| 1 |
< |
\chapter{\label{chap:md}Dipolar ordering in the ripple phases of |
| 2 |
< |
molecular-scale models of lipid membranes} |
| 1 |
> |
\chapter{\label{chap:md}DIPOLAR ORDERING IN THE RIPPLE PHASES OF |
| 2 |
> |
MOLECULAR-SCALE MODELS OF LIPID MEMBRANES} |
| 3 |
|
|
| 4 |
|
\section{Introduction} |
| 5 |
|
\label{mdsec:Int} |
| 6 |
– |
Fully hydrated lipids will aggregate spontaneously to form bilayers |
| 7 |
– |
which exhibit a variety of phases depending on their temperatures and |
| 8 |
– |
compositions. Among these phases, a periodic rippled phase |
| 9 |
– |
($P_{\beta'}$) appears as an intermediate phase between the gel |
| 10 |
– |
($L_\beta$) and fluid ($L_{\alpha}$) phases for relatively pure |
| 11 |
– |
phosphatidylcholine (PC) bilayers. The ripple phase has attracted |
| 12 |
– |
substantial experimental interest over the past 30 years. Most |
| 13 |
– |
structural information of the ripple phase has been obtained by the |
| 14 |
– |
X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
| 15 |
– |
microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
| 16 |
– |
et al.} used atomic force microscopy (AFM) to observe ripple phase |
| 17 |
– |
morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
| 18 |
– |
experimental results provide strong support for a 2-dimensional |
| 19 |
– |
hexagonal packing lattice of the lipid molecules within the ripple |
| 20 |
– |
phase. This is a notable change from the observed lipid packing |
| 21 |
– |
within the gel phase,~\cite{Cevc87} although Tenchov {\it et al.} have |
| 22 |
– |
recently observed near-hexagonal packing in some phosphatidylcholine |
| 23 |
– |
(PC) gel phases.\cite{Tenchov2001} The X-ray diffraction work by |
| 24 |
– |
Katsaras {\it et al.} showed that a rich phase diagram exhibiting both |
| 25 |
– |
{\it asymmetric} and {\it symmetric} ripples is possible for lecithin |
| 26 |
– |
bilayers.\cite{Katsaras00} |
| 6 |
|
|
| 7 |
|
A number of theoretical models have been presented to explain the |
| 8 |
|
formation of the ripple phase. Marder {\it et al.} used a |
| 12 |
|
concave portions of the membrane correspond to more solid-like |
| 13 |
|
regions. Carlson and Sethna used a packing-competition model (in |
| 14 |
|
which head groups and chains have competing packing energetics) to |
| 15 |
< |
predict the formation of a ripple-like phase. Their model predicted |
| 16 |
< |
that the high-curvature portions have lower-chain packing and |
| 17 |
< |
correspond to more fluid-like regions. Goldstein and Leibler used a |
| 18 |
< |
mean-field approach with a planar model for {\em inter-lamellar} |
| 19 |
< |
interactions to predict rippling in multilamellar |
| 15 |
> |
predict the formation of a ripple-like phase~\cite{Carlson87}. Their |
| 16 |
> |
model predicted that the high-curvature portions have lower-chain |
| 17 |
> |
packing and correspond to more fluid-like regions. Goldstein and |
| 18 |
> |
Leibler used a mean-field approach with a planar model for {\em |
| 19 |
> |
inter-lamellar} interactions to predict rippling in multilamellar |
| 20 |
|
phases.~\cite{Goldstein88} McCullough and Scott proposed that the {\em |
| 21 |
|
anisotropy of the nearest-neighbor interactions} coupled to |
| 22 |
|
hydrophobic constraining forces which restrict height differences |
| 39 |
|
described the formation of symmetric ripple-like structures using a |
| 40 |
|
coarse grained solvent-head-tail bead model.\cite{Kranenburg2005} |
| 41 |
|
Their lipids consisted of a short chain of head beads tied to the two |
| 42 |
< |
longer ``chains''. |
| 42 |
> |
longer ``chains''. |
| 43 |
|
|
| 44 |
|
In contrast, few large-scale molecular modeling studies have been |
| 45 |
|
done due to the large size of the resulting structures and the time |
| 72 |
|
driving force for ripple formation, questions about the ordering of |
| 73 |
|
the head groups in ripple phase have not been settled. |
| 74 |
|
|
| 75 |
< |
In a recent paper, we presented a simple ``web of dipoles'' spin |
| 75 |
> |
In Ch.~\ref{chap:mc}, we presented a simple ``web of dipoles'' spin |
| 76 |
|
lattice model which provides some physical insight into relationship |
| 77 |
< |
between dipolar ordering and membrane buckling.\cite{Sun2007} We found |
| 78 |
< |
that dipolar elastic membranes can spontaneously buckle, forming |
