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\def\op#1{\hat{#1}} |
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\begin{document} |
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\bibliographystyle{plain} |
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\bibliographystyle{unsrt} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%% Background %%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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{\color{ndblue} |
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A mesoscale model for phospholipids has been developed for molecular |
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dynamics simulations of lipid bilayers. The model makes several |
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dynamics simulations of phospholipid phase transitions. The model makes several |
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simplifications to both the water and the phospholipids to reduce the |
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computational cost of each force evaluation. The water was represented |
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by the soft sticky dipole model of Ichiye \emph{et |
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in the tail groups to beads representing $\mbox{CH}_{2}$ and |
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$\mbox{CH}_{3}$ unified atoms, and the replacement of the head groups |
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with a single point mass containing a centrally located dipole. The |
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model was then used to simulate micelle and bilayer formation from a |
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configuration of randomly placed phospholipids which was simulated for |
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times in excess of 30 nanoseconds. |
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model was then used to simulate micelle formation from a configuration |
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of randomly placed phospholipids which was simulated for times in |
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excess of 20 nanoseconds. |
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|
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} |
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\end{kasten} |
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\label{sec:motivation} |
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|
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|
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Simulations of phospholipid bilayers are, by necessity, quite |
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Simulations of phospholipid phases are, by necessity, quite |
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complex. The lipid molecules are large, and contain many |
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atoms. Additionally, the head groups of the lipids are often |
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zwitterions, and the large separation between charges results in a |
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\end{kasten} |
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|
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\begin{kasten} |
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\subsection{{\color{ndblue}System Simplfications}} |
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\subsection{{\color{ndblue}Our Simplifications}} |
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\begin{itemize} |
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\item Unified atoms with fixed bond lengths replace groups of atoms. |
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\item Replace charge distributions with dipoles.(Eq. \ref{eq:dipole} |
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vs. Eq. \ref{eq:coloumb}) |
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\item Charge distributions are replaced with dipoles. |
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\begin{itemize} |
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\item Relatively short range, $\frac{1}{r^3}$, interactions allow |
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the application of computational simplification algorithms, |
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ie. neighbor lists. |
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the application of neighbor lists. |
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\end{itemize} |
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\end{itemize} |
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\begin{equation} |
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{r^{5}_{ij}} \biggr] |
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\label{eq:dipole} |
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\end{equation} |
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– |
\begin{equation} |
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– |
V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) = \frac{q_{i}q_{j}}% |
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{4\pi\epsilon_{0} r_{ij}} |
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– |
\label{eq:coloumb} |
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– |
\end{equation} |
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\end{kasten} |
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|
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|
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\begin{kasten} |
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\section{{\color{red}\underline{Models}}} |
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\label{sec:model} |
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\subsection{{\color{ndblue}Water Model}} |
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\subsection{{\color{ndblue}The Water Model}} |
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\label{sec:waterModel} |
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|
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The waters in the simulation were modeled after the Soft Sticky Dipole |
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\end{wrapfigure} |
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\mbox{} |
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\begin{itemize} |
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\item $\sigma$ is the Lennard-Jones length parameter. |
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\item $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$, |
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\item $\sigma$ is the Lennard-Jones length parameter |
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\item $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$ |
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\item $\mathbf{r}_{ij}$ is the vector between molecules $i$ and $j$ |
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\item $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the Euler angles of molecule $i$ or $j$ respectively. |
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\item $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the Euler angles of molecule $i$ or $j$ respectively |
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\end{itemize} |
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|
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It's potential is as follows: |
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– |
|
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\begin{equation} |
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V_{s\!s\!d} = V_{L\!J}(r_{i\!j}) + V_{d\!p}(r_{i\!j},\Omega_{i},\Omega_{j}) |
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+ V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j}) |
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+ |
\label{eq:ssdPot} |
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\end{equation} |
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|
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|
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Where $V_{d\!p}(r_{i\!j})$ is given in Eq.~\ref{eq:dipole}, and $V_{L\!J}(r_{i\!j})$ is the Lennard-Jones potential. |
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\end{kasten} |
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|
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\begin{kasten} |
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\subsection{{\color{ndblue}Soft Sticky Potential}} |
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\label{sec:SSeq} |
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|
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Hydrogen bonding in the SSD model is described by the |
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$V_{\text{sp}}$ term in Eq.~\ref{eq:ssdPot}. Its form is as follows: |
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\begin{equation} |
