| 1 |
mmeineke |
806 |
|
| 2 |
mmeineke |
899 |
\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
| 3 |
mmeineke |
806 |
|
| 4 |
mmeineke |
899 |
\subsection{\label{sec:atomsMolecules}Atoms and Molecules} |
| 5 |
mmeineke |
806 |
|
| 6 |
gezelter |
818 |
The basic unit of an {\sc oopse} simulation is the atom. The parameters |
| 7 |
mmeineke |
806 |
describing the atom are generalized to make the atom as flexible a |
| 8 |
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representation as possible. They may represent specific atoms of an |
| 9 |
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element, or be used for collections of atoms such as a methyl |
| 10 |
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group. The atoms are also capable of having a directional component |
| 11 |
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associated with them, often in the form of a dipole. Charges on atoms |
| 12 |
gezelter |
818 |
are not currently suporrted by {\sc oopse}. |
| 13 |
mmeineke |
806 |
|
| 14 |
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The second most basic building block of a simulation is the |
| 15 |
gezelter |
818 |
molecule. The molecule is a way for {\sc oopse} to keep track of the atoms |
| 16 |
mmeineke |
806 |
in a simulation in logical manner. This particular unit will store the |
| 17 |
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identities of all the atoms associated with itself and is responsible |
| 18 |
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|
for the evaluation of its own bonded interaction (i.e.~bonds, bends, |
| 19 |
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and torsions). |
| 20 |
mmeineke |
899 |
|
| 21 |
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\subsection{\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 ({\sc duff}) was developed to |
| 25 |
|
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simulate lipid bilayers. We needed a model capable of forming |
| 26 |
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bilayers, while still being sufficiently computationally efficient to |
| 27 |
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allow simulations of large systems ($\approx$100's of phospholipids, |
| 28 |
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$\approx$1000's of waters) for long times ($\approx$10's of |
| 29 |
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nanoseconds). |
| 30 |
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With this goal in mind, we have eliminated all point charges. Charge |
| 32 |
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distributions were replaced with dipoles, and charge-neutral |
| 33 |
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distributions were reduced to Lennard-Jones interaction sites. This |
| 34 |
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simplification cuts the length scale of long range interactions from |
| 35 |
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$\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the |
| 36 |
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computationally expensive Ewald-Sum. Instead, we can use |
| 37 |
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neighbor-lists and cutoff radii for the dipolar interactions. |
| 38 |
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|
| 39 |
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As an example, lipid head groups in {\sc duff} are represented as point |
| 40 |
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dipole interaction sites.PC and PE Lipid head groups are typically |
| 41 |
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zwitterionic in nature, with charges separated by as much as |
| 42 |
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6~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group |
| 43 |
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center of mass, our model mimics the head group of PC.\cite{Cevc87} |
| 44 |
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Additionally, a Lennard-Jones site is located at the |
| 45 |
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pseudoatom's center of mass. The model is illustrated by the dark grey |
| 46 |
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atom in Fig.~\ref{fig:lipidModel}. |
| 47 |
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| 48 |
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\begin{figure} |
| 49 |
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\epsfxsize=6in |
| 50 |
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\epsfbox{lipidModel.epsi} |
| 51 |
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\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
| 52 |
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is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
| 53 |
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\label{fig:lipidModel} |
| 54 |
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\end{figure} |
| 55 |
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| 56 |
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The water model we use to complement the dipoles of the lipids is |
| 57 |
