matt_papers/canidacy_talk/canidacy_slides.backup

% temporary preamble

%\documentclass[ps,frames,final,nototal,slideColor,colorBG]{prosper}

\documentclass{seminar}
\usepackage{color}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{wrapfig}
\usepackage{epsf}
\usepackage{jurabib}

% ----------------------
% |      Title         |
% ----------------------

\title{A Mezzoscale Model for Phospholipid MD Simulations}

\author{Matthew A. Meineke\\
Department of Chemistry and Biochemistry\\ 
University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}

%-------------------------------------------------------------------
%                       Begin Document

\begin{document}

\maketitle

\bibliography{canidacy_slides}
\bibliographystyle{jurabib}


% Slide 1
\begin{slide} {Talk Outline}
\begin{itemize}

\item Discussion of the research motivation and goals

\item Methodology

\item Discussion of current research and preliminary results

\item Future research

\end{itemize}
\end{slide}


% Slide 2

\begin{slide}{Motivation A: Long Length Scales}


\begin{wrapfigure}{r}{45mm}

\epsfxsize=45mm
\epsfbox{ripple.epsi}

\end{wrapfigure}

Ripple phase:

\begin{itemize}

\item 
The ripple (~$P_{\beta'}$~) phase lies in the transition from the gel
to fluid phase.

\item 
periodicity of 100 - 200 $\mbox{\AA}$\footcite{Berne90}

\end{itemize}
\end{slide}


\begin{slide}{Motivation}

There is a strong need in phospholipid bilayer simulations for the
capability to simulate both long time and length scales. Consider the
following:

\begin{itemize}

\item Drug diffusion
        \begin{itemize} 
        \item Some drug molecules may spend an appreciable time in the 
        membrane. Long time scale dynamics are needed to observe and 
        characterize their actions.
        \end{itemize}

\item Ripple phase
        \begin{itemize}
        \item Between the bilayer gel and fluid phase there exists a ripple 
        phase. This phase has a period of about 100 - 200 $\mbox{\AA}$.
        \end{itemize}

\item Bilayer formation dynamics
        \begin{itemize}
        \item Initial simulations show that bilayers can take upwards of 
        20 ns to form completely.
        \end{itemize}

\end{itemize}
\end{slide}


% Slide 4

\begin{slide}{Length Scale Simplification}
\begin{itemize}

\item 
Replace any charged interactions of the system with dipoles.

        \begin{itemize}
        \item Allows for computational scaling approximately by $N$ for 
              dipole-dipole interactions.
        \item In contrast, the Ewald sum scales approximately by $N \log N$.
        \end{itemize}

\item
Use unified models for the water and the lipid chain.

        \begin{itemize}
        \item Drastically reduces the number of atoms to simulate.
        \item Number of water interactions alone reduced by $\frac{1}{3}$.
        \end{itemize}
\end{itemize}
\end{slide}


% Slide 5

\begin{slide}{Time Scale Simplification}
\begin{itemize}

\item
No explicit hydrogens

        \begin{itemize}
        \item Hydrogen bond vibration is normally one of the fastest time 
                events in a simulation.
        \end{itemize}

\item
Constrain all bonds to be of fixed length.

        \begin{itemize}
        \item As with the hydrogens, bond vibrations are the fastest motion in
         a simulation
        \end{itemize}

\item
Allows time steps of up to 3 fs with the current integrator.

\end{itemize}
\end{slide}


% Slide 6
\begin{slide}{Molecular Dynamics}

All of our simulations will be carried out using molecular
dynamics. This involves solving Newton's equations of motion using
the classical \emph{Hamiltonian} as follows:

\begin{equation}
H(\vec{q},\vec{p}) = T(\vec{p}) + V(\vec{q})
\end{equation}

Here $T(\vec{p})$ is the kinetic energy of the system which is a
function of momentum. In Cartesian space, $T(\vec{p})$ can be
written as:

\begin{equation}
T(\vec{p}) = \sum_{i=1}^{N} \sum_{\alpha = x,y,z} \frac{p^{2}_{i\alpha}}{2m_{i}}
\end{equation}

\end{slide}


% Slide 7
\begin{slide}{The Potential}

The main part of the simulation is then the calculation of forces from
the potential energy.

\begin{equation}
\vec{F}(\vec{q}) = - \nabla V(\vec{q})
\end{equation}

The potential itself is made of several parts.

\begin{equation}
V_{tot} = 
\overbrace{V_{l} + V_{\theta} + V_{\omega}}^{\mbox{bonded}} +
\overbrace{V_{l\!j} + V_{d\!p} + V_{s\!s\!d}}^{\mbox{non-bonded}}
\end{equation}

Where the bond interactions $V_{l}$, $V_{\theta}$, and $V_{\omega}$ are
the bond, bend, and torsion potentials, and the non-bonded
interactions $V_{l\!j}$, $V_{d\!p}$, and $V_{s\!p}$ are the
lenard-jones, dipole-dipole, and sticky potential interactions.

