--- trunk/mdRipple/mdripple.tex	2007/07/27 21:59:45	3200
+++ trunk/mdRipple/mdripple.tex	2007/08/01 16:07:12	3202
@@ -1,32 +1,48 @@
 %\documentclass[aps,pre,twocolumn,amssymb,showpacs,floatfix]{revtex4}
-\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4}
+%\documentclass[aps,pre,preprint,amssymb]{revtex4}
+\documentclass[12pt]{article}
+\usepackage{times}
+\usepackage{mathptm}
+\usepackage{tabularx}
+\usepackage{setspace}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{graphicx}
+\usepackage[ref]{overcite}
+\pagestyle{plain}
+\pagenumbering{arabic}
+\oddsidemargin 0.0cm \evensidemargin 0.0cm
+\topmargin -21pt \headsep 10pt
+\textheight 9.0in \textwidth 6.5in
+\brokenpenalty=10000
+\renewcommand{\baselinestretch}{1.2}
+\renewcommand\citemid{\ } % no comma in optional reference note
 
 \begin{document}
-\renewcommand{\thefootnote}{\fnsymbol{footnote}}
-\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
+%\renewcommand{\thefootnote}{\fnsymbol{footnote}}
+%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
 
-%\bibliographystyle{aps}
+\bibliographystyle{achemso}
 
 \title{Dipolar Ordering in Molecular-scale Models of the Ripple Phase
 in Lipid Membranes} 
-\author{Xiuquan Sun and J. Daniel Gezelter}
-\email[E-mail:]{gezelter@nd.edu}
-\affiliation{Department of Chemistry and Biochemistry,\\
+\author{Xiuquan Sun and J. Daniel Gezelter \\
+Department of Chemistry and Biochemistry,\\
 University of Notre Dame, \\ 
 Notre Dame, Indiana 46556}
 
+%\email[E-mail:]{gezelter@nd.edu}
+
 \date{\today} 
 
+\maketitle
+
 \begin{abstract}
 The ripple phase in phosphatidylcholine (PC) bilayers has never been
 completely explained. 
 \end{abstract}
 
-\pacs{}
-\maketitle
+%\maketitle
 
 \section{Introduction}
 \label{sec:Int}
@@ -178,7 +194,7 @@ $\sigma$ and $\epsilon$ parameters,
 Pechukas.\cite{Berne72} The potential is constructed in the familiar
 form of the Lennard-Jones function using orientation-dependent
 $\sigma$ and $\epsilon$ parameters,
-\begin{eqnarray*}
+\begin{equation*}
 V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat
 r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
 {\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i},
@@ -186,7 +202,7 @@ -\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_
 -\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
 {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right]
 \label{eq:gb}
-\end{eqnarray*}
+\end{equation*}
 
 The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
 \hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf
@@ -195,8 +211,7 @@ $\sigma_0$ are also governed by shape mixing and aniso
 \hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the
 intermolecular separation (${\bf \hat{r}}_{ij}$).  $\sigma$ and
 $\sigma_0$ are also governed by shape mixing and anisotropy variables,
-\begin {equation}
-\begin{array}{rcl}
+\begin {eqnarray*}
 \sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\
 \chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 -
 d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^2 +
@@ -204,12 +219,11 @@ d_j^2 \right)}\right]^{1/2},
 \alpha^2 & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 +
 d_i^2 \right)}{\left( l_j^2 - d_j^2 \right) \left(l_i^2 +
 d_j^2 \right)}\right]^{1/2},
-\end{array}
-\end{equation}
+\end{eqnarray*}
 where $l$ and $d$ describe the length and width of each uniaxial
 ellipsoid.  These shape anisotropy parameters can then be used to
 calculate the range function,
-\begin {equation}
+\begin{equation*}
 \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0}
  \left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf
 \hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf
@@ -218,7 +232,7 @@ calculate the range function,
 \hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2
 \left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}
 \right]^{-1/2}
-\end{equation}
+\end{equation*}
 
 Gay-Berne ellipsoids also have an energy scaling parameter,
 $\epsilon^s$, which describes the well depth for two identical
@@ -254,12 +268,12 @@ those obtained for the range parameter. Ref. \onlineci
 \left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\},
 \end {eqnarray*}
 although many of the quantities and derivatives are identical with
-those obtained for the range parameter. Ref. \onlinecite{Luckhurst90}
+those obtained for the range parameter. Ref. \citen{Luckhurst90}
 has a particularly good explanation of the choice of the Gay-Berne
 parameters $\mu$ and $\nu$ for modeling liquid crystal molecules. An
 excellent overview of the computational methods that can be used to
 efficiently compute forces and torques for this potential can be found
-in Ref. \onlinecite{Golubkov06}
+in Ref. \citen{Golubkov06}
 
