| 155 |
|
\label{eq:lennardJonesPot} |
| 156 |
|
\end{equation} |
| 157 |
|
where $r_{ij}$ is the distance between two $ij$ pairs, $\sigma_{ij}$ |
| 158 |
< |
scales the length of the iteraction, and $\epsilon_{ij}$ scales the |
| 158 |
> |
scales the length of the interaction, and $\epsilon_{ij}$ scales the |
| 159 |
|
energy of the potential. For SSD, $\sigma_{\text{SSD}} = 3.051 \mbox{ |
| 160 |
|
\AA}$ and $\epsilon_{\text{SSD}} = 0.152\text{ kcal/mol}$. |
| 161 |
|
$V_{\text{dp}}$ is the dipole potential: |
| 232 |
|
\end{equation} |
| 233 |
|
|
| 234 |
|
Despite the apparent complexity of Equation \ref{eq:spPot}, the SSD |
| 235 |
< |
model is still computationaly inexpensive. This is due to Equation |
| 235 |
> |
model is still computationally inexpensive. This is due to Equation |
| 236 |
|
\ref{eq:spCutoff}. With $r_{L}$ being 2.75~$\mbox{\AA}$ and $r_{U}$ |
| 237 |
|
being equal to either 3.35~$\mbox{\AA}$ for $s(r_{ij})$ or |
| 238 |
|
4.0~$\mbox{\AA}$ for $s'(r_{ij})$, the sticky potential is only active |
| 239 |
< |
over an extremly short range, and then only with other SSD |
| 239 |
> |
over an extremely short range, and then only with other SSD |
| 240 |
|
molecules. Therefore, it's predominant interaction is through it's |
| 241 |
|
point dipole and Lennard-Jones sphere. |
| 242 |
|
|
| 300 |
|
\subsection{Starting Configuration and Parameters} |
| 301 |
|
\label{sec:5x5Start} |
| 302 |
|
|
| 303 |
< |
Our first simulation was an array of 25 single chained lipids in a sea |
| 303 |
> |
Our first simulation is an array of 25 single chained lipids in a sea |
| 304 |
|
of water (Figure \ref{fig:5x5Start}). The total number of water |
| 305 |
< |
molecules was 1386, giving a final of water concentration of 70\% |
| 306 |
< |
wt. The simulation box measured 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
| 305 |
> |
molecules is 1386, giving a final of water concentration of 70\% |
| 306 |
> |
wt. The simulation box measures 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
| 307 |
|
x 39.4~$\mbox{\AA}$ with periodic boundary conditions imposed. The |
| 308 |
< |
system was simulated in the micro-canonical (NVE) ensemble with an |
| 308 |
> |
system is simulated in the micro-canonical (NVE) ensemble with an |
| 309 |
|
average temperature of 300~K. |
| 310 |
|
|
| 311 |
|
\subsection{Results} |
| 322 |
|
demonstrates a need for an isobaric-isothermal ensemble where the box |
| 323 |
|
size may relax or expand to keep the system at a 1~atm. |
| 324 |
|
|
| 325 |
< |
The simulation was analyzed using the radial distribution function, $g(r)$, which has the form: |
| 325 |
> |
The simulation was analyzed using the radial distribution function, |
| 326 |
> |
$g(r)$, which has the form: |
| 327 |
|
\begin{equation} |
| 328 |
|
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} |
| 329 |
|
\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 352 |
|
distance. |
| 353 |
|
|
| 354 |
|
Figure \ref{fig:5x5HHCorr} shows the two self correlation functions |
| 355 |
< |
for the Head groups of the lipids. The first peak at 4.03~$\mbox{\AA}$ is the |
| 356 |
< |
nearest neighbor separation of the heads of two lipids. |
| 355 |
> |
for the Head groups of the lipids. The first peak in the $g(r)$ at |
| 356 |
> |
4.03~$\mbox{\AA}$ is the nearest neighbor separation of the heads of |
| 357 |
> |
two lipids. This corresponds to a maximum in the $g_{\gamma}(r)$ which |
| 358 |
> |
means that the two neighbors on the same monolayer have their dipoles |
| 359 |
> |
aligned. The broad peak at 6.5~$\mbox{\AA}$ is the inter-bilayer |
| 360 |
> |
spacing. Here, there is a corresponding anti-alignment in the angular |
| 361 |
> |
correlation. This means that although the dipoles are aligned on the |
| 362 |
> |
same monolayer, the dipoles will orient themselves to be anti-aligned |
| 363 |
> |
on a opposite facing monolayer. With this information, the two peaks |
| 364 |
> |
at 7.0~$\mbox{\AA}$ and 7.4~$\mbox{\AA}$ are head groups on the same |
| 365 |
> |
monolayer, and they are the second nearest neighbors to the head |
| 366 |
> |
group. The peak is likely a split peak because of the small statistics |
| 367 |
> |
of this system. Finally, the peak at 8.0~$\mbox{\AA}$ is likely the |
| 368 |
> |
second nearest neighbor on the opposite monolayer because of the |
| 369 |
> |
anti-alignment evident in the angular correlation. |
| 370 |
|
|
| 371 |
+ |
Figure \ref{fig:5x5CCg} shows the radial distribution function for the |
| 372 |
+ |
$\text{CH}_2$ unified atoms. The spacing of the atoms along the tail |
| 373 |
+ |
chains accounts for the regularly spaced sharp peaks, but the broad |
| 374 |
+ |
underlying peak with its maximum at 4.6~$\mbox{\AA}$ is the |
| 375 |
+ |
distribution of chain-chain distances between two lipids. The final |
