| 62 |
|
site, as well as a point dipole. A tetrahedral potential is added to |
| 63 |
|
correct for hydrogen bonding. The following equation describes the |
| 64 |
|
interaction between two water molecules: |
| 65 |
< |
\[ |
| 65 |
> |
\begin{equation} |
| 66 |
|
V_{SSD} = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) |
| 67 |
|
+ V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) |
| 68 |
< |
\] |
| 69 |
< |
where $r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
| 68 |
> |
\label{lipidSection:ssdEquation} |
| 69 |
> |
\end{equation} |
| 70 |
> |
where$r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
| 71 |
|
$\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom |
| 72 |
< |
for molecule $i$ and molecule $j$ respectively. |
| 73 |
< |
\[ |
| 74 |
< |
V_{LJ} (r_{ij} ) = 4\varepsilon _{ij} \left[ {\left( {\frac{{\sigma |
| 75 |
< |
_{ij} }}{{r_{ij} }}} \right)^{12} - \left( {\frac{{\sigma _{ij} |
| 76 |
< |
}}{{r_{ij} }}} \right)^6 } \right] |
| 77 |
< |
\] |
| 78 |
< |
\[ |
| 79 |
< |
V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
| 80 |
< |
_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }} - |
| 81 |
< |
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
| 82 |
< |
r_{ij} } \right)}}{{r_{ij}^5 }}} \right] |
| 83 |
< |
\] |
| 84 |
< |
\[ |
| 85 |
< |
V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) = v_0 [s(r_{ij} )w(r_{ij} |
| 86 |
< |
,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i ,\Omega _j |
| 86 |
< |
)] |
| 87 |
< |
\] |
| 72 |
> |
for molecule $i$ and molecule $j$ respectively. The potential terms |
| 73 |
> |
in Eq.~\ref{lipidSection:ssdEquation} are given by : |
| 74 |
> |
\begin{eqnarray} |
| 75 |
> |
V_{LJ} (r_{ij} ) &= &4\varepsilon _{ij} \left[ {\left( |
| 76 |
> |
{\frac{{\sigma _{ij} }}{{r_{ij} }}} \right)^{12} - \left( |
| 77 |
> |
{\frac{{\sigma _{ij} |
| 78 |
> |
}}{{r_{ij} }}} \right)^6 } \right], \\ |
| 79 |
> |
V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) &= & |
| 80 |
> |
\frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 81 |
> |
\hat{u}_{i} \cdot \hat{u}_{j} - 3(\hat{u}_i \cdot \hat{\mathbf{r}}_{ij}) % |
| 82 |
> |
(\hat{u}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr],\\ |
| 83 |
> |
V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) &=& v_0 [s(r_{ij} |
| 84 |
> |
)w(r_{ij} ,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i |
| 85 |
> |
,\Omega _j )] |
| 86 |
> |
\end{eqnarray} |
| 87 |
|
where $v_0$ is a strength parameter, $s$ and $s'$ are cubic |
| 88 |
< |
switching functions, while $w$ and $w'$ are responsible for the |
| 88 |
> |
switching functions, while $w$ and $w'$ are responsible for the |
| 89 |
|
tetrahedral potential and the short-range correction to the dipolar |
| 90 |
< |
interaction respectively. |
| 91 |
< |
\[ |
| 92 |
< |
\begin{array}{l} |
| 93 |
< |
w(r_{ij} ,\Omega _i ,\Omega _j ) = \sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij} \\ |
| 94 |
< |
w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - w_0 \\ |
| 95 |
< |
\end{array} |
| 96 |
< |
\] |
| 97 |
< |
Although the dipole-dipole and sticky interactions are more |
| 90 |
> |
interaction respectively: |
| 91 |
> |
\begin{eqnarray} |
| 92 |
> |
w(r_{ij} ,\Omega _i ,\Omega _j )& = &\sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij}, \\ |
| 93 |
> |
w'(r_{ij} ,\Omega _i ,\Omega _j )& = &(\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - |
| 94 |
> |
w_0. |
| 95 |
> |
\end{eqnarray} |
| 96 |
> |
Here $\theta _{ij}$ and $\phi _{ij}$ are the spherical polar angles |
| 97 |
> |