\end{slide}


% Slide 8

\begin{slide}{Soft Sticky Dipole Model}

The Soft-Sticky model for water is a reduced model.

\begin{itemize}

\item 
The model is represented by a single point mass at the water's center
of mass.

\item 
The point mass contains a fixed dipole of 2.35 D pointing from the
oxygens toward the hydrogens.

\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})
\end{equation}
\end{slide}

% Slide 8b

\begin{slide}{SSD Diagram}

\begin{center}
\begin{figure}
\epsfxsize=50mm
\epsfbox{ssd.epsi}
\end{figure}
\end{center}

A Diagram of the SSD model.
\end{slide}

% Slide 9
\begin{slide}{Hydrogen Bonding in SSD}

It is important to note that SSD has a potential specifically to
recreate the hydrogen bonding network of water.


ICE SSD

ICE point Dipole


The importance of the hydrogen bond network is it's significant
contribution to the hydrophobic driving force of bilayer formation.
\end{slide}


% Slide 10

\begin{slide}{The Lipid Model}

To eliminate the need for charge-charge interactions, our lipid model
replaces the phospholipid head group with a single large head group
atom containing a freely oriented dipole. The tail is a simple alkane chain.

Lipid Properties:
\begin{itemize}
\item $|\vec{\mu}_{\text{HEAD}}| = 20.6\ \text{D}$
\item $m_{\text{HEAD}} = 196\ \text{amu}$
\item Tail atoms are unified CH, $\text{CH}_2$, and $\text{CH}_3$ atoms
        \begin{itemize}
        \item Alkane forcefield parameters taken from TraPPE
        \end{itemize}
\end{itemize}

\end{slide}


% Slide 11

\begin{slide}{Lipid Model}


\end{slide}


% Slide 12

\begin{slide}{Initial Runs: 25 Lipids in water}

\textbf{Simulation Parameters:}

\begin{itemize}

\item Starting Configuration:
        \begin{itemize}
        \item 25 lipid molecules arranged in a 5 x 5 square
        \item square was surrounded by a sea of 1386  waters
                \begin{itemize}
                \item final water to lipid ratio was 55.4:1
                \end{itemize}
        \end{itemize}

\item Lipid had only a single saturated chain of 16 carbons

\item Box Size: 34.5 $\mbox{\AA}$ x 39.4 $\mbox{\AA}$ x 39.4 $\mbox{\AA}$

\item dt = 2.0 - 3.0 fs

\item T = 300 K

\item NVE ensemble

\item Periodic boundary conditions
\end{itemize}

\end{slide}


% Slide 13

\begin{slide}{5x5: Initial}

\begin{center}
\begin{figure}
\epsfxsize=50mm
\epsfbox{5x5-initial.eps}
\end{figure}
\end{center}

The initial configuration

\end{slide}

\begin{slide}{5x5: Final}

\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{5x5-1.7ns.eps}
\end{figure}
\end{center}

The final configuration at 1.7 ns.

\end{slide}


% Slide 14

\begin{slide}{5x5: $g(r)$}

\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{all5x5-HEAD-HEAD-gr.eps}
\end{figure}
\end{center}


\end{slide}

\begin{slide}{5x5: $g(r)$}

\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{all5x5-HEAD-X-gr.eps}
\end{figure}
\end{center}


\end{slide}


% Slide 15

\begin{slide}{5x5: $\cos$ correlations}

\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{all5x5-HEAD-HEAD-cr.eps}
\end{figure}
\end{center}

\end{slide}

\begin{slide}{5x5: $\cos$ correlations}

\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{all5x5-HEAD-X-cr.eps}
\end{figure}
\end{center}

\end{slide}


% Slide 16

\begin{slide}{Initial Runs: 50 Lipids randomly arranged in water}

\textbf{Simulation Parameters:}

\begin{itemize}

\item Starting Configuration:
        \begin{itemize}
        \item 50 lipid molecules arranged randomly in a rectangular box
        \item The box was then filled with 1384 waters
                \begin{itemize}
                \item final water to lipid ratio was 27:1
                \end{itemize}
        \end{itemize}