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the soft sticky dipole (SSD) model of Ichiye \emph{et |
| 58 |
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al.}\cite{liu96:new_model} This model is discussed in greater detail |
| 59 |
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in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single |
| 60 |
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Lennard-Jones interaction site. The site also contains a dipole to |
| 61 |
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mimic the partial charges on the hydrogens and the oxygen. However, |
| 62 |
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what makes the SSD model unique is the inclusion of a tetrahedral |
| 63 |
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short range potential to recover the hydrogen bonding of water, an |
| 64 |
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important factor when modeling bilayers, as it has been shown that |
| 65 |
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hydrogen bond network formation is a leading contribution to the |
| 66 |
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entropic driving force towards lipid bilayer formation.\cite{Cevc87} |
| 67 |
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| 68 |
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| 69 |
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Turning to the tails of the phospholipids, we have used a set of |
| 70 |
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scalable parameters to model the alkyl groups with Lennard-Jones |
| 71 |
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sites. For this, we have used the TraPPE force field of Siepmann |
| 72 |
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\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom |
| 73 |
|
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representation of n-alkanes, which is parametrized against phase |
| 74 |
|
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equilibria using Gibbs Monte Carlo simulation |
| 75 |
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techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that |
| 76 |
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it generalizes the types of atoms in an alkyl chain to keep the number |
| 77 |
|
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of pseudoatoms to a minimum; the parameters for an atom such as |
| 78 |
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$\text{CH}_2$ do not change depending on what species are bonded to |
| 79 |
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it. |
| 80 |
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|
| 81 |
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TraPPE also constrains of all bonds to be of fixed length. Typically, |
| 82 |
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bond vibrations are the fastest motions in a molecular dynamic |
| 83 |
|
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simulation. Small time steps between force evaluations must be used to |
| 84 |
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ensure adequate sampling of the bond potential conservation of |
| 85 |
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energy. By constraining the bond lengths, larger time steps may be |
| 86 |
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used when integrating the equations of motion. |
| 87 |
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| 88 |
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| 89 |
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\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions} |
| 90 |
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| 91 |
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The total energy of function in {\sc duff} is given by the following: |
| 92 |
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\begin{equation} |
| 93 |
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V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 94 |
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+ \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}} |
| 95 |
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\label{eq:totalPotential} |
| 96 |
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\end{equation} |
| 97 |
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Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule: |
| 98 |
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\begin{equation} |
| 99 |
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V^{I}_{\text{Internal}} = |
| 100 |
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\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
| 101 |
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+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\theta_{ijkl}) |
| 102 |
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+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
| 103 |
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\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 104 |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 105 |
