1 |
gezelter |
3202 |
\documentclass[12pt]{article} |
2 |
gezelter |
3351 |
\usepackage{graphicx} |
3 |
gezelter |
3202 |
\usepackage{times} |
4 |
|
|
\usepackage{mathptm} |
5 |
|
|
\usepackage{tabularx} |
6 |
|
|
\usepackage{setspace} |
7 |
gezelter |
3199 |
\usepackage{amsmath} |
8 |
|
|
\usepackage{amssymb} |
9 |
gezelter |
3202 |
\usepackage[ref]{overcite} |
10 |
|
|
\pagestyle{plain} |
11 |
|
|
\pagenumbering{arabic} |
12 |
|
|
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
13 |
|
|
\topmargin -21pt \headsep 10pt |
14 |
|
|
\textheight 9.0in \textwidth 6.5in |
15 |
|
|
\brokenpenalty=10000 |
16 |
|
|
\renewcommand{\baselinestretch}{1.2} |
17 |
|
|
\renewcommand\citemid{\ } % no comma in optional reference note |
18 |
xsun |
3147 |
|
19 |
gezelter |
3351 |
|
20 |
xsun |
3147 |
\begin{document} |
21 |
gezelter |
3202 |
%\renewcommand{\thefootnote}{\fnsymbol{footnote}} |
22 |
|
|
%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} |
23 |
xsun |
3147 |
|
24 |
gezelter |
3202 |
\bibliographystyle{achemso} |
25 |
xsun |
3147 |
|
26 |
gezelter |
3203 |
\title{Dipolar ordering in the ripple phases of molecular-scale models |
27 |
gezelter |
3269 |
of lipid membranes} |
28 |
gezelter |
3202 |
\author{Xiuquan Sun and J. Daniel Gezelter \\ |
29 |
|
|
Department of Chemistry and Biochemistry,\\ |
30 |
gezelter |
3199 |
University of Notre Dame, \\ |
31 |
xsun |
3147 |
Notre Dame, Indiana 46556} |
32 |
|
|
|
33 |
gezelter |
3202 |
%\email[E-mail:]{gezelter@nd.edu} |
34 |
|
|
|
35 |
xsun |
3147 |
\date{\today} |
36 |
|
|
|
37 |
gezelter |
3202 |
\maketitle |
38 |
|
|
|
39 |
xsun |
3147 |
\begin{abstract} |
40 |
gezelter |
3203 |
Symmetric and asymmetric ripple phases have been observed to form in |
41 |
|
|
molecular dynamics simulations of a simple molecular-scale lipid |
42 |
|
|
model. The lipid model consists of an dipolar head group and an |
43 |
|
|
ellipsoidal tail. Within the limits of this model, an explanation for |
44 |
|
|
generalized membrane curvature is a simple mismatch in the size of the |
45 |
|
|
heads with the width of the molecular bodies. The persistence of a |
46 |
|
|
{\it bilayer} structure requires strong attractive forces between the |
47 |
|
|
head groups. One feature of this model is that an energetically |
48 |
|
|
favorable orientational ordering of the dipoles can be achieved by |
49 |
|
|
out-of-plane membrane corrugation. The corrugation of the surface |
50 |
gezelter |
3204 |
stabilizes the long range orientational ordering for the dipoles in the |
51 |
|
|
head groups which then adopt a bulk anti-ferroelectric state. We |
52 |
gezelter |
3203 |
observe a common feature of the corrugated dipolar membranes: the wave |
53 |
|
|
vectors for the surface ripples are always found to be perpendicular |
54 |
|
|
to the dipole director axis. |
55 |
xsun |
3147 |
\end{abstract} |
56 |
|
|
|
57 |
gezelter |
3202 |
%\maketitle |
58 |
gezelter |
3203 |
\newpage |
59 |
xsun |
3147 |
|
60 |
xsun |
3174 |
\section{Introduction} |
61 |
|
|
\label{sec:Int} |
62 |
gezelter |
3195 |
Fully hydrated lipids will aggregate spontaneously to form bilayers |
63 |
|
|
which exhibit a variety of phases depending on their temperatures and |
64 |
|
|
compositions. Among these phases, a periodic rippled phase |
65 |
|
|
($P_{\beta'}$) appears as an intermediate phase between the gel |
66 |
|
|
($L_\beta$) and fluid ($L_{\alpha}$) phases for relatively pure |
67 |
|
|
phosphatidylcholine (PC) bilayers. The ripple phase has attracted |
68 |
|
|
substantial experimental interest over the past 30 years. Most |
69 |
|
|
structural information of the ripple phase has been obtained by the |
70 |
|
|
X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture electron |
71 |
|
|
microscopy (FFEM).~\cite{Copeland80,Meyer96} Recently, Kaasgaard {\it |
72 |
|
|
et al.} used atomic force microscopy (AFM) to observe ripple phase |
73 |
|
|
morphology in bilayers supported on mica.~\cite{Kaasgaard03} The |
74 |
|
|
experimental results provide strong support for a 2-dimensional |
75 |
|
|
hexagonal packing lattice of the lipid molecules within the ripple |
76 |
|
|
phase. This is a notable change from the observed lipid packing |
77 |
gezelter |
3269 |
within the gel phase,~\cite{Cevc87} although Tenchov {\it et al.} have |
78 |
|
|
recently observed near-hexagonal packing in some phosphatidylcholine |
79 |
xsun |
3270 |
(PC) gel phases.\cite{Tenchov2001} The X-ray diffraction work by |
80 |
gezelter |
3204 |
Katsaras {\it et al.} showed that a rich phase diagram exhibiting both |
81 |
|
|
{\it asymmetric} and {\it symmetric} ripples is possible for lecithin |
82 |
|
|
bilayers.\cite{Katsaras00} |
83 |
xsun |
3174 |
|
84 |
gezelter |
3195 |
A number of theoretical models have been presented to explain the |
85 |
|
|
formation of the ripple phase. Marder {\it et al.} used a |
86 |
gezelter |
3204 |
curvature-dependent Landau-de~Gennes free-energy functional to predict |
87 |
gezelter |
3269 |
a rippled phase.~\cite{Marder84} This model and other related |
88 |
|
|
continuum models predict higher fluidity in convex regions and that |
89 |
|
|
concave portions of the membrane correspond to more solid-like |
90 |
|
|
regions. Carlson and Sethna used a packing-competition model (in |
91 |
|
|
which head groups and chains have competing packing energetics) to |
92 |
|
|
predict the formation of a ripple-like phase. Their model predicted |
93 |
|
|
that the high-curvature portions have lower-chain packing and |
94 |
|
|
correspond to more fluid-like regions. Goldstein and Leibler used a |
95 |
|
|
mean-field approach with a planar model for {\em inter-lamellar} |
96 |
|
|
interactions to predict rippling in multilamellar |
97 |
|
|
phases.~\cite{Goldstein88} McCullough and Scott proposed that the {\em |
98 |
|
|
anisotropy of the nearest-neighbor interactions} coupled to |
99 |
|
|
hydrophobic constraining forces which restrict height differences |
100 |
|
|
between nearest neighbors is the origin of the ripple |
101 |
|
|
phase.~\cite{McCullough90} Lubensky and MacKintosh introduced a Landau |
102 |
|
|
theory for tilt order and curvature of a single membrane and concluded |
103 |
|
|
that {\em coupling of molecular tilt to membrane curvature} is |
104 |
|
|
responsible for the production of ripples.~\cite{Lubensky93} Misbah, |
105 |
|
|
Duplat and Houchmandzadeh proposed that {\em inter-layer dipolar |
106 |
|
|
interactions} can lead to ripple instabilities.~\cite{Misbah98} |
107 |
|
|
Heimburg presented a {\em coexistence model} for ripple formation in |
108 |
|
|
which he postulates that fluid-phase line defects cause sharp |
109 |
|
|
