| 224 |
|
in the tail groups to beads representing $\mbox{CH}_{2}$ and |
| 225 |
|
$\mbox{CH}_{3}$ unified atoms, and the replacement of the head groups |
| 226 |
|
with a single point mass containing a centrally located dipole. The |
| 227 |
< |
model was then used to simulate micelle and bilayer formation from a |
| 228 |
< |
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 |
| 229 |
> |
excess of 20 nanoseconds. |
| 230 |
|
|
| 231 |
|
} |
| 232 |
|
\end{kasten} |
| 304 |
|
\end{kasten} |
| 305 |
|
|
| 306 |
|
\begin{kasten} |
| 307 |
< |
\subsection{{\color{ndblue}System Simplfications}} |
| 307 |
> |
\subsection{{\color{ndblue}System Simplifications}} |
| 308 |
|
\begin{itemize} |
| 309 |
|
\item Unified atoms with fixed bond lengths replace groups of atoms. |
| 310 |
< |
\item Replace charge distributions with dipoles.(Eq. \ref{eq:dipole} |
| 311 |
< |
vs. Eq. \ref{eq:coloumb}) |
| 310 |
> |
\item Replace charge distributions with dipoles.(Eq.~\ref{eq:dipole} |
| 311 |
> |
vs. Eq.~\ref{eq:coloumb}) |
| 312 |
|
\begin{itemize} |
| 313 |
|
\item Relatively short range, $\frac{1}{r^3}$, interactions allow |
| 314 |
|
the application of computational simplification algorithms, |
| 315 |
< |
ie. neighbor lists. |
| 315 |
> |
i.e. neighbor lists. |
| 316 |
|
\end{itemize} |
| 317 |
|
\end{itemize} |
| 318 |
|
\begin{equation} |
| 355 |
|
\begin{kasten} |
| 356 |
|
\section{{\color{red}\underline{Models}}} |
| 357 |
|
\label{sec:model} |
| 358 |
< |
\subsection{{\color{ndblue}Water Model}} |
| 358 |
> |
\subsection{{\color{ndblue}The Water Model}} |
| 359 |
|
\label{sec:waterModel} |
| 360 |
|
|
| 361 |
|
The waters in the simulation were modeled after the Soft Sticky Dipole |
| 368 |
|
\end{wrapfigure} |
| 369 |
|
\mbox{} |
| 370 |
|
\begin{itemize} |
| 371 |
< |
\item $\sigma$ is the Lennard-Jones length parameter. |
| 372 |
< |
\item $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$, |
| 371 |
> |
\item $\sigma$ is the Lennard-Jones length parameter |
| 372 |
> |
\item $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$ |
| 373 |
|
\item $\mathbf{r}_{ij}$ is the vector between molecules $i$ and $j$ |
| 374 |
< |
\item $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the Euler angles of molecule $i$ or $j$ respectively. |
| 374 |
> |
\item $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the Euler angles of molecule $i$ or $j$ respectively |
| 375 |
|
\end{itemize} |
| 376 |
|
|
| 377 |
|
It's potential is as follows: |
| 378 |
– |
|
| 378 |
|
\begin{equation} |
| 379 |
|
V_{s\!s\!d} = V_{L\!J}(r_{i\!j}) + V_{d\!p}(r_{i\!j},\Omega_{i},\Omega_{j}) |
| 380 |
|
+ V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j}) |
| 381 |
+ |
\label{eq:ssdPot} |
| 382 |
|
\end{equation} |
| 383 |
< |
|
| 383 |
> |
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: |
| 384 |
> |
\begin{equation} |
| 385 |
> |
V_{\text{LJ}} = |
| 386 |
> |
4\epsilon_{ij} \biggl[ |
| 387 |
> |
\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
| 388 |
> |
- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
| 389 |
> |
\biggr] |
| 390 |
> |
\label{eq:lennardJonesPot} |
| 391 |
> |
\end{equation} |
| 392 |
|
|
| 393 |
|
\end{kasten} |
| 394 |
|
|
| 395 |
+ |
\begin{kasten} |
| 396 |
+ |
\subsection{{\color{ndblue}Soft Sticky Potential}} |
| 397 |
+ |
\label{sec:SSeq} |
| 398 |
+ |
|
| 399 |
+ |
Hydrogen bonding in the SSD model is described by the |
| 400 |
+ |
$V_{\text{sp}}$ term in Eq.~\ref{eq:ssdPot}. Its form is as follows: |
| 401 |
+ |
\begin{equation} |
| 402 |
+ |
