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\begin{document} |
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\title{A Mesoscale Model for Phospholipid Simulations} |
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\author{Matthew A. Meineke\\ |
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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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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, typically 25 water molecules for every |
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lipid molecule. Because of these factors, many current simulations are |
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limited in both length and time scale due to to the sheer number of |
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calculations performed at every time step and the lifetime of the |
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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 are integrated for times up to 10 ns. |
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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 phenomenon 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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the numbers of atoms becomes 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 or to increase the length of |
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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 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 simulation. This is done through a |
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combination of substituting less expensive interactions for expensive |
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ones and decreasing the number of interaction sites per |
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molecule. Namely, groups of point charges are replaced with single |
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point-dipoles, and unified atoms are used in place of water, |
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phospholipid head groups, and alkyl groups. |
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The replacement of charges with dipoles allows us to replace an |
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interaction that has a relatively long range ($\frac{1}{r}$ for the |
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coulomb potential) with that of a relatively short range |
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($\frac{1}{r^{3}}$ for dipole - dipole potentials). Combined with |
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Verlet neighbor lists,\cite{allen87:csl} this should result in an |
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algorithm which scales linearly with increasing system size. This is |
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in comparison to the Ewald sum\cite{leach01:mm} needed to compute |
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periodic replicas of the coulomb interactions, which scales at |
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best\cite{darden93:pme} by $N \ln N$. |
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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:new_model,liu96:monte_carlo,chandra99:ssd_md} |
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(Section~\ref{sec:ssdModel}). For the phospholipids, a unified head |
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atom with a dipole will replace the atoms in the head group, while |
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unified $\text{CH}_2$ and $\text{CH}_3$ atoms will replace the alkyl |
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units in the tails (Section~\ref{sec:lipidModel}). This model is |
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similar in practice to that of Lipowsky and Goetz\cite{goetz98}, where |
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the whole system is reduced to attractive and repulsive Lennard-Jones |
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spheres. However, our model gives a greater level of detail to each |
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unified molecule, namely through dipole, bend, and torsion |
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interactions. |
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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 a given length of time using |
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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. 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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\subsection{The Soft Sticky Dipole Water Model} |
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\label{sec:ssdModel} |
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\begin{figure} |
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\centering |
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\includegraphics[width=50mm]{ssd.epsi} |
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\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference.} |
