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\title{Dipolar ordering in the ripple phases of molecular-scale models |
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of lipid membranes} |
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\author{Xiuquan Sun and J. Daniel Gezelter \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame, \\ |
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Notre Dame, Indiana 46556} |
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|
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%\email[E-mail:]{gezelter@nd.edu} |
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|
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\date{\today} |
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|
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\maketitle |
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|
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\begin{abstract} |
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Symmetric and asymmetric ripple phases have been observed to form in |
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molecular dynamics simulations of a simple molecular-scale lipid |
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model. The lipid model consists of an dipolar head group and an |
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ellipsoidal tail. Within the limits of this model, an explanation for |
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generalized membrane curvature is a simple mismatch in the size of the |
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heads with the width of the molecular bodies. The persistence of a |
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{\it bilayer} structure requires strong attractive forces between the |
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head groups. One feature of this model is that an energetically |
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favorable orientational ordering of the dipoles can be achieved by |
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out-of-plane membrane corrugation. The corrugation of the surface |
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stabilizes the long range orientational ordering for the dipoles in the |
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head groups which then adopt a bulk anti-ferroelectric state. We |
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observe a common feature of the corrugated dipolar membranes: the wave |
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vectors for the surface ripples are always found to be perpendicular |
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to the dipole director axis. |
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\end{abstract} |
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|
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%\maketitle |
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\newpage |
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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} The X-ray diffraction work by |
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Katsaras {\it et al.} showed that a rich phase diagram exhibiting both |
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{\it asymmetric} and {\it symmetric} ripples is possible for lecithin |
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bilayers.\cite{Katsaras00} |
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|
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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 head groups |
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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 modeling studies have been |
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done due to the large size of the resulting structures and the time |
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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 time scales in the range of 10-100 ns. One of |
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the most interesting molecular simulations was carried out by de~Vries |
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{\it et al.}~\cite{deVries05}. According to their simulation results, |
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the ripple consists of two domains, one resembling the gel bilayer, |
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while in the other, the two leaves of the bilayer are fully |
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interdigitated. The mechanism for the formation of the ripple phase |
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suggested by their work is a packing competition between the head |
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groups and the tails of the lipid molecules.\cite{Carlson87} Recently, |
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the ripple phase has also been studied by Lenz and Schmid using Monte |
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Carlo simulations.\cite{Lenz07} Their structures are similar to the De |
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Vries {\it et al.} structures except that the connection between the |
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two leaves of the bilayer is a narrow interdigitated line instead of |
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the fully interdigitated domain. The symmetric ripple phase was also |
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observed by Lenz {\it et al.}, and their work supports other claims |
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that the mismatch between the size of the head group and tail of the |
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lipid molecules is the driving force for the formation of the ripple |
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phase. Ayton and Voth have found significant undulations in |
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zero-surface-tension states of membranes simulated via dissipative |
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particle dynamics, but their results are consistent with purely |
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thermal undulations.~\cite{Ayton02} |
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|
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Although the organization of the tails of lipid molecules are |
