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% | preamble + macros and crap | |
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% ---------------------- |
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% | Title | |
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% ---------------------- |
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\title{A Mezzoscale Model for Phospholipid MD Simulations} |
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\author{Matthew A. Meineke\\ |
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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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%------------------------------------------------------------------- |
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% Begin Document |
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\begin{document} |
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%\maketitle |
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\nobibliography{canidacy_slides} |
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\bibliographystyle{jurabib} |
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% Slide 0 Title slide |
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\begin{slide} |
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\begin{center} |
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\bfseries |
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\fontsize{24pt}{30pt}\selectfont \color{Black} |
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A Mezzoscale Model for Phospholipid MD Simulations \par |
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\fontsize{16pt}{20pt}\selectfont \color{Green3} |
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Matthew A. Meineke\par |
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\fontsize{12pt}{15pt}\selectfont \color{Purple2} |
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Department of Chemistry and Biochemisty \par |
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University of Notre Dame \par |
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Notre Dame, IN 46556 \par |
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\fontsize{12pt}{15pt}\selectfont \color{Red} \date{today} \par |
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\end{center} |
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\end{slide} |
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% Slide 1 |
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\begin{slide} {\LARGE Talk Outline} |
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\begin{itemize} |
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\item Discussion of the research motivation and goals |
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\item Methodology |
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\item Discussion of current research and preliminary results |
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\item Future research |
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\end{itemize} |
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\end{slide} |
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% Slide 2 |
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mmeineke |
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\begin{slide} |
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\centerline{\LARGE Motivation A: Long Length Scales} |
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\begin{wrapfigure}{r}{60mm} |
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\epsfxsize=45mm |
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\epsfbox{ripple.epsi} |
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\end{wrapfigure} |
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mmeineke |
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%\epsfbox{ripple.epsi} |
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%\begin{floatingfigure}{0.45\linewidth} |
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% \incffig{ripple.epsi} |
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%\end{floatingfigure} |
