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# Line 11 | Line 11 | ns. In the second solvation shell, some water molecule
11   small number of water molecules are strongly held around the
12   different parts of the headgroup and are oriented by them with
13   residence times for the first hydration shell being around 0.5 - 1
14 < ns. In the second solvation shell, some water molecules are weakly
15 < bound, but are still essential for determining the properties of the
16 < system. Transport of various molecular species into living cells is
17 < one of the major functions of membranes. A thorough understanding of
18 < the underlying molecular mechanism for solute diffusion is crucial
19 < to the further studies of other related biological processes. All
20 < transport across cell membranes takes place by one of two
21 < fundamental processes: Passive transport is driven by bulk or
22 < inter-diffusion of the molecules being transported or by membrane
23 < pores which facilitate crossing. Active transport depends upon the
24 < expenditure of cellular energy in the form of ATP hydrolysis. As the
25 < central processes of membrane assembly, translocation of
26 < phospholipids across membrane bilayers requires the hydrophilic head
27 < of the phospholipid to pass through the highly hydrophobic interior
28 < of the membrane, and for the hydrophobic tails to be exposed to the
29 < aqueous environment. A number of studies indicate that the flipping
30 < of phospholipids occurs rapidly in the eukaryotic ER and the
31 < bacterial cytoplasmic membrane via a bi-directional, facilitated
32 < diffusion process requiring no metabolic energy input. Another
33 < system of interest would be the distribution of sites occupied by
34 < inhaled anesthetics in membrane. Although the physiological effects
35 < of anesthetics have been extensively studied, the controversy over
14 > ns\cite{Ho1992}. In the second solvation shell, some water molecules
15 > are weakly bound, but are still essential for determining the
16 > properties of the system. Transport of various molecular species
17 > into living cells is one of the major functions of membranes. A
18 > thorough understanding of the underlying molecular mechanism for
19 > solute diffusion is crucial to the further studies of other related
20 > biological processes. All transport across cell membranes takes
21 > place by one of two fundamental processes: Passive transport is
22 > driven by bulk or inter-diffusion of the molecules being transported
23 > or by membrane pores which facilitate crossing. Active transport
24 > depends upon the expenditure of cellular energy in the form of ATP
25 > hydrolysis. As the central processes of membrane assembly,
26 > translocation of phospholipids across membrane bilayers requires the
27 > hydrophilic head of the phospholipid to pass through the highly
28 > hydrophobic interior of the membrane, and for the hydrophobic tails
29 > to be exposed to the aqueous environment\cite{Sasaki2004}. A number
30 > of studies indicate that the flipping of phospholipids occurs
31 > rapidly in the eukaryotic endoplasmic reticulum and the bacterial
32 > cytoplasmic membrane via a bi-directional, facilitated diffusion
33 > process requiring no metabolic energy input. Another system of
34 > interest is the distribution of sites occupied by inhaled
35 > anesthetics in membrane. Although the physiological effects of
36 > anesthetics have been extensively studied, the controversy over
37   their effects on lipid bilayers still continues. Recent deuterium
38 < NMR measurements on halothane in POPC lipid bilayers suggest the
39 < anesthetics are primarily located at the hydrocarbon chain region.
40 < Infrared spectroscopy experiments suggest that halothane in DMPC
41 < lipid bilayers lives near the membrane/water interface.
38 > NMR measurements on halothane on POPC lipid bilayers suggest the
39 > anesthetics are primarily located at the hydrocarbon chain
40 > region\cite{Baber1995}. However, infrared spectroscopy experiments
41 > suggest that halothane in DMPC lipid bilayers lives near the
42 > membrane/water interface\cite{Lieb1982}.
43  
44   Molecular dynamics simulations have proven to be a powerful tool for
45   studying the functions of biological systems, providing structural,
# Line 46 | Line 48 | proposed in this chapter to overcome these difficultie
48   the range of current simulation technologies. Several schemes are
49   proposed in this chapter to overcome these difficulties.
50  
51 < \section{\label{lipidSection:model}Model}
51 > \section{\label{lipidSection:model}Model and Methodology}
52  
53   \subsection{\label{lipidSection:SSD}The Soft Sticky Dipole Water Model}
54  
55   In a typical bilayer simulation, the dominant portion of the
56   computation time will be spent calculating water-water interactions.
