--- trunk/tengDissertation/Lipid.tex	2006/06/25 17:54:42	2884
+++ trunk/tengDissertation/Lipid.tex	2006/07/17 20:01:05	2941
@@ -11,7 +11,7 @@ ns\cite{Ho1992}. In the second solvation shell, some w
 small number of water molecules are strongly held around the
 different parts of the headgroup and are oriented by them with
 residence times for the first hydration shell being around 0.5 - 1
-ns\cite{Ho1992}. In the second solvation shell, some water molecules
+ns.\cite{Ho1992} In the second solvation shell, some water molecules
 are weakly bound, but are still essential for determining the
 properties of the system. Transport of various molecular species
 into living cells is one of the major functions of membranes. A
@@ -26,7 +26,7 @@ to be exposed to the aqueous environment\cite{Sasaki20
 translocation of phospholipids across membrane bilayers requires the
 hydrophilic head of the phospholipid to pass through the highly
 hydrophobic interior of the membrane, and for the hydrophobic tails
-to be exposed to the aqueous environment\cite{Sasaki2004}. A number
+to be exposed to the aqueous environment.\cite{Sasaki2004} A number
 of studies indicate that the flipping of phospholipids occurs
 rapidly in the eukaryotic endoplasmic reticulum and the bacterial
 cytoplasmic membrane via a bi-directional, facilitated diffusion
@@ -37,16 +37,16 @@ region\cite{Baber1995}. However, infrared spectroscopy
 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
+region.\cite{Baber1995} However, infrared spectroscopy experiments
 suggest that halothane in DMPC lipid bilayers lives near the
-membrane/water interface\cite{Lieb1982}.
+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. Several schemes are
-proposed in this chapter to overcome these difficulties.
+introduced in this chapter to overcome these difficulties.
 
 \section{\label{lipidSection:model}Model and Methodology}
 
@@ -62,14 +62,16 @@ interaction between two water molecules:
 site, as well as a point dipole. A tetrahedral potential is added to
 correct for hydrogen bonding. The following equation describes the
 interaction between two water molecules:
-\[
+\begin{equation}
 V_{SSD}  = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j )
 + V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j )
-\]
+\label{lipidSection:ssdEquation}
+\end{equation}
 where $r_{ij}$ is the vector between molecule $i$ and molecule $j$,
 $\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom
-for molecule $i$ and molecule $j$ respectively.
-\begin{eqnarray*}
+for molecule $i$ and molecule $j$ respectively. The potential terms
+in Eq.~\ref{lipidSection:ssdEquation} are given by :
+\begin{eqnarray}
 V_{LJ} (r_{ij} ) &= &4\varepsilon _{ij} \left[ {\left(
 {\frac{{\sigma _{ij} }}{{r_{ij} }}} \right)^{12}  - \left(
 {\frac{{\sigma _{ij}
@@ -80,23 +82,25 @@ V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) &=& v_0 [s(
 (\hat{u}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr],\\
 V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) &=& v_0 [s(r_{ij}
 )w(r_{ij} ,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i
-,\Omega _j )].\\
-\end{eqnarray*}
+,\Omega _j )]
+\end{eqnarray}
 where $v_0$ is a strength parameter, $s$ and $s'$ are cubic
-switching functions, while $w$ and $w'$  are responsible for the
+switching functions and $w$ and $w'$  are responsible for the
 tetrahedral potential and the short-range correction to the dipolar
-interaction respectively.
-\[
-\begin{array}{l}
- w(r_{ij} ,\Omega _i ,\Omega _j ) = \sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij}  \\
- w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij}  - 0.6)^2 (\cos \theta _{ij}  + 0.8)^2  - w_0  \\
- \end{array}
-\]
-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
-more expensive interaction.
+interaction respectively:
+\begin{eqnarray}
+ w(r_{ij} ,\Omega _i ,\Omega _j )& = &\sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij},  \\
+ w'(r_{ij} ,\Omega _i ,\Omega _j )& = &(\cos \theta _{ij}  - 0.6)^2 (\cos \theta _{ij}  + 0.8)^2  -
+ w_0.
+\end{eqnarray}
+Here $\theta _{ij}$ and $\phi _{ij}$ are the spherical polar angles
+representing relative orientations of molecule $j$ in the body-fixed
+frame of molecule $i$. 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 and
+because of the "short range" nature of the more expensive
+interaction.
 
