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\begin{document} |
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%\bibliographystyle{aps} |
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\title{Dipolar Ordering in Molecular-scale Models of the Ripple Phase |
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in Lipid Membranes} |
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\author{Xiuquan Sun and J. Daniel Gezelter} |
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\email[E-mail:]{gezelter@nd.edu} |
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\affiliation{Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame, \\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\begin{abstract} |
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The ripple phase in phosphatidylcholine (PC) bilayers has never been |
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completely explained. |
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\end{abstract} |
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\pacs{} |
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\maketitle |
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\section{Introduction} |
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\label{sec:Int} |
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Fully hydrated lipids will aggregate spontaneously to form bilayers |
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which exhibit a variety of phases depending on their temperatures and |
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compositions. Among these phases, a periodic rippled phase |
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($P_{\beta'}$) appears as an intermediate phase between the gel |
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($L_\beta$) and fluid ($L_{\alpha}$) phases for relatively pure |
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phosphatidylcholine (PC) bilayers. The ripple phase has attracted |
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substantial experimental interest over the past 30 years. Most |
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structural information of the ripple phase has been obtained by the |
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X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
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microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
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et al.} used atomic force microscopy (AFM) to observe ripple phase |
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morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
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experimental results provide strong support for a 2-dimensional |
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hexagonal packing lattice of the lipid molecules within the ripple |
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phase. This is a notable change from the observed lipid packing |
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within the gel phase.~\cite{Cevc87} |
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|
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A number of theoretical models have been presented to explain the |
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formation of the ripple phase. Marder {\it et al.} used a |
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curvature-dependent Landau-de Gennes free-energy functional to predict |
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a rippled phase.~\cite{Marder84} This model and other related continuum |
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models predict higher fluidity in convex regions and that concave |
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portions of the membrane correspond to more solid-like regions. |
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Carlson and Sethna used a packing-competition model (in which head |
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groups and chains have competing packing energetics) to predict the |
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formation of a ripple-like phase. Their model predicted that the |
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high-curvature portions have lower-chain packing and correspond to |
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more fluid-like regions. Goldstein and Leibler used a mean-field |
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approach with a planar model for {\em inter-lamellar} interactions to |
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predict rippling in multilamellar phases.~\cite{Goldstein88} McCullough |
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and Scott proposed that the {\em anisotropy of the nearest-neighbor |
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interactions} coupled to hydrophobic constraining forces which |
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restrict height differences between nearest neighbors is the origin of |
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the ripple phase.~\cite{McCullough90} Lubensky and MacKintosh |
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introduced a Landau theory for tilt order and curvature of a single |
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membrane and concluded that {\em coupling of molecular tilt to membrane |
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curvature} is responsible for the production of |
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ripples.~\cite{Lubensky93} Misbah, Duplat and Houchmandzadeh proposed |
