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\chapter{\label{chap:md}Dipolar ordering in the ripple phases of |
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molecular-scale models of lipid membranes} |
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\chapter{\label{chap:md}DIPOLAR ORDERING IN THE RIPPLE PHASES OF |
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MOLECULAR-SCALE MODELS OF LIPID MEMBRANES} |
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\section{Introduction} |
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\label{mdsec:Int} |
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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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between dipolar ordering and membrane buckling.\cite{sun:031602} We |
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found 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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modeling large length-scale properties of lipid |
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bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
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was a single site model for the interactions of rigid ellipsoidal |
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molecules.\cite{Gay81} It can be thought of as a modification of the |
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molecules.\cite{Gay1981} 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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\end{figure} |
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To take into account the permanent dipolar interactions of the |
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zwitterionic head groups, we have placed 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 ($\sigma_h$) 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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zwitterionic head groups, we have placed fixed dipole moments |
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$\mu_{i}$ at one end of the Gay-Berne particles. The dipoles are |
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oriented at an angle $\theta = \pi / 2$ relative to the major axis. |
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These dipoles are protected by a head ``bead'' with a range parameter |
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($\sigma_h$) which we have varied between $1.20 d$ and $1.41 d$. The |
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head groups interact with each other using a combination of |
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Lennard-Jones, |
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\begin{equation} |
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V_{ij}(r_{ij}) = 4\epsilon_h \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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The solvent model in our simulations is similar to the one used by |
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Marrink {\it et al.} in their coarse grained simulations of lipid |
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bilayers.\cite{Marrink04} The solvent bead is a single site that |
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bilayers.\cite{Marrink2004} The solvent bead is a single site that |
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represents four water molecules (m = 72 amu) and has comparable |
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density and diffusive behavior to liquid water. However, since there |
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are no electrostatic sites on these beads, this solvent model cannot |
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replicate the dielectric properties of water. Note that although we |
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are using larger cutoff and switching radii than Marrink {\it et al.}, |
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our solvent density at 300 K remains at 0.944 g cm$^{-3}$, and the |
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solvent diffuses at 0.43 $\AA^2 ps^{-1}$ (only twice as fast as liquid |
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solvent diffuses at 0.43 \AA$^2 ps^{-1}$ (only twice as fast as liquid |
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water). |
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\begin{table*} |
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molecular dynamics runs was 25 fs. No appreciable changes in phase |
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structure were noticed upon switching to a microcanonical ensemble. |
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All simulations were performed using the {\sc oopse} molecular |
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modeling program.\cite{Meineke05} |
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modeling program.\cite{Meineke2005} |
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A switching function was applied to all potentials to smoothly turn |
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off the interactions between a range of $22$ and $25$ \AA. The |
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arrangement of the dipoles is always observed in a direction |
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perpendicular to the wave vector for the surface corrugation. This is |
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a similar finding to what we observed in our earlier work on the |
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elastic dipolar membranes.\cite{Sun2007} |
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elastic dipolar membranes.\cite{sun:031602} |
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The $P_2$ order parameters (for both the molecular bodies and the head |
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group dipoles) have been calculated to quantify the ordering in these |