\item Lipid had only a single saturated chain of 16 carbons

\item Box Size: 26.6 $\mbox{\AA}$ x 26.6 $\mbox{\AA}$ x 108.4 $\mbox{\AA}$

\item dt = 2.0 - 3.0 fs

\item T = 300 K

\item NVE ensemble

\item Periodic boundary conditions      

\end{itemize}

\end{slide}


% Slide 17

\begin{slide}{R-50: Initial}

\begin{center}
\begin{figure}
\epsfxsize=100mm
\epsfbox{r50-initial.eps}
\end{figure}
\end{center}

The initial configuration

\end{slide}

\begin{slide}{R-50: Final}

\begin{center}
\begin{figure}
\epsfxsize=100mm
\epsfbox{r50-521ps.eps}
\end{figure}
\end{center}

The fianl configuration at 521 ps

\end{slide}


% Slide 18

\begin{slide}{R-50: $g(r)$}


\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{r50-HEAD-HEAD-gr.eps}
\end{figure}
\end{center}

\end{slide}


\begin{slide}{R-50: $g(r)$}


\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{r50-HEAD-X-gr.eps}
\end{figure}
\end{center}

\end{slide}


% Slide 19

\begin{slide}{R-50: $\cos$ correlations}


\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{r50-HEAD-HEAD-cr.eps}
\end{figure}
\end{center}

\end{slide}

\begin{slide}{R-50: $\cos$ correlations}


\begin{center}
\begin{figure}
\epsfxsize=60mm
\epsfbox{r50-HEAD-X-cr.eps}
\end{figure}
\end{center}

\end{slide}


% Slide 20

\begin{slide}{Future Directions}

\begin{itemize}

\item 
Simulation of a lipid with 2 chains, or perhaps expand the current
unified chain atoms to take up greater steric bulk.

\item 
Incorporate constant pressure and constant temperature into the ensemble.

\item
Parrellize the code.

\end{itemize}
\end{slide}


% Slide 21

\begin{slide}{Acknowledgements}

\begin{itemize}

\item Dr. J. Daniel Gezelter
\item Christopher Fennel
\item Charles Vardeman
\item Teng Lin

