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\chapter{\label{chapt:lipid}LIPID MODELING} |
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\section{\label{lipidSection:introduction}Introduction} |
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Under biologically relevant conditions, phospholipids are solvated |
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in aqueous solutions at a roughly 25:1 ratio. Solvation can have a |
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tremendous impact on transport phenomena in biological membranes |
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since it can affect the dynamics of ions and molecules that are |
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transferred across membranes. Studies suggest that because of the |
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directional hydrogen bonding ability of the lipid headgroups, a |
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small number of water molecules are strongly held around the |
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different parts of the headgroup and are oriented by them with |
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residence times for the first hydration shell being around 0.5 - 1 |
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ns\cite{Ho1992}. In the second solvation shell, some water molecules |
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are weakly bound, but are still essential for determining the |
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properties of the system. Transport of various molecular species |
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into living cells is one of the major functions of membranes. A |
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thorough understanding of the underlying molecular mechanism for |
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solute diffusion is crucial to the further studies of other related |
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biological processes. All transport across cell membranes takes |
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place by one of two fundamental processes: Passive transport is |
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driven by bulk or inter-diffusion of the molecules being transported |
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or by membrane pores which facilitate crossing. Active transport |
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depends upon the expenditure of cellular energy in the form of ATP |
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hydrolysis. As the central processes of membrane assembly, |
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translocation of phospholipids across membrane bilayers requires the |
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hydrophilic head of the phospholipid to pass through the highly |
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hydrophobic interior of the membrane, and for the hydrophobic tails |
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to be exposed to the aqueous environment\cite{Sasaki2004}. A number |
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of studies indicate that the flipping of phospholipids occurs |
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rapidly in the eukaryotic ER and the bacterial cytoplasmic membrane |
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via a bi-directional, facilitated diffusion process requiring no |
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metabolic energy input. Another system of interest would be the |
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distribution of sites occupied by inhaled anesthetics in membrane. |
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Although the physiological effects of anesthetics have been |
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extensively studied, the controversy over their effects on lipid |
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bilayers still continues. Recent deuterium NMR measurements on |
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halothane in POPC lipid bilayers suggest the anesthetics are |
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primarily located at the hydrocarbon chain region\cite{Baber1995}. |
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Infrared spectroscopy experiments suggest that halothane in DMPC |
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lipid bilayers lives near the membrane/water |
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interface\cite{Lieb1982}. |
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Molecular dynamics simulations have proven to be a powerful tool for |
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studying the functions of biological systems, providing structural, |
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thermodynamic and dynamical information. Unfortunately, much of |
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biological interest happens on time and length scales well beyond |
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the range of current simulation technologies. |
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%review of coarse-grained modeling |
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Several schemes are proposed in this chapter to overcome these |
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difficulties. |
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\section{\label{lipidSection:model}Model} |
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\subsection{\label{lipidSection:SSD}The Soft Sticky Dipole Water Model} |
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In a typical bilayer simulation, the dominant portion of the |
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computation time will be spent calculating water-water interactions. |
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As an efficient solvent model, the Soft Sticky Dipole (SSD) water |
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model\cite{Chandra1999,Fennell2004} is used as the explicit solvent |
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in this project. Unlike other water models which have partial |
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charges distributed throughout the whole molecule, the SSD water |
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model consists of a single site which is a Lennard-Jones interaction |
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site, as well as a point dipole. A tetrahedral potential is added to |
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correct for hydrogen bonding. The following equation describes the |
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interaction between two water molecules: |
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\[ |