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\label{fig:ssdModel} |
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\end{figure} |
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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 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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The SSD water potential for a pair of water molecules is then given by |
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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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\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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where $\mathbf{r}_{ij}$ is the vector between molecules $i$ and $j$, |
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and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the |
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Euler angles of molecule $i$ or $j$ respectively. $V_{\text{LJ}}$ is |
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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 $\sigma_{ij}$ scales the length of the interaction, and |
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$\epsilon_{ij}$ scales the energy of the potential. For SSD, |
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$\sigma_{\text{SSD}} = 3.051 \mbox{ |
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\AA}$ 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 $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$, |
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$\boldsymbol{\mu}_i$ points along the z-axis in the body fixed |
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frame. This frame is then oriented in space by the Euler angles, |
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$\boldsymbol{\Omega}_i$. The SSD model specifies a dipole magnitude of |
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2.35~D for water. |
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The hydrogen bonding is modeled by the $V_{\text{sp}}$ |
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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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v^{\circ}[s(r_{ij})w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) |
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+ |
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s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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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 $v^\circ$ 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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are responsible for the tetrahedral potential and a correction to the |
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tetrahedral potential respectively. They are, |
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\begin{equation} |
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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:spPot2} |
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\end{equation} |
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and |
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\begin{equation} |
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\begin{split} |
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w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
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&(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij} + 0.8)^2 \\ |
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&+ (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ} |
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\end{split} |
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\label{eq:spCorrection} |
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\end{equation} |
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The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical |
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coordinates of the position of molecule $j$ in the reference frame |
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fixed on molecule $i$ 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 $180^\circ$. |
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Finally, the sticky potential is scaled by a switching 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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s(r_{ij}) = |
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\begin{cases} |
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1& \text{if $r_{ij} < r_{L}$}, \\ |
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\frac{(r_{U} - r_{ij})^2 (r_{U} + 2r_{ij} |
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- 3r_{L})}{(r_{U}-r_{L})^3}& |
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\text{if $r_{L} \leq r_{ij} \leq r_{U}$},\\ |
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0& \text{if $r_{ij} \geq r_{U}$}. |
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\end{cases} |
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\label{eq:spCutoff} |
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\end{equation} |
