--- trunk/tengDissertation/Lipid.tex 2006/06/06 20:49:05 2806 +++ trunk/tengDissertation/Lipid.tex 2006/06/23 20:21:54 2881 @@ -28,36 +28,34 @@ rapidly in the eukaryotic ER and the bacterial cytopla hydrophobic interior of the membrane, and for the hydrophobic tails to be exposed to the aqueous environment\cite{Sasaki2004}. A number of studies indicate that the flipping of phospholipids occurs -rapidly in the eukaryotic ER and the bacterial cytoplasmic membrane -via a bi-directional, facilitated diffusion process requiring no -metabolic energy input. Another system of interest would be the -distribution of sites occupied by inhaled anesthetics in membrane. -Although the physiological effects of anesthetics have been -extensively studied, the controversy over their effects on lipid -bilayers still continues. Recent deuterium NMR measurements on -halothane in POPC lipid bilayers suggest the anesthetics are -primarily located at the hydrocarbon chain region\cite{Baber1995}. -Infrared spectroscopy experiments suggest that halothane in DMPC -lipid bilayers lives near the membrane/water -interface\cite{Lieb1982}. +rapidly in the eukaryotic endoplasmic reticulum and the bacterial +cytoplasmic membrane via a bi-directional, facilitated diffusion +process requiring no metabolic energy input. Another system of +interest is the distribution of sites occupied by inhaled +anesthetics in membrane. Although the physiological effects of +anesthetics have been extensively studied, the controversy over +their effects on lipid bilayers still continues. Recent deuterium +NMR measurements on halothane on POPC lipid bilayers suggest the +anesthetics are primarily located at the hydrocarbon chain +region\cite{Baber1995}. However, infrared spectroscopy experiments +suggest that halothane in DMPC lipid bilayers lives near the +membrane/water interface\cite{Lieb1982}. Molecular dynamics simulations have proven to be a powerful tool for studying the functions of biological systems, providing structural, thermodynamic and dynamical information. Unfortunately, much of biological interest happens on time and length scales well beyond -the range of current simulation technologies. -%review of coarse-grained modeling -Several schemes are proposed in this chapter to overcome these -difficulties. +the range of current simulation technologies. Several schemes are +proposed in this chapter to overcome these difficulties. -\section{\label{lipidSection:model}Model} +\section{\label{lipidSection:model}Model and Methodology} \subsection{\label{lipidSection:SSD}The Soft Sticky Dipole Water Model} In a typical bilayer simulation, the dominant portion of the computation time will be spent calculating water-water interactions. As an efficient solvent model, the Soft Sticky Dipole (SSD) water -model\cite{Chandra1999,Fennel2004} is used as the explicit solvent +model\cite{Chandra1999,Fennell2004} is used as the explicit solvent in this project. Unlike other water models which have partial charges distributed throughout the whole molecule, the SSD water model consists of a single site which is a Lennard-Jones interaction @@ -97,11 +95,11 @@ Although dipole-dipole and sticky interactions are mor w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - w_0 \\ \end{array} \] -Although dipole-dipole and sticky interactions are more +Although the dipole-dipole and sticky interactions are more mathematically complicated than Coulomb interactions, the number of pair interactions is reduced dramatically both because the model only contains a single-point as well as "short range" nature of the -higher order interaction. +more expensive interaction. \subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model} @@ -116,8 +114,14 @@ $\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms. \begin{figure} \centering \includegraphics[width=3in]{coarse_grained.eps} -\caption[A representation of coarse-grained phospholipid model]{} -\label{lipidFigure:coarseGrained} +\caption[A representation of coarse-grained phospholipid model]{A +representation of coarse-grained phospholipid model. The lipid +headgroup consists of $\text{{\sc PO}}_4$ group (yellow), +$\text{{\sc NC}}_4$ group (blue) and a united $\text{{\sc C}}$ atom +(gray) $ with a dipole, while the glycerol backbone includes dipolar +$\text{{\sc CE}}$ (red) and $\text{{\sc CK}}$ (pink) groups. Alkyl +groups in hydrocarbon chains are simply represented by gray united +atoms.} \label{lipidFigure:coarseGrained} \end{figure} Accurate and efficient computation of electrostatics is one of the @@ -131,13 +135,12 @@ Ewald summation method mathematically transforms this \] where $N_A$ and $N_B$ are the number of point charges in the two molecular species. Originally developed to study ionic crystals, the -Ewald summation method mathematically transforms this -straightforward but conditionally convergent summation into two more -complicated but rapidly convergent sums. One summation is carried -out in reciprocal space while the other is carried out in real -space. An alternative approach is a multipole expansion, which is -based on electrostatic moments, such as charge (monopole), dipole, -quadruple etc. +Ewald sum method mathematically transforms this straightforward but +conditionally