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|
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\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
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|
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\subsection{\label{sec:atomsMolecules}Atoms and Molecules} |
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|
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The basic unit of an {\sc oopse} simulation is the atom. The parameters |
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describing the atom are generalized to make the atom as flexible a |
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representation as possible. They may represent specific atoms of an |
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element, or be used for collections of atoms such as a methyl |
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group. The atoms are also capable of having a directional component |
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associated with them, often in the form of a dipole. Charges on atoms |
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are not currently suporrted by {\sc oopse}. |
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|
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The second most basic building block of a simulation is the |
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molecule. The molecule is a way for {\sc oopse} to keep track of the atoms |
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in a simulation in logical manner. This particular unit will store the |
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identities of all the atoms associated with itself and is responsible |
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for the evaluation of its own bonded interaction (i.e.~bonds, bends, |
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and torsions). |
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|
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\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
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|
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The \underline{D}ipolar \underline{U}nified-Atom |
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\underline{F}orce \underline{F}ield ({\sc duff}) was developed to |
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simulate lipid bilayers. We needed a model capable of forming |
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bilayers, while still being sufficiently computationally efficient to |
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allow simulations of large systems ($\approx$100's of phospholipids, |
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$\approx$1000's of waters) for long times ($\approx$10's of |
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nanoseconds). |
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|
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With this goal in mind, we have eliminated all point charges. Charge |
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distributions were replaced with dipoles, and charge-neutral |
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distributions were reduced to Lennard-Jones interaction sites. This |
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simplification cuts the length scale of long range interactions from |
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$\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the |
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computationally expensive Ewald-Sum. Instead, we can use |
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neighbor-lists and cutoff radii for the dipolar interactions. |
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|
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As an example, lipid head groups in {\sc duff} are represented as point |
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dipole interaction sites.PC and PE Lipid head groups are typically |
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zwitterionic in nature, with charges separated by as much as |
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6~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group |
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center of mass, our model mimics the head group of PC.\cite{Cevc87} |
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Additionally, a Lennard-Jones site is located at the |
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pseudoatom's center of mass. The model is illustrated by the dark grey |
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atom in Fig.~\ref{fig:lipidModel}. |
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|
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\begin{figure} |
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\epsfxsize=6in |
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\epsfbox{lipidModel.epsi} |
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\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
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is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
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\label{fig:lipidModel} |
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\end{figure} |
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|
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The water model we use to complement the dipoles of the lipids is |
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the soft sticky dipole (SSD) model of Ichiye \emph{et |
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al.}\cite{liu96:new_model} This model is discussed in greater detail |
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in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single |
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Lennard-Jones interaction site. The site also contains a dipole to |
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mimic the partial charges on the hydrogens and the oxygen. However, |
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what makes the SSD model unique is the inclusion of a tetrahedral |
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short range potential to recover the hydrogen bonding of water, an |
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important factor when modeling bilayers, as it has been shown that |
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hydrogen bond network formation is a leading contribution to the |
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entropic driving force towards lipid bilayer formation.\cite{Cevc87} |
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|
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|
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Turning to the tails of the phospholipids, we have used a set of |
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scalable parameters to model the alkyl groups with Lennard-Jones |
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sites. For this, we have used the TraPPE force field of Siepmann |
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\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom |
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representation of n-alkanes, which is parametrized against phase |
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equilibria using Gibbs Monte Carlo simulation |
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techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that |
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it generalizes the types of atoms in an alkyl chain to keep the number |
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of pseudoatoms to a minimum; the parameters for an atom such as |
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$\text{CH}_2$ do not change depending on what species are bonded to |
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it. |
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|
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TraPPE also constrains of all bonds to be of fixed length. Typically, |
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bond vibrations are the fastest motions in a molecular dynamic |
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simulation. Small time steps between force evaluations must be used to |
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ensure adequate sampling of the bond potential conservation of |
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energy. By constraining the bond lengths, larger time steps may be |
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used when integrating the equations of motion. |
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|
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|
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\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions} |
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|
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The total energy of function in {\sc duff} is given by the following: |
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\begin{equation} |
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V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
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+ \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}} |
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\label{eq:totalPotential} |
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\end{equation} |
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Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule: |
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\begin{equation} |
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V^{I}_{\text{Internal}} = |
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\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
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+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\theta_{ijkl}) |
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+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
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\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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\biggr] |
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\label{eq:internalPotential} |
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\end{equation} |
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Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
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within in the molecule. $V_{\text{torsion}}$ is the torsion The |
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pairwise portions of the internal potential are excluded for pairs |
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that ar closer than three bonds, i.e.~atom pairs farther away than a |
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torsion are included in the pair-wise loop. |
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|
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The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
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as follows: |
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\begin{equation} |
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V^{IJ}_{\text{Cross}} = |
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\sum_{i \in I} \sum_{j \in J} |
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\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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+ V_{\text{sticky}} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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\biggr] |
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\label{eq:crossPotentail} |
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\end{equation} |
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Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
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$V_{\text{dipole}}$ is the dipole dipole potential, and |
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$V_{\text{sticky}}$ is the sticky potential defined by the SSD model. |
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|
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The bend potential of a molecule is represented by the following function: |
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\begin{equation} |
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V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
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\end{equation} |
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Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
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(see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium |
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bond angle. $k_{\theta}$ is the force constant which determines the |
