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\title{Dipolar Ordering in Molecular-scale Models of the Ripple Phase |
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in Lipid Membranes} |
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\author{Xiuquan Sun and J. Daniel Gezelter \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame, \\ |
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Notre Dame, Indiana 46556} |
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|
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%\email[E-mail:]{gezelter@nd.edu} |
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|
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\date{\today} |
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|
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\maketitle |
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|
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\begin{abstract} |
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The ripple phase in phosphatidylcholine (PC) bilayers has never been |
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completely explained. |
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\end{abstract} |
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|
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%\maketitle |
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|
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\section{Introduction} |
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\label{sec: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} |
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|
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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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curvature-dependent Landau-de Gennes free-energy functional to predict |
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a rippled phase.~\cite{Marder84} This model and other related continuum |
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models predict higher fluidity in convex regions and that concave |
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portions of the membrane correspond to more solid-like regions. |
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Carlson and Sethna used a packing-competition model (in which head |
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groups and chains have competing packing energetics) to predict the |
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formation of a ripple-like phase. Their model predicted that the |
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high-curvature portions have lower-chain packing and correspond to |
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more fluid-like regions. Goldstein and Leibler used a mean-field |
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approach with a planar model for {\em inter-lamellar} interactions to |
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predict rippling in multilamellar phases.~\cite{Goldstein88} McCullough |
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and Scott proposed that the {\em anisotropy of the nearest-neighbor |
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interactions} coupled to hydrophobic constraining forces which |
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restrict height differences between nearest neighbors is the origin of |
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the ripple phase.~\cite{McCullough90} Lubensky and MacKintosh |
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introduced a Landau theory for tilt order and curvature of a single |
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membrane and concluded that {\em coupling of molecular tilt to membrane |
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curvature} is responsible for the production of |
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ripples.~\cite{Lubensky93} Misbah, Duplat and Houchmandzadeh proposed |
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that {\em inter-layer dipolar interactions} can lead to ripple |
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instabilities.~\cite{Misbah98} Heimburg presented a {\em coexistence |
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model} for ripple formation in which he postulates that fluid-phase |
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line defects cause sharp curvature between relatively flat gel-phase |
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regions.~\cite{Heimburg00} Kubica has suggested that a lattice model of |
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polar head groups could be valuable in trying to understand bilayer |
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phase formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations |
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of lamellar stacks of hexagonal lattices to show that large headgroups |
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and molecular tilt with respect to the membrane normal vector can |
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cause bulk rippling.~\cite{Bannerjee02} |
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|
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In contrast, few large-scale molecular modelling studies have been |
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done due to the large size of the resulting structures and the time |
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required for the phases of interest to develop. With all-atom (and |
