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\documentclass[11pt]{article} |
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\usepackage{graphicx} |
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\usepackage{color} |
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\usepackage{floatflt} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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\section{Background and Research Goals} |
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Simulations of phospholipid bilayers are, by necessity, quite |
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complex. The lipid molecules are large molecules containing many |
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atoms,and the head group of the lipid will typically contain charge |
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separated ions which set up a large dipole within the molecule. Adding |
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to the complexity are the number of water molecules needed to properly |
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solvate the lipid bilayer. Because of these factors, many current |
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simulations are limited in both length and time scale due to to the |
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sheer number of calculations performed at every time step and the |
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lifetime of the researcher. A typical |
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simulation\cite{saiz02,lindahl00,venable00,Marrink01} will have around |
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64 phospholipids forming a bilayer approximately 40~$\mbox{\AA}$ by |
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50~$\mbox{\AA}$ with roughly 25 waters for every lipid. This means |
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there are on the order of 8,000 atoms needed to simulate these systems |
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and the trajectories in turn are integrated for times up to 10 ns. |
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|
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These limitations make it difficult to study certain biologically |
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interesting phenomena that don't fit within the short time and length |
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scale requirements. One such phenomena is the existence of the ripple |
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phase ($P_{\beta'}$) of the bilayer between the gel phase |
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($L_{\beta'}$) and the fluid phase ($L_{\alpha}$). The $P_{\beta'}$ |
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phase has been shown to have a ripple period of |
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100-200~$\mbox{\AA}$.\cite{katsaras00,sengupta00} A simulation of this |
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length scale would require approximately 1,300 lipid molecules and |
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roughly 25 waters for every lipid to fully solvate the bilayer. With |
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the large number of atoms involved in a simulation of this magnitude, |
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steps \emph{must} be taken to simplify the system to the point where |
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these numbers are reasonable. |
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Another system of interest would be drug molecule diffusion through |
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the membrane. Due to the fluid like properties of a lipid membrane, |
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not all diffusion takes place at membrane channels. It is of interest |
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to study certain molecules that may incorporate themselves directly |
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into the membrane. These molecules may then have an appreciable |
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waiting time (on the order of nanoseconds) within the |
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bilayer. Simulation of such a long time scale again requires |
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simplification of the system in order to lower the number of |
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calculations needed at each time step. |
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\section{Methodology} |
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\subsection{Length and Time Scale Simplifications} |
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The length scale simplifications are aimed at increaseing the number |
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of molecules simulated without drastically increasing the |
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The length scale simplifications are aimed at increasing the number of |
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molecules that can be simulated without drastically increasing the |
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computational cost of the system. This is done by a combination of |
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substituting less expensive interactions for expensive ones and |
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decreasing the number of interaction sites per molecule. Namely, |
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charge distributions are replaced with dipoles, and unified atoms are |
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used in place of water and phospholipid head groups. |
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point charge distributions are replaced with dipoles, and unified atoms are |
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used in place of water, phospholipid head groups, and alkyl groups. |
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The replacement of charge distributions with dipoles allows us to |
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replace an interaction that has a relatively long range, $\frac{1}{r}$ |
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for the charge charge potential, with that of a relitively short |
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range, $\frac{1}{r^{3}}$ for dipole - dipole potentials |
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(Equations~\ref{eq:dipolePot} and \ref{eq:chargePot}). This allows us |
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to use computaional simplifications algorithms such as Verlet neighbor |
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lists,\cite{allen87:csl} which gives computaional scaling by $N$. This |
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is in comparison to the Ewald sum\cite{leach01:mm} needed to compute |
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the charge - charge interactions which scales at best by $N |
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\ln N$. |
