1 |
\chapter{\label{chapt:lipid}LIPID MODELING} |
2 |
|
3 |
\section{\label{lipidSection:introduction}Introduction} |
4 |
|
5 |
Under biologically relevant conditions, phospholipids are solvated |
6 |
in aqueous solutions at a roughly 25:1 ratio. Solvation can have a |
7 |
tremendous impact on transport phenomena in biological membranes |
8 |
since it can affect the dynamics of ions and molecules that are |
9 |
transferred across membranes. Studies suggest that because of the |
10 |
directional hydrogen bonding ability of the lipid headgroups, a |
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 |
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 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, |
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 |
introduced in this chapter to overcome these difficulties. |
50 |
|
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\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 |
\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. 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 and $w$ and $w'$ are responsible for the |
89 |
tetrahedral potential and the short-range correction to the dipolar |
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 |
|
107 |
Fig.~\ref{lipidFigure:coarseGrained} shows a schematic for our |
108 |
coarse-grained phospholipid model. The lipid head group is modeled |
109 |
by a linear rigid body which consists of three Lennard-Jones spheres |
110 |
and a centrally located point-dipole. The backbone atoms in the |
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. |
114 |
\begin{figure} |
115 |
\centering |
116 |
\includegraphics[width=3in]{coarse_grained.eps} |
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 C atom (gray) with a |
121 |
dipole, while the glycerol backbone includes dipolar $\text{{\sc |
122 |
CE}}$ (red) and $\text{{\sc CK}}$ (pink) groups. Alkyl groups in |
123 |
hydrocarbon chains are simply represented by gray united atoms.} |
124 |
\label{lipidFigure:coarseGrained} |
125 |
\end{figure} |
126 |
|
127 |
Accurate and efficient computation of electrostatics is one of the |
128 |
most difficult tasks in molecular modeling. Traditionally, the |
129 |
electrostatic interaction between two molecular species is |
130 |
calculated as a sum of interactions between pairs of point charges, |
131 |
using Coulomb's law: |
132 |
\[ |
133 |
V = \sum\limits_{i = 1}^{N_A } {\sum\limits_{j = 1}^{N_B } |
134 |
{\frac{{q_i q_j }}{{4\pi \varepsilon _0 r_{ij} }}} } |
135 |
\] |
136 |
where $N_A$ and $N_B$ are the number of point charges in the two |
137 |
molecular species. Originally developed to study ionic crystals, the |
138 |
Ewald sum method mathematically transforms this straightforward but |
139 |
conditionally convergent summation into two more complicated but |
140 |
rapidly convergent sums. One summation is carried out in reciprocal |
141 |
space while the other is carried out in real space. An alternative |
142 |
approach is the multipole expansion, which is based on electrostatic |
143 |
moments, such as charge (monopole), dipole, quadrupole etc. |
144 |
|
145 |
Here we consider a linear molecule which consists of two point |
146 |
charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The |
147 |
electrostatic potential at point $P$ is given by: |
148 |
\[ |
149 |
\frac{1}{{4\pi \varepsilon _0 }}\left( {\frac{{ - q}}{{r_ - }} + |
150 |
\frac{{ + q}}{{r_ + }}} \right) = \frac{1}{{4\pi \varepsilon _0 |
151 |
}}\left( {\frac{{ - q}}{{\sqrt {r^2 + \frac{{d^2 }}{4} + rd\cos |
152 |
\theta } }} + \frac{q}{{\sqrt {r^2 + \frac{{d^2 }}{4} - rd\cos |
153 |
\theta } }}} \right) |
154 |
\] |
155 |
\begin{figure} |
156 |
\centering |
157 |
\includegraphics[width=3in]{charge_dipole.eps} |
158 |
\caption[An illustration of split-dipole |
159 |
approximation]{Electrostatic potential due to a linear molecule |
160 |
comprising two point charges with opposite charges. } |
161 |
\label{lipidFigure:chargeDipole} |
162 |
\end{figure} |
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 $r$. |
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 |
171 |
"split-dipole" approximation: |
172 |
\[ |
173 |
V(r) = \frac{1}{{4\pi \varepsilon _0 }}\frac{{r\mu \cos \theta |
174 |
}}{{R^3 }} |
175 |
\] |
176 |
where$R = \sqrt {r^2 + \frac{{d^2 }}{4}}$ Placing a dipole at point |
177 |
$P$ and applying the same strategy, the interaction between two |
178 |
split-dipoles is then given by: |
179 |
\[ |
180 |
V_{sd} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
181 |
_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{R_{ij}^3 }} - |
182 |
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
183 |
r_{ij} } \right)}}{{R_{ij}^5 }}} \right] |
184 |
\] |
185 |
where $\mu _i$ and $\mu _j$ are the dipole moment of molecule $i$ |
