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\title{Dipolar Ordering of the Ripple Phase in Lipid Membranes} |
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\author{Xiuquan Sun and J. Daniel Gezelter} |
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\email[E-mail:]{gezelter@nd.edu} |
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\affiliation{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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\date{\today} |
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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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\pacs{} |
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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 the XXX group 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 molecules into curved |
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nano-structures.\cite{Tsonchev04} |
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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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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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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{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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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. In its original form, the Gay-Berne potential was a single |
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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{eqnarray*} |
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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{eqnarray*} |
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|
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|
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|
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The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
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\hat{r}}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}}))$ parameters |
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are dependent on the relative orientations of the two molecules (${\bf |
192 |
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
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intermolecular separation (${\bf \hat{r}}$). The functional forms for |
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$\sigma({\bf |
195 |
\hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}})$ and $\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}}))$ are given elsewhere |
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and will not be repeated here. However, $\epsilon$ and $\sigma$ are |
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governed by two anisotropy parameters, |
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\begin {equation} |
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\begin{array}{rcl} |
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\chi & = & \frac |
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{(\sigma_{e}/\sigma_{s})^2-1}{(\sigma_{e}/\sigma_{s})^2+1} \\ \\ |
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\chi\prime & = & \frac {1-(\epsilon_{e}/\epsilon_{s})^{1/\mu}}{1+(\epsilon_{e}/ |
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\epsilon_{s})^{1/\mu}} |
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\end{array} |
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\end{equation} |
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In these equations, $\sigma$ and $\epsilon$ refer to the point of |
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closest contact and the depth of the well in different orientations of |
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the two molecules. The subscript $s$ refers to the {\it side-by-side} |
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configuration where $\sigma$ has it's smallest value, |
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$\sigma_{s}$, and where the potential well is $\epsilon_{s}$ deep. |
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The subscript $e$ refers to the {\it end-to-end} configuration where |
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$\sigma$ is at it's largest value, $\sigma_{e}$ and where the well |
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depth, $\epsilon_{e}$ is somewhat smaller than in the side-by-side |
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configuration. For the prolate ellipsoids we are using, we have |
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\begin{equation} |
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\begin{array}{rcl} |
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\sigma_{s} & < & \sigma_{e} \\ |
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\epsilon_{s} & > & \epsilon_{e} |
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\end{array} |
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\end{equation} |
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Ref. \onlinecite{Luckhurst90} has a particularly good explanation of the |
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choice of the Gay-Berne parameters $\mu$ and $\nu$ for modeling liquid |
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crystal molecules. |
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|
