
+space while the other is carried out in real space. An alternative
+approach is the multipole expansion, which is based on electrostatic
+moments, such as charge (monopole), dipole, quadrupole etc.
 
 Here we consider a linear molecule which consists of two point
 charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The
@@ -153,9 +156,10 @@ electrostatic potential at point $P$ is given by:
 \begin{figure}
 \centering
 \includegraphics[width=3in]{charge_dipole.eps}
-\caption[Electrostatic potential due to a linear molecule comprising
-two point charges]{Electrostatic potential due to a linear molecule
-comprising two point charges} \label{lipidFigure:chargeDipole}
+\caption[An illustration of split-dipole
+approximation]{Electrostatic potential due to a linear molecule
+comprising two point charges with opposite charges. }
+\label{lipidFigure:chargeDipole}
 \end{figure}
 
 The basic assumption of the multipole expansion is $r \gg d$ , thus,
@@ -191,14 +195,21 @@ another.
 and respectively. This approximation to the multipole expansion
 maintains the fast fall-off of the multipole potentials but lacks
 the normal divergences when two polar groups get close to one
-another.
-%description of the comparsion
+another. The comparision between different electrostatic
+approximation is shown in \ref{lipidFigure:splitDipole}. Due to the
+divergence at the central region of the headgroup introduced by
+dipole-dipole approximation, we discover that water molecules are
+locked into the central region in the simulation. This artifact can
+be corrected using split-dipole approximation or other accurate
+methods.
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{split_dipole.eps}
-\caption[Comparison between electrostatic approximation]{Electron
-density profile along the bilayer normal.}
-\label{lipidFigure:splitDipole}
+\caption[Comparison between electrostatic
+approximation]{Electrostatic potential map for two pairs of charges
+with different alignments: (a) illustration of different alignments;
+(b) charge-charge interaction; (c) dipole-dipole approximation; (d)
+split-dipole approximation.} \label{lipidFigure:splitDipole}
 \end{figure}
 
 %\section{\label{lipidSection:methods}Methods}
@@ -207,13 +218,14 @@ To exclude the inter-headgroup interaction, atomistic 
 
 \subsection{One Lipid in Sea of Water Molecules}
 
-To exclude the inter-headgroup interaction, atomistic models of one
-lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built
-and studied using atomistic molecular dynamics. The simulation was
-analyzed using a set of radial distribution functions, which give
-the probability of finding a pair of molecular species a distance
-apart, relative to the probability expected for a completely random
-distribution function at the same density.
+To tune our parameters without the inter-headgroup interactions,
+atomistic models of one lipid (DMPC or DLPE) in sea of water
+molecules (TIP3P) were built and studied using atomistic molecular
+dynamics. The simulation was analyzed using a set of radial
+distribution functions, which give the probability of finding a pair
+of molecular species a distance apart, relative to the probability
+expected for a completely random distribution function at the same
+density.
 
 \begin{equation}
 g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i
@@ -225,23 +237,33 @@ From figure 4(a), we can identify the first solvation 
 } \delta (\cos \theta _{ij}  - \cos \theta ) >
 \end{equation}
 
-From figure 4(a), we can identify the first solvation shell (3.5
-$\AA$) and the second solvation shell (5.0 $\AA$) from both plots.
-However, the corresponding orientations are different. In DLPE,
-water molecules prefer to sit around -NH3 group due to the hydrogen
-bonding. In contrast, because of the hydrophobic effect of the
--N(CH3)3 group, the preferred position of water molecules in DMPC is
-around the -PO4 Group. When the water molecules are far from the
-headgroup, the distribution of the two angles should be uniform. The
-correlation close to center of the headgroup dipole (< 5 $\AA$) in
-both plots tell us that in the closely-bound region, the dipoles of
-the water molecules are preferentially anti-aligned with the dipole
-of headgroup.
+From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first
+solvation shell (3.5 $\AA$) and the second solvation shell (5.0
+$\AA$) from both plots. However, the corresponding orientations are
+different. In DLPE, water molecules prefer to sit around $\text{{\sc
+NH}}_3$ group due to the hydrogen bonding. In contrast, because of
+the hydrophobic effect of the $ {\rm{N(CH}}_{\rm{3}}
+{\rm{)}}_{\rm{3}} $ group, the preferred position of water molecules
+in DMPC is around the $\text{{\sc PO}}_4$ Group. When the water
+molecules are far from the headgroup, the distribution of the two
+angles should be uniform. The correlation close to center of the
+headgroup dipole in both plots tells us that in the closely-bound
+region, the dipoles of the water molecules are preferentially
+anti-aligned with the dipole of headgroup. When the water molecules
+are far from the headgroup, the distribution of the two angles
+should be uniform. The correlation close to center of the headgroup
+dipole in both plots tell us that in the closely-bound region, the
+dipoles of the water molecules are preferentially aligned with the
+dipole of headgroup.
 
