--- 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