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The breadth and length of tail are $\sigma_0$, $3\sigma_0$, |
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corresponding to a shape anisotropy of 3 for the chain portion of the |
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molecule. In principle, this could be varied to allow for modeling of |
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longer or shorter chain lipid molecules. |
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|
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To take into account the permanent dipolar interactions of the |
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zwitterionic head groups, we place fixed dipole moments $\mu_{i}$ at |
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one end of the Gay-Berne particles. The dipoles will be oriented at |
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an 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 which we have |
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varied between $1.20\sigma_0$ and $1.41\sigma_0$. The head groups |
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interact with each other using a combination of Lennard-Jones, |
239 |
\begin{eqnarray*} |
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V_{ij} = 4\epsilon \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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\end{eqnarray*} |
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and dipole, |
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\begin{eqnarray*} |
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V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
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\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
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\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
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\end{eqnarray*} |
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potentials. |
250 |
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
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along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
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pointing along the inter-dipole vector $\mathbf{r}_{ij}$. |
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|
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For the interaction between nonequivalent uniaxial ellipsoids (in this |
255 |
case, between spheres and ellipsoids), the range parameter is |
256 |
generalized as\cite{Cleaver96} |
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\begin{eqnarray*} |
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\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
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{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{(\alpha{\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_i} + \alpha^{-1}{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
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u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
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\frac{(\alpha{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - \alpha^{-1}{\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
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{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
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\end{eqnarray*} |
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where $\alpha$ is given by |
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\begin{eqnarray*} |
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\alpha^2 = |
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\left[\frac{\left({\sigma_e}_i^2-{\sigma_s}_i^2\right)\left({\sigma_e}_j^2+{\sigma_s}_i^2\right)}{\left({\sigma_e}_j^2-{\sigma_s}_j^2\right)\left({\sigma_e}_i^2+{\sigma_s}_j^2\right)} |
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\right]^{\frac{1}{2}} |
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\end{eqnarray*} |
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the strength parameter has been adjusted as suggested by Cleaver {\it |
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et al.}\cite{Cleaver96} A switching function has been applied to all |
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potentials to smoothly turn off the interactions between a range of $22$ and $25$ \AA. |
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|
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The solvent model in our simulations is identical to one used by XXX |
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in their dissipative particle dynamics (DPD) simulation of lipid |
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bilayers.]cite{XXX} This solvent bead is a single site that represents |
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four water molecules (m = 72 amu) and has comparable density and |
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diffusive behavior to liquid water. However, since there are no |
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electrostatic sites on these beads, this solvent model cannot |
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replicate the dielectric properties of water. |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{} |
