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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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\author{Xiuquan Sun and J. Daniel Gezelter \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame, \\ |
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Notre Dame, Indiana 46556} |
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%\email[E-mail:]{gezelter@nd.edu} |
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\date{\today} |
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|
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\maketitle |
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|
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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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|
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\pacs{} |
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\maketitle |
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%\maketitle |
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\section{Introduction} |
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\label{sec:Int} |
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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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\begin{equation*} |
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V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
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r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
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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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\end{equation*} |
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|
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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 $(\epsilon({\bf \hat{u}}_{i},{\bf |
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\hat{u}}_{i},{\bf \hat{u}}_{j}$) 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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\begin {eqnarray*} |
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\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
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\chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 - |
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d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^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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\end{eqnarray*} |
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where $l$ and $d$ describe the length and width of each uniaxial |
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ellipsoid. These shape anisotropy parameters can then be used to |
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calculate the range function, |
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\begin {equation} |
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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} ) ({\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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\end{equation*} |
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|
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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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\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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those obtained for the range parameter. Ref. \citen{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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in Ref. \citen{Golubkov06} |
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|
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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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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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\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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\end{eqnarray*} |
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\end{equation} |
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and dipole-dipole, |
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\begin{eqnarray*} |
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\begin{equation} |
| 318 |
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V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 319 |
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\hat{r}}_{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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\end{equation} |
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potentials. |
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In these potentials, $\mathbf{\hat u}_i$ 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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different direction from the upper leaf.\label{fig:topView}} |
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\end{figure} |
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|
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The principal method for observing orientational ordering in dipolar |
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or liquid crystalline systems is the $P_2$ order parameter (defined |
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as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
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eigenvalue of the matrix, |
| 494 |
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\begin{equation} |
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{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
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\begin{array}{ccc} |
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u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
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u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
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u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
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\end{array} \right). |
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\label{eq:opmatrix} |
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\end{equation} |
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Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
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for molecule $i$. (Here, $\hat{\bf u}_i$ can refer either to the |
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principal axis of the molecular body or to the dipole on the head |