curvature between relatively flat gel-phase regions.~\cite{Heimburg00} |
110 |
|
|
Kubica has suggested that a lattice model of polar head groups could |
111 |
|
|
be valuable in trying to understand bilayer phase |
112 |
|
|
formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations of |
113 |
|
|
lamellar stacks of hexagonal lattices to show that large head groups |
114 |
gezelter |
3195 |
and molecular tilt with respect to the membrane normal vector can |
115 |
gezelter |
3269 |
cause bulk rippling.~\cite{Bannerjee02} Recently, Kranenburg and Smit |
116 |
|
|
described the formation of symmetric ripple-like structures using a |
117 |
|
|
coarse grained solvent-head-tail bead model.\cite{Kranenburg2005} |
118 |
|
|
Their lipids consisted of a short chain of head beads tied to the two |
119 |
|
|
longer ``chains''. |
120 |
xsun |
3174 |
|
121 |
gezelter |
3204 |
In contrast, few large-scale molecular modeling studies have been |
122 |
gezelter |
3195 |
done due to the large size of the resulting structures and the time |
123 |
|
|
required for the phases of interest to develop. With all-atom (and |
124 |
|
|
even unified-atom) simulations, only one period of the ripple can be |
125 |
gezelter |
3204 |
observed and only for time scales in the range of 10-100 ns. One of |
126 |
|
|
the most interesting molecular simulations was carried out by de~Vries |
127 |
gezelter |
3195 |
{\it et al.}~\cite{deVries05}. According to their simulation results, |
128 |
|
|
the ripple consists of two domains, one resembling the gel bilayer, |
129 |
|
|
while in the other, the two leaves of the bilayer are fully |
130 |
|
|
interdigitated. The mechanism for the formation of the ripple phase |
131 |
|
|
suggested by their work is a packing competition between the head |
132 |
|
|
groups and the tails of the lipid molecules.\cite{Carlson87} Recently, |
133 |
gezelter |
3199 |
the ripple phase has also been studied by Lenz and Schmid using Monte |
134 |
gezelter |
3195 |
Carlo simulations.\cite{Lenz07} Their structures are similar to the De |
135 |
|
|
Vries {\it et al.} structures except that the connection between the |
136 |
|
|
two leaves of the bilayer is a narrow interdigitated line instead of |
137 |
|
|
the fully interdigitated domain. The symmetric ripple phase was also |
138 |
|
|
observed by Lenz {\it et al.}, and their work supports other claims |
139 |
|
|
that the mismatch between the size of the head group and tail of the |
140 |
|
|
lipid molecules is the driving force for the formation of the ripple |
141 |
|
|
phase. Ayton and Voth have found significant undulations in |
142 |
|
|
zero-surface-tension states of membranes simulated via dissipative |
143 |
|
|
particle dynamics, but their results are consistent with purely |
144 |
|
|
thermal undulations.~\cite{Ayton02} |
145 |
xsun |
3174 |
|
146 |
gezelter |
3195 |
Although the organization of the tails of lipid molecules are |
147 |
|
|
addressed by these molecular simulations and the packing competition |
148 |
gezelter |
3204 |
between head groups and tails is strongly implicated as the primary |
149 |
gezelter |
3195 |
driving force for ripple formation, questions about the ordering of |
150 |
gezelter |
3203 |
the head groups in ripple phase have not been settled. |
151 |
xsun |
3174 |
|
152 |
gezelter |
3195 |
In a recent paper, we presented a simple ``web of dipoles'' spin |
153 |
|
|
lattice model which provides some physical insight into relationship |
154 |
|
|
between dipolar ordering and membrane buckling.\cite{Sun2007} We found |
155 |
|
|
that dipolar elastic membranes can spontaneously buckle, forming |
156 |
gezelter |
3203 |
ripple-like topologies. The driving force for the buckling of dipolar |
157 |
gezelter |
3204 |
elastic membranes is the anti-ferroelectric ordering of the dipoles. |
158 |
gezelter |
3203 |
This was evident in the ordering of the dipole director axis |
159 |
gezelter |
3204 |
perpendicular to the wave vector of the surface ripples. A similar |
160 |
gezelter |
3195 |
phenomenon has also been observed by Tsonchev {\it et al.} in their |
161 |
gezelter |
3199 |
work on the spontaneous formation of dipolar peptide chains into |
162 |
|
|
curved nano-structures.\cite{Tsonchev04,Tsonchev04II} |
163 |
gezelter |
3195 |
|
164 |
|
|
In this paper, we construct a somewhat more realistic molecular-scale |
165 |
|
|
lipid model than our previous ``web of dipoles'' and use molecular |
166 |
|
|
dynamics simulations to elucidate the role of the head group dipoles |
167 |
|
|
in the formation and morphology of the ripple phase. We describe our |
168 |
|
|
model and computational methodology in section \ref{sec:method}. |
169 |
|
|
Details on the simulations are presented in section |
170 |
|
|
\ref{sec:experiment}, with results following in section |
171 |
|
|
\ref{sec:results}. A final discussion of the role of dipolar heads in |
172 |
|
|
the ripple formation can be found in section |
173 |
xsun |
3174 |
\ref{sec:discussion}. |
174 |
|
|
|
175 |
gezelter |
3196 |
\section{Computational Model} |
176 |
xsun |
3174 |
\label{sec:method} |
177 |
|
|
|
178 |
gezelter |
3199 |
\begin{figure}[htb] |
179 |
|
|
\centering |
180 |
|
|
\includegraphics[width=4in]{lipidModels} |
181 |
|
|
\caption{Three different representations of DPPC lipid molecules, |
182 |
|
|
including the chemical structure, an atomistic model, and the |
183 |
|
|
head-body ellipsoidal coarse-grained model used in this |
184 |
|
|
work.\label{fig:lipidModels}} |
185 |
|
|
\end{figure} |
186 |
|
|
|
187 |
gezelter |
3195 |
Our simple molecular-scale lipid model for studying the ripple phase |
188 |
|
|
is based on two facts: one is that the most essential feature of lipid |
189 |
|
|
molecules is their amphiphilic structure with polar head groups and |
190 |
|
|
non-polar tails. Another fact is that the majority of lipid molecules |
191 |
|
|
in the ripple phase are relatively rigid (i.e. gel-like) which makes |
192 |
|
|
some fraction of the details of the chain dynamics negligible. Figure |
193 |
gezelter |
3204 |
\ref{fig:lipidModels} shows the molecular structure of a DPPC |
194 |
gezelter |
3195 |
molecule, as well as atomistic and molecular-scale representations of |
195 |
|
|
a DPPC molecule. The hydrophilic character of the head group is |
196 |
|
|
largely due to the separation of charge between the nitrogen and |
197 |
|
|
phosphate groups. The zwitterionic nature of the PC headgroups leads |
198 |
|
|
to abnormally large dipole moments (as high as 20.6 D), and this |
199 |
|
|
strongly polar head group interacts strongly with the solvating water |
200 |
|
|
layers immediately surrounding the membrane. The hydrophobic tail |