\end{itemize}

Funding by:
\begin{itemize}
\item Dreyfus New Faculty Award
\end{itemize}

\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
Revision:	63
Committed:	Wed Aug 7 19:09:03 2002 UTC (23 years, 2 months ago) by mmeineke
File size:	10054 byte(s)
Log Message:	switched in Chuck's magic slide enviroment
#	User	Rev	Content
1	mmeineke	63	% temporary preamble
2
3			%\documentclass[ps,frames,final,nototal,slideColor,colorBG]{prosper}
4
5			\documentclass{seminar}
6			\usepackage{color}
7
8			\usepackage{amsmath}
9			\usepackage{amssymb}
10			\usepackage{wrapfig}
11			\usepackage{epsf}
12			\usepackage{jurabib}
13
14			% ----------------------
15			% \| Title \|
16			% ----------------------
17
18			\title{A Mezzoscale Model for Phospholipid MD Simulations}
19
20			\author{Matthew A. Meineke\\
21			Department of Chemistry and Biochemistry\\
22			University of Notre Dame\\
23			Notre Dame, Indiana 46556}
24
25			\date{\today}
26
27			%-------------------------------------------------------------------
28			% Begin Document
29
30			\begin{document}
31
32			\maketitle
33
34			\bibliography{canidacy_slides}
35			\bibliographystyle{jurabib}
36
37
38
39
40
41			% Slide 1
42			\begin{slide} {Talk Outline}
43			\begin{itemize}
44
45			\item Discussion of the research motivation and goals
46
47			\item Methodology
48
49			\item Discussion of current research and preliminary results
50
51			\item Future research
52
53			\end{itemize}
54			\end{slide}
55
56
57			% Slide 2
58
59			\begin{slide}{Motivation A: Long Length Scales}
60
61
62
63			\begin{wrapfigure}{r}{45mm}
64
65			\epsfxsize=45mm
66			\epsfbox{ripple.epsi}
67
68			\end{wrapfigure}
69
70			Ripple phase:
71
72			\begin{itemize}
73
74			\item
75			The ripple (~$P_{\beta'}$~) phase lies in the transition from the gel
76			to fluid phase.
77
78			\item
79			periodicity of 100 - 200 $\mbox{\AA}$\footcite{Berne90}
80
81			\end{itemize}
82			\end{slide}
83
84
85
86
87
88
89			\begin{slide}{Motivation}
90
91			There is a strong need in phospholipid bilayer simulations for the
92			capability to simulate both long time and length scales. Consider the
93			following:
94
95			\begin{itemize}
96
97			\item Drug diffusion
98			\begin{itemize}
99			\item Some drug molecules may spend an appreciable time in the
100			membrane. Long time scale dynamics are needed to observe and
101			characterize their actions.
102			\end{itemize}
103
104			\item Ripple phase
105			\begin{itemize}
106			\item Between the bilayer gel and fluid phase there exists a ripple
107			phase. This phase has a period of about 100 - 200 $\mbox{\AA}$.
108			\end{itemize}
109
110			\item Bilayer formation dynamics
111			\begin{itemize}
112			\item Initial simulations show that bilayers can take upwards of
113			20 ns to form completely.
114			\end{itemize}
115
116			\end{itemize}
117			\end{slide}
118
119
120			% Slide 4
121
122			\begin{slide}{Length Scale Simplification}
123			\begin{itemize}
124
125			\item
126			Replace any charged interactions of the system with dipoles.
127
128			\begin{itemize}
129			\item Allows for computational scaling approximately by $N$ for
130			dipole-dipole interactions.
131			\item In contrast, the Ewald sum scales approximately by $N \log N$.
132			\end{itemize}
133
134			\item
135			Use unified models for the water and the lipid chain.
136
137			\begin{itemize}
138			\item Drastically reduces the number of atoms to simulate.
139			\item Number of water interactions alone reduced by $\frac{1}{3}$.
140			\end{itemize}
141			\end{itemize}
142			\end{slide}
143
144
145			% Slide 5
146
147			\begin{slide}{Time Scale Simplification}
148			\begin{itemize}
149
150			\item
151			No explicit hydrogens
152
153			\begin{itemize}
154			\item Hydrogen bond vibration is normally one of the fastest time
155			events in a simulation.
156			\end{itemize}
157
158			\item
159			Constrain all bonds to be of fixed length.
160
161			\begin{itemize}
162			\item As with the hydrogens, bond vibrations are the fastest motion in
163			a simulation
164			\end{itemize}
165
166			\item
167			Allows time steps of up to 3 fs with the current integrator.
168
169			\end{itemize}
170			\end{slide}
171
172
173			% Slide 6
174			\begin{slide}{Molecular Dynamics}
175
176			All of our simulations will be carried out using molecular
177			dynamics. This involves solving Newton's equations of motion using
