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1 < \chapter{\label{chap:md}Dipolar ordering in the ripple phases of
2 < molecular-scale models of lipid membranes}
1 > \chapter{\label{chap:md}DIPOLAR ORDERING IN THE RIPPLE PHASES OF
2 > MOLECULAR-SCALE MODELS OF LIPID MEMBRANES}
3  
4   \section{Introduction}
5   \label{mdsec:Int}
6 – Fully hydrated lipids will aggregate spontaneously to form bilayers
7 – which exhibit a variety of phases depending on their temperatures and
8 – compositions. Among these phases, a periodic rippled phase
9 – ($P_{\beta'}$) appears as an intermediate phase between the gel
10 – ($L_\beta$) and fluid ($L_{\alpha}$) phases for relatively pure
11 – phosphatidylcholine (PC) bilayers.  The ripple phase has attracted
12 – substantial experimental interest over the past 30 years. Most
13 – structural information of the ripple phase has been obtained by the
14 – X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron
15 – microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it
16 – et al.} used atomic force microscopy (AFM) to observe ripple phase
17 – morphology in bilayers supported on mica.~\cite{Kaasgaard03} The
18 – experimental results provide strong support for a 2-dimensional
19 – hexagonal packing lattice of the lipid molecules within the ripple
20 – phase.  This is a notable change from the observed lipid packing
21 – within the gel phase,~\cite{Cevc87} although Tenchov {\it et al.} have
22 – recently observed near-hexagonal packing in some phosphatidylcholine
23 – (PC) gel phases.\cite{Tenchov2001} The X-ray diffraction work by
24 – Katsaras {\it et al.} showed that a rich phase diagram exhibiting both
25 – {\it asymmetric} and {\it symmetric} ripples is possible for lecithin
26 – bilayers.\cite{Katsaras00}
6  
7   A number of theoretical models have been presented to explain the
8   formation of the ripple phase. Marder {\it et al.} used a
# Line 33 | Line 12 | predict the formation of a ripple-like phase.  Their m
12   concave portions of the membrane correspond to more solid-like
13   regions.  Carlson and Sethna used a packing-competition model (in
14   which head groups and chains have competing packing energetics) to
15 < predict the formation of a ripple-like phase.  Their model predicted
16 < that the high-curvature portions have lower-chain packing and
17 < correspond to more fluid-like regions.  Goldstein and Leibler used a
18 < mean-field approach with a planar model for {\em inter-lamellar}
19 < interactions to predict rippling in multilamellar
15 > predict the formation of a ripple-like phase~\cite{Carlson87}.  Their
16 > model predicted that the high-curvature portions have lower-chain
17 > packing and correspond to more fluid-like regions.  Goldstein and
18 > Leibler used a mean-field approach with a planar model for {\em
19 > inter-lamellar} interactions to predict rippling in multilamellar
20   phases.~\cite{Goldstein88} McCullough and Scott proposed that the {\em
21   anisotropy of the nearest-neighbor interactions} coupled to
22   hydrophobic constraining forces which restrict height differences
# Line 60 | Line 39 | longer ``chains''.
39   described the formation of symmetric ripple-like structures using a
40   coarse grained solvent-head-tail bead model.\cite{Kranenburg2005}
41   Their lipids consisted of a short chain of head beads tied to the two
42 < longer ``chains''.
42 > longer ``chains''.
43  
44   In contrast, few large-scale molecular modeling studies have been
45   done due to the large size of the resulting structures and the time
# Line 93 | Line 72 | In a recent paper, we presented a simple ``web of dipo
72   driving force for ripple formation, questions about the ordering of
73   the head groups in ripple phase have not been settled.
74  
75 < In a recent paper, we presented a simple ``web of dipoles'' spin
75 > In Ch.~\ref{chap:mc}, we presented a simple ``web of dipoles'' spin
76   lattice model which provides some physical insight into relationship
77 < between dipolar ordering and membrane buckling.\cite{Sun2007} We found
78 < that dipolar elastic membranes can spontaneously buckle, forming
77 > between dipolar ordering and membrane buckling.\cite{sun:031602} We
78 > found that dipolar elastic membranes can spontaneously buckle, forming
79   ripple-like topologies.  The driving force for the buckling of dipolar
80   elastic membranes is the anti-ferroelectric ordering of the dipoles.