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Despite the apparent complexity of Equation \ref{eq:spPot}, the SSD |
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model is still computationally 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 extremely short range, and then only with other SSD |
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molecules. Therefore, it's predominant interaction is through the |
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point dipole and the Lennard-Jones sphere. Of these, only the dipole |
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interaction can be considered ``long-range''. |
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\subsection{The Phospholipid Model} |
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\label{sec:lipidModel} |
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\begin{figure} |
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\centering |
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\includegraphics[angle=-90,width=80mm]{lipidModel.epsi} |
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\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
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\label{fig:lipidModel} |
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\end{figure} |
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|
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The lipid molecules in our simulations are unified atom models. Figure |
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\ref{fig:lipidModel} shows a schematic for one of our |
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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 the dipole moment is 20.6 D, chosen to match |
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that of DPPC\cite{Cevc87}. The tail atoms are unified $\text{CH}_2$ |
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and $\text{CH}_3$ atoms and are also modeled as Lennard-Jones |
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spheres. The total potential for the lipid is represented by Equation |
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\ref{eq:lipidModelPot}. |
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\begin{equation} |
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V_{\text{lipid}} = |
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\sum_{i}V_{i}^{\text{internal}} |
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+ \sum_i \sum_{j>i} \sum_{\alpha_i} |
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\sum_{\beta_j} |
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V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}}) |
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+\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j}) |
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\label{eq:lipidModelPot} |
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\end{equation} |
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where, |
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\begin{equation} |
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V_{i}^{\text{internal}} = |
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\sum_{\text{bends}}V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
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+ \sum_{\text{torsions}}V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
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+ \sum_{\alpha_i} \sum_{\beta_i > \alpha_i}V_{\text{LJ}} |
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(r_{\alpha_i \beta_i}) |
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\label{eq:lipidModelPotInternal} |
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\end{equation} |
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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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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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are constrained via RATTLE.\cite{leach01:mm} The bend potential is of |
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the form: |
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\begin{equation} |
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V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
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= k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \theta_0)^2}{2} |
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mmeineke |
99 |
\label{eq:bendPot} |
310 |
|
|
\end{equation} |
311 |
mmeineke |
100 |
Where $k_{\theta}$ sets the stiffness of the bend potential, and $\theta_0$ |
312 |
|
|
sets the equilibrium bend angle. The torsion potential was given by: |
313 |
|
|
\begin{equation} |
314 |
mmeineke |
110 |
V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
315 |
|
|
= c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}] |
316 |
|
|
+ c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] |
317 |
|
|
+ c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})] |
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mmeineke |
100 |
\label{eq:torsPot} |