| 376 |
+ |
Figure, Figure \ref{fig:5x5HXCorr}, includes the correlation functions |
| 377 |
+ |
between the Head group and the SSD atoms. The peak in $g(r)$ at |
| 378 |
+ |
3.6~$\mbox{\AA}$ is the distance of closest approach between the two, |
| 379 |
+ |
and $g_{\gamma}(r)$ shows that the SSD atoms will align their dipoles |
| 380 |
+ |
with the head groups at close distance. However, as one increases the |
| 381 |
+ |
distance, the SSD atoms are no longer aligned. |
| 382 |
|
|
| 383 |
+ |
\section{Second Simulation: 50 randomly oriented lipids in water} |
| 384 |
+ |
\label{sec:r50} |
| 385 |
|
|
| 386 |
+ |
\subsection{Starting Configuration and Parameters} |
| 387 |
+ |
\label{sec:r50Start} |
| 388 |
|
|
| 389 |
< |
\section{Second Simulation: 50 randomly oriented lipids in water} |
| 389 |
> |
The second simulation consists of 50 single chained lipid molecules |
| 390 |
> |
embedded in a sea of 1384 SSD waters (54\% wt.). The lipids in this |
| 391 |
> |
simulation were started with random orientation and location (Figure |
| 392 |
> |
\ref{fig:r50Start} ) The simulation box measured 34.5~$\mbox{\AA}$ x |
| 393 |
> |
39.4~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ with periodic boundary conditions |
| 394 |
> |
imposed. The simulation was run in the NVE ensemble with an average |
| 395 |
> |
temperature of 300~K. |
| 396 |
|
|
| 397 |
< |
the second simulation |
| 397 |
> |
\subsection{Results} |
| 398 |
> |
\label{sec:r50Results} |
| 399 |
|
|
| 400 |
+ |
Figure \ref{fig:r50Final} is a snapshot of the system at 2.0~ns. Here |
| 401 |
+ |
we see that the system has already aggregated into several micelles |
| 402 |
+ |
and two are even starting to merge. It will be interesting to watch as |
| 403 |
+ |
this simulation continues what the total time scale for the micelle |
| 404 |
+ |
aggregation and bilayer formation will be. |
| 405 |
+ |
|
| 406 |
+ |
Figures \ref{fig:r50HHCorr}, \ref{fig:r50CCg}, and \ref{fig:r50} are |
| 407 |
+ |
the same correlation functions for the random 50 simulation as for the |
| 408 |
+ |
previous simulation of 25 lipids. What is most interesting to note, is |
| 409 |
+ |
the high degree of similarity between the correlation functions for |
| 410 |
+ |
each system. Even though the 25 lipid simulation formed a bilayer and |
| 411 |
+ |
the random 50 simulation is still in the micelle stage, both have a |
| 412 |
+ |
inter surface spacing of 6.5~$\mbox{\AA}$ with the same characteristic |
| 413 |
+ |
anti-alignment between surfaces. Not as surprising, is the consistency |
| 414 |
+ |
of the closest packing statistics between systems. Namely, a head-head |
| 415 |
+ |
closest approach distance of 4~$\mbox{\AA}$, and similar findings for |
| 416 |
+ |
the chain-chain and head-water distributions as in the 25 lipid |
| 417 |
+ |
system. |
| 418 |
+ |
|
| 419 |
|
\section{Future Directions} |
| 420 |
|
|
| 421 |
+ |
Current simulations indicate that our model is a feasible one, however |
| 422 |
+ |
improvements will need to be made to allow the system to simulate an |
| 423 |
+ |
isobaric-isothermal ensemble. This will allow the system to relax to |
| 424 |
+ |
an equilibrium configuration at room temperature and pressure allowing |
| 425 |
+ |
us to compare our model to experimental results. Also, we plan to |
| 426 |
+ |
parallelize the code for an even greater speedup. This will allow us |
| 427 |
+ |
to simulate the size systems needed to examine phenomena such as the |
| 428 |
+ |
ripple phase and drug molecule diffusion |
| 429 |
|
|
| 430 |
< |
\pagebreak |
| 431 |
< |
\bibliographystyle{achemso} |
| 432 |
< |
\bibliography{canidacy_paper} \end{document} |
| 430 |
> |
Once the work has completed on the simulation engine, we would then |
| 431 |
> |
like to use it to explore phase diagram for our model. By |
| 432 |
> |
characterizing how our model parameters affect the bilayer properties, |
| 433 |
> |
we hope to tailor our model to more closely match real biological |
| 434 |
> |
molecules. With this information, we then hope to incorporate |
| 435 |
> |
biologically relevant molecules into the system and observe their |
| 436 |
> |
transport properties across the membrane. |
| 437 |
> |
|
| 438 |
> |
\section{Acknowledgments} |
| 439 |
> |
|
| 440 |
> |
I would like to thank Dr. J.Daniel Gezelter for his guidance on this |
| 441 |
> |
project. I would also like to acknowledge the following for their help |
| 442 |
> |
and discussions during this project: Christopher Fennell, Charles |
| 443 |
> |
Vardeman, Teng Lin, Megan Sprague, Patrick Conforti, and Dan Combest. |
| 444 |
> |
|
| 445 |
> |
\pagebreak |
| 446 |
> |
\bibliographystyle{achemso} |
| 447 |
> |
\bibliography{canidacy_paper} |
| 448 |
> |
\end{document} |