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replace an interaction that has a relatively long range, |
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$(\frac{1}{r})$, for the coulomb potential, with that of a relatively |
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short range, $(\frac{1}{r^{3}})$, for dipole - dipole |
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potentials. Combined with a computational simplification algorithm |
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such as a Verlet neighbor list,\cite{allen87:csl} this should give |
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computational scaling by $N$. This is in comparison to the Ewald |
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sum\cite{leach01:mm} needed to compute the coulomb interactions, which |
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scales at best by $N \ln N$. |
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\begin{equation} |
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V^{\text{dp}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
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\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
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- |
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\frac{3(\boldsymbol{\mu} \cdot \mathbf{r}_{ij}) % |
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(\boldsymbol{\mu} \cdot \mathbf{r}_{ij}) }{r^{5}_{ij}} \biggr] |
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\label{eq:dipolePot} |
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\end{equation} |
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|
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\begin{equation} |
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V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) = \frac{q_{i}q_{j}}% |
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{4\pi\epsilon_{0} r_{ij}} |
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\label{eq:chargePot} |
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\end{equation} |
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|
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The second step taken to simplify the number of calculationsis to |
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The second step taken to simplify the number of calculations is to |
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incorporate unified models for groups of atoms. In the case of water, |
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we use the soft sticky dipole (SSD) model developed by |
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Ichiye\cite{Liu96} (Section~\ref{sec:ssdModel}). For the phospholipids, a |
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unified head atom with a dipole will replace the atoms in the head |
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group, while unified $\text{CH}_2$ and $\text{CH}_3$ atoms will |
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replace the alkanes in the tails (Section~\ref{sec:lipidModel}). |
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replace the alkyl units in the tails (Section~\ref{sec:lipidModel}). |
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The time scale simplifications are taken so that the simulation can |
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take long time steps. By incresing the time steps taken by the |
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simulation, we are able to integrate the simulation trajectory with |
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fewer calculations. However, care must be taken to conserve the energy |
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of the simulation. This is a constraint placed upon the system by |
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simulating in the microcanonical ensemble. In practice, this means |
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taking steps small enough to resolve all motion in the system without |
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accidently moving an object too far along a repulsive energy surface |
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before it feels the affect of the surface. |
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The time scale simplifications are introduced so that we can take |
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longer time steps. By increasing the size of the time steps taken by |
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the simulation, we are able to integrate the simulation trajectory |
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with fewer calculations. However, care must be taken that any |
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simplifications used, still conserve the total energy of the |
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simulation. In practice, this means taking steps small enough to |
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resolve all motion in the system without accidently moving an object |
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too far along a repulsive energy surface before it feels the effect of |
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the surface. |
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In our simulation we have chosen to constrain all bonds to be of fixed |
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length. This means the bonds are no longer allowed to vibrate about |
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their equilibrium positions, typically the fastest periodical motion |
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in a dynamics simulation. By taking this action, we are able to take |
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time steps of 3 fs while still maintaining constant energy. This is in |
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contrast to the 1 fs time step typically needed to conserve energy when |
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bonds lengths are allowed to oscillate. |
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their equilibrium positions. Bond vibrations are typically the fastest |
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periodic motion in a dynamics simulation. By taking this action, we |
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are able to take time steps of 3 fs while still maintaining constant |
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energy. This is in contrast to the 1 fs time step typically needed to |
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conserve energy when bonds lengths are allowed to oscillate. |
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|
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\subsection{The Soft Sticky Water Model} |
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\label{sec:ssdModel} |
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|
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\begin{floatingfigure}{55mm} |
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\includegraphics[width=45mm]{ssd.epsi} |
