mattDisertation/collections/initalSimulations.tex


\section{Initial Simulations}
\label{sec:initialSimulations}

The first simulations we performed, were two lipid/water systems. The
first was a simulation of 25 lipids and 1386 waters in a cubic
box. The second simulation was 50 lipids and 1384 waters in a
rectangular box. Both systems were integrated in the microcanonical
ensemble for times in excess of 15 ns.

The 25 lipid system (scheme I) was initially started from an ordered
configuration. The lipids were aligned in a 5x5 square with a side by
side spacing of $\sim \mbox{8\AA}$ (Fig.~\ref{fig:25lipidInit}) The
box was then filled with water in an FCC lattice. The temperature was
then equilibrated to 300~K by resampling the atomic velocities from a
Maxwell-Boltzmann distribution. Once the temperature had been set, the
system was allowed to progress for $\sim 20$~ns.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{5x5-initial_pre.eps}
\caption[The initial configuration of scheme I]{The starting configuration of scheme I after thermalization.}
\label{fig:25lipidInit}
\end{figure}

The 50 lipid system (scheme II) was initially started from a randomly
generated configuration. A system box was initially filled with water
in an FCC lattice. The 50 lipid molecules were then sequentially
placed in the box with a random orientation and position. If the lipid
did not overlap with a previously placed one, its position was
accepted, and the next lipid was placed. Once all 50 lipids were
positioned, any waters overlapping with the lipids were removed
(Fig.~\ref{fig:r50Init}. The system was then equilibrated to 300~K in
the same manner as scheme I.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{r50-initial_pre.eps}
\caption[The initial configuration of scheme II]{The starting configuration of scheme II after thermalization}
\label{fig:r50Init}
\end{figure}

\subsection{Results}
\label{subSec:initSimsResults}

Scheme I was found to split into two leaves within the first
5~ns. However, as the simulation progressed, it became apparent that
the system was frustrated in its confined box. After 15~ns, the two
leaves were still unable to form a bilayer. Instead, thew were only
able to form two skewed layers (Fig.~\ref{fig:25lipidFinal}).

\begin{figure}
\centering
\includegraphics[width=\linewidth]{5x5-montage_pre.eps}
\caption[The time evolution of scheme I]{The time evolution of scheme I.}
\label{fig:25lipidFinal}
\end{figure}

Scheme II behaved similarly. Within the first 2~ns the system
aggregated into lipid and water regions. After $\sim 5$~ns the lipid
regions had become micelles. Over the course of the next 10~ns the
micelles underwent little change. At 15~ns simulation time, we switched
the ensemble to isobaric-isothermal, NPT. The new integrator had just
been written, and allowed form isometric scaling of the simulation
box. The new integrator allowed the system to relax more, and over the
next 10~ns several of the micelles started to merge. However, no
bilayer was formed. Fig.~\ref{fig:r50Final} shows the progression of
the simulation.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{r50-montage_pre.eps}
\caption[The time evolution of scheme II]{The time evolution of scheme II.}
\label{fig:25lipidFinal}
\end{figure}

\begin{figure}
\centering
        \subfigure[The self correlation of the phospholipid head groups. $g_{\text{Head,Head}}(r)$ is on the top, the bottom chart is the $\langle \cos \omega \rangle_{\text{Head,Head}}(r)$.]{%
                \label{fig:5x5HHCorr}%
                \includegraphics[angle=-90,width=0.49\linewidth]{all5x5-HEAD-HEAD.epsi}%
                }
        \subfigure[The $g_{\text{CH}_2\text{,CH}_2}(r)$ for the tail chains]{%
                \label{fig:5x5CCg}%
                \includegraphics[angle=-90,width=0.49\linewidth]{all5x5-CH2-CH2.epsi}}
        \subfigure%
        [The pair correlations between the head groups and the water]{%
                \label{fig:5x5HXCorr}%
                \includegraphics[angle=-90,width=0.49\linewidth]{all5x5-HEAD-X.epsi}}
\caption[Scheme I pair correlations]{The pair correlation functions for scheme I.}
\label{fig:5x5PairCorrs}
\end{figure}