V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
| 403 |
+ |
\boldsymbol{\Omega}_{j}) = |
| 404 |
+ |
v^{\circ}[s(r_{ij})w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 405 |
+ |
\boldsymbol{\Omega}_{j}) |
| 406 |
+ |
+ |
| 407 |
+ |
s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 408 |
+ |
\boldsymbol{\Omega}_{j})] |
| 409 |
+ |
\label{eq:spPot} |
| 410 |
+ |
\end{equation} |
| 411 |
+ |
Where $v^\circ$ scales the strength of the interaction. |
| 412 |
+ |
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
| 413 |
+ |
and |
| 414 |
+ |
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
| 415 |
+ |
are responsible for the tetrahedral potential and a correction to the |
| 416 |
+ |
tetrahedral potential respectively. They are, |
| 417 |
+ |
\begin{equation} |
| 418 |
+ |
w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
| 419 |
+ |
\sin\theta_{ij} \sin 2\theta_{ij} \cos 2\phi_{ij} |
| 420 |
+ |
+ \sin \theta_{ji} \sin 2\theta_{ji} \cos 2\phi_{ji} |
| 421 |
+ |
\label{eq:spPot2} |
| 422 |
+ |
\end{equation}o |
| 423 |
+ |
and |
| 424 |
+ |
\begin{equation} |
| 425 |
+ |
\begin{split} |
| 426 |
+ |
w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
| 427 |
+ |
&(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij} + 0.8)^2 \\ |
| 428 |
+ |
&+ (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ} |
| 429 |
+ |
\end{split} |
| 430 |
+ |
\label{eq:spCorrection} |
| 431 |
+ |
\end{equation} |
| 432 |
+ |
The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical |
| 433 |
+ |
coordinates of the position of molecule $j$ in the reference frame |
| 434 |
+ |
fixed on molecule $i$ with the z-axis aligned with the dipole moment. |
| 435 |
+ |
The correction |
| 436 |
+ |
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
| 437 |
+ |
is needed because |
| 438 |
+ |
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
| 439 |
+ |
vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. Finally, the |
| 440 |
+ |
potential is scaled by the switching function $s(r_{ij})$, |
| 441 |
+ |
which scales smoothly from 0 to 1. |
| 442 |
+ |
\begin{equation} |
| 443 |
+ |
s(r_{ij}) = |
| 444 |
+ |
\begin{cases} |
| 445 |
+ |
1& \text{if $r_{ij} < r_{L}$}, \\ |
| 446 |
+ |
\frac{(r_{U} - r_{ij})^2 (r_{U} + 2r_{ij} |
| 447 |
+ |
- 3r_{L})}{(r_{U}-r_{L})^3}& |
| 448 |
+ |
\text{if $r_{L} \leq r_{ij} \leq r_{U}$},\\ |
| 449 |
+ |
0& \text{if $r_{ij} \geq r_{U}$}. |
| 450 |
+ |
\end{cases} |
| 451 |
+ |
\label{eq:spCutoff} |
| 452 |
+ |
\end{equation} |
| 453 |
|
|
| 454 |
+ |
\end{kasten} |
| 455 |
|
|
| 456 |
|
|
| 457 |
|
\end{spalte} |
| 460 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 461 |
|
\begin{spalte} |
| 462 |
|
|
| 463 |
< |
\begin{kasten} |
| 464 |
< |
|
| 465 |
< |
\section{{\color{ndblue}Ima third column holder}} |
| 399 |
< |
|
| 400 |
< |
hello |
| 463 |
> |
\begin{kasten} |
| 464 |
> |
\subsection{{\color{ndblue}Hydrogen Bonding in SSD}} |
| 465 |
> |
\label{sec:hbonding} |
| 466 |
|
|
| 467 |
< |
\end{kasten} |
| 467 |
> |
The SSD model's $V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j})$ |
| 468 |
> |
recreates the hydrogen bonding network of water. |
| 469 |
> |
\begin{center} |
| 470 |
> |
\begin{minipage}{100mm} |
| 471 |
> |
\begin{minipage}[t]{48mm} |
| 472 |
> |
\begin{center} |
| 473 |
> |
\includegraphics[width=48mm]{iced_final.eps}\\ |
| 474 |
> |
SSD Relaxed on a diamond lattice |
| 475 |
> |
\end{center} |
| 476 |
> |
\end{minipage} |
| 477 |
> |
\hspace{4mm}% |
| 478 |
> |
\begin{minipage}[t]{48mm} |
| 479 |
> |