201 |
|
|
consists of fatty acid chains. In our molecular scale model, lipid |
202 |
|
|
molecules have been reduced to these essential features; the fatty |
203 |
|
|
acid chains are represented by an ellipsoid with a dipolar ball |
204 |
|
|
perched on one end to represent the effects of the charge-separated |
205 |
|
|
head group. In real PC lipids, the direction of the dipole is |
206 |
|
|
nearly perpendicular to the tail, so we have fixed the direction of |
207 |
|
|
the point dipole rigidly in this orientation. |
208 |
xsun |
3147 |
|
209 |
gezelter |
3195 |
The ellipsoidal portions of the model interact via the Gay-Berne |
210 |
|
|
potential which has seen widespread use in the liquid crystal |
211 |
gezelter |
3199 |
community. Ayton and Voth have also used Gay-Berne ellipsoids for |
212 |
gezelter |
3204 |
modeling large length-scale properties of lipid |
213 |
gezelter |
3199 |
bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
214 |
|
|
was a single site model for the interactions of rigid ellipsoidal |
215 |
gezelter |
3195 |
molecules.\cite{Gay81} It can be thought of as a modification of the |
216 |
|
|
Gaussian overlap model originally described by Berne and |
217 |
|
|
Pechukas.\cite{Berne72} The potential is constructed in the familiar |
218 |
|
|
form of the Lennard-Jones function using orientation-dependent |
219 |
|
|
$\sigma$ and $\epsilon$ parameters, |
220 |
gezelter |
3202 |
\begin{equation*} |
221 |
xsun |
3174 |
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
222 |
|
|
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
223 |
|
|
{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
224 |
|
|
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
225 |
|
|
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
226 |
|
|
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
227 |
gezelter |
3195 |
\label{eq:gb} |
228 |
gezelter |
3202 |
\end{equation*} |
229 |
gezelter |
3195 |
|
230 |
|
|
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
231 |
gezelter |
3199 |
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
232 |
|
|
\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
233 |
gezelter |
3195 |
are dependent on the relative orientations of the two molecules (${\bf |
234 |
|
|
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
235 |
gezelter |
3199 |
intermolecular separation (${\bf \hat{r}}_{ij}$). $\sigma$ and |
236 |
|
|
$\sigma_0$ are also governed by shape mixing and anisotropy variables, |
237 |
gezelter |
3202 |
\begin {eqnarray*} |
238 |
gezelter |
3199 |
\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
239 |
|
|
\chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 - |
240 |
|
|
d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^2 + |
241 |
|
|
d_j^2 \right)}\right]^{1/2} \\ \\ |
242 |
|
|
\alpha^2 & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 + |
243 |
|
|
d_i^2 \right)}{\left( l_j^2 - d_j^2 \right) \left(l_i^2 + |
244 |
|
|
d_j^2 \right)}\right]^{1/2}, |
245 |
gezelter |
3202 |
\end{eqnarray*} |
246 |
gezelter |
3199 |
where $l$ and $d$ describe the length and width of each uniaxial |
247 |
|
|
ellipsoid. These shape anisotropy parameters can then be used to |
248 |
|
|
calculate the range function, |
249 |
gezelter |
3202 |
\begin{equation*} |
250 |
gezelter |
3199 |
\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0} |
251 |
|
|
\left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
252 |
|
|
\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
253 |
|
|
\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
254 |
|
|
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
255 |
|
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
256 |
|
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
257 |
|
|
\right]^{-1/2} |
258 |
gezelter |
3202 |
\end{equation*} |
259 |
gezelter |
3199 |
|
260 |
|
|
Gay-Berne ellipsoids also have an energy scaling parameter, |
261 |
|
|
$\epsilon^s$, which describes the well depth for two identical |
262 |
gezelter |
3204 |
ellipsoids in a {\it side-by-side} configuration. Additionally, a well |
263 |
gezelter |
3199 |
depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes |
264 |
|
|
the ratio between the well depths in the {\it end-to-end} and |
265 |
|
|
side-by-side configurations. As in the range parameter, a set of |
266 |
|
|
mixing and anisotropy variables can be used to describe the well |
267 |
|
|
depths for dissimilar particles, |
268 |
|
|
\begin {eqnarray*} |
269 |
|
|
\epsilon_0 & = & \sqrt{\epsilon^s_i * \epsilon^s_j} \\ \\ |
270 |
|
|
\epsilon^r & = & \sqrt{\epsilon^r_i * \epsilon^r_j} \\ \\ |
271 |
|
|
\chi' & = & \frac{1 - (\epsilon^r)^{1/\mu}}{1 + (\epsilon^r)^{1/\mu}} |
272 |
|
|
\\ \\ |
273 |
|
|
\alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1} |
274 |
|
|
\end{eqnarray*} |
275 |
|
|
The form of the strength function is somewhat complicated, |
276 |
|
|
\begin {eqnarray*} |
277 |
|
|
\epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = & |
278 |
|
|
\epsilon_{0} \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j}) |
279 |
|
|
\epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
280 |
|
|
\hat{r}}_{ij}) \\ \\ |
281 |
|
|
\epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = & |
282 |
|
|
\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
283 |
|
|
\hat{u}}_{j})^{2}\right]^{-1/2} \\ \\ |
284 |
|
|
\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & |
285 |
|
|
= & |
286 |
|
|
1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
287 |
|
|
\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
288 |
|
|
\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
289 |
|
|
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
290 |
|
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
291 |
|
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
292 |
|
|
\end {eqnarray*} |
293 |
|
|
although many of the quantities and derivatives are identical with |
294 |
gezelter |
3202 |
those obtained for the range parameter. Ref. \citen{Luckhurst90} |
295 |
gezelter |
3199 |
has a particularly good explanation of the choice of the Gay-Berne |
296 |
|
|
parameters $\mu$ and $\nu$ for modeling liquid crystal molecules. An |
297 |
|
|
excellent overview of the computational methods that can be used to |
298 |
|
|
efficiently compute forces and torques for this potential can be found |
299 |
gezelter |
3202 |
in Ref. \citen{Golubkov06} |
300 |
gezelter |
3199 |
|
301 |
|
|
The choices of parameters we have used in this study correspond to a |
302 |
|
|
shape anisotropy of 3 for the chain portion of the molecule. In |
303 |
|
|
principle, this could be varied to allow for modeling of longer or |