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V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = |
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v^{\circ}[s(r_{ij})w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) |
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+ |
+ |
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s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j})] |
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\label{eq:spPot} |
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\end{equation} |
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Where $v^\circ$ scales the strength of the interaction. |
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$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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and |
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$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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are responsible for the tetrahedral potential and a correction to the |
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tetrahedral potential respectively. They are, |
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\begin{equation} |
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w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
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\sin\theta_{ij} \sin 2\theta_{ij} \cos 2\phi_{ij} |
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+ \sin \theta_{ji} \sin 2\theta_{ji} \cos 2\phi_{ji} |
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\label{eq:spPot2} |
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\end{equation} |
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and |
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\begin{equation} |
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\begin{split} |
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w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
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&(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij} + 0.8)^2 \\ |
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&+ (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ} |
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\end{split} |
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\label{eq:spCorrection} |
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\end{equation} |
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The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical |
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coordinates of the position of molecule $j$ in the reference frame |
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fixed on molecule $i$ with the z-axis aligned with the dipole moment. |
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The correction |
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$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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is needed because |
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$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. Finally, the |
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potential is scaled by the switching function $s(r_{ij})$, |
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which scales smoothly from 0 to 1. |
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\begin{equation} |
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s(r_{ij}) = |
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\begin{cases} |
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1& \text{if $r_{ij} < r_{L}$}, \\ |
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\frac{(r_{U} - r_{ij})^2 (r_{U} + 2r_{ij} |
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- 3r_{L})}{(r_{U}-r_{L})^3}& |
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\text{if $r_{L} \leq r_{ij} \leq r_{U}$},\\ |
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0& \text{if $r_{ij} \geq r_{U}$}. |
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\end{cases} |
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\label{eq:spCutoff} |
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\end{equation} |
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|
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\end{kasten} |
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|
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|
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\end{spalte} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{spalte} |
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|
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\begin{kasten} |
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|
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\section{{\color{ndblue}Ima third column holder}} |
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|
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hello |
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\begin{kasten} |
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\subsection{{\color{ndblue}Hydrogen Bonding in SSD}} |
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\label{sec:hbonding} |
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|
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< |
\end{kasten} |
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The SSD model's $V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j})$ |
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recreates the hydrogen bonding network of water. |
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\begin{center} |
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\begin{minipage}{100mm} |
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\begin{minipage}[t]{48mm} |
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\begin{center} |
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\includegraphics[width=48mm]{iced_final.eps}\\ |
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SSD Relaxed on a diamond lattice |
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\end{center} |
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\end{minipage} |
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\hspace{4mm}% |
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\begin{minipage}[t]{48mm} |
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\begin{center} |
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\includegraphics[width=48mm]{dipoled_final.eps}\\ |
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Stockmayer Spheres relaxed on a diamond lattice |
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\end{center} |
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> |
\end{minipage} |
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\end{minipage} |
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|
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\end{center} |
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|
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|
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\end{kasten} |
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|
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|
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\begin{kasten} |
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|
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\subsection{{\color{ndblue}The Lipid Model}} |
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\label{sec:lipidModel} |
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|
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\begin{center} |
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\includegraphics[width=25mm,angle=-90]{lipidModel.epsi} |
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\end{center} |
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|
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\begin{itemize} |
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\item PC \& PE head groups are replaced by a Lennard-Jones sphere containing a dipole at its center |
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\item Atoms in the tail chains modeled as unified groups of atoms |
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\item Tail group interaction parameters based on those of TraPPE\cite{Siepmann1998} |
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\end{itemize} |
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|
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The total potential is given by: |
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\begin{equation} |
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V_{\text{lipid}} = |
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\sum_{i}V_{i}^{\text{internal}} |
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+ \sum_i \sum_{j>i} \sum_{\text{$\alpha$ in $i$}} |
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\sum_{\text{$\beta$ in $j$}} |
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V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}}) |
