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\begin{document} |
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%\bibliographystyle{aps} |
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\title{} |
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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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\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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%\email[E-mail:]{gezelter@nd.edu} |
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\date{\today} |
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\begin{abstract} |
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\maketitle |
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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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\pacs{} |
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\maketitle |
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%\maketitle |
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\newpage |
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Our idea for developing a simple and reasonable lipid model to study |
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the ripple pahse of lipid bilayers is based on two facts: one is that |
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the most essential feature of lipid molecules is their amphiphilic |
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structure with polar head groups and non-polar tails. Another fact is |
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that dominant numbers of lipid molecules are very rigid in ripple |
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phase which allows the details of the lipid molecules neglectable. In |
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our model, lipid molecules are represented by rigid bodies made of one |
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head sphere with a point dipole sitting on it and one ellipsoid tail, |
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the direction of the dipole is fixed to be perpendicular to the |
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tail. The breadth and length of tail are $\sigma_0$, $3\sigma_0$. The |
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diameter of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$. The |
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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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\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 |
| 80 |
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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 |
| 87 |
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models predict higher fluidity in convex regions and that concave |
| 88 |
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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 |
| 98 |
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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 |
| 102 |
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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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Spheres interact each other with Lennard-Jones potential, ellipsoids |
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interact each other with Gay-Berne potential, dipoles interact each |
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other with typical dipole potential, spheres interact ellipsoids with |
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LJ-GB potential. All potentials are truncated at $25$ \AA and shifted |
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at $22$ \AA. |
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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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To make the simulations less expensive and to observe long-time range |
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behavior of the lipid membranes, all simulaitons were started from two |
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sepetated monolayers in the vaccum with $x-y$ anisotropic pressure |
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coupling, length of $z$ axis of the simulations was fixed to prevent |
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the shrinkage of the simulation boxes due to the free volume outside |
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of the bilayer, and a constant surface tension was applied to enable |
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the fluctuation of the surface. Periodic boundaries were used. There |
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were $480-720$ lipid molecules in simulations according to different |
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size of the heads. All the simulations were stablized for $100$ ns at |
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$300$ K. The resulted structures were solvated in the water (about |
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$6$ DPD water/lipid molecule) as the initial configurations for another |
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$30$ ns relaxation. All simulations with water were carried out at |
| 62 |
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constant pressure ($P=1$bar) by $3$D anisotropic coupling, and |
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constant surface tension ($\gamma=0.015$). Time step was |
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$50$ fs. Simulations were performed by using OOPSE package. |
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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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Snap shots show that the membrane is more corrugated with increasing |
