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%% this is my poster for the Midwest Theoretical Conference |
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\documentclass[10pt]{scrartcl} |
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% on making postscript, resizeing, and printing the poster file |
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% P. Hirschfeld 2/11/00 |
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% The recommended procedure is to first generate a ``Special Format" size poster |
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% file, which is relatively easy to manipulate and view. It can be |
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% resized later to A0 (900 x 1100 mm) full poster size, or A4 or Letter size |
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% as desired (see web site). Note the large format printers currently |
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\def\breite{390mm} % Special Format. |
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\def\op#1{\hat{#1}} |
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\begin{document} |
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\bibliographystyle{unsrt} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%% Background %%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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{\newrgbcolor{gradbegin}{0.0 0.01875 0.6992}% |
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gradbegin=gradbegin,gradmidpoint=0.1](\bgwidth,-\bgheight)} |
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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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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\begin{center} |
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{\large{\color{red} \underline{ABSTRACT} } } |
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\end{center} |
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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 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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\end{kasten} |
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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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%% \subsection{{\color{ndblue}Motivation}} |
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\label{sec:motivation} |
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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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\end{kasten} |
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\begin{kasten} |
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\subsection{{\color{ndblue}Ripple Phase}} |
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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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\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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\label{sec:ripplePhase} |
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\end{kasten} |
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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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\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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\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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\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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\item Bilayers can take up to 20 ns to form completely |
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\end{itemize} |
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\end{itemize} |
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\end{itemize} |
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\end{kasten} |
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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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\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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\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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\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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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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\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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It's potential is as follows: |
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\begin{equation} |
372 |
|
|
V_{s\!s\!d} = V_{L\!J}(r_{i\!j}) + V_{d\!p}(r_{i\!j},\Omega_{i},\Omega_{j}) |
373 |
|
|
+ V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j}) |
374 |
mmeineke |
552 |
\label{eq:ssdPot} |
375 |
mmeineke |
550 |
\end{equation} |
376 |
mmeineke |
553 |
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. |
377 |
mmeineke |
550 |
\end{kasten} |
378 |
|
|
|
379 |
mmeineke |
552 |
\begin{kasten} |
380 |
|
|
\subsection{{\color{ndblue}Soft Sticky Potential}} |
381 |
|
|
\label{sec:SSeq} |
382 |
mmeineke |
550 |
|
383 |
mmeineke |
552 |
Hydrogen bonding in the SSD model is described by the |
384 |
|
|
$V_{\text{sp}}$ term in Eq.~\ref{eq:ssdPot}. Its form is as follows: |
385 |
|
|
\begin{equation} |
386 |
|
|
V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
387 |
|
|
\boldsymbol{\Omega}_{j}) = |
388 |
|
|
