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# Line 11 | Line 11 | ns\cite{Ho1992}. In the second solvation shell, some w
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\cite{Ho1992}. In the second solvation shell, some water molecules
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
# Line 26 | Line 26 | to be exposed to the aqueous environment\cite{Sasaki20
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
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
# Line 37 | Line 37 | region\cite{Baber1995}. However, infrared spectroscopy
37   their effects on lipid bilayers still continues. Recent deuterium
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
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}.
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,
46   thermodynamic and dynamical information. Unfortunately, much of
47   biological interest happens on time and length scales well beyond
48   the range of current simulation technologies. Several schemes are
49 < proposed in this chapter to overcome these difficulties.
49 > introduced in this chapter to overcome these difficulties.
50  
51   \section{\label{lipidSection:model}Model and Methodology}
52  
# Line 62 | Line 62 | interaction between two water molecules:
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 < \[
65 > \begin{equation}
66   V_{SSD}  = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j )
67   + V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j )
68 < \]
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 < \begin{eqnarray*}
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}
# Line 80 | Line 82 | V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) &=& v_0 [s(
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*}
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 and $w$ 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 the dipole-dipole and sticky interactions are more
98 < mathematically complicated than Coulomb interactions, the number of
99 < pair interactions is reduced dramatically both because the model
100 < only contains a single-point as well as "short range" nature of the
101 < more expensive interaction.
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 of molecule $j$ in the body-fixed
98 > frame of molecule $i$. Although the dipole-dipole and sticky
99 > interactions are more mathematically complicated than Coulomb
100 > interactions, the number of pair interactions is reduced
101 > dramatically both because the model only contains a single-point and
102 > because of the "short range" nature of the more expensive
103 > interaction.
104  
105   \subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model}
106  
# Line 107 | Line 111 | $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
111   glycerol motif are modeled by Lennard-Jones spheres with dipoles.
112   Alkyl groups in hydrocarbon chains are replaced with unified
113   $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
110 –
114   \begin{figure}
115   \centering
116   \includegraphics[width=3in]{coarse_grained.eps}
# Line 149 | Line 152 | electrostatic potential at point $P$ is given by:
152   \theta } }} + \frac{q}{{\sqrt {r^2  + \frac{{d^2 }}{4} - rd\cos
153   \theta } }}} \right)
154   \]
152 –
155   \begin{figure}
156   \centering
157   \includegraphics[width=3in]{charge_dipole.eps}
# Line 158 | Line 160 | comprising two point charges with opposite charges. }
160   comprising two point charges with opposite charges. }
161   \label{lipidFigure:chargeDipole}
162   \end{figure}
161 –
163   The basic assumption of the multipole expansion is $r \gg d$ , thus,
164   $\frac{{d^2 }}{4}$ inside the square root of the denominator is
165   neglected. This is a reasonable approximation in most cases.
166   Unfortunately, in our headgroup model, the distance of charge
167 < separation $d$ (4.63 \AA  in PC headgroup) may be comparable to $r$.
167 > separation $d$ (4.63 $\rm{\AA}$ in PC headgroup) may be comparable to $r$.
168   Nevertheless, we could still assume  $ \cos \theta  \approx 0$ in
169   the central region of the headgroup. Using Taylor expansion and
170   associating appropriate terms with electric moments will result in a
# Line 195 | Line 196 | dipole-dipole approximation, we discover that water mo
196   another. The comparision between different electrostatic
197   approximation is shown in \ref{lipidFigure:splitDipole}. Due to the
198   divergence at the central region of the headgroup introduced by
199 < dipole-dipole approximation, we discover that water molecules are
200 < locked into the central region in the simulation. This artifact can
201 < be corrected using split-dipole approximation or other accurate
199 > dipole-dipole approximation, we have discovered that water molecules
200 > are locked into the central region in the simulation. This artifact
201 > can be corrected using split-dipole approximation or other accurate
202   methods.
