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# Line 28 | Line 28 | rapidly in the eukaryotic ER and the bacterial cytopla
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 ER and the bacterial cytoplasmic membrane
32 < via a bi-directional, facilitated diffusion process requiring no
33 < metabolic energy input. Another system of interest would be the
34 < distribution of sites occupied by inhaled anesthetics in membrane.
35 < Although the physiological effects of anesthetics have been
36 < extensively studied, the controversy over their effects on lipid
37 < bilayers still continues. Recent deuterium NMR measurements on
38 < halothane in POPC lipid bilayers suggest the anesthetics are
39 < primarily located at the hydrocarbon chain region\cite{Baber1995}.
40 < Infrared spectroscopy experiments suggest that halothane in DMPC
41 < lipid bilayers lives near the membrane/water
42 < interface\cite{Lieb1982}.
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.
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  
# Line 71 | Line 69 | for molecule $i$ and molecule $j$ respectively.
69   where $r_{ij}$ is the vector between molecule $i$ and molecule $j$,
70   $\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom
71   for molecule $i$ and molecule $j$ respectively.
72 < \[
73 < V_{LJ} (r_{ij} ) = 4\varepsilon _{ij} \left[ {\left( {\frac{{\sigma
74 < _{ij} }}{{r_{ij} }}} \right)^{12}  - \left( {\frac{{\sigma _{ij}
75 < }}{{r_{ij} }}} \right)^6 } \right]
76 < \]
77 < \[
78 < V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon
79 < _0 }}\left[ {\frac{{\mu _i  \cdot \mu _j }}{{r_{ij}^3 }} -
80 < \frac{{3\left( {\mu _i  \cdot r_{ij} } \right)\left( {\mu _i  \cdot
81 < r_{ij} } \right)}}{{r_{ij}^5 }}} \right]
82 < \]
83 < \[
84 < V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) = v_0 [s(r_{ij} )w(r_{ij}
87 < ,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i ,\Omega _j
88 < )]
89 < \]
72 > \begin{eqnarray*}
73 > V_{LJ} (r_{ij} ) &= &4\varepsilon _{ij} \left[ {\left(
74 > {\frac{{\sigma _{ij} }}{{r_{ij} }}} \right)^{12}  - \left(
75 > {\frac{{\sigma _{ij}
76 > }}{{r_{ij} }}} \right)^6 } \right], \\
77 > V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) &= &
78 > \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[
79 > \hat{u}_{i} \cdot \hat{u}_{j} - 3(\hat{u}_i \cdot \hat{\mathbf{r}}_{ij}) %
80 > (\hat{u}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr],\\
81 > V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) &=& v_0 [s(r_{ij}
82 > )w(r_{ij} ,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i
83 > ,\Omega _j )].\\
84 > \end{eqnarray*}
85   where $v_0$ is a strength parameter, $s$ and $s'$ are cubic
86 < switching functions, while $w$   and $w'$  are responsible for the
86 > switching functions, while $w$ and $w'$  are responsible for the
87   tetrahedral potential and the short-range correction to the dipolar
88   interaction respectively.
89   \[
# Line 97 | Line 92 | Although dipole-dipole and sticky interactions are mor
92   w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij}  - 0.6)^2 (\cos \theta _{ij}  + 0.8)^2  - w_0  \\
93   \end{array}
94   \]
95 < Although dipole-dipole and sticky interactions are more
95 > Although the dipole-dipole and sticky interactions are more
96   mathematically complicated than Coulomb interactions, the number of
97   pair interactions is reduced dramatically both because the model
98   only contains a single-point as well as "short range" nature of the
99 < higher order interaction.
99 > more expensive interaction.