v^{\circ}[s(r_{ij})w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
389 |
|
|
\boldsymbol{\Omega}_{j}) |
390 |
|
|
+ |
391 |
|
|
s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
392 |
|
|
\boldsymbol{\Omega}_{j})] |
393 |
|
|
\label{eq:spPot} |
394 |
|
|
\end{equation} |
395 |
|
|
Where $v^\circ$ scales the strength of the interaction. |
396 |
|
|
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
397 |
|
|
and |
398 |
|
|
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
399 |
|
|
are responsible for the tetrahedral potential and a correction to the |
400 |
|
|
tetrahedral potential respectively. They are, |
401 |
|
|
\begin{equation} |
402 |
|
|
w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
403 |
|
|
\sin\theta_{ij} \sin 2\theta_{ij} \cos 2\phi_{ij} |
404 |
|
|
+ \sin \theta_{ji} \sin 2\theta_{ji} \cos 2\phi_{ji} |
405 |
|
|
\label{eq:spPot2} |
406 |
mmeineke |
553 |
\end{equation} |
407 |
mmeineke |
552 |
and |
408 |
|
|
\begin{equation} |
409 |
|
|
\begin{split} |
410 |
|
|
w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
411 |
|
|
&(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij} + 0.8)^2 \\ |
412 |
|
|
&+ (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ} |
413 |
|
|
\end{split} |
414 |
|
|
\label{eq:spCorrection} |
415 |
|
|
\end{equation} |
416 |
|
|
The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical |
417 |
|
|
coordinates of the position of molecule $j$ in the reference frame |
418 |
|
|
fixed on molecule $i$ with the z-axis aligned with the dipole moment. |
419 |
|
|
The correction |
420 |
|
|
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
421 |
|
|
is needed because |
422 |
|
|
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
423 |
|
|
vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. Finally, the |
424 |
|
|
potential is scaled by the switching function $s(r_{ij})$, |
425 |
|
|
which scales smoothly from 0 to 1. |
426 |
|
|
\begin{equation} |
427 |
|
|
s(r_{ij}) = |
428 |
|
|
\begin{cases} |
429 |
|
|
1& \text{if $r_{ij} < r_{L}$}, \\ |
430 |
|
|
\frac{(r_{U} - r_{ij})^2 (r_{U} + 2r_{ij} |
431 |
|
|
- 3r_{L})}{(r_{U}-r_{L})^3}& |
432 |
|
|
\text{if $r_{L} \leq r_{ij} \leq r_{U}$},\\ |
433 |
|
|
0& \text{if $r_{ij} \geq r_{U}$}. |
434 |
|
|
\end{cases} |
435 |
|
|
\label{eq:spCutoff} |
436 |
|
|
\end{equation} |
437 |
mmeineke |
550 |
|
438 |
mmeineke |
552 |
\end{kasten} |
439 |
mmeineke |
550 |
|
440 |
mmeineke |
552 |
|
441 |
mmeineke |
546 |
\end{spalte} |
442 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
443 |
|
|
%%% third column %%% |
444 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
445 |
|
|
\begin{spalte} |
446 |
|
|
|
447 |
mmeineke |
552 |
\begin{kasten} |
448 |
|
|
\subsection{{\color{ndblue}Hydrogen Bonding in SSD}} |
449 |
|
|
\label{sec:hbonding} |
450 |
mmeineke |
546 |
|
451 |
mmeineke |
552 |
The SSD model's $V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j})$ |
452 |
|
|
recreates the hydrogen bonding network of water. |
453 |
|
|
\begin{center} |
454 |
|
|
\begin{minipage}{100mm} |
455 |
|
|
\begin{minipage}[t]{48mm} |
456 |
|
|
\begin{center} |
457 |
|
|
\includegraphics[width=48mm]{iced_final.eps}\\ |
458 |
|
|
SSD Relaxed on a diamond lattice |
459 |
|
|
\end{center} |
460 |
|
|
\end{minipage} |
461 |
|
|
\hspace{4mm}% |
462 |
|
|
\begin{minipage}[t]{48mm} |
463 |
|
|
\begin{center} |
464 |
|
|
\includegraphics[width=48mm]{dipoled_final.eps}\\ |
465 |
|
|
Stockmayer Spheres relaxed on a diamond lattice |
466 |
|
|
\end{center} |
467 |
|
|
\end{minipage} |
468 |
|
|
\end{minipage} |
469 |
mmeineke |
546 |
|
470 |
mmeineke |
552 |
\end{center} |
471 |
|
|
|
472 |
mmeineke |
546 |
|
473 |
mmeineke |
552 |
\end{kasten} |
474 |
|
|
|
475 |
|
|
|
476 |
|
|
\begin{kasten} |
477 |
|
|
|
478 |
|
|
\subsection{{\color{ndblue}The Lipid Model}} |
479 |
|
|
\label{sec:lipidModel} |
480 |
|
|
|
481 |
|
|
\begin{center} |
482 |
|
|
\includegraphics[width=25mm,angle=-90]{lipidModel.epsi} |
483 |
|
|
\end{center} |
484 |
|
|
|
485 |
|
|
\begin{itemize} |
486 |
mmeineke |
553 |
\item PC \& PE head groups are replaced by a Lennard-Jones sphere containing a dipole at its center |
487 |
mmeineke |
552 |
\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 |
mmeineke |
553 |
\label{sec:results} |
528 |
|
|
\subsection{{\color{ndblue}Simulation Snapshots:50 lipids in a sea of 1384 waters}} |
529 |
mmeineke |
552 |
\label{sec:r50snapshots} |
530 |
|
|
|
531 |
|
|
\begin{center} |
532 |
|
|
\includegraphics[width=105mm]{r50-montage.eps} |
533 |
|
|
\end{center} |
534 |
|
|
|
535 |
|
|
\end{kasten} |
536 |
|
|
|
537 |
|
|
|
538 |
mmeineke |
546 |
\end{spalte} |
539 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
540 |
|
|
%%% fourth column %%% |
541 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
542 |
|
|
\begin{spalte} |
543 |
|
|
|
544 |
mmeineke |
552 |
\begin{kasten} |
545 |
|
|
|
546 |
|
|
\subsection{{\color{ndblue}Position and Angular Correlations}} |
547 |
|
|
\label{sec:r50corr} |
548 |
mmeineke |
546 |
|
549 |
mmeineke |
552 |
\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 |
mmeineke |
546 |
|
576 |
mmeineke |
552 |
\end{kasten} |
577 |
mmeineke |
546 |
|
578 |
|
|
|
579 |
mmeineke |
552 |
\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 |
mmeineke |
553 |
parameter space of the model when the tail beads are larger than |
589 |
mmeineke |
552 |
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 |
mmeineke |
546 |
\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 |
mmeineke |
552 |
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 |
mmeineke |
546 |
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} |