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that {\em inter-layer dipolar interactions} can lead to ripple |
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instabilities.~\cite{Misbah98} Heimburg presented a {\em coexistence |
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model} for ripple formation in which he postulates that fluid-phase |
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line defects cause sharp curvature between relatively flat gel-phase |
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regions.~\cite{Heimburg00} Kubica has suggested that a lattice model of |
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polar head groups could be valuable in trying to understand bilayer |
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phase formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations |
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of lamellar stacks of hexagonal lattices to show that large headgroups |
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and molecular tilt with respect to the membrane normal vector can |
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cause bulk rippling.~\cite{Bannerjee02} |
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|
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In contrast, few large-scale molecular modelling studies have been |
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done due to the large size of the resulting structures and the time |
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required for the phases of interest to develop. With all-atom (and |
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even unified-atom) simulations, only one period of the ripple can be |
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observed and only for timescales in the range of 10-100 ns. One of |
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the most interesting molecular simulations was carried out by De Vries |
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{\it et al.}~\cite{deVries05}. According to their simulation results, |
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the ripple consists of two domains, one resembling the gel bilayer, |
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while in the other, the two leaves of the bilayer are fully |
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interdigitated. The mechanism for the formation of the ripple phase |
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suggested by their work is a packing competition between the head |
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groups and the tails of the lipid molecules.\cite{Carlson87} Recently, |
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the ripple phase has also been studied by Lenz and Schmid using Monte |
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Carlo simulations.\cite{Lenz07} Their structures are similar to the De |
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Vries {\it et al.} structures except that the connection between the |
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two leaves of the bilayer is a narrow interdigitated line instead of |
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the fully interdigitated domain. The symmetric ripple phase was also |
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observed by Lenz {\it et al.}, and their work supports other claims |
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that the mismatch between the size of the head group and tail of the |
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lipid molecules is the driving force for the formation of the ripple |
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phase. Ayton and Voth have found significant undulations in |
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zero-surface-tension states of membranes simulated via dissipative |
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particle dynamics, but their results are consistent with purely |
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thermal undulations.~\cite{Ayton02} |
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|
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Although the organization of the tails of lipid molecules are |
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addressed by these molecular simulations and the packing competition |
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between headgroups and tails is strongly implicated as the primary |
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driving force for ripple formation, questions about the ordering of |
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the head groups in ripple phase has not been settled. |
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|
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In a recent paper, we presented a simple ``web of dipoles'' spin |
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lattice model which provides some physical insight into relationship |
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between dipolar ordering and membrane buckling.\cite{Sun2007} We found |
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that dipolar elastic membranes can spontaneously buckle, forming |
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ripple-like topologies. The driving force for the buckling in dipolar |
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elastic membranes the antiferroelectric ordering of the dipoles, and |
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this was evident in the ordering of the dipole director axis |