Structural information about the simulations were calculated via the
equations in Sec.~\ref{sec:analysis}. Fig.~\ref{fig:5x5HHCorr} shows
$g_{\text{Head,Head}}(r)$ and $\langle \cos \omega
\rangle_{\text{Head,Head}}(r)$ for scheme II. The
first peak in the $g(r)$ at 4.03~$\mbox{\AA}$ is the nearest neighbor
separation of the heads of two lipids. This corresponds to a maximum
in the $\langle \cos \omega \rangle(r)$ which means that the two
neighbors on the same leaf have their dipoles aligned. The broad peak
at 6.5~$\mbox{\AA}$ is the inter-surface spacing. Here, there is a
corresponding anti-alignment in the angular correlation. This means
that although the dipoles are aligned on the same monolayer, the
dipoles will orient themselves to be anti-aligned on a opposite facing
monolayer. With this information, the two peaks at 7.0~$\mbox{\AA}$
and 7.4~$\mbox{\AA}$ are head groups on the same monolayer, and they
are the second nearest neighbors to the head group. The peak is likely
a split peak because of the small statistics of this system. Finally,
the peak at 8.0~$\mbox{\AA}$ is likely the second nearest neighbor on
the opposite monolayer because of the anti-alignment evident in the
angular correlation.

Fig.~\ref{fig:5x5CCg} shows the radial distribution function for the
$\text{CH}_2$ unified atoms. The spacing of the atoms along the tail
chains accounts for the regularly spaced sharp peaks, but the broad
underlying peak with its maximum at 4.6~$\mbox{\AA}$ is the
distribution of chain-chain distances between two lipids. The final
figure, Fig.~\ref{fig:5x5HXCorr}, includes the correlation functions
between the Head group and the SSD atoms. The peak in $g(r)$ at
3.6~$\mbox{\AA}$ is the distance of closest approach between the two,
and $\langle \cos \omega \rangle(r)$ shows that the SSD atoms will
align their dipoles with the head groups at close distance. However,
as one increases the distance, the SSD atoms are no longer aligned.

\begin{figure}
\centering
        \subfigure[The self correlation of the phospholipid head groups.]{%
                \label{fig:r50HHCorr}%
                \includegraphics[angle=-90,width=0.49\linewidth]{r50-HEAD-HEAD.epsi}%
                }
        \subfigure%
        [The pair correlations between the head groups and the water]{%
                \label{fig:r50HXCorr}%
                \includegraphics[angle=-90,width=0.49\linewidth]{r50-HEAD-X.epsi}%
                }
        \subfigure[The $g_{\text{CH}_2\text{,CH}_2}(r)$ for the tail chains]{%
                \label{fig:r50CCg}%
                \includegraphics[angle=-90,width=0.49\linewidth]{r50-CH2-CH2.epsi}}

\caption[Scheme II pair correlations]{The pair correlation functions for scheme II}
\label{fig:r50PairCorrs}
\end{figure}