\begin{center} |
| 480 |
> |
\includegraphics[width=48mm]{dipoled_final.eps}\\ |
| 481 |
> |
Stockmayer Spheres relaxed on a diamond lattice |
| 482 |
> |
\end{center} |
| 483 |
> |
\end{minipage} |
| 484 |
> |
\end{minipage} |
| 485 |
> |
|
| 486 |
> |
\end{center} |
| 487 |
> |
|
| 488 |
> |
|
| 489 |
> |
\end{kasten} |
| 490 |
> |
|
| 491 |
> |
|
| 492 |
> |
\begin{kasten} |
| 493 |
> |
|
| 494 |
> |
\subsection{{\color{ndblue}The Lipid Model}} |
| 495 |
> |
\label{sec:lipidModel} |
| 496 |
> |
|
| 497 |
> |
\begin{center} |
| 498 |
> |
\includegraphics[width=25mm,angle=-90]{lipidModel.epsi} |
| 499 |
> |
\end{center} |
| 500 |
|
|
| 501 |
+ |
\begin{itemize} |
| 502 |
+ |
\item Head group replaced by a single Lennard-Jones sphere containing a dipole at its center |
| 503 |
+ |
\item Atoms in the tail chains modeled as unified groups of atoms |
| 504 |
+ |
\item Tail group interaction parameters based on those of TraPPE\cite{Siepmann1998} |
| 505 |
+ |
\end{itemize} |
| 506 |
|
|
| 507 |
+ |
The total potential is given by: |
| 508 |
+ |
\begin{equation} |
| 509 |
+ |
V_{\text{lipid}} = |
| 510 |
+ |
\sum_{i}V_{i}^{\text{internal}} |
| 511 |
+ |
+ \sum_i \sum_{j>i} \sum_{\text{$\alpha$ in $i$}} |
| 512 |
+ |
\sum_{\text{$\beta$ in $j$}} |
| 513 |
+ |
V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}}) |
| 514 |
+ |
+\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j}) |
| 515 |
+ |
\end{equation} |
| 516 |
+ |
Where |
| 517 |
+ |
\begin{equation} |
| 518 |
+ |
V_{i}^{\text{internal}} = |
| 519 |
+ |
\sum_{\text{bends}}V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
| 520 |
+ |
+ \sum_{\text{torsions}}V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
| 521 |
+ |
+ \sum_{\alpha} \sum_{\beta>\alpha}V_{\text{LJ}}(r_{\alpha \beta}) |
| 522 |
+ |
\end{equation} |
| 523 |
+ |
The bend and torsion potentials were of the form: |
| 524 |
+ |
\begin{equation} |
| 525 |
+ |
V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
| 526 |
+ |
= k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \theta_0)^2}{2} |
| 527 |
+ |
\label{eq:bendPot} |
| 528 |
+ |
\end{equation} |
| 529 |
+ |
\begin{equation} |
| 530 |
+ |
V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
| 531 |
+ |
= c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}] |
| 532 |
+ |
+ c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] |
| 533 |
+ |
+ c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})] |
| 534 |
+ |
\label{eq:torsPot} |
| 535 |
+ |
\end{equation} |
| 536 |
+ |
|
| 537 |
+ |
|
| 538 |
+ |
\end{kasten} |
| 539 |
+ |
|
| 540 |
+ |
\begin{kasten} |
| 541 |
+ |
|
| 542 |
+ |
\section{{\color{red}\underline{Initial Results}}} |
| 543 |
+ |
\label{sec:results} |
| 544 |
+ |
\subsection{{\color{ndblue}50 lipids randomly arranged in water}} |
| 545 |
+ |
\label{sec:r50} |
| 546 |
+ |
|
| 547 |
+ |
\begin{center} |
| 548 |
+ |
\begin{minipage}{130mm} |
| 549 |
+ |
\begin{minipage}[t]{40mm} |
| 550 |
+ |
\begin{itemize} |
| 551 |
+ |
\item $N_{\mbox{lipids}} = 25$ |
| 552 |
+ |
\end{itemize} |
| 553 |
+ |
\end{minipage} |
| 554 |
+ |
\begin{minipage}[t]{40mm} |
| 555 |
+ |
\begin{itemize} |
| 556 |
+ |
\item $N_{\mbox{H}_{2}\mbox{O}} = 1386$ |
| 557 |
+ |
\end{itemize} |
| 558 |
+ |
\end{minipage} |
| 559 |
+ |
\begin{minipage}[t]{40mm} |
| 560 |
+ |
\begin{itemize} |
| 561 |
+ |
\item T = 300 K |
| 562 |
+ |
\end{itemize} |
| 563 |
+ |
\end{minipage} |
| 564 |
+ |
\end{minipage} |
| 565 |
+ |
\end{center} |
| 566 |
+ |
|
| 567 |
+ |
\end{kasten} |
| 568 |
+ |
|
| 569 |
+ |
\begin{kasten} |
