matt_papers/MWTCC03/poster.tex

%% this is my poster for the Midwest Theoretical Conference


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\def\op#1{\hat{#1}}
\begin{document}
\bibliographystyle{plain}
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      \begin{center}
        \color{ndblue}
        \textbf{\Huge {A Mesoscale Model for Phospholipid Simulations}}\\[0.5em]
        \textsc{\LARGE \underline{Matthew~A.~Meineke}, and J.~Daniel~Gezelter}\\[0.3em]
        {\large Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556, USA\\
{\tt\ mmeineke@nd.edu}
}
      \end{center}}
\hfill}}\hfill\mbox{}\\[1.cm]
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     \begin{center}
{\large{\color{red}    \underline{ABSTRACT}  } }
     \end{center}

{\color{ndblue}

A mesoscale model for phospholipids has been developed for molecular
dynamics simulations of lipid bilayers. The model makes several
simplifications to both the water and the phospholipids to reduce the
computational cost of each force evaluation. The water was represented
by the soft sticky dipole model of Ichiye \emph{et
al}.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md} The
simplifications to the phospholipids included the reduction of atoms
in the tail groups to beads representing $\mbox{CH}_{2}$ and
$\mbox{CH}_{3}$ unified atoms, and the replacement of the head groups
with a single point mass containing a centrally located dipole. The
model was then used to simulate micelle formation from a configuration
of randomly placed phospholipids which was simulated for times in
excess of 20 nanoseconds.

 }
      \end{kasten}

        
  \begin{kasten}
  \section{{\color{red}\underline{Introduction \& Background}}}
  \label{sec:intro}

%%  \subsection{{\color{ndblue}Motivation}}
  \label{sec:motivation}


Simulations of phospholipid bilayers are, by necessity, quite
complex. The lipid molecules are large, and contain many
atoms. Additionally, the head groups of the lipids are often
zwitterions, and the large separation between charges results in a
large dipole moment. Adding to the complexity are the number of water
molecules needed to properly solvate the lipid bilayer, typically 25
water molecules for every lipid molecule. These factors make it
difficult to study certain biologically interesting phenomena that
have large inherent length or time scale.

  \end{kasten}

  \begin{kasten}
  \subsection{{\color{ndblue}Ripple Phase}}

\begin{wrapfigure}{o}{60mm}
\centering
\includegraphics[width=40mm]{ripple.epsi}
\end{wrapfigure}

\mbox{}
\begin{itemize}
\item The ripple (~$P_{\beta'}$~) phase lies in the transition from the gel to fluid phase.
\item Periodicity of 100 - 200 $\mbox{\AA}$\cite{Cevc87}
\item Current simulations have box sizes ranging from 50 - 100 $\mbox{\AA}$ on a side.\cite{saiz02,lindahl00,venable00}
\end{itemize}

  \label{sec:ripplePhase}

   \end{kasten}


  \begin{kasten}
\subsection{{\color{ndblue}Diffusion \& Formation Dynamics}}
\begin{itemize}

\item
Drug Diffusion
        \begin{itemize}
        \item 
        Some drug molecules may spend appreciable amounts of time in the
        membrane

        \item
        Long time scale dynamics are need to observe and characterize their
        actions
        \end{itemize}

\item
Bilayer Formation Dynamics
        \begin{itemize}
        \item
        Current lipid simulations indicate\cite{Marrink01}:
                \begin{itemize} 
                \item Aggregation can happen as quickly as 200 ps

                \item Bilayers can take up to 20 ns to form completely
                \end{itemize}

        \end{itemize}
\end{itemize}
  \end{kasten}

  \begin{kasten}
\subsection{{\color{ndblue}System Simplifications}}
\begin{itemize}
\item Unified atoms with fixed bond lengths replace groups of atoms.
\item Replace charge distributions with dipoles.(Eq.~\ref{eq:dipole} 
        vs. Eq.~\ref{eq:coloumb})
\begin{itemize}
        \item Relatively short range, $\frac{1}{r^3}$, interactions allow
        the application of computational simplification algorithms,
        i.e. neighbor lists.
\end{itemize}
\end{itemize}
\begin{equation}
V^{\text{dp}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
        \boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[
        \frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}}
        -
        \frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) %
                (\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) }
                {r^{5}_{ij}} \biggr]
\label{eq:dipole}
\end{equation}
\begin{equation}
V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) = \frac{q_{i}q_{j}}%
        {4\pi\epsilon_{0} r_{ij}}
\label{eq:coloumb}
\end{equation} 
  \end{kasten}


     \end{spalte}
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    \begin{spalte}


  \begin{kasten}
\subsection{{\color{ndblue}Reduction in calculations}}
Unified water and lipid models and decrease the number of interactions
needed between two molecules.