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addressed by these molecular simulations and the packing competition |
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between head groups 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 have 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 of dipolar |
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elastic membranes is the anti-ferroelectric ordering of the dipoles. |
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This was evident in the ordering of the dipole director axis |
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perpendicular to the wave vector of the surface ripples. A similar |
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phenomenon has also been observed by Tsonchev {\it et al.} in their |
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work on the spontaneous formation of dipolar peptide chains into |
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curved nano-structures.\cite{Tsonchev04,Tsonchev04II} |
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|
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In this paper, we construct a somewhat more realistic molecular-scale |
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lipid model than our previous ``web of dipoles'' and use molecular |
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dynamics simulations to elucidate the role of the head group dipoles |
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in the formation and morphology of the ripple phase. We describe our |
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model and computational methodology in section \ref{sec:method}. |
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Details on the simulations are presented in section |
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\ref{sec:experiment}, with results following in section |
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\ref{sec:results}. A final discussion of the role of dipolar heads in |
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the ripple formation can be found in section |
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\ref{sec:discussion}. |
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|
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\section{Computational Model} |
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\label{sec:method} |
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|
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\begin{figure}[htb] |
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\centering |
174 |
\includegraphics[width=4in]{lipidModels} |
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\caption{Three different representations of DPPC lipid molecules, |
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including the chemical structure, an atomistic model, and the |
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head-body ellipsoidal coarse-grained model used in this |
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work.\label{fig:lipidModels}} |
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\end{figure} |
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|
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Our simple molecular-scale lipid model for studying the ripple phase |
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is based on two facts: one is that the most essential feature of lipid |
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molecules is their amphiphilic structure with polar head groups and |
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non-polar tails. Another fact is that the majority of lipid molecules |
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in the ripple phase are relatively rigid (i.e. gel-like) which makes |
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some fraction of the details of the chain dynamics negligible. Figure |
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\ref{fig:lipidModels} shows the molecular structure of a DPPC |
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molecule, as well as atomistic and molecular-scale representations of |
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a DPPC molecule. The hydrophilic character of the head group is |
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largely due to the separation of charge between the nitrogen and |
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phosphate groups. The zwitterionic nature of the PC headgroups leads |
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to abnormally large dipole moments (as high as 20.6 D), and this |
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strongly polar head group interacts strongly with the solvating water |
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layers immediately surrounding the membrane. The hydrophobic tail |
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consists of fatty acid chains. In our molecular scale model, lipid |
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molecules have been reduced to these essential features; the fatty |
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acid chains are represented by an ellipsoid with a dipolar ball |
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perched on one end to represent the effects of the charge-separated |
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head group. In real PC lipids, the direction of the dipole is |
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nearly perpendicular to the tail, so we have fixed the direction of |
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the point dipole rigidly in this orientation. |
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|
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The ellipsoidal portions of the model interact via the Gay-Berne |
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potential which has seen widespread use in the liquid crystal |
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community. Ayton and Voth have also used Gay-Berne ellipsoids for |
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modeling large length-scale properties of lipid |
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bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
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was a single site model for the interactions of rigid ellipsoidal |
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molecules.\cite{Gay81} It can be thought of as a modification of the |
