matt_papers/canidacy_talk/canidacy_slides.tex

% temporary preamble

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


\documentclass{seminar}
\usepackage{color}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{epsf}

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

\title{A Coarse Grain 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


% 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}

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 3

\begin{slide}{Research Goals}
\begin{itemize}

\item 
To develop a coarse-grain simulation model with which to simulate
phospholipid bilayers.

\item To use the model to observe:

        \begin{itemize}
        
        \item Phospholipid properties with long length scales

                \begin{itemize}
                \item The ripple phase.
                \end{itemize}

        \item Long time scale dynamics of biological relevance
                
                \begin{itemize}
                \item Trans-membrane diffusion of drug molecules
                \end{itemize}
        \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:	54
Committed:	Tue Jul 30 17:25:26 2002 UTC (22 years, 9 months ago) by mmeineke
Content type:	application/x-tex
File size:	10107 byte(s)
Log Message:	added all of the pictures. And gave a practice talk based on this version. Now begins the process of cleanup and additions of citations.
#	Content
1	% temporary preamble
2
3	%\documentclass[ps,frames,final,nototal,slideColor,colorBG]{prosper}
4
5
6	\documentclass{seminar}
7	\usepackage{color}
8
9	\usepackage{amsmath}
10	\usepackage{amssymb}
11	\usepackage{epsf}
12
13	% ----------------------
14	% \| Title \|
15	% ----------------------
16
17	\title{A Coarse Grain Model for Phospholipid MD Simulations}
18
19	\author{Matthew A. Meineke\\
20	Department of Chemistry and Biochemistry\\
21	University of Notre Dame\\
22	Notre Dame, Indiana 46556}
23
24	\date{\today}
25
26	%-------------------------------------------------------------------
27	% Begin Document
28
29	\begin{document}
30	\maketitle
31
32
33
34	% Slide 1
35	\begin{slide} {Talk Outline}
36	\begin{itemize}
37
38	\item Discussion of the research motivation and goals
39
40	\item Methodology
41
42	\item Discussion of current research and preliminary results
43
44	\item Future research
45
46	\end{itemize}
47	\end{slide}
48
49
50	% Slide 2
51
52	\begin{slide}{Motivation}
53
54	There is a strong need in phospholipid bilayer simulations for the
55	capability to simulate both long time and length scales. Consider the
56	following:
57
58	\begin{itemize}
59
60	\item Drug diffusion
61	\begin{itemize}
62	\item Some drug molecules may spend an appreciable time in the
63	membrane. Long time scale dynamics are needed to observe and
64	characterize their actions.
65	\end{itemize}
66
67	\item Ripple phase
68	\begin{itemize}
69	\item Between the bilayer gel and fluid phase there exists a ripple
70	phase. This phase has a period of about 100 - 200 $\mbox{\AA}$.
71	\end{itemize}
72
73	\item Bilayer formation dynamics
74	\begin{itemize}
75	\item Initial simulations show that bilayers can take upwards of
76	20 ns to form completely.
77	\end{itemize}
78
79	\end{itemize}
80	\end{slide}
81
82
83	% Slide 3
84
85	\begin{slide}{Research Goals}
86	\begin{itemize}
87
88	\item
89	To develop a coarse-grain simulation model with which to simulate