 The choices of parameters we have used in this study correspond to a
 shape anisotropy of 3 for the chain portion of the molecule.  In
@@ -295,17 +309,17 @@ each other using a combination of Lennard-Jones,
 are protected by a head ``bead'' with a range parameter which we have
 varied between $1.20 d$ and $1.41 d$.  The head groups interact with
 each other using a combination of Lennard-Jones,
-\begin{eqnarray*}
+\begin{equation}
 V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} -
 \left(\frac{\sigma_h}{r_{ij}}\right)^6\right],
-\end{eqnarray*}
+\end{equation}
 and dipole-dipole,
-\begin{eqnarray*}
+\begin{equation}
 V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
 \hat{r}}_{ij})) = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3}
 \left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot
 \hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right]
-\end{eqnarray*}
+\end{equation}
 potentials.  
 In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing
 along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector
@@ -473,23 +487,51 @@ Figure \ref{fig:topView} shows snapshots of bird's-eye
 different direction from the upper leaf.\label{fig:topView}} 
 \end{figure}
 
+The principal method for observing orientational ordering in dipolar
+or liquid crystalline systems is the $P_2$ order parameter (defined
+as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest
+eigenvalue of the matrix,
+\begin{equation}
+{\mathsf{S}} = \frac{1}{N} \sum_i \left(
+\begin{array}{ccc}
+        u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\
+        u^{y}_i u^{x}_i  & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\
+        u^{z}_i u^{x}_i & u^{z}_i u^{y}_i  & u^{z}_i u^{z}_i -\frac{1}{3} 
+\end{array} \right).
+\label{eq:opmatrix}
+\end{equation}
+Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector
+for molecule $i$.  (Here, $\hat{\bf u}_i$ can refer either to the
+principal axis of the molecular body or to the dipole on the head
+group of the molecule.)  $P_2$ will be $1.0$ for a perfectly-ordered
+system and near $0$ for a randomized system.  Note that this order
+parameter is {\em not} equal to the polarization of the system.  For
+example, the polarization of a perfect anti-ferroelectric arrangement
+of point dipoles is $0$, but $P_2$ for the same system is $1$.  The
+eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is
+familiar as the director axis, which can be used to determine a
+privileged axis for an orientationally-ordered system.  Since the
+molecular bodies are perpendicular to the head group dipoles, it is
+possible for the director axes for the molecular bodies and the head
+groups to be completely decoupled from each other.
+
 Figure \ref{fig:topView} shows snapshots of bird's-eye views of the
 flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.41, 1.35 d$)
 bilayers.  The directions of the dipoles on the head groups are
 represented with two colored half spheres: blue (phosphate) and yellow
 (amino).  For flat bilayers, the system exhibits signs of
-orientational frustration; some disorder in the dipolar chains is
-evident with kinks visible at the edges between different ordered
-domains.  The lipids can also move independently of lipids in the
-opposing leaf, so the ordering of the dipoles on one leaf is not
-necessarily consistant with the ordering on the other.  These two
+orientational frustration; some disorder in the dipolar head-to-tail
+chains is evident with kinks visible at the edges between differently
+ordered domains.  The lipids can also move independently of lipids in
+the opposing leaf, so the ordering of the dipoles on one leaf is not
+necessarily consistent with the ordering on the other.  These two
 factors keep the total dipolar order parameter relatively low for the
 flat phases.
 
 With increasing head group size, the surface becomes corrugated, and
 the dipoles cannot move as freely on the surface. Therefore, the
 translational freedom of lipids in one layer is dependent upon the
-position of lipids in the other layer.  As a result, the ordering of
+position of the lipids in the other layer.  As a result, the ordering of
 the dipoles on head groups in one leaf is correlated with the ordering
 in the other leaf.  Furthermore, as the membrane deforms due to the
 corrugation, the symmetry of the allowed dipolar ordering on each leaf
@@ -497,114 +539,185 @@ antiferroelectric state.   It is also notable that the
 configuration, and the dipolar order parameter increases dramatically.
 However, the total polarization of the system is still close to zero.
 This is strong evidence that the corrugated structure is an
-antiferroelectric state.   It is also notable that the head-to-tail
-arrangement of the dipoles is in a direction perpendicular to the wave
-vector for the surface corrugation.  This is a similar finding to what
-we observed in our earlier work on the elastic dipolar
-membranes.\cite{Sun07} 
+antiferroelectric state.  It is also notable that the head-to-tail
+arrangement of the dipoles is always observed in a direction
+perpendicular to the wave vector for the surface corrugation.  This is
+a similar finding to what we observed in our earlier work on the
+elastic dipolar membranes.\cite{Sun2007}
 
