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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 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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\subsection{Time Scale Simplifications} |
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\subsection{The Soft Sticky Water Model} |
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\label{sec:ssdModel} |
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\subsection{The Phospholipid Model} |
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\label{sec:lipidModel} |
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\bibliographystyle{achemso} |
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\bibliography{canidacy_paper} |
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\end{document} |