304 |
|
|
shorter chain lipid molecules. For these prolate ellipsoids, we have: |
305 |
gezelter |
3195 |
\begin{equation} |
306 |
|
|
\begin{array}{rcl} |
307 |
gezelter |
3199 |
d & < & l \\ |
308 |
|
|
\epsilon^{r} & < & 1 |
309 |
gezelter |
3195 |
\end{array} |
310 |
|
|
\end{equation} |
311 |
gezelter |
3200 |
A sketch of the various structural elements of our molecular-scale |
312 |
|
|
lipid / solvent model is shown in figure \ref{fig:lipidModel}. The |
313 |
|
|
actual parameters used in our simulations are given in table |
314 |
|
|
\ref{tab:parameters}. |
315 |
gezelter |
3195 |
|
316 |
gezelter |
3199 |
\begin{figure}[htb] |
317 |
|
|
\centering |
318 |
|
|
\includegraphics[width=4in]{2lipidModel} |
319 |
|
|
\caption{The parameters defining the behavior of the lipid |
320 |
gezelter |
3351 |
models. $\sigma_h / d$ is the ratio of the head group to body diameter. |
321 |
gezelter |
3199 |
Molecular bodies had a fixed aspect ratio of 3.0. The solvent model |
322 |
|
|
was a simplified 4-water bead ($\sigma_w \approx d$) that has been |
323 |
gezelter |
3351 |
used in other coarse-grained simulations. The dipolar strength |
324 |
gezelter |
3199 |
(and the temperature and pressure) were the only other parameters that |
325 |
|
|
were varied systematically.\label{fig:lipidModel}} |
326 |
|
|
\end{figure} |
327 |
gezelter |
3195 |
|
328 |
|
|
To take into account the permanent dipolar interactions of the |
329 |
gezelter |
3203 |
zwitterionic head groups, we have placed fixed dipole moments $\mu_{i}$ at |
330 |
gezelter |
3199 |
one end of the Gay-Berne particles. The dipoles are oriented at an |
331 |
|
|
angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
332 |
gezelter |
3203 |
are protected by a head ``bead'' with a range parameter ($\sigma_h$) which we have |
333 |
gezelter |
3199 |
varied between $1.20 d$ and $1.41 d$. The head groups interact with |
334 |
|
|
each other using a combination of Lennard-Jones, |
335 |
gezelter |
3202 |
\begin{equation} |
336 |
gezelter |
3200 |
V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
337 |
gezelter |
3195 |
\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
338 |
gezelter |
3202 |
\end{equation} |
339 |
gezelter |
3199 |
and dipole-dipole, |
340 |
gezelter |
3202 |
\begin{equation} |
341 |
gezelter |
3200 |
V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
342 |
|
|
\hat{r}}_{ij})) = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
343 |
gezelter |
3195 |
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
344 |
|
|
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
345 |
gezelter |
3202 |
\end{equation} |
346 |
gezelter |
3195 |
potentials. |
347 |
|
|
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
348 |
|
|
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
349 |
gezelter |
3199 |
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. |
350 |
gezelter |
3195 |
|
351 |
gezelter |
3269 |
Since the charge separation distance is so large in zwitterionic head |
352 |
|
|
groups (like the PC head groups), it would also be possible to use |
353 |
|
|
either point charges or a ``split dipole'' approximation, |
354 |
|
|
\begin{equation} |
355 |
|
|
V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
356 |
|
|
\hat{r}}_{ij})) = \frac{1}{{4\pi \epsilon_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{R_{ij}^3 }} - |
357 |
|
|
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
358 |
|
|
r_{ij} } \right)}}{{R_{ij}^5 }}} \right] |
359 |
|
|
\end{equation} |
360 |
|
|
where $\mu _i$ and $\mu _j$ are the dipole moments of sites $i$ and |
361 |
|
|
$j$, $r_{ij}$ is vector between the two sites, and $R_{ij}$ is given |
362 |
|
|
by, |
363 |
|
|
\begin{equation} |
364 |
|
|
R_{ij} = \sqrt {r_{ij}^2 + \frac{{d_i^2 }}{4} + \frac{{d_j^2 |
365 |
|
|
}}{4}}. |
366 |
|
|
\end{equation} |
367 |
gezelter |
3351 |
Here, $d_i$ and $d_j$ are charge separation distances associated with |
368 |
|
|
each of the two dipolar sites. This approximation to the multipole |
369 |
|
|
expansion maintains the fast fall-off of the multipole potentials but |
370 |
|
|
lacks the normal divergences when two polar groups get close to one |
371 |
|
|
another. |
372 |
gezelter |
3269 |
|
373 |
gezelter |
3195 |
For the interaction between nonequivalent uniaxial ellipsoids (in this |
374 |
gezelter |
3199 |
case, between spheres and ellipsoids), the spheres are treated as |
375 |
|
|
ellipsoids with an aspect ratio of 1 ($d = l$) and with an well depth |
376 |
gezelter |
3200 |
ratio ($\epsilon^r$) of 1 ($\epsilon^e = \epsilon^s$). The form of |
377 |
|
|
the Gay-Berne potential we are using was generalized by Cleaver {\it |
378 |
|
|
et al.} and is appropriate for dissimilar uniaxial |
379 |
|
|
ellipsoids.\cite{Cleaver96} |
380 |
xsun |
3147 |
|
381 |
gezelter |
3269 |
The solvent model in our simulations is similar to the one used by |
382 |
|
|
Marrink {\it et al.} in their coarse grained simulations of lipid |
383 |
|
|
bilayers.\cite{Marrink04} The solvent bead is a single site that |
384 |
|
|
represents four water molecules (m = 72 amu) and has comparable |
385 |
|
|
density and diffusive behavior to liquid water. However, since there |
386 |
|
|
are no electrostatic sites on these beads, this solvent model cannot |
387 |
|
|
replicate the dielectric properties of water. Note that although we |
388 |
|
|
are using larger cutoff and switching radii than Marrink {\it et al.}, |
389 |
|
|
our solvent density at 300 K remains at 0.944 g cm$^{-3}$, and the |
390 |
gezelter |
3351 |
solvent diffuses at 0.43 $\AA^2 ps^{-1}$ (only twice as fast as liquid |
391 |
|
|
water). |
392 |
gezelter |
3203 |
|
393 |
xsun |
3198 |
\begin{table*} |
394 |
|
|
\begin{minipage}{\linewidth} |
395 |
|
|
\begin{center} |
396 |
gezelter |
3199 |
\caption{Potential parameters used for molecular-scale coarse-grained |
397 |
|
|
lipid simulations} |
398 |
|
|
\begin{tabular}{llccc} |
399 |
xsun |
3198 |
\hline |
400 |
gezelter |
3199 |
& & Head & Chain & Solvent \\ |
401 |
xsun |
3198 |
\hline |
402 |
gezelter |
3200 |
$d$ (\AA) & & varied & 4.6 & 4.7 \\ |
403 |
|
|
$l$ (\AA) & & $= d$ & 13.8 & 4.7 \\ |
404 |
gezelter |
3199 |
$\epsilon^s$ (kcal/mol) & & 0.185 & 1.0 & 0.8 \\ |
405 |
|
|
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 \\ |
406 |
gezelter |
3200 |
$m$ (amu) & & 196 & 760 & 72.06 \\ |
407 |
gezelter |
3199 |
$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
408 |
|
|
\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
409 |
|
|
\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
410 |
|
|
\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