81   This was evident in the ordering of the dipole director axis
# Line 105 | Line 84 | In this paper, we construct a somewhat more realistic
84   work on the spontaneous formation of dipolar peptide chains into
85   curved nano-structures.\cite{Tsonchev04,Tsonchev04II}
86  
87 < In this paper, we construct a somewhat more realistic molecular-scale
87 > In this chapter, we construct a somewhat more realistic molecular-scale
88   lipid model than our previous ``web of dipoles'' and use molecular
89   dynamics simulations to elucidate the role of the head group dipoles
90   in the formation and morphology of the ripple phase.  We describe our
# Line 122 | Line 101 | the ripple formation can be found in section
101   \begin{figure}
102   \centering
103   \includegraphics[width=\linewidth]{./figures/mdLipidModels.pdf}
104 < \caption{Three different representations of DPPC lipid molecules,
104 > \caption[Three different representations of DPPC lipid
105 > molecules]{Three different representations of DPPC lipid molecules,
106   including the chemical structure, an atomistic model, and the
107   head-body ellipsoidal coarse-grained model used in this
108   work.\label{mdfig:lipidModels}}
# Line 156 | Line 136 | molecules.\cite{Gay81} It can be thought of as a modif
136   modeling large length-scale properties of lipid
137   bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential
138   was a single site model for the interactions of rigid ellipsoidal
139 < molecules.\cite{Gay81} It can be thought of as a modification of the
139 > molecules.\cite{Gay1981} It can be thought of as a modification of the
140   Gaussian overlap model originally described by Berne and
141   Pechukas.\cite{Berne72} The potential is constructed in the familiar
142   form of the Lennard-Jones function using orientation-dependent
143   $\sigma$ and $\epsilon$ parameters,
144 < \begin{equation*}
144 > \begin{multline}
145   V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat
146   r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
147 < {\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i},
147 > {\mathbf{\hat r}_{ij}})\left[ \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i},
148   {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12}
149 < -\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
150 < {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right]
149 > \right. \\
150 > \left. - \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i},
151 > {\mathbf{\hat u}_j}, {\mathbf{\hat
152 > r}_{ij}})+\sigma_0}\right)^6\right]
153   \label{mdeq:gb}
154 < \end{equation*}
154 > \end{multline}
155  
156   The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
157   \hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf
# Line 190 | Line 172 | calculate the range function,
172   where $l$ and $d$ describe the length and width of each uniaxial
173   ellipsoid.  These shape anisotropy parameters can then be used to
174   calculate the range function,
175 < \begin{equation*}
176 < \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0}
177 < \left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf
175 > \begin{multline}
176 > \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \\
177 > \sigma_0 \left[ 1 - \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf
178   \hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf
179   \hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf
180   \hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf
181   \hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2
182   \left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}
183   \right]^{-1/2}
184 < \end{equation*}
184 > \end{multline}
185  
186   Gay-Berne ellipsoids also have an energy scaling parameter,
187   $\epsilon^s$, which describes the well depth for two identical
# Line 217 | Line 199 | The form of the strength function is somewhat complica
199   \alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1}
200   \end{eqnarray*}
201   The form of the strength function is somewhat complicated,
202 < \begin {eqnarray*}
202 > \begin{eqnarray*}
203   \epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = &
204   \epsilon_{0}  \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j})
205   \epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
206   \hat{r}}_{ij}) \\ \\
207   \epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = &
208   \left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf
209 < \hat{u}}_{j})^{2}\right]^{-1/2} \\ \\
210 < \epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) &
211 < = &
212 < 1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf
209 > \hat{u}}_{j})^{2}\right]^{-1/2}
210 > \end{eqnarray*}
211 > \begin{multline*}
212 > \epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij})
213 > =  \\
214 > 1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf
215   \hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf
216   \hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf
217   \hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf
218   \hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2
219   \left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\},
220 < \end {eqnarray*}
220 > \end{multline*}
221   although many of the quantities and derivatives are identical with
222   those obtained for the range parameter. Ref. \citen{Luckhurst90}
223   has a particularly good explanation of the choice of the Gay-Berne
# Line 260 | Line 244 | actual parameters used in our simulations are given in
244   \begin{figure}
245   \centering
246   \includegraphics[width=\linewidth]{./figures/md2LipidModel.pdf}
247 < \caption{The parameters defining the behavior of the lipid
248 < models. $\sigma_h / d$ is the ratio of the head group to body diameter.