57   As an efficient solvent model, the Soft Sticky Dipole (SSD) water
58 < model is used as the explicit solvent in this project. Unlike other
59 < water models which have partial charges distributed throughout the
60 < whole molecule, the SSD water model consists of a single site which
61 < is a Lennard-Jones interaction site, as well as a point dipole. A
62 < tetrahedral potential is added to correct for hydrogen bonding. The
63 < following equation describes the interaction between two water
64 < molecules:
65 < \[
58 > model\cite{Chandra1999,Fennell2004} is used as the explicit solvent
59 > in this project. Unlike other water models which have partial
60 > charges distributed throughout the whole molecule, the SSD water
61 > model consists of a single site which is a Lennard-Jones interaction
62 > site, as well as a point dipole. A tetrahedral potential is added to
63 > correct for hydrogen bonding. The following equation describes the
64 > interaction between two water molecules:
65 > \begin{equation}
66   V_{SSD}  = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j )
67   + V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j )
68 < \]
69 < where $r_{ij}$ is the vector between molecule $i$ and molecule $j$,
68 > \label{lipidSection:ssdEquation}
69 > \end{equation}
70 > where$r_{ij}$ is the vector between molecule $i$ and molecule $j$,
71   $\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom
72 < for molecule $i$ and molecule $j$ respectively.
73 < \[
74 < V_{LJ} (r_{ij} ) = 4\varepsilon _{ij} \left[ {\left( {\frac{{\sigma
75 < _{ij} }}{{r_{ij} }}} \right)^{12}  - \left( {\frac{{\sigma _{ij}
76 < }}{{r_{ij} }}} \right)^6 } \right]
77 < \]
78 < \[
79 < V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon
80 < _0 }}\left[ {\frac{{\mu _i  \cdot \mu _j }}{{r_{ij}^3 }} -
81 < \frac{{3\left( {\mu _i  \cdot r_{ij} } \right)\left( {\mu _i  \cdot
82 < r_{ij} } \right)}}{{r_{ij}^5 }}} \right]
83 < \]
84 < \[
85 < V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) = v_0 [s(r_{ij} )w(r_{ij}
86 < ,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i ,\Omega _j
84 < )]
85 < \]
72 > for molecule $i$ and molecule $j$ respectively. The potential terms
73 > in Eq.~\ref{lipidSection:ssdEquation} are given by :
74 > \begin{eqnarray}
75 > V_{LJ} (r_{ij} ) &= &4\varepsilon _{ij} \left[ {\left(
76 > {\frac{{\sigma _{ij} }}{{r_{ij} }}} \right)^{12}  - \left(
77 > {\frac{{\sigma _{ij}
78 > }}{{r_{ij} }}} \right)^6 } \right], \\
79 > V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) &= &
80 > \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[
81 > \hat{u}_{i} \cdot \hat{u}_{j} - 3(\hat{u}_i \cdot \hat{\mathbf{r}}_{ij}) %
82 > (\hat{u}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr],\\
83 > V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) &=& v_0 [s(r_{ij}
84 > )w(r_{ij} ,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i
85 > ,\Omega _j )]
86 > \end{eqnarray}
87   where $v_0$ is a strength parameter, $s$ and $s'$ are cubic
88 < switching functions, while $w$   and $w'$  are responsible for the
88 > switching functions, while $w$ and $w'$  are responsible for the
89   tetrahedral potential and the short-range correction to the dipolar
90 < interaction respectively.
91 < \[
92 < \begin{array}{l}
93 < w(r_{ij} ,\Omega _i ,\Omega _j ) = \sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij}  \\
94 < w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij}  - 0.6)^2 (\cos \theta _{ij}  + 0.8)^2  - w_0  \\
95 < \end{array}
96 < \]
97 < Although dipole-dipole and sticky interactions are more
90 > interaction respectively:
91 > \begin{eqnarray}
92 > w(r_{ij} ,\Omega _i ,\Omega _j )& = &\sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij},  \\
93 > w'(r_{ij} ,\Omega _i ,\Omega _j )& = &(\cos \theta _{ij}  - 0.6)^2 (\cos \theta _{ij}  + 0.8)^2  -
94 > w_0.
95 > \end{eqnarray}
96 > Here $\theta _{ij}$ and $\phi _{ij}$ are the spherical polar angles
97 > representing relative orientations between molecule $i$ and molecule
98 > $j$. Although the dipole-dipole and sticky interactions are more
99   mathematically complicated than Coulomb interactions, the number of
100   pair interactions is reduced dramatically both because the model
101   only contains a single-point as well as "short range" nature of the
102 < higher order interaction.
