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\chapter{\label{chap:md}Dipolar ordering in the ripple phases of |
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molecular-scale models of lipid membranes} |
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\chapter{\label{chap:md}DIPOLAR ORDERING IN THE RIPPLE PHASES OF |
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MOLECULAR-SCALE MODELS OF LIPID MEMBRANES} |
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\section{Introduction} |
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\label{mdsec:Int} |
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Fully hydrated lipids will aggregate spontaneously to form bilayers |
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which exhibit a variety of phases depending on their temperatures and |
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compositions. Among these phases, a periodic rippled phase |
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($P_{\beta'}$) appears as an intermediate phase between the gel |
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($L_\beta$) and fluid ($L_{\alpha}$) phases for relatively pure |
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phosphatidylcholine (PC) bilayers. The ripple phase has attracted |
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substantial experimental interest over the past 30 years. Most |
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structural information of the ripple phase has been obtained by the |
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X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
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microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
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et al.} used atomic force microscopy (AFM) to observe ripple phase |
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morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
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experimental results provide strong support for a 2-dimensional |
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hexagonal packing lattice of the lipid molecules within the ripple |
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phase. This is a notable change from the observed lipid packing |
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within the gel phase,~\cite{Cevc87} although Tenchov {\it et al.} have |
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recently observed near-hexagonal packing in some phosphatidylcholine |
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(PC) gel phases.\cite{Tenchov2001} The X-ray diffraction work by |
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Katsaras {\it et al.} showed that a rich phase diagram exhibiting both |
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{\it asymmetric} and {\it symmetric} ripples is possible for lecithin |
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bilayers.\cite{Katsaras00} |
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A number of theoretical models have been presented to explain the |
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formation of the ripple phase. Marder {\it et al.} used a |
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concave portions of the membrane correspond to more solid-like |
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regions. Carlson and Sethna used a packing-competition model (in |
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which head groups and chains have competing packing energetics) to |
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predict the formation of a ripple-like phase. Their model predicted |
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that the high-curvature portions have lower-chain packing and |
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correspond to more fluid-like regions. Goldstein and Leibler used a |
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mean-field approach with a planar model for {\em inter-lamellar} |
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interactions to predict rippling in multilamellar |
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predict the formation of a ripple-like phase~\cite{Carlson87}. Their |
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model predicted that the high-curvature portions have lower-chain |
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packing and correspond to more fluid-like regions. Goldstein and |
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Leibler used a mean-field approach with a planar model for {\em |
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inter-lamellar} interactions to predict rippling in multilamellar |
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phases.~\cite{Goldstein88} McCullough and Scott proposed that the {\em |
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anisotropy of the nearest-neighbor interactions} coupled to |
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hydrophobic constraining forces which restrict height differences |
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described the formation of symmetric ripple-like structures using a |
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coarse grained solvent-head-tail bead model.\cite{Kranenburg2005} |
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Their lipids consisted of a short chain of head beads tied to the two |
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longer ``chains''. |
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longer ``chains''. |
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In contrast, few large-scale molecular modeling studies have been |
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done due to the large size of the resulting structures and the time |
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driving force for ripple formation, questions about the ordering of |
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the head groups in ripple phase have not been settled. |
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In a recent paper, we presented a simple ``web of dipoles'' spin |
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In Ch.~\ref{chap:mc}, we presented a simple ``web of dipoles'' spin |
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lattice model which provides some physical insight into relationship |
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between dipolar ordering and membrane buckling.\cite{Sun2007} We found |
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that dipolar elastic membranes can spontaneously buckle, forming |
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between dipolar ordering and membrane buckling.\cite{sun:031602} We |
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found that dipolar elastic membranes can spontaneously buckle, forming |
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ripple-like topologies. The driving force for the buckling of dipolar |
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elastic membranes is the anti-ferroelectric ordering of the dipoles. |
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This was evident in the ordering of the dipole director axis |
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work on the spontaneous formation of dipolar peptide chains into |
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curved nano-structures.\cite{Tsonchev04,Tsonchev04II} |
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In this paper, we construct a somewhat more realistic molecular-scale |
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In this chapter, we construct a somewhat more realistic molecular-scale |
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lipid model than our previous ``web of dipoles'' and use molecular |
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dynamics simulations to elucidate the role of the head group dipoles |
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in the formation and morphology of the ripple phase. We describe our |
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modeling large length-scale properties of lipid |
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bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
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was a single site model for the interactions of rigid ellipsoidal |