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\begin{tabular}{lccc} |
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\hline |
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N/A & Head & Chain & Solvent \\ |
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\hline |
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$\sigma_0$ (\AA) & varied & 4.6 & 4.7 \\ |
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l (aspect ratio) & N/A & 3 & N/A \\ |
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$\epsilon_0$ (kcal/mol) & 0.185 & 1.0 & 0.8 \\ |
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$\epsilon_e$ (aspect ratio) & N/A & 0.2 & N/A \\ |
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M (amu) & 196 & 760 & 72.06112 \\ |
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$I_{xx}$, $I_{yy}$, $I_{zz}$ (amu \AA$^2$) & 1125, 1125, 0 & 45000, 45000, 9000 & N/A \\ |
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$\mu$ (Debye) & varied & N/A & N/A \\ |
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\end{tabular} |
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\label{tab:parameters} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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|
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\begin{figure}[htb] |
305 |
\centering |
306 |
\includegraphics[width=\linewidth]{2lipidModel} |
307 |
\caption{The parameters defining the behavior of the lipid |
308 |
models. $\sigma_h / \sigma_0$ is the ratio of the head group to body |
309 |
diameter. Molecular bodies had a fixed aspect ratio of 3.0. The |
310 |
solvent model was a simplified 4-water bead ($\sigma_w = 1.02 |
311 |
\sigma_0$) that has been used in other coarse-grained (DPD) simulations. |
312 |
The dipolar strength (and the temperature and pressure) were the only |
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other parameters that were varied |
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systematically.\label{fig:lipidModel}} |
315 |
\end{figure} |
316 |
|
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\section{Experimental Methodology} |
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\label{sec:experiment} |
319 |
|
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To create unbiased bilayers, all simulations were started from two |
321 |
perfectly flat monolayers separated by a 20 \AA\ gap between the |
322 |
molecular bodies of the upper and lower leaves. The separated |
323 |
monolayers were evolved in a vaccum with $x-y$ anisotropic pressure |
324 |
coupling. The length of $z$ axis of the simulations was fixed and a |
325 |
constant surface tension was applied to enable real fluctuations of |
326 |
the bilayer. Periodic boundaries were used, and $480-720$ lipid |
327 |
molecules were present in the simulations depending on the size of the |
328 |
head beads. The two monolayers spontaneously collapse into bilayer |
329 |
structures within 100 ps, and following this collapse, all systems |
330 |
were equlibrated for $100$ ns at $300$ K. |
331 |
|
332 |
The resulting structures were then solvated at a ratio of $6$ DPD |
333 |
solvent beads (24 water molecules) per lipid. These configurations |
334 |
were then equilibrated for another $30$ ns. All simulations with |
335 |
solvent were carried out at constant pressure ($P=1$ atm) by $3$D |
336 |
anisotropic coupling, and constant surface tension ($\gamma=0.015$ |
337 |
UNIT). Given the absence of fast degrees of freedom in this model, a |
338 |
timestep of $50$ fs was utilized. Data collection for structural |
339 |
properties of the bilayers was carried out during a final 5 ns run |
340 |
following the solvent equilibration. All simulations were performed |
341 |
using the OOPSE molecular modeling program.\cite{Meineke05} |
342 |
|
343 |
\section{Results} |
344 |
\label{sec:results} |
345 |
|
346 |
Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
347 |
more corrugated increasing size of the head groups. The surface is |
348 |
nearly flat when $\sigma_h=1.20\sigma_0$. With |
349 |
$\sigma_h=1.28\sigma_0$, although the surface is still flat, the |
350 |
bilayer starts to splay inward; the upper leaf of the bilayer is |
351 |
connected to the lower leaf with an interdigitated line defect. Two |
352 |
periodicities with $100$ \AA\ width were observed in the |
353 |
simulation. This structure is very similiar to the structure observed |
354 |
by de Vries and Lenz {\it et al.}. The same basic structure is also |
355 |
observed when $\sigma_h=1.41\sigma_0$, but the wavelength of the |
356 |
surface corrugations depends sensitively on the size of the ``head'' |
357 |
beads. From the undulation spectrum, the corrugation is clearly |
358 |
non-thermal. |
359 |
\begin{figure}[htb] |
360 |
\centering |
361 |
\includegraphics[width=\linewidth]{phaseCartoon} |
362 |
\caption{A sketch to discribe the structure of the phases observed in |
363 |
our simulations.\label{fig:phaseCartoon}} |
364 |
\end{figure} |
365 |
|
366 |
When $\sigma_h=1.35\sigma_0$, we observed another corrugated surface |
367 |
morphology. This structure is different from the asymmetric rippled |
368 |
surface; there is no interdigitation between the upper and lower |
369 |
leaves of the bilayer. Each leaf of the bilayer is broken into several |
370 |
hemicylinderical sections, and opposite leaves are fitted together |
371 |
much like roof tiles. Unlike the surface in which the upper |
372 |
hemicylinder is always interdigitated on the leading or trailing edge |