90	phospholipid bilayers.
91
92	\item To use the model to observe:
93
94	\begin{itemize}
95
96	\item Phospholipid properties with long length scales
97
98	\begin{itemize}
99	\item The ripple phase.
100	\end{itemize}
101
102	\item Long time scale dynamics of biological relevance
103
104	\begin{itemize}
105	\item Trans-membrane diffusion of drug molecules
106	\end{itemize}
107	\end{itemize}
108	\end{itemize}
109	\end{slide}
110
111
112	% Slide 4
113
114	\begin{slide}{Length Scale Simplification}
115	\begin{itemize}
116
117	\item
118	Replace any charged interactions of the system with dipoles.
119
120	\begin{itemize}
121	\item Allows for computational scaling approximately by $N$ for
122	dipole-dipole interactions.
123	\item In contrast, the Ewald sum scales approximately by $N \log N$.
124	\end{itemize}
125
126	\item
127	Use unified models for the water and the lipid chain.
128
129	\begin{itemize}
130	\item Drastically reduces the number of atoms to simulate.
131	\item Number of water interactions alone reduced by $\frac{1}{3}$.
132	\end{itemize}
133	\end{itemize}
134	\end{slide}
135
136
137	% Slide 5
138
139	\begin{slide}{Time Scale Simplification}
140	\begin{itemize}
141
142	\item
143	No explicit hydrogens
144
145	\begin{itemize}
146	\item Hydrogen bond vibration is normally one of the fastest time
147	events in a simulation.
148	\end{itemize}
149
150	\item
151	Constrain all bonds to be of fixed length.
152
153	\begin{itemize}
154	\item As with the hydrogens, bond vibrations are the fastest motion in
155	a simulation
156	\end{itemize}
157
158	\item
159	Allows time steps of up to 3 fs with the current integrator.
160
161	\end{itemize}
162	\end{slide}
163
164
165	% Slide 6
166	\begin{slide}{Molecular Dynamics}
167
168	All of our simulations will be carried out using molecular
169	dynamics. This involves solving Newton's equations of motion using
170	the classical \emph{Hamiltonian} as follows:
171
172	\begin{equation}
173	H(\vec{q},\vec{p}) = T(\vec{p}) + V(\vec{q})
174	\end{equation}
175
176	Here $T(\vec{p})$ is the kinetic energy of the system which is a
177	function of momentum. In Cartesian space, $T(\vec{p})$ can be
178	written as:
179
180	\begin{equation}
181	T(\vec{p}) = \sum_{i=1}^{N} \sum_{\alpha = x,y,z} \frac{p^{2}_{i\alpha}}{2m_{i}}
182	\end{equation}
183
184	\end{slide}
185
186
187	% Slide 7
188	\begin{slide}{The Potential}
189
190	The main part of the simulation is then the calculation of forces from
191	the potential energy.
192
193	\begin{equation}
194	\vec{F}(\vec{q}) = - \nabla V(\vec{q})
195	\end{equation}
196
197	The potential itself is made of several parts.
198
199	\begin{equation}
200	V_{tot} =
201	\overbrace{V_{l} + V_{\theta} + V_{\omega}}^{\mbox{bonded}} +
202	\overbrace{V_{l\!j} + V_{d\!p} + V_{s\!s\!d}}^{\mbox{non-bonded}}
203	\end{equation}
204
205	Where the bond interactions $V_{l}$, $V_{\theta}$, and $V_{\omega}$ are
206	the bond, bend, and torsion potentials, and the non-bonded
207	interactions $V_{l\!j}$, $V_{d\!p}$, and $V_{s\!p}$ are the
208	lenard-jones, dipole-dipole, and sticky potential interactions.
209
210	\end{slide}
211
212
213	% Slide 8
214
215	\begin{slide}{Soft Sticky Dipole Model}
216
217	The Soft-Sticky model for water is a reduced model.
218
219	\begin{itemize}
220
221	\item
222	The model is represented by a single point mass at the water's center
