| 17 |
|
computationally expensive Ewald-Sum. Instead, we can use |
| 18 |
|
neighbor-lists and cutoff radii for the dipolar interactions. |
| 19 |
|
|
| 20 |
– |
\begin{equation} |
| 21 |
– |
V^{\text{dipole}}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 22 |
– |
\boldsymbol{\Omega}_{j}) = |
| 23 |
– |
\frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 24 |
– |
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 25 |
– |
- |
| 26 |
– |
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
| 27 |
– |
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
| 28 |
– |
{r^{5}_{ij}} \biggr]\label{eq:dipole} |
| 29 |
– |
\end{equation} |
| 30 |
– |
|
| 20 |
|
As an example, lipid head groups in DUFF are represented as point |
| 21 |
|
dipole interaction sites.PC and PE Lipid head groups are typically |
| 22 |
|
zwitterionic in nature, with charges separated by as much as |
| 33 |
|
\label{fig:lipidModel} |
| 34 |
|
\end{figure} |
| 35 |
|
|
| 36 |
+ |
The water model we use to complement the dipoles of the lipids is |
| 37 |
+ |
the soft sticky dipole (SSD) model of Ichiye \emph{et |
| 38 |
+ |
al.}\cite{liu96:new_model} This model is discussed in greater detail |
| 39 |
+ |
in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single |
| 40 |
+ |
Lennard-Jones interaction site. The site also contains a dipole to |
| 41 |
+ |
mimic the partial charges on the hydrogens and the oxygen. However, |
| 42 |
+ |
what makes the SSD model unique is the inclusion of a tetrahedral |
| 43 |
+ |
short range potential to recover the hydrogen bonding of water, an |
| 44 |
+ |
important factor when modeling bilayers, as it has been shown that |
| 45 |
+ |
hydrogen bond network formation is a leading contribution to the |
| 46 |
+ |
entropic driving force towards lipid bilayer formation.\cite{Cevc87} |
| 47 |
+ |
|
| 48 |
+ |
|
| 49 |
|
Turning to the tails of the phospholipids, we have used a set of |
| 50 |
|
scalable parameters to model the alkyl groups with Lennard-Jones |
| 51 |
|
sites. For this, we have used the TraPPE force field of Siepmann |
| 65 |
|
energy. By constraining the bond lengths, larger time steps may be |
| 66 |
|
used when integrating the equations of motion. |
| 67 |
|
|
| 66 |
– |
The water model we use to complement this the dipoles of the lipids is |
| 67 |
– |
the soft sticky dipole (SSD) model of Ichiye \emph{et |
| 68 |
– |
al.}\cite{liu96:new_model} This model is discussed in greater detail |
| 69 |
– |
in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single |
| 70 |
– |
Lennard-Jones interaction site. The site also contains a dipole to |
| 71 |
– |
mimic the partial charges on the hydrogens and the oxygen. However, |
| 72 |
– |
what makes the SSD model unique is the inclusion of a tetrahedral |
| 73 |
– |
short range potential to recover the hydrogen bonding of water, an |
| 74 |
– |
important factor when modeling bilayers, as it has been shown that |
| 75 |
– |
hydrogen bond network formation is a leading contribution to the |
| 76 |
– |
entropic driving force towards lipid bilayer formation.\cite{Cevc87} |
| 68 |
|
|
| 69 |
< |
\subsection{\label{subSec:energyFunctions}Energy Functions} |
| 69 |
> |
\subsection{\label{subSec:energyFunctions}DUFF Energy Functions} |
| 70 |
|
|
| 71 |
+ |
The total energy of function in DUFF is given by the following: |
| 72 |
|
\begin{equation} |
| 73 |
|
V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 74 |
|
+ \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}} |
| 75 |
|
\label{eq:totalPotential} |
| 76 |
|
\end{equation} |
| 77 |
< |
|
| 77 |
> |
Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule: |
| 78 |
|
\begin{equation} |
| 79 |
|
V^{I}_{\text{Internal}} = |
| 80 |
|
\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
| 85 |
|
\biggr] |
| 86 |
|
\label{eq:internalPotential} |
| 87 |
|
\end{equation} |
| 88 |
+ |
Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
| 89 |
+ |
within in the molecule. $V_{\text{torsion}}$ is the torsion The |
| 90 |
+ |
pairwise portions of the internal potential are excluded for pairs |
| 91 |
+ |
that ar closer than three bonds, i.e.~atom pairs farther away than a |
| 92 |
+ |
torsion are included in the pair-wise loop. |
| 93 |
|
|
| 94 |
+ |
The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
| 95 |
+ |
as follows: |
| 96 |
|
\begin{equation} |
| 97 |
|
V^{IJ}_{\text{Cross}} = |
| 98 |
|
\sum_{i \in I} \sum_{j \in J} |
| 103 |
|
\biggr] |
| 104 |
|
\label{eq:crossPotentail} |
| 105 |
|
\end{equation} |
| 106 |
+ |
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 107 |
+ |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 108 |
