trunk/oopsePaper/DUFF.tex


\section{\label{sec:DUFF}The DUFF Force Field}

The DUFF (\underline{D}ipolar \underline{U}nified-atom
\underline{F}orce \underline{F}ield) force field was developed to
simulate lipid bilayer formation and equilibrium dynamics. We needed a
model capable of forming bilaers, while still being sufficiently
computationally efficient allowing simulations of large systems
(\~100's of phospholipids, \~1000's of waters) for long times (\~10's
of nanoseconds).

With this goal in mind, we decided to eliminate all charged
interactions within the force field. Charge distributions were
replaced with dipolar entities, and charge neutral distributions were
reduced to Lennard-Jones interaction sites. This simplification cuts
the length scale of long range interactions from $\frac{1}{r}$ to
$\frac{1}{r^3}$ (Eq.~\ref{eq:dipole} vs.~ Eq.~\ref{eq:coloumb}).

\begin{align}
V^{\text{dipole}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
        \boldsymbol{\Omega}_{j}) &= 
        \frac{1}{4\pi\epsilon_{0}} \biggl[
        \frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}}
        -
        \frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) %
                (\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) }
                {r^{5}_{ij}} \biggr]\label{eq:dipole} \\
V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) &= \frac{q_{i}q_{j}}%
        {4\pi\epsilon_{0} r_{ij}} \label{eq:coloumb}
\end{align} 


The main energy function in OOPSE is DUFF (the Dipolar
Unified-atom Force Field). DUFF is a collection of parameters taken
from Seipmann \emph{et al.}\cite{Siepmann1998} and Ichiye \emph{et
al.}\cite{liu96:new_model} The total energy of interaction is given by
Eq.~\ref{eq:totalPotential}:
\begin{equation}
V_{\text{Total}} =
        \overbrace{V_{\theta} + V_{\phi}}^{\text{bonded}} +
        \underbrace{V_{\text{LJ}} + V_{\text{Dp}} + %
        V_{\text{SSD}}}_{\text{non-bonded}} \label{eq:totalPotential}
\end{equation}

\subsection{Bonded Interactions}
\label{subSec:bondedInteractions}

The bonded interactions in the DUFF functional set are limited to the
bend potential and the torsional potential. Bond potentials are not
calculated, instead all bond lengths are fixed to allow for large time
steps to be taken between force evaluations.

The bend functional is of the form:
\begin{equation}
V_{\theta} = \sum k_{\theta}( \theta - \theta_0 )^2 \label{eq:bendPot}
\end{equation}
$k_{\theta}$, the force constant, and $\theta_0$, the equilibrium bend
angle, were taken from the TraPPE forcefield of Siepmann.

The torsion functional has the form:
\begin{equation}
V_{\phi} =  \sum ( k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0)
\label{eq:torsionPot}
\end{equation}
Here, the authors decided to use a potential in terms of a power
expansion in $\cos \phi$ rather than the typical expansion in
$\phi$. This prevents the need for repeated trigonemtric
evaluations. Again, all $k_n$ constants were based on those in TraPPE.

\subsection{Non-Bonded Interactions}
\label{subSec:nonBondedInteractions}

\begin{equation}
V_{\text{LJ}} = \text{internal + external}
\end{equation}


Revision:	713
Committed:	Sat Aug 23 17:01:50 2003 UTC (21 years ago) by mmeineke
Content type:	application/x-tex
File size:	3001 byte(s)
Log Message:	added some work to DUFF
#	User	Rev	Content
1	mmeineke	664
2	mmeineke	713	\section{\label{sec:DUFF}The DUFF Force Field}
3	mmeineke	664
4	mmeineke	713	The DUFF (\underline{D}ipolar \underline{U}nified-atom
5			\underline{F}orce \underline{F}ield) force field was developed to
6			simulate lipid bilayer formation and equilibrium dynamics. We needed a
7			model capable of forming bilaers, while still being sufficiently
8			computationally efficient allowing simulations of large systems
9			(\~100's of phospholipids, \~1000's of waters) for long times (\~10's
10			of nanoseconds).
11	mmeineke	710
12	mmeineke	713	With this goal in mind, we decided to eliminate all charged
13			interactions within the force field. Charge distributions were
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}).
18	mmeineke	710
19	mmeineke	713	\begin{align}
20			V^{\text{dipole}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
21			\boldsymbol{\Omega}_{j}) &=
22			\frac{1}{4\pi\epsilon_{0}} \biggl[
23			\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}}
24			-
25			\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) %
26			(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) }
27			{r^{5}_{ij}} \biggr]\label{eq:dipole} \\
28			V^{\text{ch}}_{ij}(\mathbf{r}_{ij}) &= \frac{q_{i}q_{j}}%
29			{4\pi\epsilon_{0} r_{ij}} \label{eq:coloumb}
30			\end{align}
31
32
33	mmeineke	709	The main energy function in OOPSE is DUFF (the Dipolar
34	mmeineke	664	Unified-atom Force Field). DUFF is a collection of parameters taken
35	mmeineke	698	from Seipmann \emph{et al.}\cite{Siepmann1998} and Ichiye \emph{et
36	mmeineke	664	al.}\cite{liu96:new_model} The total energy of interaction is given by
37	mmeineke	666	Eq.~\ref{eq:totalPotential}:
38	mmeineke	698	\begin{equation}
39			V_{\text{Total}} =
40			\overbrace{V_{\theta} + V_{\phi}}^{\text{bonded}} +
41			\underbrace{V_{\text{LJ}} + V_{\text{Dp}} + %
42			V_{\text{SSD}}}_{\text{non-bonded}} \label{eq:totalPotential}
43			\end{equation}
44	mmeineke	666
45	mmeineke	698	\subsection{Bonded Interactions}
46			\label{subSec:bondedInteractions}
47	mmeineke	664
48	mmeineke	698	The bonded interactions in the DUFF functional set are limited to the
49			bend potential and the torsional potential. Bond potentials are not
50			calculated, instead all bond lengths are fixed to allow for large time
51			steps to be taken between force evaluations.
52	mmeineke	666
53	mmeineke	698	The bend functional is of the form:
54			\begin{equation}
55			V_{\theta} = \sum k_{\theta}( \theta - \theta_0 )^2 \label{eq:bendPot}
56			\end{equation}
57			$k_{\theta}$, the force constant, and $\theta_0$, the equilibrium bend
58			angle, were taken from the TraPPE forcefield of Siepmann.
59
60			The torsion functional has the form:
61			\begin{equation}
62	mmeineke	709	V_{\phi} = \sum ( k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0)
63	mmeineke	698	\label{eq:torsionPot}
64			\end{equation}
65			Here, the authors decided to use a potential in terms of a power
66			expansion in $\cos \phi$ rather than the typical expansion in
67			$\phi$. This prevents the need for repeated trigonemtric
68			evaluations. Again, all $k_n$ constants were based on those in TraPPE.
69
70			\subsection{Non-Bonded Interactions}
71			\label{subSec:nonBondedInteractions}
72
73			\begin{equation}
74			V_{\text{LJ}} = \text{internal + external}
75			\end{equation}
76
77