1 |
tim |
2685 |
\chapter{\label{chapt:lipid}LIPID MODELING} |
2 |
|
|
|
3 |
|
|
\section{\label{lipidSection:introduction}Introduction} |
4 |
|
|
|
5 |
tim |
2731 |
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 |
tim |
2776 |
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 |
36 |
|
|
their effects on lipid bilayers still continues. Recent deuterium |
37 |
|
|
NMR measurements on halothane in POPC lipid bilayers suggest the |
38 |
|
|
anesthetics are primarily located at the hydrocarbon chain region. |
39 |
|
|
Infrared spectroscopy experiments suggest that halothane in DMPC |
40 |
|
|
lipid bilayers lives near the membrane/water interface. |
41 |
tim |
2731 |
|
42 |
tim |
2776 |
Molecular dynamics simulations have proven to be a powerful tool for |
43 |
|
|
studying the functions of biological systems, providing structural, |
44 |
|
|
thermodynamic and dynamical information. Unfortunately, much of |
45 |
|
|
biological interest happens on time and length scales well beyond |
46 |
|
|
the range of current simulation technologies. Several schemes are |
47 |
|
|
proposed in this chapter to overcome these difficulties. |
48 |
tim |
2731 |
|
49 |
tim |
2685 |
\section{\label{lipidSection:model}Model} |
50 |
|
|
|
51 |
tim |
2776 |
\subsection{\label{lipidSection:SSD}The Soft Sticky Dipole Water Model} |
52 |
tim |
2685 |
|
53 |
tim |
2776 |
In a typical bilayer simulation, the dominant portion of the |
54 |
|
|
computation time will be spent calculating water-water interactions. |
55 |
|
|
As an efficient solvent model, the Soft Sticky Dipole (SSD) water |
56 |
|
|
model is used as the explicit solvent in this project. Unlike other |
57 |
|
|
water models which have partial charges distributed throughout the |
58 |
|
|
whole molecule, the SSD water model consists of a single site which |
59 |
|
|
is a Lennard-Jones interaction site, as well as a point dipole. A |
60 |
|
|
tetrahedral potential is added to correct for hydrogen bonding. The |
61 |
|
|
following equation describes the interaction between two water |
62 |
|
|
molecules: |
63 |
|
|
\[ |
64 |
|
|
V_{SSD} = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) |
65 |
|
|
+ V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) |
66 |
|
|
\] |
67 |
|
|
where $r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
68 |
|
|
$\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom |
69 |
|
|
for molecule $i$ and molecule $j$ respectively. |
70 |
|
|
\[ |
71 |
|
|
V_{LJ} (r_{ij} ) = 4\varepsilon _{ij} \left[ {\left( {\frac{{\sigma |
72 |
|
|
_{ij} }}{{r_{ij} }}} \right)^{12} - \left( {\frac{{\sigma _{ij} |
73 |
|
|
}}{{r_{ij} }}} \right)^6 } \right] |
74 |
|
|
\] |
75 |
|
|
\[ |
76 |
|
|
V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
77 |
|
|
_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }} - |
78 |
|
|
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
79 |
|
|
r_{ij} } \right)}}{{r_{ij}^5 }}} \right] |
80 |
|
|
\] |
81 |
|
|
\[ |
82 |
|
|
V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) = v_0 [s(r_{ij} )w(r_{ij} |
83 |
|
|
,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i ,\Omega _j |
84 |
|
|
)] |
85 |
|
|
\] |
86 |
|
|
where $v_0$ is a strength parameter, $s$ and $s'$ are cubic |
87 |
|
|
switching functions, while $w$ and $w'$ are responsible for the |
88 |
|
|
tetrahedral potential and the short-range correction to the dipolar |
89 |
|
|
interaction respectively. |
90 |
|
|
\[ |
91 |
|
|
\begin{array}{l} |
92 |
|
|
w(r_{ij} ,\Omega _i ,\Omega _j ) = \sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij} \\ |
93 |
|
|
w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - w_0 \\ |
94 |
|
|
\end{array} |
95 |
|
|
\] |
96 |
|
|
Although dipole-dipole and sticky interactions are more |
97 |
|
|
mathematically complicated than Coulomb interactions, the number of |
