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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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\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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As one of the most important components in the formation of the |
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biomembrane, lipid molecules attracted numerous studies in the past |
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several decades. Due to their amphiphilic structure, when dispersed in |
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water, lipids can self-assemble to construct a bilayer structure. The |
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phase behavior of lipid membrane is well understood. The gel-fluid |
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phase transition is known as main phase transition. However, there is |
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an intermediate phase between gel and fluid phase for some lipid (like |
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phosphatidycholine (PC)) membranes. This intermediate phase |
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distinguish itself from other phases by its corrugated membrane |
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surface, therefore this phase is termed ``ripple'' ($P_{\beta '}$) |
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phase. The phase transition between gel-fluid and ripple phase is |
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called pretransition. Since the pretransition usually occurs in room |
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temperature, there might be some important biofuntions carried by the |
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ripple phase for the living organism. |
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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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The ripple phase is observed experimentally by x-ray diffraction |
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~\cite{Sun96,Katsaras00}, freeze-fracture electron microscopy |
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(FFEM)~\cite{Copeland80,Meyer96}, and atomic force microscopy (AFM) |
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recently~\cite{Kaasgaard03}. The experimental studies suggest two |
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kinds of ripple structures: asymmetric (sawtooth like) and symmetric |
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(sinusoidal like) ripple phases. Substantial number of theoretical |
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explaination applied on the formation of the ripple |
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phases~\cite{Marder84,Carlson87,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02}. |
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In contrast, few molecular modelling have been done due to the large |
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size of the resulting structures and the time required for the phases |
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of interest to develop. One of the interesting molecular simulations |
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was carried out by De Vries and Marrink {\it et |
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al.}~\cite{deVries05}. According to their dynamic simulation results, |
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the ripple consists of two domains, one is gel bilayer, and in the |
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other domain, the upper and lower leaves of the bilayer are fully |
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interdigitated. The mechanism of the formation of the ripple phase in |
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their work suggests the theory that the packing competition between |
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head group and tail of lipid molecules is the driving force for the |
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formation of the ripple phase~\cite{Carlson87}. Recently, the ripple |
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phase is also studied by using monte carlo simulation~\cite{Lenz07}, |
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the ripple structure is similar to the results of Marrink except that |
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the connection of the upper and lower leaves of the bilayer is an |
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interdigitated line instead of the fully interdigitated |
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domain. Furthermore, the symmetric ripple phase was also observed in |
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their work. They claimed the mismatch between the size of the head |
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group and tail of the lipid molecules is the driving force for the |
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formation of the ripple phase. |
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In contrast, few large-scale molecular modelling studies have been |
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done due to the large size of the resulting structures and the time |
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required for the phases of interest to develop. With all-atom (and |
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even unified-atom) simulations, only one period of the ripple can be |
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observed and only for timescales in the range of 10-100 ns. One of |
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the most interesting molecular simulations was carried out by De Vries |
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{\it et al.}~\cite{deVries05}. According to their simulation results, |
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the ripple consists of two domains, one resembling the gel bilayer, |
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while in the other, the two leaves of the bilayer are fully |