102 > more expensive interaction.
103  
104   \subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model}
105  
# Line 108 | Line 110 | $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
110   glycerol motif are modeled by Lennard-Jones spheres with dipoles.
111   Alkyl groups in hydrocarbon chains are replaced with unified
112   $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
111 –
113   \begin{figure}
114   \centering
115 < \includegraphics[width=\linewidth]{coarse_grained.eps}
116 < \caption[A representation of coarse-grained phospholipid model]{}
115 > \includegraphics[width=3in]{coarse_grained.eps}
116 > \caption[A representation of coarse-grained phospholipid model]{A
117 > representation of coarse-grained phospholipid model. The lipid
118 > headgroup consists of $\text{{\sc PO}}_4$ group (yellow),
119 > $\text{{\sc NC}}_4$ group (blue) and a united C atom (gray) with a
120 > dipole, while the glycerol backbone includes dipolar $\text{{\sc
121 > CE}}$ (red) and $\text{{\sc CK}}$ (pink) groups. Alkyl groups in
122 > hydrocarbon chains are simply represented by gray united atoms.}
123   \label{lipidFigure:coarseGrained}
124   \end{figure}
125  
# Line 127 | Line 134 | Ewald summation method mathematically transforms this
134   \]
135   where $N_A$ and $N_B$ are the number of point charges in the two
136   molecular species. Originally developed to study ionic crystals, the
137 < Ewald summation method mathematically transforms this
138 < straightforward but conditionally convergent summation into two more
139 < complicated but rapidly convergent sums. One summation is carried
140 < out in reciprocal space while the other is carried out in real
141 < space. An alternative approach is a multipole expansion, which is
142 < based on electrostatic moments, such as charge (monopole), dipole,
136 < quadruple etc.
137 > Ewald sum method mathematically transforms this straightforward but
138 > conditionally convergent summation into two more complicated but
139 > rapidly convergent sums. One summation is carried out in reciprocal
140 > space while the other is carried out in real space. An alternative
141 > approach is the multipole expansion, which is based on electrostatic
142 > moments, such as charge (monopole), dipole, quadrupole etc.
143  
144   Here we consider a linear molecule which consists of two point
145   charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The
# Line 145 | Line 151 | electrostatic potential at point $P$ is given by:
151   \theta } }} + \frac{q}{{\sqrt {r^2  + \frac{{d^2 }}{4} - rd\cos
152   \theta } }}} \right)
153   \]
148 –
154   \begin{figure}
155   \centering
156 < \includegraphics[width=\linewidth]{charge_dipole.eps}
157 < \caption[Electrostatic potential due to a linear molecule comprising
158 < two point charges]{Electrostatic potential due to a linear molecule
159 < comprising two point charges} \label{lipidFigure:chargeDipole}
156 > \includegraphics[width=3in]{charge_dipole.eps}
157 > \caption[An illustration of split-dipole
158 > approximation]{Electrostatic potential due to a linear molecule
159 > comprising two point charges with opposite charges. }
160 > \label{lipidFigure:chargeDipole}
161   \end{figure}
156 –
162   The basic assumption of the multipole expansion is $r \gg d$ , thus,
163   $\frac{{d^2 }}{4}$ inside the square root of the denominator is
164   neglected. This is a reasonable approximation in most cases.
165   Unfortunately, in our headgroup model, the distance of charge
166 < separation $d$ (4.63 $\AA$ in PC headgroup) may be comparable to
167 < $r$. Nevertheless, we could still assume  $ \cos \theta  \approx 0$
168 < in the central region of the headgroup. Using Taylor expansion and
166 > separation $d$ (4.63 \AA  in PC headgroup) may be comparable to $r$.
167 > Nevertheless, we could still assume  $ \cos \theta  \approx 0$ in
168 > the central region of the headgroup. Using Taylor expansion and
169   associating appropriate terms with electric moments will result in a
170   "split-dipole" approximation:
171   \[
# Line 178 | Line 183 | $i$ and molecule $j$, and $R_{ij{$ is given by,
183   \]
184   where $\mu _i$  and  $\mu _j$ are the dipole moment of molecule $i$
185   and molecule $j$ respectively, $r_{ij}$ is vector between molecule
186 < $i$ and molecule $j$, and $R_{ij{$ is given by,
186 > $i$ and molecule $j$, and $R_{ij}$ is given by,
187   \[
188   R_{ij}  = \sqrt {r_{ij}^2  + \frac{{d_i^2 }}{4} + \frac{{d_j^2
189   }}{4}}
# Line 187 | Line 192 | another.
192   and respectively. This approximation to the multipole expansion
193   maintains the fast fall-off of the multipole potentials but lacks
194   the normal divergences when two polar groups get close to one
195 < another.
