| 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. |
| 49 |
< |
%review of coarse-grained modeling |
| 50 |
< |
Several schemes are proposed in this chapter to overcome these |
| 51 |
< |
difficulties. |
| 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 |
|
|
| 114 |
|
\begin{figure} |
| 115 |
|
\centering |
| 116 |
|
\includegraphics[width=3in]{coarse_grained.eps} |
| 117 |
< |
\caption[A representation of coarse-grained phospholipid model]{} |
| 118 |
< |
\label{lipidFigure:coarseGrained} |
| 117 |
> |
\caption[A representation of coarse-grained phospholipid model]{A |
| 118 |
> |
representation of coarse-grained phospholipid model. The lipid |
| 119 |
> |
headgroup consists of $\text{{\sc PO}}_4$ group (yellow), |
| 120 |
> |
$\text{{\sc NC}}_4$ group (blue) and a united $\text{{\sc C}} atom |
| 121 |
> |
(gray) $ with a dipole, while the glycerol backbone includes dipolar |
| 122 |
> |
$\text{{\sc CE}}$ (read) and $\text{{\sc CK}}$ (pink) groups. Alkyl |
| 123 |
> |
groups in hydrocarbon chains are simply represented by gray united |
| 124 |
> |
atoms.} \label{lipidFigure:coarseGrained} |
| 125 |
|
\end{figure} |
| 126 |
|
|
| 127 |
|
Accurate and efficient computation of electrostatics is one of the |
| 157 |
|
\begin{figure} |
| 158 |
|
\centering |
| 159 |
|
\includegraphics[width=3in]{charge_dipole.eps} |
| 160 |
< |
\caption[Electrostatic potential due to a linear molecule comprising |
| 161 |
< |
two point charges]{Electrostatic potential due to a linear molecule |
| 162 |
< |
comprising two point charges} \label{lipidFigure:chargeDipole} |
| 160 |
> |
\caption[An illustration of split-dipole |
| 161 |
> |
approximation]{Electrostatic potential due to a linear molecule |
| 162 |
> |
comprising two point charges with opposite charges. } |
| 163 |
> |
\label{lipidFigure:chargeDipole} |
| 164 |
|
\end{figure} |
| 165 |
|
|
| 166 |
|
The basic assumption of the multipole expansion is $r \gg d$ , thus, |
| 196 |
|
and respectively. This approximation to the multipole expansion |
| 197 |
|
maintains the fast fall-off of the multipole potentials but lacks |
| 198 |
|
the normal divergences when two polar groups get close to one |
| 199 |
< |
another. |
| 200 |
< |
%description of the comparsion |
| 199 |
> |
another. The comparision between different electrostatic |
| 200 |
> |
approximation is shown in \ref{lipidFigure:splitDipole}. Due to the |
| 201 |
> |
divergence at the central region of the headgroup introduced by |
| 202 |
> |
dipole-dipole approximation, we discover that water molecules are |
| 203 |
> |
locked into the central region in the simulation. This artifact can |
| 204 |
> |
be corrected using split-dipole approximation or other accurate |
| 205 |
> |
methods. |
| 206 |
|
\begin{figure} |
| 207 |
|
\centering |
| 208 |
|
\includegraphics[width=\linewidth]{split_dipole.eps} |
| 209 |
< |
\caption[Comparison between electrostatic approximation]{Electron |
| 210 |
< |
density profile along the bilayer normal.} |
| 211 |
< |
\label{lipidFigure:splitDipole} |
| 209 |
> |
\caption[Comparison between electrostatic |
| 210 |
> |
approximation]{Electrostatic potential map for two pairs of charges |
| 211 |
> |
with different alignments: (a) illustration of different alignments; |
| 212 |
> |
(b) charge-charge interaction; (c) dipole-dipole approximation; (d) |
| 213 |
> |
split-dipole approximation.} \label{lipidFigure:splitDipole} |
| 214 |
|
\end{figure} |
| 215 |
|
|
| 216 |
|
%\section{\label{lipidSection:methods}Methods} |
| 237 |
|
} \delta (\cos \theta _{ij} - \cos \theta ) > |
| 238 |
|
\end{equation} |
| 239 |
|
|
| 240 |
< |
From figure 4(a), we can identify the first solvation shell (3.5 |
| 241 |
< |
$\AA$) and the second solvation shell (5.0 $\AA$) from both plots. |
| 242 |
< |
However, the corresponding orientations are different. In DLPE, |
| 243 |
< |
water molecules prefer to sit around -NH3 group due to the hydrogen |
| 244 |
< |
bonding. In contrast, because of the hydrophobic effect of the |
| 245 |
< |
-N(CH3)3 group, the preferred position of water molecules in DMPC is |
| 246 |
< |
around the -PO4 Group. When the water molecules are far from the |
| 247 |
< |
headgroup, the distribution of the two angles should be uniform. The |
| 248 |
< |
correlation close to center of the headgroup dipole (< 5 $\AA$) in |
| 249 |
< |
both plots tell us that in the closely-bound region, the dipoles of |
| 250 |
< |
the water molecules are preferentially anti-aligned with the dipole |
| 251 |
< |
of headgroup. |
| 240 |
> |
From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first |
| 241 |
> |
solvation shell (3.5 $\AA$) and the second solvation shell (5.0 |
| 242 |
> |
$\AA$) from both plots. However, the corresponding orientations are |
| 243 |
> |
different. In DLPE, water molecules prefer to sit around $\text{{\sc |
| 244 |
> |
NH}}_3$ group due to the hydrogen bonding. In contrast, because of |
| 245 |
> |
the hydrophobic effect of the $ {\rm{N(CH}}_{\rm{3}} |
| 246 |
> |
{\rm{)}}_{\rm{3}} $ group, the preferred position of water molecules |
| 247 |
> |
in DMPC is around the $\text{{\sc PO}}_4$ Group. When the water |
| 248 |
> |
