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\begin{document} |
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%\bibliographystyle{aps} |
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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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\date{\today} |
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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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\pacs{} |
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\maketitle |
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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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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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\section{Methodology and Model} |
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\label{sec:method} |
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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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\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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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 (${\bf |
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\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
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intermolecular separation (${\bf \hat{r}}$). The functional forms for |
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$\sigma({\bf |
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\hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}})$ 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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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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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, |
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\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. |
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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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For the interaction between nonequivalent uniaxial ellipsoids (in this |
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case, between spheres and ellipsoids), the range parameter is |
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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 $22$ \AA\ and $25$ \AA. |
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|
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The model of the solvent in our simulations is inspired by the idea of |
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``DPD'' water. Every four water molecules are reprsented by one |
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sphere. |
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|
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\begin{figure}[htb] |
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\centering |
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\includegraphics[height=4in]{lipidModel} |
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\caption{The parameters defining the behavior of the lipid |
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models. $\sigma_h / \sigma_0$ is the ratio of the head group to body |
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diameter. Molecular bodies all had an aspect ratio of 3.0. The |
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dipolar strength (and the temperature and pressure) wer the only other |
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parameters that wer varied systematically.\label{fig:lipidModel}} |
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\end{figure} |
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|
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\section{Experiment} |
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\label{sec:experiment} |
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|
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To make the simulations less expensive and to observe long-time |
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behavior of the lipid membranes, all simulations were started from two |
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separate monolayers in the vaccum with $x-y$ anisotropic pressure |
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coupling. The length of $z$ axis of the simulations was fixed and a |
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constant surface tension was applied to enable real fluctuations of |
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the bilayer. Periodic boundaries were used. There were $480-720$ lipid |
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molecules in the simulations depending on the size of the head |
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beads. All the simulations were equlibrated for $100$ ns at $300$ |
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K. The resulting structures were solvated in water ($6$ DPD |
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water/lipid molecule). These configurations were relaxed for another |
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$30$ ns relaxation. All simulations with water were carried out at |
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constant pressure ($P=1$ atm) by $3$D anisotropic coupling, and |
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constant surface tension ($\gamma=0.015$). Given the absence of fast |
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degrees of freedom in this model, a timestep of $50$ fs was |
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utilized. Simulations were performed by using OOPSE |
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package\cite{Meineke05}. |
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|
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\section{Results and Analysis} |
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\label{sec:results} |
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|
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Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
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more corrugated increasing size of the head groups. The surface is |
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nearly flat when $\sigma_h=1.20\sigma_0$. With |
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$\sigma_h=1.28\sigma_0$, although the surface is still flat, the |
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bilayer starts to splay inward; the upper leaf of the bilayer is |
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connected to the lower leaf with an interdigitated line defect. Two |
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periodicities with $100$ \AA\ width were observed in the |
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simulation. This structure is very similiar to the structure observed |
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by de Vries and Lenz {\it et al.}. The same basic structure is also |
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observed when $\sigma_h=1.41\sigma_0$, but the wavelength of the |
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surface corrugations depends sensitively on the size of the ``head'' |
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beads. From the undulation spectrum, the corrugation is clearly |
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non-thermal. |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{phaseCartoon} |
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\caption{A sketch to discribe the structure of the phases observed in |
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our simulations.\label{fig:phaseCartoon}} |
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\end{figure} |
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|
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When $\sigma_h=1.35\sigma_0$, we observed another corrugated surface |
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morphology. This structure is different from the asymmetric rippled |
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surface; there is no interdigitation between the upper and lower |
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leaves of the bilayer. Each leaf of the bilayer is broken into several |
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hemicylinderical sections, and opposite leaves are fitted together |
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much like roof tiles. Unlike the surface in which the upper |
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hemicylinder is always interdigitated on the leading or trailing edge |
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of lower hemicylinder, the symmetric ripple has no prefered direction. |