373 |
of lower hemicylinder, the symmetric ripple has no prefered direction. |
374 |
The corresponding cartoons are shown in Figure |
375 |
\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
376 |
different phases. Figure \ref{fig:phaseCartoon} (a) is the flat phase, |
377 |
(b) is the asymmetric ripple phase corresponding to the lipid |
378 |
organization when $\sigma_h=1.28\sigma_0$ and $\sigma_h=1.41\sigma_0$, |
379 |
and (c) is the symmetric ripple phase observed when |
380 |
$\sigma_h=1.35\sigma_0$. In symmetric ripple phase, the bilayer is |
381 |
continuous everywhere on the whole membrane, however, in asymmetric |
382 |
ripple phase, the bilayer is intermittent domains connected by thin |
383 |
interdigitated monolayer which consists of upper and lower leaves of |
384 |
the bilayer. |
385 |
\begin{table*} |
386 |
\begin{minipage}{\linewidth} |
387 |
\begin{center} |
388 |
\caption{} |
389 |
\begin{tabular}{lccc} |
390 |
\hline |
391 |
$\sigma_h/\sigma_0$ & type of phase & $\lambda/\sigma_0$ & $A/\sigma_0$\\ |
392 |
\hline |
393 |
1.20 & flat & N/A & N/A \\ |
394 |
1.28 & asymmetric flat & 21.7 & N/A \\ |
395 |
1.35 & symmetric ripple & 17.2 & 2.2 \\ |
396 |
1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
397 |
\end{tabular} |
398 |
\label{tab:property} |
399 |
\end{center} |
400 |
\end{minipage} |
401 |
\end{table*} |
402 |
|
403 |
The membrane structures and the reduced wavelength $\lambda/\sigma_0$, |
404 |
reduced amplitude $A/\sigma_0$ of the ripples are summerized in Table |
405 |
\ref{tab:property}. The wavelength range is $15~21$ and the amplitude |
406 |
is $1.5$ for asymmetric ripple and $2.2$ for symmetric ripple. These |
407 |
values are consistent to the experimental results. Note, the |
408 |
amplitudes are underestimated without the melted tails in our |
409 |
simulations. |
410 |
|
411 |
\begin{figure}[htb] |
412 |
\centering |
413 |
\includegraphics[width=\linewidth]{topDown} |
414 |
\caption{Top views of the flat (upper), asymmetric ripple (middle), |
415 |
and symmetric ripple (lower) phases. Note that the head-group dipoles |
416 |
have formed head-to-tail chains in all three of these phases, but in |
417 |
the two rippled phases, the dipolar chains are all aligned |
418 |
{\it perpendicular} to the direction of the ripple. The flat membrane |
419 |
has multiple point defects in the dipolar orientational ordering, and |
420 |
the dipolar ordering on the lower leaf of the bilayer can be in a |
421 |
different direction from the upper leaf.\label{fig:topView}} |
422 |
\end{figure} |
423 |
|
424 |
The $P_2$ order paramters (for molecular bodies and head group |
425 |
dipoles) have been calculated to clarify the ordering in these phases |
426 |
quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$ |
427 |
implies orientational randomization. Figure \ref{fig:rP2} shows the |
428 |
$P_2$ order paramter of the dipoles on head group rising with |
429 |
increasing head group size. When the heads of the lipid molecules are |
430 |
small, the membrane is flat. The dipolar ordering is essentially |
431 |
frustrated on orientational ordering in this circumstance. Figure |
432 |
\ref{fig:topView} shows the snapshots of the top view for the flat system |
433 |
($\sigma_h=1.20\sigma$) and rippled system |
434 |
($\sigma_h=1.41\sigma$). The pointing direction of the dipoles on the |
435 |
head groups are represented by two colored half spheres from blue to |
436 |
yellow. For flat surfaces, the system obviously shows frustration on |
437 |
the dipolar ordering, there are kinks on the edge of defferent |
438 |
domains. Another reason is that the lipids can move independently in |
439 |
each monolayer, it is not nessasory for the direction of dipoles on |
440 |
one leaf is consistant to another layer, which makes total order |
441 |
parameter is relatively low. With increasing head group size, the |
442 |
surface is corrugated, and dipoles do not move as freely on the |
443 |
surface. Therefore, the translational freedom of lipids in one layer |
444 |
is dependent upon the position of lipids in another layer, as a |
445 |
result, the symmetry of the dipoles on head group in one layer is tied |
446 |
to the symmetry in the other layer. Furthermore, as the membrane |
447 |
deforms from two to three dimensions due to the corrugation, the |
448 |
symmetry of the ordering for the dipoles embedded on each leaf is |
449 |
broken. The dipoles then self-assemble in a head-tail configuration, |
450 |
and the order parameter increases dramaticaly. However, the total |
451 |
polarization of the system is still close to zero. This is strong |
452 |
evidence that the corrugated structure is an antiferroelectric |
453 |
state. From the snapshot in Figure \ref{}, the dipoles arrange as |
454 |
arrays along $Y$ axis and fall into head-to-tail configuration in each |
455 |
line, but every $3$ or $4$ lines of dipoles change their direction |
456 |
from neighbour lines. The system shows antiferroelectric |
457 |
charactoristic as a whole. The orientation of the dipolar is always |
458 |
perpendicular to the ripple wave vector. These results are consistent |
459 |
with our previous study on dipolar membranes. |
460 |
|
461 |
The ordering of the tails is essentially opposite to the ordering of |
462 |
the dipoles on head group. The $P_2$ order parameter decreases with |
463 |
increasing head size. This indicates the surface is more curved with |
464 |