411 |
|
|
$\mu$ (Debye) & & varied & 0 & 0 \\ |
412 |
xsun |
3198 |
\end{tabular} |
413 |
|
|
\label{tab:parameters} |
414 |
|
|
\end{center} |
415 |
|
|
\end{minipage} |
416 |
|
|
\end{table*} |
417 |
gezelter |
3195 |
|
418 |
gezelter |
3203 |
\section{Experimental Methodology} |
419 |
|
|
\label{sec:experiment} |
420 |
gezelter |
3186 |
|
421 |
gezelter |
3200 |
The parameters that were systematically varied in this study were the |
422 |
|
|
size of the head group ($\sigma_h$), the strength of the dipole moment |
423 |
|
|
($\mu$), and the temperature of the system. Values for $\sigma_h$ |
424 |
gezelter |
3351 |
ranged from 5.5 \AA\ to 6.5 \AA. If the width of the tails is taken |
425 |
gezelter |
3203 |
to be the unit of length, these head groups correspond to a range from |
426 |
|
|
$1.2 d$ to $1.41 d$. Since the solvent beads are nearly identical in |
427 |
|
|
diameter to the tail ellipsoids, all distances that follow will be |
428 |
|
|
measured relative to this unit of distance. Because the solvent we |
429 |
|
|
are using is non-polar and has a dielectric constant of 1, values for |
430 |
|
|
$\mu$ are sampled from a range that is somewhat smaller than the 20.6 |
431 |
gezelter |
3204 |
Debye dipole moment of the PC head groups. |
432 |
gezelter |
3200 |
|
433 |
gezelter |
3196 |
To create unbiased bilayers, all simulations were started from two |
434 |
gezelter |
3200 |
perfectly flat monolayers separated by a 26 \AA\ gap between the |
435 |
gezelter |
3196 |
molecular bodies of the upper and lower leaves. The separated |
436 |
gezelter |
3204 |
monolayers were evolved in a vacuum with $x-y$ anisotropic pressure |
437 |
xsun |
3174 |
coupling. The length of $z$ axis of the simulations was fixed and a |
438 |
|
|
constant surface tension was applied to enable real fluctuations of |
439 |
gezelter |
3200 |
the bilayer. Periodic boundary conditions were used, and $480-720$ |
440 |
|
|
lipid molecules were present in the simulations, depending on the size |
441 |
|
|
of the head beads. In all cases, the two monolayers spontaneously |
442 |
|
|
collapsed into bilayer structures within 100 ps. Following this |
443 |
gezelter |
3204 |
collapse, all systems were equilibrated for $100$ ns at $300$ K. |
444 |
xsun |
3147 |
|
445 |
gezelter |
3200 |
The resulting bilayer structures were then solvated at a ratio of $6$ |
446 |
gezelter |
3196 |
solvent beads (24 water molecules) per lipid. These configurations |
447 |
gezelter |
3200 |
were then equilibrated for another $30$ ns. All simulations utilizing |
448 |
|
|
the solvent were carried out at constant pressure ($P=1$ atm) with |
449 |
gezelter |
3269 |
$3$D anisotropic coupling, and small constant surface tension |
450 |
gezelter |
3203 |
($\gamma=0.015$ N/m). Given the absence of fast degrees of freedom in |
451 |
gezelter |
3204 |
this model, a time step of $50$ fs was utilized with excellent energy |
452 |
gezelter |
3200 |
conservation. Data collection for structural properties of the |
453 |
|
|
bilayers was carried out during a final 5 ns run following the solvent |
454 |
gezelter |
3269 |
equilibration. Orientational correlation functions and diffusion |
455 |
|
|
constants were computed from 30 ns simulations in the microcanonical |
456 |
|
|
(NVE) ensemble using the average volume from the end of the constant |
457 |
|
|
pressure and surface tension runs. The timestep on these final |
458 |
|
|
molecular dynamics runs was 25 fs. No appreciable changes in phase |
459 |
|
|
structure were noticed upon switching to a microcanonical ensemble. |
460 |
|
|
All simulations were performed using the {\sc oopse} molecular |
461 |
|
|
modeling program.\cite{Meineke05} |
462 |
gezelter |
3196 |
|
463 |
gezelter |
3203 |
A switching function was applied to all potentials to smoothly turn |
464 |
gezelter |
3269 |
off the interactions between a range of $22$ and $25$ \AA. The |
465 |
|
|
switching function was the standard (cubic) function, |
466 |
|
|
\begin{equation} |
467 |
|
|
s(r) = |
468 |
|
|
\begin{cases} |
469 |
|
|
1 & \text{if $r \le r_{\text{sw}}$},\\ |
470 |
|
|
\frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2} |
471 |
|
|
{(r_{\text{cut}} - r_{\text{sw}})^3} |
472 |
|
|
& \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\ |
473 |
|
|
0 & \text{if $r > r_{\text{cut}}$.} |
474 |
|
|
\end{cases} |
475 |
|
|
\label{eq:dipoleSwitching} |
476 |
|
|
\end{equation} |
477 |
gezelter |
3203 |
|
478 |
gezelter |
3196 |
\section{Results} |
479 |
xsun |
3174 |
\label{sec:results} |
480 |
xsun |
3147 |
|
481 |
gezelter |
3203 |
The membranes in our simulations exhibit a number of interesting |
482 |
|
|
bilayer phases. The surface topology of these phases depends most |
483 |
|
|
sensitively on the ratio of the size of the head groups to the width |
484 |
|
|
of the molecular bodies. With heads only slightly larger than the |
485 |
gezelter |
3204 |
bodies ($\sigma_h=1.20 d$) the membrane exhibits a flat bilayer. |
486 |
gezelter |
3203 |
|
487 |
|
|
Increasing the head / body size ratio increases the local membrane |
488 |
|
|
curvature around each of the lipids. With $\sigma_h=1.28 d$, the |
489 |
|
|
surface is still essentially flat, but the bilayer starts to exhibit |
490 |
|
|
signs of instability. We have observed occasional defects where a |
491 |
|
|
line of lipid molecules on one leaf of the bilayer will dip down to |
492 |
|
|
interdigitate with the other leaf. This gives each of the two bilayer |
493 |
|
|
leaves some local convexity near the line defect. These structures, |
494 |
|
|
once developed in a simulation, are very stable and are spaced |
495 |
|
|
approximately 100 \AA\ away from each other. |
496 |
|
|
|
497 |
|
|
With larger heads ($\sigma_h = 1.35 d$) the membrane curvature |
498 |
|
|
resolves into a ``symmetric'' ripple phase. Each leaf of the bilayer |
499 |
|
|
is broken into several convex, hemicylinderical sections, and opposite |
500 |
|
|
leaves are fitted together much like roof tiles. There is no |
501 |
|
|
interdigitation between the upper and lower leaves of the bilayer. |
502 |
|
|
|
503 |
|
|
For the largest head / body ratios studied ($\sigma_h = 1.41 d$) the |
504 |
|
|
local curvature is substantially larger, and the resulting bilayer |
505 |
|
|
structure resolves into an asymmetric ripple phase. This structure is |
506 |
gezelter |
3204 |
very similar to the structures observed by both de~Vries {\it et al.} |
507 |
gezelter |
3203 |
and Lenz {\it et al.}. For a given ripple wave vector, there are two |
508 |
|
|
possible asymmetric ripples, which is not the case for the symmetric |
509 |
|
|
phase observed when $\sigma_h = 1.35 d$. |