 The $P_2$ order parameters (for both the molecular bodies and the head
 group dipoles) have been calculated to quantify the ordering in these
-phases.  $P_2 = 1$ implies a perfectly ordered structure, and $P_2 = 0$
-implies complete orientational randomization. Figure \ref{fig:rP2}
-shows the $P_2$ order parameter for the head-group dipoles increasing
-with increasing head group size. When the heads of the lipid molecules
-are small, the membrane is nearly flat. The dipolar ordering exhibits
-frustrated orientational ordering in this circumstance. 
+phases.  Figure \ref{fig:rP2} shows that the $P_2$ order parameter for
+the head-group dipoles increases with increasing head group size. When
+the heads of the lipid molecules are small, the membrane is nearly
+flat. Since the in-plane packing is essentially a close packing of the
+head groups, the head dipoles exhibit frustration in their 
+orientational ordering.
 
-The ordering of the tails is essentially opposite to the ordering of
-the dipoles on head group. The $P_2$ order parameter {\it decreases}
-with increasing head size. This indicates that the surface is more
-curved with larger head / tail size ratios. When the surface is flat,
-all tails are pointing in the same direction (parallel to the normal
-of the surface).  This simplified model appears to be exhibiting a
-smectic A fluid phase, similar to the real $L_{\beta}$ phase.  We have
-not observed a smectic C gel phase ($L_{\beta'}$) for this model
-system.  Increasing the size of the heads, results in rapidly
-decreasing $P_2$ ordering for the molecular bodies.
+The ordering trends for the tails are essentially opposite to the
+ordering of the head group dipoles. The tail $P_2$ order parameter
+{\it decreases} with increasing head size. This indicates that the
+surface is more curved with larger head / tail size ratios. When the
+surface is flat, all tails are pointing in the same direction (normal
+to the bilayer surface).  This simplified model appears to be
+exhibiting a smectic A fluid phase, similar to the real $L_{\beta}$
+phase.  We have not observed a smectic C gel phase ($L_{\beta'}$) for
+this model system.  Increasing the size of the heads results in
+rapidly decreasing $P_2$ ordering for the molecular bodies.
 
 \begin{figure}[htb]
 \centering
 \includegraphics[width=\linewidth]{rP2} 
-\caption{The $P_2$ order parameter as a function of the ratio of
-$\sigma_h$ to $d$. \label{fig:rP2}}
+\caption{The $P_2$ order parameters for head groups (circles) and
+molecular bodies (squares) as a function of the ratio of head group
+size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{fig:rP2}}
 \end{figure}
 