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\mbox{} |
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mmeineke |
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Ripple phase: |
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\begin{itemize} |
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mmeineke |
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\item |
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mmeineke |
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The ripple (~$P_{\beta'}$~) phase lies in the transition from the gel |
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to fluid phase. |
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mmeineke |
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\item |
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Periodicity of 100 - 200 $\mbox{\AA}$\footcite{Cevc87} |
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mmeineke |
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\item |
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Current simulations have box sizes ranging from 50 - 100 $\mbox{\AA}$ |
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on a side.\footcite{Venable93}\footcite{Heller93} |
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mmeineke |
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\end{itemize} |
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\vspace{10mm} |
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\end{slide} |
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mmeineke |
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\begin{slide}{\LARGE Motivation B: Long Time Scales} |
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\begin{itemize} |
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\item |
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Drug Diffussion |
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\begin{itemize} |
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\item |
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Some drug molecules may spend appreciable amountsd of time in the |
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membrane |
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mmeineke |
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mmeineke |
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\item |
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Long time scale dynamics are need to observe and charecterize 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 bilayer simulations indicate that lipids can take nearly |
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20 ns to form completely.\footcite{Marrink01} |
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\end{itemize} |
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\end{itemize} |
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\end{slide} |
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% Slide 4 |
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\begin{slide}{\LARGE Length Scale Simplification I} |
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Replace any charged interactions of the system with dipoles. |
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\begin{itemize} |
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\item Allows for computational scaling approximately by $N$ for |
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dipole-dipole interactions. |
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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 computational simplification algorithms, |
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ie. neighbor lists. |
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\end{itemize} |
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\item In contrast, the Ewald sum, needed for calculating charge - charge |
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interactions, scales approximately by $N \log N$. |
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\end{itemize} |
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\end{slide} |