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perpendicular to the wave vector of the surface ripples. A similiar |
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phenomenon has also been observed by Tsonchev {\it et al.} in their |
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work on the spontaneous formation of dipolar peptide chains into |
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curved nano-structures.\cite{Tsonchev04,Tsonchev04II} |
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|
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In this paper, we construct a somewhat more realistic molecular-scale |
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lipid model than our previous ``web of dipoles'' and use molecular |
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dynamics simulations to elucidate the role of the head group dipoles |
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in the formation and morphology of the ripple phase. We describe our |
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model and computational methodology in section \ref{sec:method}. |
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Details on the simulations are presented in section |
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\ref{sec:experiment}, with results following in section |
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\ref{sec:results}. A final discussion of the role of dipolar heads in |
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the ripple formation can be found in section |
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\ref{sec:discussion}. |
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\section{Computational Model} |
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\label{sec:method} |
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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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Our simple molecular-scale lipid model for studying the ripple phase |
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is based on two facts: one is that the most essential feature of lipid |
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molecules is their amphiphilic structure with polar head groups and |
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non-polar tails. Another fact is that the majority of lipid molecules |
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in the ripple phase are relatively rigid (i.e. gel-like) which makes |
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some fraction of the details of the chain dynamics negligible. Figure |
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\ref{fig:lipidModels} shows the molecular strucure of a DPPC |
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molecule, as well as atomistic and molecular-scale representations of |
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a DPPC molecule. The hydrophilic character of the head group is |
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largely due to the separation of charge between the nitrogen and |
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phosphate groups. The zwitterionic nature of the PC headgroups leads |
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to abnormally large dipole moments (as high as 20.6 D), and this |
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strongly polar head group interacts strongly with the solvating water |
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layers immediately surrounding the membrane. The hydrophobic tail |
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consists of fatty acid chains. In our molecular scale model, lipid |
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molecules have been reduced to these essential features; the fatty |
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acid chains are represented by an ellipsoid with a dipolar ball |
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perched on one end to represent the effects of the charge-separated |
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head group. In real PC lipids, the direction of the dipole is |
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nearly perpendicular to the tail, so we have fixed the direction of |
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the point dipole rigidly in this orientation. |
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|
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The ellipsoidal portions of the model interact via the Gay-Berne |
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potential which has seen widespread use in the liquid crystal |
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community. Ayton and Voth have also used Gay-Berne ellipsoids for |
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modelling large length-scale properties of lipid |
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bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
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was a single site model for the interactions of rigid ellipsoidal |
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molecules.\cite{Gay81} It can be thought of as a modification of the |
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Gaussian overlap model originally described by Berne and |
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Pechukas.\cite{Berne72} The potential is constructed in the familiar |
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form of the Lennard-Jones function using orientation-dependent |
