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. |
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%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} |
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< |
\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 |
|
|
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|
Accurate and efficient computation of electrostatics is one of the |
157 |
|
\begin{figure} |
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|
\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. |