| 128 |
|
kcal/mol}$. $V_{\text{dp}}$ is the dipole potential with |
| 129 |
|
$|\mu_{\text{w}}| = 2.35\text{ D}$. |
| 130 |
|
|
| 131 |
< |
The hydrogen bonding of the model is governed by the $V_{\text{sp}}$ term of the potentail. Its form is as follows: |
| 131 |
> |
The hydrogen bonding of the model is governed by the $V_{\text{sp}}$ |
| 132 |
> |
term of the potentail. Its form is as follows: |
| 133 |
|
\begin{equation} |
| 134 |
|
V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
| 135 |
|
\boldsymbol{\Omega}_{j}) = |
| 166 |
|
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
| 167 |
|
is needed because |
| 168 |
|
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
| 169 |
< |
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. |
| 169 |
> |
vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. The angles |
| 170 |
> |
$\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical polar |
| 171 |
> |
coordinates of the position of sphere $j$ in the reference frame fixed |
| 172 |
> |
on sphere $i$ with the z-axis alligned with the dipole moment. |
| 173 |
|
|
| 174 |
< |
Finaly, the sticky potentail is scaled by a cutoff function, $s(r_{ij})$ that scales smoothly between 0 and 1. It is represented by: |
| 174 |
> |
Finaly, the sticky potentail is scaled by a cutoff function, |
| 175 |
> |
$s(r_{ij})$ that scales smoothly between 0 and 1. It is represented |
| 176 |
> |
by: |
| 177 |
|
\begin{equation} |
| 178 |
|
s(r_{ij}) = |
| 179 |
|
\begin{cases} |
| 190 |
|
\subsection{The Phospholipid Model} |
| 191 |
|
\label{sec:lipidModel} |
| 192 |
|
|
| 193 |
+ |
\begin{floatingfigure}{90mm} |
| 194 |
+ |
\includegraphics[angle=-90,width=80mm]{lipidModel.epsi} |
| 195 |
+ |
\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. \vspace{5mm}} |
| 196 |
+ |
\label{fig:lipidModel} |
| 197 |
+ |
\end{floatingfigure} |
| 198 |
|
|
| 199 |
+ |
The lipid molecules in our simulations are unified atom models. Figure |
| 200 |
+ |
\ref{fig:lipidModel} shows a template drawing for one of our |
| 201 |
+ |
lipids. The Head group of the phospholipid is replaced by a single |
| 202 |
+ |
Lennard-Jones sphere with a freely oriented dipole placed at it's |
| 203 |
+ |
center. The magnitude of it's dipole moment is 20.6 D. The tail atoms |
| 204 |
+ |
are unifed $\text{CH}_2$ and $\text{CH}_3$ atoms and are also modeled |
| 205 |
+ |
as Lennard-Jones spheres. The total potential for the lipid is |
| 206 |
+ |
represented by Equation \ref{eq:lipidModelPot}. |
| 207 |
+ |
|
| 208 |
+ |
\begin{equation} |
| 209 |
+ |
V_{\mbox{lipid}} = \overbrace{% |
| 210 |
+ |
V_{\text{bend}}(\theta_{ijk}) +% |
| 211 |
+ |
V_{\text{tors.}}(\phi_{ijkl})}^{bonded} |
| 212 |
+ |
+ \overbrace{% |
| 213 |
+ |
V_{\text{LJ}}(r_{i\!j}) + |
| 214 |
+ |
V_{\text{dp}}(r_{i\!j},\Omega_{i},\Omega_{j})% |
| 215 |
+ |
}^{non-bonded} |
| 216 |
+ |
\label{eq:lipidModelPot} |
| 217 |
+ |
\end{equation} |
| 218 |
+ |
|
| 219 |
+ |
The non bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are |
| 220 |
+ |
the Lennard-Jones and dipole-dipole interactions respectively. For the |
| 221 |
+ |
non-bonded potentials, only the bend and the torsional potentials are |
| 222 |
+ |
calculated. The bond potential is not calculated, and the bond lengths |
| 223 |
+ |
are constrained via RATTLE.\cite{leach01:mm} The bend potential is of |
| 224 |
+ |
the form: |
| 225 |
+ |
\begin{equation} |
| 226 |
+ |
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}\frac{(\theta_{ijk} - \theta_0)^2}{2} |
| 227 |
+ |
\label{eq:bendPot} |
| 228 |
+ |
\end{equation} |
| 229 |
+ |
Where $k_{\theta}$ sets the stiffness of the bend potential, and $\theta_0$ |
| 230 |
+ |
sets the equilibrium bend angle. The torsion potential was given by: |
| 231 |
+ |
\begin{equation} |
| 232 |
+ |
V_{\text{tors.}}(\phi_{ijkl})= \cos(\phi_{ijkl}) |
| 233 |
+ |
\label{eq:torsPot} |
| 234 |
+ |
\end{equation} |
| 235 |
+ |
Here, ``blank'' controls the scaling of the torsion potential, and the |
| 236 |
+ |
$c$ terms are paramterized for the $\cos$ expansion. All parameters |
| 237 |
+ |
for bonded and non-bonded potentials in the tail atoms were taken from |
| 238 |
+ |
TraPPE.\cite{Siepmann1998} The bonded interactions for the head atom |
| 239 |
+ |
were also taken from TraPPE, however it's dipole moment and mass were |
| 240 |
+ |
based on the properties of ``DMPC?'''s head group. The Lennard-Jones |
| 241 |
+ |
parameter for the Head group was chosen such that it was roughly twice |
| 242 |
+ |
the size of a $\text{CH}_3$ atom, and it's well depth was set to be |
| 243 |
+ |
aproximately equal to that of $\text{CH}_3$. |
| 244 |
+ |
|
| 245 |
+ |
\section{Simulations} |
| 246 |
+ |
\subsection{25 lipids in water} |
| 247 |
+ |
|
| 248 |
+ |
\subsection{50 randomly oriented lipids in water} |
| 249 |
+ |
|
| 250 |
+ |
\section{Preliminary Results} |
| 251 |
+ |
|
| 252 |
+ |
\section{Discussion} |
| 253 |
+ |
|
| 254 |
+ |
\section{Future Directions} |
| 255 |
+ |
|
| 256 |
+ |
|
| 257 |
+ |
\pagebreak |
| 258 |
|
\bibliographystyle{achemso} |
| 259 |
< |
\bibliography{canidacy_paper} |
| 190 |
< |
\end{document} |
| 259 |
> |
\bibliography{canidacy_paper} \end{document} |