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$\sigma$ and $\epsilon$ parameters, |
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\begin{eqnarray*} |
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V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
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r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
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{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
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-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
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\label{eq:gb} |
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\end{eqnarray*} |
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The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
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\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
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are dependent on the relative orientations of the two molecules (${\bf |
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\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
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intermolecular separation (${\bf \hat{r}}_{ij}$). $\sigma$ and |
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$\sigma_0$ are also governed by shape mixing and anisotropy variables, |
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\begin {equation} |
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\begin{array}{rcl} |
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\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
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\chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 - |
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d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^2 + |
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d_j^2 \right)}\right]^{1/2} \\ \\ |
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\alpha^2 & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 + |
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d_i^2 \right)}{\left( l_j^2 - d_j^2 \right) \left(l_i^2 + |
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d_j^2 \right)}\right]^{1/2}, |
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\end{array} |
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\end{equation} |
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where $l$ and $d$ describe the length and width of each uniaxial |
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ellipsoid. These shape anisotropy parameters can then be used to |
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calculate the range function, |
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\begin {equation} |
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\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0} |
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\left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
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\right]^{-1/2} |
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\end{equation} |
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Gay-Berne ellipsoids also have an energy scaling parameter, |
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$\epsilon^s$, which describes the well depth for two identical |
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ellipsoids in a {\it side-by-side} configuration. Additionaly, a well |
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depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes |
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the ratio between the well depths in the {\it end-to-end} and |
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side-by-side configurations. As in the range parameter, a set of |
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mixing and anisotropy variables can be used to describe the well |
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depths for dissimilar particles, |
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\begin {eqnarray*} |
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\epsilon_0 & = & \sqrt{\epsilon^s_i * \epsilon^s_j} \\ \\ |
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\epsilon^r & = & \sqrt{\epsilon^r_i * \epsilon^r_j} \\ \\ |
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\chi' & = & \frac{1 - (\epsilon^r)^{1/\mu}}{1 + (\epsilon^r)^{1/\mu}} |
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\\ \\ |
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\alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1} |
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\end{eqnarray*} |
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The form of the strength function is somewhat complicated, |
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\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 |
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\hat{r}}_{ij}) \\ \\ |
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\epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = & |
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\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
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\hat{u}}_{j})^{2}\right]^{-1/2} \\ \\ |
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\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & |
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= & |
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1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