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group of the molecule.) $P_2$ will be $1.0$ for a perfectly-ordered |
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system and near $0$ for a randomized system. Note that this order |
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parameter is {\em not} equal to the polarization of the system. For |
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example, the polarization of a perfect anti-ferroelectric arrangement |
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of point dipoles is $0$, but $P_2$ for the same system is $1$. The |
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eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is |
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familiar as the director axis, which can be used to determine a |
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privileged axis for an orientationally-ordered system. Since the |
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molecular bodies are perpendicular to the head group dipoles, it is |
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possible for the director axes for the molecular bodies and the head |
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groups to be completely decoupled from each other. |
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|
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Figure \ref{fig:topView} shows snapshots of bird's-eye views of the |
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flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.41, 1.35 d$) |
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bilayers. The directions of the dipoles on the head groups are |
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represented with two colored half spheres: blue (phosphate) and yellow |
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(amino). For flat bilayers, the system exhibits signs of |
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orientational frustration; some disorder in the dipolar chains is |
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evident with kinks visible at the edges between different ordered |
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domains. The lipids can also move independently of lipids in the |
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opposing leaf, so the ordering of the dipoles on one leaf is not |
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necessarily consistant with the ordering on the other. These two |
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orientational frustration; some disorder in the dipolar head-to-tail |
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chains is evident with kinks visible at the edges between differently |
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ordered domains. The lipids can also move independently of lipids in |
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the opposing leaf, so the ordering of the dipoles on one leaf is not |
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necessarily consistent with the ordering on the other. These two |
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factors keep the total dipolar order parameter relatively low for the |
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flat phases. |
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|
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With increasing head group size, the surface becomes corrugated, and |
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the dipoles cannot move as freely on the surface. Therefore, the |
| 533 |
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translational freedom of lipids in one layer is dependent upon the |
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position of lipids in the other layer. As a result, the ordering of |
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position of the lipids in the other layer. As a result, the ordering of |
| 535 |
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the dipoles on head groups in one leaf is correlated with the ordering |
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in the other leaf. Furthermore, as the membrane deforms due to the |
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corrugation, the symmetry of the allowed dipolar ordering on each leaf |
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configuration, and the dipolar order parameter increases dramatically. |
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However, the total polarization of the system is still close to zero. |
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This is strong evidence that the corrugated structure is an |
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antiferroelectric state. It is also notable that the head-to-tail |
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arrangement of the dipoles is in a direction perpendicular to the wave |
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vector for the surface corrugation. This is a similar finding to what |
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we observed in our earlier work on the elastic dipolar |
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membranes.\cite{Sun07} |
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antiferroelectric state. It is also notable that the head-to-tail |
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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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|
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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 |
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phases. $P_2 = 1$ implies a perfectly ordered structure, and $P_2 = 0$ |
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implies complete orientational randomization. Figure \ref{fig:rP2} |
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shows the $P_2$ order parameter for the head-group dipoles increasing |
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with increasing head group size. When the heads of the lipid molecules |
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are small, the membrane is nearly flat. The dipolar ordering exhibits |
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frustrated orientational ordering in this circumstance. |
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phases. Figure \ref{fig:rP2} shows that the $P_2$ order parameter for |
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the head-group dipoles increases with increasing head group size. When |
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the heads of the lipid molecules are small, the membrane is nearly |
