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\begin{document} |
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\bibliographystyle{unsrt} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%% Header %%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{center} |
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\color{ndblue} |
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\textbf{\Huge {A Mesoscale Model for Phospholipid Simulations}}\\[0.5em] |
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\textsc{\LARGE \underline{Matthew~A.~Meineke}, and J.~Daniel~Gezelter}\\[0.3em] |
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{\large Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556, USA\\ |
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{\tt\ mmeineke@nd.edu} |
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} |
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\end{center}} |
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\hfill}}\hfill\mbox{}\\[1.cm] |
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%\vspace*{1.3cm} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%% first column %%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{spalte} |
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\begin{kasten} |
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% |
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% This begins the first "kasten" or box |
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% |
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% |
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\begin{center} |
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{\large{\color{red} \underline{ABSTRACT} } } |
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\end{center} |
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|
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{\color{ndblue} |
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|
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A mesoscale model for phospholipids has been developed for molecular |
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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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al}.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md} The |
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simplifications to the phospholipids included the reduction of atoms |
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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 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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|
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|
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\begin{kasten} |
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\section{{\color{red}\underline{Introduction \& Background}}} |
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\label{sec:intro} |
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|
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%% \subsection{{\color{ndblue}Motivation}} |
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\label{sec:motivation} |
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|
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|
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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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large dipole moment. Adding to the complexity are the number of water |
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molecules needed to properly solvate the lipid bilayer, typically 25 |
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water molecules for every lipid molecule. These factors make it |
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difficult to study certain biologically interesting phenomena that |
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have large inherent length or time scale. |
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|
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\end{kasten} |
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|
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\begin{kasten} |
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\subsection{{\color{ndblue}Ripple Phase}} |
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|
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\begin{wrapfigure}{o}{60mm} |
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\centering |
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\includegraphics[width=40mm]{ripple.epsi} |
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\end{wrapfigure} |
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|
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\mbox{} |
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\begin{itemize} |
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\item The ripple (~$P_{\beta'}$~) phase lies in the transition from the gel to fluid phase. |
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\item Periodicity of 100 - 200 $\mbox{\AA}$\cite{Cevc87} |
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\item Current simulations have box sizes ranging from 50 - 100 $\mbox{\AA}$ on a side.\cite{saiz02,lindahl00,venable00} |
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\end{itemize} |
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|
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\label{sec:ripplePhase} |
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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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\subsection{{\color{ndblue}Diffusion \& Formation Dynamics}} |
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\begin{itemize} |
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|
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\item |
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Drug Diffusion |
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\begin{itemize} |
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\item |
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Some drug molecules may spend appreciable amounts of time in the |
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membrane |
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|
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\item |
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Long time scale dynamics are need to observe and characterize their |
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actions |
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\end{itemize} |
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|
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\item |
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Bilayer Formation Dynamics |
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\begin{itemize} |
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\item |
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Current lipid simulations indicate\cite{Marrink01}: |
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\begin{itemize} |
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\item Aggregation can happen as quickly as 200 ps |
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|
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\item Bilayers can take up to 20 ns to form completely |
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\end{itemize} |
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|
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\end{itemize} |
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\end{itemize} |
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\end{kasten} |
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|
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\begin{kasten} |
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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 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 neighbor lists. |
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\end{itemize} |
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\end{itemize} |
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\begin{equation} |
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V^{\text{dp}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