convergent summation into two more complicated but +rapidly convergent sums. One summation is carried out in reciprocal +space while the other is carried out in real space. An alternative +approach is the multipole expansion, which is based on electrostatic +moments, such as charge (monopole), dipole, quadrupole etc. Here we consider a linear molecule which consists of two point charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The @@ -153,9 +156,10 @@ electrostatic potential at point $P$ is given by: \begin{figure} \centering \includegraphics[width=3in]{charge_dipole.eps} -\caption[Electrostatic potential due to a linear molecule comprising -two point charges]{Electrostatic potential due to a linear molecule -comprising two point charges} \label{lipidFigure:chargeDipole} +\caption[An illustration of split-dipole +approximation]{Electrostatic potential due to a linear molecule +comprising two point charges with opposite charges. } +\label{lipidFigure:chargeDipole} \end{figure} The basic assumption of the multipole expansion is $r \gg d$ , thus, @@ -191,14 +195,21 @@ another. and respectively. This approximation to the multipole expansion maintains the fast fall-off of the multipole potentials but lacks the normal divergences when two polar groups get close to one -another. -%description of the comparsion +another. The comparision between different electrostatic +approximation is shown in \ref{lipidFigure:splitDipole}. Due to the +divergence at the central region of the headgroup introduced by +dipole-dipole approximation, we discover that water molecules are +locked into the central region in the simulation. This artifact can +be corrected using split-dipole approximation or other accurate +methods. \begin{figure} \centering \includegraphics[width=\linewidth]{split_dipole.eps} -\caption[Comparison between electrostatic approximation]{Electron -density profile along the bilayer normal.} -\label{lipidFigure:splitDipole} +\caption[Comparison between electrostatic +approximation]{Electrostatic potential map for two pairs of charges +with different alignments: (a) illustration of different alignments; +(b) charge-charge interaction; (c) dipole-dipole approximation; (d) +split-dipole approximation.} \label{lipidFigure:splitDipole} \end{figure} %\section{\label{lipidSection:methods}Methods} @@ -207,13 +218,14 @@ To exclude the inter-headgroup interaction, atomistic \subsection{One Lipid in Sea of Water Molecules} -To exclude the inter-headgroup interaction, atomistic models of one -lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built -and studied using atomistic molecular dynamics. The simulation was -analyzed using a set of radial distribution functions, which give -the probability of finding a pair of molecular species a distance -apart, relative to the probability expected for a completely random -distribution function at the same density. +To tune our parameters without the inter-headgroup interactions, +atomistic models of one lipid (DMPC or DLPE) in sea of water +molecules (TIP3P) were built and studied using atomistic molecular +dynamics. The simulation was analyzed using a set of radial +distribution functions, which give the probability of finding a pair +of molecular species a distance apart, relative to the probability +expected for a completely random distribution function at the same +density. \begin{equation} g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i @@ -225,23 +237,33 @@ From figure 4(a), we can identify the first solvation } \delta (\cos \theta _{ij} - \cos \theta ) > \end{equation} -From figure 4(a), we can identify the first solvation shell (3.5 -$\AA$) and the second solvation shell (5.0 $\AA$) from both plots. -However, the corresponding orientations are different. In DLPE, -water molecules prefer to sit around -NH3 group due to the hydrogen -bonding. In contrast, because of the hydrophobic effect of the --N(CH3)3 group, the preferred position of water molecules in DMPC is -around the -PO4 Group. When the water molecules are far from the -headgroup, the distribution of the two angles should be uniform. The -correlation close to center of the headgroup dipole (< 5 $\AA$) in -both plots tell us that in the closely-bound region, the dipoles of -the water molecules are preferentially anti-aligned with the dipole -of headgroup. +From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first +solvation shell (3.5 $\AA$) and the second solvation shell (5.0 +$\AA$) from both plots. However, the corresponding orientations are +different. In DLPE, water molecules prefer to sit around $\text{{\sc +NH}}_3$ group due to the hydrogen bonding. In contrast, because of +the hydrophobic effect of the $ {\rm{N(CH}}_{\rm{3}} +{\rm{)}}_{\rm{3}} $ group, the preferred position of water molecules +in DMPC is around the $\text{{\sc PO}}_4$ Group. When the water +molecules are far from the headgroup, the distribution of the two +angles should be uniform. The correlation close to center of the +headgroup dipole in both plots tells us that in the closely-bound +region, the dipoles of the water molecules are preferentially +anti-aligned with the dipole of headgroup. When the water molecules +are far from the headgroup, the distribution of the two angles +should be uniform. The correlation close to center of the headgroup +dipole in both plots tell us that in the closely-bound region, the +dipoles of the water molecules are preferentially aligned with the +dipole of headgroup. \begin{figure} \centering \includegraphics[width=\linewidth]{g_atom.eps} -\caption[The pair correlation functions for atomistic models]{} +\caption[The pair correlation functions for atomistic models]{The +pair correlation functions for atomistic models: (a)$g(r,\cos \theta +)$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE; (c)$g(r,\cos \omega +)$ for DMPC; (d)$g(r,\cos \omega )$ for DLPE; (e)$g(\cos \theta,\cos +\omega)$ for DMPC; (f)$g(\cos \theta,\cos \omega)$ for DMLPE.