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flat. Since the in-plane packing is essentially a close packing of the |
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head groups, the head dipoles exhibit frustration in their |
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orientational ordering. |
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|
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The ordering of the tails is essentially opposite to the ordering of |
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the dipoles on head group. The $P_2$ order parameter {\it decreases} |
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with increasing head size. This indicates that the surface is more |
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curved with larger head / tail size ratios. When the surface is flat, |
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all tails are pointing in the same direction (parallel to the normal |
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of the surface). This simplified model appears to be exhibiting a |
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smectic A fluid phase, similar to the real $L_{\beta}$ phase. We have |
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not observed a smectic C gel phase ($L_{\beta'}$) for this model |
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system. Increasing the size of the heads, results in rapidly |
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decreasing $P_2$ ordering for the molecular bodies. |
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The ordering trends for the tails are essentially opposite to the |
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ordering of the head group dipoles. The tail $P_2$ order parameter |
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{\it decreases} with increasing head size. This indicates that the |
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surface is more curved with larger head / tail size ratios. When the |
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surface is flat, all tails are pointing in the same direction (normal |
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to the bilayer surface). This simplified model appears to be |
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exhibiting a smectic A fluid phase, similar to the real $L_{\beta}$ |
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phase. We have not observed a smectic C gel phase ($L_{\beta'}$) for |
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this model system. Increasing the size of the heads results in |
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rapidly decreasing $P_2$ ordering for the molecular bodies. |
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|
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{rP2} |
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\caption{The $P_2$ order parameter as a function of the ratio of |
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$\sigma_h$ to $d$. \label{fig:rP2}} |
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\caption{The $P_2$ order parameters for head groups (circles) and |
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molecular bodies (squares) as a function of the ratio of head group |
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size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{fig:rP2}} |
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\end{figure} |
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|
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We studied the effects of the interactions between head groups on the |
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structure of lipid bilayer by changing the strength of the dipole. |
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Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with |
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increasing strength of the dipole. Generally the dipoles on the head |
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group are more ordered by increase in the strength of the interaction |
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between heads and are more disordered by decreasing the interaction |
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stength. When the interaction between the heads is weak enough, the |
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bilayer structure does not persist; all lipid molecules are solvated |
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directly in the water. The critial value of the strength of the dipole |
| 585 |
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depends on the head size. The perfectly flat surface melts at $5$ |
| 586 |
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$0.03$ debye, the asymmetric rippled surfaces melt at $8$ $0.04$ |
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$0.03$ debye, the symmetric rippled surfaces melt at $10$ $0.04$ |
| 588 |
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debye. The ordering of the tails is the same as the ordering of the |
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dipoles except for the flat phase. Since the surface is already |
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perfect flat, the order parameter does not change much until the |
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strength of the dipole is $15$ debye. However, the order parameter |
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decreases quickly when the strength of the dipole is further |
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increased. The head groups of the lipid molecules are brought closer |
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by stronger interactions between them. For a flat surface, a large |
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amount of free volume between the head groups is available, but when |
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the head groups are brought closer, the tails will splay outward, |
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forming an inverse micelle. When $\sigma_h=1.28\sigma_0$, the $P_2$ |
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order parameter decreases slightly after the strength of the dipole is |
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increased to $16$ debye. For rippled surfaces, there is less free |
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volume available between the head groups. Therefore there is little |
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effect on the structure of the membrane due to increasing dipolar |