100  
101   \subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model}
102  
# Line 116 | Line 111 | $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
111   \begin{figure}
112   \centering
113   \includegraphics[width=3in]{coarse_grained.eps}
114 < \caption[A representation of coarse-grained phospholipid model]{}
114 > \caption[A representation of coarse-grained phospholipid model]{A
115 > representation of coarse-grained phospholipid model. The lipid
116 > headgroup consists of $\text{{\sc PO}}_4$ group (yellow),
117 > $\text{{\sc NC}}_4$ group (blue) and a united C atom (gray) with a
118 > dipole, while the glycerol backbone includes dipolar $\text{{\sc
119 > CE}}$ (red) and $\text{{\sc CK}}$ (pink) groups. Alkyl groups in
120 > hydrocarbon chains are simply represented by gray united atoms.}
121   \label{lipidFigure:coarseGrained}
122   \end{figure}
123  
# Line 131 | Line 132 | Ewald summation method mathematically transforms this
132   \]
133   where $N_A$ and $N_B$ are the number of point charges in the two
134   molecular species. Originally developed to study ionic crystals, the
135 < Ewald summation method mathematically transforms this
136 < straightforward but conditionally convergent summation into two more
137 < complicated but rapidly convergent sums. One summation is carried
138 < out in reciprocal space while the other is carried out in real
139 < space. An alternative approach is a multipole expansion, which is
140 < based on electrostatic moments, such as charge (monopole), dipole,
140 < quadruple etc.
135 > Ewald sum method mathematically transforms this straightforward but
136 > conditionally convergent summation into two more complicated but
137 > rapidly convergent sums. One summation is carried out in reciprocal
138 > space while the other is carried out in real space. An alternative
139 > approach is the multipole expansion, which is based on electrostatic
140 > moments, such as charge (monopole), dipole, quadrupole etc.
141  
142   Here we consider a linear molecule which consists of two point
143   charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The
# Line 153 | Line 153 | electrostatic potential at point $P$ is given by:
153   \begin{figure}
154   \centering
155   \includegraphics[width=3in]{charge_dipole.eps}
156 < \caption[Electrostatic potential due to a linear molecule comprising
157 < two point charges]{Electrostatic potential due to a linear molecule
158 < comprising two point charges} \label{lipidFigure:chargeDipole}
156 > \caption[An illustration of split-dipole
157 > approximation]{Electrostatic potential due to a linear molecule
158 > comprising two point charges with opposite charges. }
159 > \label{lipidFigure:chargeDipole}
160   \end{figure}
161  
162   The basic assumption of the multipole expansion is $r \gg d$ , thus,
163   $\frac{{d^2 }}{4}$ inside the square root of the denominator is
164   neglected. This is a reasonable approximation in most cases.
165   Unfortunately, in our headgroup model, the distance of charge
166 < separation $d$ (4.63 $\AA$ in PC headgroup) may be comparable to
167 < $r$. Nevertheless, we could still assume  $ \cos \theta  \approx 0$
168 < in the central region of the headgroup. Using Taylor expansion and
166 > separation $d$ (4.63 \AA  in PC headgroup) may be comparable to $r$.
167 > Nevertheless, we could still assume  $ \cos \theta  \approx 0$ in
168 > the central region of the headgroup. Using Taylor expansion and
169   associating appropriate terms with electric moments will result in a
170   "split-dipole" approximation:
171   \[
# Line 191 | Line 192 | another.
192   and respectively. This approximation to the multipole expansion
193   maintains the fast fall-off of the multipole potentials but lacks
194   the normal divergences when two polar groups get close to one
195 < another.
196 < %description of the comparsion
195 > another. The comparision between different electrostatic
196 > approximation is shown in \ref{lipidFigure:splitDipole}. Due to the
197 > divergence at the central region of the headgroup introduced by
198 > dipole-dipole approximation, we discover that water molecules are
199 > locked into the central region in the simulation. This artifact can
200 > be corrected using split-dipole approximation or other accurate
201 > methods.
202   \begin{figure}
203   \centering
204   \includegraphics[width=\linewidth]{split_dipole.eps}
205 < \caption[Comparison between electrostatic approximation]{Electron
206 < density profile along the bilayer normal.}
207 < \label{lipidFigure:splitDipole}
205 > \caption[Comparison between electrostatic
206 > approximation]{Electrostatic potential map for two pairs of charges
207 > with different alignments: (a) illustration of different alignments;
208 > (b) charge-charge interaction; (c) dipole-dipole approximation; (d)
209 > split-dipole approximation.} \label{lipidFigure:splitDipole}
210   \end{figure}
211  
212   %\section{\label{lipidSection:methods}Methods}
# Line 207 | Line 215 | To exclude the inter-headgroup interaction, atomistic
215  
216   \subsection{One Lipid in Sea of Water Molecules}
217  
218 < To exclude the inter-headgroup interaction, atomistic models of one
219 < lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built
220 < and studied using atomistic molecular dynamics. The simulation was
221 < analyzed using a set of radial distribution functions, which give
222 < the probability of finding a pair of molecular species a distance
223 < apart, relative to the probability expected for a completely random
224 < distribution function at the same density.