223	of mass.
224
225	\item
226	The point mass contains a fixed dipole of 2.35 D pointing from the
227	oxygens toward the hydrogens.
228
229	\end{itemize}
230
231	It's potential is as follows:
232
233	\begin{equation}
234	V_{s\!s\!d} = V_{l\!j}(r_{i\!j}) + V_{d\!p}(r_{i\!j},\Omega_{i},\Omega_{j})
235	+ V_{s\!p}(r_{i\!j},\Omega_{i},\Omega_{j})
236	\end{equation}
237	\end{slide}
238
239	% Slide 8b
240
241	\begin{slide}{SSD Diagram}
242
243	\begin{center}
244	\begin{figure}
245	\epsfxsize=50mm
246	\epsfbox{ssd.epsi}
247	\end{figure}
248	\end{center}
249
250	A Diagram of the SSD model.
251	\end{slide}
252
253	% Slide 9
254	\begin{slide}{Hydrogen Bonding in SSD}
255
256	It is important to note that SSD has a potential specifically to
257	recreate the hydrogen bonding network of water.
258
259
260	ICE SSD
261
262	ICE point Dipole
263
264
265	The importance of the hydrogen bond network is it's significant
266	contribution to the hydrophobic driving force of bilayer formation.
267	\end{slide}
268
269
270	% Slide 10
271
272	\begin{slide}{The Lipid Model}
273
274	To eliminate the need for charge-charge interactions, our lipid model
275	replaces the phospholipid head group with a single large head group
276	atom containing a freely oriented dipole. The tail is a simple alkane chain.
277
278	Lipid Properties:
279	\begin{itemize}
280	\item $\|\vec{\mu}_{\text{HEAD}}\| = 20.6\ \text{D}$
281	\item $m_{\text{HEAD}} = 196\ \text{amu}$
282	\item Tail atoms are unified CH, $\text{CH}_2$, and $\text{CH}_3$ atoms
283	\begin{itemize}
284	\item Alkane forcefield parameters taken from TraPPE
285	\end{itemize}
286	\end{itemize}
287
288	\end{slide}
289
290
291	% Slide 11
292
293	\begin{slide}{Lipid Model}
294
295
296
297	\end{slide}
298
299
300	% Slide 12
301
302	\begin{slide}{Initial Runs: 25 Lipids in water}
303
304	\textbf{Simulation Parameters:}
305
306	\begin{itemize}
307
308	\item Starting Configuration:
309	\begin{itemize}
310	\item 25 lipid molecules arranged in a 5 x 5 square
311	\item square was surrounded by a sea of 1386 waters
312	\begin{itemize}
313	\item final water to lipid ratio was 55.4:1
314	\end{itemize}
315	\end{itemize}
316
317	\item Lipid had only a single saturated chain of 16 carbons
318
319	\item Box Size: 34.5 $\mbox{\AA}$ x 39.4 $\mbox{\AA}$ x 39.4 $\mbox{\AA}$
320
321	\item dt = 2.0 - 3.0 fs
322
323	\item T = 300 K
324
325	\item NVE ensemble
326
327	\item Periodic boundary conditions
328	\end{itemize}
329
330	\end{slide}
331
332
333	% Slide 13
334
335	\begin{slide}{5x5: Initial}
336
337	\begin{center}
338	\begin{figure}
339	\epsfxsize=50mm
340	\epsfbox{5x5-initial.eps}
341	\end{figure}
342	\end{center}
343
344	The initial configuration
345
346	\end{slide}
347
348	\begin{slide}{5x5: Final}
349
350	\begin{center}
351	\begin{figure}
352	\epsfxsize=60mm
353	\epsfbox{5x5-1.7ns.eps}
354	\end{figure}
355	\end{center}
356
357	The final configuration at 1.7 ns.
358
359	\end{slide}
360
361
362	% Slide 14
363
364	\begin{slide}{5x5: $g(r)$}
365
366	\begin{center}
367	\begin{figure}
368	\epsfxsize=60mm
369	\epsfbox{all5x5-HEAD-HEAD-gr.eps}
370	\end{figure}
371	\end{center}
372
373
374	\end{slide}
375
376	\begin{slide}{5x5: $g(r)$}
377
378	\begin{center}
379	\begin{figure}
380	\epsfxsize=60mm
381	\epsfbox{all5x5-HEAD-X-gr.eps}
382	\end{figure}
383	\end{center}
384
385
