| 140 |
|
Pechukas.\cite{Berne72} The potential is constructed in the familiar |
| 141 |
|
form of the Lennard-Jones function using orientation-dependent |
| 142 |
|
$\sigma$ and $\epsilon$ parameters, |
| 143 |
< |
\begin{equation} |
| 144 |
< |
\begin{split} |
| 143 |
> |
\begin{multline} |
| 144 |
|
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 145 |
< |
r}_{ij}}) = & 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 145 |
> |
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 146 |
|
{\mathbf{\hat r}_{ij}})\left[ \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 147 |
< |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} \right.\\ |
| 148 |
< |
&\left. -\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 147 |
> |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
| 148 |
> |
\right. \\ |
| 149 |
> |
\left. - \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 150 |
|
{\mathbf{\hat u}_j}, {\mathbf{\hat |
| 151 |
|
r}_{ij}})+\sigma_0}\right)^6\right] |
| 152 |
– |
\end{split} |
| 152 |
|
\label{mdeq:gb} |
| 153 |
< |
\end{equation} |
| 153 |
> |
\end{multline} |
| 154 |
|
|
| 155 |
|
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 156 |
|
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
| 171 |
|
where $l$ and $d$ describe the length and width of each uniaxial |
| 172 |
|
ellipsoid. These shape anisotropy parameters can then be used to |
| 173 |
|
calculate the range function, |
| 174 |
< |
\begin{equation} |
| 175 |
< |
\begin{split} |
| 176 |
< |
& \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = |
| 178 |
< |
\sigma_{0} \times \\ |
| 179 |
< |
& \left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
| 174 |
> |
\begin{multline} |
| 175 |
> |
\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \\ |
| 176 |
> |
\sigma_0 \left[ 1 - \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
| 177 |
|
\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
| 178 |
|
\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
| 179 |
|
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
| 180 |
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
| 181 |
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
| 182 |
< |
\right]^{-1/2} |
| 183 |
< |
\end{split} |
| 187 |
< |
\end{equation} |
| 182 |
> |
\right]^{-1/2} |
| 183 |
> |
\end{multline} |
| 184 |
|
|
| 185 |
|
Gay-Berne ellipsoids also have an energy scaling parameter, |
| 186 |
|
$\epsilon^s$, which describes the well depth for two identical |
| 207 |
|
\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
| 208 |
|
\hat{u}}_{j})^{2}\right]^{-1/2} |
| 209 |
|
\end{eqnarray*} |
| 210 |
< |
\begin{equation*} |
| 211 |
< |
\begin{split} |
| 212 |
< |
& \epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) |
| 213 |
< |
= 1 - \\ |
| 218 |
< |
& \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 210 |
> |
\begin{multline*} |
| 211 |
> |
\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) |
| 212 |
> |
= \\ |
| 213 |
> |
1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 214 |
|
\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
| 215 |
|
\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 216 |
|
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
| 217 |
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
| 218 |
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
| 219 |
< |
\end{split} |
| 225 |
< |
\end{equation*} |
| 219 |
> |
\end{multline*} |
| 220 |
|
although many of the quantities and derivatives are identical with |
| 221 |
|
those obtained for the range parameter. Ref. \citen{Luckhurst90} |
| 222 |
|
has a particularly good explanation of the choice of the Gay-Berne |
| 459 |
|
|
| 460 |
|
It is reasonable to ask how well the parameters we used can produce |
| 461 |
|
bilayer properties that match experimentally known values for real |
| 462 |
< |
lipid bilayers. Using a value of $l = 13.8$ \AA for the ellipsoidal |
| 462 |
> |
lipid bilayers. Using a value of $l = 13.8$ \AA~for the ellipsoidal |
| 463 |
|
tails and the fixed ellipsoidal aspect ratio of 3, our values for the |
| 464 |
|