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\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
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\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
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\end {eqnarray*} |
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although many of the quantities and derivatives are identical with |
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those obtained for the range parameter. Ref. \onlinecite{Luckhurst90} |
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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 |
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in Ref. \onlinecite{Golubkov06} |
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The choices of parameters we have used in this study correspond to a |
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shape anisotropy of 3 for the chain portion of the molecule. In |
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principle, this could be varied to allow for modeling of longer or |
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shorter chain lipid molecules. For these prolate ellipsoids, we have: |
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\begin{equation} |
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\begin{array}{rcl} |
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d & < & l \\ |
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\epsilon^{r} & < & 1 |
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\end{array} |
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\end{equation} |
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|
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=4in]{2lipidModel} |
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\caption{The parameters defining the behavior of the lipid |
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models. $l / d$ is the ratio of the head group to body diameter. |
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Molecular bodies had a fixed aspect ratio of 3.0. The solvent model |
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was a simplified 4-water bead ($\sigma_w \approx d$) that has been |
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used in other coarse-grained (DPD) simulations. The dipolar strength |
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(and the temperature and pressure) were the only other parameters that |
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were varied systematically.\label{fig:lipidModel}} |
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\end{figure} |
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|
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To take into account the permanent dipolar interactions of the |
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zwitterionic head groups, we place fixed dipole moments $\mu_{i}$ at |
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one end of the Gay-Berne particles. The dipoles are oriented at an |
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angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
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are protected by a head ``bead'' with a range parameter which we have |
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varied between $1.20 d$ and $1.41 d$. The head groups interact with |
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each other using a combination of Lennard-Jones, |
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\begin{eqnarray*} |
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V_{ij} = 4\epsilon \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
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\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
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\end{eqnarray*} |
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and dipole-dipole, |
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\begin{eqnarray*} |
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gezelter |
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V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
301 |
|
|
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
302 |
|
|
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
303 |
xsun |
3174 |
\end{eqnarray*} |
304 |
gezelter |
3195 |
potentials. |
305 |
|
|
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
306 |
|
|
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
307 |
gezelter |
3199 |
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. |
308 |
gezelter |
3195 |
|
309 |
|
|
For the interaction between nonequivalent uniaxial ellipsoids (in this |
310 |
gezelter |
3199 |
case, between spheres and ellipsoids), the spheres are treated as |
311 |
|
|
ellipsoids with an aspect ratio of 1 ($d = l$) and with an well depth |
312 |
|
|
ratio of 1 ($\epsilon^e = \epsilon^s$). The form of the Gay-Berne |
313 |
|
|
potential we are using was generalized by Cleaver {\it et al.} and is |
314 |
|
|
appropriate for dissimilar uniaxial ellipsoids.\cite{Cleaver96} |
315 |
xsun |
3147 |
|
316 |
gezelter |
3199 |
The solvent model in our simulations is identical to one used by |
317 |
|
|
Marrink {\it et al.} in their dissipative particle dynamics (DPD) |
318 |
|
|
simulation of lipid bilayers.\cite{Marrink04} This solvent bead is a single |
319 |
|
|
site that represents four water molecules (m = 72 amu) and has |
320 |
|
|
comparable density and diffusive behavior to liquid water. However, |
321 |
|
|
since there are no electrostatic sites on these beads, this solvent |
322 |
|
|