Fig.~ \ref{fig:r50HHCorr}, \ref{fig:r50HXCorr}, and \ref{fig:r50CCg}
are the same correlation functions for scheme II as for scheme I. What
is most interesting to note, is the high degree of similarity between
the correlation functions between systems. Even though scheme I formed
a skewed bilayer and scheme II is still in the micelle stage, both
have an inter-surface spacing of 6.5~$\mbox{\AA}$ with the same
characteristic anti-alignment between surfaces. Not as surprising, is
the consistency of the closest packing statistics between
systems. Namely, a head-head closest approach distance of
4~$\mbox{\AA}$, and similar findings for the chain-chain and
head-water distributions as in scheme I.
Revision:	753
Committed:	Tue Sep 9 16:35:31 2003 UTC (20 years, 10 months ago) by mmeineke
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Log Message:	a rough draft of the intial simulations
#	Content
1
2	\section{Initial Simulations}
3	\label{sec:initialSimulations}
4
5	The first simulations we performed, were two lipid/water systems. The
6	first was a simulation of 25 lipids and 1386 waters in a cubic
7	box. The second simulation was 50 lipids and 1384 waters in a
8	rectangular box. Both systems were integrated in the microcanonical
9	ensemble for times in excess of 15 ns.
10
11	The 25 lipid system (scheme I) was initially started from an ordered
12	configuration. The lipids were aligned in a 5x5 square with a side by
13	side spacing of $\sim \mbox{8\AA}$ (Fig.~\ref{fig:25lipidInit}) The
14	box was then filled with water in an FCC lattice. The temperature was
15	then equilibrated to 300~K by resampling the atomic velocities from a
16	Maxwell-Boltzmann distribution. Once the temperature had been set, the
17	system was allowed to progress for $\sim 20$~ns.
18
19	\begin{figure}
20	\centering
21	\includegraphics[width=\linewidth]{5x5-initial_pre.eps}
22	\caption[The initial configuration of scheme I]{The starting configuration of scheme I after thermalization.}
23	\label{fig:25lipidInit}
24	\end{figure}
25
26	The 50 lipid system (scheme II) was initially started from a randomly
27	generated configuration. A system box was initially filled with water
28	in an FCC lattice. The 50 lipid molecules were then sequentially
29	placed in the box with a random orientation and position. If the lipid
30	did not overlap with a previously placed one, its position was
31	accepted, and the next lipid was placed. Once all 50 lipids were
32	positioned, any waters overlapping with the lipids were removed
33	(Fig.~\ref{fig:r50Init}. The system was then equilibrated to 300~K in
34	the same manner as scheme I.
35
36	\begin{figure}
37	\centering
38	\includegraphics[width=\linewidth]{r50-initial_pre.eps}
39	\caption[The initial configuration of scheme II]{The starting configuration of scheme II after thermalization}
40	\label{fig:r50Init}
41	\end{figure}
42
43	\subsection{Results}
44	\label{subSec:initSimsResults}
45
46	Scheme I was found to split into two leaves within the first
47	5~ns. However, as the simulation progressed, it became apparent that
48	the system was frustrated in its confined box. After 15~ns, the two
49	leaves were still unable to form a bilayer. Instead, thew were only
50	able to form two skewed layers (Fig.~\ref{fig:25lipidFinal}).
51
52	\begin{figure}
53	\centering
54	\includegraphics[width=\linewidth]{5x5-montage_pre.eps}
55	\caption[The time evolution of scheme I]{The time evolution of scheme I.}
56	\label{fig:25lipidFinal}
57	\end{figure}
58
59	Scheme II behaved similarly. Within the first 2~ns the system
60	aggregated into lipid and water regions. After $\sim 5$~ns the lipid
61	regions had become micelles. Over the course of the next 10~ns the
62	micelles underwent little change. At 15~ns simulation time, we switched
63	the ensemble to isobaric-isothermal, NPT. The new integrator had just
64	been written, and allowed form isometric scaling of the simulation
65	box. The new integrator allowed the system to relax more, and over the
66	next 10~ns several of the micelles started to merge. However, no
67	bilayer was formed. Fig.~\ref{fig:r50Final} shows the progression of
68	the simulation.
69
70	\begin{figure}
71	\centering
72	\includegraphics[width=\linewidth]{r50-montage_pre.eps}
73	\caption[The time evolution of scheme II]{The time evolution of scheme II.}
74	\label{fig:25lipidFinal}