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\begin{slide}{\LARGE Length Scale Simplification II} |
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Use unified models for the water and the lipid chain. |
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\begin{itemize} |
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\item |
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Drastically reduces the number of atoms to simulate. |
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\end{itemize} |
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\begin{center} |
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\begin{figure} |
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\epsfxsize=30mm |
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\epsfbox[angle=-90]{reduction.epsi} |
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\end{figure} |
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\end{center} |
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\end{slide} |
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% Slide 5 |
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\begin{slide}{Time Scale Simplification} |
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\begin{itemize} |
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\item |
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No explicit hydrogens |
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mmeineke |
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\begin{itemize} |
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\item Hydrogen bond vibration is normally one of the fastest time |
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events in a simulation. |
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\end{itemize} |
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mmeineke |
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\item |
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Constrain all bonds to be of fixed length. |
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mmeineke |
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\begin{itemize} |
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\item As with the hydrogens, bond vibrations are the fastest motion in |
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a simulation |
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\end{itemize} |
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mmeineke |
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\item |
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Allows time steps of up to 3 fs with the current integrator. |
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\end{itemize} |
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\end{slide} |
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% Slide 6 |
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\begin{slide}{Molecular Dynamics} |
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mmeineke |
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All of our simulations will be carried out using molecular |
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dynamics. This involves solving Newton's equations of motion using |
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the classical \emph{Hamiltonian} as follows: |
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\begin{equation} |
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H(\vec{q},\vec{p}) = T(\vec{p}) + V(\vec{q}) |
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\end{equation} |
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Here $T(\vec{p})$ is the kinetic energy of the system which is a |
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function of momentum. In Cartesian space, $T(\vec{p})$ can be |
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written as: |
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\begin{equation} |
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T(\vec{p}) = \sum_{i=1}^{N} \sum_{\alpha = x,y,z} \frac{p^{2}_{i\alpha}}{2m_{i}} |
393 |
|
|
\end{equation} |
394 |
|
|
|
395 |
|
|
\end{slide} |
396 |
|
|
|
397 |
|
|
|
398 |
|
|
% Slide 7 |