319 |
|
|
\end{equation} |
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mmeineke |
102 |
All parameters for bonded and non-bonded potentials in the tail atoms |
321 |
|
|
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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mmeineke |
106 |
and mass were based on the properties of the phosphatidylcholine head |
324 |
|
|
group. The Lennard-Jones parameter for the head group was chosen such |
325 |
|
|
that it was roughly twice the size of a $\text{CH}_3$ atom, and it's |
326 |
|
|
well depth was set to be approximately equal to that of $\text{CH}_3$. |
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mmeineke |
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|
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mmeineke |
102 |
\section{Initial Simulation: 25 lipids in water} |
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|
|
\label{sec:5x5} |
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mmeineke |
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|
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mmeineke |
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\subsection{Starting Configuration and Parameters} |
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mmeineke |
102 |
\label{sec:5x5Start} |
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mmeineke |
101 |
|
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mmeineke |
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\begin{figure} |
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mmeineke |
111 |
\centering |
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|
|
\mbox{ |
337 |
|
|
\subfigure[The starting configuration of the 25 lipid system.]{% |
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|
|
\label{fig:5x5Start}% |
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|
|
\includegraphics[width=70mm]{5x5-initial.eps}}\quad |
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|
|
\subfigure[The 25 lipid system at 6.27~ns.]{% |
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|
|
\label{fig:5x5Final}% |
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|
|
\includegraphics[width=70mm]{5x5-6.27ns.eps}} |
343 |
|
|
} |
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|
|
\caption{Snapshots of the 25 lipid system. Carbon tail atoms are drawn in gray, the phospholipid head groups are colored blue, and the waters are scaled down for visibility. A box has been drawn around the periodic image.} |
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mmeineke |
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\end{figure} |
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|
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|
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|
Our first simulation is an array of 25 single chain lipids in a sea |
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mmeineke |
102 |
of water (Figure \ref{fig:5x5Start}). The total number of water |
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mmeineke |
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molecules is 1386, giving a final of water concentration of 70\% |
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|
|
wt. The simulation box measures 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
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mmeineke |
107 |
x 39.4~$\mbox{\AA}$ with periodic boundary conditions imposed. The |
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mmeineke |
108 |
system is simulated in the micro-canonical (NVE) ensemble with an |
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mmeineke |
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average temperature of 300~K. |
354 |
|
|
|
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|
|
\subsection{Results} |
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|
|
\label{sec:5x5Results} |
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|
|
|
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mmeineke |
111 |
\begin{figure} |
359 |
|
|
\centering |
360 |
|
|
\subfigure[The self correlation of the phospholipid head groups. $g(r)$ is on the top, the bottom chart is the $g_\gamma(r)$.]{% |
361 |
|
|
\label{fig:5x5HHCorr}% |
362 |
|
|
\includegraphics[angle=-90,width=80mm]{all5x5-HEAD-HEAD.epsi}% |
363 |
|
|
} |
364 |
|
|
\subfigure[The $g(r)$ for the $\text{CH}_2$ molecules in the chain tails]{% |
365 |
|
|
\label{fig:5x5CCg}% |
366 |
|
|
\includegraphics[angle=-90,width=80mm]{all5x5-CH2-CH2.epsi}} |
367 |
|
|
\subfigure% |
368 |
|
|
[The pair correlations between the head groups and the water]{% |
369 |
|
|
\label{fig:5x5HXCorr}% |
370 |
|
|
\includegraphics[angle=-90,width=80mm]{all5x5-HEAD-X.epsi}} |
371 |
|
|
\caption{The pair correlation functions for the 25 lipid system} |
372 |
|
|
\end{figure} |
373 |
|
|
|
374 |
|
|
|
375 |
mmeineke |
102 |
Figure \ref{fig:5x5Final} shows a snapshot of the system at |
376 |
mmeineke |
110 |
6.27~ns. Note that the system has spontaneously self assembled into a |
377 |
mmeineke |
107 |
bilayer. Discussion of the length scales of the bilayer will follow in |
378 |
|
|