203   \begin{figure}
204   \centering
# Line 209 | Line 210 | split-dipole approximation.} \label{lipidFigure:splitD
210   split-dipole approximation.} \label{lipidFigure:splitDipole}
211   \end{figure}
212  
212 – %\section{\label{lipidSection:methods}Methods}
213 –
213   \section{\label{lipidSection:resultDiscussion}Results and Discussion}
214  
215   \subsection{One Lipid in Sea of Water Molecules}
# Line 222 | Line 221 | density.
221   distribution functions, which give the probability of finding a pair
222   of molecular species a distance apart, relative to the probability
223   expected for a completely random distribution function at the same
224 < density.
225 <
226 < \begin{equation}
227 < g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i
228 < \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} }  >
229 < \end{equation}
231 < \begin{equation}
232 < g_{AB} (r,\cos \theta ) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
224 > density
225 > \begin{eqnarray}
226 > g_{AB} (r) & = & \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
227 > \sum\limits_{i
228 > \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} }  >, \\
229 > g_{AB} (r,\cos \theta ) & = & \frac{1}{{\rho _B }}\frac{1}{{N_A }} <
230   \sum\limits_{i \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )}
231 < } \delta (\cos \theta _{ij}  - \cos \theta ) >
232 < \end{equation}
236 <
231 > } \delta (\cos \theta _{ij}  - \cos \theta ) >.
232 > \end{eqnarray}
233   From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first
234 < solvation shell (3.5 \AA) and the second solvation shell (5.0 \AA)
234 > solvation shell (3.5 $\rm{\AA}$) and the second solvation shell (5.0 \AA)
235   from both plots. However, the corresponding orientations are
236   different. In DLPE, water molecules prefer to sit around $\text{{\sc
237   NH}}_3$ group due to the hydrogen bonding. In contrast, because of
# Line 246 | Line 242 | anti-aligned with the dipole of headgroup. When the wa
242   angles should be uniform. The correlation close to center of the
243   headgroup dipole in both plots tells us that in the closely-bound
244   region, the dipoles of the water molecules are preferentially
245 < anti-aligned with the dipole of headgroup. When the water molecules
245 > anti-aligned with the dipole of the headgroup. When the water molecules
246   are far from the headgroup, the distribution of the two angles
247   should be uniform. The correlation close to center of the headgroup
248   dipole in both plots tell us that in the closely-bound region, the
249   dipoles of the water molecules are preferentially aligned with the
250 < dipole of headgroup.
255 <
250 > dipole of the headgroup.
251   \begin{figure}
252   \centering
253   \includegraphics[width=\linewidth]{g_atom.eps}
# Line 275 | Line 270 | atoms.
270   Fig.~\ref{lipidFigure:PCFCoarse}. It is clear that the phosphate end
271   in DMPC and the amine end in DMPE are the two most heavily solvated
272   atoms.
278 –
273   \begin{figure}
274   \centering
275   \includegraphics[width=\linewidth]{g_coarse.eps}
# Line 284 | Line 278 | models]{The pair correlation functions for coarse-grai
278   (a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.}
279   \label{lipidFigure:PCFCoarse}
280   \end{figure}
287 –
281   \begin{figure}
282   \centering
283   \includegraphics[width=\linewidth]{EWD_coarse.eps}
# Line 294 | Line 287 | phospholipids.} \label{lipidFigure:EWDCoarse}
287   \end{figure}
288  
289   \begin{table}
290 < \caption{The Parameters For Coarse-grained Phospholipids}
290 > \caption{THE PARAMETERS FOR COARSE-GRAINED PHOSPHOLIPIDS}
291   \label{lipidTable:parameter}
292   \begin{center}
293 < \begin{tabular}{|l|c|c|c|c|c|}
293 > \begin{tabular}{lccccc}
294    \hline
295    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
296    Atom type & Mass & $\sigma$ & $\epsilon$ & charge & Dipole \\
297 +  \hline
298    $\text{{\sc CH}}_2$ & 14.03  & 3.95 & 0.0914 & 0 & 0 \\
299    $\text{{\sc CH}}_3$ & 15.04  & 3.75 & 0.195  & 0 & 0 \\
300    $\text{{\sc CE}}$   & 28.01  & 3.427& 0.294  & 0 & 1.693 \\
# Line 326 | Line 320 | different properties were evaluated over a production
320   was relaxed by performing under NPT conditions using Nos\'{e}-Hoover
321   extended system isothermal-isobaric dynamics. After equilibration,
322   different properties were evaluated over a production run of 5 ns.