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The corresponding cartoons are shown in Figure |
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\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
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different phases. Figure \ref{fig:phaseCartoon} (a) is the flat phase, |
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(b) is the asymmetric ripple phase corresponding to the lipid |
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organization when $\sigma_h=1.28\sigma_0$ and $\sigma_h=1.41\sigma_0$, |
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and (c) is the symmetric ripple phase observed when |
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$\sigma_h=1.35\sigma_0$. In symmetric ripple phase, the bilayer is |
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continuous everywhere on the whole membrane, however, in asymmetric |
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ripple phase, the bilayer is intermittent domains connected by thin |
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interdigitated monolayer which consists of upper and lower leaves of |
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the bilayer. |
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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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$\sigma_h/\sigma_0$ & type of phase & $\lambda/\sigma_0$ & $A/\sigma_0$\\ |
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\hline |
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1.20 & flat & N/A & N/A \\ |
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1.28 & asymmetric flat & 21.7 & N/A \\ |
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1.35 & symmetric ripple & 17.2 & 2.2 \\ |
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1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
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\end{tabular} |
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\label{tab:property} |
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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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The membrane structures and the reduced wavelength $\lambda/\sigma_0$, |
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reduced amplitude $A/\sigma_0$ of the ripples are summerized in Table |
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\ref{tab:property}. The wavelength range is $15~21$ and the amplitude |
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is $1.5$ for asymmetric ripple and $2.2$ for symmetric ripple. These |
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values are consistent to the experimental results. Note, the |
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amplitudes are underestimated without the melted tails in our |
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simulations. |
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|
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gezelter |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{topDown} |
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\caption{Top views of the flat (upper), asymmetric ripple (middle), |
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and symmetric ripple (lower) phases. Note that the head-group dipoles |
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have formed head-to-tail chains in all three of these phases, but in |
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the two rippled phases, the dipolar chains are all aligned |
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{\it perpendicular} to the direction of the ripple. The flat membrane |
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has multiple point defects in the dipolar orientational ordering, and |
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the dipolar ordering on the lower leaf of the bilayer can be in a |
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different direction from the upper leaf.\label{fig:topView}} |
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\end{figure} |
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|
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The $P_2$ order paramters (for molecular bodies and head group |
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dipoles) have been calculated to clarify the ordering in these phases |
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quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$ |
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implies orientational randomization. Figure \ref{fig:rP2} shows the |
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$P_2$ order paramter of the dipoles on head group rising with |
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increasing head group size. When the heads of the lipid molecules are |
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small, the membrane is flat. The dipolar ordering is essentially |
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frustrated on orientational ordering in this circumstance. Figure |
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\ref{fig:topView} shows the snapshots of the top view for the flat system |
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($\sigma_h=1.20\sigma$) and rippled system |
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($\sigma_h=1.41\sigma$). The pointing direction of the dipoles on the |
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head groups are represented by two colored half spheres from blue to |
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yellow. For flat surfaces, the system obviously shows frustration on |
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the dipolar ordering, there are kinks on the edge of defferent |
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domains. Another reason is that the lipids can move independently in |
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each monolayer, it is not nessasory for the direction of dipoles on |
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one leaf is consistant to another layer, which makes total order |
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parameter is relatively low. With increasing head group size, the |
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surface is corrugated, and dipoles do not move as freely on the |
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surface. Therefore, the translational freedom of lipids in one layer |
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is dependent upon the position of lipids in another layer, as a |
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result, the symmetry of the dipoles on head group in one layer is tied |
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to the symmetry in the other layer. Furthermore, as the membrane |
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deforms from two to three dimensions due to the corrugation, the |
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symmetry of the ordering for the dipoles embedded on each leaf is |
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broken. The dipoles then self-assemble in a head-tail configuration, |
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and the order parameter increases dramaticaly. However, the total |
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polarization of the system is still close to zero. This is strong |
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evidence that the corrugated structure is an antiferroelectric |
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state. From the snapshot in Figure \ref{}, the dipoles arrange as |
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arrays along $Y$ axis and fall into head-to-tail configuration in each |
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line, but every $3$ or $4$ lines of dipoles change their direction |
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from neighbour lines. The system shows antiferroelectric |
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charactoristic as a whole. The orientation of the dipolar is always |
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perpendicular to the ripple wave vector. These results are consistent |
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with our previous study on dipolar membranes. |
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|
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The ordering of the tails is essentially opposite to the ordering of |
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the dipoles on head group. The $P_2$ order parameter decreases with |
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increasing head size. This indicates the surface is more curved with |
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larger head groups. When the surface is flat, all tails are pointing |
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in the same direction; in this case, all tails are parallel to the |
