--- trunk/mdRipple/mdripple.tex	2007/08/02 15:57:11	3203
+++ trunk/mdRipple/mdripple.tex	2007/08/02 21:20:07	3204
@@ -48,8 +48,8 @@ stablizes the long range orientational ordering for th
 head groups.  One feature of this model is that an energetically
 favorable orientational ordering of the dipoles can be achieved by
 out-of-plane membrane corrugation.  The corrugation of the surface
-stablizes the long range orientational ordering for the dipoles in the
-head groups which then adopt a bulk antiferroelectric state. We
+stabilizes the long range orientational ordering for the dipoles in the
+head groups which then adopt a bulk anti-ferroelectric state. We
 observe a common feature of the corrugated dipolar membranes: the wave
 vectors for the surface ripples are always found to be perpendicular
 to the dipole director axis.  
@@ -75,11 +75,14 @@ within the gel phase.~\cite{Cevc87}
 experimental results provide strong support for a 2-dimensional
 hexagonal packing lattice of the lipid molecules within the ripple
 phase.  This is a notable change from the observed lipid packing
-within the gel phase.~\cite{Cevc87}
+within the gel phase.~\cite{Cevc87} The X-ray diffraction work by
+Katsaras {\it et al.} showed that a rich phase diagram exhibiting both
+{\it asymmetric} and {\it symmetric} ripples is possible for lecithin
+bilayers.\cite{Katsaras00}
 
 A number of theoretical models have been presented to explain the
 formation of the ripple phase. Marder {\it et al.} used a
-curvature-dependent Landau-de Gennes free-energy functional to predict
+curvature-dependent Landau-de~Gennes free-energy functional to predict
 a rippled phase.~\cite{Marder84} This model and other related continuum
 models predict higher fluidity in convex regions and that concave
 portions of the membrane correspond to more solid-like regions.
@@ -105,16 +108,16 @@ of lamellar stacks of hexagonal lattices to show that 
 regions.~\cite{Heimburg00} Kubica has suggested that a lattice model of
 polar head groups could be valuable in trying to understand bilayer
 phase formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations
-of lamellar stacks of hexagonal lattices to show that large headgroups
+of lamellar stacks of hexagonal lattices to show that large head groups
 and molecular tilt with respect to the membrane normal vector can
 cause bulk rippling.~\cite{Bannerjee02}
 
-In contrast, few large-scale molecular modelling studies have been
+In contrast, few large-scale molecular modeling studies have been
 done due to the large size of the resulting structures and the time
 required for the phases of interest to develop.  With all-atom (and
 even unified-atom) simulations, only one period of the ripple can be
-observed and only for timescales in the range of 10-100 ns.  One of
-the most interesting molecular simulations was carried out by De Vries
+observed and only for time scales in the range of 10-100 ns.  One of
+the most interesting molecular simulations was carried out by de~Vries
 {\it et al.}~\cite{deVries05}. According to their simulation results,
 the ripple consists of two domains, one resembling the gel bilayer,
 while in the other, the two leaves of the bilayer are fully
@@ -136,7 +139,7 @@ between headgroups and tails is strongly implicated as
 
 Although the organization of the tails of lipid molecules are
 addressed by these molecular simulations and the packing competition
-between headgroups and tails is strongly implicated as the primary
+between head groups and tails is strongly implicated as the primary
 driving force for ripple formation, questions about the ordering of
 the head groups in ripple phase have not been settled.
 
@@ -145,9 +148,9 @@ elastic membranes is the antiferroelectric ordering of
 between dipolar ordering and membrane buckling.\cite{Sun2007} We found
 that dipolar elastic membranes can spontaneously buckle, forming
 ripple-like topologies.  The driving force for the buckling of dipolar
-elastic membranes is the antiferroelectric ordering of the dipoles.
+elastic membranes is the anti-ferroelectric ordering of the dipoles.
 This was evident in the ordering of the dipole director axis
-perpendicular to the wave vector of the surface ripples.  A similiar
+perpendicular to the wave vector of the surface ripples.  A similar
 phenomenon has also been observed by Tsonchev {\it et al.} in their
 work on the spontaneous formation of dipolar peptide chains into
 curved nano-structures.\cite{Tsonchev04,Tsonchev04II}
@@ -181,7 +184,7 @@ some fraction of the details of the chain dynamics neg
 non-polar tails. Another fact is that the majority of lipid molecules
 in the ripple phase are relatively rigid (i.e. gel-like) which makes
 some fraction of the details of the chain dynamics negligible.  Figure
-\ref{fig:lipidModels} shows the molecular strucure of a DPPC
+\ref{fig:lipidModels} shows the molecular structure of a DPPC
 molecule, as well as atomistic and molecular-scale representations of
 a DPPC molecule.  The hydrophilic character of the head group is
 largely due to the separation of charge between the nitrogen and
@@ -200,7 +203,7 @@ modelling large length-scale properties of lipid
 The ellipsoidal portions of the model interact via the Gay-Berne
 potential which has seen widespread use in the liquid crystal
 community.  Ayton and Voth have also used Gay-Berne ellipsoids for
-modelling large length-scale properties of lipid
+modeling large length-scale properties of lipid
 bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential
 was a single site model for the interactions of rigid ellipsoidal
 molecules.\cite{Gay81} It can be thought of as a modification of the
@@ -250,7 +253,7 @@ ellipsoids in a {\it side-by-side} configuration.  Add
 