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V_{SSD} = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) |
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+ V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) |
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\] |
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where $r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
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$\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom |
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for molecule $i$ and molecule $j$ respectively. |
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\[ |
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V_{LJ} (r_{ij} ) = 4\varepsilon _{ij} \left[ {\left( {\frac{{\sigma |
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_{ij} }}{{r_{ij} }}} \right)^{12} - \left( {\frac{{\sigma _{ij} |
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}}{{r_{ij} }}} \right)^6 } \right] |
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\] |
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\[ |
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V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
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_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }} - |
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\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
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r_{ij} } \right)}}{{r_{ij}^5 }}} \right] |
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\] |
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\[ |
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V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) = v_0 [s(r_{ij} )w(r_{ij} |
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,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i ,\Omega _j |
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)] |
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\] |
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where $v_0$ is a strength parameter, $s$ and $s'$ are cubic |
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switching functions, while $w$ and $w'$ are responsible for the |
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tetrahedral potential and the short-range correction to the dipolar |
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interaction respectively. |
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\[ |
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\begin{array}{l} |
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w(r_{ij} ,\Omega _i ,\Omega _j ) = \sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij} \\ |
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w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - w_0 \\ |
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\end{array} |
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\] |
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Although dipole-dipole and sticky interactions are more |
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mathematically complicated than Coulomb interactions, the number of |
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pair interactions is reduced dramatically both because the model |
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only contains a single-point as well as "short range" nature of the |
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higher order interaction. |
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\subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model} |
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Fig.~\ref{lipidFigure:coarseGrained} shows a schematic for our |
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coarse-grained phospholipid model. The lipid head group is modeled |
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by a linear rigid body which consists of three Lennard-Jones spheres |
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and a centrally located point-dipole. The backbone atoms in the |
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glycerol motif are modeled by Lennard-Jones spheres with dipoles. |
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Alkyl groups in hydrocarbon chains are replaced with unified |
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$\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms. |
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\begin{figure} |
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\centering |
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\includegraphics[width=3in]{coarse_grained.eps} |
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\caption[A representation of coarse-grained phospholipid model]{} |
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\label{lipidFigure:coarseGrained} |
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\end{figure} |
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Accurate and efficient computation of electrostatics is one of the |
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most difficult tasks in molecular modeling. Traditionally, the |
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electrostatic interaction between two molecular species is |
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calculated as a sum of interactions between pairs of point charges, |
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using Coulomb's law: |
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\[ |
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V = \sum\limits_{i = 1}^{N_A } {\sum\limits_{j = 1}^{N_B } |
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{\frac{{q_i q_j }}{{4\pi \varepsilon _0 r_{ij} }}} } |
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\] |
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where $N_A$ and $N_B$ are the number of point charges in the two |
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molecular species. Originally developed to study ionic crystals, the |
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Ewald summation method mathematically transforms this |
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straightforward but conditionally convergent summation into two more |
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complicated but rapidly convergent sums. One summation is carried |
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out in reciprocal space while the other is carried out in real |
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space. An alternative approach is a multipole expansion, which is |
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based on electrostatic moments, such as charge (monopole), dipole, |
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quadruple etc. |
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Here we consider a linear molecule which consists of two point |
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charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The |