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even unified-atom) simulations, only one period of the ripple can be |
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observed and only for timescales in the range of 10-100 ns. One of |
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the most interesting molecular simulations was carried out by De Vries |
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{\it et al.}~\cite{deVries05}. According to their simulation results, |
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the ripple consists of two domains, one resembling the gel bilayer, |
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while in the other, the two leaves of the bilayer are fully |
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interdigitated. The mechanism for the formation of the ripple phase |
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suggested by their work is a packing competition between the head |
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groups and the tails of the lipid molecules.\cite{Carlson87} Recently, |
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the ripple phase has also been studied by Lenz and Schmid using Monte |
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Carlo simulations.\cite{Lenz07} Their structures are similar to the De |
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Vries {\it et al.} structures except that the connection between the |
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two leaves of the bilayer is a narrow interdigitated line instead of |
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the fully interdigitated domain. The symmetric ripple phase was also |
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observed by Lenz {\it et al.}, and their work supports other claims |
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that the mismatch between the size of the head group and tail of the |
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lipid molecules is the driving force for the formation of the ripple |
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phase. Ayton and Voth have found significant undulations in |
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zero-surface-tension states of membranes simulated via dissipative |
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particle dynamics, but their results are consistent with purely |
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thermal undulations.~\cite{Ayton02} |
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|
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Although the organization of the tails of lipid molecules are |
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addressed by these molecular simulations and the packing competition |
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between headgroups and tails is strongly implicated as the primary |
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driving force for ripple formation, questions about the ordering of |
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the head groups in ripple phase has not been settled. |
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|
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In a recent paper, 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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ripple-like topologies. The driving force for the buckling in dipolar |
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elastic membranes the antiferroelectric ordering of the dipoles, and |
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this was evident in the ordering of the dipole director axis |
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perpendicular to the wave vector of the surface ripples. A similiar |
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phenomenon has also been observed by Tsonchev {\it et al.} in their |
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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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|
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In this paper, 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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model and computational methodology in section \ref{sec:method}. |
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Details on the simulations are presented in section |
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\ref{sec:experiment}, with results following in section |
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\ref{sec:results}. A final discussion of the role of dipolar heads in |
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the ripple formation can be found in section |
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\ref{sec:discussion}. |
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|
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\section{Computational Model} |
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\label{sec:method} |
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|
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=4in]{lipidModels} |
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\caption{Three different representations of DPPC lipid molecules, |
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including the chemical structure, an atomistic model, and the |
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head-body ellipsoidal coarse-grained model used in this |
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work.\label{fig:lipidModels}} |
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\end{figure} |