218 > To tune our parameters without the inter-headgroup interactions,
219 > atomistic models of one lipid (DMPC or DLPE) in sea of water
220 > molecules (TIP3P) were built and studied using atomistic molecular
221 > dynamics. The simulation was analyzed using a set of radial
222 > distribution functions, which give the probability of finding a pair
223 > of molecular species a distance apart, relative to the probability
224 > expected for a completely random distribution function at the same
225 > density.
226  
227   \begin{equation}
228   g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i
# Line 225 | Line 234 | From figure 4(a), we can identify the first solvation
234   } \delta (\cos \theta _{ij}  - \cos \theta ) >
235   \end{equation}
236  
237 < From figure 4(a), we can identify the first solvation shell (3.5
238 < $\AA$) and the second solvation shell (5.0 $\AA$) from both plots.
239 < However, the corresponding orientations are different. In DLPE,
240 < water molecules prefer to sit around -NH3 group due to the hydrogen
241 < bonding. In contrast, because of the hydrophobic effect of the
242 < -N(CH3)3 group, the preferred position of water molecules in DMPC is
243 < around the -PO4 Group. When the water molecules are far from the
244 < headgroup, the distribution of the two angles should be uniform. The
245 < correlation close to center of the headgroup dipole (< 5 $\AA$) in
246 < both plots tell us that in the closely-bound region, the dipoles of
247 < the water molecules are preferentially anti-aligned with the dipole
248 < of headgroup.
237 > From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first
238 > solvation shell (3.5 \AA) and the second solvation shell (5.0 \AA)
239 > from both plots. However, the corresponding orientations are
240 > different. In DLPE, water molecules prefer to sit around $\text{{\sc
241 > NH}}_3$ group due to the hydrogen bonding. In contrast, because of
242 > the hydrophobic effect of the $ {\rm{N(CH}}_{\rm{3}}
243 > {\rm{)}}_{\rm{3}} $ group, the preferred position of water molecules
244 > in DMPC is around the $\text{{\sc PO}}_4$ Group. When the water
245 > molecules are far from the headgroup, the distribution of the two
246 > angles should be uniform. The correlation close to center of the
247 > headgroup dipole in both plots tells us that in the closely-bound
248 > region, the dipoles of the water molecules are preferentially
249 > anti-aligned with the dipole of headgroup. When the water molecules
250 > are far from the headgroup, the distribution of the two angles
251 > should be uniform. The correlation close to center of the headgroup
252 > dipole in both plots tell us that in the closely-bound region, the
253 > dipoles of the water molecules are preferentially aligned with the
254 > dipole of headgroup.
255  
256   \begin{figure}
257   \centering
258   \includegraphics[width=\linewidth]{g_atom.eps}
259 < \caption[The pair correlation functions for atomistic models]{}
259 > \caption[The pair correlation functions for atomistic models]{The
260 > pair correlation functions for atomistic models: (a)$g(r,\cos \theta
261 > )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE; (c)$g(r,\cos \omega