249 < Molecular bodies had a fixed aspect ratio of 3.0.  The solvent model
250 < was a simplified 4-water bead ($\sigma_w \approx d$) that has been
251 < used in other coarse-grained simulations.  The dipolar strength
252 < (and the temperature and pressure) were the only other parameters that
253 < were varied systematically.\label{mdfig:lipidModel}}
247 > \caption[The parameters defining the behavior of the lipid
248 > models]{The parameters defining the behavior of the lipid
249 > models. $\sigma_h / d$ is the ratio of the head group to body
250 > diameter.  Molecular bodies had a fixed aspect ratio of 3.0.  The
251 > solvent model was a simplified 4-water bead ($\sigma_w \approx d$)
252 > that has been used in other coarse-grained simulations.  The dipolar
253 > strength (and the temperature and pressure) were the only other
254 > parameters that were varied systematically.\label{mdfig:lipidModel}}
255   \end{figure}
256  
257   To take into account the permanent dipolar interactions of the
258 < zwitterionic head groups, we have placed fixed dipole moments $\mu_{i}$ at
259 < one end of the Gay-Berne particles.  The dipoles are oriented at an
260 < angle $\theta = \pi / 2$ relative to the major axis.  These dipoles
261 < are protected by a head ``bead'' with a range parameter ($\sigma_h$) which we have
262 < varied between $1.20 d$ and $1.41 d$.  The head groups interact with
263 < each other using a combination of Lennard-Jones,
258 > zwitterionic head groups, we have placed fixed dipole moments
259 > $\mu_{i}$ at one end of the Gay-Berne particles.  The dipoles are
260 > oriented at an angle $\theta = \pi / 2$ relative to the major axis.
261 > These dipoles are protected by a head ``bead'' with a range parameter
262 > ($\sigma_h$) which we have varied between $1.20 d$ and $1.41 d$.  The
263 > head groups interact with each other using a combination of
264 > Lennard-Jones,
265   \begin{equation}
266   V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} -
267   \left(\frac{\sigma_h}{r_{ij}}\right)^6\right],
# Line 324 | Line 310 | bilayers.\cite{Marrink04} The solvent bead is a single
310  
311   The solvent model in our simulations is similar to the one used by
312   Marrink {\it et al.}  in their coarse grained simulations of lipid
313 < bilayers.\cite{Marrink04} The solvent bead is a single site that
313 > bilayers.\cite{Marrink2004} The solvent bead is a single site that
314   represents four water molecules (m = 72 amu) and has comparable
315   density and diffusive behavior to liquid water.  However, since there
316   are no electrostatic sites on these beads, this solvent model cannot
317   replicate the dielectric properties of water.  Note that although we
318   are using larger cutoff and switching radii than Marrink {\it et al.},
319   our solvent density at 300 K remains at 0.944 g cm$^{-3}$, and the
320 < solvent diffuses at 0.43 $\AA^2 ps^{-1}$ (only twice as fast as liquid
320 > solvent diffuses at 0.43 \AA$^2 ps^{-1}$ (only twice as fast as liquid
321   water).