\begin{center}
\includegraphics[width=50mm,angle=-90]{reduction.epsi}
\end{center}
     \end{kasten}
        

  \begin{kasten}
\section{{\color{red}\underline{Models}}}
\label{sec:model}
\subsection{{\color{ndblue}The Water Model}}
\label{sec:waterModel}

The waters in the simulation were modeled after the Soft Sticky Dipole
(SSD) model of Ichiye.\cite{liu96:new_model} Where:

\begin{wrapfigure}[10]{o}{60mm}
\begin{center}
\includegraphics[width=40mm]{ssd.epsi}
\end{center}
\end{wrapfigure}
\mbox{}
\begin{itemize}
\item $\sigma$ is the Lennard-Jones length parameter
\item $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$
\item $\mathbf{r}_{ij}$ is the vector between molecules $i$ and $j$
\item $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the Euler angles of molecule $i$ or $j$ respectively
\end{itemize}

It's potential is as follows:
\begin{equation}
V_{s\!s\!d} = V_{L\!J}(r_{i\!j}) + V_{d\!p}(r_{i\!j},\Omega_{i},\Omega_{j})
        + V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j})
\label{eq:ssdPot}
\end{equation}
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:
\begin{equation}
V_{\text{LJ}} = 
        4\epsilon_{ij} \biggl[
        \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12}
        - \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6}
        \biggr]
\label{eq:lennardJonesPot}
\end{equation}

  \end{kasten}

  \begin{kasten}
        \subsection{{\color{ndblue}Soft Sticky Potential}}
        \label{sec:SSeq}

        Hydrogen bonding in the SSD model is described by the
        $V_{\text{sp}}$ term in Eq.~\ref{eq:ssdPot}. Its form is as follows:
\begin{equation}
V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i},
        \boldsymbol{\Omega}_{j}) =
        v^{\circ}[s(r_{ij})w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
                \boldsymbol{\Omega}_{j})
        +
        s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
                \boldsymbol{\Omega}_{j})]
\label{eq:spPot}
\end{equation}
Where $v^\circ$ scales the strength of the interaction.
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
and
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
are responsible for the tetrahedral potential and a correction to the
tetrahedral potential respectively. They are,
\begin{equation}
w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) =
        \sin\theta_{ij} \sin 2\theta_{ij} \cos 2\phi_{ij}
        + \sin \theta_{ji} \sin 2\theta_{ji} \cos 2\phi_{ji}
\label{eq:spPot2}
\end{equation}o
and
\begin{equation}
\begin{split}
w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) =
        &(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij} + 0.8)^2 \\
        &+ (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ}
\end{split}
\label{eq:spCorrection}
\end{equation}
The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical
coordinates of the position of molecule $j$ in the reference frame
fixed on molecule $i$ with the z-axis aligned with the dipole moment.
The correction
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
is needed because
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$
vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. Finally, the 
potential is scaled by the switching function $s(r_{ij})$, 
which scales smoothly from 0 to 1.
\begin{equation}
s(r_{ij}) = 
        \begin{cases}
        1&      \text{if $r_{ij} < r_{L}$}, \\
        \frac{(r_{U} - r_{ij})^2 (r_{U} + 2r_{ij} 
                - 3r_{L})}{(r_{U}-r_{L})^3}&
                \text{if $r_{L} \leq r_{ij} \leq r_{U}$},\\
        0&      \text{if $r_{ij} \geq r_{U}$}.
        \end{cases}
\label{eq:spCutoff}
\end{equation}

  \end{kasten}


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    \begin{spalte}

\begin{kasten}
        \subsection{{\color{ndblue}Hydrogen Bonding in SSD}}
        \label{sec:hbonding}

        The SSD model's $V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j})$
        recreates the hydrogen bonding network of water.
        \begin{center}
        \begin{minipage}{100mm}
          \begin{minipage}[t]{48mm}
                \begin{center}
                \includegraphics[width=48mm]{iced_final.eps}\\
                SSD Relaxed on a diamond lattice
                \end{center}
          \end{minipage}
          \hspace{4mm}%
          \begin{minipage}[t]{48mm}
                \begin{center}
                \includegraphics[width=48mm]{dipoled_final.eps}\\
                Stockmayer Spheres relaxed on a diamond lattice
                \end{center}
          \end{minipage}
        \end{minipage}

        \end{center}
        

  \end{kasten}


  \begin{kasten}

        \subsection{{\color{ndblue}The Lipid Model}}
        \label{sec:lipidModel}

        \begin{center}
        \includegraphics[width=25mm,angle=-90]{lipidModel.epsi}
        \end{center}

        \begin{itemize}
        \item Head group replaced by a single Lennard-Jones sphere containing a dipole at its center
        \item Atoms in the tail chains modeled as unified groups of atoms
        \item Tail group interaction parameters based on those of TraPPE\cite{Siepmann1998}
        \end{itemize}