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\biggr] |
| 106 |
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\label{eq:internalPotential} |
| 107 |
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\end{equation} |
| 108 |
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Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
| 109 |
|
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within in the molecule. $V_{\text{torsion}}$ is the torsion The |
| 110 |
|
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pairwise portions of the internal potential are excluded for pairs |
| 111 |
|
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that ar closer than three bonds, i.e.~atom pairs farther away than a |
| 112 |
|
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torsion are included in the pair-wise loop. |
| 113 |
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|
| 114 |
|
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The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
| 115 |
|
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as follows: |
| 116 |
|
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\begin{equation} |
| 117 |
|
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V^{IJ}_{\text{Cross}} = |
| 118 |
|
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\sum_{i \in I} \sum_{j \in J} |
| 119 |
|
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\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 120 |
|
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 121 |
|
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+ V_{\text{sticky}} |
| 122 |
|
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 123 |
|
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\biggr] |
| 124 |
|
|
\label{eq:crossPotentail} |
| 125 |
|
|
\end{equation} |
| 126 |
|
|
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 127 |
|
|
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 128 |
|
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$V_{\text{sticky}}$ is the sticky potential defined by the SSD model. |
| 129 |
|
|
|
| 130 |
|
|
The bend potential of a molecule is represented by the following function: |
| 131 |
|
|
\begin{equation} |
| 132 |
|
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V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
| 133 |
|
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\end{equation} |
| 134 |
|
|
Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
| 135 |
|
|
(see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium |
| 136 |
|
|
bond angle. $k_{\theta}$ is the force constant which determines the |
| 137 |
|
|
strength of the harmonic bend. The parameters for $k_{\theta}$ and |
| 138 |
|
|
$\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998} |
| 139 |
|
|
|
| 140 |
|
|
The torsion potential and parameters are also taken from TraPPE. It is |
| 141 |
|
|
of the form: |
| 142 |
|
|
\begin{equation} |
| 143 |
|
|
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] |
| 144 |
|
|
+ c_2[1 + \cos(2\phi)] |
| 145 |
|
|
+ c_3[1 + \cos(3\phi)] |
| 146 |
|
|
\label{eq:origTorsionPot} |
| 147 |
|
|
\end{equation} |
| 148 |
|
|
Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$, |
| 149 |
|
|
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). However, |
| 150 |
|
|
for computaional efficency, the torsion potentail has been recast |
| 151 |
|
|
after the method of CHARMM\cite{charmm1983} whereby the angle series |
| 152 |
|
|
is converted to a power series of the form: |
| 153 |
|
|
\begin{equation} |
| 154 |
|
|
V_{\text{torsion}}(\phi_{ijkl}) = |
| 155 |
|
|
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 156 |
|
|
\label{eq:torsionPot} |
| 157 |
|
|
\end{equation} |
| 158 |
|
|
Where: |
| 159 |
|
|
\begin{align*} |
| 160 |
|
|
k_0 &= c_1 + c_3 \\ |
| 161 |
|
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k_1 &= c_1 - 3c_3 \\ |
| 162 |
|
|
k_2 &= 2 c_2 \\ |
| 163 |
|
|
k_3 &= 4c_3 |
| 164 |
|
|
\end{align*} |
| 165 |
|
|
By recasting the equation to a power series, repeated trigonometric |
| 166 |
|
|
evaluations are avoided during the calculation of the potential. |
| 167 |
|
|
|
| 168 |
|
|
The Lennard-Jones potential is given by: |
| 169 |
|
|
\begin{equation} |
| 170 |
|
|
V_{\text{LJ}}(r_{ij}) = |
| 171 |
|
|
4\epsilon_{ij} \biggl[ |
| 172 |
|
|
\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
| 173 |
|
|
- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
| 174 |
|
|
\biggr] |
| 175 |
|
|
\label{eq:lennardJonesPot} |
| 176 |
|
|
\end{equation} |
| 177 |
|
|
Where $r_ij$ is the distance between atoms $i$ and $j$, $\sigma_{ij}$ |
| 178 |
|
|
scales the length of the interaction, and $\epsilon_{ij}$ scales the |
| 179 |
|
|
energy of the potential. |
| 180 |
|
|
|
| 181 |
|
|
The dipole-dipole potential has the following form: |
| 182 |
|
|
\begin{equation} |
| 183 |
|
|
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 184 |
|
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\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 185 |
|
|
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 186 |
|
|
- |
| 187 |
|
|
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
| 188 |
|
|
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
