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\documentclass[11pt]{article} |
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\usepackage{graphicx} |
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\usepackage{amsmath} |
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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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\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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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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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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\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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The second step taken to simplify the number of calculationsis 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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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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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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\subsection{The Soft Sticky Water Model} |
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\label{sec:ssdModel} |
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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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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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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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\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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$V_{\text{LJ}}$ 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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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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\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 $v^\circ$ is responsible for scaling the strength of the |
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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:apPot2} |
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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 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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Finaly, the sticky potentail 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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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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\subsection{The Phospholipid Model} |
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\label{sec:lipidModel} |
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\begin{floatingfigure}{90mm} |
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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. \vspace{5mm}} |
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\label{fig:lipidModel} |
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\end{floatingfigure} |
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The lipid molecules in our simulations are unified atom models. Figure |
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\ref{fig:lipidModel} shows a template drawing 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 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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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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\begin{equation} |
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V_{\mbox{lipid}} = \overbrace{% |
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V_{\text{bend}}(\theta_{ijk}) +% |
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V_{\text{tors.}}(\phi_{ijkl})}^{bonded} |
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+ \overbrace{% |
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V_{\text{LJ}}(r_{i\!j}) + |
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V_{\text{dp}}(r_{i\!j},\Omega_{i},\Omega_{j})% |
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}^{non-bonded} |
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\label{eq:lipidModelPot} |
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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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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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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_{ijk}) = k_{\theta}\frac{(\theta_{ijk} - \theta_0)^2}{2} |
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\label{eq:bendPot} |
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\end{equation} |
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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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\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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\section{Initial Simulation: 25 lipids in water} |
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\label{sec:5x5} |
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\subsection{Starting Configurtion and Parameters} |
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\label{sec:5x5Start} |
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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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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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system was simulated in the micro-canonical (NVE) ensemble with an |
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average temperature of 300~K. |
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yada |
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yada |
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yada |
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yya |
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d |
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uada |
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adsd |
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asfa |
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asfdads |
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\subsection{Results} |
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\label{sec:5x5Results} |
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Figure \ref{fig:5x5Final} shows a snapshot of the system at |
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3.6~ns. Here the bylayer has formed with the lipid chains tilted to |
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the bylayer normal. |
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yada |
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yada |
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yada |
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yada |
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%\begin{floatingfigure}{85mm} |
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%\includegraphics[angle=-90,width=75mm]{5x5-3.6ns.epsi} |
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%\caption{A snapshot of the system at 3.6~ns.\vspace{5mm}} |
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%\label{fig:5x5Final} |
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%\end{floatingfigure} |
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\section{Second Simulation: 50 randomly oriented lipids in water} |
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the second simulation |
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\section{Preliminary Results} |
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\section{Discussion} |
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\section{Future Directions} |
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\pagebreak |
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\bibliographystyle{achemso} |
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\bibliography{canidacy_paper} \end{document} |