model cannot replicate the dielectric properties of water. |
323 |
xsun |
3198 |
\begin{table*} |
324 |
|
|
\begin{minipage}{\linewidth} |
325 |
|
|
\begin{center} |
326 |
gezelter |
3199 |
\caption{Potential parameters used for molecular-scale coarse-grained |
327 |
|
|
lipid simulations} |
328 |
|
|
\begin{tabular}{llccc} |
329 |
xsun |
3198 |
\hline |
330 |
gezelter |
3199 |
& & Head & Chain & Solvent \\ |
331 |
xsun |
3198 |
\hline |
332 |
gezelter |
3199 |
$d$ (\AA) & & varied & 4.6 & 4.7 \\ |
333 |
|
|
$l$ (\AA) & & 1 & 3 & 1 \\ |
334 |
|
|
$\epsilon^s$ (kcal/mol) & & 0.185 & 1.0 & 0.8 \\ |
335 |
|
|
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 \\ |
336 |
|
|
$m$ (amu) & & 196 & 760 & 72.06112 \\ |
337 |
|
|
$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
338 |
|
|
\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
339 |
|
|
\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
340 |
|
|
\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
341 |
|
|
$\mu$ (Debye) & & varied & 0 & 0 \\ |
342 |
xsun |
3198 |
\end{tabular} |
343 |
|
|
\label{tab:parameters} |
344 |
|
|
\end{center} |
345 |
|
|
\end{minipage} |
346 |
|
|
\end{table*} |
347 |
gezelter |
3195 |
|
348 |
gezelter |
3199 |
A switching function has been applied to all potentials to smoothly |
349 |
|
|
turn off the interactions between a range of $22$ and $25$ \AA. |
350 |
gezelter |
3186 |
|
351 |
gezelter |
3196 |
\section{Experimental Methodology} |
352 |
xsun |
3174 |
\label{sec:experiment} |
353 |
xsun |
3147 |
|
354 |
gezelter |
3196 |
To create unbiased bilayers, all simulations were started from two |
355 |
|
|
perfectly flat monolayers separated by a 20 \AA\ gap between the |
356 |
|
|
molecular bodies of the upper and lower leaves. The separated |
357 |
|
|
monolayers were evolved in a vaccum with $x-y$ anisotropic pressure |
358 |
xsun |
3174 |
coupling. The length of $z$ axis of the simulations was fixed and a |
359 |
|
|
constant surface tension was applied to enable real fluctuations of |
360 |
gezelter |
3196 |
the bilayer. Periodic boundaries were used, and $480-720$ lipid |
361 |
|
|
molecules were present in the simulations depending on the size of the |
362 |
|
|
head beads. The two monolayers spontaneously collapse into bilayer |
363 |
|
|
structures within 100 ps, and following this collapse, all systems |
364 |
|
|
were equlibrated for $100$ ns at $300$ K. |
365 |
xsun |
3147 |
|
366 |
gezelter |
3196 |
The resulting structures were then solvated at a ratio of $6$ DPD |
367 |
|
|
solvent beads (24 water molecules) per lipid. These configurations |
368 |
|
|
were then equilibrated for another $30$ ns. All simulations with |
369 |
|
|
solvent were carried out at constant pressure ($P=1$ atm) by $3$D |
370 |
|
|
anisotropic coupling, and constant surface tension ($\gamma=0.015$ |
371 |
|
|
UNIT). Given the absence of fast degrees of freedom in this model, a |
372 |
|
|
timestep of $50$ fs was utilized. Data collection for structural |
373 |
|
|
properties of the bilayers was carried out during a final 5 ns run |
374 |
|
|
following the solvent equilibration. All simulations were performed |
375 |
|
|
using the OOPSE molecular modeling program.\cite{Meineke05} |
376 |
|
|
|
377 |
|
|
\section{Results} |
378 |
xsun |
3174 |
\label{sec:results} |
379 |
xsun |
3147 |
|
380 |
xsun |
3174 |
Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
381 |
|
|
more corrugated increasing size of the head groups. The surface is |
382 |
|
|
nearly flat when $\sigma_h=1.20\sigma_0$. With |
383 |
|
|
$\sigma_h=1.28\sigma_0$, although the surface is still flat, the |
384 |
|
|
bilayer starts to splay inward; the upper leaf of the bilayer is |
385 |
|
|
connected to the lower leaf with an interdigitated line defect. Two |
386 |
|
|
periodicities with $100$ \AA\ width were observed in the |
387 |
|
|
simulation. This structure is very similiar to the structure observed |
388 |
|
|
by de Vries and Lenz {\it et al.}. The same basic structure is also |
389 |
|
|
observed when $\sigma_h=1.41\sigma_0$, but the wavelength of the |
390 |
|
|
surface corrugations depends sensitively on the size of the ``head'' |
391 |
|
|
beads. From the undulation spectrum, the corrugation is clearly |
392 |
|
|
non-thermal. |
393 |
|
|
\begin{figure}[htb] |
394 |
|
|
\centering |
395 |
gezelter |
3199 |
\includegraphics[width=4in]{phaseCartoon} |
396 |
xsun |
3174 |
\caption{A sketch to discribe the structure of the phases observed in |
397 |
|
|
our simulations.\label{fig:phaseCartoon}} |
398 |
|
|
\end{figure} |
399 |
xsun |
3147 |
|
400 |
xsun |
3174 |
When $\sigma_h=1.35\sigma_0$, we observed another corrugated surface |
401 |
|
|
morphology. This structure is different from the asymmetric rippled |
402 |
|
|
surface; there is no interdigitation between the upper and lower |
403 |
|
|
leaves of the bilayer. Each leaf of the bilayer is broken into several |
404 |
|
|
hemicylinderical sections, and opposite leaves are fitted together |
405 |
|
|
much like roof tiles. Unlike the surface in which the upper |
406 |
|
|
hemicylinder is always interdigitated on the leading or trailing edge |
407 |
|
|
of lower hemicylinder, the symmetric ripple has no prefered direction. |
408 |
|
|
The corresponding cartoons are shown in Figure |
409 |
|
|
\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
410 |
|
|
different phases. Figure \ref{fig:phaseCartoon} (a) is the flat phase, |
411 |
|
|
(b) is the asymmetric ripple phase corresponding to the lipid |
412 |
|
|
organization when $\sigma_h=1.28\sigma_0$ and $\sigma_h=1.41\sigma_0$, |
413 |
|
|
and (c) is the symmetric ripple phase observed when |
414 |
|
|
$\sigma_h=1.35\sigma_0$. In symmetric ripple phase, the bilayer is |
415 |
|
|