this section. However, it is interesting to note a key qualitative |
379 |
|
|
property of the system revealed by this snapshot, the tail chains are |
380 |
|
|
tilted to the bilayer normal. This is usually indicative of the gel |
381 |
|
|
($L_{\beta'}$) phase. In this system, the box size is probably too |
382 |
|
|
small for the bilayer to relax to the fluid ($P_{\alpha}$) phase. This |
383 |
|
|
demonstrates a need for an isobaric-isothermal ensemble where the box |
384 |
mmeineke |
111 |
size may relax or expand to keep the system at 1~atm. |
385 |
mmeineke |
102 |
|
386 |
mmeineke |
108 |
The simulation was analyzed using the radial distribution function, |
387 |
|
|
$g(r)$, which has the form: |
388 |
mmeineke |
106 |
\begin{equation} |
389 |
mmeineke |
107 |
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} |
390 |
mmeineke |
106 |
\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle |
391 |
|
|
\label{eq:gofr} |
392 |
|
|
\end{equation} |
393 |
|
|
Equation \ref{eq:gofr} gives us information about the spacing of two |
394 |
|
|
species as a function of radius. Essentially, if the observer were |
395 |
|
|
located at atom $i$ and were looking out in all directions, $g(r)$ |
396 |
|
|
shows the relative density of atom $j$ at any given radius, $r$, |
397 |
|
|
normalized by the expected density of atom $j$ at $r$. In a |
398 |
|
|
homogeneously mixed fluid, $g(r)$ will approach 1 at large $r$, as a |
399 |
|
|
fluid contains no long range structure to contribute peaks in the |
400 |
|
|
number density. |
401 |
mmeineke |
102 |
|
402 |
mmeineke |
111 |
For the species containing dipoles, a second pair-wise distribution |
403 |
mmeineke |
106 |
function was used, $g_{\gamma}(r)$. It is of the form: |
404 |
|
|
\begin{equation} |
405 |
mmeineke |
110 |
g_{\gamma}(r) = \langle \sum_i \sum_{j>i} |
406 |
|
|
(\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle |
407 |
mmeineke |
106 |
\label{eq:gammaofr} |
408 |
|
|
\end{equation} |
409 |
|
|
Where $\gamma_{ij}$ is the angle between the dipole of atom $j$ with |
410 |
|
|
respect to the dipole of atom $i$. This correlation will vary between |
411 |
|
|
$+1$ and $-1$ when the two dipoles are perfectly aligned and |
412 |
|
|
anti-aligned respectively. This then gives us information about how |
413 |
|
|
directional species are aligned with each other as a function of |
414 |
|
|
distance. |
415 |
mmeineke |
102 |
|
416 |
mmeineke |
106 |
Figure \ref{fig:5x5HHCorr} shows the two self correlation functions |
417 |
mmeineke |
108 |
for the Head groups of the lipids. The first peak in the $g(r)$ at |
418 |
|
|
4.03~$\mbox{\AA}$ is the nearest neighbor separation of the heads of |
419 |
|
|
two lipids. This corresponds to a maximum in the $g_{\gamma}(r)$ which |
420 |
mmeineke |
110 |
means that the two neighbors on the same leaf have their dipoles |
421 |
|
|
aligned. The broad peak at 6.5~$\mbox{\AA}$ is the inter-surface |
422 |
mmeineke |
108 |
spacing. Here, there is a corresponding anti-alignment in the angular |
423 |
|
|
correlation. This means that although the dipoles are aligned on the |
424 |
|
|
same monolayer, the dipoles will orient themselves to be anti-aligned |
425 |
|
|
on a opposite facing monolayer. With this information, the two peaks |
426 |
|
|
at 7.0~$\mbox{\AA}$ and 7.4~$\mbox{\AA}$ are head groups on the same |
427 |
|
|
monolayer, and they are the second nearest neighbors to the head |
428 |
|
|
group. The peak is likely a split peak because of the small statistics |
429 |
|
|
of this system. Finally, the peak at 8.0~$\mbox{\AA}$ is likely the |
430 |
|
|
second nearest neighbor on the opposite monolayer because of the |
431 |
|
|
anti-alignment evident in the angular correlation. |
432 |
mmeineke |
102 |
|
433 |
mmeineke |
108 |
Figure \ref{fig:5x5CCg} shows the radial distribution function for the |
434 |
|
|
$\text{CH}_2$ unified atoms. The spacing of the atoms along the tail |
435 |
|
|
chains accounts for the regularly spaced sharp peaks, but the broad |
436 |
|
|
underlying peak with its maximum at 4.6~$\mbox{\AA}$ is the |
437 |
|
|
distribution of chain-chain distances between two lipids. The final |
438 |
|
|
Figure, Figure \ref{fig:5x5HXCorr}, includes the correlation functions |
439 |
|
|
between the Head group and the SSD atoms. The peak in $g(r)$ at |
440 |
|
|
3.6~$\mbox{\AA}$ is the distance of closest approach between the two, |
441 |
|
|
and $g_{\gamma}(r)$ shows that the SSD atoms will align their dipoles |
442 |
|
|
with the head groups at close distance. However, as one increases the |
443 |
|
|
distance, the SSD atoms are no longer aligned. |
444 |
mmeineke |
102 |
|
445 |
mmeineke |
108 |
\section{Second Simulation: 50 randomly oriented lipids in water} |
446 |
|
|
\label{sec:r50} |
447 |
mmeineke |
102 |
|
448 |
mmeineke |
108 |
\subsection{Starting Configuration and Parameters} |
449 |
|
|
\label{sec:r50Start} |
450 |
mmeineke |
102 |
|
451 |
mmeineke |
110 |
\begin{figure} |
452 |
mmeineke |
111 |
\centering |
453 |
|
|
\mbox{ |
454 |
|
|
\subfigure[The starting configuration of the 50 lipid system.]{% |
455 |
|
|
\label{fig:r50Start}% |
456 |
|
|
\includegraphics[width=70mm]{r50-initial.eps}}\quad |
457 |
|
|
\subfigure[The 50 lipid system at 2.21~ns]{% |