larger head groups. When the surface is flat, all tails are pointing |
465 |
in the same direction; in this case, all tails are parallel to the |
466 |
normal of the surface,(making this structure remindcent of the |
467 |
$L_{\beta}$ phase. Increasing the size of the heads, results in |
468 |
rapidly decreasing $P_2$ ordering for the molecular bodies. |
469 |
\begin{figure}[htb] |
470 |
\centering |
471 |
\includegraphics[width=\linewidth]{rP2} |
472 |
\caption{The $P_2$ order parameter as a funtion of the ratio of |
473 |
$\sigma_h$ to $\sigma_0$.\label{fig:rP2}} |
474 |
\end{figure} |
475 |
|
476 |
We studied the effects of the interactions between head groups on the |
477 |
structure of lipid bilayer by changing the strength of the dipole. |
478 |
Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with |
479 |
increasing strength of the dipole. Generally the dipoles on the head |
480 |
group are more ordered by increase in the strength of the interaction |
481 |
between heads and are more disordered by decreasing the interaction |
482 |
stength. When the interaction between the heads is weak enough, the |
483 |
bilayer structure does not persist; all lipid molecules are solvated |
484 |
directly in the water. The critial value of the strength of the dipole |
485 |
depends on the head size. The perfectly flat surface melts at $5$ |
486 |
$0.03$ debye, the asymmetric rippled surfaces melt at $8$ $0.04$ |
487 |
$0.03$ debye, the symmetric rippled surfaces melt at $10$ $0.04$ |
488 |
debye. The ordering of the tails is the same as the ordering of the |
489 |
dipoles except for the flat phase. Since the surface is already |
490 |
perfect flat, the order parameter does not change much until the |
491 |
strength of the dipole is $15$ debye. However, the order parameter |
492 |
decreases quickly when the strength of the dipole is further |
493 |
increased. The head groups of the lipid molecules are brought closer |
494 |
by stronger interactions between them. For a flat surface, a large |
495 |
amount of free volume between the head groups is available, but when |
496 |
the head groups are brought closer, the tails will splay outward, |
497 |
forming an inverse micelle. When $\sigma_h=1.28\sigma_0$, the $P_2$ |
498 |
order parameter decreases slightly after the strength of the dipole is |
499 |
increased to $16$ debye. For rippled surfaces, there is less free |
500 |
volume available between the head groups. Therefore there is little |
501 |
effect on the structure of the membrane due to increasing dipolar |
502 |
strength. However, the increase of the $P_2$ order parameter implies |
503 |
the membranes are flatten by the increase of the strength of the |
504 |
dipole. Unlike other systems that melt directly when the interaction |
505 |
is weak enough, for $\sigma_h=1.41\sigma_0$, part of the membrane |
506 |
melts into itself first. The upper leaf of the bilayer becomes totally |
507 |
interdigitated with the lower leaf. This is different behavior than |
508 |
what is exhibited with the interdigitated lines in the rippled phase |
509 |
where only one interdigitated line connects the two leaves of bilayer. |
510 |
\begin{figure}[htb] |
511 |
\centering |
512 |
\includegraphics[width=\linewidth]{sP2} |
513 |
\caption{The $P_2$ order parameter as a funtion of the strength of the |
514 |
dipole.\label{fig:sP2}} |
515 |
\end{figure} |
516 |
|
517 |
Figure \ref{fig:tP2} shows the dependence of the order parameter on |
518 |
temperature. The behavior of the $P_2$ order paramter is |
519 |
straightforward. Systems are more ordered at low temperature, and more |
520 |
disordered at high temperatures. When the temperature is high enough, |
521 |
the membranes are instable. For flat surfaces ($\sigma_h=1.20\sigma_0$ |
522 |
and $\sigma_h=1.28\sigma_0$), when the temperature is increased to |
523 |
$310$, the $P_2$ order parameter increases slightly instead of |
524 |
decreases like ripple surface. This is an evidence of the frustration |
525 |
of the dipolar ordering in each leaf of the lipid bilayer, at low |
526 |
temperature, the systems are locked in a local minimum energy state, |
527 |
with increase of the temperature, the system can jump out the local |
528 |
energy well to find the lower energy state which is the longer range |
529 |
orientational ordering. Like the dipolar ordering of the flat |
530 |
surfaces, the ordering of the tails of the lipid molecules for ripple |
531 |
membranes ($\sigma_h=1.35\sigma_0$ and $\sigma_h=1.41\sigma_0$) also |
532 |
show some nonthermal characteristic. With increase of the temperature, |
533 |
the $P_2$ order parameter decreases firstly, and increases afterward |
534 |
when the temperature is greater than $290 K$. The increase of the |
535 |
$P_2$ order parameter indicates a more ordered structure for the tails |
536 |
of the lipid molecules which corresponds to a more flat surface. Since |
537 |
our model lacks the detailed information on lipid tails, we can not |
538 |
simulate the fluid phase with melted fatty acid chains. Moreover, the |
539 |
formation of the tilted $L_{\beta'}$ phase also depends on the |
540 |
organization of fatty groups on tails. |
541 |
\begin{figure}[htb] |
542 |
\centering |
543 |
\includegraphics[width=\linewidth]{tP2} |
544 |
\caption{The $P_2$ order parameter as a funtion of |
545 |
temperature.\label{fig:tP2}} |
546 |
\end{figure} |
547 |
|
548 |
\section{Discussion} |
549 |
\label{sec:discussion} |
550 |
|
551 |
\bibliography{mdripple} |
552 |
\end{document} |