98 |
|
|
pair interactions is reduced dramatically both because the model |
99 |
|
|
only contains a single-point as well as "short range" nature of the |
100 |
|
|
higher order interaction. |
101 |
|
|
|
102 |
|
|
\subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model} |
103 |
|
|
|
104 |
|
|
Figure 1 shows a schematic for our coarse-grained phospholipid |
105 |
|
|
model. The lipid head group is modeled by a linear rigid body which |
106 |
|
|
consists of three Lennard-Jones spheres and a centrally located |
107 |
|
|
point-dipole. The backbone atoms in the glycerol motif are modeled |
108 |
|
|
by Lennard-Jones spheres with dipoles. Alkyl groups in hydrocarbon |
109 |
|
|
chains are replaced with unified CH2 or CH3 atoms. |
110 |
|
|
|
111 |
|
|
Accurate and efficient computation of electrostatics is one of the |
112 |
|
|
most difficult tasks in molecular modeling. Traditionally, the |
113 |
|
|
electrostatic interaction between two molecular species is |
114 |
|
|
calculated as a sum of interactions between pairs of point charges, |
115 |
|
|
using Coulomb's law: |
116 |
|
|
\[ |
117 |
|
|
V = \sum\limits_{i = 1}^{N_A } {\sum\limits_{j = 1}^{N_B } |
118 |
|
|
{\frac{{q_i q_j }}{{4\pi \varepsilon _0 r_{ij} }}} } |
119 |
|
|
\] |
120 |
|
|
where $N_A$ and $N_B$ are the number of point charges in the two |
121 |
|
|
molecular species. Originally developed to study ionic crystals, the |
122 |
|
|
Ewald summation method mathematically transforms this |
123 |
|
|
straightforward but conditionally convergent summation into two more |
124 |
|
|
complicated but rapidly convergent sums. One summation is carried |
125 |
|
|
out in reciprocal space while the other is carried out in real |
126 |
|
|
space. An alternative approach is a multipole expansion, which is |
127 |
|
|
based on electrostatic moments, such as charge (monopole), dipole, |
128 |
|
|
quadruple etc. |
129 |
|
|
|
130 |
|
|
Here we consider a linear molecule which consists of two point |
131 |
|
|
charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The |
132 |
|
|
electrostatic potential at point $P$ is given by: |
133 |
|
|
\[ |
134 |
|
|
\frac{1}{{4\pi \varepsilon _0 }}\left( {\frac{{ - q}}{{r_ - }} + |
135 |
|
|
\frac{{ + q}}{{r_ + }}} \right) = \frac{1}{{4\pi \varepsilon _0 |
136 |
|
|
}}\left( {\frac{{ - q}}{{\sqrt {r^2 + \frac{{d^2 }}{4} + rd\cos |
137 |
|
|
\theta } }} + \frac{q}{{\sqrt {r^2 + \frac{{d^2 }}{4} - rd\cos |
138 |
|
|
\theta } }}} \right) |
139 |
|
|
\] |
140 |
|
|
|
141 |
|
|
The basic assumption of the multipole expansion is $r \gg d$ , thus, |
142 |
|
|
$\frac{{d^2 }}{4}$ inside the square root of the denominator is |
143 |
|
|
neglected. This is a reasonable approximation in most cases. |
144 |
|
|
Unfortunately, in our headgroup model, the distance of charge |
145 |
|
|
separation $d$ (4.63 $\AA$ in PC headgroup) may be comparable to |
146 |
|
|
$r$. Nevertheless, we could still assume $ \cos \theta \approx 0$ |
147 |
|
|
in the central region of the headgroup. Using Taylor expansion and |
148 |
|
|
associating appropriate terms with electric moments will result in a |
149 |
|
|
"split-dipole" approximation: |
150 |
|
|
\[ |
151 |
|
|
V(r) = \frac{1}{{4\pi \varepsilon _0 }}\frac{{r\mu \cos \theta |
152 |
|
|
}}{{R^3 }} |
153 |
|
|
\] |
154 |
|
|
where$R = \sqrt {r^2 + \frac{{d^2 }}{4}}$ Placing a dipole at point |
155 |
|
|
$P$ and applying the same strategy, the interaction between two |
156 |
|
|
split-dipoles is then given by: |
157 |
|
|
\[ |
158 |
|
|
V_{sd} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
159 |
|
|
_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{R_{ij}^3 }} - |
160 |
|
|
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
161 |
|
|
r_{ij} } \right)}}{{R_{ij}^5 }}} \right] |
162 |
|
|
\] |
163 |
|
|
where $\mu _i$ and $\mu _j$ are the dipole moment of molecule $i$ |