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+\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j}) |
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\end{equation} |
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Where |
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+ |
\begin{equation} |
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V_{i}^{\text{internal}} = |
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+ |
\sum_{\text{bends}}V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
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+ |
+ \sum_{\text{torsions}}V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
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+ |
+ \sum_{\alpha} \sum_{\beta>\alpha}V_{\text{LJ}}(r_{\alpha \beta}) |
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+ |
\end{equation} |
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The bend and torsion potentials were of the form: |
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\begin{equation} |
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+ |
V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
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+ |
= k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \theta_0)^2}{2} |
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+ |
\label{eq:bendPot} |
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+ |
\end{equation} |
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\begin{equation} |
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V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
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= c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}] |
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+ |
+ c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] |
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+ |
+ c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})] |
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+ |
\label{eq:torsPot} |
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\end{equation} |
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|
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|
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\end{kasten} |
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|
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\begin{kasten} |
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|
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\section{{\color{red}\underline{Initial Results}}} |
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\label{sec:results} |
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\subsection{{\color{ndblue}Simulation Snapshots:50 lipids in a sea of 1384 waters}} |
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\label{sec:r50snapshots} |
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|
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\begin{center} |
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\includegraphics[width=105mm]{r50-montage.eps} |
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+ |
\end{center} |
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+ |
|
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+ |
\end{kasten} |
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|
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|
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\end{spalte} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%% fourth column %%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
\begin{spalte} |
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|
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\begin{kasten} |
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|
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\subsection{{\color{ndblue}Position and Angular Correlations}} |
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\label{sec:r50corr} |
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|
|
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+ |
\begin{center} |
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+ |
\begin{minipage}{110mm} |
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+ |
\begin{minipage}[t]{55mm} |
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+ |
\begin{center} |
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+ |
\includegraphics[width=36mm,angle=-90]{r50-HEAD-HEAD.epsi}\\ |
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+ |
The self correlation of the head groups |
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+ |
\end{center} |
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+ |
\end{minipage} |
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+ |
\begin{minipage}[t]{55mm} |
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\begin{center} |
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+ |
\includegraphics[width=36mm,angle=-90]{r50-CH2-CH2.epsi}\\ |
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The self correlation of the tail beads. |
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+ |
\end{center} |
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+ |
\end{minipage} |
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+ |
\end{minipage} |
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+ |
\end{center} |
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+ |
\begin{equation} |
| 566 |
+ |
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} |
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+ |
\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 568 |
+ |
\label{eq:gofr} |
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+ |
\end{equation} |
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+ |
\begin{equation} |
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+ |
g_{\gamma}(r) = \langle \sum_i \sum_{j>i} |
| 572 |
+ |
(\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle |
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+ |
\label{eq:gammaofr} |
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+ |
\end{equation} |
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|
|
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+ |
\end{kasten} |
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|
|
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|
|
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\begin{kasten} |
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+ |
|
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\subsection{{\color{red}\underline{Discussion}}} |
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+ |
\label{sec:discussion} |
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+ |
|
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+ |
The initial results show much promise for the model. The |
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+ |
system of 50 lipids was able to form micelles quickly, however |
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+ |
bilayer formation was not seen on the time scale of the |
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+ |
current simulation. Current simulations are exploring the |
| 588 |
+ |
parameter space of the model when the tail beads are larger than |
| 589 |
+ |
the head group. This should help to drive the system toward a |
| 590 |
+ |
bilayer rather than a micelle. Work is also being done on the |
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+ |
simulation engine to allow for the box size of the system to |
| 592 |
+ |
be adjustable in all three dimensions to allow for constant |
| 593 |
+ |
pressure. |
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+ |
|
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+ |
\end{kasten} |
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|
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|
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\begin{kasten} |
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|
\begin{center} |
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|
{\large{\color{red} \underline{Acknowledgments}}} |
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The authors would like to acknowledge Charles Vardeman, Christopher |
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Fennell, and Teng lin for their contributions to the simulation |
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|
engine. MAM would also like to extend a special thank you to Charles |
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< |
Vardeman for his help with the TeX formatting of this |
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< |
poster. Computaion time was provided on the Bunch-of-Boxes (B.o.B.) |
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> |
Vardeman for his help with the \TeX formatting of this |
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> |
poster. Computation time was provided on the Notre Dame Bunch-of-Boxes (B.o.B.) |
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cluster under NSF grant DMR 00 79647. The authors acknowledge support |
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under NSF grant CHE-0134881. |
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|