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|
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Our simple molecular-scale lipid model for studying the ripple phase |
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is based on two facts: one is that the most essential feature of lipid |
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molecules is their amphiphilic structure with polar head groups and |
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non-polar tails. Another fact is that the majority of lipid molecules |
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in the ripple phase are relatively rigid (i.e. gel-like) which makes |
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some fraction of the details of the chain dynamics negligible. Figure |
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\ref{fig:lipidModels} shows the molecular strucure of a DPPC |
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molecule, as well as atomistic and molecular-scale representations of |
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a DPPC molecule. The hydrophilic character of the head group is |
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largely due to the separation of charge between the nitrogen and |
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phosphate groups. The zwitterionic nature of the PC headgroups leads |
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to abnormally large dipole moments (as high as 20.6 D), and this |
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strongly polar head group interacts strongly with the solvating water |
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layers immediately surrounding the membrane. The hydrophobic tail |
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consists of fatty acid chains. In our molecular scale model, lipid |
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molecules have been reduced to these essential features; the fatty |
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acid chains are represented by an ellipsoid with a dipolar ball |
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perched on one end to represent the effects of the charge-separated |
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head group. In real PC lipids, the direction of the dipole is |
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nearly perpendicular to the tail, so we have fixed the direction of |
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the point dipole rigidly in this orientation. |
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|
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The ellipsoidal portions of the model interact via the Gay-Berne |
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potential which has seen widespread use in the liquid crystal |
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community. Ayton and Voth have also used Gay-Berne ellipsoids for |
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modelling 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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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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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 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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\label{eq:gb} |
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\end{equation*} |
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|
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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 $(\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
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are dependent on the relative orientations of the two molecules (${\bf |
211 |
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
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intermolecular separation (${\bf \hat{r}}_{ij}$). $\sigma$ and |
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$\sigma_0$ are also governed by shape mixing and anisotropy variables, |
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\begin {eqnarray*} |
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\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
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\chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 - |
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d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^2 + |
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d_j^2 \right)}\right]^{1/2} \\ \\ |
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\alpha^2 & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 + |
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d_i^2 \right)}{\left( l_j^2 - d_j^2 \right) \left(l_i^2 + |
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d_j^2 \right)}\right]^{1/2}, |
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\end{eqnarray*} |
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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 |
229 |
\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
230 |
\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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|
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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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ellipsoids in a {\it side-by-side} configuration. Additionaly, a well |
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depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes |
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the ratio between the well depths in the {\it end-to-end} and |
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side-by-side configurations. As in the range parameter, a set of |
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mixing and anisotropy variables can be used to describe the well |
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depths for dissimilar particles, |
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\begin {eqnarray*} |
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\epsilon_0 & = & \sqrt{\epsilon^s_i * \epsilon^s_j} \\ \\ |
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\epsilon^r & = & \sqrt{\epsilon^r_i * \epsilon^r_j} \\ \\ |
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\chi' & = & \frac{1 - (\epsilon^r)^{1/\mu}}{1 + (\epsilon^r)^{1/\mu}} |
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\\ \\ |
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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, |
253 |
\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 |
257 |
\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 |
260 |
\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{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
265 |
\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
266 |
\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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although many of the quantities and derivatives are identical with |
271 |
those obtained for the range parameter. Ref. \citen{Luckhurst90} |
272 |
has a particularly good explanation of the choice of the Gay-Berne |
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parameters $\mu$ and $\nu$ for modeling liquid crystal molecules. An |
274 |
excellent overview of the computational methods that can be used to |
275 |
efficiently compute forces and torques for this potential can be found |
276 |
in Ref. \citen{Golubkov06} |
277 |
|
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The choices of parameters we have used in this study correspond to a |
279 |
shape anisotropy of 3 for the chain portion of the molecule. In |
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principle, this could be varied to allow for modeling of longer or |
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shorter chain lipid molecules. For these prolate ellipsoids, we have: |
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\begin{equation} |
283 |
\begin{array}{rcl} |
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d & < & l \\ |
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\epsilon^{r} & < & 1 |
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\end{array} |
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\end{equation} |
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A sketch of the various structural elements of our molecular-scale |
289 |
lipid / solvent model is shown in figure \ref{fig:lipidModel}. The |
290 |
actual parameters used in our simulations are given in table |
291 |
\ref{tab:parameters}. |
292 |
|
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\begin{figure}[htb] |
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\centering |
295 |
\includegraphics[width=4in]{2lipidModel} |
296 |
\caption{The parameters defining the behavior of the lipid |
297 |
models. $l / d$ is the ratio of the head group to body diameter. |
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Molecular bodies had a fixed aspect ratio of 3.0. The solvent model |
299 |
was a simplified 4-water bead ($\sigma_w \approx d$) that has been |
300 |
used in other coarse-grained (DPD) simulations. The dipolar strength |
301 |
(and the temperature and pressure) were the only other parameters that |
302 |
were varied systematically.\label{fig:lipidModel}} |
303 |
\end{figure} |
304 |
|
305 |
To take into account the permanent dipolar interactions of the |
306 |
zwitterionic head groups, we place fixed dipole moments $\mu_{i}$ at |
307 |
one end of the Gay-Berne particles. The dipoles are oriented at an |
308 |
angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
309 |
are protected by a head ``bead'' with a range parameter which we have |
310 |
varied between $1.20 d$ and $1.41 d$. The head groups interact with |
311 |
each other using a combination of Lennard-Jones, |
312 |
\begin{equation} |
313 |
V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
314 |
\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
315 |
\end{equation} |
316 |
and dipole-dipole, |
317 |
\begin{equation} |
318 |
V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
319 |
\hat{r}}_{ij})) = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
320 |
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
321 |
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
322 |
\end{equation} |
323 |
potentials. |
324 |
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
325 |
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
326 |
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. |
327 |
|
328 |
For the interaction between nonequivalent uniaxial ellipsoids (in this |
329 |
case, between spheres and ellipsoids), the spheres are treated as |
330 |
ellipsoids with an aspect ratio of 1 ($d = l$) and with an well depth |
331 |
ratio ($\epsilon^r$) of 1 ($\epsilon^e = \epsilon^s$). The form of |
332 |
the Gay-Berne potential we are using was generalized by Cleaver {\it |
333 |
et al.} and is appropriate for dissimilar uniaxial |
334 |
ellipsoids.\cite{Cleaver96} |
335 |
|
336 |
The solvent model in our simulations is identical to one used by |
337 |
Marrink {\it et al.} in their dissipative particle dynamics (DPD) |
338 |
simulation of lipid bilayers.\cite{Marrink04} This solvent bead is a single |
