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\section{Initial Simulations} |
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\label{sec:initialSimulations} |
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The first simulations we performed, were two lipid/water systems. The |
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first was a simulation of 25 lipids and 1386 waters in a cubic |
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box. The second simulation was 50 lipids and 1384 waters in a |
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rectangular box. Both systems were integrated in the microcanonical |
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ensemble for times in excess of 15 ns. |
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The 25 lipid system (scheme I) was initially started from an ordered |
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configuration. The lipids were aligned in a 5x5 square with a side by |
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side spacing of $\sim \mbox{8\AA}$ (Fig.~\ref{fig:25lipidInit}) The |
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box was then filled with water in an FCC lattice. The temperature was |
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then equilibrated to 300~K by resampling the atomic velocities from a |
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Maxwell-Boltzmann distribution. Once the temperature had been set, the |
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system was allowed to progress for $\sim 20$~ns. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{5x5-initial_pre.eps} |
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\caption[The initial configuration of scheme I]{The starting configuration of scheme I after thermalization.} |
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\label{fig:25lipidInit} |
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\end{figure} |
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The 50 lipid system (scheme II) was initially started from a randomly |
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generated configuration. A system box was initially filled with water |
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in an FCC lattice. The 50 lipid molecules were then sequentially |
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placed in the box with a random orientation and position. If the lipid |
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did not overlap with a previously placed one, its position was |
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accepted, and the next lipid was placed. Once all 50 lipids were |
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positioned, any waters overlapping with the lipids were removed |
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(Fig.~\ref{fig:r50Init}. The system was then equilibrated to 300~K in |
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the same manner as scheme I. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{r50-initial_pre.eps} |
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\caption[The initial configuration of scheme II]{The starting configuration of scheme II after thermalization} |
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\label{fig:r50Init} |
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\end{figure} |
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\subsection{Results} |
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\label{subSec:initSimsResults} |
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Scheme I was found to split into two leaves within the first |
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5~ns. However, as the simulation progressed, it became apparent that |
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the system was frustrated in its confined box. After 15~ns, the two |
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leaves were still unable to form a bilayer. Instead, thew were only |
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able to form two skewed layers (Fig.~\ref{fig:25lipidFinal}). |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{5x5-montage_pre.eps} |
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\caption[The time evolution of scheme I]{The time evolution of scheme I.} |
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\label{fig:25lipidFinal} |
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\end{figure} |
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Scheme II behaved similarly. Within the first 2~ns the system |
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aggregated into lipid and water regions. After $\sim 5$~ns the lipid |
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regions had become micelles. Over the course of the next 10~ns the |
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micelles underwent little change. At 15~ns simulation time, we switched |
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the ensemble to isobaric-isothermal, NPT. The new integrator had just |
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been written, and allowed form isometric scaling of the simulation |
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box. The new integrator allowed the system to relax more, and over the |
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next 10~ns several of the micelles started to merge. However, no |
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bilayer was formed. Fig.~\ref{fig:r50Final} shows the progression of |
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the simulation. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{r50-montage_pre.eps} |
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\caption[The time evolution of scheme II]{The time evolution of scheme II.} |
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\label{fig:25lipidFinal} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\subfigure[The self correlation of the phospholipid head groups. $g_{\text{Head,Head}}(r)$ is on the top, the bottom chart is the $\langle \cos \omega \rangle_{\text{Head,Head}}(r)$.]{% |
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\label{fig:5x5HHCorr}% |
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\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-HEAD-HEAD.epsi}% |
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} |
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\subfigure[The $g_{\text{CH}_2\text{,CH}_2}(r)$ for the tail chains]{% |
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\label{fig:5x5CCg}% |
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\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-CH2-CH2.epsi}} |
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\subfigure% |
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[The pair correlations between the head groups and the water]{% |
