--- trunk/oopsePaper/EmpericalEnergy.tex	2003/10/24 21:27:59	818
+++ trunk/oopsePaper/EmpericalEnergy.tex	2004/01/06 18:53:58	899
@@ -1,7 +1,7 @@
 
-\section{The Emperical Energy Functions}
+\section{\label{sec:empericalEnergy}The Emperical Energy Functions}
 
-\subsection{Atoms and Molecules}
+\subsection{\label{sec:atomsMolecules}Atoms and Molecules}
 
 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
@@ -17,3 +17,264 @@ and torsions).
 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).
+
+\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}
+\epsfxsize=6in
+\epsfbox{lipidModel.epsi}
+\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 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 atoms $i$ and $j$, $\sigma_{ij}$
+scales the length of the interaction, and $\epsilon_{ij}$ scales the
+energy 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}Water Model: SSD and Derivatives}
+
+In the interest of computational efficiency, the native 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}
+u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
+(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
+u_{ij}^{sp}
+(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
+\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, dipole,
+and sticky parts of the potential are giving by the following
+equations,
+\begin{equation}
+u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right],
+\end{equation}
+\begin{equation}
+u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ ,
+\end{equation}
+\begin{equation}
+\begin{split}
+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)\\
+& \quad \ +
+s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
+\end{split}
+\end{equation}
+where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole
+unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D,
+$\nu_0$ scales the strength of the overall sticky potential, $s$ and
+$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
+functions take the following forms,
+\begin{equation}
+w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
+\end{equation}
+\begin{equation}
+w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
+\end{equation}
+where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive
+term that promotes hydrogen bonding orientations within the first
+solvation shell, and $w^\prime$ is a dipolar repulsion term that
+repels unrealistic dipolar arrangements within the first solvation
+shell. A more detailed description of the functional parts and
+variables in this potential can be found in other
+articles.\cite{liu96:new_model,chandra99:ssd_md}
+
+Since SSD is a one-site point dipole model, the force calculations are
+simplified significantly from a computational standpoint, in that the
+number of long-range interactions is dramatically reduced. In the
+original Monte Carlo simulations using this model, Ichiye \emph{et
+al.} reported a calculation speed up of up to an order of magnitude
+over other comparable models while maintaining the structural behavior
+of water.\cite{liu96:new_model} In the original molecular dynamics studies of
+SSD, it was shown that it actually improves upon the prediction of
+water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md}
+
+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} Reparameterizations of the
+original SSD have resulted in improved density behavior, as well as
+alterations in the water structure and transport behavior. {\sc oopse} is
+easily modified to impliment these new potential parameter sets for
+the derivative water models: SSD1, SSD/E, and SSD/RF. All of the
+variable parameters are listed in the accompanying BASS file, and
+these parameters simply need to be changed to the updated values.
+
+
+\subsection{\label{sec:eam}Embedded Atom Model}
+
+here there be Monsters