339 |
site that represents four water molecules (m = 72 amu) and has |
340 |
comparable density and diffusive behavior to liquid water. However, |
341 |
since there are no electrostatic sites on these beads, this solvent |
342 |
model cannot replicate the dielectric properties of water. |
343 |
\begin{table*} |
344 |
\begin{minipage}{\linewidth} |
345 |
\begin{center} |
346 |
\caption{Potential parameters used for molecular-scale coarse-grained |
347 |
lipid simulations} |
348 |
\begin{tabular}{llccc} |
349 |
\hline |
350 |
& & Head & Chain & Solvent \\ |
351 |
\hline |
352 |
$d$ (\AA) & & varied & 4.6 & 4.7 \\ |
353 |
$l$ (\AA) & & $= d$ & 13.8 & 4.7 \\ |
354 |
$\epsilon^s$ (kcal/mol) & & 0.185 & 1.0 & 0.8 \\ |
355 |
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 \\ |
356 |
$m$ (amu) & & 196 & 760 & 72.06 \\ |
357 |
$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
358 |
\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
359 |
\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
360 |
\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
361 |
$\mu$ (Debye) & & varied & 0 & 0 \\ |
362 |
\end{tabular} |
363 |
\label{tab:parameters} |
364 |
\end{center} |
365 |
\end{minipage} |
366 |
\end{table*} |
367 |
|
368 |
A switching function has been applied to all potentials to smoothly |
369 |
turn off the interactions between a range of $22$ and $25$ \AA. |
370 |
|
371 |
The parameters that were systematically varied in this study were the |
372 |
size of the head group ($\sigma_h$), the strength of the dipole moment |
373 |
($\mu$), and the temperature of the system. Values for $\sigma_h$ |
374 |
ranged from 5.5 \AA\ to 6.5 \AA\ . If the width of the tails is |
375 |
taken to be the unit of length, these head groups correspond to a |
376 |
range from $1.2 d$ to $1.41 d$. Since the solvent beads are nearly |
377 |
identical in diameter to the tail ellipsoids, all distances that |
378 |
follow will be measured relative to this unit of distance. |
379 |
|
380 |
\section{Experimental Methodology} |
381 |
\label{sec:experiment} |
382 |
|
383 |
To create unbiased bilayers, all simulations were started from two |
384 |
perfectly flat monolayers separated by a 26 \AA\ gap between the |
385 |
molecular bodies of the upper and lower leaves. The separated |
386 |
monolayers were evolved in a vaccum with $x-y$ anisotropic pressure |
387 |
coupling. The length of $z$ axis of the simulations was fixed and a |
388 |
constant surface tension was applied to enable real fluctuations of |
389 |
the bilayer. Periodic boundary conditions were used, and $480-720$ |
390 |
lipid molecules were present in the simulations, depending on the size |
391 |
of the head beads. In all cases, the two monolayers spontaneously |
392 |
collapsed into bilayer structures within 100 ps. Following this |
393 |
collapse, all systems were equlibrated for $100$ ns at $300$ K. |
394 |
|
395 |
The resulting bilayer structures were then solvated at a ratio of $6$ |
396 |
solvent beads (24 water molecules) per lipid. These configurations |
397 |
were then equilibrated for another $30$ ns. All simulations utilizing |
398 |
the solvent were carried out at constant pressure ($P=1$ atm) with |
399 |
$3$D anisotropic coupling, and constant surface tension |
400 |
($\gamma=0.015$ UNIT). Given the absence of fast degrees of freedom in |
401 |
this model, a timestep of $50$ fs was utilized with excellent energy |
402 |
conservation. Data collection for structural properties of the |
403 |
bilayers was carried out during a final 5 ns run following the solvent |
404 |
equilibration. All simulations were performed using the OOPSE |
405 |
molecular modeling program.\cite{Meineke05} |
406 |
|
407 |
\section{Results} |
408 |
\label{sec:results} |
409 |
|
410 |
Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
411 |
more corrugated with increasing size of the head groups. The surface |
412 |
is nearly flat when $\sigma_h=1.20 d$. With $\sigma_h=1.28 d$, |
413 |
although the surface is still flat, the bilayer starts to splay |
414 |
inward; the upper leaf of the bilayer is connected to the lower leaf |
415 |
with an interdigitated line defect. Two periodicities with $100$ \AA\ |
416 |
wavelengths were observed in the simulation. This structure is very |
417 |
similiar to the structure observed by de Vries and Lenz {\it et |
418 |
al.}. The same basic structure is also observed when $\sigma_h=1.41 |
419 |
d$, but the wavelength of the surface corrugations depends sensitively |
420 |
on the size of the ``head'' beads. From the undulation spectrum, the |
421 |
corrugation is clearly non-thermal. |
422 |
\begin{figure}[htb] |
423 |
\centering |
424 |
\includegraphics[width=4in]{phaseCartoon} |
425 |
\caption{A sketch to discribe the structure of the phases observed in |
426 |
our simulations.\label{fig:phaseCartoon}} |
427 |
\end{figure} |
428 |
|
429 |
When $\sigma_h=1.35 d$, we observed another corrugated surface |
430 |
morphology. This structure is different from the asymmetric rippled |
431 |
surface; there is no interdigitation between the upper and lower |
432 |
leaves of the bilayer. Each leaf of the bilayer is broken into several |
433 |
hemicylinderical sections, and opposite leaves are fitted together |
434 |
much like roof tiles. Unlike the surface in which the upper |
435 |
hemicylinder is always interdigitated on the leading or trailing edge |
436 |
of lower hemicylinder, this ``symmetric'' ripple has no prefered |
437 |
direction. The corresponding structures are shown in Figure |
438 |
\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
439 |
different phases. The top panel in figure \ref{fig:phaseCartoon} is |
440 |