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\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
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- |
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\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
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(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
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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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\end{kasten} |
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\end{spalte} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%% second column %%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{spalte} |
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\begin{kasten} |
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\subsection{{\color{ndblue}Reduction in calculations}} |
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Unified water and lipid models and decrease the number of interactions |
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needed between two molecules. |
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|
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\begin{center} |
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\includegraphics[width=50mm,angle=-90]{reduction.epsi} |
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\end{center} |
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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}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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(SSD) model of Ichiye.\cite{liu96:new_model} Where: |
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|
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\begin{wrapfigure}[10]{o}{60mm} |
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\begin{center} |
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\includegraphics[width=40mm]{ssd.epsi} |
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\end{center} |
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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 $\mathbf{r}_{ij}$ 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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\end{itemize} |
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|
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It's potential is as follows: |
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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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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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\end{kasten} |
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\end{spalte} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%% third column %%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{spalte} |
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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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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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\end{kasten} |
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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} |
| 484 |
|
| 485 |
\begin{itemize} |
| 486 |
\item PC \& PE head groups are replaced by a Lennard-Jones sphere containing a dipole at its center |
| 487 |
\item Atoms in the tail chains modeled as unified groups of atoms |
| 488 |
\item Tail group interaction parameters based on those of TraPPE\cite{Siepmann1998} |
| 489 |
\end{itemize} |
| 490 |
|
| 491 |
The total potential is given by: |
| 492 |
\begin{equation} |
| 493 |
V_{\text{lipid}} = |
| 494 |
\sum_{i}V_{i}^{\text{internal}} |
| 495 |
+ \sum_i \sum_{j>i} \sum_{\text{$\alpha$ in $i$}} |
| 496 |
\sum_{\text{$\beta$ in $j$}} |
| 497 |
V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}}) |
| 498 |
+\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j}) |
| 499 |
\end{equation} |
| 500 |
Where |
| 501 |
\begin{equation} |
| 502 |
V_{i}^{\text{internal}} = |
| 503 |
\sum_{\text{bends}}V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
| 504 |
+ \sum_{\text{torsions}}V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
| 505 |
+ \sum_{\alpha} \sum_{\beta>\alpha}V_{\text{LJ}}(r_{\alpha \beta}) |
| 506 |
\end{equation} |
| 507 |
The bend and torsion potentials were of the form: |
| 508 |
\begin{equation} |
| 509 |
V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
| 510 |
= k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \theta_0)^2}{2} |
| 511 |
\label{eq:bendPot} |
| 512 |
\end{equation} |
| 513 |
\begin{equation} |
| 514 |
V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
| 515 |
= c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}] |
| 516 |
+ c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] |
| 517 |
+ c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})] |
| 518 |
\label{eq:torsPot} |
| 519 |
\end{equation} |
| 520 |
|
| 521 |
|
| 522 |
\end{kasten} |
| 523 |
|
| 524 |
\begin{kasten} |
| 525 |
|
| 526 |
\section{{\color{red}\underline{Initial Results}}} |
| 527 |
\label{sec:results} |
| 528 |
\subsection{{\color{ndblue}Simulation Snapshots:50 lipids in a sea of 1384 waters}} |
| 529 |
\label{sec:r50snapshots} |
| 530 |
|
| 531 |
\begin{center} |
| 532 |
\includegraphics[width=105mm]{r50-montage.eps} |
| 533 |
\end{center} |
| 534 |
|
| 535 |
\end{kasten} |
| 536 |
|
| 537 |
|
| 538 |
\end{spalte} |
| 539 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 540 |
%%% fourth column %%% |
| 541 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 542 |
\begin{spalte} |
| 543 |
|
| 544 |
\begin{kasten} |
| 545 |
|
| 546 |
\subsection{{\color{ndblue}Position and Angular Correlations}} |
| 547 |
\label{sec:r50corr} |
| 548 |
|
| 549 |
\begin{center} |
| 550 |
\begin{minipage}{110mm} |
| 551 |
\begin{minipage}[t]{55mm} |
| 552 |
\begin{center} |
| 553 |
\includegraphics[width=36mm,angle=-90]{r50-HEAD-HEAD.epsi}\\ |
| 554 |
The self correlation of the head groups |
| 555 |
\end{center} |
| 556 |
\end{minipage} |
| 557 |
\begin{minipage}[t]{55mm} |
| 558 |
\begin{center} |
| 559 |
\includegraphics[width=36mm,angle=-90]{r50-CH2-CH2.epsi}\\ |
| 560 |
The self correlation of the tail beads. |
| 561 |
\end{center} |
| 562 |
\end{minipage} |
| 563 |
\end{minipage} |
| 564 |
\end{center} |
| 565 |
\begin{equation} |
| 566 |
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} |
| 567 |
\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 568 |
\label{eq:gofr} |
| 569 |
\end{equation} |
| 570 |
\begin{equation} |
| 571 |
g_{\gamma}(r) = \langle \sum_i \sum_{j>i} |
| 572 |
(\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 573 |
\label{eq:gammaofr} |
| 574 |
\end{equation} |
| 575 |
|
| 576 |
\end{kasten} |
| 577 |
|
| 578 |
|
| 579 |
\begin{kasten} |
| 580 |
|
| 581 |
\subsection{{\color{red}\underline{Discussion}}} |
| 582 |
\label{sec:discussion} |
| 583 |
|
| 584 |
The initial results show much promise for the model. The |
| 585 |
system of 50 lipids was able to form micelles quickly, however |
| 586 |
bilayer formation was not seen on the time scale of the |
| 587 |
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 |
| 591 |
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. |
| 594 |
|
| 595 |
\end{kasten} |
| 596 |
|
| 597 |
|
| 598 |
\begin{kasten} |
| 599 |
\begin{center} |
| 600 |
{\large{\color{red} \underline{Acknowledgments}}} |
| 601 |
\end{center} |
| 602 |
|
| 603 |
The authors would like to acknowledge Charles Vardeman, Christopher |
| 604 |
Fennell, and Teng lin for their contributions to the simulation |
| 605 |
engine. MAM would also like to extend a special thank you to Charles |
| 606 |
Vardeman for his help with the \TeX formatting of this |
| 607 |
poster. Computation time was provided on the Notre Dame Bunch-of-Boxes (B.o.B.) |
| 608 |
cluster under NSF grant DMR 00 79647. The authors acknowledge support |
| 609 |
under NSF grant CHE-0134881. |
| 610 |
|
| 611 |
\end{kasten} |
| 612 |
|
| 613 |
\vspace{0.5cm} |
| 614 |
\begin{kasten} |
| 615 |
{\small |
| 616 |
\bibliography{poster} |
| 617 |
} |
| 618 |
\end{kasten} |
| 619 |
\end{spalte} |
| 620 |
} |
| 621 |
\end{lrbox} |
| 622 |
\resizebox*{0.98\textwidth}{!}{% |
| 623 |
\usebox{\spalten}}\hfill\mbox{}\vfill |
| 624 |
\end{document} |