continuous everywhere on the whole membrane, however, in asymmetric |
416 |
|
|
ripple phase, the bilayer is intermittent domains connected by thin |
417 |
|
|
interdigitated monolayer which consists of upper and lower leaves of |
418 |
|
|
the bilayer. |
419 |
|
|
\begin{table*} |
420 |
|
|
\begin{minipage}{\linewidth} |
421 |
|
|
\begin{center} |
422 |
|
|
\caption{} |
423 |
|
|
\begin{tabular}{lccc} |
424 |
|
|
\hline |
425 |
|
|
$\sigma_h/\sigma_0$ & type of phase & $\lambda/\sigma_0$ & $A/\sigma_0$\\ |
426 |
|
|
\hline |
427 |
|
|
1.20 & flat & N/A & N/A \\ |
428 |
|
|
1.28 & asymmetric flat & 21.7 & N/A \\ |
429 |
|
|
1.35 & symmetric ripple & 17.2 & 2.2 \\ |
430 |
|
|
1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
431 |
|
|
\end{tabular} |
432 |
|
|
\label{tab:property} |
433 |
|
|
\end{center} |
434 |
|
|
\end{minipage} |
435 |
|
|
\end{table*} |
436 |
xsun |
3147 |
|
437 |
xsun |
3174 |
The membrane structures and the reduced wavelength $\lambda/\sigma_0$, |
438 |
|
|
reduced amplitude $A/\sigma_0$ of the ripples are summerized in Table |
439 |
|
|
\ref{tab:property}. The wavelength range is $15~21$ and the amplitude |
440 |
|
|
is $1.5$ for asymmetric ripple and $2.2$ for symmetric ripple. These |
441 |
|
|
values are consistent to the experimental results. Note, the |
442 |
|
|
amplitudes are underestimated without the melted tails in our |
443 |
|
|
simulations. |
444 |
|
|
|
445 |
gezelter |
3195 |
\begin{figure}[htb] |
446 |
|
|
\centering |
447 |
gezelter |
3199 |
\includegraphics[width=4in]{topDown} |
448 |
gezelter |
3195 |
\caption{Top views of the flat (upper), asymmetric ripple (middle), |
449 |
|
|
and symmetric ripple (lower) phases. Note that the head-group dipoles |
450 |
|
|
have formed head-to-tail chains in all three of these phases, but in |
451 |
|
|
the two rippled phases, the dipolar chains are all aligned |
452 |
|
|
{\it perpendicular} to the direction of the ripple. The flat membrane |
453 |
|
|
has multiple point defects in the dipolar orientational ordering, and |
454 |
|
|
the dipolar ordering on the lower leaf of the bilayer can be in a |
455 |
|
|
different direction from the upper leaf.\label{fig:topView}} |
456 |
|
|
\end{figure} |
457 |
|
|
|
458 |
xsun |
3174 |
The $P_2$ order paramters (for molecular bodies and head group |
459 |
|
|
dipoles) have been calculated to clarify the ordering in these phases |
460 |
|
|
quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$ |
461 |
|
|
implies orientational randomization. Figure \ref{fig:rP2} shows the |
462 |
|
|
$P_2$ order paramter of the dipoles on head group rising with |
463 |
|
|
increasing head group size. When the heads of the lipid molecules are |
464 |
|
|
small, the membrane is flat. The dipolar ordering is essentially |
465 |
xsun |
3189 |
frustrated on orientational ordering in this circumstance. Figure |
466 |
gezelter |
3195 |
\ref{fig:topView} shows the snapshots of the top view for the flat system |
467 |
xsun |
3189 |
($\sigma_h=1.20\sigma$) and rippled system |
468 |
|
|
($\sigma_h=1.41\sigma$). The pointing direction of the dipoles on the |
469 |
|
|
head groups are represented by two colored half spheres from blue to |
470 |
|
|
yellow. For flat surfaces, the system obviously shows frustration on |
471 |
|
|
the dipolar ordering, there are kinks on the edge of defferent |
472 |
|
|
domains. Another reason is that the lipids can move independently in |
473 |
|
|
each monolayer, it is not nessasory for the direction of dipoles on |
474 |
|
|
one leaf is consistant to another layer, which makes total order |
475 |
|
|
parameter is relatively low. With increasing head group size, the |
476 |
|
|
surface is corrugated, and dipoles do not move as freely on the |
477 |
xsun |
3174 |
surface. Therefore, the translational freedom of lipids in one layer |
478 |
xsun |
3147 |
is dependent upon the position of lipids in another layer, as a |
479 |
xsun |
3174 |
result, the symmetry of the dipoles on head group in one layer is tied |
480 |
|
|
to the symmetry in the other layer. Furthermore, as the membrane |
481 |
|
|
deforms from two to three dimensions due to the corrugation, the |
482 |
|
|
symmetry of the ordering for the dipoles embedded on each leaf is |
483 |
|
|
broken. The dipoles then self-assemble in a head-tail configuration, |
484 |
|
|
and the order parameter increases dramaticaly. However, the total |
485 |
|
|
polarization of the system is still close to zero. This is strong |
486 |
|
|
evidence that the corrugated structure is an antiferroelectric |
487 |
xsun |
3189 |
state. From the snapshot in Figure \ref{}, the dipoles arrange as |
488 |
|
|
arrays along $Y$ axis and fall into head-to-tail configuration in each |
489 |
|
|
line, but every $3$ or $4$ lines of dipoles change their direction |
490 |
|
|
from neighbour lines. The system shows antiferroelectric |
491 |
|
|
charactoristic as a whole. The orientation of the dipolar is always |
492 |
|
|
perpendicular to the ripple wave vector. These results are consistent |
493 |
|
|
with our previous study on dipolar membranes. |
494 |
xsun |
3147 |
|
495 |
xsun |
3174 |
The ordering of the tails is essentially opposite to the ordering of |
496 |
|
|
the dipoles on head group. The $P_2$ order parameter decreases with |
497 |
|
|
increasing head size. This indicates the surface is more curved with |
498 |
|
|
larger head groups. When the surface is flat, all tails are pointing |
499 |
|
|
in the same direction; in this case, all tails are parallel to the |
500 |
|
|
normal of the surface,(making this structure remindcent of the |
501 |
|
|
$L_{\beta}$ phase. Increasing the size of the heads, results in |
502 |
|
|
rapidly decreasing $P_2$ ordering for the molecular bodies. |