458 |
|
|
\label{fig:r50Final}% |
459 |
|
|
\includegraphics[width=70mm]{r50-2.21ns.eps}} |
460 |
|
|
} |
461 |
|
|
\caption{Snapshots of the 50 lipid system} |
462 |
mmeineke |
110 |
\end{figure} |
463 |
|
|
|
464 |
mmeineke |
108 |
The second simulation consists of 50 single chained lipid molecules |
465 |
|
|
embedded in a sea of 1384 SSD waters (54\% wt.). The lipids in this |
466 |
|
|
simulation were started with random orientation and location (Figure |
467 |
mmeineke |
110 |
\ref{fig:r50Start} ) The simulation box measured 26.6~$\mbox{\AA}$ x |
468 |
|
|
26.6~$\mbox{\AA}$ x 108.4~$\mbox{\AA}$ with periodic boundary conditions |
469 |
mmeineke |
108 |
imposed. The simulation was run in the NVE ensemble with an average |
470 |
|
|
temperature of 300~K. |
471 |
mmeineke |
102 |
|
472 |
mmeineke |
108 |
\subsection{Results} |
473 |
|
|
\label{sec:r50Results} |
474 |
mmeineke |
102 |
|
475 |
mmeineke |
111 |
\begin{figure} |
476 |
|
|
\centering |
477 |
|
|
\subfigure[The self correlation of the phospholipid head groups.]{% |
478 |
|
|
\label{fig:r50HHCorr}% |
479 |
|
|
\includegraphics[angle=-90,width=80mm]{r50-HEAD-HEAD.epsi}% |
480 |
|
|
} |
481 |
|
|
\subfigure% |
482 |
|
|
[The pair correlations between the head groups and the water]{% |
483 |
|
|
\label{fig:r50HXCorr}% |
484 |
|
|
\includegraphics[angle=-90,width=80mm]{r50-HEAD-X.epsi}% |
485 |
|
|
} |
486 |
|
|
\subfigure[The $g(r)$ for the $\text{CH}_2$ molecules in the chain tails]{% |
487 |
|
|
\label{fig:r50CCg}% |
488 |
|
|
\includegraphics[angle=-90,width=80mm]{r50-CH2-CH2.epsi}} |
489 |
|
|
|
490 |
|
|
\caption{The pair correlation functions for the 50 lipid system} |
491 |
|
|
\end{figure} |
492 |
|
|
|
493 |
|
|
Figure \ref{fig:r50Final} is a snapshot of the system at 2.21~ns. Here |
494 |
mmeineke |
108 |
we see that the system has already aggregated into several micelles |
495 |
|
|
and two are even starting to merge. It will be interesting to watch as |
496 |
|
|
this simulation continues what the total time scale for the micelle |
497 |
mmeineke |
111 |
aggregation and bilayer formation will be, in Marrink's\cite{Marrink01} |
498 |
|
|
simulation, bilayer aggregation is predicted to occur around 10~ns. |
499 |
mmeineke |
108 |
|
500 |
mmeineke |
111 |
Figures \ref{fig:r50HHCorr}, \ref{fig:r50HXCorr}, and \ref{fig:r50CCg} are |
501 |
mmeineke |
108 |
the same correlation functions for the random 50 simulation as for the |
502 |
|
|
previous simulation of 25 lipids. What is most interesting to note, is |
503 |
mmeineke |
110 |
the high degree of similarity between the correlation functions |
504 |
|
|
between systems. Even though the 25 lipid simulation formed a bilayer |
505 |
|
|
and the random 50 simulation is still in the micelle stage, both have |
506 |
|
|
an inter-surface spacing of 6.5~$\mbox{\AA}$ with the same |
507 |
|
|
characteristic anti-alignment between surfaces. Not as surprising, is |
508 |
|
|
the consistency of the closest packing statistics between |
509 |
|
|
systems. Namely, a head-head closest approach distance of |
510 |
|
|
4~$\mbox{\AA}$, and similar findings for the chain-chain and |
511 |
|
|
head-water distributions as in the 25 lipid system. |
512 |
mmeineke |
108 |
|
513 |
mmeineke |
101 |
\section{Future Directions} |
514 |
|
|
|
515 |
mmeineke |
108 |
Current simulations indicate that our model is a feasible one, however |
516 |
mmeineke |
110 |
improvements will need to be made to allow the system to be simulated |
517 |
|
|
in the isobaric-isothermal ensemble. This will relax the system to an |
518 |
|
|
equilibrium configuration at room temperature and pressure allowing us |
519 |
|
|
to compare our model to experimental results. Also, we are in the |
520 |
mmeineke |
111 |
process of parallelizing the code for an even greater speedup. This |
521 |
|
|
will allow us to simulate large enough systems to examine phenomena |
522 |
mmeineke |
110 |
such as the ripple phase and drug molecule diffusion |
523 |
mmeineke |
101 |
|
524 |
mmeineke |
110 |
Once the work has been completed on the simulation engine, we will |
525 |
|
|
then use it to explore the phase diagram for our model. By |
526 |
mmeineke |
108 |
characterizing how our model parameters affect the bilayer properties, |
527 |
mmeineke |
110 |
we will tailor our model to more closely match real biological |
528 |
|
|
molecules. With this information, we will then incorporate |
529 |
mmeineke |
108 |
biologically relevant molecules into the system and observe their |
530 |
|
|
transport properties across the membrane. |
531 |
|
|
|
532 |
|
|
\section{Acknowledgments} |
533 |
|
|
|
534 |
mmeineke |
111 |
I would like to thank Dr.~J.~Daniel Gezelter for his guidance on this |
535 |
mmeineke |
108 |
project. I would also like to acknowledge the following for their help |
536 |
|
|
and discussions during this project: Christopher Fennell, Charles |
537 |
mmeineke |
110 |
Vardeman, Teng Lin, Megan Sprague, Patrick Conforti, and Dan |
538 |
|
|
Combest. Funding for this project came from the National Science |
539 |
|
|
Foundation. |
540 |
mmeineke |
108 |
|
541 |
|
|
\pagebreak |
542 |
|
|
\bibliographystyle{achemso} |
543 |
|
|
\bibliography{canidacy_paper} |
544 |
|
|
\end{document} |