178			the classical \emph{Hamiltonian} as follows:
179
180			\begin{equation}
181			H(\vec{q},\vec{p}) = T(\vec{p}) + V(\vec{q})
182			\end{equation}
183
184			Here $T(\vec{p})$ is the kinetic energy of the system which is a
185			function of momentum. In Cartesian space, $T(\vec{p})$ can be
186			written as:
187
188			\begin{equation}
189			T(\vec{p}) = \sum_{i=1}^{N} \sum_{\alpha = x,y,z} \frac{p^{2}_{i\alpha}}{2m_{i}}
190			\end{equation}
191
192			\end{slide}
193
194
195			% Slide 7
196			\begin{slide}{The Potential}
197
198			The main part of the simulation is then the calculation of forces from
199			the potential energy.
200
201			\begin{equation}
202			\vec{F}(\vec{q}) = - \nabla V(\vec{q})
203			\end{equation}
204
205			The potential itself is made of several parts.
206
207			\begin{equation}
208			V_{tot} =
209			\overbrace{V_{l} + V_{\theta} + V_{\omega}}^{\mbox{bonded}} +
210			\overbrace{V_{l\!j} + V_{d\!p} + V_{s\!s\!d}}^{\mbox{non-bonded}}
211			\end{equation}
212
213			Where the bond interactions $V_{l}$, $V_{\theta}$, and $V_{\omega}$ are
214			the bond, bend, and torsion potentials, and the non-bonded
215			interactions $V_{l\!j}$, $V_{d\!p}$, and $V_{s\!p}$ are the
216			lenard-jones, dipole-dipole, and sticky potential interactions.
217
218			\end{slide}
219
220
221			% Slide 8
222
223			\begin{slide}{Soft Sticky Dipole Model}
224
225			The Soft-Sticky model for water is a reduced model.
226
227			\begin{itemize}
228
229			\item
230			The model is represented by a single point mass at the water's center
231			of mass.
232
233			\item
234			The point mass contains a fixed dipole of 2.35 D pointing from the
235			oxygens toward the hydrogens.
236
237			\end{itemize}
238
239			It's potential is as follows:
240
241			\begin{equation}
242			V_{s\!s\!d} = V_{l\!j}(r_{i\!j}) + V_{d\!p}(r_{i\!j},\Omega_{i},\Omega_{j})
243			+ V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j})
244			\end{equation}
245			\end{slide}
246
247			% Slide 8b
248
249			\begin{slide}{SSD Diagram}
250
251			\begin{center}
252			\begin{figure}
253			\epsfxsize=50mm
254			\epsfbox{ssd.epsi}
255			\end{figure}
256			\end{center}
257
258			A Diagram of the SSD model.
259			\end{slide}
260
261			% Slide 9
262			\begin{slide}{Hydrogen Bonding in SSD}
263
264			It is important to note that SSD has a potential specifically to
265			recreate the hydrogen bonding network of water.
266
267
268			ICE SSD
269
270			ICE point Dipole
271
272
273			The importance of the hydrogen bond network is it's significant
274			contribution to the hydrophobic driving force of bilayer formation.
275			\end{slide}
276
277
278			% Slide 10
279
280			\begin{slide}{The Lipid Model}
281
282			To eliminate the need for charge-charge interactions, our lipid model
283			replaces the phospholipid head group with a single large head group
284			atom containing a freely oriented dipole. The tail is a simple alkane chain.
285
286			Lipid Properties:
287			\begin{itemize}
288			\item $\|\vec{\mu}_{\text{HEAD}}\| = 20.6\ \text{D}$
289			\item $m_{\text{HEAD}} = 196\ \text{amu}$
290			\item Tail atoms are unified CH, $\text{CH}_2$, and $\text{CH}_3$ atoms
291			\begin{itemize}
292			\item Alkane forcefield parameters taken from TraPPE
293			\end{itemize}
294			\end{itemize}
295
296			\end{slide}
297
298
299			% Slide 11
300
301			\begin{slide}{Lipid Model}
302
303
304
305			\end{slide}
306
307
308			% Slide 12
309
310			\begin{slide}{Initial Runs: 25 Lipids in water}
311
312			\textbf{Simulation Parameters:}
313
314			\begin{itemize}
315
316			\item Starting Configuration:
317			\begin{itemize}
318			\item 25 lipid molecules arranged in a 5 x 5 square
319			\item square was surrounded by a sea of 1386 waters
320			\begin{itemize}
321			\item final water to lipid ratio was 55.4:1
322			\end{itemize}
323			\end{itemize}
324
325			\item Lipid had only a single saturated chain of 16 carbons
326
327			\item Box Size: 34.5 $\mbox{\AA}$ x 39.4 $\mbox{\AA}$ x 39.4 $\mbox{\AA}$
328
329			\item dt = 2.0 - 3.0 fs
330
331			\item T = 300 K
332
333			\item NVE ensemble
334
335			\item Periodic boundary conditions
336			\end{itemize}
337
338			\end{slide}
339
340
341			% Slide 13
342
343			\begin{slide}{5x5: Initial}
344
345			\begin{center}
346			\begin{figure}
347			\epsfxsize=50mm
348			\epsfbox{5x5-initial.eps}