503 |
gezelter |
3199 |
|
504 |
xsun |
3174 |
\begin{figure}[htb] |
505 |
|
|
\centering |
506 |
|
|
\includegraphics[width=\linewidth]{rP2} |
507 |
|
|
\caption{The $P_2$ order parameter as a funtion of the ratio of |
508 |
|
|
$\sigma_h$ to $\sigma_0$.\label{fig:rP2}} |
509 |
|
|
\end{figure} |
510 |
xsun |
3147 |
|
511 |
xsun |
3174 |
We studied the effects of the interactions between head groups on the |
512 |
|
|
structure of lipid bilayer by changing the strength of the dipole. |
513 |
|
|
Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with |
514 |
|
|
increasing strength of the dipole. Generally the dipoles on the head |
515 |
|
|
group are more ordered by increase in the strength of the interaction |
516 |
|
|
between heads and are more disordered by decreasing the interaction |
517 |
|
|
stength. When the interaction between the heads is weak enough, the |
518 |
|
|
bilayer structure does not persist; all lipid molecules are solvated |
519 |
|
|
directly in the water. The critial value of the strength of the dipole |
520 |
|
|
depends on the head size. The perfectly flat surface melts at $5$ |
521 |
xsun |
3182 |
$0.03$ debye, the asymmetric rippled surfaces melt at $8$ $0.04$ |
522 |
|
|
$0.03$ debye, the symmetric rippled surfaces melt at $10$ $0.04$ |
523 |
|
|
debye. The ordering of the tails is the same as the ordering of the |
524 |
|
|
dipoles except for the flat phase. Since the surface is already |
525 |
|
|
perfect flat, the order parameter does not change much until the |
526 |
|
|
strength of the dipole is $15$ debye. However, the order parameter |
527 |
|
|
decreases quickly when the strength of the dipole is further |
528 |
|
|
increased. The head groups of the lipid molecules are brought closer |
529 |
|
|
by stronger interactions between them. For a flat surface, a large |
530 |
|
|
amount of free volume between the head groups is available, but when |
531 |
|
|
the head groups are brought closer, the tails will splay outward, |
532 |
|
|
forming an inverse micelle. When $\sigma_h=1.28\sigma_0$, the $P_2$ |
533 |
|
|
order parameter decreases slightly after the strength of the dipole is |
534 |
|
|
increased to $16$ debye. For rippled surfaces, there is less free |
535 |
|
|
volume available between the head groups. Therefore there is little |
536 |
|
|
effect on the structure of the membrane due to increasing dipolar |
537 |
|
|
strength. However, the increase of the $P_2$ order parameter implies |
538 |
|
|
the membranes are flatten by the increase of the strength of the |
539 |
|
|
dipole. Unlike other systems that melt directly when the interaction |
540 |
|
|
is weak enough, for $\sigma_h=1.41\sigma_0$, part of the membrane |
541 |
|
|
melts into itself first. The upper leaf of the bilayer becomes totally |
542 |
|
|
interdigitated with the lower leaf. This is different behavior than |
543 |
|
|
what is exhibited with the interdigitated lines in the rippled phase |
544 |
|
|
where only one interdigitated line connects the two leaves of bilayer. |
545 |
xsun |
3174 |
\begin{figure}[htb] |
546 |
|
|
\centering |
547 |
|
|
\includegraphics[width=\linewidth]{sP2} |
548 |
|
|
\caption{The $P_2$ order parameter as a funtion of the strength of the |
549 |
|
|
dipole.\label{fig:sP2}} |
550 |
|
|
\end{figure} |
551 |
xsun |
3147 |
|
552 |
xsun |
3174 |
Figure \ref{fig:tP2} shows the dependence of the order parameter on |
553 |
|
|
temperature. The behavior of the $P_2$ order paramter is |
554 |
|
|
straightforward. Systems are more ordered at low temperature, and more |
555 |
|
|
disordered at high temperatures. When the temperature is high enough, |
556 |
xsun |
3182 |
the membranes are instable. For flat surfaces ($\sigma_h=1.20\sigma_0$ |
557 |
|
|
and $\sigma_h=1.28\sigma_0$), when the temperature is increased to |
558 |
|
|
$310$, the $P_2$ order parameter increases slightly instead of |
559 |
|
|
decreases like ripple surface. This is an evidence of the frustration |
560 |
|
|
of the dipolar ordering in each leaf of the lipid bilayer, at low |
561 |
|
|
temperature, the systems are locked in a local minimum energy state, |
562 |
|
|
with increase of the temperature, the system can jump out the local |
563 |
|
|
energy well to find the lower energy state which is the longer range |
564 |
|
|
orientational ordering. Like the dipolar ordering of the flat |
565 |
|
|
surfaces, the ordering of the tails of the lipid molecules for ripple |
566 |
|
|
membranes ($\sigma_h=1.35\sigma_0$ and $\sigma_h=1.41\sigma_0$) also |
567 |
|
|
show some nonthermal characteristic. With increase of the temperature, |
568 |
|
|
the $P_2$ order parameter decreases firstly, and increases afterward |
569 |
|
|
when the temperature is greater than $290 K$. The increase of the |
570 |
|
|
$P_2$ order parameter indicates a more ordered structure for the tails |
571 |
|
|
of the lipid molecules which corresponds to a more flat surface. Since |
572 |
|
|
our model lacks the detailed information on lipid tails, we can not |
573 |
|
|
simulate the fluid phase with melted fatty acid chains. Moreover, the |
574 |
|
|
formation of the tilted $L_{\beta'}$ phase also depends on the |
575 |
|
|
organization of fatty groups on tails. |
576 |
xsun |
3174 |
\begin{figure}[htb] |
577 |
|
|
\centering |
578 |
|
|
\includegraphics[width=\linewidth]{tP2} |
579 |
|
|
\caption{The $P_2$ order parameter as a funtion of |
580 |
|
|
temperature.\label{fig:tP2}} |
581 |
|
|
\end{figure} |
582 |
xsun |
3147 |
|
583 |
xsun |
3174 |
\section{Discussion} |
584 |
|
|
\label{sec:discussion} |
585 |
xsun |
3147 |
|
586 |
gezelter |
3199 |
\newpage |
587 |
xsun |
3147 |
\bibliography{mdripple} |
588 |
|
|
\end{document} |