75	\end{figure}
76
77	\begin{figure}
78	\centering
79	\subfigure[The self correlation of the phospholipid head groups. $g_{\text{Head,Head}}(r)$ is on the top, the bottom chart is the $\langle \cos \omega \rangle_{\text{Head,Head}}(r)$.]{%
80	\label{fig:5x5HHCorr}%
81	\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-HEAD-HEAD.epsi}%
82	}
83	\subfigure[The $g_{\text{CH}_2\text{,CH}_2}(r)$ for the tail chains]{%
84	\label{fig:5x5CCg}%
85	\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-CH2-CH2.epsi}}
86	\subfigure%
87	[The pair correlations between the head groups and the water]{%
88	\label{fig:5x5HXCorr}%
89	\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-HEAD-X.epsi}}
90	\caption[Scheme I pair correlations]{The pair correlation functions for scheme I.}
91	\label{fig:5x5PairCorrs}
92	\end{figure}
93
94	Structural information about the simulations were calculated via the
95	equations in Sec.~\ref{sec:analysis}. Fig.~\ref{fig:5x5HHCorr} shows
96	$g_{\text{Head,Head}}(r)$ and $\langle \cos \omega
97	\rangle_{\text{Head,Head}}(r)$ for scheme II. The
98	first peak in the $g(r)$ at 4.03~$\mbox{\AA}$ is the nearest neighbor
99	separation of the heads of two lipids. This corresponds to a maximum
100	in the $\langle \cos \omega \rangle(r)$ which means that the two
101	neighbors on the same leaf have their dipoles aligned. The broad peak
102	at 6.5~$\mbox{\AA}$ is the inter-surface spacing. Here, there is a
103	corresponding anti-alignment in the angular correlation. This means
104	that although the dipoles are aligned on the same monolayer, the
105	dipoles will orient themselves to be anti-aligned on a opposite facing
106	monolayer. With this information, the two peaks at 7.0~$\mbox{\AA}$
107	and 7.4~$\mbox{\AA}$ are head groups on the same monolayer, and they
108	are the second nearest neighbors to the head group. The peak is likely
109	a split peak because of the small statistics of this system. Finally,
110	the peak at 8.0~$\mbox{\AA}$ is likely the second nearest neighbor on
111	the opposite monolayer because of the anti-alignment evident in the
112	angular correlation.
113
114	Fig.~\ref{fig:5x5CCg} shows the radial distribution function for the
115	$\text{CH}_2$ unified atoms. The spacing of the atoms along the tail
116	chains accounts for the regularly spaced sharp peaks, but the broad
117	underlying peak with its maximum at 4.6~$\mbox{\AA}$ is the
118	distribution of chain-chain distances between two lipids. The final
119	figure, Fig.~\ref{fig:5x5HXCorr}, includes the correlation functions
120	between the Head group and the SSD atoms. The peak in $g(r)$ at
121	3.6~$\mbox{\AA}$ is the distance of closest approach between the two,
122	and $\langle \cos \omega \rangle(r)$ shows that the SSD atoms will
123	align their dipoles with the head groups at close distance. However,
124	as one increases the distance, the SSD atoms are no longer aligned.
125
126	\begin{figure}
127	\centering
128	\subfigure[The self correlation of the phospholipid head groups.]{%
129	\label{fig:r50HHCorr}%
130	\includegraphics[angle=-90,width=0.49\linewidth]{r50-HEAD-HEAD.epsi}%
131	}
132	\subfigure%
133	[The pair correlations between the head groups and the water]{%
134	\label{fig:r50HXCorr}%
135	\includegraphics[angle=-90,width=0.49\linewidth]{r50-HEAD-X.epsi}%
136	}
137	\subfigure[The $g_{\text{CH}_2\text{,CH}_2}(r)$ for the tail chains]{%
138	\label{fig:r50CCg}%
139	\includegraphics[angle=-90,width=0.49\linewidth]{r50-CH2-CH2.epsi}}
140
141	\caption[Scheme II pair correlations]{The pair correlation functions for scheme II}
142	\label{fig:r50PairCorrs}
143	\end{figure}
144
145	Fig.~ \ref{fig:r50HHCorr}, \ref{fig:r50HXCorr}, and \ref{fig:r50CCg}
146	are the same correlation functions for scheme II as for scheme I. What
147	is most interesting to note, is the high degree of similarity between
148	the correlation functions between systems. Even though scheme I formed
149	a skewed bilayer and scheme II is still in the micelle stage, both
150	have an inter-surface spacing of 6.5~$\mbox{\AA}$ with the same
151	characteristic anti-alignment between surfaces. Not as surprising, is
152	the consistency of the closest packing statistics between
153	systems. Namely, a head-head closest approach distance of
154	4~$\mbox{\AA}$, and similar findings for the chain-chain and
155	head-water distributions as in scheme I.