| 733 |
|
strength of the head group dipole moment |
| 734 |
|
($\mu$).\label{fig:phaseDiagram}} |
| 735 |
|
\end{figure} |
| 736 |
– |
|
| 736 |
|
|
| 737 |
< |
We have also computed orientational diffusion constants for the head |
| 738 |
< |
groups from the relaxation of the second-order Legendre polynomial |
| 739 |
< |
correlation function, |
| 737 |
> |
We have computed translational diffusion constants for lipid molecules |
| 738 |
> |
from the mean-square displacement, |
| 739 |
> |
\begin{equation} |
| 740 |
> |
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
| 741 |
> |
\end{equation} |
| 742 |
> |
of the lipid bodies. Translational diffusion constants for the |
| 743 |
> |
different head-to-tail size ratios (all at 300 K) are shown in table |
| 744 |
> |
\ref{tab:relaxation}. We have also computed orientational diffusion |
| 745 |
> |
constants for the head groups from the relaxation of the second-order |
| 746 |
> |
Legendre polynomial correlation function, |
| 747 |
|
\begin{eqnarray} |
| 748 |
|
C_{\ell}(t) & = & \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 749 |
|
\mu}_{i}(0) \right) \rangle \\ \\ |
| 750 |
|
& \approx & e^{-\ell(\ell + 1) \theta t}, |
| 751 |
|
\end{eqnarray} |
| 752 |
< |
of the head group dipoles. In this last line, we have used a simple |
| 753 |
< |
``Debye''-like model for the relaxation of the correlation function, |
| 754 |
< |
specifically in the case when $\ell = 2$. The computed orientational |
| 755 |
< |
diffusion constants are given in table \ref{tab:relaxation}. The |
| 756 |
< |
notable feature we observe is that the orientational diffusion |
| 757 |
< |
constant for the head group exhibits an order of magnitude decrease |
| 758 |
< |
upon entering the rippled phase. Our orientational correlation times |
| 753 |
< |
are substantially in excess of those provided by... |
| 752 |
> |
of the head group dipoles. In this last line, we have assumed a |
| 753 |
> |
simple Debye-like model for the relaxation of the correlation |
| 754 |
> |
function, specifically in the case when $\ell = 2$. The computed |
| 755 |
> |
orientational diffusion constants are given in table |
| 756 |
> |
\ref{tab:relaxation}. We observe that the head group orientational diffusion |
| 757 |
> |
constant exhibits an order of magnitude decrease upon entering the |
| 758 |
> |
rippled phase. |
| 759 |
|
|
| 760 |
+ |
Sparrman and Westlund used $T_1$ and $T_2$ measurements in analyzing |
| 761 |
+ |
NMR lineshapes for gel, fluid, and ripple phases and obtained |
| 762 |
+ |
estimates of a correlation time for water translational diffusion |
| 763 |
+ |
($\tau_c$) of 20 ns.\cite{Sparrman2003} This correlation time |
| 764 |
+ |
corresponds to water bound to small regions of the lipid membrane. |
| 765 |
+ |
Sparrman and Westlund further assume that the lipids can explore only |
| 766 |
+ |
a single period of the ripple (essentially moving in an almost |
| 767 |
+ |
one-dimensional path to do so), and the correlation time can therefore |
| 768 |
+ |
be used to estimate a value for the translational diffusion constant |
| 769 |
+ |
of $2.25 \times 10^{-11} m^2 s^{-1}$. The translational diffusion |
| 770 |
+ |
constants we obtain are in reasonable agreement with this |
| 771 |
+ |
experimentally determined value. |
| 772 |
|
|
| 773 |
+ |
Our orientational correlation times are substantially in excess of |
| 774 |
+ |
those provided by. |
| 775 |
+ |
|
| 776 |
+ |
|
| 777 |
|
\begin{table*} |
| 778 |
|
\begin{minipage}{\linewidth} |
| 779 |
|
\begin{center} |
| 780 |
|
\caption{Rotational diffusion constants for the head groups |
| 781 |
< |
($\theta_h$) and molecular bodies ($\theta_b$) as a function of the |
| 782 |
< |
head-to-body width ratio. The orientational mobility of the head |
| 781 |
> |
($\theta_h$) and molecular bodies ($\theta_b$) as well as the |
| 782 |
> |
translational diffusion coefficients for the molecule as a function of |
| 783 |
> |
the head-to-body width ratio. The orientational mobility of the head |
| 784 |
|
groups experiences an {\it order of magnitude decrease} upon entering |
| 785 |
|
the rippled phase, which suggests that the rippling is tied to a |
| 786 |
|
freezing out of head group orientational freedom. Uncertainties in |
| 787 |
|
the last digit are indicated by the values in parentheses.} |
| 788 |
< |
\begin{tabular}{lcc} |
| 788 |
> |
\begin{tabular}{lccc} |
| 789 |
|
\hline |
| 790 |
< |
$\sigma_h / d$ & $\theta_h (\mu s^{-1})$ & $\theta_b (1/fs)$ \\ |
| 790 |
> |
$\sigma_h / d$ & $\theta_h (\mu s^{-1})$ & $\theta_b (1/fs)$ & $D ( |
| 791 |
> |
\times 10^{-11} m^2 s^{-1})$ \\ |
| 792 |
|
\hline |
| 793 |
< |
1.20 & $0.206(1) $ & $0.0175(5) $ \\ |
| 794 |
< |
1.28 & $0.179(2) $ & $0.055(2) $ \\ |
| 795 |
< |
1.35 & $0.025(1) $ & $0.195(3) $ \\ |
| 796 |
< |
1.41 & $0.023(1) $ & $0.024(3) $ \\ |
| 793 |
> |
1.20 & $0.206(1) $ & $0.0175(5) $ & $0.43(1)$ \\ |
| 794 |
> |
1.28 & $0.179(2) $ & $0.055(2) $ & $5.91(3)$ \\ |
| 795 |
> |
1.35 & $0.025(1) $ & $0.195(3) $ & $3.42(1)$ \\ |
| 796 |
> |
1.41 & $0.023(1) $ & $0.024(3) $ & $7.16(1)$ \\ |
| 797 |
|
\end{tabular} |
| 798 |
|
\label{tab:relaxation} |
| 799 |
|
\end{center} |