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{g_atom.eps}
-\caption[The pair correlation functions for atomistic models]{}
+\caption[The pair correlation functions for atomistic models]{The
+pair correlation functions for atomistic models: (a)$g(r,\cos \theta
+)$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE; (c)$g(r,\cos \omega
+)$ for DMPC; (d)$g(r,\cos \omega )$ for DLPE; (e)$g(\cos \theta,\cos
+\omega)$ for DMPC; (f)$g(\cos \theta,\cos \omega)$ for DMLPE.}
 \label{lipidFigure:PCFAtom}
 \end{figure}
 
@@ -260,15 +282,18 @@ atoms.
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{g_coarse.eps}
-\caption[The pair correlation functions for coarse-grained models]{}
+\caption[The pair correlation functions for coarse-grained
+models]{The pair correlation functions for coarse-grained models:
+(a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.}
 \label{lipidFigure:PCFCoarse}
 \end{figure}
 
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{EWD_coarse.eps}
-\caption[Excess water density of coarse-grained phospholipids]{ }
-\label{lipidFigure:EWDCoarse}
+\caption[Excess water density of coarse-grained
+phospholipids]{Excess water density of coarse-grained
+phospholipids.} \label{lipidFigure:EWDCoarse}
 \end{figure}
 
 \begin{table}
@@ -295,16 +320,15 @@ molecules has been constructed from an atomistic coord
 \subsection{Bilayer Simulations Using Coarse-grained Model}
 
 A bilayer system consisting of 128 DMPC lipids and 3655 water
-molecules has been constructed from an atomistic coordinate
-file.[15] The MD simulation is performed at constant temperature, T
-= 300K, and constant pressure, p = 1 atm, and consisted of an
-equilibration period of 2 ns. During the equilibration period, the
-system was initially simulated at constant volume for 1ns. Once the
-system was equilibrated at constant volume, the cell dimensions of
-the system was relaxed by performing under NPT conditions using
-Nos��-Hoover extended system isothermal-isobaric dynamics. After
-equilibration, different properties were evaluated over a production
-run of 5 ns.
+molecules has been constructed from an atomistic coordinate file.
+The MD simulation is performed at constant temperature, T = 300K,
+and constant pressure, p = 1 atm, and consisted of an equilibration
+period of 2 ns. During the equilibration period, the system was
+initially simulated at constant volume for 1 ns. Once the system was
+equilibrated at constant volume, the cell dimensions of the system
+was relaxed by performing under NPT conditions using Nos\'{e}-Hoover
+extended system isothermal-isobaric dynamics. After equilibration,
+different properties were evaluated over a production run of 5 ns.
 
 \begin{figure}
 \centering
@@ -315,7 +339,7 @@ molecules.}
 \label{lipidFigure:bilayer}
 \end{figure}
 
-\subsubsection{Electron Density Profile (EDP)}
+\subsubsection{\textbf{Electron Density Profile (EDP)}}
 
 Assuming a gaussian distribution of electrons on each atomic center
 with a variance estimated from the size of the van der Waals radius,
@@ -351,7 +375,7 @@ and total density due to DMPC in blue.}
 \label{lipidFigure:electronDensity}
 \end{figure}
 
-\subsubsection{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}
+\subsubsection{\textbf{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}}
 
 Measuring deuterium order parameters by NMR is a useful technique to
 study the orientation of hydrocarbon chains in phospholipids. The
@@ -392,8 +416,29 @@ of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grain
 \includegraphics[width=\linewidth]{scd.eps}
 \caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison
 of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model
-(blue) and DMPC\cite{petrache00} (black) near 300~K.}
+(blue) and DMPC\cite{Petrache2000} (black) near 300~K.}
 \label{lipidFigure:Scd}
 \end{figure}
 
 %\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model}
+
+\section{\label{lipidSection:Conclusion}Conclusion}
+
+Atomistic simulations have been used in this study to determine the
+preferred orientation and location of water molecules relative to
+the location and orientation of the PC and PE lipid headgroups.
+Based on the results from our all-atom simulations, we developed a
+simple coarse-grained model which captures the essential features of
+the headgroup solvation which is crucial to transport process in
+membrane system. In addition, the model has been explored in a
+bilayer system was shown to have reasonable electron density
+profiles, $\text{S}_{\text{{\sc cd}}}$ order parameter and other
+structural properties. The accuracy of this model is achieved by
+matching atomistic result. It is also easy to represent different
+phospholipids by changing a few parameters of the model. Another
+important characteristic of this model distinguishing itself from
+other models\cite{Goetz1998,Marrink2004} is the computational speed
+gained by introducing a short range electrostatic approximation.
+Further studies of this system using z-constraint method could
+extend the time length of the simulations to study transport
+phenomena in large-scale membrane systems.
