| 1 |
mmeineke |
971 |
|
| 2 |
|
|
|
| 3 |
|
|
\chapter{\label{chapt:lipid}Phospholipid Simulations} |
| 4 |
|
|
|
| 5 |
|
|
\section{\label{lipidSec:Intro}Introduction} |
| 6 |
|
|
|
| 7 |
|
|
\section{\label{lipidSec:Methods}Methods} |
| 8 |
|
|
|
| 9 |
|
|
\subsection{\label{lipidSec:lipidMedel}The Lipid Model} |
| 10 |
|
|
|
| 11 |
|
|
\begin{figure} |
| 12 |
|
|
|
| 13 |
|
|
\caption{Schematic diagram of the single chain phospholipid model} |
| 14 |
|
|
|
| 15 |
|
|
\label{lipidFig:lipidModel} |
| 16 |
|
|
|
| 17 |
|
|
\end{figure} |
| 18 |
|
|
|
| 19 |
|
|
The phospholipid model used in these simulations is based on the |
| 20 |
|
|
design illustrated in Fig.~\ref{lipidFig:lipidModel}. The head group |
| 21 |
|
|
of the phospholipid is replaced by a single Lennard-Jones sphere of |
| 22 |
|
|
diameter $fix$, with $fix$ scaling the well depth of its van der Walls |
| 23 |
|
|
interaction. This sphere also contains a single dipole of magnitude |
| 24 |
|
|
$fix$, where $fix$ can be varied to mimic the charge separation of a |
| 25 |
|
|
given phospholipid head group. The atoms of the tail region are |
| 26 |
|
|
modeled by unified atom beads. They are free of partial charges or |
| 27 |
|
|
dipoles, containing only Lennard-Jones interaction sites at their |
| 28 |
|
|
centers of mass. As with the head groups, their potentials can be |
| 29 |
|
|
scaled by $fix$ and $fix$. |
| 30 |
|
|
|
| 31 |
|
|
The long range interactions between lipids are given by the following: |
| 32 |
|
|
\begin{equation} |
| 33 |
|
|
EQ Here |
| 34 |
|
|
\label{lipidEq:LJpot} |
| 35 |
|
|
\end{equation} |
| 36 |
|
|
and |
| 37 |
|
|
\begin{equation} |
| 38 |
|
|
EQ Here |
| 39 |
|
|
\label{lipidEq:dipolePot} |
| 40 |
|
|
\end{equation} |
| 41 |
|
|
Where $V_{\text{LJ}}$ is the Lennard-Jones potential and |
| 42 |
|
|
$V_{\text{dipole}}$ is the dipole-dipole potential. As previously |
| 43 |
|
|
stated, $\sigma_{ij}$ and $\epsilon_{ij}$ are the Lennard-Jones |
| 44 |
|
|
parameters which scale the length and depth of the interaction |
| 45 |
|
|
respectively, and $r_{ij}$ is the distance between beads $i$ and $j$. |
| 46 |
|
|
In $V_{\text{dipole}}$, $\mathbf{r}_{ij}$ is the vector starting at |
| 47 |
|
|
bead$i$ and pointing towards bead $j$. Vectors $\mathbf{\Omega}_i$ |
| 48 |
|
|
and $\mathbf{\Omega}_j$ are the orientational degrees of freedom for |
| 49 |
|
|
beads $i$ and $j$. $|\mu_i|$ is the magnitude of the dipole moment of |
| 50 |
|
|
$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 51 |
|
|
vector of $\boldsymbol{\Omega}_i$. |
| 52 |
|
|
|
| 53 |
|
|
The model also allows for the bonded interactions of bonds, bends, and |
| 54 |
|
|
torsions. The bonds between two beads on a chain are of fixed length, |
| 55 |
|
|
and are maintained according to the {\sc rattle} algorithm. \cite{fix} |
| 56 |
|
|
The bends are subject to a harmonic potential: |
| 57 |
|
|
\begin{equation} |
| 58 |
|
|
eq here |
| 59 |
|
|
\label{lipidEq:bendPot} |
| 60 |
|
|
\end{equation} |
| 61 |
|
|
where $fix$ scales the strength of the harmonic well, and $fix$ is the |
| 62 |
|
|
angle between bond vectors $fix$ and $fix$. The torsion potential is |
| 63 |
|
|
given by: |
| 64 |
|
|
\begin{equation} |
| 65 |
|
|
eq here |
| 66 |
|
|
\label{lipidEq:torsionPot} |
| 67 |
|
|
\end{equation} |
| 68 |
|
|
Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine |
| 69 |
|
|
power series to the desired torsion potential surface, and $\phi$ is |
| 70 |
|
|
the angle between bondvectors $fix$ and $fix$ along the vector $fix$ |
| 71 |
|
|
(see Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as |
| 72 |
|
|
the Lennard-Jones potential are excluded for bead pairs involved in |
| 73 |
|
|
the same bond, bend, or torsion. However, internal interactions not |
| 74 |
|
|
directly involved in a bonded pair are calculated. |
| 75 |
|
|
|
| 76 |
|
|
|