322  
323   \begin{table*}
324   \begin{minipage}{\linewidth}
325   \begin{center}
326 < \caption{Potential parameters used for molecular-scale coarse-grained
327 < lipid simulations}
326 > \caption{POTENTIAL PARAMETERS USED FOR MOLECULAR SCALE COARSE-GRAINED
327 > LIPID SIMULATIONS}
328   \begin{tabular}{llccc}
329   \hline
330    & &  Head & Chain & Solvent \\
# Line 359 | Line 345 | $\mu$ (Debye) & & varied & 0 & 0 \\
345   \end{minipage}
346   \end{table*}
347  
348 < \section{Experimental Methodology}
349 < \label{mdsec:experiment}
348 > \section{Simulation Methodology}
349 > \label{mdsec:simulation}
350  
351   The parameters that were systematically varied in this study were the
352   size of the head group ($\sigma_h$), the strength of the dipole moment
# Line 402 | Line 388 | modeling program.\cite{Meineke05}
388   molecular dynamics runs was 25 fs.  No appreciable changes in phase
389   structure were noticed upon switching to a microcanonical ensemble.
390   All simulations were performed using the {\sc oopse} molecular
391 < modeling program.\cite{Meineke05}
391 > modeling program.\cite{Meineke2005}
392  
393   A switching function was applied to all potentials to smoothly turn
394   off the interactions between a range of $22$ and $25$ \AA.  The
# Line 455 | Line 441 | phase observed when $\sigma_h = 1.35 d$.
441   \begin{figure}
442   \centering
443   \includegraphics[width=\linewidth]{./figures/mdPhaseCartoon.pdf}
444 < \caption{The role of the ratio between the head group size and the
445 < width of the molecular bodies is to increase the local membrane
446 < curvature.  With strong attractive interactions between the head
447 < groups, this local curvature can be maintained in bilayer structures
448 < through surface corrugation.  Shown above are three phases observed in
449 < these simulations.  With $\sigma_h = 1.20 d$, the bilayer maintains a
450 < flat topology.  For larger heads ($\sigma_h = 1.35 d$) the local
451 < curvature resolves into a symmetrically rippled phase with little or
452 < no interdigitation between the upper and lower leaves of the membrane.
453 < The largest heads studied ($\sigma_h = 1.41 d$) resolve into an
454 < asymmetric rippled phases with interdigitation between the two
455 < leaves.\label{mdfig:phaseCartoon}}
444 > \caption[ three phases observed in the simulations]{The role of the
445 > ratio between the head group size and the width of the molecular
446 > bodies is to increase the local membrane curvature.  With strong
447 > attractive interactions between the head groups, this local curvature
448 > can be maintained in bilayer structures through surface corrugation.
449 > Shown above are three phases observed in these simulations.  With
450 > $\sigma_h = 1.20 d$, the bilayer maintains a flat topology.  For
451 > larger heads ($\sigma_h = 1.35 d$) the local curvature resolves into a
452 > symmetrically rippled phase with little or no interdigitation between
453 > the upper and lower leaves of the membrane.  The largest heads studied
454 > ($\sigma_h = 1.41 d$) resolve into an asymmetric rippled phases with
455 > interdigitation between the two leaves.\label{mdfig:phaseCartoon}}
456   \end{figure}
457  
458   Sample structures for the flat ($\sigma_h = 1.20 d$), symmetric
# Line 475 | Line 461 | lipid bilayers.  Using a value of $l = 13.8$ \AA for t
461  
462   It is reasonable to ask how well the parameters we used can produce
463   bilayer properties that match experimentally known values for real
464 < lipid bilayers.  Using a value of $l = 13.8$ \AA for the ellipsoidal
464 > lipid bilayers.  Using a value of $l = 13.8$ \AA~for the ellipsoidal
465   tails and the fixed ellipsoidal aspect ratio of 3, our values for the
466   area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend
467   entirely on the size of the head bead relative to the molecular body.
# Line 492 | Line 478 | for PE head groups.
478   for PE head groups.