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interdigitated. The mechanism for the formation of the ripple phase |
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suggested by their work is a packing competition between the head |
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groups and the tails of the lipid molecules.\cite{Carlson87} Recently, |
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the ripple phase has also been studied by the XXX group using Monte |
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Carlo simulations.\cite{Lenz07} Their structures are similar to the De |
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Vries {\it et al.} structures except that the connection between the |
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two leaves of the bilayer is a narrow interdigitated line instead of |
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the fully interdigitated domain. The symmetric ripple phase was also |
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observed by Lenz {\it et al.}, and their work supports other claims |
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that the mismatch between the size of the head group and tail of the |
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lipid molecules is the driving force for the formation of the ripple |
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phase. Ayton and Voth have found significant undulations in |
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zero-surface-tension states of membranes simulated via dissipative |
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particle dynamics, but their results are consistent with purely |
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thermal undulations.~\cite{Ayton02} |
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|
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Although the organizations of the tails of lipid molecules are |
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addressed by these molecular simulations, the ordering of the head |
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group in ripple phase is still not settlement. We developed a simple |
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``web of dipoles'' spin lattice model which provides some physical |
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insight in our previous studies~\cite{Sun2007}, we found the dipoles |
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on head groups of the lipid molecules are ordered in an |
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antiferroelectric state. The similiar phenomenon is also observed by |
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Tsonchev {\it et al.} when they studied the formation of the |
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nanotube\cite{Tsonchev04}. |
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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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In this paper, we made a more realistic coarse-grained lipid model to |
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understand the primary driving force for membrane corrugation and to |
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elucidate the organization of the anisotropic interacting head group |
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via molecular dynamics simulation. We will talk about our model and |
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methodology in section \ref{sec:method}, and details of the simulation |
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in section \ref{sec:experiment}. The results are shown in section |
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\ref{sec:results}. At last, we will discuss the results in section |
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In a recent paper, we presented a simple ``web of dipoles'' spin |
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lattice model which provides some physical insight into relationship |
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between dipolar ordering and membrane buckling.\cite{Sun2007} We found |
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that dipolar elastic membranes can spontaneously buckle, forming |
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ripple-like topologies. The driving force for the buckling in dipolar |
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elastic membranes the antiferroelectric ordering of the dipoles, and |
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this was evident in the ordering of the dipole director axis |
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perpendicular to the wave vector of the surface ripples. A similiar |
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phenomenon has also been observed by Tsonchev {\it et al.} in their |
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work on the spontaneous formation of dipolar molecules into curved |
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nano-structures.\cite{Tsonchev04} |
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|
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In this paper, we construct a somewhat more realistic molecular-scale |
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lipid model than our previous ``web of dipoles'' and use molecular |
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dynamics simulations to elucidate the role of the head group dipoles |
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in the formation and morphology of the ripple phase. We describe our |
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model and computational methodology in section \ref{sec:method}. |
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Details on the simulations are presented in section |
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\ref{sec:experiment}, with results following in section |
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\ref{sec:results}. A final discussion of the role of dipolar heads in |
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the ripple formation can be found in section |
