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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, Charles F. Vardeman II, and J. Daniel Gezelter\\ |
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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{Model and Methodology} |
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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=170mm]{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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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 + 4)}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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\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_{\alpha\beta\gamma\zeta}) |
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= c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}] |
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+ c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] |
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+ c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})] |
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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 the phosphatidylcholine head |
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group. The Lennard-Jones parameter for the head group was chosen such |
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that it was roughly twice the size of a $\text{CH}_3$ atom, and it's |
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well depth was set to be approximately equal to that of $\text{CH}_3$. |
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\end{document} |
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