510 |
|
|
|
511 |
xsun |
3174 |
\begin{figure}[htb] |
512 |
|
|
\centering |
513 |
gezelter |
3199 |
\includegraphics[width=4in]{phaseCartoon} |
514 |
gezelter |
3203 |
\caption{The role of the ratio between the head group size and the |
515 |
|
|
width of the molecular bodies is to increase the local membrane |
516 |
|
|
curvature. With strong attractive interactions between the head |
517 |
|
|
groups, this local curvature can be maintained in bilayer structures |
518 |
|
|
through surface corrugation. Shown above are three phases observed in |
519 |
|
|
these simulations. With $\sigma_h = 1.20 d$, the bilayer maintains a |
520 |
|
|
flat topology. For larger heads ($\sigma_h = 1.35 d$) the local |
521 |
|
|
curvature resolves into a symmetrically rippled phase with little or |
522 |
|
|
no interdigitation between the upper and lower leaves of the membrane. |
523 |
|
|
The largest heads studied ($\sigma_h = 1.41 d$) resolve into an |
524 |
|
|
asymmetric rippled phases with interdigitation between the two |
525 |
|
|
leaves.\label{fig:phaseCartoon}} |
526 |
xsun |
3174 |
\end{figure} |
527 |
xsun |
3147 |
|
528 |
gezelter |
3203 |
Sample structures for the flat ($\sigma_h = 1.20 d$), symmetric |
529 |
|
|
($\sigma_h = 1.35 d$, and asymmetric ($\sigma_h = 1.41 d$) ripple |
530 |
|
|
phases are shown in Figure \ref{fig:phaseCartoon}. |
531 |
|
|
|
532 |
gezelter |
3204 |
It is reasonable to ask how well the parameters we used can produce |
533 |
|
|
bilayer properties that match experimentally known values for real |
534 |
|
|
lipid bilayers. Using a value of $l = 13.8$ \AA for the ellipsoidal |
535 |
|
|
tails and the fixed ellipsoidal aspect ratio of 3, our values for the |
536 |
|
|
area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend |
537 |
|
|
entirely on the size of the head bead relative to the molecular body. |
538 |
|
|
These values are tabulated in table \ref{tab:property}. Kucera {\it |
539 |
|
|
et al.} have measured values for the head group spacings for a number |
540 |
gezelter |
3351 |
of PC lipid bilayers that range from 30.8 \AA\ (DLPC) to 37.8 \AA\ (DPPC). |
541 |
gezelter |
3204 |
They have also measured values for the area per lipid that range from |
542 |
|
|
60.6 |
543 |
|
|
\AA$^2$ (DMPC) to 64.2 \AA$^2$ |
544 |
|
|
(DPPC).\cite{NorbertKucerka04012005,NorbertKucerka06012006} Only the |
545 |
|
|
largest of the head groups we modeled ($\sigma_h = 1.41 d$) produces |
546 |
|
|
bilayers (specifically the area per lipid) that resemble real PC |
547 |
|
|
bilayers. The smaller head beads we used are perhaps better models |
548 |
|
|
for PE head groups. |
549 |
|
|
|
550 |
xsun |
3174 |
\begin{table*} |
551 |
|
|
\begin{minipage}{\linewidth} |
552 |
|
|
\begin{center} |
553 |
gezelter |
3204 |
\caption{Phase, bilayer spacing, area per lipid, ripple wavelength |
554 |
|
|
and amplitude observed as a function of the ratio between the head |
555 |
|
|
beads and the diameters of the tails. Ripple wavelengths and |
556 |
|
|
amplitudes are normalized to the diameter of the tail ellipsoids.} |
557 |
|
|
\begin{tabular}{lccccc} |
558 |
xsun |
3174 |
\hline |
559 |
gezelter |
3204 |
$\sigma_h / d$ & type of phase & bilayer spacing (\AA) & area per |
560 |
|
|
lipid (\AA$^2$) & $\lambda / d$ & $A / d$\\ |
561 |
xsun |
3174 |
\hline |
562 |
gezelter |
3204 |
1.20 & flat & 33.4 & 49.6 & N/A & N/A \\ |
563 |
|
|
1.28 & flat & 33.7 & 54.7 & N/A & N/A \\ |
564 |
|
|
1.35 & symmetric ripple & 42.9 & 51.7 & 17.2 & 2.2 \\ |
565 |
|
|
1.41 & asymmetric ripple & 37.1 & 63.1 & 15.4 & 1.5 \\ |
566 |
xsun |
3174 |
\end{tabular} |
567 |
|
|
\label{tab:property} |
568 |
|
|
\end{center} |
569 |
|
|
\end{minipage} |
570 |
|
|
\end{table*} |
571 |
xsun |
3147 |
|
572 |
gezelter |
3200 |
The membrane structures and the reduced wavelength $\lambda / d$, |
573 |
|
|
reduced amplitude $A / d$ of the ripples are summarized in Table |
574 |
gezelter |
3203 |
\ref{tab:property}. The wavelength range is $15 - 17$ molecular bodies |
575 |
gezelter |
3200 |
and the amplitude is $1.5$ molecular bodies for asymmetric ripple and |
576 |
gezelter |
3203 |
$2.2$ for symmetric ripple. These values are reasonably consistent |
577 |
|
|
with experimental measurements.\cite{Sun96,Katsaras00,Kaasgaard03} |
578 |
|
|
Note, that given the lack of structural freedom in the tails of our |
579 |
|
|
model lipids, the amplitudes observed from these simulations are |
580 |
|
|
likely to underestimate of the true amplitudes. |
581 |
xsun |
3174 |
|
582 |
gezelter |
3195 |
\begin{figure}[htb] |
583 |
|
|
\centering |
584 |
gezelter |
3199 |
\includegraphics[width=4in]{topDown} |
585 |
gezelter |
3203 |
\caption{Top views of the flat (upper), symmetric ripple (middle), |
586 |
|
|
and asymmetric ripple (lower) phases. Note that the head-group |
587 |
|
|
dipoles have formed head-to-tail chains in all three of these phases, |
588 |
|
|
but in the two rippled phases, the dipolar chains are all aligned {\it |
589 |
|
|
perpendicular} to the direction of the ripple. Note that the flat |
590 |
|
|
membrane has multiple vortex defects in the dipolar ordering, and the |
591 |
|
|
ordering on the lower leaf of the bilayer can be in an entirely |
592 |
|
|
different direction from the upper leaf.\label{fig:topView}} |
593 |
gezelter |
3195 |
\end{figure} |
594 |
|
|
|
595 |
gezelter |
3202 |
The principal method for observing orientational ordering in dipolar |
596 |
|
|
or liquid crystalline systems is the $P_2$ order parameter (defined |
597 |
|
|
as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
598 |
|
|
eigenvalue of the matrix, |
599 |
|
|
\begin{equation} |
600 |
|
|
{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
601 |
|
|
\begin{array}{ccc} |
602 |
|
|
u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
603 |
|
|
u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
604 |
|
|
u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
605 |
|
|
\end{array} \right). |
606 |
|
|
\label{eq:opmatrix} |
607 |
|
|
\end{equation} |
608 |
|
|
Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
609 |
|
|
for molecule $i$. (Here, $\hat{\bf u}_i$ can refer either to the |
610 |
|
|
principal axis of the molecular body or to the dipole on the head |
611 |
|
|
group of the molecule.) $P_2$ will be $1.0$ for a perfectly-ordered |
612 |
|
|
system and near $0$ for a randomized system. Note that this order |
613 |
|
|
parameter is {\em not} equal to the polarization of the system. For |
614 |
|
|
example, the polarization of a perfect anti-ferroelectric arrangement |
615 |
|
|
of point dipoles is $0$, but $P_2$ for the same system is $1$. The |
616 |
|
|
eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is |
617 |
|
|
familiar as the director axis, which can be used to determine a |
618 |
|
|
privileged axis for an orientationally-ordered system. Since the |
619 |
|
|
molecular bodies are perpendicular to the head group dipoles, it is |
620 |
|
|
possible for the director axes for the molecular bodies and the head |
621 |
|
|
groups to be completely decoupled from each other. |
622 |
|
|
|
623 |
gezelter |
3200 |
Figure \ref{fig:topView} shows snapshots of bird's-eye views of the |
624 |
gezelter |
3203 |
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$) |
625 |
gezelter |
3200 |
bilayers. The directions of the dipoles on the head groups are |
626 |
|
|
represented with two colored half spheres: blue (phosphate) and yellow |
627 |
|
|
(amino). For flat bilayers, the system exhibits signs of |
628 |
gezelter |
3202 |
orientational frustration; some disorder in the dipolar head-to-tail |
629 |
|
|
chains is evident with kinks visible at the edges between differently |
630 |
|
|
ordered domains. The lipids can also move independently of lipids in |
631 |
|
|
the opposing leaf, so the ordering of the dipoles on one leaf is not |
632 |
|
|
necessarily consistent with the ordering on the other. These two |
633 |
gezelter |
3200 |
factors keep the total dipolar order parameter relatively low for the |
634 |
|
|
flat phases. |
635 |
xsun |
3147 |
|
636 |
gezelter |
3200 |
With increasing head group size, the surface becomes corrugated, and |
637 |
|
|
the dipoles cannot move as freely on the surface. Therefore, the |
638 |
|
|
translational freedom of lipids in one layer is dependent upon the |
639 |
gezelter |
3202 |
position of the lipids in the other layer. As a result, the ordering of |
640 |
gezelter |
3200 |
the dipoles on head groups in one leaf is correlated with the ordering |
641 |
|
|
in the other leaf. Furthermore, as the membrane deforms due to the |
642 |
|
|
corrugation, the symmetry of the allowed dipolar ordering on each leaf |
643 |
|
|
is broken. The dipoles then self-assemble in a head-to-tail |
644 |
|
|
configuration, and the dipolar order parameter increases dramatically. |
645 |
|
|
However, the total polarization of the system is still close to zero. |
646 |
|
|
This is strong evidence that the corrugated structure is an |
647 |
gezelter |
3204 |
anti-ferroelectric state. It is also notable that the head-to-tail |
648 |
gezelter |
3202 |
arrangement of the dipoles is always observed in a direction |
649 |
|
|
perpendicular to the wave vector for the surface corrugation. This is |
650 |
|
|
a similar finding to what we observed in our earlier work on the |
651 |
|
|
elastic dipolar membranes.\cite{Sun2007} |
652 |
gezelter |
3200 |
|
653 |
|
|
The $P_2$ order parameters (for both the molecular bodies and the head |
654 |
|
|
group dipoles) have been calculated to quantify the ordering in these |
655 |
gezelter |
3202 |
phases. Figure \ref{fig:rP2} shows that the $P_2$ order parameter for |
656 |
|
|
the head-group dipoles increases with increasing head group size. When |
657 |
|
|
the heads of the lipid molecules are small, the membrane is nearly |
658 |
|
|
flat. Since the in-plane packing is essentially a close packing of the |
659 |
|
|
head groups, the head dipoles exhibit frustration in their |
660 |
|
|
orientational ordering. |
661 |
gezelter |
3200 |
|
662 |
gezelter |
3202 |
The ordering trends for the tails are essentially opposite to the |
663 |
|
|
ordering of the head group dipoles. The tail $P_2$ order parameter |
664 |
|
|
{\it decreases} with increasing head size. This indicates that the |
665 |
|
|
surface is more curved with larger head / tail size ratios. When the |
666 |
|
|
surface is flat, all tails are pointing in the same direction (normal |
667 |
|
|
to the bilayer surface). This simplified model appears to be |
668 |
|
|
exhibiting a smectic A fluid phase, similar to the real $L_{\beta}$ |
669 |
|
|
phase. We have not observed a smectic C gel phase ($L_{\beta'}$) for |
670 |
|
|
this model system. Increasing the size of the heads results in |
671 |
|
|
rapidly decreasing $P_2$ ordering for the molecular bodies. |
672 |
gezelter |
3199 |
|
673 |
xsun |
3174 |
\begin{figure}[htb] |
674 |
|
|
\centering |
675 |
|
|
\includegraphics[width=\linewidth]{rP2} |
676 |
gezelter |
3202 |
\caption{The $P_2$ order parameters for head groups (circles) and |
677 |
|
|
molecular bodies (squares) as a function of the ratio of head group |
678 |
|
|
size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{fig:rP2}} |
679 |
xsun |
3174 |
\end{figure} |
680 |
xsun |
3147 |
|
681 |
gezelter |
3202 |
In addition to varying the size of the head groups, we studied the |
682 |
|
|
effects of the interactions between head groups on the structure of |
683 |
|
|
lipid bilayer by changing the strength of the dipoles. Figure |
684 |
|
|
\ref{fig:sP2} shows how the $P_2$ order parameter changes with |
685 |
|
|
increasing strength of the dipole. Generally, the dipoles on the head |
686 |
|
|
groups become more ordered as the strength of the interaction between |
687 |
|
|
heads is increased and become more disordered by decreasing the |
688 |
gezelter |
3204 |
interaction strength. When the interaction between the heads becomes |
689 |
gezelter |
3202 |
too weak, the bilayer structure does not persist; all lipid molecules |
690 |
|
|
become dispersed in the solvent (which is non-polar in this |
691 |
gezelter |
3204 |
molecular-scale model). The critical value of the strength of the |
692 |
gezelter |
3202 |
dipole depends on the size of the head groups. The perfectly flat |
693 |
|
|
surface becomes unstable below $5$ Debye, while the rippled |
694 |
|
|
surfaces melt at $8$ Debye (asymmetric) or $10$ Debye (symmetric). |
695 |
|
|
|
696 |
|
|
The ordering of the tails mirrors the ordering of the dipoles {\it |
697 |
|
|
except for the flat phase}. Since the surface is nearly flat in this |
698 |
|
|
phase, the order parameters are only weakly dependent on dipolar |
699 |
|
|
strength until it reaches $15$ Debye. Once it reaches this value, the |
700 |
|
|
head group interactions are strong enough to pull the head groups |
701 |
|
|
close to each other and distort the bilayer structure. For a flat |
702 |
|
|
surface, a substantial amount of free volume between the head groups |
703 |
|
|
is normally available. When the head groups are brought closer by |
704 |
gezelter |
3203 |
dipolar interactions, the tails are forced to splay outward, first forming |
705 |
|
|
curved bilayers, and then inverted micelles. |
706 |
gezelter |
3202 |
|
707 |
|
|
When $\sigma_h=1.28 d$, the $P_2$ order parameter decreases slightly |