164 |
|
|
and molecule $j$ respectively, $r_{ij}$ is vector between molecule |
165 |
|
|
$i$ and molecule $j$, and $R_{ij{$ is given by, |
166 |
|
|
\[ |
167 |
|
|
R_{ij} = \sqrt {r_{ij}^2 + \frac{{d_i^2 }}{4} + \frac{{d_j^2 |
168 |
|
|
}}{4}} |
169 |
|
|
\] |
170 |
|
|
where $d_i$ and $d_j$ are the charge separation distance of dipole |
171 |
|
|
and respectively. This approximation to the multipole expansion |
172 |
|
|
maintains the fast fall-off of the multipole potentials but lacks |
173 |
|
|
the normal divergences when two polar groups get close to one |
174 |
|
|
another. |
175 |
|
|
|
176 |
|
|
\begin{figure} |
177 |
|
|
\centering |
178 |
|
|
\includegraphics[width=\linewidth]{split_dipole.eps} |
179 |
|
|
\caption[Comparison between electrostatic approximation]{Electron |
180 |
|
|
density profile along the bilayer normal.} |
181 |
|
|
\label{lipidFigure:splitDipole} |
182 |
|
|
\end{figure} |
183 |
|
|
|
184 |
|
|
%\section{\label{lipidSection:methods}Methods} |
185 |
|
|
|
186 |
tim |
2685 |
\section{\label{lipidSection:resultDiscussion}Results and Discussion} |
187 |
tim |
2776 |
|
188 |
|
|
\subsection{One Lipid in Sea of Water Molecules} |
189 |
|
|
|
190 |
|
|
To exclude the inter-headgroup interaction, atomistic models of one |
191 |
|
|
lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built |
192 |
|
|
and studied using atomistic molecular dynamics. The simulation was |
193 |
|
|
analyzed using a set of radial distribution functions, which give |
194 |
|
|
the probability of finding a pair of molecular species a distance |
195 |
|
|
apart, relative to the probability expected for a completely random |
196 |
|
|
distribution function at the same density. |
197 |
|
|
|
198 |
|
|
\begin{equation} |
199 |
|
|
g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i |
200 |
|
|
\in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} } > |
201 |
|
|
\end{equation} |
202 |
|
|
\begin{equation} |
203 |
|
|
g_{AB} (r,\cos \theta ) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < |
204 |
|
|
\sum\limits_{i \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} |
205 |
|
|
} \delta (\cos \theta _{ij} - \cos \theta ) > |
206 |
|
|
\end{equation} |
207 |
|
|
|
208 |
|
|
From figure 4(a), we can identify the first solvation shell (3.5 |
209 |
|
|
$\AA$) and the second solvation shell (5.0 $\AA$) from both plots. |
210 |
|
|
However, the corresponding orientations are different. In DLPE, |
211 |
|
|
water molecules prefer to sit around -NH3 group due to the hydrogen |
212 |
|
|
bonding. In contrast, because of the hydrophobic effect of the |
213 |
|
|
-N(CH3)3 group, the preferred position of water molecules in DMPC is |
214 |
|
|
around the -PO4 Group. When the water molecules are far from the |
215 |
|
|
headgroup, the distribution of the two angles should be uniform. The |
216 |
|
|
correlation close to center of the headgroup dipole (< 5 $\AA$) in |
217 |
|
|
both plots tell us that in the closely-bound region, the dipoles of |
218 |
|
|
the water molecules are preferentially anti-aligned with the dipole |
219 |
|
|
of headgroup. |
220 |
|
|
|
221 |
|
|
\begin{figure} |
222 |
|
|
\centering |
223 |
|
|
\includegraphics[width=\linewidth]{g_atom.eps} |
224 |
|
|
\caption[The pair correlation functions for atomistic models]{} |
225 |
|
|
\label{lipidFigure:PCFAtom} |
226 |
|
|
\end{figure} |
227 |
|
|
|
228 |
|
|
The initial configurations of coarse-grained systems are constructed |
229 |
|
|
from the previous atomistic ones. The parameters for the |
230 |
|
|
coarse-grained model in Table~\ref{lipidTable:parameter} are |
231 |
|
|
estimated and tuned using isothermal-isobaric molecular dynamics. |
232 |
|
|
Pair distribution functions calculated from coarse-grained models |
233 |
|
|