186 |
and molecule $j$ respectively, $r_{ij}$ is vector between molecule |
187 |
$i$ and molecule $j$, and $R_{ij}$ is given by, |
188 |
\[ |
189 |
R_{ij} = \sqrt {r_{ij}^2 + \frac{{d_i^2 }}{4} + \frac{{d_j^2 |
190 |
}}{4}} |
191 |
\] |
192 |
where $d_i$ and $d_j$ are the charge separation distance of dipole |
193 |
and respectively. This approximation to the multipole expansion |
194 |
maintains the fast fall-off of the multipole potentials but lacks |
195 |
the normal divergences when two polar groups get close to one |
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 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 |
205 |
\includegraphics[width=\linewidth]{split_dipole.eps} |
206 |
\caption[Comparison between electrostatic |
207 |
approximation]{Electrostatic potential map for two pairs of charges |
208 |
with different alignments: (a) illustration of different alignments; |
209 |
(b) charge-charge interaction; (c) dipole-dipole approximation; (d) |
210 |
split-dipole approximation.} \label{lipidFigure:splitDipole} |
211 |
\end{figure} |
212 |
|
213 |
\section{\label{lipidSection:resultDiscussion}Results and Discussion} |
214 |
|
215 |
\subsection{One Lipid in Sea of Water Molecules} |
216 |
|
217 |
To tune our parameters without the inter-headgroup interactions, |
218 |
atomistic models of one lipid (DMPC or DLPE) in sea of water |
219 |
molecules (TIP3P) were built and studied using atomistic molecular |
220 |
dynamics. The simulation was analyzed using a set of radial |
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 |
\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{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) |
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 |
238 |
the hydrophobic effect of the $ {\rm{N(CH}}_{\rm{3}} |
239 |
{\rm{)}}_{\rm{3}} $ group, the preferred position of water molecules |
240 |
in DMPC is around the $\text{{\sc PO}}_4$ Group. When the water |
241 |
molecules are far from the headgroup, the distribution of the two |
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 |
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. |
251 |
\begin{figure} |
252 |
\centering |
253 |
\includegraphics[width=\linewidth]{g_atom.eps} |
254 |
\caption[The pair correlation functions for atomistic models]{The |
255 |
pair correlation functions for atomistic models: (a)$g(r,\cos \theta |
256 |
)$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE; (c)$g(r,\cos \omega |
257 |
)$ for DMPC; (d)$g(r,\cos \omega )$ for DLPE; (e)$g(\cos \theta,\cos |
258 |
\omega)$ for DMPC; (f)$g(\cos \theta,\cos \omega)$ for DMLPE.} |
259 |
\label{lipidFigure:PCFAtom} |
260 |
\end{figure} |
261 |
|
262 |
The initial configurations of coarse-grained systems are constructed |
263 |
from the previous atomistic ones. The parameters for the |
264 |
coarse-grained model in Table~\ref{lipidTable:parameter} are |
265 |
estimated and tuned using isothermal-isobaric molecular dynamics. |
266 |
Pair distribution functions calculated from coarse-grained models |
267 |
preserve the basic characteristics of the atomistic simulations. The |
268 |
water density, measured in a head-group-fixed reference frame, |
269 |
surrounding two phospholipid headgroups is shown in |
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. |
273 |
\begin{figure} |
274 |
\centering |
275 |
\includegraphics[width=\linewidth]{g_coarse.eps} |
276 |
\caption[The pair correlation functions for coarse-grained |
277 |
models]{The pair correlation functions for coarse-grained models: |
278 |
(a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.} |
279 |
\label{lipidFigure:PCFCoarse} |
280 |
\end{figure} |
281 |
\begin{figure} |
282 |
\centering |
283 |
\includegraphics[width=\linewidth]{EWD_coarse.eps} |
284 |
\caption[Excess water density of coarse-grained |
285 |
phospholipids]{Excess water density of coarse-grained |
286 |
phospholipids.} \label{lipidFigure:EWDCoarse} |
287 |
\end{figure} |
288 |
|
289 |
\begin{table} |
290 |
\caption{THE PARAMETERS FOR COARSE-GRAINED PHOSPHOLIPIDS} |
291 |
\label{lipidTable:parameter} |
292 |
\begin{center} |
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 \\ |
301 |
$\text{{\sc CK}}$ & 28.01 & 3.592& 0.311 & 0 & 2.478 \\ |
302 |
$\text{{\sc PO}}_4$ & 108.995& 3.9 & 1.88 & -1& 0 \\ |
303 |
$\text{{\sc HDP}}$ & 14.03 & 4.0 & 0.13 & 0 & 0 \\ |
304 |
$\text{{\sc NC}}_4$ & 73.137 & 4.9 & 0.88 & +1& 0 \\ |
305 |
$\text{{\sc NH}}_3$ & 17.03 & 3.25 & 0.17 & +1& 0\\ |
306 |
\hline |
307 |
\end{tabular} |
308 |
\end{center} |
309 |
\end{table} |
310 |
|
311 |
\subsection{Bilayer Simulations Using Coarse-grained Model} |
312 |
|
313 |
A bilayer system consisting of 128 DMPC lipids and 3655 water |
314 |
molecules has been constructed from an atomistic coordinate file. |
