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# Line 11 | Line 11 | ns. In the second solvation shell, some water molecule
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. 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.
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 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 interface.
41 > lipid bilayers lives near the membrane/water
42 > 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,
# Line 46 | Line 48 | proposed in this chapter to overcome these difficultie
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 is used as the explicit solvent in this project. Unlike other
59 < water models which have partial charges distributed throughout the
60 < whole molecule, the SSD water model consists of a single site which
61 < is a Lennard-Jones interaction site, as well as a point dipole. A
62 < tetrahedral potential is added to correct for hydrogen bonding. The
63 < following equation describes the interaction between two water
64 < molecules:
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   \[
66   V_{SSD}  = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j )
67   + V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j )
# Line 111 | Line 113 | $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms.
113  
114   \begin{figure}
115   \centering
116 < \includegraphics[width=\linewidth]{coarse_grained.eps}
117 < \caption[A representation of coarse-grained phospholipid model]{}
118 < \label{lipidFigure:coarseGrained}
116 > \includegraphics[width=3in]{coarse_grained.eps}
117 > \caption[A representation of coarse-grained phospholipid model]{A
118 > representation of coarse-grained phospholipid model. The lipid
119 > headgroup consists of $\text{{\sc PO}}_4$ group (yellow),
120 > $\text{{\sc NC}}_4$ group (blue) and a united $\text{{\sc C}} atom
121 > (gray) $ with a dipole, while the glycerol backbone includes dipolar
122 > $\text{{\sc CE}}$ (read) and $\text{{\sc CK}}$ (pink) groups. Alkyl
123 > groups in hydrocarbon chains are simply represented by gray united
124 > atoms.} \label{lipidFigure:coarseGrained}
125   \end{figure}
126  
127   Accurate and efficient computation of electrostatics is one of the
# Line 148 | Line 156 | electrostatic potential at point $P$ is given by:
156  
157   \begin{figure}
158   \centering
159 < \includegraphics[width=\linewidth]{charge_dipole.eps}
160 < \caption[Electrostatic potential due to a linear molecule comprising
161 < two point charges]{Electrostatic potential due to a linear molecule
162 < comprising two point charges} \label{lipidFigure:chargeDipole}
159 > \includegraphics[width=3in]{charge_dipole.eps}
160 > \caption[An illustration of split-dipole
161 > approximation]{Electrostatic potential due to a linear molecule
162 > comprising two point charges with opposite charges. }
163 > \label{lipidFigure:chargeDipole}
164   \end{figure}
165  
166   The basic assumption of the multipole expansion is $r \gg d$ , thus,
# Line 178 | Line 187 | $i$ and molecule $j$, and $R_{ij{$ is given by,
187   \]
188   where $\mu _i$  and  $\mu _j$ are the dipole moment of molecule $i$
189   and molecule $j$ respectively, $r_{ij}$ is vector between molecule
190 < $i$ and molecule $j$, and $R_{ij{$ is given by,
190 > $i$ and molecule $j$, and $R_{ij}$ is given by,
191   \[
192   R_{ij}  = \sqrt {r_{ij}^2  + \frac{{d_i^2 }}{4} + \frac{{d_j^2
193   }}{4}}
# Line 187 | Line 196 | another.
196   and respectively. This approximation to the multipole expansion
197   maintains the fast fall-off of the multipole potentials but lacks
198   the normal divergences when two polar groups get close to one
199 < another.
200 <
199 > another. The comparision between different electrostatic
200 > approximation is shown in \ref{lipidFigure:splitDipole}. Due to the
201 > divergence at the central region of the headgroup introduced by
202 > dipole-dipole approximation, we discover that water molecules are
203 > locked into the central region in the simulation. This artifact can
204 > be corrected using split-dipole approximation or other accurate
205 > methods.
206   \begin{figure}
207   \centering
208 < \includegraphics[width=\linewidth]{split_dipole.eps}
209 < \caption[Comparison between electrostatic approximation]{Electron
210 < density profile along the bilayer normal.}
211 < \label{lipidFigure:splitDipole}
208 > \includegraphics[width=3in]{split_dipole.eps}
209 > \caption[Comparison between electrostatic
210 > approximation]{Electrostatic potential map for two pairs of charges
211 > with different alignments: (a) illustration of different alignments;
212 > (b) charge-charge interaction; (c) dipole-dipole approximation; (d)
213 > split-dipole approximation.} \label{lipidFigure:splitDipole}
214   \end{figure}
215  
216   %\section{\label{lipidSection:methods}Methods}
# Line 221 | 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 tell 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 256 | 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 291 | 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
303 < 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¨¦-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 311 | 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,
346   the EDPs which are proportional to the density profiles measured
347   along the bilayer normal obtained by x-ray scattering experiment,
348 < can be expressed by
348 > can be expressed by\cite{Tu1995}
349   \begin{equation}
350   \rho _{x - ray} (z)dz \propto \sum\limits_{i = 1}^N {\frac{{n_i
351   }}{V}\frac{1}{{\sqrt {2\pi \sigma ^2 } }}e^{ - (z - z_i )^2 /2\sigma
# Line 334 | Line 362 | $\AA$.
362   , is defined as the distance between two peaks in the electron
363   density profile, calculated from our simulations to be 34.1 $\AA$.
364   This value is close to the x-ray diffraction experimental value 34.4
365 < $\AA$.
365 > $\AA$\cite{Petrache1998}.
366  
367   \begin{figure}
368   \centering
# Line 347 | 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 368 | Line 396 | at each point of the chain
396   \end{itemize}
397   In coarse-grained model, although there are no explicit hydrogens,
398   the order parameter can still be written in terms of carbon ordering
399 < at each point of the chain
399 > at each point of the chain\cite{Egberts1988}
400   \begin{equation}
401   S_{ij}  = \frac{1}{2} < 3\cos \theta _i \cos \theta _j  - \delta
402   _{ij}  >.
# Line 376 | Line 404 | shown are the experimental data of Tiburu. The fact th
404  
405   Fig.~\ref{lipidFigure:Scd} shows the order parameter profile
406   calculated for our coarse-grained DMPC bilayer system at 300K. Also
407 < shown are the experimental data of Tiburu. The fact that
407 > shown are the experimental data of Tu\cite{Tu1995}. The fact that
408   $\text{S}_{\text{{\sc cd}}}$ order parameters calculated from
409   simulation are higher than the experimental ones is ascribed to the
410   assumption of the locations of implicit hydrogen atoms which are
# Line 388 | 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 are 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 result from all-atom simulations, we developed a simple
431 + coarse-grained model capturing essential features of the headgroup
432 + solvation which is crucial to transport process in membrane system.
433 + In addition, the model has been explored in a bilayer system which
434 + is shown to have reasonable electron density profile,
435 + $\text{S}_{\text{{\sc cd}}}$ order parameter and other structural
436 + properties. The accuracy of this model is achieved by matching
437 + atomistic result. It is also easy to represent different
438 + phosphorlipids 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 + gaining by introducing 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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