399 |
|
|
\begin{slide}{The Potential} |
400 |
|
|
|
401 |
|
|
The main part of the simulation is then the calculation of forces from |
402 |
|
|
the potential energy. |
403 |
|
|
|
404 |
|
|
\begin{equation} |
405 |
|
|
\vec{F}(\vec{q}) = - \nabla V(\vec{q}) |
406 |
|
|
\end{equation} |
407 |
|
|
|
408 |
|
|
The potential itself is made of several parts. |
409 |
|
|
|
410 |
|
|
\begin{equation} |
411 |
mmeineke |
63 |
V_{tot} = |
412 |
mmeineke |
49 |
\overbrace{V_{l} + V_{\theta} + V_{\omega}}^{\mbox{bonded}} + |
413 |
|
|
\overbrace{V_{l\!j} + V_{d\!p} + V_{s\!s\!d}}^{\mbox{non-bonded}} |
414 |
|
|
\end{equation} |
415 |
|
|
|
416 |
|
|
Where the bond interactions $V_{l}$, $V_{\theta}$, and $V_{\omega}$ are |
417 |
|
|
the bond, bend, and torsion potentials, and the non-bonded |
418 |
mmeineke |
51 |
interactions $V_{l\!j}$, $V_{d\!p}$, and $V_{s\!p}$ are the |
419 |
|
|
lenard-jones, dipole-dipole, and sticky potential interactions. |
420 |
mmeineke |
49 |
|
421 |
|
|
\end{slide} |
422 |
|
|
|
423 |
|
|
|
424 |
mmeineke |
51 |
% Slide 8 |
425 |
mmeineke |
49 |
|
426 |
mmeineke |
51 |
\begin{slide}{Soft Sticky Dipole Model} |
427 |
mmeineke |
49 |
|
428 |
mmeineke |
52 |
The Soft-Sticky model for water is a reduced model. |
429 |
mmeineke |
49 |
|
430 |
mmeineke |
52 |
\begin{itemize} |
431 |
mmeineke |
49 |
|
432 |
mmeineke |
63 |
\item |
433 |
mmeineke |
52 |
The model is represented by a single point mass at the water's center |
434 |
|
|
of mass. |
435 |
mmeineke |
49 |
|
436 |
mmeineke |
63 |
\item |
437 |
mmeineke |
52 |
The point mass contains a fixed dipole of 2.35 D pointing from the |
438 |
mmeineke |
53 |
oxygens toward the hydrogens. |
439 |
mmeineke |
51 |
|
440 |
mmeineke |
52 |
\end{itemize} |
441 |
mmeineke |
51 |
|
442 |
mmeineke |
52 |
It's potential is as follows: |
443 |
|
|
|
444 |
|
|
\begin{equation} |
445 |
|
|
V_{s\!s\!d} = V_{l\!j}(r_{i\!j}) + V_{d\!p}(r_{i\!j},\Omega_{i},\Omega_{j}) |
446 |
mmeineke |
63 |
+ V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j}) |
447 |
mmeineke |
52 |
\end{equation} |
448 |
|
|
\end{slide} |
449 |
|
|
|
450 |
mmeineke |
54 |
% Slide 8b |
451 |
mmeineke |
52 |
|
452 |
mmeineke |
54 |
\begin{slide}{SSD Diagram} |
453 |
|
|
|
454 |
|
|
\begin{center} |
455 |
|
|
\begin{figure} |
456 |
|
|
\epsfxsize=50mm |
457 |
|
|
\epsfbox{ssd.epsi} |
458 |
|
|
\end{figure} |
459 |
|
|
\end{center} |
460 |
|
|
|
461 |
|
|
A Diagram of the SSD model. |
462 |
|
|
\end{slide} |
463 |
|
|
|
464 |
mmeineke |
52 |
% Slide 9 |
465 |
|
|
\begin{slide}{Hydrogen Bonding in SSD} |
466 |
|
|
|
467 |
|
|
It is important to note that SSD has a potential specifically to |
468 |
mmeineke |
53 |
recreate the hydrogen bonding network of water. |
469 |
mmeineke |
52 |
|
470 |
mmeineke |
54 |
|
471 |
mmeineke |
52 |
ICE SSD |
472 |
|
|
|
473 |
|
|
ICE point Dipole |
474 |
|
|
|
475 |
mmeineke |
54 |
|
476 |
mmeineke |
53 |
The importance of the hydrogen bond network is it's significant |
477 |
mmeineke |
52 |
contribution to the hydrophobic driving force of bilayer formation. |
478 |
|
|
\end{slide} |
479 |
|
|
|
480 |
|
|
|
481 |
|
|
% Slide 10 |
482 |
|
|
|
483 |
|
|
\begin{slide}{The Lipid Model} |
484 |
|
|
|
485 |
mmeineke |
53 |
To eliminate the need for charge-charge interactions, our lipid model |
486 |
|
|
replaces the phospholipid head group with a single large head group |
487 |
|
|
atom containing a freely oriented dipole. The tail is a simple alkane chain. |
488 |
|
|
|
489 |
|
|
Lipid Properties: |
490 |
|
|
\begin{itemize} |
491 |
|
|
\item $|\vec{\mu}_{\text{HEAD}}| = 20.6\ \text{D}$ |
492 |
|
|
\item $m_{\text{HEAD}} = 196\ \text{amu}$ |
493 |
|
|
\item Tail atoms are unified CH, $\text{CH}_2$, and $\text{CH}_3$ atoms |
494 |
mmeineke |
63 |
\begin{itemize} |
495 |
|
|
\item Alkane forcefield parameters taken from TraPPE |
496 |
|
|
\end{itemize} |
497 |
mmeineke |
53 |
\end{itemize} |
498 |
|
|
|
499 |
|
|
\end{slide} |
500 |
|
|
|
501 |
|
|
|
502 |
|
|
% Slide 11 |
503 |
|
|
|
504 |
|
|
\begin{slide}{Lipid Model} |
505 |
|
|
|
506 |
mmeineke |
52 |
|
507 |
mmeineke |
63 |
|
508 |
mmeineke |
52 |
\end{slide} |
509 |
|
|
|
510 |
|
|
|
511 |
mmeineke |
53 |