 Gay-Berne ellipsoids also have an energy scaling parameter,
 $\epsilon^s$, which describes the well depth for two identical
-ellipsoids in a {\it side-by-side} configuration.  Additionaly, a well
+ellipsoids in a {\it side-by-side} configuration.  Additionally, a well
 depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes
 the ratio between the well depths in the {\it end-to-end} and
 side-by-side configurations.  As in the range parameter, a set of
@@ -393,19 +396,19 @@ Debye dipole moment of the PC headgroups.
 measured relative to this unit of distance.  Because the solvent we
 are using is non-polar and has a dielectric constant of 1, values for
 $\mu$ are sampled from a range that is somewhat smaller than the 20.6
-Debye dipole moment of the PC headgroups.
+Debye dipole moment of the PC head groups.
 
 To create unbiased bilayers, all simulations were started from two
 perfectly flat monolayers separated by a 26 \AA\ gap between the
 molecular bodies of the upper and lower leaves.  The separated
-monolayers were evolved in a vaccum with $x-y$ anisotropic pressure
+monolayers were evolved in a vacuum with $x-y$ anisotropic pressure
 coupling. The length of $z$ axis of the simulations was fixed and a
 constant surface tension was applied to enable real fluctuations of
 the bilayer. Periodic boundary conditions were used, and $480-720$
 lipid molecules were present in the simulations, depending on the size
 of the head beads.  In all cases, the two monolayers spontaneously
 collapsed into bilayer structures within 100 ps. Following this
-collapse, all systems were equlibrated for $100$ ns at $300$ K.
+collapse, all systems were equilibrated for $100$ ns at $300$ K.
 
 The resulting bilayer structures were then solvated at a ratio of $6$
 solvent beads (24 water molecules) per lipid. These configurations
@@ -413,7 +416,7 @@ this model, a timestep of $50$ fs was utilized with ex
 the solvent were carried out at constant pressure ($P=1$ atm) with
 $3$D anisotropic coupling, and constant surface tension
 ($\gamma=0.015$ N/m). Given the absence of fast degrees of freedom in
-this model, a timestep of $50$ fs was utilized with excellent energy
+this model, a time step of $50$ fs was utilized with excellent energy
 conservation.  Data collection for structural properties of the
 bilayers was carried out during a final 5 ns run following the solvent
 equilibration.  All simulations were performed using the OOPSE
@@ -429,10 +432,7 @@ bodies ($\sigma_h=1.20 d$) the membrane exhibits a fla
 bilayer phases.  The surface topology of these phases depends most
 sensitively on the ratio of the size of the head groups to the width
 of the molecular bodies.  With heads only slightly larger than the
-bodies ($\sigma_h=1.20 d$) the membrane exhibits a flat bilayer.  The
-mean spacing between the head groups is XXX \AA, and the mean
-area per lipid in this phase is   \AA$^2$.  This corresponds
-reasonably well to a bilayer of DPPC.\cite{XXX}
+bodies ($\sigma_h=1.20 d$) the membrane exhibits a flat bilayer. 
 
 Increasing the head / body size ratio increases the local membrane
 curvature around each of the lipids.  With $\sigma_h=1.28 d$, the
@@ -453,7 +453,7 @@ very similiar to the structures observed by both de Vr
 For the largest head / body ratios studied ($\sigma_h = 1.41 d$) the
 local curvature is substantially larger, and the resulting bilayer
 structure resolves into an asymmetric ripple phase.  This structure is
-very similiar to the structures observed by both de Vries {\it et al.}
+very similar to the structures observed by both de~Vries {\it et al.}
 and Lenz {\it et al.}.  For a given ripple wave vector, there are two
 possible asymmetric ripples, which is not the case for the symmetric
 phase observed when $\sigma_h = 1.35 d$.
@@ -479,21 +479,40 @@ phases are shown in Figure \ref{fig:phaseCartoon}.  
 ($\sigma_h = 1.35 d$, and asymmetric ($\sigma_h = 1.41 d$) ripple
 phases are shown in Figure \ref{fig:phaseCartoon}.  
 