| 77 |
> |
between dipolar ordering and membrane buckling.\cite{sun:031602} We |
| 78 |
> |
found that dipolar elastic membranes can spontaneously buckle, forming |
| 79 |
|
ripple-like topologies. The driving force for the buckling of dipolar |
| 80 |
|
elastic membranes is the anti-ferroelectric ordering of the dipoles. |
| 81 |
|
This was evident in the ordering of the dipole director axis |
| 84 |
|
work on the spontaneous formation of dipolar peptide chains into |
| 85 |
|
curved nano-structures.\cite{Tsonchev04,Tsonchev04II} |
| 86 |
|
|
| 87 |
< |
In this paper, we construct a somewhat more realistic molecular-scale |
| 87 |
> |
In this chapter, we construct a somewhat more realistic molecular-scale |
| 88 |
|
lipid model than our previous ``web of dipoles'' and use molecular |
| 89 |
|
dynamics simulations to elucidate the role of the head group dipoles |
| 90 |
|
in the formation and morphology of the ripple phase. We describe our |
| 135 |
|
modeling large length-scale properties of lipid |
| 136 |
|
bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
| 137 |
|
was a single site model for the interactions of rigid ellipsoidal |
| 138 |
< |
molecules.\cite{Gay81} It can be thought of as a modification of the |
| 138 |
> |
molecules.\cite{Gay1981} It can be thought of as a modification of the |
| 139 |
|
Gaussian overlap model originally described by Berne and |
| 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*} |
| 143 |
> |
\begin{equation} |
| 144 |
> |
\begin{split} |
| 145 |
|
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 146 |
< |
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 147 |
< |
{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 148 |
< |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
| 149 |
< |
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 150 |
< |
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 146 |
> |
r}_{ij}}) = & 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 147 |
> |
{\mathbf{\hat r}_{ij}})\left[ \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 148 |
> |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} \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} |
| 153 |
|
\label{mdeq:gb} |
| 154 |
< |
\end{equation*} |
| 154 |
> |
\end{equation} |
| 155 |
|
|
| 156 |
|
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 157 |
|
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
| 172 |
|
where $l$ and $d$ describe the length and width of each uniaxial |
| 173 |
|
ellipsoid. These shape anisotropy parameters can then be used to |
| 174 |
|
calculate the range function, |
| 175 |
< |
\begin{equation*} |
| 176 |
< |
\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0} |
| 177 |
< |
\left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
| 175 |
> |
\begin{equation} |
| 176 |
> |
\begin{split} |
| 177 |
> |
& \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 |
| 180 |
|
\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
| 181 |
|
\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
| 182 |
|
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
| 183 |
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
| 184 |
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
| 185 |
< |
\right]^{-1/2} |
| 186 |
< |
\end{equation*} |
| 185 |
> |
\right]^{-1/2} |
| 186 |
> |
\end{split} |
| 187 |
> |
\end{equation} |
| 188 |
|
|
| 189 |
|
Gay-Berne ellipsoids also have an energy scaling parameter, |
| 190 |
|
$\epsilon^s$, which describes the well depth for two identical |
| 202 |
|
\alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1} |
| 203 |
|
\end{eqnarray*} |
| 204 |
|
The form of the strength function is somewhat complicated, |
| 205 |
< |
\begin {eqnarray*} |
| 205 |
> |
\begin{eqnarray*} |
| 206 |
|
\epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = & |
| 207 |
|
\epsilon_{0} \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j}) |
| 208 |
|
\epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 209 |
|
\hat{r}}_{ij}) \\ \\ |
| 210 |
|
\epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = & |
| 211 |
|
\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
| 212 |
< |
\hat{u}}_{j})^{2}\right]^{-1/2} \\ \\ |
| 213 |
< |
\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & |
| 214 |
< |
= & |