479  
480   \begin{table*}
495 – \begin{minipage}{\linewidth}
481   \begin{center}
482 < \caption{Phase, bilayer spacing, area per lipid, ripple wavelength
483 < and amplitude observed as a function of the ratio between the head
484 < beads and the diameters of the tails.  Ripple wavelengths and
500 < amplitudes are normalized to the diameter of the tail ellipsoids.}
482 > \caption{PHASE, BILAYER SPACING, AREA PER LIPID, RIPPLE WAVELENGTH AND
483 > AMPLITUDE OBSERVED AS A FUNCTION OF THE RATIO BETWEEN THE HEAD BEADS
484 > AND THE DIAMETERS OF THE TAILS}
485   \begin{tabular}{lccccc}
486   \hline
487   $\sigma_h / d$ & type of phase & bilayer spacing (\AA) & area per
# Line 508 | Line 492 | lipid (\AA$^2$) & $\lambda / d$ & $A / d$\\
492   1.35 & symmetric ripple & 42.9 & 51.7 & 17.2 & 2.2 \\
493   1.41 & asymmetric ripple & 37.1 & 63.1 & 15.4 & 1.5 \\
494   \end{tabular}
495 + \begin{minipage}{\linewidth}
496 + %\centering
497 + \vspace{2mm}    
498 + Ripple wavelengths and amplitudes are normalized to the diameter of
499 + the tail ellipsoids.
500   \label{mdtab:property}
512 – \end{center}
501   \end{minipage}
502 + \end{center}
503   \end{table*}
504  
505   The membrane structures and the reduced wavelength $\lambda / d$,
# Line 526 | Line 515 | likely to underestimate of the true amplitudes.
515   \begin{figure}
516   \centering
517   \includegraphics[width=\linewidth]{./figures/mdTopDown.pdf}
518 < \caption{Top views of the flat (upper), symmetric ripple (middle),
519 < and asymmetric ripple (lower) phases.  Note that the head-group
520 < dipoles have formed head-to-tail chains in all three of these phases,
521 < but in the two rippled phases, the dipolar chains are all aligned {\it
522 < perpendicular} to the direction of the ripple.  Note that the flat
523 < membrane has multiple vortex defects in the dipolar ordering, and the
524 < ordering on the lower leaf of the bilayer can be in an entirely
525 < different direction from the upper leaf.\label{mdfig:topView}}
518 > \caption[Top views of the flat, symmetric ripple, and asymmetric
519 > ripple phases]{Top views of the flat (upper), symmetric ripple
520 > (middle), and asymmetric ripple (lower) phases.  Note that the
521 > head-group dipoles have formed head-to-tail chains in all three of
522 > these phases, but in the two rippled phases, the dipolar chains are
523 > all aligned {\it perpendicular} to the direction of the ripple.  Note
524 > that the flat membrane has multiple vortex defects in the dipolar
525 > ordering, and the ordering on the lower leaf of the bilayer can be in
526 > an entirely different direction from the upper
527 > leaf.\label{mdfig:topView}}
528   \end{figure}
529  
530 < The principal method for observing orientational ordering in dipolar
531 < or liquid crystalline systems is the $P_2$ order parameter (defined
532 < as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest
542 < eigenvalue of the matrix,
543 < \begin{equation}
544 < {\mathsf{S}} = \frac{1}{N} \sum_i \left(
545 < \begin{array}{ccc}
546 <        u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\
547 <        u^{y}_i u^{x}_i  & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\
548 <        u^{z}_i u^{x}_i & u^{z}_i u^{y}_i  & u^{z}_i u^{z}_i -\frac{1}{3}
549 < \end{array} \right).
550 < \label{mdeq:opmatrix}
551 < \end{equation}
552 < Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector
553 < for molecule $i$.  (Here, $\hat{\bf u}_i$ can refer either to the
530 > The orientational ordering in the system is observed by $P_2$ order
531 > parameter, which is calculated from Eq.~\ref{mceq:opmatrix}
532 > in Ch.~\ref{chap:mc}. Here, $\hat{\bf u}_i$ can refer either to the
533   principal axis of the molecular body or to the dipole on the head
534 < group of the molecule.)  $P_2$ will be $1.0$ for a perfectly-ordered
535 < system and near $0$ for a randomized system.  Note that this order
536 < parameter is {\em not} equal to the polarization of the system.  For
537 < example, the polarization of a perfect anti-ferroelectric arrangement
559 < of point dipoles is $0$, but $P_2$ for the same system is $1$.  The
560 < eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is
561 < familiar as the director axis, which can be used to determine a
562 < privileged axis for an orientationally-ordered system.  Since the
563 < molecular bodies are perpendicular to the head group dipoles, it is
564 < possible for the director axes for the molecular bodies and the head
565 < groups to be completely decoupled from each other.