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Gaussian overlap model originally described by Berne and |
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Pechukas.\cite{Berne72} The potential is constructed in the familiar |
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form of the Lennard-Jones function using orientation-dependent |
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$\sigma$ and $\epsilon$ parameters, |
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\begin{equation*} |
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V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
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r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
218 |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
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-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
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\label{eq:gb} |
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\end{equation*} |
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|
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The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
225 |
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
227 |
are dependent on the relative orientations of the two molecules (${\bf |
228 |
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
229 |
intermolecular separation (${\bf \hat{r}}_{ij}$). $\sigma$ and |
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$\sigma_0$ are also governed by shape mixing and anisotropy variables, |
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\begin {eqnarray*} |
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\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
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\chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 - |
234 |
d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^2 + |
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d_j^2 \right)}\right]^{1/2} \\ \\ |
236 |
\alpha^2 & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 + |
237 |
d_i^2 \right)}{\left( l_j^2 - d_j^2 \right) \left(l_i^2 + |
238 |
d_j^2 \right)}\right]^{1/2}, |
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\end{eqnarray*} |
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where $l$ and $d$ describe the length and width of each uniaxial |
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ellipsoid. These shape anisotropy parameters can then be used to |
242 |
calculate the range function, |
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\begin{equation*} |
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\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0} |
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\left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
246 |
\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
247 |
\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
248 |
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
249 |
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
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\right]^{-1/2} |
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\end{equation*} |
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|
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Gay-Berne ellipsoids also have an energy scaling parameter, |
255 |
$\epsilon^s$, which describes the well depth for two identical |
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ellipsoids in a {\it side-by-side} configuration. Additionally, a well |
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depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes |
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the ratio between the well depths in the {\it end-to-end} and |
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side-by-side configurations. As in the range parameter, a set of |
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mixing and anisotropy variables can be used to describe the well |
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depths for dissimilar particles, |
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\begin {eqnarray*} |
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\epsilon_0 & = & \sqrt{\epsilon^s_i * \epsilon^s_j} \\ \\ |
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\epsilon^r & = & \sqrt{\epsilon^r_i * \epsilon^r_j} \\ \\ |
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\chi' & = & \frac{1 - (\epsilon^r)^{1/\mu}}{1 + (\epsilon^r)^{1/\mu}} |
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\\ \\ |
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\alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1} |
268 |
\end{eqnarray*} |
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The form of the strength function is somewhat complicated, |
270 |
\begin {eqnarray*} |
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\epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = & |
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\epsilon_{0} \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j}) |
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\epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
274 |
\hat{r}}_{ij}) \\ \\ |
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\epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = & |
276 |
\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
277 |
\hat{u}}_{j})^{2}\right]^{-1/2} \\ \\ |
278 |
\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & |
279 |
= & |
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1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
281 |
\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
282 |
\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
283 |
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
284 |
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
286 |
\end {eqnarray*} |
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although many of the quantities and derivatives are identical with |
288 |
those obtained for the range parameter. Ref. \citen{Luckhurst90} |
289 |
has a particularly good explanation of the choice of the Gay-Berne |
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parameters $\mu$ and $\nu$ for modeling liquid crystal molecules. An |
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excellent overview of the computational methods that can be used to |
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efficiently compute forces and torques for this potential can be found |