} \label{lipidFigure:PCFAtom} \end{figure} @@ -260,15 +282,18 @@ atoms. \begin{figure} \centering \includegraphics[width=\linewidth]{g_coarse.eps} -\caption[The pair correlation functions for coarse-grained models]{} +\caption[The pair correlation functions for coarse-grained +models]{The pair correlation functions for coarse-grained models: +(a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.} \label{lipidFigure:PCFCoarse} \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{EWD_coarse.eps} -\caption[Excess water density of coarse-grained phospholipids]{ } -\label{lipidFigure:EWDCoarse} +\caption[Excess water density of coarse-grained +phospholipids]{Excess water density of coarse-grained +phospholipids.} \label{lipidFigure:EWDCoarse} \end{figure} \begin{table} @@ -295,16 +320,15 @@ molecules has been constructed from an atomistic coord \subsection{Bilayer Simulations Using Coarse-grained Model} A bilayer system consisting of 128 DMPC lipids and 3655 water -molecules has been constructed from an atomistic coordinate -file.[15] The MD simulation is performed at constant temperature, T -= 300K, and constant pressure, p = 1 atm, and consisted of an -equilibration period of 2 ns. During the equilibration period, the -system was initially simulated at constant volume for 1ns. Once the -system was equilibrated at constant volume, the cell dimensions of -the system was relaxed by performing under NPT conditions using -Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After -equilibration, different properties were evaluated over a production -run of 5 ns. +molecules has been constructed from an atomistic coordinate file. +The MD simulation is performed at constant temperature, T = 300K, +and constant pressure, p = 1 atm, and consisted of an equilibration +period of 2 ns. During the equilibration period, the system was +initially simulated at constant volume for 1 ns. Once the system was +equilibrated at constant volume, the cell dimensions of the system +was relaxed by performing under NPT conditions using Nos\'{e}-Hoover +extended system isothermal-isobaric dynamics. After equilibration, +different properties were evaluated over a production run of 5 ns. \begin{figure} \centering @@ -315,7 +339,7 @@ molecules.} \label{lipidFigure:bilayer} \end{figure} -\subsubsection{Electron Density Profile (EDP)} +\subsubsection{\textbf{Electron Density Profile (EDP)}} Assuming a gaussian distribution of electrons on each atomic center with a variance estimated from the size of the van der Waals radius, @@ -351,7 +375,7 @@ and total density due to DMPC in blue.} \label{lipidFigure:electronDensity} \end{figure} -\subsubsection{$\text{S}_{\text{{\sc cd}}}$ Order Parameter} +\subsubsection{\textbf{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}} Measuring deuterium order parameters by NMR is a useful technique to study the orientation of hydrocarbon chains in phospholipids. The @@ -392,8 +416,29 @@ of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grain \includegraphics[width=\linewidth]{scd.eps} \caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model -(blue) and DMPC\cite{petrache00} (black) near 300~K.} +(blue) and DMPC\cite{Petrache2000} (black) near 300~K.} \label{lipidFigure:Scd} \end{figure} %\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model} + +\section{\label{lipidSection:Conclusion}Conclusion} + +Atomistic simulations have been used in this study to determine the +preferred orientation and location of water molecules relative to +the location and orientation of the PC and PE lipid headgroups. +Based on the results from our all-atom simulations, we developed a +simple coarse-grained model which captures the essential features of +the headgroup solvation which is crucial to transport process in +membrane system. In addition, the model has been explored in a +bilayer system was shown to have reasonable electron density +profiles, $\text{S}_{\text{{\sc cd}}}$ order parameter and other +structural properties. The accuracy of this model is achieved by +matching atomistic result. It is also easy to represent different +phospholipids by changing a few parameters of the model. Another +important characteristic of this model distinguishing itself from +other models\cite{Goetz1998,Marrink2004} is the computational speed +gained by introducing a short range electrostatic approximation. +Further studies of this system using z-constraint method could +extend the time length of the simulations to study transport +phenomena in large-scale membrane systems.