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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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\usepackage{subfigure} |
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\usepackage{palatino} |
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\usepackage[ref]{overcite} |
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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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\section{Background and Research Goals} |
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\section{Methodology} |
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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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\subsection{Length and Time Scale Simplifications} |
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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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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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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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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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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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\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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\section{Methodology} |
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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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\subsection{Length and Time Scale Simplifications} |
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The second step taken to simplify the number of calculationsis to |
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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} (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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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 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 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, 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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\subsection{The Soft Sticky Water Model} |
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\subsection{The Soft Sticky Dipole Water Model} |
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\label{sec:ssdModel} |
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The water model used in our simulations is |
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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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\label{eq:ssdTotPot} |
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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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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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\label{eq:spPot} |
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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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+ |
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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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\label{eq:apPot2} |
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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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|
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and |
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\begin{equation} |
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\label{eq:spCorrection} |
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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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&\phantom{=} + (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ} |
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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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|
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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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\label{eq:spCutoff} |
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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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\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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|
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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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|
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\subsection{The Phospholipid Model} |
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\label{sec:lipidModel} |
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|
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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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\bibliographystyle{achemso} |
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\bibliography{canidacy_paper} |
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\end{document} |
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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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|
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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}) |
| 295 |
> |
+ \sum_{\alpha_i} \sum_{\beta_i > (\alpha_i + 4)}V_{\text{LJ}} |
| 296 |
> |
(r_{\alpha_i \beta_i}) |
| 297 |
> |
\label{eq:lipidModelPotInternal} |
| 298 |
> |
\end{equation} |
| 299 |
> |
|
| 300 |
> |
The non-bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are |
| 301 |
> |
the Lennard-Jones and dipole-dipole interactions respectively. For the |
| 302 |
> |
bonded potentials, only the bend and the torsional potentials are |
| 303 |
> |
calculated. The bond potential is not calculated, and the bond lengths |
| 304 |
> |
are constrained via RATTLE.\cite{leach01:mm} The bend potential is of |
| 305 |
> |
the form: |