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electrostatic potential at point $P$ is given by: |
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\[ |
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\frac{1}{{4\pi \varepsilon _0 }}\left( {\frac{{ - q}}{{r_ - }} + |
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\frac{{ + q}}{{r_ + }}} \right) = \frac{1}{{4\pi \varepsilon _0 |
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}}\left( {\frac{{ - q}}{{\sqrt {r^2 + \frac{{d^2 }}{4} + rd\cos |
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\theta } }} + \frac{q}{{\sqrt {r^2 + \frac{{d^2 }}{4} - rd\cos |
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\theta } }}} \right) |
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\] |
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\begin{figure} |
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\centering |
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\includegraphics[width=3in]{charge_dipole.eps} |
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\caption[Electrostatic potential due to a linear molecule comprising |
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two point charges]{Electrostatic potential due to a linear molecule |
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comprising two point charges} \label{lipidFigure:chargeDipole} |
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\end{figure} |
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The basic assumption of the multipole expansion is $r \gg d$ , thus, |
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$\frac{{d^2 }}{4}$ inside the square root of the denominator is |
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neglected. This is a reasonable approximation in most cases. |
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Unfortunately, in our headgroup model, the distance of charge |
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separation $d$ (4.63 $\AA$ in PC headgroup) may be comparable to |
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$r$. Nevertheless, we could still assume $ \cos \theta \approx 0$ |
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in the central region of the headgroup. Using Taylor expansion and |
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associating appropriate terms with electric moments will result in a |
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"split-dipole" approximation: |
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\[ |
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V(r) = \frac{1}{{4\pi \varepsilon _0 }}\frac{{r\mu \cos \theta |
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}}{{R^3 }} |
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\] |
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where$R = \sqrt {r^2 + \frac{{d^2 }}{4}}$ Placing a dipole at point |
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$P$ and applying the same strategy, the interaction between two |
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split-dipoles is then given by: |
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\[ |
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V_{sd} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
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_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{R_{ij}^3 }} - |
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\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
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r_{ij} } \right)}}{{R_{ij}^5 }}} \right] |
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\] |
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where $\mu _i$ and $\mu _j$ are the dipole moment of molecule $i$ |
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and molecule $j$ respectively, $r_{ij}$ is vector between molecule |
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$i$ and molecule $j$, and $R_{ij}$ is given by, |
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\[ |
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R_{ij} = \sqrt {r_{ij}^2 + \frac{{d_i^2 }}{4} + \frac{{d_j^2 |
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}}{4}} |
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\] |
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where $d_i$ and $d_j$ are the charge separation distance of dipole |
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and respectively. This approximation to the multipole expansion |
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maintains the fast fall-off of the multipole potentials but lacks |
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the normal divergences when two polar groups get close to one |
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another. |
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%description of the comparsion |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{split_dipole.eps} |
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\caption[Comparison between electrostatic approximation]{Electron |
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density profile along the bilayer normal.} |
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\label{lipidFigure:splitDipole} |
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\end{figure} |
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%\section{\label{lipidSection:methods}Methods} |
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\section{\label{lipidSection:resultDiscussion}Results and Discussion} |
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\subsection{One Lipid in Sea of Water Molecules} |
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To exclude the inter-headgroup interaction, atomistic models of one |
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lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built |
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and studied using atomistic molecular dynamics. The simulation was |
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analyzed using a set of radial distribution functions, which give |
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the probability of finding a pair of molecular species a distance |
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apart, relative to the probability expected for a completely random |
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distribution function at the same density. |
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\begin{equation} |
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g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i |
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\in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} } > |
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\end{equation} |
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\begin{equation} |
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g_{AB} (r,\cos \theta ) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < |