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strength of the harmonic bend. The parameters for $k_{\theta}$ and |
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$\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998} |
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|
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The torsion potential and parameters are also taken from TraPPE. It is |
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of the form: |
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\begin{equation} |
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V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] |
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+ c_2[1 + \cos(2\phi)] |
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+ c_3[1 + \cos(3\phi)] |
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\label{eq:origTorsionPot} |
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\end{equation} |
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Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$, |
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$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). However, |
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for computaional efficency, the torsion potentail has been recast |
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after the method of CHARMM\cite{charmm1983} whereby the angle series |
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is converted to a power series of the form: |
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\begin{equation} |
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V_{\text{torsion}}(\phi_{ijkl}) = |
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k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
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\label{eq:torsionPot} |
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\end{equation} |
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Where: |
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\begin{align*} |
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k_0 &= c_1 + c_3 \\ |
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k_1 &= c_1 - 3c_3 \\ |
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k_2 &= 2 c_2 \\ |
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k_3 &= 4c_3 |
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\end{align*} |
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By recasting the equation to a power series, repeated trigonometric |
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evaluations are avoided during the calculation of the potential. |
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|
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The Lennard-Jones potential is given by: |
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\begin{equation} |
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V_{\text{LJ}}(r_{ij}) = |
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4\epsilon_{ij} \biggl[ |
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\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
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- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
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\biggr] |
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\label{eq:lennardJonesPot} |
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\end{equation} |
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Where $r_ij$ is the distance between atoms $i$ and $j$, $\sigma_{ij}$ |
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scales the length of the interaction, and $\epsilon_{ij}$ scales the |
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energy of the potential. |
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|
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The dipole-dipole potential has the following form: |
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\begin{equation} |
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V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
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\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
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- |
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\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
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(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
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{r^{5}_{ij}} \biggr] |
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\label{eq:dipolePot} |
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\end{equation} |
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Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
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towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
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are the Euler angles of atom $i$ and $j$ |
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respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom |
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$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
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|
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|
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\subsection{\label{sec:SSD}Water Model: SSD and Derivatives} |
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|
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In the interest of computational efficiency, the native solvent used |
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in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was |
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developed by Ichiye \emph{et al.} as a modified form of the |
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hard-sphere water model proposed by Bratko, Blum, and |
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Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
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with a Lennard-Jones core and a sticky potential that directs the |
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particles to assume the proper hydrogen bond orientation in the first |
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solvation shell. Thus, the interaction between two SSD water molecules |
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\emph{i} and \emph{j} is given by the potential |
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\begin{equation} |
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u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
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\end{equation} |
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where the $\mathbf{r}_{ij}$ is the position vector between molecules |
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\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and |
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$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
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orientations of the respective molecules. The Lennard-Jones, dipole, |
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and sticky parts of the potential are giving by the following |
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equations, |
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\begin{equation} |
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u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right], |
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\end{equation} |
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\begin{equation} |
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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}\ , |
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\end{equation} |
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\begin{equation} |
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\begin{split} |
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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) |
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&= |
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\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\ |
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& \quad \ + |
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s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
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\end{split} |
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\end{equation} |
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where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
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unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
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$\nu_0$ scales the strength of the overall sticky potential, $s$ and |
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$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
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functions take the following forms, |
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\begin{equation} |
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w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
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\end{equation} |
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\begin{equation} |
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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, |
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\end{equation} |
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where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive |
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term that promotes hydrogen bonding orientations within the first |
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solvation shell, and $w^\prime$ is a dipolar repulsion term that |
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repels unrealistic dipolar arrangements within the first solvation |
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shell. A more detailed description of the functional parts and |
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variables in this potential can be found in other |
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articles.\cite{liu96:new_model,chandra99:ssd_md} |
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|
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Since SSD is a one-site point dipole model, the force calculations are |
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simplified significantly from a computational standpoint, in that the |
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number of long-range interactions is dramatically reduced. In the |
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original Monte Carlo simulations using this model, Ichiye \emph{et |
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al.} reported a calculation speed up of up to an order of magnitude |
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over other comparable models while maintaining the structural behavior |
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of water.\cite{liu96:new_model} In the original molecular dynamics studies of |
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SSD, it was shown that it actually improves upon the prediction of |
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water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md} |
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|
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Recent constant pressure simulations revealed issues in the original |
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SSD model that led to lower than expected densities at all target |
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pressures.\cite{Ichiye03,Gezelter04} Reparameterizations of the |
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original SSD have resulted in improved density behavior, as well as |
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alterations in the water structure and transport behavior. {\sc oopse} is |
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easily modified to impliment these new potential parameter sets for |
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the derivative water models: SSD1, SSD/E, and SSD/RF. All of the |
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variable parameters are listed in the accompanying BASS file, and |
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these parameters simply need to be changed to the updated values. |
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|
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|
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\subsection{\label{sec:eam}Embedded Atom Model} |
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|
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here there be Monsters |
