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 |
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 |
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 |
|
|
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 $j$, |
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 : |
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 |
|
\begin{eqnarray} |
94 |
|
w_0. |
95 |
|
\end{eqnarray} |
96 |
|
Here $\theta _{ij}$ and $\phi _{ij}$ are the spherical polar angles |
97 |
< |
representing relative orientations between molecule $i$ and molecule |
98 |
< |
$j$. Although the dipole-dipole and sticky interactions are more |
99 |
< |
mathematically complicated than Coulomb interactions, the number of |
100 |
< |
pair interactions is reduced dramatically both because the model |
101 |
< |
only contains a single-point as well as "short range" nature of the |
102 |
< |
more expensive interaction. |
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 |
|
|
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. |
113 |
– |
|
114 |
|
\begin{figure} |
115 |
|
\centering |
116 |
|
\includegraphics[width=3in]{coarse_grained.eps} |
152 |
|
\theta } }} + \frac{q}{{\sqrt {r^2 + \frac{{d^2 }}{4} - rd\cos |
153 |
|
\theta } }}} \right) |
154 |
|
\] |
155 |
– |
|
155 |
|
\begin{figure} |
156 |
|
\centering |
157 |
|
\includegraphics[width=3in]{charge_dipole.eps} |
160 |
|
comprising two point charges with opposite charges. } |
161 |
|
\label{lipidFigure:chargeDipole} |
162 |
|
\end{figure} |
164 |
– |
|
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 |
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 |
210 |
|
split-dipole approximation.} \label{lipidFigure:splitDipole} |
211 |
|
\end{figure} |
212 |
|
|
215 |
– |
%\section{\label{lipidSection:methods}Methods} |
216 |
– |
|
213 |
|
\section{\label{lipidSection:resultDiscussion}Results and Discussion} |
214 |
|
|
215 |
|
\subsection{One Lipid in Sea of Water Molecules} |
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} |
230 |
< |
\begin{equation} |
231 |
< |
g_{AB} (r,\cos \theta ) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < |
232 |
< |
\sum\limits_{i \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} |
237 |
< |
} \delta (\cos \theta _{ij} - \cos \theta ) > |
238 |
< |
\end{equation} |
239 |
< |
|
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) |
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 |
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. |
258 |
< |
|
250 |
> |
dipole of the headgroup. |
251 |
|
\begin{figure} |
252 |
|
\centering |
253 |
|
\includegraphics[width=\linewidth]{g_atom.eps} |
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. |
281 |
– |
|
273 |
|
\begin{figure} |
274 |
|
\centering |
275 |
|
\includegraphics[width=\linewidth]{g_coarse.eps} |
278 |
|
(a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.} |
279 |
|
\label{lipidFigure:PCFCoarse} |
280 |
|
\end{figure} |
290 |
– |
|
281 |
|
\begin{figure} |
282 |
|
\centering |
283 |
|
\includegraphics[width=\linewidth]{EWD_coarse.eps} |
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 \\ |
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. |
332 |
– |
|
323 |
|
\begin{figure} |
324 |
|
\centering |
325 |
|
\includegraphics[width=\linewidth]{bilayer.eps} |
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 |
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 |
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 |
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 |