534 > group of the molecule. Since the molecular bodies are perpendicular to
535 > the head group dipoles, it is possible for the director axes for the
536 > molecular bodies and the head groups to be completely decoupled from
537 > each other.
538  
539   Figure \ref{mdfig:topView} shows snapshots of bird's-eye views of the
540   flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$)
# Line 592 | Line 564 | elastic dipolar membranes.\cite{Sun2007}
564   arrangement of the dipoles is always observed in a direction
565   perpendicular to the wave vector for the surface corrugation.  This is
566   a similar finding to what we observed in our earlier work on the
567 < elastic dipolar membranes.\cite{Sun2007}
567 > elastic dipolar membranes.\cite{sun:031602}
568  
569   The $P_2$ order parameters (for both the molecular bodies and the head
570   group dipoles) have been calculated to quantify the ordering in these
# Line 617 | Line 589 | rapidly decreasing $P_2$ ordering for the molecular bo
589   \begin{figure}
590   \centering
591   \includegraphics[width=\linewidth]{./figures/mdRP2.pdf}
592 < \caption{The $P_2$ order parameters for head groups (circles) and
593 < molecular bodies (squares) as a function of the ratio of head group
594 < size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{mdfig:rP2}}
592 > \caption[The $P_2$ order parameters as a function of the ratio of head group
593 > size to the width of the molecular bodies]{The $P_2$ order parameters
594 > for head groups (circles) and molecular bodies (squares) as a function
595 > of the ratio of head group size ($\sigma_h$) to the width of the
596 > molecular bodies ($d$). \label{mdfig:rP2}}
597   \end{figure}
598  
599   In addition to varying the size of the head groups, we studied the
# Line 668 | Line 642 | vector.
642   \begin{figure}
643   \centering
644   \includegraphics[width=\linewidth]{./figures/mdSP2.pdf}
645 < \caption{The $P_2$ order parameters for head group dipoles (a) and
646 < molecular bodies (b) as a function of the strength of the dipoles.
645 > \caption[The $P_2$ order parameters as a function of the strength of
646 > the dipoles.]{The $P_2$ order parameters for head group dipoles (a)
647 > and molecular bodies (b) as a function of the strength of the dipoles.
648   These order parameters are shown for four values of the head group /
649   molecular width ratio ($\sigma_h / d$). \label{mdfig:sP2}}
650   \end{figure}
# Line 700 | Line 675 | the ripple to gel ($L_{\beta'}$) phase transition.
675   \begin{figure}
676   \centering
677   \includegraphics[width=\linewidth]{./figures/mdTP2.pdf}
678 < \caption{The $P_2$ order parameters for head group dipoles (a) and
679 < molecular bodies (b) as a function of temperature.
680 < These order parameters are shown for four values of the head group /
681 < molecular width ratio ($\sigma_h / d$).\label{mdfig:tP2}}
678 > \caption[The $P_2$ order parameters as a function of temperature]{The
679 > $P_2$ order parameters for head group dipoles (a) and molecular bodies
680 > (b) as a function of temperature.  These order parameters are shown
681 > for four values of the head group / molecular width ratio ($\sigma_h /
682 > d$).\label{mdfig:tP2}}
683   \end{figure}
684  
685   Fig. \ref{mdfig:phaseDiagram} shows a phase diagram for the model as a
# Line 722 | Line 698 | dipole region of this diagram.
698   \begin{figure}
699   \centering
700   \includegraphics[width=\linewidth]{./figures/mdPhaseDiagram.pdf}
701 < \caption{Phase diagram for the simple molecular model as a function
702 < of the head group / molecular width ratio ($\sigma_h / d$) and the
703 < strength of the head group dipole moment
704 < ($\mu$).\label{mdfig:phaseDiagram}}
701 > \caption[Phase diagram for the simple molecular model]{Phase diagram
702 > for the simple molecular model as a function of the head group /
703 > molecular width ratio ($\sigma_h / d$) and the strength of the head
704 > group dipole moment ($\mu$).\label{mdfig:phaseDiagram}}
705   \end{figure}
706  
707   We have computed translational diffusion constants for lipid molecules
708   from the mean-square displacement,
709   \begin{equation}
710 < D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle,
710 > D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf
711 > r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle,
712 > \label{mdeq:msdisplacement}
713   \end{equation}
714   of the lipid bodies. Translational diffusion constants for the
715   different head-to-tail size ratios (all at 300 K) are shown in table
# Line 772 | Line 750 | times that are too fast when compared with experimenta
750   times that are too fast when compared with experimental measurements.
