trunk/oopsePaper/EmpericalEnergy.tex


\section{\label{sec:empericalEnergy}The Emperical Energy Functions}

\subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies}

The basic unit of an {\sc oopse} simulation is the atom. The parameters
describing the atom are generalized to make the atom as flexible a
representation as possible. They may represent specific atoms of an
element, or be used for collections of atoms such as a methyl
group. The atoms are also capable of having a directional component
associated with them, often in the form of a dipole. Charges on atoms
are not currently suporrted by {\sc oopse}.

The second most basic building block of a simulation is the
molecule. The molecule is a way for {\sc oopse} to keep track of the
atoms in a simulation in logical manner. This particular unit will
store the identities of all the atoms associated with itself and is
responsible for the evaluation of its own bonded interaction
(i.e.~bonds, bends, and torsions).

As stated previously, one of the features that sets {\sc OOPSE} apart
from most of the current molecular simulation packages is the ability
to handle rigid body dynamics. Rigid bodies are non-spherical
particles or collections of particles that have a constant internal
potential and move collectively.\cite{Goldstein01} They are not
included in most simulation packages because of the need to
consider orientational degrees of freedom and include an integrator
that accurately propagates these motions in time.

Moving a rigid body involves determination of both the force and
torque applied by the surroundings, which directly affect the
translation and rotation in turn. In order to accumulate the total
force on a rigid body, the external forces must first be calculated
for all the internal particles. The total force on the rigid body is
simply the sum of these external forces.  Accumulation of the total
torque on the rigid body is more complex than the force in that it is
the torque applied on the center of mass that dictates rotational
motion. The summation of this torque is given by
\begin{equation}
\mathbf{\tau}_i=
        \sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia},
\label{eq:torqueAccumulate}
\end{equation}
where $\mathbf{\tau}_i$ and $\mathbf{r}_i$ are the torque about and
position of the center of mass respectively, while $\mathbf{f}_{ia}$
and $\mathbf{r}_{ia}$ are the force on and position of the component
particles of the rigid body.\cite{allen87:csl}

The application of the total torque is done in the body fixed axis of
the rigid body. In order to move between the space fixed and body
fixed coordinate axes, parameters describing the orientation must be
maintained for each rigid body. At a minimum, the rotation matrix
(\textbf{A}) can be described and propagated by the three Euler angles
($\phi, \theta, \text{and} \psi$), where \textbf{A} is composed of
trigonometric operations involving $\phi, \theta,$ and
$\psi$.\cite{Goldstein01} In order to avoid rotational limitations
inherent in using the Euler angles, the four parameter ``quaternion''
scheme can be used instead, where \textbf{A} is composed of arithmetic
operations involving the four components of a quaternion ($q_0, q_1,
q_2, \text{and} q_3$).\cite{allen87:csl} Use of quaternions also leads
to performance enhancements, particularly for very small
systems.\cite{Evans77}

{\sc OOPSE} utilizes a relatively new scheme that uses the entire nine
parameter rotation matrix internally. Further discussion on this
choice can be found in Sec.~\ref{sec:integrate}.

\subsection{\label{sec:LJPot}The Lennard Jones Potential}

The most basic force field implemented in OOPSE is the Lennard-Jones
potential. The Lennard-Jones potential mimics the attractive forces
two charge neutral particles experience when spontaneous dipoles are
induced on each other. This is the predominant intermolecular force in
systems of such as noble gases and simple alkanes.

The Lennard-Jones potential is given by:
\begin{equation}
V_{\text{LJ}}(r_{ij}) = 
        4\epsilon_{ij} \biggl[
        \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12}
        - \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6}
        \biggr]
\label{eq:lennardJonesPot}
\end{equation}
Where $r_ij$ is the distance between particle $i$ and $j$, $\sigma_{ij}$
scales the length of the interaction, and $\epsilon_{ij}$ scales the
energy well depth of the potential.

Because the potential is calculated between all pairs, the force
evaluation can become computationally expensive for large systems. To
keep the pair evaluation to a manegable number, OOPSE employs the use
of a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be
$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest self self length
parameter in the system. Truncating the calculation at
$r_{\text{cut}}$ introduces a discontinuity into the potential
energy. To offset this discontinuity, the energy value at
$r_{\text{cut}}$ is subtracted from the entire potential. This causes
the equation to go to zero at the cut-off radius.