293 |
in Ref. \citen{Golubkov06} |
294 |
|
295 |
The choices of parameters we have used in this study correspond to a |
296 |
shape anisotropy of 3 for the chain portion of the molecule. In |
297 |
principle, this could be varied to allow for modeling of longer or |
298 |
shorter chain lipid molecules. For these prolate ellipsoids, we have: |
299 |
\begin{equation} |
300 |
\begin{array}{rcl} |
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d & < & l \\ |
302 |
\epsilon^{r} & < & 1 |
303 |
\end{array} |
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\end{equation} |
305 |
A sketch of the various structural elements of our molecular-scale |
306 |
lipid / solvent model is shown in figure \ref{fig:lipidModel}. The |
307 |
actual parameters used in our simulations are given in table |
308 |
\ref{tab:parameters}. |
309 |
|
310 |
\begin{figure}[htb] |
311 |
\centering |
312 |
\includegraphics[width=4in]{2lipidModel} |
313 |
\caption{The parameters defining the behavior of the lipid |
314 |
models. $l / d$ is the ratio of the head group to body diameter. |
315 |
Molecular bodies had a fixed aspect ratio of 3.0. The solvent model |
316 |
was a simplified 4-water bead ($\sigma_w \approx d$) that has been |
317 |
used in other coarse-grained (DPD) simulations. The dipolar strength |
318 |
(and the temperature and pressure) were the only other parameters that |
319 |
were varied systematically.\label{fig:lipidModel}} |
320 |
\end{figure} |
321 |
|
322 |
To take into account the permanent dipolar interactions of the |
323 |
zwitterionic head groups, we have placed fixed dipole moments $\mu_{i}$ at |
324 |
one end of the Gay-Berne particles. The dipoles are oriented at an |
325 |
angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
326 |
are protected by a head ``bead'' with a range parameter ($\sigma_h$) which we have |
327 |
varied between $1.20 d$ and $1.41 d$. The head groups interact with |
328 |
each other using a combination of Lennard-Jones, |
329 |
\begin{equation} |
330 |
V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
331 |
\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
332 |
\end{equation} |
333 |
and dipole-dipole, |
334 |
\begin{equation} |
335 |
V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
336 |
\hat{r}}_{ij})) = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
337 |
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
338 |
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
339 |
\end{equation} |
340 |
potentials. |
341 |
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
342 |
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
343 |
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. |
344 |
|
345 |
For the interaction between nonequivalent uniaxial ellipsoids (in this |
346 |
case, between spheres and ellipsoids), the spheres are treated as |
347 |
ellipsoids with an aspect ratio of 1 ($d = l$) and with an well depth |
348 |
ratio ($\epsilon^r$) of 1 ($\epsilon^e = \epsilon^s$). The form of |
349 |
the Gay-Berne potential we are using was generalized by Cleaver {\it |
350 |
et al.} and is appropriate for dissimilar uniaxial |
351 |
ellipsoids.\cite{Cleaver96} |
352 |
|
353 |
The solvent model in our simulations is identical to one used by |
354 |
Marrink {\it et al.} in their dissipative particle dynamics (DPD) |
355 |
simulation of lipid bilayers.\cite{Marrink04} This solvent bead is a |
356 |
single site that represents four water molecules (m = 72 amu) and has |
357 |
comparable density and diffusive behavior to liquid water. However, |
358 |
since there are no electrostatic sites on these beads, this solvent |
359 |
model cannot replicate the dielectric properties of water. |
360 |
|
361 |
\begin{table*} |
362 |
\begin{minipage}{\linewidth} |
363 |
\begin{center} |
364 |
\caption{Potential parameters used for molecular-scale coarse-grained |
365 |
lipid simulations} |
366 |
\begin{tabular}{llccc} |
367 |
\hline |
368 |
& & Head & Chain & Solvent \\ |
369 |
\hline |
370 |
$d$ (\AA) & & varied & 4.6 & 4.7 \\ |
371 |
$l$ (\AA) & & $= d$ & 13.8 & 4.7 \\ |
372 |
$\epsilon^s$ (kcal/mol) & & 0.185 & 1.0 & 0.8 \\ |
373 |
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 \\ |
374 |
$m$ (amu) & & 196 & 760 & 72.06 \\ |
375 |
$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
376 |
\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
377 |
\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
378 |
\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
379 |
$\mu$ (Debye) & & varied & 0 & 0 \\ |
380 |
\end{tabular} |
381 |
\label{tab:parameters} |
382 |
\end{center} |
383 |
\end{minipage} |
384 |
\end{table*} |
385 |
|
386 |
\section{Experimental Methodology} |
387 |
\label{sec:experiment} |
388 |
|
389 |
The parameters that were systematically varied in this study were the |
390 |
size of the head group ($\sigma_h$), the strength of the dipole moment |
391 |
($\mu$), and the temperature of the system. Values for $\sigma_h$ |
392 |
ranged from 5.5 \AA\ to 6.5 \AA\ . If the width of the tails is taken |
393 |
to be the unit of length, these head groups correspond to a range from |
394 |
$1.2 d$ to $1.41 d$. Since the solvent beads are nearly identical in |
395 |
diameter to the tail ellipsoids, all distances that follow will be |
396 |
measured relative to this unit of distance. Because the solvent we |
397 |
are using is non-polar and has a dielectric constant of 1, values for |
398 |
$\mu$ are sampled from a range that is somewhat smaller than the 20.6 |
399 |
Debye dipole moment of the PC head groups. |
400 |
|
401 |
To create unbiased bilayers, all simulations were started from two |
402 |
perfectly flat monolayers separated by a 26 \AA\ gap between the |
403 |
molecular bodies of the upper and lower leaves. The separated |
404 |
monolayers were evolved in a vacuum with $x-y$ anisotropic pressure |
405 |
coupling. The length of $z$ axis of the simulations was fixed and a |
406 |
constant surface tension was applied to enable real fluctuations of |
407 |
the bilayer. Periodic boundary conditions were used, and $480-720$ |
408 |
lipid molecules were present in the simulations, depending on the size |
409 |
of the head beads. In all cases, the two monolayers spontaneously |
410 |
collapsed into bilayer structures within 100 ps. Following this |
411 |
collapse, all systems were equilibrated for $100$ ns at $300$ K. |
412 |
|
413 |
The resulting bilayer structures were then solvated at a ratio of $6$ |
414 |
solvent beads (24 water molecules) per lipid. These configurations |
415 |