349			\end{figure}
350			\end{center}
351
352			The initial configuration
353
354			\end{slide}
355
356			\begin{slide}{5x5: Final}
357
358			\begin{center}
359			\begin{figure}
360			\epsfxsize=60mm
361			\epsfbox{5x5-1.7ns.eps}
362			\end{figure}
363			\end{center}
364
365			The final configuration at 1.7 ns.
366
367			\end{slide}
368
369
370			% Slide 14
371
372			\begin{slide}{5x5: $g(r)$}
373
374			\begin{center}
375			\begin{figure}
376			\epsfxsize=60mm
377			\epsfbox{all5x5-HEAD-HEAD-gr.eps}
378			\end{figure}
379			\end{center}
380
381
382			\end{slide}
383
384			\begin{slide}{5x5: $g(r)$}
385
386			\begin{center}
387			\begin{figure}
388			\epsfxsize=60mm
389			\epsfbox{all5x5-HEAD-X-gr.eps}
390			\end{figure}
391			\end{center}
392
393
394			\end{slide}
395
396
397			% Slide 15
398
399			\begin{slide}{5x5: $\cos$ correlations}
400
401			\begin{center}
402			\begin{figure}
403			\epsfxsize=60mm
404			\epsfbox{all5x5-HEAD-HEAD-cr.eps}
405			\end{figure}
406			\end{center}
407
408			\end{slide}
409
410			\begin{slide}{5x5: $\cos$ correlations}
411
412			\begin{center}
413			\begin{figure}
414			\epsfxsize=60mm
415			\epsfbox{all5x5-HEAD-X-cr.eps}
416			\end{figure}
417			\end{center}
418
419			\end{slide}
420
421
422			% Slide 16
423
424			\begin{slide}{Initial Runs: 50 Lipids randomly arranged in water}
425
426			\textbf{Simulation Parameters:}
427
428			\begin{itemize}
429
430			\item Starting Configuration:
431			\begin{itemize}
432			\item 50 lipid molecules arranged randomly in a rectangular box
433			\item The box was then filled with 1384 waters
434			\begin{itemize}
435			\item final water to lipid ratio was 27:1
436			\end{itemize}
437			\end{itemize}
438
439			\item Lipid had only a single saturated chain of 16 carbons
440
441			\item Box Size: 26.6 $\mbox{\AA}$ x 26.6 $\mbox{\AA}$ x 108.4 $\mbox{\AA}$
442
443			\item dt = 2.0 - 3.0 fs
444
445			\item T = 300 K
446
447			\item NVE ensemble
448
449			\item Periodic boundary conditions
450
451			\end{itemize}
452
453			\end{slide}
454
455
456			% Slide 17
457
458			\begin{slide}{R-50: Initial}
459
460			\begin{center}
461			\begin{figure}
462			\epsfxsize=100mm
463			\epsfbox{r50-initial.eps}
464			\end{figure}
465			\end{center}
466
467			The initial configuration
468
469			\end{slide}
470
471			\begin{slide}{R-50: Final}
472
473			\begin{center}
474			\begin{figure}
475			\epsfxsize=100mm
476			\epsfbox{r50-521ps.eps}
477			\end{figure}
478			\end{center}
479
480			The fianl configuration at 521 ps
481
482			\end{slide}
483
484
485			% Slide 18
486
487			\begin{slide}{R-50: $g(r)$}
488
489
490			\begin{center}
491			\begin{figure}
492			\epsfxsize=60mm
493			\epsfbox{r50-HEAD-HEAD-gr.eps}
494			\end{figure}
495			\end{center}
496
497			\end{slide}
498
499
500			\begin{slide}{R-50: $g(r)$}
501
502
503			\begin{center}
504			\begin{figure}
505			\epsfxsize=60mm
506			\epsfbox{r50-HEAD-X-gr.eps}
507			\end{figure}
508			\end{center}
509
510			\end{slide}
511
512
513			% Slide 19
514
515			\begin{slide}{R-50: $\cos$ correlations}
516
517
518			\begin{center}
519			\begin{figure}
520			\epsfxsize=60mm
521			\epsfbox{r50-HEAD-HEAD-cr.eps}
522			\end{figure}
523			\end{center}
524
525			\end{slide}
526
527			\begin{slide}{R-50: $\cos$ correlations}
528
529
530			\begin{center}
531			\begin{figure}
532			\epsfxsize=60mm
533			\epsfbox{r50-HEAD-X-cr.eps}
534			\end{figure}
535			\end{center}
536
537			\end{slide}
538
539
540			% Slide 20
541
542			\begin{slide}{Future Directions}
543
544			\begin{itemize}
545
546			\item
547			Simulation of a lipid with 2 chains, or perhaps expand the current
548			unified chain atoms to take up greater steric bulk.
549
550			\item
551			Incorporate constant pressure and constant temperature into the ensemble.
552
553			\item
554			Parrellize the code.
555
556			\end{itemize}
557			\end{slide}
558
559
560			% Slide 21
561
562			\begin{slide}{Acknowledgements}
563
564			\begin{itemize}
565
566			\item Dr. J. Daniel Gezelter
567			\item Christopher Fennel
568			\item Charles Vardeman
569			\item Teng Lin
570
571			\end{itemize}
572
573			Funding by:
574			\begin{itemize}
575			\item Dreyfus New Faculty Award
576			\end{itemize}
577
578			\end{slide}
579
580
581
582
583
584
585
586
587			%%%%%%%%%%%%%%%%%%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
588
589			\end{document}