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\label{fig:5x5HXCorr}% |
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\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-HEAD-X.epsi}} |
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\caption[Scheme I pair correlations]{The pair correlation functions for scheme I.} |
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\label{fig:5x5PairCorrs} |
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\end{figure} |
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Structural information about the simulations were calculated via the |
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equations in Sec.~\ref{sec:analysis}. Fig.~\ref{fig:5x5HHCorr} shows |
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$g_{\text{Head,Head}}(r)$ and $\langle \cos \omega |
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\rangle_{\text{Head,Head}}(r)$ for scheme II. The |
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first peak in the $g(r)$ at 4.03~$\mbox{\AA}$ is the nearest neighbor |
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separation of the heads of two lipids. This corresponds to a maximum |
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in the $\langle \cos \omega \rangle(r)$ which means that the two |
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neighbors on the same leaf have their dipoles aligned. The broad peak |
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at 6.5~$\mbox{\AA}$ is the inter-surface spacing. Here, there is a |
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corresponding anti-alignment in the angular correlation. This means |
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that although the dipoles are aligned on the same monolayer, the |
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dipoles will orient themselves to be anti-aligned on a opposite facing |
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monolayer. With this information, the two peaks at 7.0~$\mbox{\AA}$ |
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and 7.4~$\mbox{\AA}$ are head groups on the same monolayer, and they |
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are the second nearest neighbors to the head group. The peak is likely |
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a split peak because of the small statistics of this system. Finally, |
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the peak at 8.0~$\mbox{\AA}$ is likely the second nearest neighbor on |
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the opposite monolayer because of the anti-alignment evident in the |
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angular correlation. |
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Fig.~\ref{fig:5x5CCg} shows the radial distribution function for the |
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$\text{CH}_2$ unified atoms. The spacing of the atoms along the tail |
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chains accounts for the regularly spaced sharp peaks, but the broad |
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underlying peak with its maximum at 4.6~$\mbox{\AA}$ is the |
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distribution of chain-chain distances between two lipids. The final |
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figure, Fig.~\ref{fig:5x5HXCorr}, includes the correlation functions |
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between the Head group and the SSD atoms. The peak in $g(r)$ at |
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3.6~$\mbox{\AA}$ is the distance of closest approach between the two, |
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and $\langle \cos \omega \rangle(r)$ shows that the SSD atoms will |
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align their dipoles with the head groups at close distance. However, |
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as one increases the distance, the SSD atoms are no longer aligned. |
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\begin{figure} |
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\centering |
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\subfigure[The self correlation of the phospholipid head groups.]{% |
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\label{fig:r50HHCorr}% |
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\includegraphics[angle=-90,width=0.49\linewidth]{r50-HEAD-HEAD.epsi}% |
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} |
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\subfigure% |
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[The pair correlations between the head groups and the water]{% |
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\label{fig:r50HXCorr}% |
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\includegraphics[angle=-90,width=0.49\linewidth]{r50-HEAD-X.epsi}% |
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} |
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\subfigure[The $g_{\text{CH}_2\text{,CH}_2}(r)$ for the tail chains]{% |
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\label{fig:r50CCg}% |
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\includegraphics[angle=-90,width=0.49\linewidth]{r50-CH2-CH2.epsi}} |
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\caption[Scheme II pair correlations]{The pair correlation functions for scheme II} |
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\label{fig:r50PairCorrs} |
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\end{figure} |
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Fig.~ \ref{fig:r50HHCorr}, \ref{fig:r50HXCorr}, and \ref{fig:r50CCg} |
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are the same correlation functions for scheme II as for scheme I. What |
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is most interesting to note, is the high degree of similarity between |
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the correlation functions between systems. Even though scheme I formed |
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a skewed bilayer and scheme II is still in the micelle stage, both |
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have an inter-surface spacing of 6.5~$\mbox{\AA}$ with the same |
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characteristic anti-alignment between surfaces. Not as surprising, is |
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the consistency of the closest packing statistics between |
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systems. Namely, a head-head closest approach distance of |
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4~$\mbox{\AA}$, and similar findings for the chain-chain and |
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head-water distributions as in scheme I. |