were then equilibrated for another $30$ ns. All simulations utilizing |
416 |
the solvent were carried out at constant pressure ($P=1$ atm) with |
417 |
$3$D anisotropic coupling, and constant surface tension |
418 |
($\gamma=0.015$ N/m). Given the absence of fast degrees of freedom in |
419 |
this model, a time step of $50$ fs was utilized with excellent energy |
420 |
conservation. Data collection for structural properties of the |
421 |
bilayers was carried out during a final 5 ns run following the solvent |
422 |
equilibration. All simulations were performed using the OOPSE |
423 |
molecular modeling program.\cite{Meineke05} |
424 |
|
425 |
A switching function was applied to all potentials to smoothly turn |
426 |
off the interactions between a range of $22$ and $25$ \AA. |
427 |
|
428 |
\section{Results} |
429 |
\label{sec:results} |
430 |
|
431 |
The membranes in our simulations exhibit a number of interesting |
432 |
bilayer phases. The surface topology of these phases depends most |
433 |
sensitively on the ratio of the size of the head groups to the width |
434 |
of the molecular bodies. With heads only slightly larger than the |
435 |
bodies ($\sigma_h=1.20 d$) the membrane exhibits a flat bilayer. |
436 |
|
437 |
Increasing the head / body size ratio increases the local membrane |
438 |
curvature around each of the lipids. With $\sigma_h=1.28 d$, the |
439 |
surface is still essentially flat, but the bilayer starts to exhibit |
440 |
signs of instability. We have observed occasional defects where a |
441 |
line of lipid molecules on one leaf of the bilayer will dip down to |
442 |
interdigitate with the other leaf. This gives each of the two bilayer |
443 |
leaves some local convexity near the line defect. These structures, |
444 |
once developed in a simulation, are very stable and are spaced |
445 |
approximately 100 \AA\ away from each other. |
446 |
|
447 |
With larger heads ($\sigma_h = 1.35 d$) the membrane curvature |
448 |
resolves into a ``symmetric'' ripple phase. Each leaf of the bilayer |
449 |
is broken into several convex, hemicylinderical sections, and opposite |
450 |
leaves are fitted together much like roof tiles. There is no |
451 |
interdigitation between the upper and lower leaves of the bilayer. |
452 |
|
453 |
For the largest head / body ratios studied ($\sigma_h = 1.41 d$) the |
454 |
local curvature is substantially larger, and the resulting bilayer |
455 |
structure resolves into an asymmetric ripple phase. This structure is |
456 |
very similar to the structures observed by both de~Vries {\it et al.} |
457 |
and Lenz {\it et al.}. For a given ripple wave vector, there are two |
458 |
possible asymmetric ripples, which is not the case for the symmetric |
459 |
phase observed when $\sigma_h = 1.35 d$. |
460 |
|
461 |
\begin{figure}[htb] |
462 |
\centering |
463 |
\includegraphics[width=4in]{phaseCartoon} |
464 |
\caption{The role of the ratio between the head group size and the |
465 |
width of the molecular bodies is to increase the local membrane |
466 |
curvature. With strong attractive interactions between the head |
467 |
groups, this local curvature can be maintained in bilayer structures |
468 |
through surface corrugation. Shown above are three phases observed in |
469 |
these simulations. With $\sigma_h = 1.20 d$, the bilayer maintains a |
470 |
flat topology. For larger heads ($\sigma_h = 1.35 d$) the local |
471 |
curvature resolves into a symmetrically rippled phase with little or |
472 |
no interdigitation between the upper and lower leaves of the membrane. |
473 |
The largest heads studied ($\sigma_h = 1.41 d$) resolve into an |
474 |
asymmetric rippled phases with interdigitation between the two |
475 |
leaves.\label{fig:phaseCartoon}} |
476 |
\end{figure} |
477 |
|
478 |
Sample structures for the flat ($\sigma_h = 1.20 d$), symmetric |
479 |
($\sigma_h = 1.35 d$, and asymmetric ($\sigma_h = 1.41 d$) ripple |
480 |
phases are shown in Figure \ref{fig:phaseCartoon}. |
481 |
|
482 |
It is reasonable to ask how well the parameters we used can produce |
483 |
bilayer properties that match experimentally known values for real |
484 |
lipid bilayers. Using a value of $l = 13.8$ \AA for the ellipsoidal |
485 |
tails and the fixed ellipsoidal aspect ratio of 3, our values for the |
486 |
area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend |
487 |
entirely on the size of the head bead relative to the molecular body. |
488 |
These values are tabulated in table \ref{tab:property}. Kucera {\it |
489 |
et al.} have measured values for the head group spacings for a number |
490 |
of PC lipid bilayers that range from 30.8 \AA (DLPC) to 37.8 (DPPC). |
491 |
They have also measured values for the area per lipid that range from |
492 |
60.6 |
493 |
\AA$^2$ (DMPC) to 64.2 \AA$^2$ |
494 |
(DPPC).\cite{NorbertKucerka04012005,NorbertKucerka06012006} Only the |
495 |
largest of the head groups we modeled ($\sigma_h = 1.41 d$) produces |
496 |
bilayers (specifically the area per lipid) that resemble real PC |
497 |
bilayers. The smaller head beads we used are perhaps better models |
498 |
for PE head groups. |
499 |
|
500 |
\begin{table*} |
501 |
\begin{minipage}{\linewidth} |
502 |
\begin{center} |
503 |
\caption{Phase, bilayer spacing, area per lipid, ripple wavelength |
504 |
and amplitude observed as a function of the ratio between the head |
505 |
beads and the diameters of the tails. Ripple wavelengths and |
506 |
amplitudes are normalized to the diameter of the tail ellipsoids.} |
507 |
\begin{tabular}{lccccc} |
508 |
\hline |
509 |
$\sigma_h / d$ & type of phase & bilayer spacing (\AA) & area per |
510 |
lipid (\AA$^2$) & $\lambda / d$ & $A / d$\\ |
511 |
\hline |
512 |
1.20 & flat & 33.4 & 49.6 & N/A & N/A \\ |
513 |
1.28 & flat & 33.7 & 54.7 & N/A & N/A \\ |
514 |
1.35 & symmetric ripple & 42.9 & 51.7 & 17.2 & 2.2 \\ |
515 |
1.41 & asymmetric ripple & 37.1 & 63.1 & 15.4 & 1.5 \\ |
516 |
\end{tabular} |
517 |
\label{tab:property} |
518 |
\end{center} |
519 |
\end{minipage} |
520 |
\end{table*} |
521 |
|
522 |
The membrane structures and the reduced wavelength $\lambda / d$, |
523 |
reduced amplitude $A / d$ of the ripples are summarized in Table |
524 |
\ref{tab:property}. The wavelength range is $15 - 17$ molecular bodies |
525 |
and the amplitude is $1.5$ molecular bodies for asymmetric ripple and |
526 |
$2.2$ for symmetric ripple. These values are reasonably consistent |
527 |
with experimental measurements.\cite{Sun96,Katsaras00,Kaasgaard03} |
528 |
Note, that given the lack of structural freedom in the tails of our |
529 |