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the size of the head groups. The surface is nearly perfect flat when |
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$\sigma_h$ is $1.20\sigma_0$. At $1.28\sigma_0$, although the surface |
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is still flat, the bilayer starts to splay inward, the upper leaf of |
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the bilayer is connected to the lower leaf with a interdigitated line |
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defect. Two periodicities with $100$\AA width were observed in the |
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simulation. This structure is very similiar to OTHER PAPER. The same |
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structure was also observed when $\sigma_h=1.41\sigma_0$. However, the |
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surface of the membrane is corrugated, and the periodicity of the |
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connection between upper and lower leaf membrane is shorter. From the |
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undulation spectrum of the surface (the exact form is in OUR PREVIOUS |
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PAPER), the corrugation is non-thermal fluctuation, and we are |
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confident to identify it as the ripple phase. The width of one ripple |
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is about $71$ \AA, and amplitude is about $7$ \AA. When |
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$\sigma_h=1.35\sigma_0$, we observed another corrugated surface with |
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$79$ \AA width and $10$ \AA amplitude. This structure is different to |
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the previous rippled surface, there is no connection between upper and |
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lower leaf of the bilayer. Each leaf of the bilayer is broken to |
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several curved pieces, the broken position is mounted into the center |
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of opposite piece in another leaf. Unlike another corrugated surface |
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in which the upper leaf of the surface is always connected to the |
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lower leaf from one direction, this ripple of this surface is |
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isotropic. Therefore, we claim this is a symmetric ripple phase. |
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\section{Computational Model} |
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\label{sec:method} |
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|
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=4in]{lipidModels} |
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\caption{Three different representations of DPPC lipid molecules, |
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including the chemical structure, an atomistic model, and the |
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head-body ellipsoidal coarse-grained model used in this |
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work.\label{fig:lipidModels}} |
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\end{figure} |
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|
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The $P_2$ order paramter is calculated to understand the phase |
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behavior quantatively. $P_2=1$ means a perfect ordered structure, and |
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$P_2=0$ means a random structure. The method can be found in OUR |
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PAPER. Fig. shows $P_2$ order paramter of the dipoles on head group |
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raises with increasing the size of the head group. When head of lipid |
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molecule is small, the membrane is flat and shows strong two |
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dimensional characters, dipoles are frustrated on orientational |
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ordering in this circumstance. Another reason is that the lipids can |
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move independently in each monolayer, it is not nessasory for the |
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direction of dipoles on one leaf is consistant to another layer, which |
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makes total order parameter is relatively low. With increasing the |
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size of head group, the surface is being more corrugated, dipoles are |
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not allowed to move freely on the surface, they are |
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localized. 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 |
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consistant to the symmetry in another layer. Furthermore, the membrane |
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tranlates from a two dimensional system to a three dimensional system |
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by the corrugation, the symmetry of the ordering for the two |
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dimensional dipoles on the head group of lipid molecules is broken, on |
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a distorted lattice, dipoles are ordered on a head to tail energy |
| 113 |
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state, the order parameter is increased dramaticly. However, the total |
| 114 |