preserve the basic characteristics of the atomistic simulations. The |
234 |
|
|
water density, measured in a head-group-fixed reference frame, |
235 |
|
|
surrounding two phospholipid headgroups is shown in |
236 |
|
|
Fig.~\ref{lipidFigure:PCFCoarse}. It is clear that the phosphate end |
237 |
|
|
in DMPC and the amine end in DMPE are the two most heavily solvated |
238 |
|
|
atoms. |
239 |
|
|
|
240 |
|
|
\begin{figure} |
241 |
|
|
\centering |
242 |
|
|
\includegraphics[width=\linewidth]{g_coarse.eps} |
243 |
|
|
\caption[The pair correlation functions for coarse-grained models]{} |
244 |
|
|
\label{lipidFigure:PCFCoarse} |
245 |
|
|
\end{figure} |
246 |
|
|
|
247 |
|
|
\begin{figure} |
248 |
|
|
\centering |
249 |
|
|
\includegraphics[width=\linewidth]{EWD_coarse.eps} |
250 |
|
|
\caption[Excess water density of coarse-grained phospholipids]{ } |
251 |
|
|
\label{lipidFigure:EWDCoarse} |
252 |
|
|
\end{figure} |
253 |
|
|
|
254 |
|
|
\begin{table} |
255 |
|
|
\caption{The Parameters For Coarse-grained Phospholipids} |
256 |
|
|
\label{lipidTable:parameter} |
257 |
|
|
\begin{center} |
258 |
|
|
\begin{tabular}{|l|c|c|c|c|c|} |
259 |
|
|
\hline |
260 |
|
|
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
261 |
|
|
Atom type & Mass & $\sigma$ & $\epsilon$ & charge & Dipole \\ |
262 |
|
|
|
263 |
|
|
$\text{{\sc CH}}_2$ & 14.03 & 3.95 & 0.0914 & 0 & 0 \\ |
264 |
|
|
$\text{{\sc CH}}_3$ & 15.04 & 3.75 & 0.195 & 0 & 0 \\ |
265 |
|
|
$\text{{\sc CE}}$ & 28.01 & 3.427& 0.294 & 0 & 1.693 |
266 |
|
|
$\text{{\sc CK}}$ & 28.01 & 3.592& 0.311 & 0 & 2.478 \\ |
267 |
|
|
$\text{{\sc PO}}_4$ & 108.995& 3.9 & 1.88 & -1& 0 \\ |
268 |
|
|
$\text{{\sc HDP}}$ & 14.03 & 4.0 & 0.13 & 0 & 0 \\ |
269 |
|
|
$\text{{\sc NC}}_4$ & 73.137 & 4.9 & 0.88 & +1& 0 \\ |
270 |
|
|
$\text{{\sc NH}}_3$ & 17.03 & 3.25 & 0.17 & +1& 0\\ |
271 |
|
|
\hline |
272 |
|
|
\end{tabular} |
273 |
|
|
\end{center} |
274 |
|
|
\end{table} |
275 |
|
|
|
276 |
|
|
\subsection{Bilayer Simulations Using Coarse-grained Model} |
277 |
|
|
|
278 |
|
|
A bilayer system consisting of 128 DMPC lipids and 3655 water |
279 |
|
|
molecules has been constructed from an atomistic coordinate |
280 |
|
|
file.[15] The MD simulation is performed at constant temperature, T |
281 |
|
|
= 300K, and constant pressure, p = 1 atm, and consisted of an |
282 |
|
|
equilibration period of 2 ns. During the equilibration period, the |
283 |
|
|
system was initially simulated at constant volume for 1ns. Once the |
284 |
|
|
system was equilibrated at constant volume, the cell dimensions of |
285 |
|
|
the system was relaxed by performing under NPT conditions using |
286 |
|
|
Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After |
287 |
|
|
equilibration, different properties were evaluated over a production |
288 |
|
|
run of 5 ns. |
289 |
|
|
|
290 |
|
|
\begin{figure} |
291 |
|
|
\centering |
292 |
|
|
\includegraphics[width=\linewidth]{bilayer.eps} |
293 |
|
|
\caption[Image of a coarse-grained bilayer system]{A coarse-grained |
294 |
|
|
bilayer system consisting of 128 DMPC lipids and 3655 SSD water |
295 |
|
|
molecules.} |
296 |
|
|
\label{lipidFigure:bilayer} |
297 |
|
|
\end{figure} |
298 |
|
|
|
299 |
|
|
\subsubsection{Electron Density Profile (EDP)} |
300 |
|
|
|
301 |
|
|
Assuming a gaussian distribution of electrons on each atomic center |
302 |
|
|
with a variance estimated from the size of the van der Waals radius, |
303 |
|
|
the EDPs which are proportional to the density profiles measured |
304 |
|
|
along the bilayer normal obtained by x-ray scattering experiment, |
305 |
|
|
can be expressed by |
306 |
|
|
\begin{equation} |
307 |
|
|
\rho _{x - ray} (z)dz \propto \sum\limits_{i = 1}^N {\frac{{n_i |
308 |
|
|
}}{V}\frac{1}{{\sqrt {2\pi \sigma ^2 } }}e^{ - (z - z_i )^2 /2\sigma |
309 |
|
|
^2 } dz}, |
310 |
|
|
\end{equation} |
311 |
|
|