        The total potential is given by:
        \begin{equation}
V_{\text{lipid}} =
        \sum_{i}V_{i}^{\text{internal}}
        + \sum_i \sum_{j>i} \sum_{\text{$\alpha$ in $i$}} 
        \sum_{\text{$\beta$ in $j$}}
        V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}})
        +\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j})
\end{equation}
Where
\begin{equation}
V_{i}^{\text{internal}} = 
        \sum_{\text{bends}}V_{\text{bend}}(\theta_{\alpha\beta\gamma})
        + \sum_{\text{torsions}}V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta})
        + \sum_{\alpha} \sum_{\beta>\alpha}V_{\text{LJ}}(r_{\alpha \beta})
\end{equation}
The bend and torsion potentials were of the form:
\begin{equation}
V_{\text{bend}}(\theta_{\alpha\beta\gamma}) 
        = k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \theta_0)^2}{2}
\label{eq:bendPot}
\end{equation}
\begin{equation}
V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta})
        = c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}]
        + c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] 
        + c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})]
\label{eq:torsPot}
\end{equation}
        

  \end{kasten}

  \begin{kasten}

        \section{{\color{red}\underline{Initial Results}}}
        \label{sec:results}
        \subsection{{\color{ndblue}50 lipids randomly arranged in water}}
        \label{sec:r50}

        \begin{center}
        \begin{minipage}{130mm}
                \begin{minipage}[t]{40mm}
        \begin{itemize}
        \item $N_{\mbox{lipids}} = 25$
        \end{itemize}
                \end{minipage}
                \begin{minipage}[t]{40mm}
        \begin{itemize}
        \item $N_{\mbox{H}_{2}\mbox{O}} = 1386$
        \end{itemize}
                \end{minipage}
                \begin{minipage}[t]{40mm}
        \begin{itemize}
        \item T = 300 K
        \end{itemize}
                \end{minipage}
        \end{minipage}
        \end{center}

  \end{kasten}

  \begin{kasten}
        
        \subsection{{\color{ndblue}Simulation Snapshots}}
        \label{sec:r50snapshots}
        
        \begin{center}
        \includegraphics[width=105mm]{r50-montage.eps}
        \end{center}

  \end{kasten}


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  \begin{kasten}
        
        \subsection{{\color{ndblue}Position and Angular Correlations}}
        \label{sec:r50corr}

        \begin{center}
        \begin{minipage}{110mm}
                \begin{minipage}[t]{55mm}
                \begin{center}
                \includegraphics[width=36mm,angle=-90]{r50-HEAD-HEAD.epsi}\\
                The self correlation of the head groups
                \end{center}
                \end{minipage}
                \begin{minipage}[t]{55mm}
                \begin{center}
                \includegraphics[width=36mm,angle=-90]{r50-CH2-CH2.epsi}\\
                The self correlation of the tail beads.
                \end{center}
                \end{minipage}
        \end{minipage}
        \end{center}
\begin{equation}
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} 
        \delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle
\label{eq:gofr}
\end{equation}
\begin{equation}
g_{\gamma}(r) = \langle \sum_i \sum_{j>i}
        (\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle
\label{eq:gammaofr}
\end{equation}

  \end{kasten}


  \begin{kasten}

        \subsection{{\color{red}\underline{Discussion}}}
        \label{sec:discussion}
        
        The initial results show much promise for the model. The
        system of 50 lipids was able to form micelles quickly, however
        bilayer formation was not seen on the time scale of the
        current simulation. Current simulations are exploring the
        phase space of the model when the tail beads are larger than
        the head group. This should help to drive the system toward a
        bilayer rather than a micelle. Work is also being done on the
        simulation engine to allow for the box size of the system to
        be adjustable in all three dimensions to allow for constant
        pressure.

  \end{kasten}


     \begin{kasten}
        \begin{center}  
        {\large{\color{red} \underline{Acknowledgments}}}
     \end{center}

The authors would like to acknowledge Charles Vardeman, Christopher
Fennell, and Teng lin for their contributions to the simulation
engine. MAM would also like to extend a special thank you to Charles
Vardeman for his help with the \TeX formatting of this
poster. Computation time was provided on the Notre Dame Bunch-of-Boxes (B.o.B.)
cluster under NSF grant DMR 00 79647. The authors acknowledge support
under NSF grant CHE-0134881.

     \end{kasten}

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           }
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\end{document}
Revision:	552
Committed:	Mon Jun 9 21:19:02 2003 UTC (21 years, 3 months ago) by mmeineke
Content type:	application/x-tex
File size:	19679 byte(s)
Log Message:	First completed rough draft (spell checked even).