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\ref{sec:discussion}. |
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|
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\section{Methodology and Model} |
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\label{sec:method} |
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Our idea for developing a simple and reasonable lipid model to study |
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the ripple phase 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. The |
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lipid model is shown in Figure \ref{fig:lipidMM}. Figure |
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\ref{fig:lipidMM}a shows the molecular strucure of a DPPC molecule. The |
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hydrophilic character of the head group is the effect of the strong |
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dipole composed by a positive charge sitting on the nitrogen and a |
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negative charge on the phosphate. The hydrophobic tail consists of |
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fatty acid chains. In our model shown in Figure \ref{fig:lipidMM}b, |
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lipid molecules are represented by rigid bodies made of one head |
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sphere with a point dipole sitting on it and one ellipsoid tail, the |
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direction of the dipole is fixed to be perpendicular to the tail. The |
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breadth and length of tail are $\sigma_0$, $3\sigma_0$. The diameter |
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of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$. The model of |
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the solvent in our simulations is inspired by the idea of ``DPD'' |
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water. Every four water molecules are reprsented by one sphere. |
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Our simple molecular-scale lipid model for studying the ripple phase |
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is based on two facts: one is that the most essential feature of lipid |
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molecules is their amphiphilic structure with polar head groups and |
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non-polar tails. Another fact is that the majority of lipid molecules |
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in the ripple phase are relatively rigid (i.e. gel-like) which makes |
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some fraction of the details of the chain dynamics negligible. Figure |
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\ref{fig:lipidModels} shows the molecular strucure of a DPPC |
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molecule, as well as atomistic and molecular-scale representations of |
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a DPPC molecule. The hydrophilic character of the head group is |
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largely due to the separation of charge between the nitrogen and |
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phosphate groups. The zwitterionic nature of the PC headgroups leads |
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to abnormally large dipole moments (as high as 20.6 D), and this |
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strongly polar head group interacts strongly with the solvating water |
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layers immediately surrounding the membrane. The hydrophobic tail |
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consists of fatty acid chains. In our molecular scale model, lipid |
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molecules have been reduced to these essential features; the fatty |
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acid chains are represented by an ellipsoid with a dipolar ball |
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perched on one end to represent the effects of the charge-separated |
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head group. In real PC lipids, the direction of the dipole is |
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nearly perpendicular to the tail, so we have fixed the direction of |
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the point dipole rigidly in this orientation. |
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|
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\begin{figure}[htb] |
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\centering |
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work.\label{fig:lipidModels}} |
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\end{figure} |
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|
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Spheres interact each other with Lennard-Jones potential |
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The ellipsoidal portions of the model interact via the Gay-Berne |
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potential which has seen widespread use in the liquid crystal |
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community. In its original form, the Gay-Berne potential was a single |
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site model for the interactions of rigid ellipsoidal |
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molecules.\cite{Gay81} It can be thought of as a modification of the |
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Gaussian overlap model originally described by Berne and |
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Pechukas.\cite{Berne72} The potential is constructed in the familiar |
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form of the Lennard-Jones function using orientation-dependent |
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$\sigma$ and $\epsilon$ parameters, |
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|
\begin{eqnarray*} |