386	\end{slide}
387
388
389	% Slide 15
390
391	\begin{slide}{5x5: $\cos$ correlations}
392
393	\begin{center}
394	\begin{figure}
395	\epsfxsize=60mm
396	\epsfbox{all5x5-HEAD-HEAD-cr.eps}
397	\end{figure}
398	\end{center}
399
400	\end{slide}
401
402	\begin{slide}{5x5: $\cos$ correlations}
403
404	\begin{center}
405	\begin{figure}
406	\epsfxsize=60mm
407	\epsfbox{all5x5-HEAD-X-cr.eps}
408	\end{figure}
409	\end{center}
410
411	\end{slide}
412
413
414	% Slide 16
415
416	\begin{slide}{Initial Runs: 50 Lipids randomly arranged in water}
417
418	\textbf{Simulation Parameters:}
419
420	\begin{itemize}
421
422	\item Starting Configuration:
423	\begin{itemize}
424	\item 50 lipid molecules arranged randomly in a rectangular box
425	\item The box was then filled with 1384 waters
426	\begin{itemize}
427	\item final water to lipid ratio was 27:1
428	\end{itemize}
429	\end{itemize}
430
431	\item Lipid had only a single saturated chain of 16 carbons
432
433	\item Box Size: 26.6 $\mbox{\AA}$ x 26.6 $\mbox{\AA}$ x 108.4 $\mbox{\AA}$
434
435	\item dt = 2.0 - 3.0 fs
436
437	\item T = 300 K
438
439	\item NVE ensemble
440
441	\item Periodic boundary conditions
442
443	\end{itemize}
444
445	\end{slide}
446
447
448	% Slide 17
449
450	\begin{slide}{R-50: Initial}
451
452	\begin{center}
453	\begin{figure}
454	\epsfxsize=100mm
455	\epsfbox{r50-initial.eps}
456	\end{figure}
457	\end{center}
458
459	The initial configuration
460
461	\end{slide}
462
463	\begin{slide}{R-50: Final}
464
465	\begin{center}
466	\begin{figure}
467	\epsfxsize=100mm
468	\epsfbox{r50-521ps.eps}
469	\end{figure}
470	\end{center}
471
472	The fianl configuration at 521 ps
473
474	\end{slide}
475
476
477	% Slide 18
478
479	\begin{slide}{R-50: $g(r)$}
480
481
482	\begin{center}
483	\begin{figure}
484	\epsfxsize=60mm
485	\epsfbox{r50-HEAD-HEAD-gr.eps}
486	\end{figure}
487	\end{center}
488
489	\end{slide}
490
491
492	\begin{slide}{R-50: $g(r)$}
493
494
495	\begin{center}
496	\begin{figure}
497	\epsfxsize=60mm
498	\epsfbox{r50-HEAD-X-gr.eps}
499	\end{figure}
500	\end{center}
501
502	\end{slide}
503
504
505	% Slide 19
506
507	\begin{slide}{R-50: $\cos$ correlations}
508
509
510	\begin{center}
511	\begin{figure}
512	\epsfxsize=60mm
513	\epsfbox{r50-HEAD-HEAD-cr.eps}
514	\end{figure}
515	\end{center}
516
517	\end{slide}
518
519	\begin{slide}{R-50: $\cos$ correlations}
520
521
522	\begin{center}
523	\begin{figure}
524	\epsfxsize=60mm
525	\epsfbox{r50-HEAD-X-cr.eps}
526	\end{figure}
527	\end{center}
528
529	\end{slide}
530
531
532	% Slide 20
533
534	\begin{slide}{Future Directions}
535
536	\begin{itemize}
537
538	\item
539	Simulation of a lipid with 2 chains, or perhaps expand the current
540	unified chain atoms to take up greater steric bulk.
541
542	\item
543	Incorporate constant pressure and constant temperature into the ensemble.
544
545	\item
546	Parrellize the code.
547
548	\end{itemize}
549	\end{slide}
550
551
552	% Slide 21
553
554	\begin{slide}{Acknowledgements}
555
556	\begin{itemize}
557
558	\item Dr. J. Daniel Gezelter
559	\item Christopher Fennel
560	\item Charles Vardeman
561	\item Teng Lin
562
563	\end{itemize}
564
565	Funding by:
566	\begin{itemize}
567	\item Dreyfus New Faculty Award
568	\end{itemize}
569
570	\end{slide}
571
572
573
574
575
576
577
578
579	%%%%%%%%%%%%%%%%%%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
580
581	\end{document}