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molecules.\cite{Gay81} It can be thought of as a modification of the |
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molecules.\cite{Gay1981} It can be thought of as a modification of the |
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Gaussian overlap model originally described by Berne and |
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Pechukas.\cite{Berne72} The potential is constructed in the familiar |
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form of the Lennard-Jones function using orientation-dependent |
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$\sigma$ and $\epsilon$ parameters, |
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\begin{equation*} |
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\begin{multline} |
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V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
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r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
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{\mathbf{\hat r}_{ij}})\left[ \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
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{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
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-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
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\right. \\ |
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\left. - \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
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{\mathbf{\hat u}_j}, {\mathbf{\hat |
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r}_{ij}})+\sigma_0}\right)^6\right] |
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\label{mdeq:gb} |
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\end{equation*} |
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\end{multline} |
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The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
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\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
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where $l$ and $d$ describe the length and width of each uniaxial |
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ellipsoid. These shape anisotropy parameters can then be used to |
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calculate the range function, |
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\begin{equation*} |
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\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0} |
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\left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
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\begin{multline} |
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\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \\ |
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\sigma_0 \left[ 1 - \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
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\right]^{-1/2} |
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\end{equation*} |
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\end{multline} |
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Gay-Berne ellipsoids also have an energy scaling parameter, |
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$\epsilon^s$, which describes the well depth for two identical |
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\alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1} |
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\end{eqnarray*} |
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The form of the strength function is somewhat complicated, |
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\begin {eqnarray*} |
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\begin{eqnarray*} |
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\epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = & |
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\epsilon_{0} \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j}) |
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\epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
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\hat{r}}_{ij}) \\ \\ |
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\epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = & |
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\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
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\hat{u}}_{j})^{2}\right]^{-1/2} \\ \\ |
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\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & |
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= & |
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1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{u}}_{j})^{2}\right]^{-1/2} |
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\end{eqnarray*} |
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\begin{multline*} |
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\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) |
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= \\ |
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1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
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\end {eqnarray*} |
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\end{multline*} |
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although many of the quantities and derivatives are identical with |
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those obtained for the range parameter. Ref. \citen{Luckhurst90} |
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has a particularly good explanation of the choice of the Gay-Berne |
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\end{figure} |
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To take into account the permanent dipolar interactions of the |
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zwitterionic head groups, we have placed fixed dipole moments $\mu_{i}$ at |
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one end of the Gay-Berne particles. The dipoles are oriented at an |
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angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
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are protected by a head ``bead'' with a range parameter ($\sigma_h$) which we have |
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varied between $1.20 d$ and $1.41 d$. The head groups interact with |
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each other using a combination of Lennard-Jones, |
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zwitterionic head groups, we have placed fixed dipole moments |
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$\mu_{i}$ at one end of the Gay-Berne particles. The dipoles are |
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oriented at an angle $\theta = \pi / 2$ relative to the major axis. |
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These dipoles are protected by a head ``bead'' with a range parameter |
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($\sigma_h$) which we have varied between $1.20 d$ and $1.41 d$. The |
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head groups interact with each other using a combination of |
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Lennard-Jones, |
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\begin{equation} |
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V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
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\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
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The solvent model in our simulations is similar to the one used by |
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Marrink {\it et al.} in their coarse grained simulations of lipid |
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bilayers.\cite{Marrink04} The solvent bead is a single site that |
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bilayers.\cite{Marrink2004} The solvent bead is a single site that |
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represents four water molecules (m = 72 amu) and has comparable |
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density and diffusive behavior to liquid water. However, since there |