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\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference. \vspace{5mm}} |
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% The dipole magnitude is 2.35 D and the Lennard-Jones parameters are $\sigma = 3.051 \mbox{\AA}$ and $\epsilon = 0.152$ kcal/mol.} |
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\label{fig:ssdModel} |
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\end{floatingfigure} |
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%\begin{floatingfigure}{55mm} |
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%\includegraphics[width=45mm]{ssd.epsi} |
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%\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference. \vspace{5mm}} |
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%\label{fig:ssdModel} |
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%\end{floatingfigure} |
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The water model used in our simulations is a modified soft Stockmayer |
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sphere model. Like the soft Stockmayer sphere, the SSD |
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The water model used in our simulations is a modified soft |
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Stockmayer-sphere model.\cite{stevens95} Like the Stockmayer-sphere, the SSD |
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model\cite{Liu96} consists of a Lennard-Jones interaction site and a |
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dipole both located at the water's center of mass (Figure |
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\ref{fig:ssdModel}). However, the SSD model extends this by adding a |
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tetrahedral potential to correct for hydrogen bonding. |
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This SSD water's motion is then governed by the following potential: |
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The SSD water potential is then given by the following equation: |
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\begin{equation} |
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V_{\text{ssd}} = V_{\text{LJ}}(r_{i\!j}) + V_{\text{dp}}(\mathbf{r}_{i\!j}, |
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V_{\text{SSD}} = V_{\text{LJ}}(r_{i\!j}) + V_{\text{dp}}(\mathbf{r}_{i\!j}, |
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\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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+ V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) |
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\label{eq:ssdTotPot} |
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\end{equation} |
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$V_{\text{LJ}}$ is the Lennard-Jones potential with $\sigma_{\text{w}} |
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= 3.051 \mbox{ \AA}$ and $\epsilon_{\text{w}} = 0.152\text{ |
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kcal/mol}$. $V_{\text{dp}}$ is the dipole potential with |
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$|\mu_{\text{w}}| = 2.35\text{ D}$. |
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$V_{\text{LJ}}$ is the Lennard-Jones potential: |
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\begin{equation} |
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V_{\text{LJ}} = |
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4\epsilon_{ij} \biggl[ |
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\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
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- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
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\biggr] |
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\label{eq:lennardJonesPot} |
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\end{equation} |
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where $r_{ij}$ is the distance between two $ij$ pairs, $\sigma_{ij}$ |
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scales the length of the iteraction, and $\epsilon_{ij}$ scales the |
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energy of the potential. For SSD, $\sigma_{\text{SSD}} = 3.051 \mbox{ |
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\AA}$ and $\epsilon_{\text{SSD}} = 0.152\text{ kcal/mol}$. |
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$V_{\text{dp}}$ is the dipole potential: |
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\begin{equation} |
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V_{\text{dp}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
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\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
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- |
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\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
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(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
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{r^{5}_{ij}} \biggr] |
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\label{eq:dipolePot} |
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\end{equation} |
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where $\mathbf{r}_{ij}$ is the vector between $i$ and $j$, |
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$\boldsymbol{\Omega}$ is the orientation of the species, and |
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$\boldsymbol{\mu}$ is the dipole vector. The SSD model specifies a dipole |
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magnitude of 2.35~D for water. |
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|
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The hydrogen bonding of the model is governed by the $V_{\text{sp}}$ |
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term of the potentail. Its form is as follows: |
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term of the potential. Its form is as follows: |
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\begin{equation} |
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V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = |
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\boldsymbol{\Omega}_{j})] |
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\label{eq:spPot} |
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\end{equation} |
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Where $v^\circ$ is responsible for scaling the strength of the |
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interaction. |
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Where $v^\circ$ scales strength of the interaction. |
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$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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and |
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$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
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\sin\theta_{ij} \sin 2\theta_{ij} \cos 2\phi_{ij} |
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+ \sin \theta_{ji} \sin 2\theta_{ji} \cos 2\phi_{ji} |
