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\usepackage{palatino} |
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\usepackage[latin1]{inputenc} |
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\usepackage{epsf} |
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\usepackage{latexsym} |
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\usepackage{calc} |
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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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%%{\newrgbcolor{gradbegin}{0.87 0.82 0.59}% |
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gradbegin=gradbegin,gradmidpoint=0.1](\bgwidth,-\bgheight)} |
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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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{\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 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 |
| 220 |
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computational cost of each force evaluation. The water was represented |
| 221 |
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by the soft sticky dipole model of Ichiye \emph{et |
| 224 |
|
in the tail groups to beads representing $\mbox{CH}_{2}$ and |
| 225 |
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$\mbox{CH}_{3}$ unified atoms, and the replacement of the head groups |
| 226 |
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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 |
| 229 |
< |
times in excess of 30 nanoseconds. |
| 227 |
> |
model was then used to simulate micelle formation from a configuration |
| 228 |
> |
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 |
| 244 |
+ |
complex. The lipid molecules are large, and contain many |
| 245 |
+ |
atoms. Additionally, the head groups of the lipids are often |
| 246 |
+ |
zwitterions, and the large separation between charges results in a |
| 247 |
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large dipole moment. Adding to the complexity are the number of water |
| 248 |
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molecules needed to properly solvate the lipid bilayer, typically 25 |
| 249 |
+ |
water molecules for every lipid molecule. These factors make it |
| 250 |
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difficult to study certain biologically interesting phenomena that |
| 251 |
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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{} |
| 264 |
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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. |
| 266 |
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\item Periodicity of 100 - 200 $\mbox{\AA}$\cite{Cevc87} |
| 267 |
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\item Current simulations have box sizes ranging from 50 - 100 $\mbox{\AA}$ on a side.\cite{saiz02,lindahl00,venable00} |
| 268 |
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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}} |
| 277 |
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\begin{itemize} |
| 278 |
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|
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\item |
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Drug Diffusion |
| 281 |
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\begin{itemize} |
| 282 |
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\item |
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Some drug molecules may spend appreciable amounts of time in the |
| 284 |
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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 |
| 288 |
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actions |
| 289 |
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\end{itemize} |
| 290 |
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|
| 291 |
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\item |
| 292 |
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Bilayer Formation Dynamics |
| 293 |
+ |
\begin{itemize} |
| 294 |
+ |
\item |
| 295 |
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Current lipid simulations indicate\cite{Marrink01}: |
| 296 |
+ |
\begin{itemize} |
| 297 |
+ |
\item Aggregation can happen as quickly as 200 ps |
| 298 |
+ |
|
| 299 |
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\item Bilayers can take up to 20 ns to form completely |
| 300 |
+ |
\end{itemize} |
| 301 |
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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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|
| 306 |
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\begin{kasten} |
| 307 |
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\subsection{{\color{ndblue}Our Simplifications}} |
| 308 |
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\begin{itemize} |
| 309 |
+ |
\item Unified atoms with fixed bond lengths replace groups of atoms. |
| 310 |
+ |
\item Charge distributions are replaced with dipoles. |
| 311 |
+ |
\begin{itemize} |
| 312 |
+ |
\item Relatively short range, $\frac{1}{r^3}$, interactions allow |
| 313 |
+ |
the application of neighbor lists. |
| 314 |
+ |
\end{itemize} |
| 315 |
+ |
\end{itemize} |
| 316 |
+ |
\begin{equation} |
| 317 |
+ |
V^{\text{dp}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 318 |
+ |
\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 319 |
+ |
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 320 |
+ |
- |
| 321 |
+ |
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
| 322 |
+ |
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
| 323 |
+ |
{r^{5}_{ij}} \biggr] |
| 324 |
+ |
\label{eq:dipole} |
| 325 |
+ |
\end{equation} |
| 326 |
+ |
\end{kasten} |
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|
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|
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|
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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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|
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\begin{kasten} |
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\section{{\color{ndblue}Ima second column holder}} |
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|
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hello |
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|
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\begin{kasten} |
| 338 |
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\subsection{{\color{ndblue}Reduction in calculations}} |
| 339 |
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Unified water and lipid models and decrease the number of interactions |
| 340 |
+ |
needed between two molecules. |
| 341 |
+ |
|
| 342 |
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\begin{center} |
| 343 |
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\includegraphics[width=50mm,angle=-90]{reduction.epsi} |
| 344 |
+ |
\end{center} |
| 345 |
|
\end{kasten} |
| 346 |
|
|
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+ |
|
| 348 |
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\begin{kasten} |
| 349 |
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\section{{\color{red}\underline{Models}}} |
| 350 |
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\label{sec:model} |
| 351 |
+ |
\subsection{{\color{ndblue}The Water Model}} |
| 352 |
+ |
\label{sec:waterModel} |
| 353 |
+ |
|
| 354 |
+ |
The waters in the simulation were modeled after the Soft Sticky Dipole |
| 355 |
+ |
(SSD) model of Ichiye.\cite{liu96:new_model} Where: |
| 356 |
+ |
|
| 357 |
+ |
\begin{wrapfigure}[10]{o}{60mm} |
| 358 |
+ |
\begin{center} |
| 359 |
+ |
\includegraphics[width=40mm]{ssd.epsi} |
| 360 |
+ |
\end{center} |
| 361 |
+ |
\end{wrapfigure} |
| 362 |
+ |
\mbox{} |
| 363 |
+ |
\begin{itemize} |
| 364 |
+ |
\item $\sigma$ is the Lennard-Jones length parameter |
| 365 |
+ |
\item $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$ |
| 366 |
+ |
\item $\mathbf{r}_{ij}$ is the vector between molecules $i$ and $j$ |
| 367 |
+ |
\item $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the Euler angles of molecule $i$ or $j$ respectively |
| 368 |
+ |
\end{itemize} |
| 369 |
+ |
|
| 370 |
+ |
It's potential is as follows: |
| 371 |
+ |
\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 |
+ |
\label{eq:ssdPot} |
| 375 |
+ |
\end{equation} |
| 376 |
+ |
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 |
+ |
\end{kasten} |
| 378 |
+ |
|
| 379 |
+ |
\begin{kasten} |
| 380 |
+ |
\subsection{{\color{ndblue}Soft Sticky Potential}} |
| 381 |
+ |
\label{sec:SSeq} |
| 382 |
+ |
|
| 383 |
+ |
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 |
+ |
\end{equation} |
| 407 |
+ |
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 |
|
|
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+ |
\end{kasten} |
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|
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|
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|
\end{spalte} |
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 445 |
|
\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}} |
| 270 |
< |
|
| 271 |
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hello |
| 447 |
> |
\begin{kasten} |
| 448 |
> |
\subsection{{\color{ndblue}Hydrogen Bonding in SSD}} |
| 449 |
> |
\label{sec:hbonding} |
| 450 |
|
|
| 451 |
< |
\end{kasten} |
| 451 |
> |
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 |
|
|
| 470 |
+ |
\end{center} |
| 471 |
+ |
|
| 472 |
|
|
| 473 |
+ |
\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 |
+ |
\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}}} |
| 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. Computaion time was provided on the Bunch-of-Boxes (B.o.B.) |
| 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 |
|
|