| 189 |
|
|
{r^{5}_{ij}} \biggr] |
| 190 |
|
|
\label{eq:dipolePot} |
| 191 |
|
|
\end{equation} |
| 192 |
|
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 193 |
|
|
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 194 |
|
|
are the Euler angles of atom $i$ and $j$ |
| 195 |
|
|
respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom |
| 196 |
|
|
$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
| 197 |
|
|
|
| 198 |
|
|
|
| 199 |
|
|
\subsection{\label{sec:SSD}Water Model: SSD and Derivatives} |
| 200 |
|
|
|
| 201 |
|
|
In the interest of computational efficiency, the native solvent used |
| 202 |
|
|
in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was |
| 203 |
|
|
developed by Ichiye \emph{et al.} as a modified form of the |
| 204 |
|
|
hard-sphere water model proposed by Bratko, Blum, and |
| 205 |
|
|
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
| 206 |
|
|
with a Lennard-Jones core and a sticky potential that directs the |
| 207 |
|
|
particles to assume the proper hydrogen bond orientation in the first |
| 208 |
|
|
solvation shell. Thus, the interaction between two SSD water molecules |
| 209 |
|
|
\emph{i} and \emph{j} is given by the potential |
| 210 |
|
|
\begin{equation} |
| 211 |
|
|
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
| 212 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
| 213 |
|
|
u_{ij}^{sp} |
| 214 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
| 215 |
|
|
\end{equation} |
| 216 |
|
|
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
| 217 |
|
|
\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and |
| 218 |
|
|
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
| 219 |
|
|
orientations of the respective molecules. The Lennard-Jones, dipole, |
| 220 |
|
|
and sticky parts of the potential are giving by the following |
| 221 |
|
|
equations, |
| 222 |
|
|
\begin{equation} |
| 223 |
|
|
u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right], |
| 224 |
|
|
\end{equation} |
| 225 |
|
|
\begin{equation} |
| 226 |
|
|
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}\ , |
| 227 |
|
|
\end{equation} |
| 228 |
|
|
\begin{equation} |
| 229 |
|
|
\begin{split} |
| 230 |
|
|
u_{ij}^{sp} |
| 231 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) |
| 232 |
|
|
&= |
| 233 |
|
|
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\ |
| 234 |
|
|
& \quad \ + |
| 235 |
|
|
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
| 236 |
|
|
\end{split} |
| 237 |
|
|
\end{equation} |
| 238 |
|
|
where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
| 239 |
|
|
unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
| 240 |
|
|
$\nu_0$ scales the strength of the overall sticky potential, $s$ and |
| 241 |
|
|
$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
| 242 |
|
|
functions take the following forms, |
| 243 |
|
|
\begin{equation} |
| 244 |
|
|
w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
| 245 |
|
|
\end{equation} |
| 246 |
|
|
\begin{equation} |
| 247 |
|
|
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, |
| 248 |
|
|
\end{equation} |
| 249 |
|
|
where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive |
| 250 |
|
|
term that promotes hydrogen bonding orientations within the first |
| 251 |
|
|
solvation shell, and $w^\prime$ is a dipolar repulsion term that |
| 252 |
|
|
repels unrealistic dipolar arrangements within the first solvation |
| 253 |
|
|
shell. A more detailed description of the functional parts and |
| 254 |
|
|
variables in this potential can be found in other |
| 255 |
|
|
articles.\cite{liu96:new_model,chandra99:ssd_md} |
| 256 |
|
|
|
| 257 |
|
|
Since SSD is a one-site point dipole model, the force calculations are |
| 258 |
|
|
simplified significantly from a computational standpoint, in that the |
| 259 |
|
|
number of long-range interactions is dramatically reduced. In the |
| 260 |
|
|
original Monte Carlo simulations using this model, Ichiye \emph{et |
| 261 |
|
|
al.} reported a calculation speed up of up to an order of magnitude |
| 262 |
|
|
over other comparable models while maintaining the structural behavior |
| 263 |
|
|
of water.\cite{liu96:new_model} In the original molecular dynamics studies of |
| 264 |
|
|
SSD, it was shown that it actually improves upon the prediction of |
| 265 |
|
|
water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md} |
| 266 |
|
|
|
| 267 |
|
|
Recent constant pressure simulations revealed issues in the original |
| 268 |
|
|
SSD model that led to lower than expected densities at all target |
| 269 |
|
|
pressures.\cite{Ichiye03,Gezelter04} Reparameterizations of the |
| 270 |
|
|
original SSD have resulted in improved density behavior, as well as |
| 271 |
|
|
alterations in the water structure and transport behavior. {\sc oopse} is |
| 272 |
|
|
easily modified to impliment these new potential parameter sets for |
| 273 |
|
|
the derivative water models: SSD1, SSD/E, and SSD/RF. All of the |
| 274 |
|
|
variable parameters are listed in the accompanying BASS file, and |
| 275 |
|
|
these parameters simply need to be changed to the updated values. |
| 276 |
|
|
|
| 277 |
|
|
|
| 278 |
|
|
\subsection{\label{sec:eam}Embedded Atom Model} |
| 279 |
|
|
|
| 280 |
|
|
here there be Monsters |