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strength. However, the increase of the $P_2$ order parameter implies |
| 603 |
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the membranes are flatten by the increase of the strength of the |
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dipole. Unlike other systems that melt directly when the interaction |
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is weak enough, for $\sigma_h=1.41\sigma_0$, part of the membrane |
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melts into itself first. The upper leaf of the bilayer becomes totally |
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interdigitated with the lower leaf. This is different behavior than |
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what is exhibited with the interdigitated lines in the rippled phase |
| 609 |
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where only one interdigitated line connects the two leaves of bilayer. |
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In addition to varying the size of the head groups, we studied the |
| 577 |
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effects of the interactions between head groups on the structure of |
| 578 |
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lipid bilayer by changing the strength of the dipoles. Figure |
| 579 |
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\ref{fig:sP2} shows how the $P_2$ order parameter changes with |
| 580 |
> |
increasing strength of the dipole. Generally, the dipoles on the head |
| 581 |
> |
groups become more ordered as the strength of the interaction between |
| 582 |
> |
heads is increased and become more disordered by decreasing the |
| 583 |
> |
interaction stength. When the interaction between the heads becomes |
| 584 |
> |
too weak, the bilayer structure does not persist; all lipid molecules |
| 585 |
> |
become dispersed in the solvent (which is non-polar in this |
| 586 |
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molecular-scale model). The critial value of the strength of the |
| 587 |
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dipole depends on the size of the head groups. The perfectly flat |
| 588 |
> |
surface becomes unstable below $5$ Debye, while the rippled |
| 589 |
> |
surfaces melt at $8$ Debye (asymmetric) or $10$ Debye (symmetric). |
| 590 |
> |
|
| 591 |
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The ordering of the tails mirrors the ordering of the dipoles {\it |
| 592 |
> |
except for the flat phase}. Since the surface is nearly flat in this |
| 593 |
> |
phase, the order parameters are only weakly dependent on dipolar |
| 594 |
> |
strength until it reaches $15$ Debye. Once it reaches this value, the |
| 595 |
> |
head group interactions are strong enough to pull the head groups |
| 596 |
> |
close to each other and distort the bilayer structure. For a flat |
| 597 |
> |
surface, a substantial amount of free volume between the head groups |
| 598 |
> |
is normally available. When the head groups are brought closer by |
| 599 |
> |
dipolar interactions, the tails are forced to splay outward, forming |
| 600 |
> |
first curved bilayers, and then inverted micelles. |
| 601 |
> |
|
| 602 |
> |
When $\sigma_h=1.28 d$, the $P_2$ order parameter decreases slightly |
| 603 |
> |
when the strength of the dipole is increased above $16$ debye. For |
| 604 |
> |
rippled bilayers, there is less free volume available between the head |
| 605 |
> |
groups. Therefore increasing dipolar strength weakly influences the |
| 606 |
> |
structure of the membrane. However, the increase in the body $P_2$ |
| 607 |
> |
order parameters implies that the membranes are being slightly |
| 608 |
> |
flattened due to the effects of increasing head-group attraction. |
| 609 |
> |
|
| 610 |
> |
A very interesting behavior takes place when the head groups are very |
| 611 |
> |
large relative to the molecular bodies ($\sigma_h = 1.41 d$) and the |
| 612 |
> |
dipolar strength is relatively weak ($\mu < 9$ Debye). In this case, |
| 613 |
> |
the two leaves of the bilayer become totally interdigitated with each |
| 614 |
> |
other in large patches of the membrane. With higher dipolar |
| 615 |
> |
strength, the interdigitation is limited to single lines that run |
| 616 |
> |
through the bilayer in a direction perpendicular to the ripple wave |
| 617 |
> |
vector. |
| 618 |
> |
|
| 619 |
|
\begin{figure}[htb] |
| 620 |
|
\centering |
| 621 |
|
\includegraphics[width=\linewidth]{sP2} |
| 622 |
< |
\caption{The $P_2$ order parameter as a funtion of the strength of the |
| 623 |
< |
dipole.\label{fig:sP2}} |
| 622 |
> |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
| 623 |
> |
molecular bodies (b) as a function of the strength of the dipoles. |
| 624 |
> |
These order parameters are shown for four values of the head group / |
| 625 |
> |
molecular width ratio ($\sigma_h / d$). \label{fig:sP2}} |
| 626 |
|
\end{figure} |
| 627 |
|
|
| 628 |
< |
Figure \ref{fig:tP2} shows the dependence of the order parameter on |
| 629 |
< |
temperature. The behavior of the $P_2$ order paramter is |
| 630 |
< |
straightforward. Systems are more ordered at low temperature, and more |
| 631 |
< |
disordered at high temperatures. When the temperature is high enough, |
| 632 |
< |
the membranes are instable. For flat surfaces ($\sigma_h=1.20\sigma_0$ |
| 633 |
< |
and $\sigma_h=1.28\sigma_0$), when the temperature is increased to |
| 634 |
< |
$310$, the $P_2$ order parameter increases slightly instead of |
| 635 |
< |
decreases like ripple surface. This is an evidence of the frustration |
| 636 |
< |
of the dipolar ordering in each leaf of the lipid bilayer, at low |
| 637 |
< |
temperature, the systems are locked in a local minimum energy state, |
| 638 |
< |
with increase of the temperature, the system can jump out the local |
| 639 |
< |
energy well to find the lower energy state which is the longer range |
| 640 |
< |
orientational ordering. Like the dipolar ordering of the flat |
| 641 |
< |
surfaces, the ordering of the tails of the lipid molecules for ripple |
| 642 |
< |
membranes ($\sigma_h=1.35\sigma_0$ and $\sigma_h=1.41\sigma_0$) also |
| 643 |
< |
show some nonthermal characteristic. With increase of the temperature, |
| 644 |
< |
the $P_2$ order parameter decreases firstly, and increases afterward |
| 645 |
< |
when the temperature is greater than $290 K$. The increase of the |
| 646 |
< |
$P_2$ order parameter indicates a more ordered structure for the tails |
| 647 |
< |
of the lipid molecules which corresponds to a more flat surface. Since |
| 648 |
< |
our model lacks the detailed information on lipid tails, we can not |
| 649 |
< |
simulate the fluid phase with melted fatty acid chains. Moreover, the |