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\sum\limits_{i \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} |
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} \delta (\cos \theta _{ij} - \cos \theta ) > |
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\end{equation} |
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From figure 4(a), we can identify the first solvation shell (3.5 |
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$\AA$) and the second solvation shell (5.0 $\AA$) from both plots. |
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However, the corresponding orientations are different. In DLPE, |
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water molecules prefer to sit around -NH3 group due to the hydrogen |
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bonding. In contrast, because of the hydrophobic effect of the |
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-N(CH3)3 group, the preferred position of water molecules in DMPC is |
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around the -PO4 Group. When the water molecules are far from the |
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headgroup, the distribution of the two angles should be uniform. The |
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correlation close to center of the headgroup dipole (< 5 $\AA$) in |
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both plots tell us that in the closely-bound region, the dipoles of |
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the water molecules are preferentially anti-aligned with the dipole |
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of headgroup. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{g_atom.eps} |
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\caption[The pair correlation functions for atomistic models]{} |
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\label{lipidFigure:PCFAtom} |
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\end{figure} |
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The initial configurations of coarse-grained systems are constructed |
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from the previous atomistic ones. The parameters for the |
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coarse-grained model in Table~\ref{lipidTable:parameter} are |
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estimated and tuned using isothermal-isobaric molecular dynamics. |
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Pair distribution functions calculated from coarse-grained models |
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preserve the basic characteristics of the atomistic simulations. The |
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water density, measured in a head-group-fixed reference frame, |
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surrounding two phospholipid headgroups is shown in |
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Fig.~\ref{lipidFigure:PCFCoarse}. It is clear that the phosphate end |
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in DMPC and the amine end in DMPE are the two most heavily solvated |
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atoms. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{g_coarse.eps} |
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\caption[The pair correlation functions for coarse-grained models]{} |
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\label{lipidFigure:PCFCoarse} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{EWD_coarse.eps} |
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\caption[Excess water density of coarse-grained phospholipids]{ } |
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\label{lipidFigure:EWDCoarse} |
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\end{figure} |
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\begin{table} |
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\caption{The Parameters For Coarse-grained Phospholipids} |
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\label{lipidTable:parameter} |
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\begin{center} |
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\begin{tabular}{|l|c|c|c|c|c|} |
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\hline |
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% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
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Atom type & Mass & $\sigma$ & $\epsilon$ & charge & Dipole \\ |
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$\text{{\sc CH}}_2$ & 14.03 & 3.95 & 0.0914 & 0 & 0 \\ |
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$\text{{\sc CH}}_3$ & 15.04 & 3.75 & 0.195 & 0 & 0 \\ |
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$\text{{\sc CE}}$ & 28.01 & 3.427& 0.294 & 0 & 1.693 \\ |
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$\text{{\sc CK}}$ & 28.01 & 3.592& 0.311 & 0 & 2.478 \\ |
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$\text{{\sc PO}}_4$ & 108.995& 3.9 & 1.88 & -1& 0 \\ |
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$\text{{\sc HDP}}$ & 14.03 & 4.0 & 0.13 & 0 & 0 \\ |
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$\text{{\sc NC}}_4$ & 73.137 & 4.9 & 0.88 & +1& 0 \\ |
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$\text{{\sc NH}}_3$ & 17.03 & 3.25 & 0.17 & +1& 0\\ |
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\hline |
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\end{tabular} |
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\end{center} |
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\end{table} |
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\subsection{Bilayer Simulations Using Coarse-grained Model} |
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A bilayer system consisting of 128 DMPC lipids and 3655 water |
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molecules has been constructed from an atomistic coordinate |
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file.[15] The MD simulation is performed at constant temperature, T |
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= 300K, and constant pressure, p = 1 atm, and consisted of an |
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equilibration period of 2 ns. During the equilibration period, the |
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system was initially simulated at constant volume for 1ns. Once the |
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system was equilibrated at constant volume, the cell dimensions of |
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the system was relaxed by performing under NPT conditions using |
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Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After |
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equilibration, different properties were evaluated over a production |