| 306 |
> |
\begin{equation} |
| 307 |
> |
V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
| 308 |
> |
= k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \theta_0)^2}{2} |
| 309 |
> |
\label{eq:bendPot} |
| 310 |
> |
\end{equation} |
| 311 |
> |
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 |
> |
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})] |
| 318 |
> |
\label{eq:torsPot} |
| 319 |
> |
\end{equation} |
| 320 |
> |
All parameters for bonded and non-bonded potentials in the tail atoms |
| 321 |
> |
were taken from TraPPE.\cite{Siepmann1998} The bonded interactions for |
| 322 |
> |
the head atom were also taken from TraPPE, however it's dipole moment |
| 323 |
> |
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 $\text{CH}_3$. |
| 327 |
> |
|
| 328 |
> |
\section{Initial Simulation: 25 lipids in water} |
| 329 |
> |
\label{sec:5x5} |
| 330 |
> |
|
| 331 |
> |
\subsection{Starting Configuration and Parameters} |
| 332 |
> |
\label{sec:5x5Start} |
| 333 |
> |
|
| 334 |
> |
\begin{figure} |
| 335 |
> |
\centering |
| 336 |
> |
\mbox{ |
| 337 |
> |
\subfigure[The starting configuration of the 25 lipid system.]{% |
| 338 |
> |
\label{fig:5x5Start}% |
| 339 |
> |
\includegraphics[width=70mm]{5x5-initial.eps}}\quad |
| 340 |
> |
\subfigure[The 25 lipid system at 6.27~ns.]{% |
| 341 |
> |
\label{fig:5x5Final}% |
| 342 |
> |
\includegraphics[width=70mm]{5x5-6.27ns.eps}} |
| 343 |
> |
} |
| 344 |
> |
\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.} |
| 345 |
> |
\end{figure} |
| 346 |
> |
|
| 347 |
> |
Our first simulation is an array of 25 single chain lipids in a sea |
| 348 |
> |
of water (Figure \ref{fig:5x5Start}). The total number of water |
| 349 |
> |
molecules is 1386, giving a final of water concentration of 70\% |
| 350 |
> |
wt. The simulation box measures 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
| 351 |
> |
x 39.4~$\mbox{\AA}$ with periodic boundary conditions imposed. The |
| 352 |
> |
system is simulated in the micro-canonical (NVE) ensemble with an |
| 353 |
> |
average temperature of 300~K. |
| 354 |
> |
|
| 355 |
> |
\subsection{Results} |
| 356 |
> |
\label{sec:5x5Results} |
| 357 |
> |
|
| 358 |
> |
\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 |
> |
Figure \ref{fig:5x5Final} shows a snapshot of the system at |
| 376 |
> |
6.27~ns. Note that the system has spontaneously self assembled into a |
| 377 |
> |
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 |
> |
size may relax or expand to keep the system at 1~atm. |
| 385 |
> |
|
| 386 |
> |
The simulation was analyzed using the radial distribution function, |
| 387 |
> |
$g(r)$, which has the form: |
| 388 |
> |
\begin{equation} |
| 389 |
> |
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} |
| 390 |
> |
\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 |
> |
|
| 402 |
> |
For the species containing dipoles, a second pair-wise distribution |
| 403 |
> |
function was used, $g_{\gamma}(r)$. It is of the form: |
| 404 |
> |
\begin{equation} |
| 405 |
> |
g_{\gamma}(r) = \langle \sum_i \sum_{j>i} |
| 406 |
> |
(\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 407 |
> |
\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 |
> |
|
| 416 |
> |
Figure \ref{fig:5x5HHCorr} shows the two self correlation functions |
| 417 |
> |
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 |
> |
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 |
> |
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 |
> |
|
| 433 |
> |
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 |
> |
|
| 445 |
> |
\section{Second Simulation: 50 randomly oriented lipids in water} |
| 446 |
> |
\label{sec:r50} |
| 447 |
> |
|
| 448 |
> |
\subsection{Starting Configuration and Parameters} |
| 449 |
> |
\label{sec:r50Start} |
| 450 |
> |
|
| 451 |
> |
\begin{figure} |
| 452 |
> |
\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 |
> |
\end{figure} |
| 463 |
> |
|
| 464 |
> |
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 |
> |
\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 |
> |
imposed. The simulation was run in the NVE ensemble with an average |
| 470 |
> |
temperature of 300~K. |
| 471 |
> |
|
| 472 |
> |
\subsection{Results} |
| 473 |
> |
\label{sec:r50Results} |
| 474 |
> |
|
| 475 |
> |
\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 |
> |
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 |
> |
aggregation and bilayer formation will be, in Marrink's\cite{Marrink01} |
| 498 |
> |
simulation, bilayer aggregation is predicted to occur around 10~ns. |
| 499 |
> |
|
| 500 |
> |
Figures \ref{fig:r50HHCorr}, \ref{fig:r50HXCorr}, and \ref{fig:r50CCg} are |
| 501 |
> |
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 |
> |
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 |
> |
|
| 513 |
> |
\section{Future Directions} |
| 514 |
> |
|
| 515 |
> |
Current simulations indicate that our model is a feasible one, however |
| 516 |
> |
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 |
> |
process of parallelizing the code for an even greater speedup. This |
| 521 |
> |
will allow us to simulate large enough systems to examine phenomena |
| 522 |
> |
such as the ripple phase and drug molecule diffusion |
| 523 |
> |
|
| 524 |
> |
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 |
> |
characterizing how our model parameters affect the bilayer properties, |
| 527 |
> |
we will tailor our model to more closely match real biological |
| 528 |
> |
molecules. With this information, we will then incorporate |
| 529 |
> |
biologically relevant molecules into the system and observe their |
| 530 |
> |
transport properties across the membrane. |
| 531 |
> |
|
| 532 |
> |
\section{Acknowledgments} |
| 533 |
> |
|
| 534 |
> |
I would like to thank Dr.~J.~Daniel Gezelter for his guidance on this |
| 535 |
> |
project. I would also like to acknowledge the following for their help |
| 536 |
> |
and discussions during this project: Christopher Fennell, Charles |
| 537 |
> |
Vardeman, Teng Lin, Megan Sprague, Patrick Conforti, and Dan |
| 538 |
> |
Combest. Funding for this project came from the National Science |
| 539 |
> |
Foundation. |
| 540 |
> |
|
| 541 |
> |
\pagebreak |
| 542 |
> |
\bibliographystyle{achemso} |
| 543 |
> |
\bibliography{canidacy_paper} |
| 544 |
> |
\end{document} |