Interactions between dissimilar particles requires the generation of
cross term parameters for $\sigma$ and $\epsilon$. These are
calculated through the Lorentz-Berthelot mixing
rules:\cite{allen87:csl}
\begin{equation}
\sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}]
\label{eq:sigmaMix}
\end{equation}
and
\begin{equation}
\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}
\label{eq:epsilonMix}
\end{equation}


\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field}

The \underline{D}ipolar \underline{U}nified-Atom
\underline{F}orce \underline{F}ield ({\sc duff}) was developed to
simulate lipid bilayers. We needed a model capable of forming
bilayers, while still being sufficiently computationally efficient to
allow simulations of large systems ($\approx$100's of phospholipids,
$\approx$1000's of waters) for long times ($\approx$10's of
nanoseconds).

With this goal in mind, we have eliminated all point charges. Charge
distributions were replaced with dipoles, 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 $\frac{1}{r^3}$, allowing us to avoid the
computationally expensive Ewald-Sum. Instead, we can use
neighbor-lists and cutoff radii for the dipolar interactions.

As an example, lipid head groups in {\sc duff} are represented as point
dipole interaction sites.PC and PE Lipid head groups are typically
zwitterionic in nature, with charges separated by as much as
6~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group
center of mass, our model mimics the head group of PC.\cite{Cevc87}
Additionally, a Lennard-Jones site is located at the
pseudoatom's center of mass. The model is illustrated by the dark grey
atom in Fig.~\ref{fig:lipidModel}.

\begin{figure}
\epsfbox{lipidModel.eps}
\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.}
\label{fig:lipidModel}
\end{figure}

The water model we use to complement the dipoles of the lipids is
the soft sticky dipole (SSD) model of Ichiye \emph{et
al.}\cite{liu96:new_model} This model is discussed in greater detail
in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single
Lennard-Jones interaction site. The site also contains a dipole to
mimic the partial charges on the hydrogens and the oxygen. However,
what makes the SSD model unique is the inclusion of a tetrahedral
short range potential to recover the hydrogen bonding of water, an
important factor when modeling bilayers, as it has been shown that
hydrogen bond network formation is a leading contribution to the
entropic driving force towards lipid bilayer formation.\cite{Cevc87}


Turning to the tails of the phospholipids, we have used a set of
scalable parameters to model the alkyl groups with Lennard-Jones
sites. For this, we have used the TraPPE force field of Siepmann
\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom
representation of n-alkanes, which is parametrized against phase
equilibria using Gibbs Monte Carlo simulation
techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that
it generalizes the types of atoms in an alkyl chain to keep the number
of pseudoatoms to a minimum; the parameters for an atom such as
$\text{CH}_2$ do not change depending on what species are bonded to
it.

TraPPE also constrains of all bonds to be of fixed length. Typically,
bond vibrations are the fastest motions in a molecular dynamic
simulation. Small time steps between force evaluations must be used to
ensure adequate sampling of the bond potential conservation of
energy. By constraining the bond lengths, larger time steps may be
used when integrating the equations of motion.


\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions}

The total energy of function in {\sc duff} is given by the following:
\begin{equation}
V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}}
        + \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}}
\label{eq:totalPotential}
\end{equation}
Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule:
\begin{equation}
 V^{I}_{\text{Internal}} = 
        \sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk})
        + \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\theta_{ijkl})
        + \sum_{i \in I} \sum_{(j>i+4) \in I} 
        \biggl[ V_{\text{LJ}}(r_{ij}) +  V_{\text{dipole}}
        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
        \biggr]
\label{eq:internalPotential}
\end{equation}
Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs
within in the molecule. $V_{\text{torsion}}$ is the torsion The
pairwise portions of the internal potential are excluded for pairs
that ar closer than three bonds, i.e.~atom pairs farther away than a
torsion are included in the pair-wise loop. 