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run of 5 ns. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{bilayer.eps} |
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\caption[Image of a coarse-grained bilayer system]{A coarse-grained |
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bilayer system consisting of 128 DMPC lipids and 3655 SSD water |
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molecules.} |
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\label{lipidFigure:bilayer} |
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\end{figure} |
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\subsubsection{Electron Density Profile (EDP)} |
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Assuming a gaussian distribution of electrons on each atomic center |
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with a variance estimated from the size of the van der Waals radius, |
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the EDPs which are proportional to the density profiles measured |
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along the bilayer normal obtained by x-ray scattering experiment, |
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can be expressed by\cite{Tu1995} |
325 |
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\begin{equation} |
326 |
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\rho _{x - ray} (z)dz \propto \sum\limits_{i = 1}^N {\frac{{n_i |
327 |
|
|
}}{V}\frac{1}{{\sqrt {2\pi \sigma ^2 } }}e^{ - (z - z_i )^2 /2\sigma |
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|
^2 } dz}, |
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|
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\end{equation} |
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|
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where $\sigma$ is the variance equal to the van der Waals radius, |
331 |
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$n_i$ is the atomic number of site $i$ and $V$ is the volume of the |
332 |
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slab between $z$ and $z+dz$ . The highest density of total EDP |
333 |
|
|
appears at the position of lipid-water interface corresponding to |
334 |
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headgroup, glycerol, and carbonyl groups of the lipids and the |
335 |
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distribution of water locked near the head groups, while the lowest |
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electron density is in the hydrocarbon region. As a good |
337 |
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approximation to the thickness of the bilayer, the headgroup spacing |
338 |
|
|
, is defined as the distance between two peaks in the electron |
339 |
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density profile, calculated from our simulations to be 34.1 $\AA$. |
340 |
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This value is close to the x-ray diffraction experimental value 34.4 |
341 |
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$\AA$\cite{Petrache1998}. |
342 |
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|
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|
|
\begin{figure} |
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|
|
\centering |
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|
|
\includegraphics[width=\linewidth]{electron_density.eps} |
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|
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\caption[The density profile of the lipid bilayers]{Electron density |
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|
|
profile along the bilayer normal. The water density is shown in red, |
348 |
|
|
the density due to the headgroups in green, the glycerol backbone in |
349 |
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brown, $\text{{\sc CH}}_2$ in yellow, $\text{{\sc CH}}_3$ in cyan, |
350 |
|
|
and total density due to DMPC in blue.} |
351 |
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\label{lipidFigure:electronDensity} |
352 |
|
|
\end{figure} |
353 |
|
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|
354 |
|
|
\subsubsection{$\text{S}_{\text{{\sc cd}}}$ Order Parameter} |
355 |
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|
356 |
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Measuring deuterium order parameters by NMR is a useful technique to |
357 |
|
|
study the orientation of hydrocarbon chains in phospholipids. The |
358 |
|
|
order parameter tensor $S$ is defined by: |
359 |
|
|
\begin{equation} |
360 |
|
|
S_{ij} = \frac{1}{2} < 3\cos \theta _i \cos \theta _j - \delta |
361 |
|
|
_{ij} > |
362 |
|
|
\end{equation} |
363 |
|
|
where $\theta$ is the angle between the $i$th molecular axis and |
364 |
|
|
the bilayer normal ($z$ axis). The brackets denote an average over |
365 |
|
|
time and molecules. The molecular axes are defined: |
366 |
|
|
\begin{itemize} |
367 |
|
|
\item $\mathbf{\hat{z}}$ is the vector from $C_{n-1}$ to $C_{n+1}$. |
368 |
|
|
\item $\mathbf{\hat{y}}$ is the vector that is perpendicular to $z$ and |
369 |
|
|
in the plane through $C_{n-1}$, $C_{n}$, and $C_{n+1}$. |
370 |
|
|
\item $\mathbf{\hat{x}}$ is the vector perpendicular to |
371 |
|
|
$\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$. |
372 |
|
|
\end{itemize} |
373 |
|
|
In coarse-grained model, although there are no explicit hydrogens, |
374 |
|
|
the order parameter can still be written in terms of carbon ordering |
375 |
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at each point of the chain\cite{Egberts1988} |
376 |
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\begin{equation} |
377 |
|
|
S_{ij} = \frac{1}{2} < 3\cos \theta _i \cos \theta _j - \delta |
378 |
|
|
_{ij} >. |
379 |
|
|
\end{equation} |
380 |
|
|
|
381 |
|
|
Fig.~\ref{lipidFigure:Scd} shows the order parameter profile |
382 |
|
|
calculated for our coarse-grained DMPC bilayer system at 300K. Also |
383 |
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shown are the experimental data of Tu\cite{Tu1995}. The fact that |
384 |
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$\text{S}_{\text{{\sc cd}}}$ order parameters calculated from |
385 |
|
|
simulation are higher than the experimental ones is ascribed to the |
386 |
|
|
assumption of the locations of implicit hydrogen atoms which are |
387 |
|
|
fixed in coarse-grained models at positions relative to the CC |
388 |
|
|
vector. |
389 |
|
|
|
390 |
|
|
\begin{figure} |
391 |
|
|
\centering |
392 |
|
|
\includegraphics[width=\linewidth]{scd.eps} |
393 |
|
|
\caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison |
394 |
|
|
of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model |
395 |
|
|
(blue) and DMPC\cite{petrache00} (black) near 300~K.} |
396 |
|
|
\label{lipidFigure:Scd} |
397 |
|
|
\end{figure} |
398 |
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|
399 |
|
|
%\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model} |