751  
752   \begin{table*}
753 < \begin{minipage}{\linewidth}
754 < \begin{center}
755 < \caption{Fit values for the rotational correlation times for the head
756 < groups ($\tau^h$) and molecular bodies ($\tau^b$) as well as the
757 < translational diffusion constants for the molecule as a function of
780 < the head-to-body width ratio.  All correlation functions and transport
781 < coefficients were computed from microcanonical simulations with an
782 < average temperture of 300 K.  In all of the phases, the head group
783 < correlation functions decay with an fast librational contribution ($12
784 < \pm 1$ ps).  There are additional moderate ($\tau^h_{\rm mid}$) and
785 < slow $\tau^h_{\rm slow}$ contributions to orientational decay that
786 < depend strongly on the phase exhibited by the lipids.  The symmetric
787 < ripple phase ($\sigma_h / d = 1.35$) appears to exhibit the slowest
788 < molecular reorientation.}
753 > \begin{center}
754 > \caption{FIT VALUES FOR THE ROTATIONAL CORRELATION TIMES FOR THE HEAD
755 > GROUPS ($\tau^h$) AND MOLECULAR BODIES ($\tau^b$) AS WELL AS THE
756 > TRANSLATIONAL DIFFUSION CONSTANTS FOR THE MOL\-E\-CULE AS A FUNCTION
757 > OF THE HEAD-TO-BODY WIDTH RATIO}
758   \begin{tabular}{lcccc}
759   \hline
760   $\sigma_h / d$ & $\tau^h_{\rm mid} (ns)$ & $\tau^h_{\rm
# Line 796 | Line 765 | slow} (\mu s)$ & $\tau^b (\mu s)$ & $D (\times 10^{-11
765   1.35 & $3.2$ &  $4.0$ & $0.9$ & $3.42(1)$ \\
766   1.41 & $0.3$ & $23.8$ & $6.9$ & $7.16(1)$ \\
767   \end{tabular}
768 + \begin{minipage}{\linewidth}
769 + %\centering
770 + \vspace{2mm}
771 + All correlation functions and transport coefficients were computed
772 + from microcanonical simulations with an average temperture of 300 K.
773 + In all of the phases, the head group correlation functions decay with
774 + an fast librational contribution ($12 \pm 1$ ps).  There are
775 + additional moderate ($\tau^h_{\rm mid}$) and slow $\tau^h_{\rm slow}$
776 + contributions to orientational decay that depend strongly on the phase
777 + exhibited by the lipids.  The symmetric ripple phase ($\sigma_h / d =
778 + 1.35$) appears to exhibit the slowest molecular reorientation.
779   \label{mdtab:relaxation}
800 – \end{center}
780   \end{minipage}
781 + \end{center}
782   \end{table*}
783  
784   \section{Discussion}
# Line 872 | Line 852 | have recently used axially-specific chromophores
852   orientations of the membrane dipoles may be available from
853   fluorescence detected linear dichroism (LD).  Benninger {\it et al.}
854   have recently used axially-specific chromophores
855 < 2-(4,4-difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-pentanoyl)-1-hexadecanoyl-sn-glycero-3-phospocholine
856 < ($\beta$-BODIPY FL C5-HPC or BODIPY-PC) and 3,3'
855 > 2-(4,4-difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-pentanoyl)-1-hexadecanoyl-sn-glycero-3-\\
856 > phospocholine ($\beta$-BODIPY FL C5-HPC or BODIPY-PC) and 3,3'
857   dioctadecyloxacarbocyanine perchlorate (DiO) in their
858   fluorescence-detected linear dichroism (LD) studies of plasma
859   membranes of living cells.\cite{Benninger:2005qy} The DiO dye aligns

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