196 <
195 > another. The comparision between different electrostatic
196 > approximation is shown in \ref{lipidFigure:splitDipole}. Due to the
197 > divergence at the central region of the headgroup introduced by
198 > dipole-dipole approximation, we discover that water molecules are
199 > locked into the central region in the simulation. This artifact can
200 > be corrected using split-dipole approximation or other accurate
201 > methods.
202   \begin{figure}
203   \centering
204   \includegraphics[width=\linewidth]{split_dipole.eps}
205 < \caption[Comparison between electrostatic approximation]{Electron
206 < density profile along the bilayer normal.}
207 < \label{lipidFigure:splitDipole}
205 > \caption[Comparison between electrostatic
206 > approximation]{Electrostatic potential map for two pairs of charges
207 > with different alignments: (a) illustration of different alignments;
208 > (b) charge-charge interaction; (c) dipole-dipole approximation; (d)
209 > split-dipole approximation.} \label{lipidFigure:splitDipole}
210   \end{figure}
211  
200 – %\section{\label{lipidSection:methods}Methods}
201 –
212   \section{\label{lipidSection:resultDiscussion}Results and Discussion}
213  
214   \subsection{One Lipid in Sea of Water Molecules}
215  
216 < To exclude the inter-headgroup interaction, atomistic models of one
217 < lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built
218 < and studied using atomistic molecular dynamics. The simulation was
219 < analyzed using a set of radial distribution functions, which give
220 < the probability of finding a pair of molecular species a distance
221 < apart, relative to the probability expected for a completely random
222 < distribution function at the same density.
223 <
224 < \begin{equation}
225 < g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i
226 < \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} }  >
227 < \end{equation}
228 < \begin{equation}
219 < g_{AB} (r,\cos \theta ) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
216 > To tune our parameters without the inter-headgroup interactions,
217 > atomistic models of one lipid (DMPC or DLPE) in sea of water
218 > molecules (TIP3P) were built and studied using atomistic molecular
219 > dynamics. The simulation was analyzed using a set of radial
220 > distribution functions, which give the probability of finding a pair
221 > of molecular species a distance apart, relative to the probability
222 > expected for a completely random distribution function at the same
223 > density
224 > \begin{eqnarray}
225 > g_{AB} (r) & = & \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
226 > \sum\limits_{i
227 > \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} }  >, \\
228 > g_{AB} (r,\cos \theta ) & = & \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
229   \sum\limits_{i \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )}
230 < } \delta (\cos \theta _{ij}  - \cos \theta ) >
231 < \end{equation}
232 <
233 < From figure 4(a), we can identify the first solvation shell (3.5
234 < $\AA$) and the second solvation shell (5.0 $\AA$) from both plots.
235 < However, the corresponding orientations are different. In DLPE,
236 < water molecules prefer to sit around -NH3 group due to the hydrogen
237 < bonding. In contrast, because of the hydrophobic effect of the
238 < -N(CH3)3 group, the preferred position of water molecules in DMPC is
239 < around the -PO4 Group. When the water molecules are far from the
240 < headgroup, the distribution of the two angles should be uniform. The
241 < correlation close to center of the headgroup dipole (< 5 $\AA$) in
242 < both plots tell us that in the closely-bound region, the dipoles of
243 < the water molecules are preferentially anti-aligned with the dipole
244 < of headgroup.
245 <
230 > } \delta (\cos \theta _{ij}  - \cos \theta ) >.
231 > \end{eqnarray}
232 > From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first
233 > solvation shell (3.5 \AA) and the second solvation shell (5.0 \AA)
234 > from both plots. However, the corresponding orientations are
235 > different. In DLPE, water molecules prefer to sit around $\text{{\sc
236 > NH}}_3$ group due to the hydrogen bonding. In contrast, because of
237 > the hydrophobic effect of the $ {\rm{N(CH}}_{\rm{3}}
238 > {\rm{)}}_{\rm{3}} $ group, the preferred position of water molecules
239 > in DMPC is around the $\text{{\sc PO}}_4$ Group. When the water
240 > molecules are far from the headgroup, the distribution of the two
241 > angles should be uniform. The correlation close to center of the
242 > headgroup dipole in both plots tells us that in the closely-bound
243 > region, the dipoles of the water molecules are preferentially
244 > anti-aligned with the dipole of headgroup. When the water molecules
245 > are far from the headgroup, the distribution of the two angles
246 > should be uniform. The correlation close to center of the headgroup
247 > dipole in both plots tell us that in the closely-bound region, the
248 > dipoles of the water molecules are preferentially aligned with the
249 > dipole of headgroup.