-We studied the effects of the interactions between head groups on the
-structure of lipid bilayer by changing the strength of the dipole.
-Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with
-increasing strength of the dipole. Generally the dipoles on the head
-group are more ordered by increase in the strength of the interaction
-between heads and are more disordered by decreasing the interaction
-stength. When the interaction between the heads is weak enough, the
-bilayer structure does not persist; all lipid molecules are solvated
-directly in the water. The critial value of the strength of the dipole
-depends on the head size. The perfectly flat surface melts at $5$
-$0.03$ debye, the asymmetric rippled surfaces melt at $8$ $0.04$
-$0.03$ debye, the symmetric rippled surfaces melt at $10$ $0.04$
-debye. The ordering of the tails is the same as the ordering of the
-dipoles except for the flat phase. Since the surface is already
-perfect flat, the order parameter does not change much until the
-strength of the dipole is $15$ debye. However, the order parameter
-decreases quickly when the strength of the dipole is further
-increased. The head groups of the lipid molecules are brought closer
-by stronger interactions between them. For a flat surface, a large
-amount of free volume between the head groups is available, but when
-the head groups are brought closer, the tails will splay outward,
-forming an inverse micelle. When $\sigma_h=1.28\sigma_0$, the $P_2$
-order parameter decreases slightly after the strength of the dipole is
-increased to $16$ debye. For rippled surfaces, there is less free
-volume available between the head groups. Therefore there is little
-effect on the structure of the membrane due to increasing dipolar
-strength. However, the increase of the $P_2$ order parameter implies
-the membranes are flatten by the increase of the strength of the
-dipole. Unlike other systems that melt directly when the interaction
-is weak enough, for $\sigma_h=1.41\sigma_0$, part of the membrane
-melts into itself first. The upper leaf of the bilayer becomes totally
-interdigitated with the lower leaf. This is different behavior than
-what is exhibited with the interdigitated lines in the rippled phase
-where only one interdigitated line connects the two leaves of bilayer.
+In addition to varying the size of the head groups, we studied the
+effects of the interactions between head groups on the structure of
+lipid bilayer by changing the strength of the dipoles.  Figure
+\ref{fig:sP2} shows how the $P_2$ order parameter changes with
+increasing strength of the dipole.  Generally, the dipoles on the head
+groups become more ordered as the strength of the interaction between
+heads is increased and become more disordered by decreasing the
+interaction stength.  When the interaction between the heads becomes
+too weak, the bilayer structure does not persist; all lipid molecules
+become dispersed in the solvent (which is non-polar in this
+molecular-scale model).  The critial value of the strength of the
+dipole depends on the size of the head groups.  The perfectly flat
+surface becomes unstable below $5$ Debye, while the  rippled
+surfaces melt at $8$ Debye (asymmetric) or $10$ Debye (symmetric). 
+
+The ordering of the tails mirrors the ordering of the dipoles {\it
+except for the flat phase}. Since the surface is nearly flat in this
+phase, the order parameters are only weakly dependent on dipolar
+strength until it reaches $15$ Debye.  Once it reaches this value, the
+head group interactions are strong enough to pull the head groups
+close to each other and distort the bilayer structure. For a flat
+surface, a substantial amount of free volume between the head groups
+is normally available.  When the head groups are brought closer by
+dipolar interactions, the tails are forced to splay outward, forming
+first curved bilayers, and then inverted micelles. 
+
+When $\sigma_h=1.28 d$, the $P_2$ order parameter decreases slightly
+when the strength of the dipole is increased above $16$ debye. For
+rippled bilayers, there is less free volume available between the head
+groups. Therefore increasing dipolar strength weakly influences the
+structure of the membrane.  However, the increase in the body $P_2$
+order parameters implies that the membranes are being slightly
+flattened due to the effects of increasing head-group attraction.
+
+A very interesting behavior takes place when the head groups are very
+large relative to the molecular bodies ($\sigma_h = 1.41 d$) and the
+dipolar strength is relatively weak ($\mu < 9$ Debye). In this case,
+the two leaves of the bilayer become totally interdigitated with each
+other in large patches of the membrane.   With higher dipolar
+strength, the interdigitation is limited to single lines that run
+through the bilayer in a direction perpendicular to the ripple wave
+vector. 
+
 \begin{figure}[htb]
 \centering
 \includegraphics[width=\linewidth]{sP2} 
-\caption{The $P_2$ order parameter as a funtion of the strength of the
-dipole.\label{fig:sP2}}
+\caption{The $P_2$ order parameters for head group dipoles (a) and
+molecular bodies (b) as a function of the strength of the dipoles.
+These order parameters are shown for four values of the head group /
+molecular width ratio ($\sigma_h / d$). \label{fig:sP2}}
 \end{figure}
 
-Figure \ref{fig:tP2} shows the dependence of the order parameter on
-temperature. The behavior of the $P_2$ order paramter is
-straightforward. Systems are more ordered at low temperature, and more
-disordered at high temperatures. When the temperature is high enough,
-the membranes are instable. For flat surfaces ($\sigma_h=1.20\sigma_0$
-and $\sigma_h=1.28\sigma_0$), when the temperature is increased to
-$310$, the $P_2$ order parameter increases slightly instead of
-decreases like ripple surface. This is an evidence of the frustration
-of the dipolar ordering in each leaf of the lipid bilayer, at low
-temperature, the systems are locked in a local minimum energy state,
-with increase of the temperature, the system can jump out the local
-energy well to find the lower energy state which is the longer range
-orientational ordering. Like the dipolar ordering of the flat
-surfaces, the ordering of the tails of the lipid molecules for ripple
-membranes ($\sigma_h=1.35\sigma_0$ and $\sigma_h=1.41\sigma_0$) also
-show some nonthermal characteristic. With increase of the temperature,
-the $P_2$ order parameter decreases firstly, and increases afterward
-when the temperature is greater than $290 K$. The increase of the
-$P_2$ order parameter indicates a more ordered structure for the tails
-of the lipid molecules which corresponds to a more flat surface. Since
-our model lacks the detailed information on lipid tails, we can not
-simulate the fluid phase with melted fatty acid chains. Moreover, the
-formation of the tilted $L_{\beta'}$ phase also depends on the
-organization of fatty groups on tails.
+Figure \ref{fig:tP2} shows the dependence of the order parameters on
+temperature.  As expected, systems are more ordered at low
+temperatures, and more disordered at high temperatures.  All of the
+bilayers we studied can become unstable if the temperature becomes
+high enough.  The only interesting feature of the temperature
+dependence is in the flat surfaces ($\sigma_h=1.20 d$ and
+$\sigma_h=1.28 d$).  Here, when the temperature is increased above
+$310$K, there is enough jostling of the head groups to allow the
+dipolar frustration to resolve into more ordered states.  This results
+in a slight increase in the $P_2$ order parameter above this
+temperature.
+
+For the rippled surfaces ($\sigma_h=1.35 d$ and $\sigma_h=1.41 d$),
+there is a slightly increased orientational ordering in the molecular
+bodies above $290$K.  Since our model lacks the detailed information
+about the behavior of the lipid tails, this is the closest the model
+can come to depicting the ripple ($P_{\beta'}$) to fluid
+($L_{\alpha}$) phase transition.  What we are observing is a
+flattening of the rippled structures made possible by thermal
+expansion of the tightly-packed head groups.  The lack of detailed
+chain configurations also makes it impossible for this model to depict
+the ripple to gel ($L_{\beta'}$) phase transition.
+
 \begin{figure}[htb]
 \centering
 \includegraphics[width=\linewidth]{tP2} 
-\caption{The $P_2$ order parameter as a funtion of
-temperature.\label{fig:tP2}}
+\caption{The $P_2$ order parameters for head group dipoles (a) and
+molecular bodies (b) as a function of temperature.
+These order parameters are shown for four values of the head group /
+molecular width ratio ($\sigma_h / d$).\label{fig:tP2}}
 \end{figure}
 