329 –
323   \begin{figure}
324   \centering
325   \includegraphics[width=\linewidth]{bilayer.eps}
# Line 345 | Line 338 | can be expressed by\cite{Tu1995}
338   can be expressed by\cite{Tu1995}
339   \begin{equation}
340   \rho _{x - ray} (z)dz \propto \sum\limits_{i = 1}^N {\frac{{n_i
341 < }}{V}\frac{1}{{\sqrt {2\pi \sigma ^2 } }}e^{ - (z - z_i )^2 /2\sigma
342 < ^2 } dz},
341 > }}{V}\frac{1}{{\sqrt {2\pi \sigma _i ^2 } }}e^{ - (z - z_i )^2
342 > /2\sigma _i ^2 } dz},
343   \end{equation}
344 < where $\sigma$ is the variance equal to the van der Waals radius,
344 > where $\sigma _i$ is the variance equal to the van der Waals radius,
345   $n_i$ is the atomic number of site $i$ and $V$ is the volume of the
346   slab between $z$ and $z+dz$ . The highest density of total EDP
347   appears at the position of lipid-water interface corresponding to
# Line 356 | Line 349 | approximation to the thickness of the bilayer, the hea
349   distribution of water locked near the head groups, while the lowest
350   electron density is in the hydrocarbon region. As a good
351   approximation to the thickness of the bilayer, the headgroup spacing
352 < , is defined as the distance between two peaks in the electron
353 < density profile, calculated from our simulations to be 34.1 \AA.
354 < This value is close to the x-ray diffraction experimental value 34.4
355 < \AA\cite{Petrache1998}.
352 > $d$ , is defined as the distance between two peaks in the electron
353 > density profile, calculated from our simulations to be 34.1
354 > $\rm{\AA}$. This value is close to the x-ray diffraction
355 > experimental value 34.4 $\rm{\AA}$.\cite{Petrache1998}
356  
357   \begin{figure}
358   \centering
# Line 381 | Line 374 | where $\theta$ is the angle between the  $i$th molecul
374   S_{ij}  = \frac{1}{2} < 3\cos \theta _i \cos \theta _j  - \delta
375   _{ij}  >
376   \end{equation}
377 < where $\theta$ is the angle between the  $i$th molecular axis and
377 > where $\theta _i$ is the angle between the  $i$th molecular axis and
378   the bilayer normal ($z$ axis). The brackets denote an average over
379 < time and molecules. The molecular axes are defined:
379 > time and lipid molecules. The molecular axes are defined:
380   \begin{itemize}
381   \item $\mathbf{\hat{z}}$ is the vector from $C_{n-1}$ to $C_{n+1}$.
382   \item $\mathbf{\hat{y}}$ is the vector that is perpendicular to $z$ and
# Line 391 | Line 384 | In coarse-grained model, although there are no explici
384   \item $\mathbf{\hat{x}}$ is the vector perpendicular to
385   $\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$.
386   \end{itemize}
387 < In coarse-grained model, although there are no explicit hydrogens,
388 < the order parameter can still be written in terms of carbon ordering
389 < at each point of the chain\cite{Egberts1988}
387 > In our coarse-grained model, although there are no explicit
388 > hydrogens, the order parameter can still be written in terms of
389 > carbon ordering at each point of the chain\cite{Egberts1988}
390   \begin{equation}
391   S_{ij}  = \frac{1}{2} < 3\cos \theta _i \cos \theta _j  - \delta
392   _{ij}  >.
393   \end{equation}
394  
395   Fig.~\ref{lipidFigure:Scd} shows the order parameter profile
396 < calculated for our coarse-grained DMPC bilayer system at 300K. Also
397 < shown are the experimental data of Tu\cite{Tu1995}. The fact that
396 > calculated for our coarse-grained DMPC bilayer system at 300K as
397 > well as the experimental data.\cite{Tu1995} The fact that
398   $\text{S}_{\text{{\sc cd}}}$ order parameters calculated from
399   simulation are higher than the experimental ones is ascribed to the
400   assumption of the locations of implicit hydrogen atoms which are

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