250   \begin{figure}
251   \centering
252   \includegraphics[width=\linewidth]{g_atom.eps}
253 < \caption[The pair correlation functions for atomistic models]{}
253 > \caption[The pair correlation functions for atomistic models]{The
254 > pair correlation functions for atomistic models: (a)$g(r,\cos \theta
255 > )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE; (c)$g(r,\cos \omega
256 > )$ for DMPC; (d)$g(r,\cos \omega )$ for DLPE; (e)$g(\cos \theta,\cos
257 > \omega)$ for DMPC; (f)$g(\cos \theta,\cos \omega)$ for DMLPE.}
258   \label{lipidFigure:PCFAtom}
259   \end{figure}
260  
# Line 252 | Line 269 | atoms.
269   Fig.~\ref{lipidFigure:PCFCoarse}. It is clear that the phosphate end
270   in DMPC and the amine end in DMPE are the two most heavily solvated
271   atoms.
255 –
272   \begin{figure}
273   \centering
274   \includegraphics[width=\linewidth]{g_coarse.eps}
275 < \caption[The pair correlation functions for coarse-grained models]{}
275 > \caption[The pair correlation functions for coarse-grained
276 > models]{The pair correlation functions for coarse-grained models:
277 > (a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.}
278   \label{lipidFigure:PCFCoarse}
279   \end{figure}
262 –
280   \begin{figure}
281   \centering
282   \includegraphics[width=\linewidth]{EWD_coarse.eps}
283 < \caption[Excess water density of coarse-grained phospholipids]{ }
284 < \label{lipidFigure:EWDCoarse}
283 > \caption[Excess water density of coarse-grained
284 > phospholipids]{Excess water density of coarse-grained
285 > phospholipids.} \label{lipidFigure:EWDCoarse}
286   \end{figure}
287  
288   \begin{table}
289 < \caption{The Parameters For Coarse-grained Phospholipids}
289 > \caption{THE PARAMETERS FOR COARSE-GRAINED PHOSPHOLIPIDS}
290   \label{lipidTable:parameter}
291   \begin{center}
292 < \begin{tabular}{|l|c|c|c|c|c|}
292 > \begin{tabular}{lccccc}
293    \hline
294    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
295    Atom type & Mass & $\sigma$ & $\epsilon$ & charge & Dipole \\
296 +  \hline
297    $\text{{\sc CH}}_2$ & 14.03  & 3.95 & 0.0914 & 0 & 0 \\
298    $\text{{\sc CH}}_3$ & 15.04  & 3.75 & 0.195  & 0 & 0 \\
299    $\text{{\sc CE}}$   & 28.01  & 3.427& 0.294  & 0 & 1.693 \\
# Line 291 | Line 310 | molecules has been constructed from an atomistic coord
310   \subsection{Bilayer Simulations Using Coarse-grained Model}
311  
312   A bilayer system consisting of 128 DMPC lipids and 3655 water
313 < molecules has been constructed from an atomistic coordinate
314 < file.[15] The MD simulation is performed at constant temperature, T
315 < = 300K, and constant pressure, p = 1 atm, and consisted of an
316 < equilibration period of 2 ns. During the equilibration period, the
317 < system was initially simulated at constant volume for 1ns. Once the
318 < system was equilibrated at constant volume, the cell dimensions of
319 < the system was relaxed by performing under NPT conditions using
320 < Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After
321 < equilibration, different properties were evaluated over a production
303 < run of 5 ns.
304 <
313 > molecules has been constructed from an atomistic coordinate file.
314 > The MD simulation is performed at constant temperature, T = 300K,
315 > and constant pressure, p = 1 atm, and consisted of an equilibration
316 > period of 2 ns. During the equilibration period, the system was
317 > initially simulated at constant volume for 1 ns. Once the system was
318 > equilibrated at constant volume, the cell dimensions of the system
319 > was relaxed by performing under NPT conditions using Nos\'{e}-Hoover
320 > extended system isothermal-isobaric dynamics. After equilibration,
321 > different properties were evaluated over a production run of 5 ns.