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polarization of the system is close to zero, which is a strong |
| 115 |
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evidence it is a antiferroelectric state. The orientation of the |
| 116 |
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dipole ordering is alway perpendicular to the ripple vector. These |
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results are consistant to our previous study on similar system. The |
| 118 |
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ordering of the tails are opposite to the ordering of the dipoles on |
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head group, the $P_2$ order parameter decreases with increasing the |
| 120 |
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size of head. This indicates the surface is more curved with larger |
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head. When surface is flat, all tails are pointing to the same |
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direction, in this case, all tails are parallal to the normal of the |
| 123 |
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surface, which shares the same structure with $L_{\beta}$ phase. For the |
| 124 |
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size of head being $1.28\sigma_0$, the surface starts to splay inward, |
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however, the surface is still flat, therefore, although the order |
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parameter is lower, it still indicates a very flat surface. Further |
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increasing the size of the head, the order parameter drops dramaticly, |
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the surface is rippled. |
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Our simple molecular-scale lipid model for studying the ripple phase |
| 182 |
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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 |
| 184 |
> |
non-polar tails. Another fact is that the majority of lipid molecules |
| 185 |
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in the ripple phase are relatively rigid (i.e. gel-like) which makes |
| 186 |
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some fraction of the details of the chain dynamics negligible. Figure |
| 187 |
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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 |
| 189 |
> |
a DPPC molecule. The hydrophilic character of the head group is |
| 190 |
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largely due to the separation of charge between the nitrogen and |
| 191 |
> |
phosphate groups. The zwitterionic nature of the PC headgroups leads |
| 192 |
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to abnormally large dipole moments (as high as 20.6 D), and this |
| 193 |
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strongly polar head group interacts strongly with the solvating water |
| 194 |
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layers immediately surrounding the membrane. The hydrophobic tail |
| 195 |
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consists of fatty acid chains. In our molecular scale model, lipid |
| 196 |
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molecules have been reduced to these essential features; the fatty |
| 197 |
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acid chains are represented by an ellipsoid with a dipolar ball |
| 198 |
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perched on one end to represent the effects of the charge-separated |
| 199 |
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head group. In real PC lipids, the direction of the dipole is |
| 200 |
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nearly perpendicular to the tail, so we have fixed the direction of |
| 201 |
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the point dipole rigidly in this orientation. |
| 202 |
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|
| 203 |
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The ellipsoidal portions of the model interact via the Gay-Berne |
| 204 |
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potential which has seen widespread use in the liquid crystal |
| 205 |
+ |
community. Ayton and Voth have also used Gay-Berne ellipsoids for |
| 206 |
+ |
modeling large length-scale properties of lipid |
| 207 |
+ |
bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
| 208 |
+ |
was a single site model for the interactions of rigid ellipsoidal |
| 209 |
+ |
molecules.\cite{Gay81} It can be thought of as a modification of the |
| 210 |
+ |
Gaussian overlap model originally described by Berne and |
| 211 |
+ |
Pechukas.\cite{Berne72} The potential is constructed in the familiar |
| 212 |
+ |
form of the Lennard-Jones function using orientation-dependent |
| 213 |
+ |
$\sigma$ and $\epsilon$ parameters, |
| 214 |
+ |
\begin{equation*} |
| 215 |
+ |
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 216 |
+ |
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 217 |
+ |
{\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} |
| 219 |
+ |
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 220 |
+ |
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 221 |
+ |
\label{eq:gb} |
| 222 |
+ |
\end{equation*} |
| 223 |
|
|
| 224 |
< |
We studied the effects of interaction between head groups on the |
| 225 |
< |
structure of lipid bilayer by changing the strength of the dipole. The |
| 226 |
< |
fig. shows the $P_2$ order parameter changing with strength of the |
| 227 |
< |
dipole. Generally the dipoles on the head group are more ordered with |
| 228 |
< |
increasing the interaction between heads and more disordered with |
| 229 |
< |
decreasing the interaction between heads. When the interaction between |
| 230 |
< |
heads is weak enough, the bilayer structure is not persisted any more, |
