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\chapter{\label{chapt:lipid}Phospholipid Simulations} |
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\section{\label{lipidSec:Intro}Introduction} |
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\section{\label{lipidSec:Methods}Methods} |
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\subsection{\label{lipidSec:lipidMedel}The Lipid Model} |
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\begin{figure} |
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\caption{Schematic diagram of the single chain phospholipid model} |
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\label{lipidFig:lipidModel} |
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\end{figure} |
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The phospholipid model used in these simulations is based on the |
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design illustrated in Fig.~\ref{lipidFig:lipidModel}. The head group |
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of the phospholipid is replaced by a single Lennard-Jones sphere of |
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diameter $fix$, with $fix$ scaling the well depth of its van der Walls |
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interaction. This sphere also contains a single dipole of magnitude |
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$fix$, where $fix$ can be varied to mimic the charge separation of a |
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given phospholipid head group. The atoms of the tail region are |
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modeled by unified atom beads. They are free of partial charges or |
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dipoles, containing only Lennard-Jones interaction sites at their |
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centers of mass. As with the head groups, their potentials can be |
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scaled by $fix$ and $fix$. |
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The long range interactions between lipids are given by the following: |
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\begin{equation} |
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EQ Here |
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\label{lipidEq:LJpot} |
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\end{equation} |
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and |
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\begin{equation} |
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EQ Here |
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\label{lipidEq:dipolePot} |
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\end{equation} |
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Where $V_{\text{LJ}}$ is the Lennard-Jones potential and |
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$V_{\text{dipole}}$ is the dipole-dipole potential. As previously |
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stated, $\sigma_{ij}$ and $\epsilon_{ij}$ are the Lennard-Jones |
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parameters which scale the length and depth of the interaction |
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respectively, and $r_{ij}$ is the distance between beads $i$ and $j$. |
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In $V_{\text{dipole}}$, $\mathbf{r}_{ij}$ is the vector starting at |
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bead$i$ and pointing towards bead $j$. Vectors $\mathbf{\Omega}_i$ |
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and $\mathbf{\Omega}_j$ are the orientational degrees of freedom for |
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beads $i$ and $j$. $|\mu_i|$ is the magnitude of the dipole moment of |
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$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
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vector of $\boldsymbol{\Omega}_i$. |
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The model also allows for the bonded interactions of bonds, bends, and |
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torsions. The bonds between two beads on a chain are of fixed length, |
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and are maintained according to the {\sc rattle} algorithm. \cite{fix} |
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The bends are subject to a harmonic potential: |
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\begin{equation} |
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eq here |
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\label{lipidEq:bendPot} |
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\end{equation} |
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where $fix$ scales the strength of the harmonic well, and $fix$ is the |
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angle between bond vectors $fix$ and $fix$. The torsion potential is |
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given by: |
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\begin{equation} |
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eq here |
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\label{lipidEq:torsionPot} |
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\end{equation} |
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Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine |
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power series to the desired torsion potential surface, and $\phi$ is |
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the angle between bondvectors $fix$ and $fix$ along the vector $fix$ |
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(see Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as |
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the Lennard-Jones potential are excluded for bead pairs involved in |
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the same bond, bend, or torsion. However, internal interactions not |
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directly involved in a bonded pair are calculated. |
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