The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is
as follows:
\begin{equation}
V^{IJ}_{\text{Cross}} = 
        \sum_{i \in I} \sum_{j \in J}
        \biggl[ V_{\text{LJ}}(r_{ij}) +  V_{\text{dipole}}
        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
        + V_{\text{sticky}}
        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
        \biggr]
\label{eq:crossPotentail}
\end{equation}
Where $V_{\text{LJ}}$ is the Lennard Jones potential,
$V_{\text{dipole}}$ is the dipole dipole potential, and
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model.

The bend potential of a molecule is represented by the following function:
\begin{equation}
V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot}
\end{equation}
Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$
(see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium
bond angle. $k_{\theta}$ is the force constant which determines the
strength of the harmonic bend. The parameters for $k_{\theta}$ and
$\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998}

The torsion potential and parameters are also taken from TraPPE. It is
of the form:
\begin{equation}
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] 
        + c_2[1 + \cos(2\phi)] 
        + c_3[1 + \cos(3\phi)]
\label{eq:origTorsionPot}
\end{equation}
Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$,
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}).  However,
for computaional efficency, the torsion potentail has been recast
after the method of CHARMM\cite{charmm1983} whereby the angle series
is converted to a power series of the form:
\begin{equation}
V_{\text{torsion}}(\phi_{ijkl}) =  
        k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0
\label{eq:torsionPot}
\end{equation}
Where:
\begin{align*}
k_0 &= c_1 + c_3 \\
k_1 &= c_1 - 3c_3 \\
k_2 &= 2 c_2 \\
k_3 &= 4c_3
\end{align*}
By recasting the equation to a power series, repeated trigonometric
evaluations are avoided during the calculation of the potential.


The dipole-dipole potential has the following form:
\begin{equation}
V_{\text{dipole}}(\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:dipolePot}
\end{equation}
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$
are the Euler angles of atom $i$ and $j$
respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom
$i$ it takes its orientation from $\boldsymbol{\Omega}_i$.


\subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF}

In the interest of computational efficiency, the default solvent used
in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was
developed by Ichiye \emph{et al.} as a modified form of the
hard-sphere water model proposed by Bratko, Blum, and
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole
with a Lennard-Jones core and a sticky potential that directs the
particles to assume the proper hydrogen bond orientation in the first
solvation shell. Thus, the interaction between two SSD water molecules
\emph{i} and \emph{j} is given by the potential
\begin{equation}
V_{ij} = 
        V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp}
        (\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
        V_{ij}^{sp}
        (\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
\label{eq:ssdPot}
\end{equation}
where the $\mathbf{r}_{ij}$ is the position vector between molecules
\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the
orientations of the respective molecules. The Lennard-Jones and dipole
parts of the potential are given by equations \ref{eq:lennardJonesPot}
and \ref{eq:dipolePot} respectively. The sticky part is described by
the following,
\begin{equation}
u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=
        \frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},
        \boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) +
        s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},
        \boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
\label{eq:stickyPot}
\end{equation}
where $\nu_0$ is a strength parameter for the sticky potential, and
$s$ and $s^\prime$ are cubic switching functions which turn off the
sticky interaction beyond the first solvation shell. The $w$ function
can be thought of as an attractive potential with tetrahedral
geometry:
\begin{equation}
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=
        \sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
\label{eq:stickyW}
\end{equation}
while the $w^\prime$ function counters the normal aligned and
anti-aligned structures favored by point dipoles:
\begin{equation}
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=
        (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
\label{eq:stickyWprime}
\end{equation}
It should be noted that $w$ is proportional to the sum of the $Y_3^2$
and $Y_3^{-2}$ spherical harmonics (a linear combination which
enhances the tetrahedral geometry for hydrogen bonded structures),
while $w^\prime$ is a purely empirical function.  A more detailed
description of the functional parts and variables in this potential
can be found in the original SSD
articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}

Since SSD is a single-point {\it dipolar} model, the force
calculations are simplified significantly relative to the standard
{\it charged} multi-point models. In the original Monte Carlo
simulations using this model, Ichiye {\it et al.} reported that using
SSD decreased computer time by a factor of 6-7 compared to other
models.\cite{Ichiye96} What is most impressive is that this savings
did not come at the expense of accurate depiction of the liquid state
properties.  Indeed, SSD maintains reasonable agreement with the Soper
data for the structural features of liquid
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
exhibited by SSD agree with experiment better than those of more
computationally expensive models (like TIP3P and
SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
of solvent properties makes SSD a very attractive model for the
simulation of large scale biochemical simulations.