molecules are far from the headgroup, the distribution of the two |
| 249 |
> |
angles should be uniform. The correlation close to center of the |
| 250 |
> |
headgroup dipole in both plots tell us that in the closely-bound |
| 251 |
> |
region, the dipoles of the water molecules are preferentially |
| 252 |
> |
anti-aligned with the dipole of headgroup. When the water molecules |
| 253 |
> |
are far from the headgroup, the distribution of the two angles |
| 254 |
> |
should be uniform. The correlation close to center of the headgroup |
| 255 |
> |
dipole in both plots tell us that in the closely-bound region, the |
| 256 |
> |
dipoles of the water molecules are preferentially aligned with the |
| 257 |
> |
dipole of headgroup. |
| 258 |
|
|
| 259 |
|
\begin{figure} |
| 260 |
|
\centering |
| 261 |
|
\includegraphics[width=\linewidth]{g_atom.eps} |
| 262 |
< |
\caption[The pair correlation functions for atomistic models]{} |
| 262 |
> |
\caption[The pair correlation functions for atomistic models]{The |
| 263 |
> |
pair correlation functions for atomistic models: (a)$g(r,\cos \theta |
| 264 |
> |
)$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE; (c)$g(r,\cos \omega |
| 265 |
> |
)$ for DMPC; (d)$g(r,\cos \omega )$ for DLPE; (e)$g(\cos \theta,\cos |
| 266 |
> |
\omega)$ for DMPC; (f)$g(\cos \theta,\cos \omega)$ for DMLPE.} |
| 267 |
|
\label{lipidFigure:PCFAtom} |
| 268 |
|
\end{figure} |
| 269 |
|
|
| 282 |
|
\begin{figure} |
| 283 |
|
\centering |
| 284 |
|
\includegraphics[width=\linewidth]{g_coarse.eps} |
| 285 |
< |
\caption[The pair correlation functions for coarse-grained models]{} |
| 285 |
> |
\caption[The pair correlation functions for coarse-grained |
| 286 |
> |
models]{The pair correlation functions for coarse-grained models: |
| 287 |
> |
(a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.} |
| 288 |
|
\label{lipidFigure:PCFCoarse} |
| 289 |
|
\end{figure} |
| 290 |
|
|
| 319 |
|
\subsection{Bilayer Simulations Using Coarse-grained Model} |
| 320 |
|
|
| 321 |
|
A bilayer system consisting of 128 DMPC lipids and 3655 water |
| 322 |
< |
molecules has been constructed from an atomistic coordinate |
| 323 |
< |
file.[15] The MD simulation is performed at constant temperature, T |
| 324 |
< |
= 300K, and constant pressure, p = 1 atm, and consisted of an |
| 325 |
< |
equilibration period of 2 ns. During the equilibration period, the |
| 326 |
< |
system was initially simulated at constant volume for 1ns. Once the |
| 327 |
< |
system was equilibrated at constant volume, the cell dimensions of |
| 328 |
< |
the system was relaxed by performing under NPT conditions using |
| 329 |
< |
Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After |
| 330 |
< |
equilibration, different properties were evaluated over a production |
| 307 |
< |
run of 5 ns. |
| 322 |
> |
molecules has been constructed from an atomistic coordinate file. |
| 323 |
> |
The MD simulation is performed at constant temperature, T = 300K, |
| 324 |
> |
and constant pressure, p = 1 atm, and consisted of an equilibration |
| 325 |
> |
period of 2 ns. During the equilibration period, the system was |
| 326 |
> |
initially simulated at constant volume for 1 ns. Once the system was |
| 327 |
> |
equilibrated at constant volume, the cell dimensions of the system |
| 328 |
> |
was relaxed by performing under NPT conditions using Nos¨¦-Hoover |
| 329 |
> |
extended system isothermal-isobaric dynamics. After equilibration, |
| 330 |
> |
different properties were evaluated over a production run of 5 ns. |
| 331 |
|
|
| 332 |
|
\begin{figure} |
| 333 |
|
\centering |
| 420 |
|
\end{figure} |
| 421 |
|
|
| 422 |
|
%\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model} |
| 423 |
+ |
|
| 424 |
+ |
\section{\label{lipidSection:Conclusion}Conclusion} |
| 425 |
+ |
|
| 426 |
+ |
Atomistic simulations are used in this study to determine the |
| 427 |
+ |
preferred orientation and location of water molecules relative to |
| 428 |
+ |
the location and orientation of the PC and PE lipid headgroups. |
| 429 |
+ |
Based on the result from all-atom simulations, we developed a simple |
| 430 |
+ |
coarse-grained model capturing essential features of the headgroup |
| 431 |
+ |
solvation which is crucial to transport process in membrane system. |
| 432 |
+ |
In addition, the model has been explored in a bilayer system which |
| 433 |
+ |
is shown to have reasonable electron density profile, |
| 434 |
+ |
$\text{S}_{\text{{\sc cd}}}$ order parameter and other structural |
| 435 |
+ |
properties. The accuracy of this model is achieved by matching |
| 436 |
+ |
atomistic result. It is also easy to represent different |
| 437 |
+ |
phosphorlipids by changing a few parameters of the model. Another |
| 438 |
+ |
important characteristic of this model distinguishing itself from |
| 439 |
+ |
other models\cite{Goetz1998,Marrink2004} is the computational speed |
| 440 |
+ |
gaining by introducing short range electrostatic approximation. |
| 441 |
+ |
Further studies of this system using z-constraint method could |
| 442 |
+ |
extend the time length of the simulations to study transport |
| 443 |
+ |
phenomena in large-scale membrane systems. |