model lipids, the amplitudes observed from these simulations are |
530 |
likely to underestimate of the true amplitudes. |
531 |
|
532 |
\begin{figure}[htb] |
533 |
\centering |
534 |
\includegraphics[width=4in]{topDown} |
535 |
\caption{Top views of the flat (upper), symmetric ripple (middle), |
536 |
and asymmetric ripple (lower) phases. Note that the head-group |
537 |
dipoles have formed head-to-tail chains in all three of these phases, |
538 |
but in the two rippled phases, the dipolar chains are all aligned {\it |
539 |
perpendicular} to the direction of the ripple. Note that the flat |
540 |
membrane has multiple vortex defects in the dipolar ordering, and the |
541 |
ordering on the lower leaf of the bilayer can be in an entirely |
542 |
different direction from the upper leaf.\label{fig:topView}} |
543 |
\end{figure} |
544 |
|
545 |
The principal method for observing orientational ordering in dipolar |
546 |
or liquid crystalline systems is the $P_2$ order parameter (defined |
547 |
as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
548 |
eigenvalue of the matrix, |
549 |
\begin{equation} |
550 |
{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
551 |
\begin{array}{ccc} |
552 |
u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
553 |
u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
554 |
u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
555 |
\end{array} \right). |
556 |
\label{eq:opmatrix} |
557 |
\end{equation} |
558 |
Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
559 |
for molecule $i$. (Here, $\hat{\bf u}_i$ can refer either to the |
560 |
principal axis of the molecular body or to the dipole on the head |
561 |
group of the molecule.) $P_2$ will be $1.0$ for a perfectly-ordered |
562 |
system and near $0$ for a randomized system. Note that this order |
563 |
parameter is {\em not} equal to the polarization of the system. For |
564 |
example, the polarization of a perfect anti-ferroelectric arrangement |
565 |
of point dipoles is $0$, but $P_2$ for the same system is $1$. The |
566 |
eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is |
567 |
familiar as the director axis, which can be used to determine a |
568 |
privileged axis for an orientationally-ordered system. Since the |
569 |
molecular bodies are perpendicular to the head group dipoles, it is |
570 |
possible for the director axes for the molecular bodies and the head |
571 |
groups to be completely decoupled from each other. |
572 |
|
573 |
Figure \ref{fig:topView} shows snapshots of bird's-eye views of the |
574 |
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$) |
575 |
bilayers. The directions of the dipoles on the head groups are |
576 |
represented with two colored half spheres: blue (phosphate) and yellow |
577 |
(amino). For flat bilayers, the system exhibits signs of |
578 |
orientational frustration; some disorder in the dipolar head-to-tail |
579 |
chains is evident with kinks visible at the edges between differently |
580 |
ordered domains. The lipids can also move independently of lipids in |
581 |
the opposing leaf, so the ordering of the dipoles on one leaf is not |
582 |
necessarily consistent with the ordering on the other. These two |
583 |
factors keep the total dipolar order parameter relatively low for the |
584 |
flat phases. |
585 |
|
586 |
With increasing head group size, the surface becomes corrugated, and |
587 |
the dipoles cannot move as freely on the surface. Therefore, the |
588 |
translational freedom of lipids in one layer is dependent upon the |
589 |
position of the lipids in the other layer. As a result, the ordering of |
590 |
the dipoles on head groups in one leaf is correlated with the ordering |
591 |
in the other leaf. Furthermore, as the membrane deforms due to the |
592 |
corrugation, the symmetry of the allowed dipolar ordering on each leaf |
593 |
is broken. The dipoles then self-assemble in a head-to-tail |
594 |
configuration, and the dipolar order parameter increases dramatically. |
595 |
However, the total polarization of the system is still close to zero. |
596 |
This is strong evidence that the corrugated structure is an |
597 |
anti-ferroelectric state. It is also notable that the head-to-tail |
598 |
arrangement of the dipoles is always observed in a direction |
599 |
perpendicular to the wave vector for the surface corrugation. This is |
600 |
a similar finding to what we observed in our earlier work on the |
601 |
elastic dipolar membranes.\cite{Sun2007} |
602 |
|
603 |
The $P_2$ order parameters (for both the molecular bodies and the head |
604 |
group dipoles) have been calculated to quantify the ordering in these |
605 |
phases. Figure \ref{fig:rP2} shows that the $P_2$ order parameter for |
606 |
the head-group dipoles increases with increasing head group size. When |
607 |
the heads of the lipid molecules are small, the membrane is nearly |
608 |
flat. Since the in-plane packing is essentially a close packing of the |
609 |
head groups, the head dipoles exhibit frustration in their |
610 |
orientational ordering. |
611 |
|
612 |
The ordering trends for the tails are essentially opposite to the |
613 |
ordering of the head group dipoles. The tail $P_2$ order parameter |
614 |
{\it decreases} with increasing head size. This indicates that the |
615 |
surface is more curved with larger head / tail size ratios. When the |
616 |
surface is flat, all tails are pointing in the same direction (normal |
617 |
to the bilayer surface). This simplified model appears to be |
618 |
exhibiting a smectic A fluid phase, similar to the real $L_{\beta}$ |
619 |
phase. We have not observed a smectic C gel phase ($L_{\beta'}$) for |
620 |
this model system. Increasing the size of the heads results in |
621 |
rapidly decreasing $P_2$ ordering for the molecular bodies. |
622 |
|
623 |
\begin{figure}[htb] |
624 |
\centering |
625 |
\includegraphics[width=\linewidth]{rP2} |
626 |
\caption{The $P_2$ order parameters for head groups (circles) and |
627 |
molecular bodies (squares) as a function of the ratio of head group |
628 |
size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{fig:rP2}} |
629 |
\end{figure} |
630 |
|
631 |
In addition to varying the size of the head groups, we studied the |
632 |
effects of the interactions between head groups on the structure of |
633 |
lipid bilayer by changing the strength of the dipoles. Figure |
634 |
\ref{fig:sP2} shows how the $P_2$ order parameter changes with |
635 |
increasing strength of the dipole. Generally, the dipoles on the head |
636 |
groups become more ordered as the strength of the interaction between |