315 |
The MD simulation is performed at constant temperature, T = 300K, |
316 |
and constant pressure, p = 1 atm, and consisted of an equilibration |
317 |
period of 2 ns. During the equilibration period, the system was |
318 |
initially simulated at constant volume for 1 ns. Once the system was |
319 |
equilibrated at constant volume, the cell dimensions of the system |
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. |
323 |
\begin{figure} |
324 |
\centering |
325 |
\includegraphics[width=\linewidth]{bilayer.eps} |
326 |
\caption[Image of a coarse-grained bilayer system]{A coarse-grained |
327 |
bilayer system consisting of 128 DMPC lipids and 3655 SSD water |
328 |
molecules.} |
329 |
\label{lipidFigure:bilayer} |
330 |
\end{figure} |
331 |
|
332 |
\subsubsection{\textbf{Electron Density Profile (EDP)}} |
333 |
|
334 |
Assuming a gaussian distribution of electrons on each atomic center |
335 |
with a variance estimated from the size of the van der Waals radius, |
336 |
the EDPs which are proportional to the density profiles measured |
337 |
along the bilayer normal obtained by x-ray scattering experiment, |
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 _i ^2 } }}e^{ - (z - z_i )^2 |
342 |
/2\sigma _i ^2 } dz}, |
343 |
\end{equation} |
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 |
348 |
headgroup, glycerol, and carbonyl groups of the lipids and the |
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}. |
356 |
|
357 |
\begin{figure} |
358 |
\centering |
359 |
\includegraphics[width=\linewidth]{electron_density.eps} |
360 |
\caption[The density profile of the lipid bilayers]{Electron density |
361 |
profile along the bilayer normal. The water density is shown in red, |
362 |
the density due to the headgroups in green, the glycerol backbone in |
363 |
brown, $\text{{\sc CH}}_2$ in yellow, $\text{{\sc CH}}_3$ in cyan, |
364 |
and total density due to DMPC in blue.} |
365 |
\label{lipidFigure:electronDensity} |
366 |
\end{figure} |
367 |
|
368 |
\subsubsection{\textbf{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}} |
369 |
|
370 |
Measuring deuterium order parameters by NMR is a useful technique to |
371 |
study the orientation of hydrocarbon chains in phospholipids. The |
372 |
order parameter tensor $S$ is defined by: |
373 |
\begin{equation} |
374 |
S_{ij} = \frac{1}{2} < 3\cos \theta _i \cos \theta _j - \delta |
375 |
_{ij} > |
376 |
\end{equation} |
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 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 |
383 |
in the plane through $C_{n-1}$, $C_{n}$, and $C_{n+1}$. |
384 |
\item $\mathbf{\hat{x}}$ is the vector perpendicular to |
385 |
$\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$. |
386 |
\end{itemize} |
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 |
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 |
401 |
fixed in coarse-grained models at positions relative to the CC |
402 |
vector. |
403 |
|
404 |
\begin{figure} |
405 |
\centering |
406 |
\includegraphics[width=\linewidth]{scd.eps} |
407 |
\caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison |
408 |
of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model |
409 |
(blue) and DMPC\cite{Petrache2000} (black) near 300~K.} |
410 |
\label{lipidFigure:Scd} |
411 |
\end{figure} |
412 |
|
413 |
%\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model} |
414 |
|
415 |
\section{\label{lipidSection:Conclusion}Conclusion} |
416 |
|
417 |
Atomistic simulations have been used in this study to determine the |
418 |
preferred orientation and location of water molecules relative to |
419 |
the location and orientation of the PC and PE lipid headgroups. |
420 |
Based on the results from our all-atom simulations, we developed a |
421 |
simple coarse-grained model which captures the essential features of |
422 |
the headgroup solvation which is crucial to transport process in |
423 |
membrane system. In addition, the model has been explored in a |
424 |
bilayer system was shown to have reasonable electron density |
425 |
profiles, $\text{S}_{\text{{\sc cd}}}$ order parameter and other |
426 |
structural properties. The accuracy of this model is achieved by |
427 |
matching atomistic result. It is also easy to represent different |
428 |
phospholipids by changing a few parameters of the model. Another |
429 |
important characteristic of this model distinguishing itself from |
430 |
other models\cite{Goetz1998,Marrink2004} is the computational speed |
431 |
gained by introducing a short range electrostatic approximation. |
432 |
Further studies of this system using z-constraint method could |
433 |
extend the time length of the simulations to study transport |
434 |
phenomena in large-scale membrane systems. |