representing relative orientations between molecule $i$ and molecule |
| 98 |
> |
$j$. Although the dipole-dipole and sticky interactions are more |
| 99 |
|
mathematically complicated than Coulomb interactions, the number of |
| 100 |
|
pair interactions is reduced dramatically both because the model |
| 101 |
|
only contains a single-point as well as "short range" nature of the |
| 117 |
|
\caption[A representation of coarse-grained phospholipid model]{A |
| 118 |
|
representation of coarse-grained phospholipid model. The lipid |
| 119 |
|
headgroup consists of $\text{{\sc PO}}_4$ group (yellow), |
| 120 |
< |
$\text{{\sc NC}}_4$ group (blue) and a united $\text{{\sc C}}$ atom |
| 121 |
< |
(gray) $ with a dipole, while the glycerol backbone includes dipolar |
| 122 |
< |
$\text{{\sc CE}}$ (red) and $\text{{\sc CK}}$ (pink) groups. Alkyl |
| 123 |
< |
groups in hydrocarbon chains are simply represented by gray united |
| 124 |
< |
atoms.} \label{lipidFigure:coarseGrained} |
| 120 |
> |
$\text{{\sc NC}}_4$ group (blue) and a united C atom (gray) with a |
| 121 |
> |
dipole, while the glycerol backbone includes dipolar $\text{{\sc |
| 122 |
> |
CE}}$ (red) and $\text{{\sc CK}}$ (pink) groups. Alkyl groups in |
| 123 |
> |
hydrocarbon chains are simply represented by gray united atoms.} |
| 124 |
> |
\label{lipidFigure:coarseGrained} |
| 125 |
|
\end{figure} |
| 126 |
|
|
| 127 |
|
Accurate and efficient computation of electrostatics is one of the |
| 166 |
|
$\frac{{d^2 }}{4}$ inside the square root of the denominator is |
| 167 |
|
neglected. This is a reasonable approximation in most cases. |
| 168 |
|
Unfortunately, in our headgroup model, the distance of charge |
| 169 |
< |
separation $d$ (4.63 $\AA$ in PC headgroup) may be comparable to |
| 170 |
< |
$r$. Nevertheless, we could still assume $ \cos \theta \approx 0$ |
| 171 |
< |
in the central region of the headgroup. Using Taylor expansion and |
| 169 |
> |
separation $d$ (4.63 \AA in PC headgroup) may be comparable to $r$. |
| 170 |
> |
Nevertheless, we could still assume $ \cos \theta \approx 0$ in |
| 171 |
> |
the central region of the headgroup. Using Taylor expansion and |
| 172 |
|
associating appropriate terms with electric moments will result in a |
| 173 |
|
"split-dipole" approximation: |
| 174 |
|
\[ |
| 238 |
|
\end{equation} |
| 239 |
|
|
| 240 |
|
From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first |
| 241 |
< |
solvation shell (3.5 $\AA$) and the second solvation shell (5.0 |
| 242 |
< |
$\AA$) from both plots. However, the corresponding orientations are |
| 241 |
> |
solvation shell (3.5 \AA) and the second solvation shell (5.0 \AA) |
| 242 |
> |
from both plots. However, the corresponding orientations are |
| 243 |
|
different. In DLPE, water molecules prefer to sit around $\text{{\sc |
| 244 |
|
NH}}_3$ group due to the hydrogen bonding. In contrast, because of |
| 245 |
|
the hydrophobic effect of the $ {\rm{N(CH}}_{\rm{3}} |
| 360 |
|
electron density is in the hydrocarbon region. As a good |
| 361 |
|
approximation to the thickness of the bilayer, the headgroup spacing |
| 362 |
|
, is defined as the distance between two peaks in the electron |
| 363 |
< |
density profile, calculated from our simulations to be 34.1 $\AA$. |
| 363 |
> |
density profile, calculated from our simulations to be 34.1 \AA. |
| 364 |
|
This value is close to the x-ray diffraction experimental value 34.4 |
| 365 |
< |
$\AA$\cite{Petrache1998}. |
| 365 |
> |
\AA\cite{Petrache1998}. |
| 366 |
|
|
| 367 |
|
\begin{figure} |
| 368 |
|
\centering |