+It is reasonable to ask how well the parameters we used can produce
+bilayer properties that match experimentally known values for real
+lipid bilayers.  Using a value of $l = 13.8$ \AA for the ellipsoidal
+tails and the fixed ellipsoidal aspect ratio of 3, our values for the
+area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend
+entirely on the size of the head bead relative to the molecular body.
+These values are tabulated in table \ref{tab:property}.  Kucera {\it
+et al.}  have measured values for the head group spacings for a number
+of PC lipid bilayers that range from 30.8 \AA (DLPC) to 37.8 (DPPC).
+They have also measured values for the area per lipid that range from
+60.6
+\AA$^2$ (DMPC) to 64.2 \AA$^2$
+(DPPC).\cite{NorbertKucerka04012005,NorbertKucerka06012006} Only the
+largest of the head groups we modeled ($\sigma_h = 1.41 d$) produces
+bilayers (specifically the area per lipid) that resemble real PC
+bilayers.  The smaller head beads we used are perhaps better models
+for PE head groups.
+
 \begin{table*}
 \begin{minipage}{\linewidth}
 \begin{center}
-\caption{Phases, ripple wavelengths and amplitudes observed as a
-function of the ratio between the head beads and the diameters of the
-tails.  All lengths are normalized to the diameter of the tail
-ellipsoids.}
-\begin{tabular}{lccc}
+\caption{Phase, bilayer spacing, area per lipid, ripple wavelength
+and amplitude observed as a function of the ratio between the head
+beads and the diameters of the tails.  Ripple wavelengths and
+amplitudes are normalized to the diameter of the tail ellipsoids.}
+\begin{tabular}{lccccc}
 \hline 
-$\sigma_h / d$ & type of phase & $\lambda / d$ & $A / d$\\
+$\sigma_h / d$ & type of phase & bilayer spacing (\AA) & area per
+lipid (\AA$^2$) & $\lambda / d$ & $A / d$\\
 \hline 
-1.20 & flat & N/A & N/A \\
-1.28 & flat & N/A & N/A \\
-1.35 & symmetric ripple & 17.2 & 2.2 \\
-1.41 & asymmetric ripple & 15.4 & 1.5 \\
+1.20 & flat & 33.4 & 49.6 & N/A & N/A \\
+1.28 & flat & 33.7 & 54.7 & N/A & N/A \\
+1.35 & symmetric ripple & 42.9 & 51.7 & 17.2 & 2.2 \\
+1.41 & asymmetric ripple & 37.1 & 63.1 & 15.4 & 1.5 \\
 \end{tabular}
 \label{tab:property}
 \end{center}
@@ -575,7 +594,7 @@ 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
+anti-ferroelectric 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
@@ -616,10 +635,10 @@ interaction stength.  When the interaction between the
 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
+interaction strength.  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
+molecular-scale model).  The critical 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). 
@@ -636,7 +655,7 @@ when the strength of the dipole is increased above $16
 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
+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 $P_2$
@@ -751,10 +770,10 @@ plane, making vortex defects somewhat more expensive, 
 One feature of this model is that an energetically favorable
 orientational ordering of the dipoles can be achieved by forming
 ripples.  The corrugation of the surface breaks the symmetry of the
-plane, making vortex defects somewhat more expensive, and stablizing
+plane, making vortex defects somewhat more expensive, and stabilizing
 the long range orientational ordering for the dipoles in the head
 groups.  Most of the rows of the head-to-tail dipoles are parallel to
-each other and the system adopts a bulk antiferroelectric state.  We
+each other and the system adopts a bulk anti-ferroelectric state.  We
 believe that this is the first time the organization of the head
 groups in ripple phases has been addressed.
 
@@ -778,7 +797,7 @@ sizes, 50 fs timesteps) allow us to rapidly explore th
 we allowed bending motions between the dipoles and the molecular
 bodies, and if we replaced the rigid ellipsoids with ball-and-chain
 tails.  However, the advantages of this simple model (large system
-sizes, 50 fs timesteps) allow us to rapidly explore the phase diagram
+sizes, 50 fs time steps) allow us to rapidly explore the phase diagram
 for a wide range of parameters.  Our explanation of this rippling
 phenomenon will help us design more accurate molecular models for
 corrugated membranes and experiments to test whether or not