the flat phase, the middle panel shows the asymmetric ripple phase |
441 |
corresponding to $\sigma_h = 1.41 d$ and the lower panel shows the |
442 |
symmetric ripple phase observed when $\sigma_h=1.35 d$. In the |
443 |
symmetric ripple, the bilayer is continuous over the whole membrane, |
444 |
however, in asymmetric ripple phase, the bilayer domains are connected |
445 |
by thin interdigitated monolayers that share molecules between the |
446 |
upper and lower leaves. |
447 |
\begin{table*} |
448 |
\begin{minipage}{\linewidth} |
449 |
\begin{center} |
450 |
\caption{Phases, ripple wavelengths and amplitudes observed as a |
451 |
function of the ratio between the head beads and the diameters of the |
452 |
tails. All lengths are normalized to the diameter of the tail |
453 |
ellipsoids.} |
454 |
\begin{tabular}{lccc} |
455 |
\hline |
456 |
$\sigma_h / d$ & type of phase & $\lambda / d$ & $A / d$\\ |
457 |
\hline |
458 |
1.20 & flat & N/A & N/A \\ |
459 |
1.28 & asymmetric ripple or flat & 21.7 & N/A \\ |
460 |
1.35 & symmetric ripple & 17.2 & 2.2 \\ |
461 |
1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
462 |
\end{tabular} |
463 |
\label{tab:property} |
464 |
\end{center} |
465 |
\end{minipage} |
466 |
\end{table*} |
467 |
|
468 |
The membrane structures and the reduced wavelength $\lambda / d$, |
469 |
reduced amplitude $A / d$ of the ripples are summarized in Table |
470 |
\ref{tab:property}. The wavelength range is $15~21$ molecular bodies |
471 |
and the amplitude is $1.5$ molecular bodies for asymmetric ripple and |
472 |
$2.2$ for symmetric ripple. These values are consistent to the |
473 |
experimental results. Note, that given the lack of structural freedom |
474 |
in the tails of our model lipids, the amplitudes observed from these |
475 |
simulations are likely to underestimate of the true amplitudes. |
476 |
|
477 |
\begin{figure}[htb] |
478 |
\centering |
479 |
\includegraphics[width=4in]{topDown} |
480 |
\caption{Top views of the flat (upper), asymmetric ripple (middle), |
481 |
and symmetric ripple (lower) phases. Note that the head-group dipoles |
482 |
have formed head-to-tail chains in all three of these phases, but in |
483 |
the two rippled phases, the dipolar chains are all aligned |
484 |
{\it perpendicular} to the direction of the ripple. The flat membrane |
485 |
has multiple point defects in the dipolar orientational ordering, and |
486 |
the dipolar ordering on the lower leaf of the bilayer can be in a |
487 |
different direction from the upper leaf.\label{fig:topView}} |
488 |
\end{figure} |
489 |
|
490 |
The principal method for observing orientational ordering in dipolar |
491 |
or liquid crystalline systems is the $P_2$ order parameter (defined |
492 |
as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
493 |
eigenvalue of the matrix, |
494 |
\begin{equation} |
495 |
{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
496 |
\begin{array}{ccc} |
497 |
u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
498 |
u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
499 |
u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
500 |
\end{array} \right). |
501 |
\label{eq:opmatrix} |
502 |
\end{equation} |
503 |
Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
504 |
for molecule $i$. (Here, $\hat{\bf u}_i$ can refer either to the |
505 |
principal axis of the molecular body or to the dipole on the head |
506 |
group of the molecule.) $P_2$ will be $1.0$ for a perfectly-ordered |
507 |
system and near $0$ for a randomized system. Note that this order |
508 |
parameter is {\em not} equal to the polarization of the system. For |
509 |
example, the polarization of a perfect anti-ferroelectric arrangement |
510 |
of point dipoles is $0$, but $P_2$ for the same system is $1$. The |
511 |
eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is |
512 |
familiar as the director axis, which can be used to determine a |
513 |
privileged axis for an orientationally-ordered system. Since the |
514 |
molecular bodies are perpendicular to the head group dipoles, it is |
515 |
possible for the director axes for the molecular bodies and the head |
516 |
groups to be completely decoupled from each other. |
517 |
|
518 |
Figure \ref{fig:topView} shows snapshots of bird's-eye views of the |
519 |
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.41, 1.35 d$) |
520 |
bilayers. The directions of the dipoles on the head groups are |
521 |
represented with two colored half spheres: blue (phosphate) and yellow |
522 |
(amino). For flat bilayers, the system exhibits signs of |
523 |
orientational frustration; some disorder in the dipolar head-to-tail |
524 |
chains is evident with kinks visible at the edges between differently |
525 |
ordered domains. The lipids can also move independently of lipids in |
526 |
the opposing leaf, so the ordering of the dipoles on one leaf is not |
527 |
necessarily consistent with the ordering on the other. These two |
528 |
factors keep the total dipolar order parameter relatively low for the |
529 |
flat phases. |
530 |
|
531 |
With increasing head group size, the surface becomes corrugated, and |
532 |
the dipoles cannot move as freely on the surface. Therefore, the |
533 |
translational freedom of lipids in one layer is dependent upon the |
534 |
position of the lipids in the other layer. As a result, the ordering of |
535 |
the dipoles on head groups in one leaf is correlated with the ordering |
536 |
in the other leaf. Furthermore, as the membrane deforms due to the |
537 |
corrugation, the symmetry of the allowed dipolar ordering on each leaf |
538 |
is broken. The dipoles then self-assemble in a head-to-tail |
539 |
configuration, and the dipolar order parameter increases dramatically. |
