14 |
|
replaced with dipolar entities, and charge neutral distributions were |
15 |
|
reduced to Lennard-Jones interaction sites. This simplification cuts |
16 |
|
the length scale of long range interactions from $\frac{1}{r}$ to |
17 |
< |
$\frac{1}{r^3}$ (Eq.~\ref{eq:dipole} vs.~ Eq.~\ref{eq:coloumb}). |
17 |
> |
$\frac{1}{r^3}$ (Eq.~\ref{eq:dipole} vs.~ Eq.~\ref{eq:coloumb}), |
18 |
> |
allowing us to avoid the computationally expensive Ewald-Sum. Instead, |
19 |
> |
we can use neighbor-lists and cutoff radii for the dipolar |
20 |
> |
interactions. |
21 |
|
|
22 |
|
\begin{align} |
23 |
|
V^{\text{dipole}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
32 |
|
{4\pi\epsilon_{0} r_{ij}} \label{eq:coloumb} |
33 |
|
\end{align} |
34 |
|
|
35 |
+ |
Applying this standard to the lipid model, we decided to represent the |
36 |
+ |
lipid model as a point dipole interaction site. Lipid head groups are |
37 |
+ |
typically zwitterionic in nature, with sometimes full integer charges |
38 |
+ |
seperated by only 5 to 6~$\mbox{\AA}$. By placing a dipole of |
39 |
+ |
20.6~Debye at the head groups center of mass, our model mimics the |
40 |
+ |
dipole of DMPC.\cite{Cevc87} Then, to account for the steric henderanc |
41 |
+ |
of the head group, a Lennard-Jones interaction site is also oacted at |
42 |
+ |
the psuedoatom's center of mass. The model is illustrated in |
43 |
+ |
Fig.~\ref{fig:lipidModel}. |
44 |
|
|
45 |
< |
The main energy function in OOPSE is DUFF (the Dipolar |
46 |
< |
Unified-atom Force Field). DUFF is a collection of parameters taken |
47 |
< |
from Seipmann \emph{et al.}\cite{Siepmann1998} and Ichiye \emph{et |
45 |
> |
\begin{figure} |
46 |
> |
\includegraphics[angle=-90,width=\linewidth]{lipidModel.epsi} |
47 |
> |
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
48 |
> |
is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
49 |
> |
\label{fig:lipidModel} |
50 |
> |
\end{figure} |
51 |
> |
|
52 |
> |
Turning to the tail chains of the phospholipids, we needed a set of |
53 |
> |
scalable parameters to model the alkyl groups as Lennard-Jones |
54 |
> |
interaction sites. For this, we used the TraPPE force field of |
55 |
> |
Siepmann \emph{et al}.\cite{Siepmann1998} The force field is a |
56 |
> |
unified-atom representation of n-alkanes. It is parametrized against |
57 |
> |
phase equilibria using Gibbs Monte Carlo simulation techniques. One of |
58 |
> |
the advantages of TraPPE is that is generalizes the types of atoms in |
59 |
> |
an alkyl chain to keep the number of pseudoatoms to a minimum. |
60 |
> |
%( $ \mbox{CH_3} $ %-$\mathbf{\mbox{CH_2}}$-$\mbox{CH_3}$ is the same as |
61 |
> |
|
62 |
> |
Another advantage of using TraPPE is the constraining of all bonds to |
63 |
> |
be of fixed length. Typically, bond vibrations are the motions in a |
64 |
> |
molecular dynamic simulation. This neccesitates a small time step |
65 |
> |
between force evaluations be used to ensure adequate sampling of the |
66 |
> |
bond potential. Failure to do so will result in loss of energy |
67 |
> |
conservation within the microcanonical ensemble. By constraining this |
68 |
> |
degree of freedom, time steps larger than were previously allowable |
69 |
> |
are able to be used when integrating the equations of motion. |
70 |
> |
|
71 |
> |
The main energy function in OOPSE is DUFF (the Dipolar Unified-atom |
72 |
> |
Force Field). DUFF is a collection of parameters taken from Seipmann |
73 |
> |
and Ichiye \emph{et |
74 |
|
al.}\cite{liu96:new_model} The total energy of interaction is given by |
75 |
|
Eq.~\ref{eq:totalPotential}: |
76 |
|
\begin{equation} |