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# Line 3 | Line 3 | Fully hydrated lipids will aggregate spontaneously to
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{sun:031602} We
78   found that dipolar elastic membranes can spontaneously buckle, forming
# 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 161 | Line 140 | $\sigma$ and $\epsilon$ parameters,
140   Pechukas.\cite{Berne72} The potential is constructed in the familiar
141   form of the Lennard-Jones function using orientation-dependent
142   $\sigma$ and $\epsilon$ parameters,
143 < \begin{equation*}
143 > \begin{equation}
144 > \begin{split}
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},
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]
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},
148 > {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} \right.\\
149 > &\left. -\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i},
150 > {\mathbf{\hat u}_j}, {\mathbf{\hat
151 > r}_{ij}})+\sigma_0}\right)^6\right]
152 > \end{split}
153   \label{mdeq:gb}
154 < \end{equation*}
154 > \end{equation}
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{equation}
176 > \begin{split}
177 > & \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) =
178 > \sigma_{0} \times  \\
179 > & \left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf
180   \hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf
181   \hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf
182   \hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf
183   \hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2
184   \left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}
185 < \right]^{-1/2}
186 < \end{equation*}
185 > \right]^{-1/2}
186 > \end{split}
187 > \end{equation}
188  
189   Gay-Berne ellipsoids also have an energy scaling parameter,
190   $\epsilon^s$, which describes the well depth for two identical
# Line 217 | Line 202 | The form of the strength function is somewhat complica
202   \alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1}
203   \end{eqnarray*}
204   The form of the strength function is somewhat complicated,
205 < \begin {eqnarray*}
205 > \begin{eqnarray*}
206   \epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = &
207   \epsilon_{0}  \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j})
208   \epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
209   \hat{r}}_{ij}) \\ \\
210   \epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = &
211   \left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf
212 < \hat{u}}_{j})^{2}\right]^{-1/2} \\ \\
213 < \epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) &
214 < = &
215 < 1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf
212 > \hat{u}}_{j})^{2}\right]^{-1/2}
213 > \end{eqnarray*}
214 > \begin{equation*}
215 > \begin{split}
216 > & \epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij})
217 > = 1 - \\
218 > & \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf
219   \hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf
220   \hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf
221   \hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf
222   \hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2
223   \left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\},
224 < \end {eqnarray*}
224 > \end{split}
225 > \end{equation*}
226   although many of the quantities and derivatives are identical with
227   those obtained for the range parameter. Ref. \citen{Luckhurst90}
228   has a particularly good explanation of the choice of the Gay-Berne
# Line 476 | Line 465 | lipid bilayers.  Using a value of $l = 13.8$ \AA for t
465  
466   It is reasonable to ask how well the parameters we used can produce
467   bilayer properties that match experimentally known values for real
468 < lipid bilayers.  Using a value of $l = 13.8$ \AA for the ellipsoidal
468 > lipid bilayers.  Using a value of $l = 13.8$ \AA~for the ellipsoidal
469   tails and the fixed ellipsoidal aspect ratio of 3, our values for the
470   area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend
471   entirely on the size of the head bead relative to the molecular body.
# Line 537 | Line 526 | The principal method for observing orientational order
526   different direction from the upper leaf.\label{mdfig:topView}}
527   \end{figure}
528  
529 < The principal method for observing orientational ordering in dipolar
530 < or liquid crystalline systems is the $P_2$ order parameter (defined
531 < as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest
543 < eigenvalue of the matrix,
544 < \begin{equation}
545 < {\mathsf{S}} = \frac{1}{N} \sum_i \left(
546 < \begin{array}{ccc}
547 <        u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\
548 <        u^{y}_i u^{x}_i  & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\
549 <        u^{z}_i u^{x}_i & u^{z}_i u^{y}_i  & u^{z}_i u^{z}_i -\frac{1}{3}
550 < \end{array} \right).
551 < \label{mdeq:opmatrix}
552 < \end{equation}
553 < Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector
554 < for molecule $i$.  (Here, $\hat{\bf u}_i$ can refer either to the
529 > The orientational ordering in the system is observed by $P_2$ order
530 > parameter, which is calculated from Eq.~\ref{mceq:opmatrix}
531 > in Ch.~\ref{chap:mc}. Here, $\hat{\bf u}_i$ can refer either to the
532   principal axis of the molecular body or to the dipole on the head
533 < group of the molecule.)  $P_2$ will be $1.0$ for a perfectly-ordered
534 < system and near $0$ for a randomized system.  Note that this order
535 < parameter is {\em not} equal to the polarization of the system.  For
536 < example, the polarization of a perfect anti-ferroelectric arrangement
560 < of point dipoles is $0$, but $P_2$ for the same system is $1$.  The
561 < eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is
562 < familiar as the director axis, which can be used to determine a
563 < privileged axis for an orientationally-ordered system.  Since the
564 < molecular bodies are perpendicular to the head group dipoles, it is
565 < possible for the director axes for the molecular bodies and the head
566 < groups to be completely decoupled from each other.
533 > group of the molecule. Since the molecular bodies are perpendicular to
534 > the head group dipoles, it is possible for the director axes for the
535 > molecular bodies and the head groups to be completely decoupled from
536 > each other.
537  
538   Figure \ref{mdfig:topView} shows snapshots of bird's-eye views of the
539   flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$)

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