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V_{ij} = 4\epsilon \left[\left(\frac{\sigma_0}{r_{ij}}\right)^{12} - |
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\left(\frac{\sigma_0}{r_{ij}}\right)^6\right] |
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– |
\end{eqnarray*} |
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here, $\sigma_0$ is diameter of the spherical particle. $r_{ij}$ is |
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the distance between two spheres. $\epsilon$ is the well depth. |
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Dipoles interact each other with typical dipole potential |
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\begin{eqnarray*} |
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V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
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\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
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\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
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– |
\end{eqnarray*} |
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In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
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along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
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pointing along the inter-dipole vector $\mathbf{r}_{ij}$. Identical |
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ellipsoids interact each other with Gay-Berne potential. |
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– |
\begin{eqnarray*} |
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|
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 178 |
|
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}, |
| 180 |
|
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
| 181 |
|
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 182 |
|
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
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< |
\end{eqnarray*} |
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< |
where $\sigma_0 = \sqrt 2\sigma_s$, and orietation-dependent range |
| 150 |
< |
parameter is given by |
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< |
\begin{eqnarray*} |
| 152 |
< |
\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
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< |
{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{({\mathbf{\hat r}_{ij}} |
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< |
\cdot {\mathbf{\hat u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 155 |
< |
u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
| 156 |
< |
\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}} |
| 157 |
< |
\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
| 158 |
< |
{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
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< |
\end{eqnarray*} |
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< |
and the strength anisotropy function is, |
| 161 |
< |
\begin{eqnarray*} |
| 162 |
< |
\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 163 |
< |
{\epsilon^\nu}({\mathbf{\hat u}_i}, {\mathbf{\hat |
| 164 |
< |
u}_j}){\epsilon'^\mu}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 165 |
< |
{\mathbf{\hat r}_{ij}}) |
| 166 |
< |
\end{eqnarray*} |
| 167 |
< |
with $\nu$ and $\mu$ being adjustable exponent, and |
| 168 |
< |
$\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j})$, |
| 169 |
< |
$\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 170 |
< |
r}_{ij}})$ defined as |
| 171 |
< |
\begin{eqnarray*} |
| 172 |
< |
\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}) = |
| 173 |
< |
\epsilon_0\left[1-\chi^2({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
| 174 |
< |
u}_j})^2\right]^{-\frac{1}{2}} |
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> |
\label{eq:gb} |
| 184 |
|
\end{eqnarray*} |
| 185 |
< |
\begin{eqnarray*} |
| 186 |
< |
\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 187 |
< |
1-\frac{1}{2}\chi'\left[\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 188 |
< |
u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 189 |
< |
u}_j})^2}{1+\chi'({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
| 190 |
< |
\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}} |
| 191 |
< |
\cdot {\mathbf{\hat u}_j})^2}{1-\chi'({\mathbf{\hat u}_i} \cdot |
| 192 |
< |
{\mathbf{\hat u}_j})} \right] |
| 193 |
< |
\end{eqnarray*} |
| 194 |
< |
the diameter dependent parameter $\chi$ is given by |
| 195 |
< |
\begin{eqnarray*} |
| 196 |
< |
\chi = \frac{({\sigma_s}^2 - |
| 197 |
< |
{\sigma_e}^2)}{({\sigma_s}^2 + {\sigma_e}^2)} |
| 198 |
< |
\end{eqnarray*} |
| 199 |
< |
and the well depth dependent parameter $\chi'$ is given by |
| 200 |
< |
\begin{eqnarray*} |
| 201 |
< |
\chi' = \frac{({\epsilon_s}^{\frac{1}{\mu}} - |
| 202 |
< |
{\epsilon_e}^{\frac{1}{\mu}})}{({\epsilon_s}^{\frac{1}{\mu}} + |
| 203 |
< |
{\epsilon_e}^{\frac{1}{\mu}})} |
| 185 |
> |
|
| 186 |
> |
|
| 187 |
> |
|
| 188 |
> |
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 189 |
> |
\hat{r}}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
| 190 |
> |
\hat{u}}_{j},{\bf \hat{r}}))$ parameters |
| 191 |
> |
are dependent on the relative orientations of the two molecules (${\bf |
| 192 |
> |
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
| 193 |
> |
intermolecular separation (${\bf \hat{r}}$). The functional forms for |
| 194 |
> |
$\sigma({\bf |
| 195 |
> |
\hat{u}}_{i},{\bf |
| 196 |
> |
\hat{u}}_{j},{\bf \hat{r}})$ and $\epsilon({\bf \hat{u}}_{i},{\bf |
| 197 |
> |
\hat{u}}_{j},{\bf \hat{r}}))$ are given elsewhere |
| 198 |
> |
and will not be repeated here. However, $\epsilon$ and $\sigma$ are |
| 199 |
> |
governed by two anisotropy parameters, |
| 200 |
> |
\begin {equation} |
| 201 |
> |
\begin{array}{rcl} |
| 202 |
> |
\chi & = & \frac |
| 203 |
> |
{(\sigma_{e}/\sigma_{s})^2-1}{(\sigma_{e}/\sigma_{s})^2+1} \\ \\ |
| 204 |
> |
\chi\prime & = & \frac {1-(\epsilon_{e}/\epsilon_{s})^{1/\mu}}{1+(\epsilon_{e}/ |
| 205 |
> |
\epsilon_{s})^{1/\mu}} |
| 206 |
> |
\end{array} |
| 207 |
> |
\end{equation} |
| 208 |
> |
In these equations, $\sigma$ and $\epsilon$ refer to the point of |
| 209 |
> |