262 > )$ for DMPC; (d)$g(r,\cos \omega )$ for DLPE; (e)$g(\cos \theta,\cos
263 > \omega)$ for DMPC; (f)$g(\cos \theta,\cos \omega)$ for DMLPE.}
264   \label{lipidFigure:PCFAtom}
265   \end{figure}
266  
# Line 260 | Line 279 | atoms.
279   \begin{figure}
280   \centering
281   \includegraphics[width=\linewidth]{g_coarse.eps}
282 < \caption[The pair correlation functions for coarse-grained models]{}
282 > \caption[The pair correlation functions for coarse-grained
283 > models]{The pair correlation functions for coarse-grained models:
284 > (a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.}
285   \label{lipidFigure:PCFCoarse}
286   \end{figure}
287  
288   \begin{figure}
289   \centering
290   \includegraphics[width=\linewidth]{EWD_coarse.eps}
291 < \caption[Excess water density of coarse-grained phospholipids]{ }
292 < \label{lipidFigure:EWDCoarse}
291 > \caption[Excess water density of coarse-grained
292 > phospholipids]{Excess water density of coarse-grained
293 > phospholipids.} \label{lipidFigure:EWDCoarse}
294   \end{figure}
295  
296   \begin{table}
# Line 295 | Line 317 | molecules has been constructed from an atomistic coord
317   \subsection{Bilayer Simulations Using Coarse-grained Model}
318  
319   A bilayer system consisting of 128 DMPC lipids and 3655 water
320 < molecules has been constructed from an atomistic coordinate
321 < file.[15] The MD simulation is performed at constant temperature, T
322 < = 300K, and constant pressure, p = 1 atm, and consisted of an
323 < equilibration period of 2 ns. During the equilibration period, the
324 < system was initially simulated at constant volume for 1ns. Once the
325 < system was equilibrated at constant volume, the cell dimensions of
326 < the system was relaxed by performing under NPT conditions using
327 < Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After
328 < equilibration, different properties were evaluated over a production
307 < run of 5 ns.
320 > molecules has been constructed from an atomistic coordinate file.
321 > The MD simulation is performed at constant temperature, T = 300K,
322 > and constant pressure, p = 1 atm, and consisted of an equilibration
323 > period of 2 ns. During the equilibration period, the system was
324 > initially simulated at constant volume for 1 ns. Once the system was
325 > equilibrated at constant volume, the cell dimensions of the system
326 > was relaxed by performing under NPT conditions using Nos\'{e}-Hoover
327 > extended system isothermal-isobaric dynamics. After equilibration,
328 > different properties were evaluated over a production run of 5 ns.
329  
330   \begin{figure}
331   \centering
# Line 336 | Line 357 | density profile, calculated from our simulations to be
357   electron density is in the hydrocarbon region. As a good
358   approximation to the thickness of the bilayer, the headgroup spacing
359   , is defined as the distance between two peaks in the electron
360 < density profile, calculated from our simulations to be 34.1 $\AA$.
360 > density profile, calculated from our simulations to be 34.1 \AA.
361   This value is close to the x-ray diffraction experimental value 34.4
362 < $\AA$\cite{Petrache1998}.
362 > \AA\cite{Petrache1998}.
363  
364   \begin{figure}
365   \centering
# Line 392 | Line 413 | of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grain
413   \includegraphics[width=\linewidth]{scd.eps}
414   \caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison
415   of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model
416 < (blue) and DMPC\cite{petrache00} (black) near 300~K.}
416 > (blue) and DMPC\cite{Petrache2000} (black) near 300~K.}
417   \label{lipidFigure:Scd}
418   \end{figure}
419  
420   %\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model}
421 +
422 + \section{\label{lipidSection:Conclusion}Conclusion}
423 +
424 + Atomistic simulations have been used in this study to determine the
425 + preferred orientation and location of water molecules relative to
426 + the location and orientation of the PC and PE lipid headgroups.
427 + Based on the results from our all-atom simulations, we developed a
428 + simple coarse-grained model which captures the essential features of
429 + the headgroup solvation which is crucial to transport process in
430 + membrane system. In addition, the model has been explored in a
431 + bilayer system was shown to have reasonable electron density
432 + profiles, $\text{S}_{\text{{\sc cd}}}$ order parameter and other
433 + structural properties. The accuracy of this model is achieved by
434 + matching atomistic result. It is also easy to represent different
435 + phospholipids by changing a few parameters of the model. Another
436 + important characteristic of this model distinguishing itself from
437 + other models\cite{Goetz1998,Marrink2004} is the computational speed
438 + gained by introducing a short range electrostatic approximation.
439 + Further studies of this system using z-constraint method could
440 + extend the time length of the simulations to study transport
441 + phenomena in large-scale membrane systems.

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