322   \begin{figure}
323   \centering
324   \includegraphics[width=\linewidth]{bilayer.eps}
# Line 311 | Line 328 | molecules.}
328   \label{lipidFigure:bilayer}
329   \end{figure}
330  
331 < \subsubsection{Electron Density Profile (EDP)}
331 > \subsubsection{\textbf{Electron Density Profile (EDP)}}
332  
333   Assuming a gaussian distribution of electrons on each atomic center
334   with a variance estimated from the size of the van der Waals radius,
335   the EDPs which are proportional to the density profiles measured
336   along the bilayer normal obtained by x-ray scattering experiment,
337 < can be expressed by
337 > can be expressed by\cite{Tu1995}
338   \begin{equation}
339   \rho _{x - ray} (z)dz \propto \sum\limits_{i = 1}^N {\frac{{n_i
340   }}{V}\frac{1}{{\sqrt {2\pi \sigma ^2 } }}e^{ - (z - z_i )^2 /2\sigma
# Line 332 | Line 349 | density profile, calculated from our simulations to be
349   electron density is in the hydrocarbon region. As a good
350   approximation to the thickness of the bilayer, the headgroup spacing
351   , is defined as the distance between two peaks in the electron
352 < density profile, calculated from our simulations to be 34.1 $\AA$.
352 > density profile, calculated from our simulations to be 34.1 \AA.
353   This value is close to the x-ray diffraction experimental value 34.4
354 < $\AA$.
354 > \AA\cite{Petrache1998}.
355  
356   \begin{figure}
357   \centering
# Line 347 | Line 364 | and total density due to DMPC in blue.}
364   \label{lipidFigure:electronDensity}
365   \end{figure}
366  
367 < \subsubsection{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}
367 > \subsubsection{\textbf{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}}
368  
369   Measuring deuterium order parameters by NMR is a useful technique to
370   study the orientation of hydrocarbon chains in phospholipids. The
# Line 368 | Line 385 | at each point of the chain
385   \end{itemize}
386   In coarse-grained model, although there are no explicit hydrogens,
387   the order parameter can still be written in terms of carbon ordering
388 < at each point of the chain
388 > at each point of the chain\cite{Egberts1988}
389   \begin{equation}
390   S_{ij}  = \frac{1}{2} < 3\cos \theta _i \cos \theta _j  - \delta
391   _{ij}  >.
# Line 376 | Line 393 | shown are the experimental data of Tiburu. The fact th
393  
394   Fig.~\ref{lipidFigure:Scd} shows the order parameter profile
395   calculated for our coarse-grained DMPC bilayer system at 300K. Also
396 < shown are the experimental data of Tiburu. The fact that
396 > shown are the experimental data of Tu\cite{Tu1995}. The fact that
397   $\text{S}_{\text{{\sc cd}}}$ order parameters calculated from
398   simulation are higher than the experimental ones is ascribed to the
399   assumption of the locations of implicit hydrogen atoms which are
# Line 388 | Line 405 | of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grain
405   \includegraphics[width=\linewidth]{scd.eps}
406   \caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison
407   of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model
408 < (blue) and DMPC\cite{petrache00} (black) near 300~K.}
408 > (blue) and DMPC\cite{Petrache2000} (black) near 300~K.}
409   \label{lipidFigure:Scd}
410   \end{figure}
411  
412   %\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model}
413 +
414 + \section{\label{lipidSection:Conclusion}Conclusion}
415 +
416 + Atomistic simulations have been used in this study to determine the
417 + preferred orientation and location of water molecules relative to
418 + the location and orientation of the PC and PE lipid headgroups.
419 + Based on the results from our all-atom simulations, we developed a
420 + simple coarse-grained model which captures the essential features of
421 + the headgroup solvation which is crucial to transport process in
422 + membrane system. In addition, the model has been explored in a
423 + bilayer system was shown to have reasonable electron density
424 + profiles, $\text{S}_{\text{{\sc cd}}}$ order parameter and other
425 + structural properties. The accuracy of this model is achieved by
426 + matching atomistic result. It is also easy to represent different
427 + phospholipids by changing a few parameters of the model. Another
428 + important characteristic of this model distinguishing itself from
429 + other models\cite{Goetz1998,Marrink2004} is the computational speed
430 + gained by introducing a short range electrostatic approximation.
431 + Further studies of this system using z-constraint method could
432 + extend the time length of the simulations to study transport
433 + phenomena in large-scale membrane systems.

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