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\label{eq:apPot2} |
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\label{eq:spPot2} |
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\end{equation} |
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and |
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\begin{equation} |
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\end{split} |
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\label{eq:spCorrection} |
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\end{equation} |
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The correction |
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The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical |
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polar coordinates of the position of sphere $j$ in the reference frame |
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fixed on sphere $i$ with the z-axis aligned with the dipole moment. |
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The correction |
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$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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is needed because |
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$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. The angles |
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$\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical polar |
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coordinates of the position of sphere $j$ in the reference frame fixed |
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on sphere $i$ with the z-axis alligned with the dipole moment. |
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vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. |
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|
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Finaly, the sticky potentail is scaled by a cutoff function, |
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Finally, the sticky potential is scaled by a cutoff function, |
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$s(r_{ij})$ that scales smoothly between 0 and 1. It is represented |
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by: |
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\begin{equation} |
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\label{eq:spCutoff} |
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\end{equation} |
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|
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Despite the apparent complexity of Equation \ref{eq:spPot}, the SSD |
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model is still computationaly inexpensive. This is due to Equation |
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\ref{eq:spCutoff}. With $r_{L}$ being 2.75~$\mbox\AA}$ and $r_{U}$ |
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being equal to either 3.35~$\mbox{\AA}$ for $s(r_{ij})$ or |
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4.0~$\mbox{\AA}$ for $s'(r_{ij})$, the sticky potential is only active |
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over an extremly short range, and then only with other SSD |
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molecules. Therefore, it's predominant interaction is through it's |
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point dipole and Lennard-Jones sphere. |
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|
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\subsection{The Phospholipid Model} |
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\label{sec:lipidModel} |
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lipids. The Head group of the phospholipid is replaced by a single |
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Lennard-Jones sphere with a freely oriented dipole placed at it's |
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center. The magnitude of it's dipole moment is 20.6 D. The tail atoms |
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are unifed $\text{CH}_2$ and $\text{CH}_3$ atoms and are also modeled |
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are unified $\text{CH}_2$ and $\text{CH}_3$ atoms and are also modeled |
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as Lennard-Jones spheres. The total potential for the lipid is |
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represented by Equation \ref{eq:lipidModelPot}. |
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|
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\label{eq:lipidModelPot} |
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\end{equation} |
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|
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The non bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are |
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The non-bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are |
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the Lennard-Jones and dipole-dipole interactions respectively. For the |
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non-bonded potentials, only the bend and the torsional potentials are |
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calculated. The bond potential is not calculated, and the bond lengths |
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Where $k_{\theta}$ sets the stiffness of the bend potential, and $\theta_0$ |
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sets the equilibrium bend angle. The torsion potential was given by: |
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\begin{equation} |
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V_{\text{tors.}}(\phi_{ijkl})= c_1[1+\cos\phi] |
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+ c_2 [1 - \cos(2\phi)] + c_3[1 + \cos(3\phi)] |
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V_{\text{tors.}}(\phi_{ijkl})= c_1[1+\cos\phi_{ijkl}] |
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+ c_2 [1 - \cos(2\phi_{ijkl})] + c_3[1 + \cos(3\phi_{ijkl})] |
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\label{eq:torsPot} |
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\end{equation} |
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All parameters for bonded and non-bonded potentials in the tail atoms |
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were taken from TraPPE.\cite{Siepmann1998} The bonded interactions for |
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the head atom were also taken from TraPPE, however it's dipole moment |
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and mass were based on the properties of ``DMPC?'''s head group. The |
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Lennard-Jones parameter for the head group was chosen such that it was |
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roughly twice the size of a $\text{CH}_3$ atom, and it's well depth |
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was set to be aproximately equal to that of $\text{CH}_3$. |
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and mass were based on the properties of the phosphatidylcholine head |