| 570 |
+ |
|
| 571 |
+ |
\subsection{{\color{ndblue}Simulation Snapshots}} |
| 572 |
+ |
\label{sec:r50snapshots} |
| 573 |
+ |
|
| 574 |
+ |
\begin{center} |
| 575 |
+ |
\includegraphics[width=105mm]{r50-montage.eps} |
| 576 |
+ |
\end{center} |
| 577 |
+ |
|
| 578 |
+ |
\end{kasten} |
| 579 |
+ |
|
| 580 |
+ |
|
| 581 |
|
\end{spalte} |
| 582 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 583 |
|
%%% fourth column %%% |
| 584 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 585 |
|
\begin{spalte} |
| 586 |
|
|
| 587 |
+ |
\begin{kasten} |
| 588 |
+ |
|
| 589 |
+ |
\subsection{{\color{ndblue}Position and Angular Correlations}} |
| 590 |
+ |
\label{sec:r50corr} |
| 591 |
|
|
| 592 |
+ |
\begin{center} |
| 593 |
+ |
\begin{minipage}{110mm} |
| 594 |
+ |
\begin{minipage}[t]{55mm} |
| 595 |
+ |
\begin{center} |
| 596 |
+ |
\includegraphics[width=36mm,angle=-90]{r50-HEAD-HEAD.epsi}\\ |
| 597 |
+ |
The self correlation of the head groups |
| 598 |
+ |
\end{center} |
| 599 |
+ |
\end{minipage} |
| 600 |
+ |
\begin{minipage}[t]{55mm} |
| 601 |
+ |
\begin{center} |
| 602 |
+ |
\includegraphics[width=36mm,angle=-90]{r50-CH2-CH2.epsi}\\ |
| 603 |
+ |
The self correlation of the tail beads. |
| 604 |
+ |
\end{center} |
| 605 |
+ |
\end{minipage} |
| 606 |
+ |
\end{minipage} |
| 607 |
+ |
\end{center} |
| 608 |
+ |
\begin{equation} |
| 609 |
+ |
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} |
| 610 |
+ |
\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 611 |
+ |
\label{eq:gofr} |
| 612 |
+ |
\end{equation} |
| 613 |
+ |
\begin{equation} |
| 614 |
+ |
g_{\gamma}(r) = \langle \sum_i \sum_{j>i} |
| 615 |
+ |
(\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 616 |
+ |
\label{eq:gammaofr} |
| 617 |
+ |
\end{equation} |
| 618 |
|
|
| 619 |
+ |
\end{kasten} |
| 620 |
|
|
| 621 |
|
|
| 622 |
+ |
\begin{kasten} |
| 623 |
+ |
|
| 624 |
+ |
\subsection{{\color{red}\underline{Discussion}}} |
| 625 |
+ |
\label{sec:discussion} |
| 626 |
+ |
|
| 627 |
+ |
The initial results show much promise for the model. The |
| 628 |
+ |
system of 50 lipids was able to form micelles quickly, however |
| 629 |
+ |
bilayer formation was not seen on the time scale of the |
| 630 |
+ |
current simulation. Current simulations are exploring the |
| 631 |
+ |
phase space of the model when the tail beads are larger than |
| 632 |
+ |
the head group. This should help to drive the system toward a |
| 633 |
+ |
bilayer rather than a micelle. Work is also being done on the |
| 634 |
+ |
simulation engine to allow for the box size of the system to |
| 635 |
+ |
be adjustable in all three dimensions to allow for constant |
| 636 |
+ |
pressure. |
| 637 |
+ |
|
| 638 |
+ |
\end{kasten} |
| 639 |
+ |
|
| 640 |
+ |
|
| 641 |
|
\begin{kasten} |
| 642 |
|
\begin{center} |
| 643 |
|
{\large{\color{red} \underline{Acknowledgments}}} |
| 646 |
|
The authors would like to acknowledge Charles Vardeman, Christopher |
| 647 |
|
Fennell, and Teng lin for their contributions to the simulation |
| 648 |
|
engine. MAM would also like to extend a special thank you to Charles |
| 649 |
< |
Vardeman for his help with the TeX formatting of this |
| 650 |
< |
poster. Computaion time was provided on the Bunch-of-Boxes (B.o.B.) |
| 649 |
> |
Vardeman for his help with the \TeX formatting of this |
| 650 |
> |
poster. Computation time was provided on the Notre Dame Bunch-of-Boxes (B.o.B.) |
| 651 |
|
cluster under NSF grant DMR 00 79647. The authors acknowledge support |
| 652 |
|
under NSF grant CHE-0134881. |
| 653 |
|
|
