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\section{The DUFF Energy Function} |
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\label{sec:energyFunctionals} |
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\section{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
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The \underline{D}ipolar \underline{U}nified-Atom |
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\underline{F}orce \underline{F}ield (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$10's of |
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nanoseconds). |
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With this goal in mind, we have eliminated all point charges. Charge |
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distributions were replaced with dipoles, and charge-neutral |
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distributions were reduced to Lennard-Jones interaction sites. This |
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simplification cuts the length scale of long range interactions from |
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$\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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The main energy function in OOPSE is DUFF (the Dipolar |
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Unified-atom Force Field). DUFF is a collection of parameters taken |
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from Seipmann \emph{et al.}\cite{Siepmann1998} and Ichiye \emph{et |
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al.}\cite{liu96:new_model} The total energy of interaction is given by |
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Eq.~\ref{eq:totalPotential}: |
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As an example, lipid head groups in 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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\begin{figure} |
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\includegraphics[angle=-90,width=\linewidth]{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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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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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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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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\subsection{\label{subSec:energyFunctions}DUFF Energy Functions} |
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The total energy of function in DUFF is given by the following: |
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\begin{equation} |
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V_{\text{Total}} = |
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\overbrace{V_{\theta} + V_{\phi}}^{\text{bonded}} + |
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\underbrace{V_{\text{LJ}} + V_{\text{Dp}} + % |
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V_{\text{SSD}}}_{\text{non-bonded}} \label{eq:totalPotential} |
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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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\subsection{Bonded Interactions} |
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\label{subSec:bondedInteractions} |
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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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The bonded interactions in the DUFF functional set are limited to the |
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bend potential and the torsional potential. Bond potentials are not |
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calculated, instead all bond lengths are fixed to allow for large time |
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steps to be taken between force evaluations. |
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The bend functional is of the form: |
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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} = \sum k_{\theta}( \theta - \theta_0 )^2 \label{eq:bendPot} |
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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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$k_{\theta}$, the force constant, and $\theta_0$, the equilibrium bend |
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angle, were taken from the TraPPE forcefield of Siepmann. |
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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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The torsion functional has the form: |
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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_{\phi} = \sum ( k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0) |
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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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Here, the authors decided to use a potential in terms of a power |
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expansion in $\cos \phi$ rather than the typical expansion in |
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$\phi$. This prevents the need for repeated trigonemtric |
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evaluations. Again, all $k_n$ constants were based on those in TraPPE. |
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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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\subsection{Non-Bonded Interactions} |
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\label{subSec:nonBondedInteractions} |
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The Lennard-Jones potential is given by: |
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\begin{equation} |
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V_{\text{LJ}} = \text{internal + external} |
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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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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 $\boldsymbol{\Omega}_i$. |