637 |
heads is increased and become more disordered by decreasing the |
638 |
interaction strength. When the interaction between the heads becomes |
639 |
too weak, the bilayer structure does not persist; all lipid molecules |
640 |
become dispersed in the solvent (which is non-polar in this |
641 |
molecular-scale model). The critical value of the strength of the |
642 |
dipole depends on the size of the head groups. The perfectly flat |
643 |
surface becomes unstable below $5$ Debye, while the rippled |
644 |
surfaces melt at $8$ Debye (asymmetric) or $10$ Debye (symmetric). |
645 |
|
646 |
The ordering of the tails mirrors the ordering of the dipoles {\it |
647 |
except for the flat phase}. Since the surface is nearly flat in this |
648 |
phase, the order parameters are only weakly dependent on dipolar |
649 |
strength until it reaches $15$ Debye. Once it reaches this value, the |
650 |
head group interactions are strong enough to pull the head groups |
651 |
close to each other and distort the bilayer structure. For a flat |
652 |
surface, a substantial amount of free volume between the head groups |
653 |
is normally available. When the head groups are brought closer by |
654 |
dipolar interactions, the tails are forced to splay outward, first forming |
655 |
curved bilayers, and then inverted micelles. |
656 |
|
657 |
When $\sigma_h=1.28 d$, the $P_2$ order parameter decreases slightly |
658 |
when the strength of the dipole is increased above $16$ Debye. For |
659 |
rippled bilayers, there is less free volume available between the head |
660 |
groups. Therefore increasing dipolar strength weakly influences the |
661 |
structure of the membrane. However, the increase in the body $P_2$ |
662 |
order parameters implies that the membranes are being slightly |
663 |
flattened due to the effects of increasing head-group attraction. |
664 |
|
665 |
A very interesting behavior takes place when the head groups are very |
666 |
large relative to the molecular bodies ($\sigma_h = 1.41 d$) and the |
667 |
dipolar strength is relatively weak ($\mu < 9$ Debye). In this case, |
668 |
the two leaves of the bilayer become totally interdigitated with each |
669 |
other in large patches of the membrane. With higher dipolar |
670 |
strength, the interdigitation is limited to single lines that run |
671 |
through the bilayer in a direction perpendicular to the ripple wave |
672 |
vector. |
673 |
|
674 |
\begin{figure}[htb] |
675 |
\centering |
676 |
\includegraphics[width=\linewidth]{sP2} |
677 |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
678 |
molecular bodies (b) as a function of the strength of the dipoles. |
679 |
These order parameters are shown for four values of the head group / |
680 |
molecular width ratio ($\sigma_h / d$). \label{fig:sP2}} |
681 |
\end{figure} |
682 |
|
683 |
Figure \ref{fig:tP2} shows the dependence of the order parameters on |
684 |
temperature. As expected, systems are more ordered at low |
685 |
temperatures, and more disordered at high temperatures. All of the |
686 |
bilayers we studied can become unstable if the temperature becomes |
687 |
high enough. The only interesting feature of the temperature |
688 |
dependence is in the flat surfaces ($\sigma_h=1.20 d$ and |
689 |
$\sigma_h=1.28 d$). Here, when the temperature is increased above |
690 |
$310$K, there is enough jostling of the head groups to allow the |
691 |
dipolar frustration to resolve into more ordered states. This results |
692 |
in a slight increase in the $P_2$ order parameter above this |
693 |
temperature. |
694 |
|
695 |
For the rippled surfaces ($\sigma_h=1.35 d$ and $\sigma_h=1.41 d$), |
696 |
there is a slightly increased orientational ordering in the molecular |
697 |
bodies above $290$K. Since our model lacks the detailed information |
698 |
about the behavior of the lipid tails, this is the closest the model |
699 |
can come to depicting the ripple ($P_{\beta'}$) to fluid |
700 |
($L_{\alpha}$) phase transition. What we are observing is a |
701 |
flattening of the rippled structures made possible by thermal |
702 |
expansion of the tightly-packed head groups. The lack of detailed |
703 |
chain configurations also makes it impossible for this model to depict |
704 |
the ripple to gel ($L_{\beta'}$) phase transition. |
705 |
|
706 |
\begin{figure}[htb] |
707 |
\centering |
708 |
\includegraphics[width=\linewidth]{tP2} |
709 |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
710 |
molecular bodies (b) as a function of temperature. |
711 |
These order parameters are shown for four values of the head group / |
712 |
molecular width ratio ($\sigma_h / d$).\label{fig:tP2}} |
713 |
\end{figure} |
714 |
|
715 |
Fig. \ref{fig:phaseDiagram} shows a phase diagram for the model as a |
716 |
function of the head group / molecular width ratio ($\sigma_h / d$) |
717 |
and the strength of the head group dipole moment ($\mu$). Note that |
718 |
the specific form of the bilayer phase is governed almost entirely by |
719 |
the head group / molecular width ratio, while the strength of the |
720 |
dipolar interactions between the head groups governs the stability of |
721 |
the bilayer phase. Weaker dipoles result in unstable bilayer phases, |
722 |
while extremely strong dipoles can shift the equilibrium to an |
723 |
inverted micelle phase when the head groups are small. Temperature |
724 |
has little effect on the actual bilayer phase observed, although higher |
725 |
temperatures can cause the unstable region to grow into the higher |
726 |
dipole region of this diagram. |
727 |
|
728 |
\begin{figure}[htb] |
729 |
\centering |
730 |
\includegraphics[width=\linewidth]{phaseDiagram} |
731 |
\caption{Phase diagram for the simple molecular model as a function |
732 |
of the head group / molecular width ratio ($\sigma_h / d$) and the |
733 |
strength of the head group dipole moment |
734 |
($\mu$).\label{fig:phaseDiagram}} |
735 |
\end{figure} |
736 |
|
737 |
We have computed translational diffusion coefficients for lipid |
738 |
molecules from the mean square displacement, |
739 |
\begin{eqnarray} |
740 |
\langle {|\left({\bf r}_{i}(t) - {\bt r}_{i}(0) \right)|}^2 \rangle \\ \\ |
741 |
& = & 6Dt |
742 |
\end{eqnarray} |
743 |
of the lipid bodies. The values of the translational diffusion |
744 |
coefficient for different head-to-tail size ratio are shown in table |
745 |
\ref{tab:relaxation}. |
746 |
|
747 |
We have also computed orientational diffusion constants for the head |
748 |
groups from the relaxation of the second-order Legendre polynomial |
749 |
correlation function, |
