--- trunk/xDissertation/md.tex	2008/03/05 19:53:46	3359
+++ trunk/xDissertation/md.tex	2008/03/14 21:38:07	3370
@@ -1,29 +1,8 @@
-\chapter{\label{chap:md}Dipolar ordering in the ripple phases of
-molecular-scale models of lipid membranes}
+\chapter{\label{chap:md}DIPOLAR ORDERING IN THE RIPPLE PHASES OF
+MOLECULAR-SCALE MODELS OF LIPID MEMBRANES}
 
 \section{Introduction}
 \label{mdsec:Int}
-Fully hydrated lipids will aggregate spontaneously to form bilayers
-which exhibit a variety of phases depending on their temperatures and
-compositions. Among these phases, a periodic rippled phase
-($P_{\beta'}$) appears as an intermediate phase between the gel
-($L_\beta$) and fluid ($L_{\alpha}$) phases for relatively pure
-phosphatidylcholine (PC) bilayers.  The ripple phase has attracted
-substantial experimental interest over the past 30 years. Most
-structural information of the ripple phase has been obtained by the
-X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron
-microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it
-et al.} used atomic force microscopy (AFM) to observe ripple phase
-morphology in bilayers supported on mica.~\cite{Kaasgaard03} The
-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} although Tenchov {\it et al.} have
-recently observed near-hexagonal packing in some phosphatidylcholine
-(PC) gel phases.\cite{Tenchov2001} 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
@@ -33,11 +12,11 @@ predict the formation of a ripple-like phase.  Their m
 concave portions of the membrane correspond to more solid-like
 regions.  Carlson and Sethna used a packing-competition model (in
 which head groups and chains have competing packing energetics) to
-predict the formation of a ripple-like phase.  Their model predicted
-that the high-curvature portions have lower-chain packing and
-correspond to more fluid-like regions.  Goldstein and Leibler used a
-mean-field approach with a planar model for {\em inter-lamellar}
-interactions to predict rippling in multilamellar
+predict the formation of a ripple-like phase~\cite{Carlson87}.  Their
+model predicted that the high-curvature portions have lower-chain
+packing and correspond to more fluid-like regions.  Goldstein and
+Leibler used a mean-field approach with a planar model for {\em
+inter-lamellar} interactions to predict rippling in multilamellar
 phases.~\cite{Goldstein88} McCullough and Scott proposed that the {\em
 anisotropy of the nearest-neighbor interactions} coupled to
 hydrophobic constraining forces which restrict height differences
@@ -60,7 +39,7 @@ longer ``chains''. 
 described the formation of symmetric ripple-like structures using a
 coarse grained solvent-head-tail bead model.\cite{Kranenburg2005}
 Their lipids consisted of a short chain of head beads tied to the two
-longer ``chains''. 
+longer ``chains''.
 
 In contrast, few large-scale molecular modeling studies have been
 done due to the large size of the resulting structures and the time
@@ -93,7 +72,7 @@ In a recent paper, we presented a simple ``web of dipo
 driving force for ripple formation, questions about the ordering of
 the head groups in ripple phase have not been settled.
 
-In a recent paper, we presented a simple ``web of dipoles'' spin
+In Ch.~\ref{chap:mc}, we presented a simple ``web of dipoles'' spin
 lattice model which provides some physical insight into relationship
 between dipolar ordering and membrane buckling.\cite{sun:031602} We
 found that dipolar elastic membranes can spontaneously buckle, forming
@@ -105,7 +84,7 @@ In this paper, we construct a somewhat more realistic 
 work on the spontaneous formation of dipolar peptide chains into
 curved nano-structures.\cite{Tsonchev04,Tsonchev04II}
 
-In this paper, we construct a somewhat more realistic molecular-scale
+In this chapter, we construct a somewhat more realistic molecular-scale
 lipid model than our previous ``web of dipoles'' and use molecular
 dynamics simulations to elucidate the role of the head group dipoles
 in the formation and morphology of the ripple phase.  We describe our
@@ -161,15 +140,17 @@ $\sigma$ 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{equation*}
+\begin{multline}
 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},
+{\mathbf{\hat r}_{ij}})\left[ \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i},
 {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12}
--\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
-{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right]
+\right. \\
+\left. - \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{mdeq:gb}
-\end{equation*}
+\end{multline}
 
 The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
 \hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf
@@ -190,16 +171,16 @@ calculate the range function,
 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*}
-\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
+\begin{multline}
+\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
 \hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf
 \hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf
 \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{multline}
 
 Gay-Berne ellipsoids also have an energy scaling parameter,
 $\epsilon^s$, which describes the well depth for two identical
@@ -217,23 +198,25 @@ The form of the strength function is somewhat complica
 \alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1}
 \end{eqnarray*}
 The form of the strength function is somewhat complicated,
-\begin {eqnarray*}
+\begin{eqnarray*}
 \epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = &
 \epsilon_{0}  \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j})
  \epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
 \hat{r}}_{ij}) \\ \\
 \epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = &
 \left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf
-\hat{u}}_{j})^{2}\right]^{-1/2} \\ \\
-\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) &
-= &
- 1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf
+\hat{u}}_{j})^{2}\right]^{-1/2}
+\end{eqnarray*}
+\begin{multline*}
+\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij})
+=  \\
+1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf
 \hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf
 \hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf
 \hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf
 \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\},
-\end {eqnarray*}
+\end{multline*}
 although many of the quantities and derivatives are identical with
 those obtained for the range parameter. Ref. \citen{Luckhurst90}
 has a particularly good explanation of the choice of the Gay-Berne
@@ -476,7 +459,7 @@ lipid bilayers.  Using a value of $l = 13.8$ \AA for t
 
 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
+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.
@@ -537,33 +520,14 @@ The principal method for observing orientational order
 different direction from the upper leaf.\label{mdfig: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{mdeq: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
+The orientational ordering in the system is observed by $P_2$ order
+parameter, which is calculated from Eq.~\ref{mceq:opmatrix}
+in Ch.~\ref{chap:mc}. 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.
+group of the molecule. 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{mdfig:topView} shows snapshots of bird's-eye views of the
 flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$)
@@ -732,7 +696,9 @@ D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle 
 We have computed translational diffusion constants for lipid molecules
 from the mean-square displacement,
 \begin{equation}
-D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle,
+D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf
+r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle,
+\label{mdeq:msdisplacement}
 \end{equation}
 of the lipid bodies. Translational diffusion constants for the
 different head-to-tail size ratios (all at 300 K) are shown in table