| 215 |
< |
1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 212 |
> |
\hat{u}}_{j})^{2}\right]^{-1/2} |
| 213 |
> |
\end{eqnarray*} |
| 214 |
> |
\begin{equation*} |
| 215 |
> |
\begin{split} |
| 216 |
> |
& \epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) |
| 217 |
> |
= 1 - \\ |
| 218 |
> |
& \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 219 |
|
\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
| 220 |
|
\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
| 221 |
|
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
| 222 |
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
| 223 |
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
| 224 |
< |
\end {eqnarray*} |
| 224 |
> |
\end{split} |
| 225 |
> |
\end{equation*} |
| 226 |
|
although many of the quantities and derivatives are identical with |
| 227 |
|
those obtained for the range parameter. Ref. \citen{Luckhurst90} |
| 228 |
|
has a particularly good explanation of the choice of the Gay-Berne |
| 259 |
|
\end{figure} |
| 260 |
|
|
| 261 |
|
To take into account the permanent dipolar interactions of the |
| 262 |
< |
zwitterionic head groups, we have placed fixed dipole moments $\mu_{i}$ at |
| 263 |
< |
one end of the Gay-Berne particles. The dipoles are oriented at an |
| 264 |
< |
angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
| 265 |
< |
are protected by a head ``bead'' with a range parameter ($\sigma_h$) which we have |
| 266 |
< |
varied between $1.20 d$ and $1.41 d$. The head groups interact with |
| 267 |
< |
each other using a combination of Lennard-Jones, |
| 262 |
> |
zwitterionic head groups, we have placed fixed dipole moments |
| 263 |
> |
$\mu_{i}$ at one end of the Gay-Berne particles. The dipoles are |
| 264 |
> |
oriented at an angle $\theta = \pi / 2$ relative to the major axis. |
| 265 |
> |
These dipoles are protected by a head ``bead'' with a range parameter |
| 266 |
> |
($\sigma_h$) which we have varied between $1.20 d$ and $1.41 d$. The |
| 267 |
> |
head groups interact with each other using a combination of |
| 268 |
> |
Lennard-Jones, |
| 269 |
|
\begin{equation} |
| 270 |
|
V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
| 271 |
|
\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
| 314 |
|
|
| 315 |
|
The solvent model in our simulations is similar to the one used by |
| 316 |
|
Marrink {\it et al.} in their coarse grained simulations of lipid |
| 317 |
< |
bilayers.\cite{Marrink04} The solvent bead is a single site that |
| 317 |
> |
bilayers.\cite{Marrink2004} The solvent bead is a single site that |
| 318 |
|
represents four water molecules (m = 72 amu) and has comparable |
| 319 |
|
density and diffusive behavior to liquid water. However, since there |
| 320 |
|
are no electrostatic sites on these beads, this solvent model cannot |
| 321 |
|
replicate the dielectric properties of water. Note that although we |
| 322 |
|
are using larger cutoff and switching radii than Marrink {\it et al.}, |
| 323 |
|
our solvent density at 300 K remains at 0.944 g cm$^{-3}$, and the |
| 324 |
< |
solvent diffuses at 0.43 $\AA^2 ps^{-1}$ (only twice as fast as liquid |
| 324 |
> |
solvent diffuses at 0.43 \AA$^2 ps^{-1}$ (only twice as fast as liquid |
| 325 |
|
water). |
| 326 |
|
|
| 327 |
|
\begin{table*} |
| 392 |
|
molecular dynamics runs was 25 fs. No appreciable changes in phase |
| 393 |
|
structure were noticed upon switching to a microcanonical ensemble. |
| 394 |
|
All simulations were performed using the {\sc oopse} molecular |
| 395 |
< |
modeling program.\cite{Meineke05} |
| 395 |
> |
modeling program.\cite{Meineke2005} |
| 396 |
|
|
| 397 |
|
A switching function was applied to all potentials to smoothly turn |
| 398 |
|
off the interactions between a range of $22$ and $25$ \AA. The |
| 582 |
|
arrangement of the dipoles is always observed in a direction |
| 583 |
|
perpendicular to the wave vector for the surface corrugation. This is |
| 584 |
|
a similar finding to what we observed in our earlier work on the |
| 585 |
< |
elastic dipolar membranes.\cite{Sun2007} |
| 585 |
> |
elastic dipolar membranes.\cite{sun:031602} |
| 586 |
|
|
| 587 |
|
The $P_2$ order parameters (for both the molecular bodies and the head |
| 588 |
|
group dipoles) have been calculated to quantify the ordering in these |