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normal of the surface,(making this structure remindcent of the |
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$L_{\beta}$ phase. Increasing the size of the heads, results in |
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rapidly decreasing $P_2$ ordering for the molecular bodies. |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{rP2} |
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\caption{The $P_2$ order parameter as a funtion of the ratio of |
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$\sigma_h$ to $\sigma_0$.\label{fig:rP2}} |
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\end{figure} |
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|
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We studied the effects of the interactions between head groups on the |
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structure of lipid bilayer by changing the strength of the dipole. |
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Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with |
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increasing strength of the dipole. Generally the dipoles on the head |
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group are more ordered by increase in the strength of the interaction |
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between heads and are more disordered by decreasing the interaction |
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stength. When the interaction between the heads is weak enough, the |
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bilayer structure does not persist; all lipid molecules are solvated |
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directly in the water. The critial value of the strength of the dipole |
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depends on the head size. The perfectly flat surface melts at $5$ |
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$0.03$ debye, the asymmetric rippled surfaces melt at $8$ $0.04$ |
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$0.03$ debye, the symmetric rippled surfaces melt at $10$ $0.04$ |
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debye. The ordering of the tails is the same as the ordering of the |
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dipoles except for the flat phase. Since the surface is already |
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perfect flat, the order parameter does not change much until the |
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strength of the dipole is $15$ debye. However, the order parameter |
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decreases quickly when the strength of the dipole is further |
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increased. The head groups of the lipid molecules are brought closer |
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by stronger interactions between them. For a flat surface, a large |
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amount of free volume between the head groups is available, but when |
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the head groups are brought closer, the tails will splay outward, |
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forming an inverse micelle. When $\sigma_h=1.28\sigma_0$, the $P_2$ |
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order parameter decreases slightly after the strength of the dipole is |
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increased to $16$ debye. For rippled surfaces, there is less free |
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volume available between the head groups. Therefore there is little |
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effect on the structure of the membrane due to increasing dipolar |
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strength. However, the increase of the $P_2$ order parameter implies |
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the membranes are flatten by the increase of the strength of the |
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dipole. Unlike other systems that melt directly when the interaction |
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is weak enough, for $\sigma_h=1.41\sigma_0$, part of the membrane |
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melts into itself first. The upper leaf of the bilayer becomes totally |
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interdigitated with the lower leaf. This is different behavior than |
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what is exhibited with the interdigitated lines in the rippled phase |
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where only one interdigitated line connects the two leaves of bilayer. |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{sP2} |
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\caption{The $P_2$ order parameter as a funtion of the strength of the |
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dipole.\label{fig:sP2}} |
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\end{figure} |
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|
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Figure \ref{fig:tP2} shows the dependence of the order parameter on |
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temperature. The behavior of the $P_2$ order paramter is |
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straightforward. Systems are more ordered at low temperature, and more |
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disordered at high temperatures. When the temperature is high enough, |
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the membranes are instable. For flat surfaces ($\sigma_h=1.20\sigma_0$ |
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and $\sigma_h=1.28\sigma_0$), when the temperature is increased to |
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$310$, the $P_2$ order parameter increases slightly instead of |
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decreases like ripple surface. This is an evidence of the frustration |
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of the dipolar ordering in each leaf of the lipid bilayer, at low |
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temperature, the systems are locked in a local minimum energy state, |
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with increase of the temperature, the system can jump out the local |
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energy well to find the lower energy state which is the longer range |
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orientational ordering. Like the dipolar ordering of the flat |
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surfaces, the ordering of the tails of the lipid molecules for ripple |
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membranes ($\sigma_h=1.35\sigma_0$ and $\sigma_h=1.41\sigma_0$) also |
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show some nonthermal characteristic. With increase of the temperature, |
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the $P_2$ order parameter decreases firstly, and increases afterward |
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when the temperature is greater than $290 K$. The increase of the |
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$P_2$ order parameter indicates a more ordered structure for the tails |
503 |
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of the lipid molecules which corresponds to a more flat surface. Since |
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our model lacks the detailed information on lipid tails, we can not |
505 |
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simulate the fluid phase with melted fatty acid chains. Moreover, the |
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formation of the tilted $L_{\beta'}$ phase also depends on the |
507 |
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organization of fatty groups on tails. |
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\begin{figure}[htb] |
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\centering |
510 |
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\includegraphics[width=\linewidth]{tP2} |
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\caption{The $P_2$ order parameter as a funtion of |
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temperature.\label{fig:tP2}} |
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\end{figure} |
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|
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\section{Discussion} |
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\label{sec:discussion} |
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|
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\bibliography{mdripple} |
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\end{document} |