closest contact and the depth of the well in different orientations of |
| 210 |
> |
the two molecules. The subscript $s$ refers to the {\it side-by-side} |
| 211 |
> |
configuration where $\sigma$ has it's smallest value, |
| 212 |
> |
$\sigma_{s}$, and where the potential well is $\epsilon_{s}$ deep. |
| 213 |
> |
The subscript $e$ refers to the {\it end-to-end} configuration where |
| 214 |
> |
$\sigma$ is at it's largest value, $\sigma_{e}$ and where the well |
| 215 |
> |
depth, $\epsilon_{e}$ is somewhat smaller than in the side-by-side |
| 216 |
> |
configuration. For the prolate ellipsoids we are using, we have |
| 217 |
> |
\begin{equation} |
| 218 |
> |
\begin{array}{rcl} |
| 219 |
> |
\sigma_{s} & < & \sigma_{e} \\ |
| 220 |
> |
\epsilon_{s} & > & \epsilon_{e} |
| 221 |
> |
\end{array} |
| 222 |
> |
\end{equation} |
| 223 |
> |
Ref. \onlinecite{Luckhurst90} has a particularly good explanation of the |
| 224 |
> |
choice of the Gay-Berne parameters $\mu$ and $\nu$ for modeling liquid |
| 225 |
> |
crystal molecules. |
| 226 |
> |
|
| 227 |
> |
The breadth and length of tail are $\sigma_0$, $3\sigma_0$, |
| 228 |
> |
corresponding to a shape anisotropy of 3 for the chain portion of the |
| 229 |
> |
molecule. In principle, this could be varied to allow for modeling of |
| 230 |
> |
longer or shorter chain lipid molecules. |
| 231 |
> |
|
| 232 |
> |
To take into account the permanent dipolar interactions of the |
| 233 |
> |
zwitterionic head groups, we place fixed dipole moments $\mu_{i}$ at |
| 234 |
> |
one end of the Gay-Berne particles. The dipoles will be oriented at |
| 235 |
> |
an angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
| 236 |
> |
are protected by a head ``bead'' with a range parameter which we have |
| 237 |
> |
varied between $1.20\sigma_0$ and $1.41\sigma_0$. The head groups |
| 238 |
> |
interact with each other using a combination of Lennard-Jones, |
| 239 |
> |
\begin{eqnarray*} |
| 240 |
> |
V_{ij} = 4\epsilon \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
| 241 |
> |
\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
| 242 |
|
\end{eqnarray*} |
| 243 |
< |
$\sigma_s$ is the side-by-side width, and $\sigma_e$ is the end-to-end |
| 197 |
< |
length. $\epsilon_s$ is the side-by-side well depth, and $\epsilon_e$ |
| 198 |
< |
is the end-to-end well depth. For the interaction between |
| 199 |
< |
nonequivalent uniaxial ellipsoids (in this case, between spheres and |
| 200 |
< |
ellipsoids), the range parameter is generalized as\cite{Cleaver96} |
| 243 |
> |
and dipole, |
| 244 |
|
\begin{eqnarray*} |
| 245 |
+ |
V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
| 246 |
+ |
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
| 247 |
+ |
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
| 248 |
+ |
\end{eqnarray*} |
| 249 |
+ |
potentials. |
| 250 |
+ |
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 251 |
+ |
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
| 252 |
+ |
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. |
| 253 |
+ |
|
| 254 |
+ |
For the interaction between nonequivalent uniaxial ellipsoids (in this |
| 255 |
+ |
case, between spheres and ellipsoids), the range parameter is |
| 256 |
+ |
generalized as\cite{Cleaver96} |
| 257 |
+ |
\begin{eqnarray*} |
| 258 |
|
\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 259 |
|
{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{(\alpha{\mathbf{\hat r}_{ij}} |
| 260 |
|
\cdot {\mathbf{\hat u}_i} + \alpha^{-1}{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 269 |
|
\left[\frac{\left({\sigma_e}_i^2-{\sigma_s}_i^2\right)\left({\sigma_e}_j^2+{\sigma_s}_i^2\right)}{\left({\sigma_e}_j^2-{\sigma_s}_j^2\right)\left({\sigma_e}_i^2+{\sigma_s}_j^2\right)} |
| 270 |
|
\right]^{\frac{1}{2}} |
| 271 |
|
\end{eqnarray*} |
| 272 |
< |
the strength parameter is adjusted by the suggestion of |
| 273 |
< |
\cite{Cleaver96}. All potentials are truncated at $25$ \AA\ and |
| 274 |
< |
shifted at $22$ \AA. |
| 272 |
> |
the strength parameter has been adjusted as suggested by Cleaver {\it |
| 273 |
> |
et al.}\cite{Cleaver96} A switching function has been applied to all |
| 274 |
> |
potentials to smoothly turn off the interactions between a range of $22$ \AA\ and $25$ \AA. |
| 275 |
|
|
| 276 |
+ |
The model of the solvent in our simulations is inspired by the idea of |
| 277 |
+ |
``DPD'' water. Every four water molecules are reprsented by one |
| 278 |
+ |
sphere. |
| 279 |
+ |
|
| 280 |
|
\begin{figure}[htb] |
| 281 |
|
\centering |
| 282 |
|
\includegraphics[height=4in]{lipidModel} |
| 375 |
|
amplitudes are underestimated without the melted tails in our |
| 376 |
|
simulations. |
| 377 |
|
|
| 378 |
+ |
\begin{figure}[htb] |
| 379 |
+ |
\centering |
| 380 |
+ |
\includegraphics[width=\linewidth]{topDown} |
| 381 |
+ |
\caption{Top views of the flat (upper), asymmetric ripple (middle), |
| 382 |
+ |
and symmetric ripple (lower) phases. Note that the head-group dipoles |
| 383 |
+ |
have formed head-to-tail chains in all three of these phases, but in |
| 384 |
+ |
the two rippled phases, the dipolar chains are all aligned |
| 385 |
+ |
{\it perpendicular} to the direction of the ripple. The flat membrane |
| 386 |
+ |
has multiple point defects in the dipolar orientational ordering, and |
| 387 |
+ |
the dipolar ordering on the lower leaf of the bilayer can be in a |
| 388 |
+ |
different direction from the upper leaf.\label{fig:topView}} |
| 389 |
+ |
\end{figure} |
| 390 |
+ |
|
| 391 |
|
The $P_2$ order paramters (for molecular bodies and head group |
| 392 |
|
dipoles) have been calculated to clarify the ordering in these phases |
| 393 |
|
quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$ |
| 396 |
|
increasing head group size. When the heads of the lipid molecules are |
| 397 |
|
small, the membrane is flat. The dipolar ordering is essentially |
| 398 |
|
frustrated on orientational ordering in this circumstance. Figure |
| 399 |
< |
\ref{} shows the snapshots of the top view for the flat system |
| 399 |
> |
\ref{fig:topView} shows the snapshots of the top view for the flat system |
| 400 |
|
($\sigma_h=1.20\sigma$) and rippled system |
| 401 |
|
($\sigma_h=1.41\sigma$). The pointing direction of the dipoles on the |
| 402 |
|
head groups are represented by two colored half spheres from blue to |
