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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  
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,Fennel2004} is used as the explicit solvent
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
# Line 97 | Line 95 | Although dipole-dipole and sticky interactions are mor
95   w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij}  - 0.6)^2 (\cos \theta _{ij}  + 0.8)^2  - w_0  \\
96   \end{array}
97   \]
98 < Although dipole-dipole and sticky interactions are more
98 > 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 < higher order interaction.
102 > more expensive interaction.
103  
104   \subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model}
105  
# Line 116 | Line 114 | $\text{{\sc CH}}_2$ 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]{}
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 > $\text{{\sc CE}}$ (red) 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  
127   Accurate and efficient computation of electrostatics is one of the
# Line 131 | Line 135 | Ewald summation method mathematically transforms this
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 summation method mathematically transforms this
139 < straightforward but conditionally convergent summation into two more
140 < complicated but rapidly convergent sums. One summation is carried
141 < out in reciprocal space while the other is carried out in real
142 < space. An alternative approach is a multipole expansion, which is
143 < based on electrostatic moments, such as charge (monopole), dipole,
140 < quadruple etc.
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
# Line 153 | Line 156 | electrostatic potential at point $P$ is given by:
156   \begin{figure}
157   \centering
158   \includegraphics[width=3in]{charge_dipole.eps}
159 < \caption[Electrostatic potential due to a linear molecule comprising
160 < two point charges]{Electrostatic potential due to a linear molecule
161 < comprising two point charges} \label{lipidFigure:chargeDipole}
159 > \caption[An illustration of split-dipole
160 > approximation]{Electrostatic potential due to a linear molecule
161 > comprising two point charges with opposite charges. }
162 > \label{lipidFigure:chargeDipole}
163   \end{figure}
164  
165   The basic assumption of the multipole expansion is $r \gg d$ , thus,
# Line 191 | Line 195 | another.
195   and respectively. This approximation to the multipole expansion
196   maintains the fast fall-off of the multipole potentials but lacks
197   the normal divergences when two polar groups get close to one
198 < another.
199 < %description of the comparsion
198 > another. The comparision between different electrostatic
199 > approximation is shown in \ref{lipidFigure:splitDipole}. Due to the
200 > divergence at the central region of the headgroup introduced by
201 > dipole-dipole approximation, we discover that water molecules are
202 > locked into the central region in the simulation. This artifact can
203 > be corrected using split-dipole approximation or other accurate
204 > methods.
205   \begin{figure}
206   \centering
207   \includegraphics[width=\linewidth]{split_dipole.eps}
208 < \caption[Comparison between electrostatic approximation]{Electron
209 < density profile along the bilayer normal.}
210 < \label{lipidFigure:splitDipole}
208 > \caption[Comparison between electrostatic
209 > approximation]{Electrostatic potential map for two pairs of charges
210 > with different alignments: (a) illustration of different alignments;
211 > (b) charge-charge interaction; (c) dipole-dipole approximation; (d)
212 > split-dipole approximation.} \label{lipidFigure:splitDipole}
213   \end{figure}
214  
215   %\section{\label{lipidSection:methods}Methods}
# Line 207 | Line 218 | To exclude the inter-headgroup interaction, atomistic
218  
219   \subsection{One Lipid in Sea of Water Molecules}
220  
221 < To exclude the inter-headgroup interaction, atomistic models of one
222 < lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built
223 < and studied using atomistic molecular dynamics. The simulation was
224 < analyzed using a set of radial distribution functions, which give
225 < the probability of finding a pair of molecular species a distance
226 < apart, relative to the probability expected for a completely random
227 < distribution function at the same density.
221 > To tune our parameters without the inter-headgroup interactions,
222 > atomistic models of one lipid (DMPC or DLPE) in sea of water
223 > molecules (TIP3P) were built and studied using atomistic molecular
224 > dynamics. The simulation was analyzed using a set of radial
225 > distribution functions, which give the probability of finding a pair
226 > of molecular species a distance apart, relative to the probability
227 > expected for a completely random distribution function at the same
228 > density.