540 |
However, the total polarization of the system is still close to zero. |
541 |
This is strong evidence that the corrugated structure is an |
542 |
antiferroelectric state. It is also notable that the head-to-tail |
543 |
arrangement of the dipoles is always observed in a direction |
544 |
perpendicular to the wave vector for the surface corrugation. This is |
545 |
a similar finding to what we observed in our earlier work on the |
546 |
elastic dipolar membranes.\cite{Sun2007} |
547 |
|
548 |
The $P_2$ order parameters (for both the molecular bodies and the head |
549 |
group dipoles) have been calculated to quantify the ordering in these |
550 |
phases. Figure \ref{fig:rP2} shows that the $P_2$ order parameter for |
551 |
the head-group dipoles increases with increasing head group size. When |
552 |
the heads of the lipid molecules are small, the membrane is nearly |
553 |
flat. Since the in-plane packing is essentially a close packing of the |
554 |
head groups, the head dipoles exhibit frustration in their |
555 |
orientational ordering. |
556 |
|
557 |
The ordering trends for the tails are essentially opposite to the |
558 |
ordering of the head group dipoles. The tail $P_2$ order parameter |
559 |
{\it decreases} with increasing head size. This indicates that the |
560 |
surface is more curved with larger head / tail size ratios. When the |
561 |
surface is flat, all tails are pointing in the same direction (normal |
562 |
to the bilayer surface). This simplified model appears to be |
563 |
exhibiting a smectic A fluid phase, similar to the real $L_{\beta}$ |
564 |
phase. We have not observed a smectic C gel phase ($L_{\beta'}$) for |
565 |
this model system. Increasing the size of the heads results in |
566 |
rapidly decreasing $P_2$ ordering for the molecular bodies. |
567 |
|
568 |
\begin{figure}[htb] |
569 |
\centering |
570 |
\includegraphics[width=\linewidth]{rP2} |
571 |
\caption{The $P_2$ order parameters for head groups (circles) and |
572 |
molecular bodies (squares) as a function of the ratio of head group |
573 |
size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{fig:rP2}} |
574 |
\end{figure} |
575 |
|
576 |
In addition to varying the size of the head groups, we studied the |
577 |
effects of the interactions between head groups on the structure of |
578 |
lipid bilayer by changing the strength of the dipoles. Figure |
579 |
\ref{fig:sP2} shows how the $P_2$ order parameter changes with |
580 |
increasing strength of the dipole. Generally, the dipoles on the head |
581 |
groups become more ordered as the strength of the interaction between |
582 |
heads is increased and become more disordered by decreasing the |
583 |
interaction stength. When the interaction between the heads becomes |
584 |
too weak, the bilayer structure does not persist; all lipid molecules |
585 |
become dispersed in the solvent (which is non-polar in this |
586 |
molecular-scale model). The critial value of the strength of the |
587 |
dipole depends on the size of the head groups. The perfectly flat |
588 |
surface becomes unstable below $5$ Debye, while the rippled |
589 |
surfaces melt at $8$ Debye (asymmetric) or $10$ Debye (symmetric). |
590 |
|
591 |
The ordering of the tails mirrors the ordering of the dipoles {\it |
592 |
except for the flat phase}. Since the surface is nearly flat in this |
593 |
phase, the order parameters are only weakly dependent on dipolar |
594 |
strength until it reaches $15$ Debye. Once it reaches this value, the |
595 |
head group interactions are strong enough to pull the head groups |
596 |
close to each other and distort the bilayer structure. For a flat |
597 |
surface, a substantial amount of free volume between the head groups |
598 |
is normally available. When the head groups are brought closer by |
599 |
dipolar interactions, the tails are forced to splay outward, forming |
600 |
first curved bilayers, and then inverted micelles. |
601 |
|
602 |
When $\sigma_h=1.28 d$, the $P_2$ order parameter decreases slightly |
603 |
when the strength of the dipole is increased above $16$ debye. For |
604 |
rippled bilayers, there is less free volume available between the head |
605 |
groups. Therefore increasing dipolar strength weakly influences the |
606 |
structure of the membrane. However, the increase in the body $P_2$ |
607 |
order parameters implies that the membranes are being slightly |
608 |
flattened due to the effects of increasing head-group attraction. |
609 |
|
610 |
A very interesting behavior takes place when the head groups are very |
611 |
large relative to the molecular bodies ($\sigma_h = 1.41 d$) and the |
612 |
dipolar strength is relatively weak ($\mu < 9$ Debye). In this case, |
613 |
the two leaves of the bilayer become totally interdigitated with each |
614 |
other in large patches of the membrane. With higher dipolar |
615 |
strength, the interdigitation is limited to single lines that run |
616 |
through the bilayer in a direction perpendicular to the ripple wave |
617 |
vector. |
618 |
|
619 |
\begin{figure}[htb] |
620 |
\centering |
621 |
\includegraphics[width=\linewidth]{sP2} |
622 |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
623 |
molecular bodies (b) as a function of the strength of the dipoles. |
624 |
These order parameters are shown for four values of the head group / |
625 |
molecular width ratio ($\sigma_h / d$). \label{fig:sP2}} |
626 |
\end{figure} |
627 |
|
628 |
Figure \ref{fig:tP2} shows the dependence of the order parameters on |
629 |
temperature. As expected, systems are more ordered at low |
630 |
temperatures, and more disordered at high temperatures. All of the |
631 |
bilayers we studied can become unstable if the temperature becomes |