750 |
\begin{eqnarray} |
751 |
C_{\ell}(t) & = & \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
752 |
\mu}_{i}(0) \right) \rangle \\ \\ |
753 |
& \approx & e^{-\ell(\ell + 1) \theta t}, |
754 |
\end{eqnarray} |
755 |
of the head group dipoles. In this last line, we have used a simple |
756 |
``Debye''-like model for the relaxation of the correlation function, |
757 |
specifically in the case when $\ell = 2$. The computed orientational |
758 |
diffusion constants are given in table \ref{tab:relaxation}. The |
759 |
notable feature we observe is that the orientational diffusion |
760 |
constant for the head group exhibits an order of magnitude decrease |
761 |
upon entering the rippled phase. Our orientational correlation times |
762 |
are substantially in excess of those provided by... |
763 |
|
764 |
|
765 |
\begin{table*} |
766 |
\begin{minipage}{\linewidth} |
767 |
\begin{center} |
768 |
\caption{Rotational diffusion constants for the head groups |
769 |
($\theta_h$) and molecular bodies ($\theta_b$) as well as the |
770 |
translational diffusion coefficients for the molecule as a function of |
771 |
the head-to-body width ratio. The orientational mobility of the head |
772 |
groups experiences an {\it order of magnitude decrease} upon entering |
773 |
the rippled phase, which suggests that the rippling is tied to a |
774 |
freezing out of head group orientational freedom. Uncertainties in |
775 |
the last digit are indicated by the values in parentheses.} |
776 |
\begin{tabular}{lccc} |
777 |
\hline |
778 |
$\sigma_h / d$ & $\theta_h (\mu s^{-1})$ & $\theta_b (1/fs)$ & $D ( |
779 |
\times 10^{-11} m^2 s^{-1} \\ |
780 |
\hline |
781 |
1.20 & $0.206(1) $ & $0.0175(5) $ & $0.43(1)$ \\ |
782 |
1.28 & $0.179(2) $ & $0.055(2) $ & $5.91(3)$ \\ |
783 |
1.35 & $0.025(1) $ & $0.195(3) $ & $3.42(1)$ \\ |
784 |
1.41 & $0.023(1) $ & $0.024(3) $ & $7.16(1)$ \\ |
785 |
\end{tabular} |
786 |
\label{tab:relaxation} |
787 |
\end{center} |
788 |
\end{minipage} |
789 |
\end{table*} |
790 |
|
791 |
\section{Discussion} |
792 |
\label{sec:discussion} |
793 |
|
794 |
Symmetric and asymmetric ripple phases have been observed to form in |
795 |
our molecular dynamics simulations of a simple molecular-scale lipid |
796 |
model. The lipid model consists of an dipolar head group and an |
797 |
ellipsoidal tail. Within the limits of this model, an explanation for |
798 |
generalized membrane curvature is a simple mismatch in the size of the |
799 |
heads with the width of the molecular bodies. With heads |
800 |
substantially larger than the bodies of the molecule, this curvature |
801 |
should be convex nearly everywhere, a requirement which could be |
802 |
resolved either with micellar or cylindrical phases. |
803 |
|
804 |
The persistence of a {\it bilayer} structure therefore requires either |
805 |
strong attractive forces between the head groups or exclusionary |
806 |
forces from the solvent phase. To have a persistent bilayer structure |
807 |
with the added requirement of convex membrane curvature appears to |
808 |
result in corrugated structures like the ones pictured in |
809 |
Fig. \ref{fig:phaseCartoon}. In each of the sections of these |
810 |
corrugated phases, the local curvature near a most of the head groups |
811 |
is convex. These structures are held together by the extremely strong |
812 |
and directional interactions between the head groups. |
813 |
|
814 |
Dipolar head groups are key for the maintaining the bilayer structures |
815 |
exhibited by this model. The dipoles are likely to form head-to-tail |
816 |
configurations even in flat configurations, but the temperatures are |
817 |
high enough that vortex defects become prevalent in the flat phase. |
818 |
The flat phase we observed therefore appears to be substantially above |
819 |
the Kosterlitz-Thouless transition temperature for a planar system of |
820 |
dipoles with this set of parameters. For this reason, it would be |
821 |
interesting to observe the thermal behavior of the flat phase at |
822 |
substantially lower temperatures. |
823 |
|
824 |
One feature of this model is that an energetically favorable |
825 |
orientational ordering of the dipoles can be achieved by forming |
826 |
ripples. The corrugation of the surface breaks the symmetry of the |
827 |
plane, making vortex defects somewhat more expensive, and stabilizing |
828 |
the long range orientational ordering for the dipoles in the head |
829 |
groups. Most of the rows of the head-to-tail dipoles are parallel to |
830 |
each other and the system adopts a bulk anti-ferroelectric state. We |
831 |
believe that this is the first time the organization of the head |
832 |
groups in ripple phases has been addressed. |
833 |
|
834 |
Although the size-mismatch between the heads and molecular bodies |
835 |
appears to be the primary driving force for surface convexity, the |
836 |
persistence of the bilayer through the use of rippled structures is a |
837 |
function of the strong, attractive interactions between the heads. |
838 |
One important prediction we can make using the results from this |
839 |
simple model is that if the dipole-dipole interaction is the leading |
840 |
contributor to the head group attractions, the wave vectors for the |
841 |
ripples should always be found {\it perpendicular} to the dipole |
842 |
director axis. This echoes the prediction we made earlier for simple |
843 |
elastic dipolar membranes, and may suggest experimental designs which |
844 |
will test whether this is really the case in the phosphatidylcholine |
845 |
$P_{\beta'}$ phases. The dipole director axis should also be easily |
846 |
computable for the all-atom and coarse-grained simulations that have |
847 |
been published in the literature.\cite{deVries05} |
848 |
|
849 |
Although our model is simple, it exhibits some rich and unexpected |
850 |
behaviors. It would clearly be a closer approximation to reality if |
851 |
we allowed bending motions between the dipoles and the molecular |
852 |
bodies, and if we replaced the rigid ellipsoids with ball-and-chain |
853 |
tails. However, the advantages of this simple model (large system |
854 |
sizes, 50 fs time steps) allow us to rapidly explore the phase diagram |
855 |
for a wide range of parameters. Our explanation of this rippling |
856 |
phenomenon will help us design more accurate molecular models for |
857 |
corrugated membranes and experiments to test whether or not |
858 |
dipole-dipole interactions exert an influence on membrane rippling. |
859 |
\newpage |
860 |
\bibliography{mdripple} |
861 |
\end{document} |