where $\sigma$ is the variance equal to the van der Waals radius, |
312 |
|
|
$n_i$ is the atomic number of site $i$ and $V$ is the volume of the |
313 |
|
|
slab between $z$ and $z+dz$ . The highest density of total EDP |
314 |
|
|
appears at the position of lipid-water interface corresponding to |
315 |
|
|
headgroup, glycerol, and carbonyl groups of the lipids and the |
316 |
|
|
distribution of water locked near the head groups, while the lowest |
317 |
|
|
electron density is in the hydrocarbon region. As a good |
318 |
|
|
approximation to the thickness of the bilayer, the headgroup spacing |
319 |
|
|
, is defined as the distance between two peaks in the electron |
320 |
|
|
density profile, calculated from our simulations to be 34.1 $\AA$. |
321 |
|
|
This value is close to the x-ray diffraction experimental value 34.4 |
322 |
|
|
$\AA$ |
323 |
|
|
|
324 |
|
|
\begin{figure} |
325 |
|
|
\centering |
326 |
|
|
\includegraphics[width=\linewidth]{electron_density.eps} |
327 |
|
|
\caption[The density profile of the lipid bilayers]{Electron density |
328 |
|
|
profile along the bilayer normal. The water density is shown in red, |
329 |
|
|
the density due to the headgroups in green, the glycerol backbone in |
330 |
|
|
brown, $\text{{\sc CH}}_2$ in yellow, $\text{{\sc CH}}_3$ in cyan, |
331 |
|
|
and total density due to DMPC in blue.} |
332 |
|
|
\label{lipidFigure:electronDensity} |
333 |
|
|
\end{figure} |
334 |
|
|
|
335 |
|
|
\subsubsection{$\text{S}_{\text{{\sc cd}}}$ Order Parameter} |
336 |
|
|
|
337 |
|
|
Measuring deuterium order parameters by NMR is a useful technique to |
338 |
|
|
study the orientation of hydrocarbon chains in phospholipids. The |
339 |
|
|
order parameter tensor $S$ is defined by: |
340 |
|
|
\begin{equation} |
341 |
|
|
S_{ij} = \frac{1}{2} < 3\cos \theta _i \cos \theta _j - \delta |
342 |
|
|
_{ij} > |
343 |
|
|
\end{equation} |
344 |
|
|
where $\theta$ is the angle between the $i$th molecular axis and |
345 |
|
|
the bilayer normal ($z$ axis). The brackets denote an average over |
346 |
|
|
time and molecules. The molecular axes are defined: |
347 |
|
|
\begin{itemize} |
348 |
|
|
\item $\mathbf{\hat{z}}$ is the vector from $C_{n-1}$ to $C_{n+1}$. |
349 |
|
|
\item $\mathbf{\hat{y}}$ is the vector that is perpendicular to $z$ and |
350 |
|
|
in the plane through $C_{n-1}$, $C_{n}$, and $C_{n+1}$. |
351 |
|
|
\item $\mathbf{\hat{x}}$ is the vector perpendicular to |
352 |
|
|
$\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$. |
353 |
|
|
\end{itemize} |
354 |
|
|
In coarse-grained model, although there are no explicit hydrogens, |
355 |
|
|
the order parameter can still be written in terms of carbon ordering |
356 |
|
|
at each point of the chain |
357 |
|
|
\begin{equation} |
358 |
|
|
S_{ij} = \frac{1}{2} < 3\cos \theta _i \cos \theta _j - \delta |
359 |
|
|
_{ij} >. |
360 |
|
|
\end{equation} |
361 |
|
|
|
362 |
|
|
Fig.~\ref{lipidFigure:Scd} shows the order parameter profile |
363 |
|
|
calculated for our coarse-grained DMPC bilayer system at 300K. Also |
364 |
|
|
shown are the experimental data of Tiburu. The fact that |
365 |
|
|
$\text{S}_{\text{{\sc cd}}}$ order parameters calculated from |
366 |
|
|
simulation are higher than the experimental ones is ascribed to the |
367 |
|
|
assumption of the locations of implicit hydrogen atoms which are |
368 |
|
|
fixed in coarse-grained models at positions relative to the CC |
369 |
|
|
vector. |
370 |
|
|
|
371 |
|
|
\begin{figure} |
372 |
|
|
\centering |
373 |
|
|
\includegraphics[width=\linewidth]{scd.eps} |
374 |
|
|
\caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison |
375 |
|
|
of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model |
376 |
|
|
(blue) and DMPC\cite{petrache00} (black) near 300~K.} |
377 |
|
|
\label{lipidFigure:Scd} |
378 |
|
|
\end{figure} |