632 |
high enough. The only interesting feature of the temperature |
633 |
dependence is in the flat surfaces ($\sigma_h=1.20 d$ and |
634 |
$\sigma_h=1.28 d$). Here, when the temperature is increased above |
635 |
$310$K, there is enough jostling of the head groups to allow the |
636 |
dipolar frustration to resolve into more ordered states. This results |
637 |
in a slight increase in the $P_2$ order parameter above this |
638 |
temperature. |
639 |
|
640 |
For the rippled surfaces ($\sigma_h=1.35 d$ and $\sigma_h=1.41 d$), |
641 |
there is a slightly increased orientational ordering in the molecular |
642 |
bodies above $290$K. Since our model lacks the detailed information |
643 |
about the behavior of the lipid tails, this is the closest the model |
644 |
can come to depicting the ripple ($P_{\beta'}$) to fluid |
645 |
($L_{\alpha}$) phase transition. What we are observing is a |
646 |
flattening of the rippled structures made possible by thermal |
647 |
expansion of the tightly-packed head groups. The lack of detailed |
648 |
chain configurations also makes it impossible for this model to depict |
649 |
the ripple to gel ($L_{\beta'}$) phase transition. |
650 |
|
651 |
\begin{figure}[htb] |
652 |
\centering |
653 |
\includegraphics[width=\linewidth]{tP2} |
654 |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
655 |
molecular bodies (b) as a function of temperature. |
656 |
These order parameters are shown for four values of the head group / |
657 |
molecular width ratio ($\sigma_h / d$).\label{fig:tP2}} |
658 |
\end{figure} |
659 |
|
660 |
\section{Discussion} |
661 |
\label{sec:discussion} |
662 |
|
663 |
The ripple phases have been observed in our molecular dynamic |
664 |
simulations using a simple molecular lipid model. The lipid model |
665 |
consists of an anisotropic interacting dipolar head group and an |
666 |
ellipsoid shape tail. According to our simulations, the explanation of |
667 |
the formation for the ripples are originated in the size mismatch |
668 |
between the head groups and the tails. The ripple phases are only |
669 |
observed in the studies using larger head group lipid models. However, |
670 |
there is a mismatch betweent the size of the head groups and the size |
671 |
of the tails in the simulations of the flat surface. This indicates |
672 |
the competition between the anisotropic dipolar interaction and the |
673 |
packing of the tails also plays a major role for formation of the |
674 |
ripple phase. The larger head groups provide more free volume for the |
675 |
tails, while these hydrophobic ellipsoids trying to be close to each |
676 |
other, this gives the origin of the spontanous curvature of the |
677 |
surface, which is believed as the beginning of the ripple phases. The |
678 |
lager head groups cause the spontanous curvature inward for both of |
679 |
leaves of the bilayer. This results in a steric strain when the tails |
680 |
of two leaves too close to each other. The membrane has to be broken |
681 |
to release this strain. There are two ways to arrange these broken |
682 |
curvatures: symmetric and asymmetric ripples. Both of the ripple |
683 |
phases have been observed in our studies. The difference between these |
684 |
two ripples is that the bilayer is continuum in the symmetric ripple |
685 |
phase and is disrupt in the asymmetric ripple phase. |
686 |
|
687 |
Dipolar head groups are the key elements for the maintaining of the |
688 |
bilayer structure. The lipids are solvated in water when lowering the |
689 |
the strength of the dipole on the head groups. The long range |
690 |
orientational ordering of the dipoles can be achieved by forming the |
691 |
ripples, although the dipoles are likely to form head-to-tail |
692 |
configurations even in flat surface, the frustration prevents the |
693 |
formation of the long range orientational ordering for dipoles. The |
694 |
corrugation of the surface breaks the frustration and stablizes the |
695 |
long range oreintational ordering for the dipoles in the head groups |
696 |
of the lipid molecules. Many rows of the head-to-tail dipoles are |
697 |
parallel to each other and adopt the antiferroelectric state as a |
698 |
whole. This is the first time the organization of the head groups in |
699 |
ripple phases of the lipid bilayer has been addressed. |
700 |
|
701 |
The most important prediction we can make using the results from this |
702 |
simple model is that if dipolar ordering is driving the surface |
703 |
corrugation, the wave vectors for the ripples should always found to |
704 |
be {\it perpendicular} to the dipole director axis. This prediction |
705 |
should suggest experimental designs which test whether this is really |
706 |
true in the phosphatidylcholine $P_{\beta'}$ phases. The dipole |
707 |
director axis should also be easily computable for the all-atom and |
708 |
coarse-grained simulations that have been published in the literature. |
709 |
|
710 |
Although our model is simple, it exhibits some rich and unexpected |
711 |
behaviors. It would clearly be a closer approximation to the reality |
712 |
if we allowed greater translational freedom to the dipoles and |
713 |
replaced the somewhat artificial lattice packing and the harmonic |
714 |
elastic tension with more realistic molecular modeling potentials. |
715 |
What we have done is to present a simple model which exhibits bulk |
716 |
non-thermal corrugation, and our explanation of this rippling |
717 |
phenomenon will help us design more accurate molecular models for |
718 |
corrugated membranes and experiments to test whether rippling is |
719 |
dipole-driven or not. |
720 |
|
721 |
\newpage |
722 |
\bibliography{mdripple} |
723 |
\end{document} |