 \subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model}
 
@@ -107,7 +111,6 @@ $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
 glycerol motif are modeled by Lennard-Jones spheres with dipoles.
 Alkyl groups in hydrocarbon chains are replaced with unified
 $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
-
 \begin{figure}
 \centering
 \includegraphics[width=3in]{coarse_grained.eps}
@@ -149,7 +152,6 @@ electrostatic potential at point $P$ is given by:
 \theta } }} + \frac{q}{{\sqrt {r^2  + \frac{{d^2 }}{4} - rd\cos
 \theta } }}} \right)
 \]
-
 \begin{figure}
 \centering
 \includegraphics[width=3in]{charge_dipole.eps}
@@ -158,12 +160,11 @@ comprising two point charges with opposite charges. }
 comprising two point charges with opposite charges. }
 \label{lipidFigure:chargeDipole}
 \end{figure}
-
 The basic assumption of the multipole expansion is $r \gg d$ , thus,
 $\frac{{d^2 }}{4}$ inside the square root of the denominator is
 neglected. This is a reasonable approximation in most cases.
 Unfortunately, in our headgroup model, the distance of charge
-separation $d$ (4.63 \AA  in PC headgroup) may be comparable to $r$.
+separation $d$ (4.63 $\rm{\AA}$ in PC headgroup) may be comparable to $r$.
 Nevertheless, we could still assume  $ \cos \theta  \approx 0$ in
 the central region of the headgroup. Using Taylor expansion and
 associating appropriate terms with electric moments will result in a
@@ -195,9 +196,9 @@ dipole-dipole approximation, we discover that water mo
 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
+dipole-dipole approximation, we have discovered 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
@@ -209,8 +210,6 @@ split-dipole approximation.} \label{lipidFigure:splitD
 split-dipole approximation.} \label{lipidFigure:splitDipole}
 \end{figure}
 
-%\section{\label{lipidSection:methods}Methods}
-
 \section{\label{lipidSection:resultDiscussion}Results and Discussion}
 
 \subsection{One Lipid in Sea of Water Molecules}
@@ -222,20 +221,17 @@ density.
 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
-\in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} }  >
-\end{equation}
-\begin{equation}
-g_{AB} (r,\cos \theta ) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
+density
+\begin{eqnarray}
+g_{AB} (r) & = & \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
+\sum\limits_{i
+\in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} }  >, \\
+g_{AB} (r,\cos \theta ) & = & \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
 \sum\limits_{i \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )}
-} \delta (\cos \theta _{ij}  - \cos \theta ) >
-\end{equation}
-
+} \delta (\cos \theta _{ij}  - \cos \theta ) >.
+\end{eqnarray}
 From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first
-solvation shell (3.5 \AA) and the second solvation shell (5.0 \AA)
+solvation shell (3.5 $\rm{\AA}$) 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
@@ -246,13 +242,12 @@ anti-aligned with the dipole of headgroup. When the wa
 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
+anti-aligned with the dipole of the 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.
-
+dipole of the headgroup.
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{g_atom.eps}
@@ -275,7 +270,6 @@ atoms.
 Fig.~\ref{lipidFigure:PCFCoarse}. It is clear that the phosphate end
 in DMPC and the amine end in DMPE are the two most heavily solvated
 atoms.
-
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{g_coarse.eps}
@@ -284,7 +278,6 @@ models]{The pair correlation functions for coarse-grai
 (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}
@@ -294,13 +287,14 @@ phospholipids.} \label{lipidFigure:EWDCoarse}
 \end{figure}
 