Recent constant pressure simulations revealed issues in the original
SSD model that led to lower than expected densities at all target
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse}
is SSD/E, a density corrected derivative of SSD that exhibits improved
liquid structure and transport behavior. If the use of a reaction
field long-range interaction correction is desired, it is recommended
that the parameters be modified to those of the SSD/RF model. Solvent
parameters can be easily modified in an accompanying {\sc BASS} file
as illustrated in the scheme below. A table of the parameter values
and the drawbacks and benefits of the different density corrected SSD
models can be found in reference \ref{Gezelter04}.

!!!Place a {\sc BASS} scheme showing SSD parameters around here!!!

\subsection{\label{sec:eam}Embedded Atom Model}

Several molecular dynamics codes\cite{dynamo86} exist which have the
capacity to simulate metallic systems, including some that have
parallel computational abilities\cite{plimpton93}. Potentials that
describe bonding transition metal
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a
attractive interaction which models the stabilization of ``Embedding''
a positively charged metal ion in an electron density created by the
free valance ``sea'' of electrons created by the surrounding atoms in
the system. A mostly repulsive pairwise part of the potential
describes the interaction of the positively charged metal core ions
with one another. A particular potential description called the
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}(EAM) that has
particularly wide adoption has been selected for inclusion in OOPSE. A
good review of EAM and other metallic potential formulations was done
by Voter.\cite{voter}

The {\sc eam} potential has the form:
\begin{eqnarray}
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i}
\phi_{ij}({\bf r}_{ij})  \\
\rho_{i}  & = & \sum_{j \neq i} f_{j}({\bf r}_{ij})
\end{eqnarray}

where $\phi_{ij}$ is a primarily repulsive pairwise interaction
between atoms $i$ and $j$.In the origional formulation of
EAM\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however
in later refinements to EAM have shown that nonuniqueness between $F$
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} The
embedding function $F_{i}$ is the energy required to embedded an
positively-charged core ion $i$ into a linear supeposition of
sperically averaged atomic electron densities given by
$\rho_{i}$. There is a cutoff distance, $r_{cut}$, which limits the
summations in the {\sc eam} equation to the few dozen atoms
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$
interactions.

\subsection{\label{Sec:pbc}Periodic Boundary Conditions} 
 
\textit{Periodic boundary conditions} are widely used to simulate truly 
macroscopic systems with a relatively small number of particles. Simulation 
box is replicated throughout space to form an infinite lattice. During the 
simulation, when a particle moves in the primary cell, its periodic image 
particles in other boxes move in exactly the same direction with exactly the 
same orientation.Thus, as a particle leaves the primary cell, one of its 
images will enter through the opposite face.If the simulation box is large 
enough to avoid "feeling" the symmetric of the periodic lattice,the surface 
effect could be ignored. Cubic and parallelepiped are the available periodic 
cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe 
the property of the simulation box. Therefore, not only the size of the 
simulation box could be changed during the simulation, but also the shape of 
it. The transformation from box space vector $\overrightarrow{s}$ to its 
corresponding real space vector $\overrightarrow{r}$ is defined by 
\begin{equation} 
\overrightarrow{r}=H\overrightarrow{s}% 
\end{equation} 
 
 
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three 
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the 
simulation box respectively. 
 
To find the minimum image, we need to convert the real vector to its 
corresponding vector in box space first, \bigskip% 
\begin{equation} 
\overrightarrow{s}=H^{-1}\overrightarrow{r}% 
\end{equation} 
And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5 
to 0.5, 
\begin{equation} 
s_{i}^{\prime}=s_{i}-round(s_{i}) 
\end{equation} 
where% 
 
\begin{equation} 
round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ 
}x\geqslant0 
\end{equation} 
% 
 
\begin{equation} 
round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0 
\end{equation} 
 
 
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. 
 
Finally, we could get the minimum image by transforming back to real space,% 
 
\begin{equation} 
\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}% 
\end{equation} 
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