+ |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model. |
| 109 |
|
|
| 110 |
+ |
The bend potential of a molecule is represented by the following function: |
| 111 |
|
\begin{equation} |
| 112 |
|
V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
| 113 |
|
\end{equation} |
| 114 |
+ |
Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
| 115 |
+ |
(see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium |
| 116 |
+ |
bond angle. $k_{\theta}$ is the force constant which determines the |
| 117 |
+ |
strength of the harmonic bend. The parameters for $k_{\theta}$ and |
| 118 |
+ |
$\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998} |
| 119 |
|
|
| 120 |
+ |
The torsion potential and parameters are also taken from TraPPE. It is |
| 121 |
+ |
of the form: |
| 122 |
|
\begin{equation} |
| 123 |
< |
V_{\phi_{ijkl}} = ( k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0) |
| 123 |
> |
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] |
| 124 |
> |
+ c_2[1 + \cos(2\phi)] |
| 125 |
> |
+ c_3[1 + \cos(3\phi)] |
| 126 |
> |
\label{eq:origTorsionPot} |
| 127 |
> |
\end{equation} |
| 128 |
> |
Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$, |
| 129 |
> |
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). However, |
| 130 |
> |
for computaional efficency, the torsion potentail has been recast |
| 131 |
> |
after the method of CHARMM\cite{charmm1983} whereby the angle series |
| 132 |
> |
is converted to a power series of the form: |
| 133 |
> |
\begin{equation} |
| 134 |
> |
V_{\text{torsion}}(\phi_{ijkl}) = |
| 135 |
> |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 136 |
|
\label{eq:torsionPot} |
| 137 |
|
\end{equation} |
| 138 |
+ |
Where: |
| 139 |
+ |
\begin{align*} |
| 140 |
+ |
k_0 &= c_1 + c_3 \\ |
| 141 |
+ |
k_1 &= c_1 - 3c_3 \\ |
| 142 |
+ |
k_2 &= 2 c_2 \\ |
| 143 |
+ |
k_3 &= 4c_3 |
| 144 |
+ |
\end{align*} |
| 145 |
+ |
By recasting the equation to a power series, repeated trigonometric |
| 146 |
+ |
evaluations are avoided during the calculation of the potential. |
| 147 |
|
|
| 148 |
< |
|
| 118 |
< |
The bonded interactions in the DUFF functional set are limited to the |
| 119 |
< |
bend potential and the torsional potential. Bond potentials are not |
| 120 |
< |
calculated, instead all bond lengths are fixed to allow for large time |
| 121 |
< |
steps to be taken between force evaluations. |
| 122 |
< |
|
| 123 |
< |
The bend functional is of the form: |
| 124 |
< |
|
| 125 |
< |
$k_{\theta}$, the force constant, and $\theta_0$, the equilibrium bend |
| 126 |
< |
angle, were taken from the TraPPE forcefield of Siepmann. |
| 127 |
< |
|
| 128 |
< |
The torsion functional has the form: |
| 129 |
< |
|
| 130 |
< |
Here, the authors decided to use a potential in terms of a power |
| 131 |
< |
expansion in $\cos \phi$ rather than the typical expansion in |
| 132 |
< |
$\phi$. This prevents the need for repeated trigonometric |
| 133 |
< |
evaluations. Again, all $k_n$ constants were based on those in TraPPE. |
| 134 |
< |
|
| 135 |
< |
\subsection{Non-Bonded Interactions} |
| 136 |
< |
\label{subSec:nonBondedInteractions} |
| 137 |
< |
|
| 148 |
> |
The Lennard-Jones potential is given by: |
| 149 |
|
\begin{equation} |
| 150 |
< |
V_{\text{LJ}} = \text{internal + external} |
| 150 |
> |
V_{\text{LJ}}(r_{ij}) = |
| 151 |
> |
4\epsilon_{ij} \biggl[ |
| 152 |
> |
\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
| 153 |
> |
- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
| 154 |
> |
\biggr] |
| 155 |
> |
\label{eq:lennardJonesPot} |
| 156 |
|
\end{equation} |
| 157 |
+ |
Where $r_ij$ is the distance between atoms $i$ and $j$, $\sigma_{ij}$ |
| 158 |
+ |
scales the length of the interaction, and $\epsilon_{ij}$ scales the |
| 159 |
+ |
energy of the potential. |
| 160 |
|
|
| 161 |
< |
|
| 161 |
> |
The dipole-dipole potential has the following form: |
| 162 |
> |
\begin{equation} |
| 163 |
> |
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 164 |
> |
\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 165 |
> |
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 166 |
> |
- |
| 167 |
> |
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
| 168 |
> |
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
| 169 |
> |
{r^{5}_{ij}} \biggr] |
| 170 |
> |
\label{eq:dipolePot} |
| 171 |
> |
\end{equation} |
| 172 |
> |
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 173 |
> |
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 174 |
> |
are the Euler angles of atom $i$ and $j$ |
| 175 |
> |
respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom |
| 176 |
> |
$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