 \begin{table}
-\caption{The Parameters For Coarse-grained Phospholipids}
+\caption{THE PARAMETERS FOR COARSE-GRAINED PHOSPHOLIPIDS}
 \label{lipidTable:parameter}
 \begin{center}
-\begin{tabular}{|l|c|c|c|c|c|}
+\begin{tabular}{lccccc}
   \hline
   % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
   Atom type & Mass & $\sigma$ & $\epsilon$ & charge & Dipole \\
+  \hline
   $\text{{\sc CH}}_2$ & 14.03  & 3.95 & 0.0914 & 0 & 0 \\
   $\text{{\sc CH}}_3$ & 15.04  & 3.75 & 0.195  & 0 & 0 \\
   $\text{{\sc CE}}$   & 28.01  & 3.427& 0.294  & 0 & 1.693 \\
@@ -326,7 +320,6 @@ different properties were evaluated over a production 
 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
 \includegraphics[width=\linewidth]{bilayer.eps}
@@ -345,10 +338,10 @@ can be expressed by\cite{Tu1995}
 can be expressed by\cite{Tu1995}
 \begin{equation}
 \rho _{x - ray} (z)dz \propto \sum\limits_{i = 1}^N {\frac{{n_i
-}}{V}\frac{1}{{\sqrt {2\pi \sigma ^2 } }}e^{ - (z - z_i )^2 /2\sigma
-^2 } dz},
+}}{V}\frac{1}{{\sqrt {2\pi \sigma _i ^2 } }}e^{ - (z - z_i )^2
+/2\sigma _i ^2 } dz},
 \end{equation}
-where $\sigma$ is the variance equal to the van der Waals radius,
+where $\sigma _i$ is the variance equal to the van der Waals radius,
 $n_i$ is the atomic number of site $i$ and $V$ is the volume of the
 slab between $z$ and $z+dz$ . The highest density of total EDP
 appears at the position of lipid-water interface corresponding to
@@ -356,10 +349,10 @@ approximation to the thickness of the bilayer, the hea
 distribution of water locked near the head groups, while the lowest
 electron density is in the hydrocarbon region. As a good
 approximation to the thickness of the bilayer, the headgroup spacing
-, is defined as the distance between two peaks in the electron
-density profile, calculated from our simulations to be 34.1 \AA.
-This value is close to the x-ray diffraction experimental value 34.4
-\AA\cite{Petrache1998}.
+$d$ , is defined as the distance between two peaks in the electron
+density profile, calculated from our simulations to be 34.1
+$\rm{\AA}$. This value is close to the x-ray diffraction
+experimental value 34.4 $\rm{\AA}$.\cite{Petrache1998}
 
 \begin{figure}
 \centering
@@ -381,9 +374,9 @@ where $\theta$ is the angle between the  $i$th molecul
 S_{ij}  = \frac{1}{2} < 3\cos \theta _i \cos \theta _j  - \delta
 _{ij}  >
 \end{equation}
-where $\theta$ is the angle between the  $i$th molecular axis and
+where $\theta _i$ is the angle between the  $i$th molecular axis and
 the bilayer normal ($z$ axis). The brackets denote an average over
-time and molecules. The molecular axes are defined:
+time and lipid molecules. The molecular axes are defined:
 \begin{itemize}
 \item $\mathbf{\hat{z}}$ is the vector from $C_{n-1}$ to $C_{n+1}$.
 \item $\mathbf{\hat{y}}$ is the vector that is perpendicular to $z$ and
@@ -391,17 +384,17 @@ In coarse-grained model, although there are no explici
 \item $\mathbf{\hat{x}}$ is the vector perpendicular to
 $\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$.
 \end{itemize}
-In coarse-grained model, although there are no explicit hydrogens,
-the order parameter can still be written in terms of carbon ordering
-at each point of the chain\cite{Egberts1988}
+In our coarse-grained model, although there are no explicit
+hydrogens, the order parameter can still be written in terms of
+carbon ordering at each point of the chain\cite{Egberts1988}
 \begin{equation}
 S_{ij}  = \frac{1}{2} < 3\cos \theta _i \cos \theta _j  - \delta
 _{ij}  >.
 \end{equation}
 
 Fig.~\ref{lipidFigure:Scd} shows the order parameter profile
-calculated for our coarse-grained DMPC bilayer system at 300K. Also
-shown are the experimental data of Tu\cite{Tu1995}. The fact that
+calculated for our coarse-grained DMPC bilayer system at 300K as
+well as the experimental data.\cite{Tu1995} The fact that
 $\text{S}_{\text{{\sc cd}}}$ order parameters calculated from
 simulation are higher than the experimental ones is ascribed to the
 assumption of the locations of implicit hydrogen atoms which are