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are no electrostatic sites on these beads, this solvent model cannot |
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replicate the dielectric properties of water. Note that although we |
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are using larger cutoff and switching radii than Marrink {\it et al.}, |
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our solvent density at 300 K remains at 0.944 g cm$^{-3}$, and the |
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solvent diffuses at 0.43 $\AA^2 ps^{-1}$ (only twice as fast as liquid |
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solvent diffuses at 0.43 \AA$^2 ps^{-1}$ (only twice as fast as liquid |
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water). |
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|
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\begin{table*} |
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molecular dynamics runs was 25 fs. No appreciable changes in phase |
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structure were noticed upon switching to a microcanonical ensemble. |
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All simulations were performed using the {\sc oopse} molecular |
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modeling program.\cite{Meineke05} |
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modeling program.\cite{Meineke2005} |
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|
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A switching function was applied to all potentials to smoothly turn |
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off the interactions between a range of $22$ and $25$ \AA. The |
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It is reasonable to ask how well the parameters we used can produce |
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bilayer properties that match experimentally known values for real |
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lipid bilayers. Using a value of $l = 13.8$ \AA for the ellipsoidal |
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lipid bilayers. Using a value of $l = 13.8$ \AA~for the ellipsoidal |
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tails and the fixed ellipsoidal aspect ratio of 3, our values for the |
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area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend |
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entirely on the size of the head bead relative to the molecular body. |
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different direction from the upper leaf.\label{mdfig:topView}} |
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\end{figure} |
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The principal method for observing orientational ordering in dipolar |
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or liquid crystalline systems is the $P_2$ order parameter (defined |
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as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
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eigenvalue of the matrix, |
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\begin{equation} |
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{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
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\begin{array}{ccc} |
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u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
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u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
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u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
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\end{array} \right). |
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\label{mdeq:opmatrix} |
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\end{equation} |
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Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
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for molecule $i$. (Here, $\hat{\bf u}_i$ can refer either to the |
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The orientational ordering in the system is observed by $P_2$ order |
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parameter, which is calculated from Eq.~\ref{mceq:opmatrix} |
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in Ch.~\ref{chap:mc}. Here, $\hat{\bf u}_i$ can refer either to the |
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principal axis of the molecular body or to the dipole on the head |
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group of the molecule.) $P_2$ will be $1.0$ for a perfectly-ordered |
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system and near $0$ for a randomized system. Note that this order |
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parameter is {\em not} equal to the polarization of the system. For |
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example, the polarization of a perfect anti-ferroelectric arrangement |
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of point dipoles is $0$, but $P_2$ for the same system is $1$. The |
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eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is |
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familiar as the director axis, which can be used to determine a |
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privileged axis for an orientationally-ordered system. Since the |
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molecular bodies are perpendicular to the head group dipoles, it is |
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possible for the director axes for the molecular bodies and the head |
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groups to be completely decoupled from each other. |
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group of the molecule. Since the molecular bodies are perpendicular to |
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the head group dipoles, it is possible for the director axes for the |
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molecular bodies and the head groups to be completely decoupled from |
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each other. |
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|
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Figure \ref{mdfig:topView} shows snapshots of bird's-eye views of the |
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flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$) |
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arrangement of the dipoles is always observed in a direction |
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perpendicular to the wave vector for the surface corrugation. This is |
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a similar finding to what we observed in our earlier work on the |
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elastic dipolar membranes.\cite{Sun2007} |
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elastic dipolar membranes.\cite{sun:031602} |
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|
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The $P_2$ order parameters (for both the molecular bodies and the head |
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group dipoles) have been calculated to quantify the ordering in these |
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We have computed translational diffusion constants for lipid molecules |
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from the mean-square displacement, |
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\begin{equation} |
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D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
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D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf |
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r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
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\label{mdeq:msdisplacement} |
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\end{equation} |
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of the lipid bodies. Translational diffusion constants for the |
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different head-to-tail size ratios (all at 300 K) are shown in table |