% Slide 12 |
512 |
mmeineke |
52 |
|
513 |
|
|
\begin{slide}{Initial Runs: 25 Lipids in water} |
514 |
|
|
|
515 |
mmeineke |
53 |
\textbf{Simulation Parameters:} |
516 |
mmeineke |
52 |
|
517 |
mmeineke |
53 |
\begin{itemize} |
518 |
|
|
|
519 |
|
|
\item Starting Configuration: |
520 |
mmeineke |
63 |
\begin{itemize} |
521 |
|
|
\item 25 lipid molecules arranged in a 5 x 5 square |
522 |
|
|
\item square was surrounded by a sea of 1386 waters |
523 |
|
|
\begin{itemize} |
524 |
|
|
\item final water to lipid ratio was 55.4:1 |
525 |
|
|
\end{itemize} |
526 |
|
|
\end{itemize} |
527 |
mmeineke |
53 |
|
528 |
|
|
\item Lipid had only a single saturated chain of 16 carbons |
529 |
|
|
|
530 |
|
|
\item Box Size: 34.5 $\mbox{\AA}$ x 39.4 $\mbox{\AA}$ x 39.4 $\mbox{\AA}$ |
531 |
|
|
|
532 |
|
|
\item dt = 2.0 - 3.0 fs |
533 |
|
|
|
534 |
|
|
\item T = 300 K |
535 |
|
|
|
536 |
|
|
\item NVE ensemble |
537 |
|
|
|
538 |
|
|
\item Periodic boundary conditions |
539 |
|
|
\end{itemize} |
540 |
|
|
|
541 |
mmeineke |
52 |
\end{slide} |
542 |
|
|
|
543 |
|
|
|
544 |
mmeineke |
53 |
% Slide 13 |
545 |
mmeineke |
52 |
|
546 |
mmeineke |
54 |
\begin{slide}{5x5: Initial} |
547 |
mmeineke |
52 |
|
548 |
mmeineke |
54 |
\begin{center} |
549 |
|
|
\begin{figure} |
550 |
|
|
\epsfxsize=50mm |
551 |
|
|
\epsfbox{5x5-initial.eps} |
552 |
|
|
\end{figure} |
553 |
|
|
\end{center} |
554 |
mmeineke |
52 |
|
555 |
mmeineke |
54 |
The initial configuration |
556 |
mmeineke |
52 |
|
557 |
|
|
\end{slide} |
558 |
|
|
|
559 |
mmeineke |
54 |
\begin{slide}{5x5: Final} |
560 |
mmeineke |
52 |
|
561 |
mmeineke |
54 |
\begin{center} |
562 |
|
|
\begin{figure} |
563 |
|
|
\epsfxsize=60mm |
564 |
|
|
\epsfbox{5x5-1.7ns.eps} |
565 |
|
|
\end{figure} |
566 |
|
|
\end{center} |
567 |
|
|
|
568 |
|
|
The final configuration at 1.7 ns. |
569 |
|
|
|
570 |
|
|
\end{slide} |
571 |
|
|
|
572 |
|
|
|
573 |
mmeineke |
53 |
% Slide 14 |
574 |
mmeineke |
52 |
|
575 |
|
|
\begin{slide}{5x5: $g(r)$} |
576 |
|
|
|
577 |
mmeineke |
54 |
\begin{center} |
578 |
|
|
\begin{figure} |
579 |
|
|
\epsfxsize=60mm |
580 |
|
|
\epsfbox{all5x5-HEAD-HEAD-gr.eps} |
581 |
|
|
\end{figure} |
582 |
|
|
\end{center} |
583 |
mmeineke |
52 |
|
584 |
|
|
|
585 |
mmeineke |
54 |
\end{slide} |
586 |
mmeineke |
52 |
|
587 |
mmeineke |
54 |
\begin{slide}{5x5: $g(r)$} |
588 |
|
|
|
589 |
|
|
\begin{center} |
590 |
|
|
\begin{figure} |
591 |
|
|
\epsfxsize=60mm |
592 |
|
|
\epsfbox{all5x5-HEAD-X-gr.eps} |
593 |
|
|
\end{figure} |
594 |
|
|
\end{center} |
595 |
|
|
|
596 |
|
|
|
597 |
mmeineke |
52 |
\end{slide} |
598 |
|
|
|
599 |
|
|
|
600 |
mmeineke |
53 |
% Slide 15 |
601 |
mmeineke |
52 |
|
602 |
|
|
\begin{slide}{5x5: $\cos$ correlations} |
603 |
|
|
|
604 |
mmeineke |
54 |
\begin{center} |
605 |
|
|
\begin{figure} |
606 |
|
|
\epsfxsize=60mm |
607 |
|
|
\epsfbox{all5x5-HEAD-HEAD-cr.eps} |
608 |
|
|
\end{figure} |
609 |
|
|
\end{center} |
610 |
mmeineke |
52 |
|
611 |
|
|
\end{slide} |
612 |
|
|
|
613 |
mmeineke |
54 |
\begin{slide}{5x5: $\cos$ correlations} |
614 |
mmeineke |
52 |
|
615 |
mmeineke |
54 |
\begin{center} |
616 |
|
|
\begin{figure} |
617 |
|
|
\epsfxsize=60mm |
618 |
|
|
\epsfbox{all5x5-HEAD-X-cr.eps} |
619 |
|
|
\end{figure} |
620 |
|
|
\end{center} |
621 |
|
|
|
622 |
|
|
\end{slide} |
623 |
|
|
|
624 |
|
|
|
625 |
mmeineke |
53 |
% Slide 16 |
626 |
mmeineke |
52 |
|
627 |
mmeineke |
53 |
\begin{slide}{Initial Runs: 50 Lipids randomly arranged in water} |
628 |
mmeineke |
52 |
|
629 |
mmeineke |
53 |
\textbf{Simulation Parameters:} |
630 |
mmeineke |
52 |
|
631 |
mmeineke |
53 |
\begin{itemize} |
632 |
|
|
|
633 |
|
|
\item Starting Configuration: |
634 |
mmeineke |
63 |
\begin{itemize} |
635 |
|
|
\item 50 lipid molecules arranged randomly in a rectangular box |
636 |
|
|
\item The box was then filled with 1384 waters |
637 |
|
|
\begin{itemize} |
638 |
|
|
\item final water to lipid ratio was 27:1 |
639 |
|
|
\end{itemize} |
640 |
|
|
\end{itemize} |
641 |
mmeineke |
53 |
|
642 |
|
|
\item Lipid had only a single saturated chain of 16 carbons |
643 |
|
|
|
644 |
|
|
\item Box Size: 26.6 $\mbox{\AA}$ x 26.6 $\mbox{\AA}$ x 108.4 $\mbox{\AA}$ |
645 |