| 231 |
< |
all lipid molecules are melted in the water. The critial value of the |
| 232 |
< |
strength of the dipole is various for different system. The perfect |
| 233 |
< |
flat surface melts at $5$ debye, the asymmetric rippled surfaces melt |
| 234 |
< |
at $8$ debye, the symmetric rippled surfaces melt at $10$ debye. This |
| 235 |
< |
indicates that the flat phase is the most stable state, the asymmetric |
| 236 |
< |
ripple phase is second stalbe state, and the symmetric ripple phase is |
| 237 |
< |
the most unstable state. The ordering of the tails is the same as the |
| 238 |
< |
ordering of the dipoles except for the flat phase. Since the surface |
| 239 |
< |
is already perfect flat, the order parameter does not change much |
| 240 |
< |
until the strength of the dipole is $15$ debye. However, the order |
| 241 |
< |
parameter decreases quickly when the strength of the dipole is further |
| 242 |
< |
increased. The head group of the lipid molecules are brought closer by |
| 243 |
< |
strenger interaction between them. For a flat surface, a mount of free |
| 244 |
< |
volume between head groups is available, when the head groups are |
| 245 |
< |
brought closer, the surface will splay outward to be a inverse |
| 246 |
< |
micelle. For rippled surfaces, there is few free volume available on |
| 247 |
< |
between the head groups, they can be closer, therefore there are |
| 248 |
< |
little effect on the structure of the membrane. Another interesting |
| 249 |
< |
fact, unlike other systems melting directly when the interaction is |
| 250 |
< |
weak enough, for $\sigma_h$ is $1.41\sigma_0$, part of the membrane |
| 251 |
< |
melts into itself first, the upper leaf of the bilayer is totally |
| 252 |
< |
interdigitated with the lower leaf, this is different with the |
| 160 |
< |
interdigitated lines in rippled phase where only one interdigitated |
| 161 |
< |
line connects the two leaves of bilayer. |
| 224 |
> |
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 225 |
> |
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
| 226 |
> |
\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ 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 |
| 230 |
> |
$\sigma_0$ are also governed by shape mixing and anisotropy variables, |
| 231 |
> |
\begin {eqnarray*} |
| 232 |
> |
\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
| 233 |
> |
\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 + |
| 235 |
> |
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}, |
| 239 |
> |
\end{eqnarray*} |
| 240 |
> |
where $l$ and $d$ describe the length and width of each uniaxial |
| 241 |
> |
ellipsoid. These shape anisotropy parameters can then be used to |
| 242 |
> |
calculate the range function, |
| 243 |
> |
\begin{equation*} |
| 244 |
> |
\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0} |
| 245 |
> |
\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 |
| 250 |
> |
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
| 251 |
> |
\right]^{-1/2} |
| 252 |
> |
\end{equation*} |
| 253 |
|
|
| 254 |
+ |
Gay-Berne ellipsoids also have an energy scaling parameter, |
| 255 |
+ |
$\epsilon^s$, which describes the well depth for two identical |
| 256 |
+ |
ellipsoids in a {\it side-by-side} configuration. Additionally, a well |
| 257 |
+ |
depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes |
| 258 |
+ |
the ratio between the well depths in the {\it end-to-end} and |
| 259 |
+ |
side-by-side configurations. As in the range parameter, a set of |
| 260 |
+ |
mixing and anisotropy variables can be used to describe the well |
| 261 |
+ |
depths for dissimilar particles, |
| 262 |
+ |
\begin {eqnarray*} |
| 263 |
+ |
\epsilon_0 & = & \sqrt{\epsilon^s_i * \epsilon^s_j} \\ \\ |
| 264 |
+ |
\epsilon^r & = & \sqrt{\epsilon^r_i * \epsilon^r_j} \\ \\ |
| 265 |
+ |
\chi' & = & \frac{1 - (\epsilon^r)^{1/\mu}}{1 + (\epsilon^r)^{1/\mu}} |
| 266 |
+ |
\\ \\ |
| 267 |
+ |
\alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1} |
| 268 |
+ |
\end{eqnarray*} |
| 269 |
+ |
The form of the strength function is somewhat complicated, |
| 270 |
+ |
\begin {eqnarray*} |
| 271 |
+ |
\epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = & |
| 272 |
+ |
\epsilon_{0} \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j}) |
| 273 |
+ |
\epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 274 |
+ |
\hat{r}}_{ij}) \\ \\ |
| 275 |
+ |
\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 |
+ |
= & |
| 280 |
+ |
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 |
| 285 |
+ |
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
| 286 |
+ |
\end {eqnarray*} |
| 287 |
+ |
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 |
| 290 |
+ |
parameters $\mu$ and $\nu$ for modeling liquid crystal molecules. An |
| 291 |
+ |
excellent overview of the computational methods that can be used to |
| 292 |
+ |
efficiently compute forces and torques for this potential can be found |
| 293 |
+ |
in Ref. \citen{Golubkov06} |
| 294 |
|
|
| 295 |
< |
Fig. shows the changing of the order parameter with temperature. The |
| 296 |
< |
behavior of the $P_2$ orderparamter is straightforword. Systems are |
| 297 |
< |
more ordered at low temperature, and more disordered at high |
| 298 |
< |
temperature. When the temperature is high enough, the membranes are |
| 299 |
< |
discontinuted. The structures are stable during the changing of the |
| 300 |
< |
temperature. Since our model lacks the detail information for tails of |
| 301 |
< |
lipid molecules, we did not simulate the fluid phase with a melted |
| 302 |
< |
fatty chains. Moreover, the formation of the tilted $L_{\beta'}$ phase |
| 303 |
< |
also depends on the organization of fatty groups on tails, we did not |
| 304 |
< |
observe it either. |
| 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} |
| 301 |
> |
d & < & l \\ |
| 302 |
> |
\epsilon^{r} & < & 1 |
| 303 |
> |
\end{array} |
| 304 |
> |
\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} |