| 650 |
< |
formation of the tilted $L_{\beta'}$ phase also depends on the |
| 597 |
< |
organization of fatty groups on tails. |
| 628 |
> |
Figure \ref{fig:tP2} shows the dependence of the order parameters on |
| 629 |
> |
temperature. As expected, systems are more ordered at low |
| 630 |
> |
temperatures, and more disordered at high temperatures. All of the |
| 631 |
> |
bilayers we studied can become unstable if the temperature becomes |
| 632 |
> |
high enough. The only interesting feature of the temperature |
| 633 |
> |
dependence is in the flat surfaces ($\sigma_h=1.20 d$ and |
| 634 |
> |
$\sigma_h=1.28 d$). Here, when the temperature is increased above |
| 635 |
> |
$310$K, there is enough jostling of the head groups to allow the |
| 636 |
> |
dipolar frustration to resolve into more ordered states. This results |
| 637 |
> |
in a slight increase in the $P_2$ order parameter above this |
| 638 |
> |
temperature. |
| 639 |
> |
|
| 640 |
> |
For the rippled surfaces ($\sigma_h=1.35 d$ and $\sigma_h=1.41 d$), |
| 641 |
> |
there is a slightly increased orientational ordering in the molecular |
| 642 |
> |
bodies above $290$K. Since our model lacks the detailed information |
| 643 |
> |
about the behavior of the lipid tails, this is the closest the model |
| 644 |
> |
can come to depicting the ripple ($P_{\beta'}$) to fluid |
| 645 |
> |
($L_{\alpha}$) phase transition. What we are observing is a |
| 646 |
> |
flattening of the rippled structures made possible by thermal |
| 647 |
> |
expansion of the tightly-packed head groups. The lack of detailed |
| 648 |
> |
chain configurations also makes it impossible for this model to depict |
| 649 |
> |
the ripple to gel ($L_{\beta'}$) phase transition. |
| 650 |
> |
|
| 651 |
|
\begin{figure}[htb] |
| 652 |
|
\centering |
| 653 |
|
\includegraphics[width=\linewidth]{tP2} |
| 654 |
< |
\caption{The $P_2$ order parameter as a funtion of |
| 655 |
< |
temperature.\label{fig:tP2}} |
| 654 |
> |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
| 655 |
> |
molecular bodies (b) as a function of temperature. |
| 656 |
> |
These order parameters are shown for four values of the head group / |
| 657 |
> |
molecular width ratio ($\sigma_h / d$).\label{fig:tP2}} |
| 658 |
|
\end{figure} |
| 659 |
|
|
| 660 |
|
\section{Discussion} |
| 661 |
|
\label{sec:discussion} |
| 662 |
|
|
| 663 |
+ |
The ripple phases have been observed in our molecular dynamic |
| 664 |
+ |
simulations using a simple molecular lipid model. The lipid model |
| 665 |
+ |
consists of an anisotropic interacting dipolar head group and an |
| 666 |
+ |
ellipsoid shape tail. According to our simulations, the explanation of |
| 667 |
+ |
the formation for the ripples are originated in the size mismatch |
| 668 |
+ |
between the head groups and the tails. The ripple phases are only |
| 669 |
+ |
observed in the studies using larger head group lipid models. However, |
| 670 |
+ |
there is a mismatch betweent the size of the head groups and the size |
| 671 |
+ |
of the tails in the simulations of the flat surface. This indicates |
| 672 |
+ |
the competition between the anisotropic dipolar interaction and the |
| 673 |
+ |
packing of the tails also plays a major role for formation of the |
| 674 |
+ |
ripple phase. The larger head groups provide more free volume for the |
| 675 |
+ |
tails, while these hydrophobic ellipsoids trying to be close to each |
| 676 |
+ |
other, this gives the origin of the spontanous curvature of the |
| 677 |
+ |
surface, which is believed as the beginning of the ripple phases. The |
| 678 |
+ |
lager head groups cause the spontanous curvature inward for both of |
| 679 |
+ |
leaves of the bilayer. This results in a steric strain when the tails |
| 680 |
+ |
of two leaves too close to each other. The membrane has to be broken |
| 681 |
+ |
to release this strain. There are two ways to arrange these broken |
| 682 |
+ |
curvatures: symmetric and asymmetric ripples. Both of the ripple |
| 683 |
+ |
phases have been observed in our studies. The difference between these |
| 684 |
+ |
two ripples is that the bilayer is continuum in the symmetric ripple |
| 685 |
+ |
phase and is disrupt in the asymmetric ripple phase. |
| 686 |
+ |
|
| 687 |
+ |
Dipolar head groups are the key elements for the maintaining of the |
| 688 |
+ |
bilayer structure. The lipids are solvated in water when lowering the |
| 689 |
+ |
the strength of the dipole on the head groups. The long range |
| 690 |
+ |
orientational ordering of the dipoles can be achieved by forming the |
| 691 |
+ |
ripples, although the dipoles are likely to form head-to-tail |
| 692 |
+ |
configurations even in flat surface, the frustration prevents the |
| 693 |
+ |
formation of the long range orientational ordering for dipoles. The |
| 694 |
+ |
corrugation of the surface breaks the frustration and stablizes the |
| 695 |
+ |
long range oreintational ordering for the dipoles in the head groups |
| 696 |
+ |
of the lipid molecules. Many rows of the head-to-tail dipoles are |
| 697 |
+ |
parallel to each other and adopt the antiferroelectric state as a |
| 698 |
+ |
whole. This is the first time the organization of the head groups in |
| 699 |
+ |
ripple phases of the lipid bilayer has been addressed. |
| 700 |
+ |
|
| 701 |
+ |
The most important prediction we can make using the results from this |
| 702 |
+ |
simple model is that if dipolar ordering is driving the surface |
| 703 |
+ |
corrugation, the wave vectors for the ripples should always found to |
| 704 |
+ |
be {\it perpendicular} to the dipole director axis. This prediction |
| 705 |
+ |
should suggest experimental designs which test whether this is really |
| 706 |
+ |
true in the phosphatidylcholine $P_{\beta'}$ phases. The dipole |
| 707 |
+ |
director axis should also be easily computable for the all-atom and |
| 708 |
+ |
coarse-grained simulations that have been published in the literature. |
| 709 |
+ |
|
| 710 |
+ |
Although our model is simple, it exhibits some rich and unexpected |
| 711 |
+ |
behaviors. It would clearly be a closer approximation to the reality |
| 712 |
+ |
if we allowed greater translational freedom to the dipoles and |
| 713 |
+ |
replaced the somewhat artificial lattice packing and the harmonic |
| 714 |
+ |
elastic tension with more realistic molecular modeling potentials. |
| 715 |
+ |
What we have done is to present a simple model which exhibits bulk |
| 716 |
+ |
non-thermal corrugation, and our explanation of this rippling |
| 717 |
+ |
phenomenon will help us design more accurate molecular models for |
| 718 |
+ |
corrugated membranes and experiments to test whether rippling is |
| 719 |
+ |
dipole-driven or not. |
| 720 |
+ |
|
| 721 |
|
\newpage |
| 722 |
|
\bibliography{mdripple} |
| 723 |
|
\end{document} |