229  
230   \begin{equation}
231   g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i
# Line 225 | Line 237 | From figure 4(a), we can identify the first solvation
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 tells 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  
# Line 260 | Line 282 | atoms.
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  
291   \begin{figure}
292   \centering
293   \includegraphics[width=\linewidth]{EWD_coarse.eps}
294 < \caption[Excess water density of coarse-grained phospholipids]{ }
295 < \label{lipidFigure:EWDCoarse}
294 > \caption[Excess water density of coarse-grained
295 > phospholipids]{Excess water density of coarse-grained
296 > phospholipids.} \label{lipidFigure:EWDCoarse}
297   \end{figure}
298  
299   \begin{table}
# Line 295 | Line 320 | molecules has been constructed from an atomistic coord
320   \subsection{Bilayer Simulations Using Coarse-grained Model}
321  
322   A bilayer system consisting of 128 DMPC lipids and 3655 water
323 < molecules has been constructed from an atomistic coordinate
324 < file.[15] The MD simulation is performed at constant temperature, T
325 < = 300K, and constant pressure, p = 1 atm, and consisted of an
326 < equilibration period of 2 ns. During the equilibration period, the
327 < system was initially simulated at constant volume for 1ns. Once the
328 < system was equilibrated at constant volume, the cell dimensions of
329 < the system was relaxed by performing under NPT conditions using
330 < Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After
331 < equilibration, different properties were evaluated over a production
307 < run of 5 ns.
323 > molecules has been constructed from an atomistic coordinate file.
324 > The MD simulation is performed at constant temperature, T = 300K,
325 > and constant pressure, p = 1 atm, and consisted of an equilibration
326 > period of 2 ns. During the equilibration period, the system was
327 > initially simulated at constant volume for 1 ns. Once the system was
328 > equilibrated at constant volume, the cell dimensions of the system
329 > was relaxed by performing under NPT conditions using Nos\'{e}-Hoover
330 > extended system isothermal-isobaric dynamics. After equilibration,
331 > different properties were evaluated over a production run of 5 ns.
332  
333   \begin{figure}
334   \centering
# Line 315 | Line 339 | molecules.}
339   \label{lipidFigure:bilayer}
340   \end{figure}
341  
342 < \subsubsection{Electron Density Profile (EDP)}
342 > \subsubsection{\textbf{Electron Density Profile (EDP)}}
343  
344   Assuming a gaussian distribution of electrons on each atomic center
345   with a variance estimated from the size of the van der Waals radius,
# Line 351 | Line 375 | and total density due to DMPC in blue.}
375   \label{lipidFigure:electronDensity}
376   \end{figure}
377  
378 < \subsubsection{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}
378 > \subsubsection{\textbf{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}}
379  
380   Measuring deuterium order parameters by NMR is a useful technique to
381   study the orientation of hydrocarbon chains in phospholipids. The
# Line 392 | Line 416 | of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grain
416   \includegraphics[width=\linewidth]{scd.eps}
417   \caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison
418   of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model
419 < (blue) and DMPC\cite{petrache00} (black) near 300~K.}
419 > (blue) and DMPC\cite{Petrache2000} (black) near 300~K.}
420   \label{lipidFigure:Scd}
421   \end{figure}
422  
423   %\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model}
424 +
425 + \section{\label{lipidSection:Conclusion}Conclusion}
426 +
427 + Atomistic simulations have been used in this study to determine the
428 + preferred orientation and location of water molecules relative to
429 + the location and orientation of the PC and PE lipid headgroups.
430 + Based on the results from our all-atom simulations, we developed a
431 + simple coarse-grained model which captures the essential features of
432 + the headgroup solvation which is crucial to transport process in
433 + membrane system. In addition, the model has been explored in a
434 + bilayer system was shown to have reasonable electron density
435 + profiles, $\text{S}_{\text{{\sc cd}}}$ order parameter and other
436 + structural properties. The accuracy of this model is achieved by
437 + matching atomistic result. It is also easy to represent different
438 + phospholipids by changing a few parameters of the model. Another
439 + important characteristic of this model distinguishing itself from
440 + other models\cite{Goetz1998,Marrink2004} is the computational speed
441 + gained by introducing a short range electrostatic approximation.
442 + Further studies of this system using z-constraint method could
443 + extend the time length of the simulations to study transport
444 + phenomena in large-scale membrane systems.

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