area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend |
| 465 |
|
entirely on the size of the head bead relative to the molecular body. |
| 520 |
|
different direction from the upper leaf.\label{mdfig:topView}} |
| 521 |
|
\end{figure} |
| 522 |
|
|
| 523 |
< |
The principal method for observing orientational ordering in dipolar |
| 524 |
< |
or liquid crystalline systems is the $P_2$ order parameter (defined |
| 525 |
< |
as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
| 532 |
< |
eigenvalue of the matrix, |
| 533 |
< |
\begin{equation} |
| 534 |
< |
{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
| 535 |
< |
\begin{array}{ccc} |
| 536 |
< |
u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
| 537 |
< |
u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
| 538 |
< |
u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
| 539 |
< |
\end{array} \right). |
| 540 |
< |
\label{mdeq:opmatrix} |
| 541 |
< |
\end{equation} |
| 542 |
< |
Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
| 543 |
< |
for molecule $i$. (Here, $\hat{\bf u}_i$ can refer either to the |
| 523 |
> |
The orientational ordering in the system is observed by $P_2$ order |
| 524 |
> |
parameter, which is calculated from Eq.~\ref{mceq:opmatrix} |
| 525 |
> |
in Ch.~\ref{chap:mc}. Here, $\hat{\bf u}_i$ can refer either to the |
| 526 |
|
principal axis of the molecular body or to the dipole on the head |
| 527 |
< |
group of the molecule.) $P_2$ will be $1.0$ for a perfectly-ordered |
| 528 |
< |
system and near $0$ for a randomized system. Note that this order |
| 529 |
< |
parameter is {\em not} equal to the polarization of the system. For |
| 530 |
< |
example, the polarization of a perfect anti-ferroelectric arrangement |
| 549 |
< |
of point dipoles is $0$, but $P_2$ for the same system is $1$. The |
| 550 |
< |
eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is |
| 551 |
< |
familiar as the director axis, which can be used to determine a |
| 552 |
< |
privileged axis for an orientationally-ordered system. Since the |
| 553 |
< |
molecular bodies are perpendicular to the head group dipoles, it is |
| 554 |
< |
possible for the director axes for the molecular bodies and the head |
| 555 |
< |
groups to be completely decoupled from each other. |
| 527 |
> |
group of the molecule. Since the molecular bodies are perpendicular to |
| 528 |
> |
the head group dipoles, it is possible for the director axes for the |
| 529 |
> |
molecular bodies and the head groups to be completely decoupled from |
| 530 |
> |
each other. |
| 531 |
|
|
| 532 |
|
Figure \ref{mdfig:topView} shows snapshots of bird's-eye views of the |
| 533 |
|
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$) |
| 696 |
|
We have computed translational diffusion constants for lipid molecules |
| 697 |
|
from the mean-square displacement, |
| 698 |
|
\begin{equation} |
| 699 |
< |
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
| 699 |
> |
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf |
| 700 |
> |
r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
| 701 |
> |
\label{mdeq:msdisplacement} |
| 702 |
|
\end{equation} |
| 703 |
|
of the lipid bodies. Translational diffusion constants for the |
| 704 |
|
different head-to-tail size ratios (all at 300 K) are shown in table |
| 839 |
|
orientations of the membrane dipoles may be available from |
| 840 |
|
fluorescence detected linear dichroism (LD). Benninger {\it et al.} |
| 841 |
|
have recently used axially-specific chromophores |
| 842 |
< |
2-(4,4-difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-pentanoyl)-1-hexadecanoyl-sn-glycero-3-phospocholine |
| 843 |
< |
($\beta$-BODIPY FL C5-HPC or BODIPY-PC) and 3,3' |
| 842 |
> |
2-(4,4-difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-pentanoyl)-1-hexadecanoyl-sn-glycero-3-\\ |
| 843 |
> |
phospocholine ($\beta$-BODIPY FL C5-HPC or BODIPY-PC) and 3,3' |
| 844 |
|
dioctadecyloxacarbocyanine perchlorate (DiO) in their |
| 845 |
|
fluorescence-detected linear dichroism (LD) studies of plasma |
| 846 |
|
membranes of living cells.\cite{Benninger:2005qy} The DiO dye aligns |
