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\documentclass[11pt]{article} |
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\usepackage{graphicx} |
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\usepackage{floatflt} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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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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\subsection{The Soft Sticky 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{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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\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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\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 $V_{\text{sp}}$ term of the potentail. 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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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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\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:apPot2} |
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\end{equation} |
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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 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 $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical polar coordinates of the position of sphere $j$ in the reference frame fixed 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, $s(r_{ij})$ that scales smoothly between 0 and 1. It is represented 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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\subsection{The Phospholipid Model} |
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\label{sec:lipidModel} |
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