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group. The Lennard-Jones parameter for the head group was chosen such |
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that it was roughly twice the size of a $\text{CH}_3$ atom, and it's |
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well depth was set to be approximately equal to that of $\text{CH}_3$. |
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|
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\section{Initial Simulation: 25 lipids in water} |
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\label{sec:5x5} |
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|
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\subsection{Starting Configurtion and Parameters} |
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\subsection{Starting Configuration and Parameters} |
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\label{sec:5x5Start} |
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|
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\begin{floatingfigure}{85mm} |
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\includegraphics[width=75mm]{5x5-initial.eps} |
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\caption{A snapshot of the initial configuration of the 25 lipid simulation.} |
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\label{fig:5x5Start} |
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\end{floatingfigure} |
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|
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Our first simulation was an array of 25 single chained lipids in a sea |
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of water (Figure \ref{fig:5x5Start}). The total number of water |
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molecules was 1386, giving a final of water concentration of 70\% |
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wt. The simulation box measured 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
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x 39.4~$\mbox{\AA}$ with periodic boundry connditions invoked. The |
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x 39.4~$\mbox{\AA}$ with periodic boundary conditions invoked. The |
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system was simulated in the micro-canonical (NVE) ensemble with an |
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average temperature of 300~K. |
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|
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|
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yada |
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|
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– |
yada |
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– |
|
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yada |
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yya |
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d |
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– |
|
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uada |
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– |
|
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adsd |
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– |
|
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asfa |
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– |
|
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asfdads |
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\subsection{Results} |
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\label{sec:5x5Results} |
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|
| 284 |
– |
|
| 314 |
|
Figure \ref{fig:5x5Final} shows a snapshot of the system at |
| 315 |
< |
3.6~ns. Here the bylayer has formed with the lipid chains tilted to |
| 316 |
< |
the bylayer normal. |
| 315 |
> |
3.6~ns. Here it is exciting to note that the system has spontaneously |
| 316 |
> |
self assembled into a bilayer. Discussion of the length scales of the |
| 317 |
> |
bilayer will follow in this section. However, it is interesting to |
| 318 |
> |
note several qualitative properties of the system revealed by this |
| 319 |
> |
snapshot. First, is how the tail chains are tilted to the bilayer |
| 320 |
> |
normal. This is usually indicative of the gel ($L_{\beta'}$) |
| 321 |
> |
phase. Likely, the box size is too small for the bilayer to relax to |
| 322 |
> |
the fluid ($P_{\alpha}$ phase. Showing the need for a constant |
| 323 |
> |
pressure simulation versus the constant volume of the current |
| 324 |
> |
ensemble. |
| 325 |
|
|
| 326 |
< |
yada |
| 326 |
> |
The trajectory of the simulation was analyzed using the pair-wise |
| 327 |
> |
radial distribution function, $g(r)$, which has the form: |
| 328 |
> |
\begin{equation} |
| 329 |
> |
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j \neq i} |
| 330 |
> |
\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 331 |
> |
\label{eq:gofr} |
| 332 |
> |
\end{equation} |
| 333 |
> |
Equation \ref{eq:gofr} gives us information about the spacing of two |
| 334 |
> |
species as a function of radius. Essentially, if the observer were |
| 335 |
> |
located at atom $i$ and were looking out in all directions, $g(r)$ |
| 336 |
> |
shows the relative density of atom $j$ at any given radius, $r$, |
| 337 |
> |
normalized by the expected density of atom $j$ at $r$. In a |
| 338 |
> |
homogeneously mixed fluid, $g(r)$ will approach 1 at large $r$, as a |
| 339 |
> |
fluid contains no long range structure to contribute peaks in the |
| 340 |
> |
number density. |
| 341 |
|
|
| 342 |
< |
yada |
| 342 |
> |
For the species containing dipoles, a second pair wise distribution |
| 343 |
> |
function was used, $g_{\gamma}(r)$. It is of the form: |
| 344 |
> |
\begin{equation} |
| 345 |
> |
g_{\gamma}(r) = foobar |
| 346 |
> |
\label{eq:gammaofr} |
| 347 |
> |
\end{equation} |
| 348 |
> |
Where $\gamma_{ij}$ is the angle between the dipole of atom $j$ with |
| 349 |
> |
respect to the dipole of atom $i$. This correlation will vary between |
| 350 |
> |
$+1$ and $-1$ when the two dipoles are perfectly aligned and |
| 351 |
> |
anti-aligned respectively. This then gives us information about how |
| 352 |
> |
directional species are aligned with each other as a function of |
| 353 |
> |
distance. |
| 354 |
|
|
| 355 |
< |
yada |
| 355 |
> |
Figure \ref{fig:5x5HHCorr} shows the two self correlation functions |
| 356 |
> |
for the Head groups of the lipids. The first peak at 4.03~$\mbox{\AA}$ is the |
| 357 |
> |
nearest neighbor separation of the heads of two lipids. |
| 358 |
|
|
| 295 |
– |
yada |
| 359 |
|
|
| 297 |
– |
%\begin{floatingfigure}{85mm} |
| 298 |
– |
%\includegraphics[angle=-90,width=75mm]{5x5-3.6ns.epsi} |
| 299 |
– |
%\caption{A snapshot of the system at 3.6~ns.\vspace{5mm}} |
| 300 |
– |
%\label{fig:5x5Final} |
| 301 |
– |
%\end{floatingfigure} |
| 360 |
|
|
| 361 |
|
|
| 304 |
– |
|
| 305 |
– |
|
| 362 |
|
\section{Second Simulation: 50 randomly oriented lipids in water} |
| 363 |
|
|
| 364 |
|
the second simulation |
| 365 |
|
|
| 310 |
– |
\section{Preliminary Results} |
| 311 |
– |
|
| 312 |
– |
\section{Discussion} |
| 313 |
– |
|
| 366 |
|
\section{Future Directions} |
| 367 |
|
|
| 368 |
|
|