 \section{Discussion}
 \label{sec:discussion}
 
+The ripple phases have been observed in our molecular dynamic
+simulations using a simple molecular lipid model. The lipid model
+consists of an anisotropic interacting dipolar head group and an
+ellipsoid shape tail. According to our simulations, the explanation of
+the formation for the ripples are originated in the size mismatch
+between the head groups and the tails. The ripple phases are only
+observed in the studies using larger head group lipid models. However,
+there is a mismatch betweent the size of the head groups and the size
+of the tails in the simulations of the flat surface. This indicates
+the competition between the anisotropic dipolar interaction and the
+packing of the tails also plays a major role for formation of the
+ripple phase. The larger head groups provide more free volume for the
+tails, while these hydrophobic ellipsoids trying to be close to each
+other, this gives the origin of the spontanous curvature of the
+surface, which is believed as the beginning of the ripple phases. The
+lager head groups cause the spontanous curvature inward for both of
+leaves of the bilayer. This results in a steric strain when the tails
+of two leaves too close to each other. The membrane has to be broken
+to release this strain. There are two ways to arrange these broken
+curvatures: symmetric and asymmetric ripples. Both of the ripple
+phases have been observed in our studies. The difference between these
+two ripples is that the bilayer is continuum in the symmetric ripple
+phase and is disrupt in the asymmetric ripple phase.
+
+Dipolar head groups are the key elements for the maintaining of the
+bilayer structure. The lipids are solvated in water when lowering the
+the strength of the dipole on the head groups. The long range
+orientational ordering of the dipoles can be achieved by forming the
+ripples, although the dipoles are likely to form head-to-tail
+configurations even in flat surface, the frustration prevents the
+formation of the long range orientational ordering for dipoles. The
+corrugation of the surface breaks the frustration and stablizes the
+long range oreintational ordering for the dipoles in the head groups
+of the lipid molecules. Many rows of the head-to-tail dipoles are
+parallel to each other and adopt the antiferroelectric state as a
+whole. This is the first time the organization of the head groups in
+ripple phases of the lipid bilayer has been addressed.
+
+The most important prediction we can make using the results from this
+simple model is that if dipolar ordering is driving the surface
+corrugation, the wave vectors for the ripples should always found to
+be {\it perpendicular} to the dipole director axis.  This prediction
+should suggest experimental designs which test whether this is really
+true in the phosphatidylcholine $P_{\beta'}$ phases.  The dipole
+director axis should also be easily computable for the all-atom and
+coarse-grained simulations that have been published in the literature.
+
+Although our model is simple, it exhibits some rich and unexpected
+behaviors.  It would clearly be a closer approximation to the reality
+if we allowed greater translational freedom to the dipoles and
+replaced the somewhat artificial lattice packing and the harmonic
+elastic tension with more realistic molecular modeling potentials.
+What we have done is to present a simple model which exhibits bulk
+non-thermal corrugation, and our explanation of this rippling
+phenomenon will help us design more accurate molecular models for
+corrugated membranes and experiments to test whether rippling is
+dipole-driven or not.
+
 \newpage
 \bibliography{mdripple}
 \end{document}