|
|
|
646 |
|
|
\item dt = 2.0 - 3.0 fs |
647 |
|
|
|
648 |
|
|
\item T = 300 K |
649 |
|
|
|
650 |
|
|
\item NVE ensemble |
651 |
|
|
|
652 |
mmeineke |
63 |
\item Periodic boundary conditions |
653 |
mmeineke |
53 |
|
654 |
|
|
\end{itemize} |
655 |
|
|
|
656 |
mmeineke |
52 |
\end{slide} |
657 |
|
|
|
658 |
|
|
|
659 |
mmeineke |
53 |
% Slide 17 |
660 |
mmeineke |
52 |
|
661 |
mmeineke |
54 |
\begin{slide}{R-50: Initial} |
662 |
mmeineke |
52 |
|
663 |
mmeineke |
54 |
\begin{center} |
664 |
|
|
\begin{figure} |
665 |
|
|
\epsfxsize=100mm |
666 |
|
|
\epsfbox{r50-initial.eps} |
667 |
|
|
\end{figure} |
668 |
|
|
\end{center} |
669 |
mmeineke |
52 |
|
670 |
mmeineke |
54 |
The initial configuration |
671 |
mmeineke |
52 |
|
672 |
|
|
\end{slide} |
673 |
|
|
|
674 |
mmeineke |
54 |
\begin{slide}{R-50: Final} |
675 |
mmeineke |
52 |
|
676 |
mmeineke |
54 |
\begin{center} |
677 |
|
|
\begin{figure} |
678 |
|
|
\epsfxsize=100mm |
679 |
|
|
\epsfbox{r50-521ps.eps} |
680 |
|
|
\end{figure} |
681 |
|
|
\end{center} |
682 |
|
|
|
683 |
|
|
The fianl configuration at 521 ps |
684 |
|
|
|
685 |
|
|
\end{slide} |
686 |
|
|
|
687 |
|
|
|
688 |
mmeineke |
53 |
% Slide 18 |
689 |
mmeineke |
52 |
|
690 |
|
|
\begin{slide}{R-50: $g(r)$} |
691 |
|
|
|
692 |
|
|
|
693 |
mmeineke |
54 |
\begin{center} |
694 |
|
|
\begin{figure} |
695 |
|
|
\epsfxsize=60mm |
696 |
|
|
\epsfbox{r50-HEAD-HEAD-gr.eps} |
697 |
|
|
\end{figure} |
698 |
|
|
\end{center} |
699 |
mmeineke |
52 |
|
700 |
mmeineke |
54 |
\end{slide} |
701 |
mmeineke |
52 |
|
702 |
mmeineke |
54 |
|
703 |
|
|
\begin{slide}{R-50: $g(r)$} |
704 |
|
|
|
705 |
|
|
|
706 |
|
|
\begin{center} |
707 |
|
|
\begin{figure} |
708 |
|
|
\epsfxsize=60mm |
709 |
|
|
\epsfbox{r50-HEAD-X-gr.eps} |
710 |
|
|
\end{figure} |
711 |
|
|
\end{center} |
712 |
|
|
|
713 |
mmeineke |
52 |
\end{slide} |
714 |
|
|
|
715 |
|
|
|
716 |
mmeineke |
53 |
% Slide 19 |
717 |
mmeineke |
52 |
|
718 |
|
|
\begin{slide}{R-50: $\cos$ correlations} |
719 |
|
|
|
720 |
|
|
|
721 |
mmeineke |
54 |
\begin{center} |
722 |
|
|
\begin{figure} |
723 |
|
|
\epsfxsize=60mm |
724 |
|
|
\epsfbox{r50-HEAD-HEAD-cr.eps} |
725 |
|
|
\end{figure} |
726 |
|
|
\end{center} |
727 |
|
|
|
728 |
mmeineke |
52 |
\end{slide} |
729 |
|
|
|
730 |
mmeineke |
54 |
\begin{slide}{R-50: $\cos$ correlations} |
731 |
mmeineke |
52 |
|
732 |
mmeineke |
54 |
|
733 |
|
|
\begin{center} |
734 |
|
|
\begin{figure} |
735 |
|
|
\epsfxsize=60mm |
736 |
|
|
\epsfbox{r50-HEAD-X-cr.eps} |
737 |
|
|
\end{figure} |
738 |
|
|
\end{center} |
739 |
|
|
|
740 |
|
|
\end{slide} |
741 |
|
|
|
742 |
|
|
|
743 |
mmeineke |
53 |
% Slide 20 |
744 |
mmeineke |
52 |
|
745 |
|
|
\begin{slide}{Future Directions} |
746 |
|
|
|
747 |
mmeineke |
53 |
\begin{itemize} |
748 |
mmeineke |
52 |
|
749 |
mmeineke |
63 |
\item |
750 |
mmeineke |
53 |
Simulation of a lipid with 2 chains, or perhaps expand the current |
751 |
|
|
unified chain atoms to take up greater steric bulk. |
752 |
|
|
|
753 |
mmeineke |
63 |
\item |
754 |
mmeineke |
53 |
Incorporate constant pressure and constant temperature into the ensemble. |
755 |
|
|
|
756 |
|
|
\item |
757 |
|
|
Parrellize the code. |
758 |
|
|
|
759 |
|
|
\end{itemize} |
760 |
mmeineke |
52 |
\end{slide} |
761 |
|
|
|
762 |
|
|
|
763 |
mmeineke |
53 |
% Slide 21 |
764 |
mmeineke |
52 |
|
765 |
|
|
\begin{slide}{Acknowledgements} |
766 |
|
|
|
767 |
mmeineke |
53 |
\begin{itemize} |
768 |
mmeineke |
52 |
|
769 |
mmeineke |
53 |
\item Dr. J. Daniel Gezelter |
770 |
|
|
\item Christopher Fennel |
771 |
|
|
\item Charles Vardeman |
772 |
|
|
\item Teng Lin |
773 |
mmeineke |
64 |
\item Megan Sprauge |
774 |
|
|
\item Patrick Conforti |
775 |
|
|
\item Dan Combest |
776 |
mmeineke |
52 |
|
777 |
mmeineke |
53 |
\end{itemize} |
778 |
|
|
|
779 |
|
|
Funding by: |
780 |
|
|
\begin{itemize} |
781 |
|
|
\item Dreyfus New Faculty Award |
782 |
|
|
\end{itemize} |
783 |
|
|
|
784 |
mmeineke |
52 |
\end{slide} |
785 |
|
|
|
786 |
|
|
|
787 |
|
|
|
788 |
|
|
|
789 |
|
|
|
790 |
|
|
|
791 |
|
|
|
792 |
|
|
|
793 |
mmeineke |
49 |
%%%%%%%%%%%%%%%%%%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
794 |
|
|
|
795 |
|
|
\end{document} |