708 |
gezelter |
3204 |
when the strength of the dipole is increased above $16$ Debye. For |
709 |
gezelter |
3202 |
rippled bilayers, there is less free volume available between the head |
710 |
|
|
groups. Therefore increasing dipolar strength weakly influences the |
711 |
|
|
structure of the membrane. However, the increase in the body $P_2$ |
712 |
|
|
order parameters implies that the membranes are being slightly |
713 |
|
|
flattened due to the effects of increasing head-group attraction. |
714 |
|
|
|
715 |
|
|
A very interesting behavior takes place when the head groups are very |
716 |
|
|
large relative to the molecular bodies ($\sigma_h = 1.41 d$) and the |
717 |
|
|
dipolar strength is relatively weak ($\mu < 9$ Debye). In this case, |
718 |
|
|
the two leaves of the bilayer become totally interdigitated with each |
719 |
|
|
other in large patches of the membrane. With higher dipolar |
720 |
|
|
strength, the interdigitation is limited to single lines that run |
721 |
|
|
through the bilayer in a direction perpendicular to the ripple wave |
722 |
|
|
vector. |
723 |
|
|
|
724 |
xsun |
3174 |
\begin{figure}[htb] |
725 |
|
|
\centering |
726 |
|
|
\includegraphics[width=\linewidth]{sP2} |
727 |
gezelter |
3202 |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
728 |
|
|
molecular bodies (b) as a function of the strength of the dipoles. |
729 |
|
|
These order parameters are shown for four values of the head group / |
730 |
|
|
molecular width ratio ($\sigma_h / d$). \label{fig:sP2}} |
731 |
xsun |
3174 |
\end{figure} |
732 |
xsun |
3147 |
|
733 |
gezelter |
3202 |
Figure \ref{fig:tP2} shows the dependence of the order parameters on |
734 |
|
|
temperature. As expected, systems are more ordered at low |
735 |
|
|
temperatures, and more disordered at high temperatures. All of the |
736 |
|
|
bilayers we studied can become unstable if the temperature becomes |
737 |
|
|
high enough. The only interesting feature of the temperature |
738 |
|
|
dependence is in the flat surfaces ($\sigma_h=1.20 d$ and |
739 |
|
|
$\sigma_h=1.28 d$). Here, when the temperature is increased above |
740 |
|
|
$310$K, there is enough jostling of the head groups to allow the |
741 |
|
|
dipolar frustration to resolve into more ordered states. This results |
742 |
|
|
in a slight increase in the $P_2$ order parameter above this |
743 |
|
|
temperature. |
744 |
|
|
|
745 |
|
|
For the rippled surfaces ($\sigma_h=1.35 d$ and $\sigma_h=1.41 d$), |
746 |
|
|
there is a slightly increased orientational ordering in the molecular |
747 |
|
|
bodies above $290$K. Since our model lacks the detailed information |
748 |
|
|
about the behavior of the lipid tails, this is the closest the model |
749 |
|
|
can come to depicting the ripple ($P_{\beta'}$) to fluid |
750 |
|
|
($L_{\alpha}$) phase transition. What we are observing is a |
751 |
|
|
flattening of the rippled structures made possible by thermal |
752 |
|
|
expansion of the tightly-packed head groups. The lack of detailed |
753 |
|
|
chain configurations also makes it impossible for this model to depict |
754 |
|
|
the ripple to gel ($L_{\beta'}$) phase transition. |
755 |
|
|
|
756 |
xsun |
3174 |
\begin{figure}[htb] |
757 |
|
|
\centering |
758 |
|
|
\includegraphics[width=\linewidth]{tP2} |
759 |
gezelter |
3202 |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
760 |
|
|
molecular bodies (b) as a function of temperature. |
761 |
|
|
These order parameters are shown for four values of the head group / |
762 |
|
|
molecular width ratio ($\sigma_h / d$).\label{fig:tP2}} |
763 |
xsun |
3174 |
\end{figure} |
764 |
xsun |
3147 |
|
765 |
gezelter |
3203 |
Fig. \ref{fig:phaseDiagram} shows a phase diagram for the model as a |
766 |
|
|
function of the head group / molecular width ratio ($\sigma_h / d$) |
767 |
|
|
and the strength of the head group dipole moment ($\mu$). Note that |
768 |
|
|
the specific form of the bilayer phase is governed almost entirely by |
769 |
|
|
the head group / molecular width ratio, while the strength of the |
770 |
|
|
dipolar interactions between the head groups governs the stability of |
771 |
|
|
the bilayer phase. Weaker dipoles result in unstable bilayer phases, |
772 |
|
|
while extremely strong dipoles can shift the equilibrium to an |
773 |
|
|
inverted micelle phase when the head groups are small. Temperature |
774 |
|
|
has little effect on the actual bilayer phase observed, although higher |
775 |
|
|
temperatures can cause the unstable region to grow into the higher |
776 |
|
|
dipole region of this diagram. |
777 |
|
|
|
778 |
|
|
\begin{figure}[htb] |
779 |
|
|
\centering |
780 |
|
|
\includegraphics[width=\linewidth]{phaseDiagram} |
781 |
|
|
\caption{Phase diagram for the simple molecular model as a function |
782 |
|
|
of the head group / molecular width ratio ($\sigma_h / d$) and the |
783 |
|
|
strength of the head group dipole moment |
784 |
|
|
($\mu$).\label{fig:phaseDiagram}} |
785 |
|
|
\end{figure} |
786 |
|
|
|
787 |
gezelter |
3268 |
We have computed translational diffusion constants for lipid molecules |
788 |
|
|
from the mean-square displacement, |
789 |
|
|
\begin{equation} |
790 |
gezelter |
3351 |
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
791 |
gezelter |
3268 |
\end{equation} |
792 |
|
|
of the lipid bodies. Translational diffusion constants for the |
793 |
|
|
different head-to-tail size ratios (all at 300 K) are shown in table |
794 |
gezelter |
3269 |
\ref{tab:relaxation}. We have also computed orientational correlation |
795 |
|
|
times for the head groups from fits of the second-order Legendre |
796 |
|
|
polynomial correlation function, |
797 |
|
|
\begin{equation} |
798 |
|
|
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
799 |
gezelter |
3351 |
\mu}_{i}(0) \right) \rangle |
800 |
gezelter |
3269 |
\end{equation} |
801 |
|
|
of the head group dipoles. The orientational correlation functions |
802 |
|
|
appear to have multiple components in their decay: a fast ($12 \pm 2$ |
803 |
|
|
ps) decay due to librational motion of the head groups, as well as |
804 |
|
|
moderate ($\tau^h_{\rm mid}$) and slow ($\tau^h_{\rm slow}$) |
805 |
|
|
components. The fit values for the moderate and slow correlation |
806 |
|
|
times are listed in table \ref{tab:relaxation}. Standard deviations |
807 |
|
|
in the fit time constants are quite large (on the order of the values |
808 |
|
|
themselves). |
809 |
gezelter |
3264 |
|
810 |
gezelter |
3269 |
Sparrman and Westlund used a multi-relaxation model for NMR lineshapes |
811 |
|
|
observed in gel, fluid, and ripple phases of DPPC and obtained |
812 |
gezelter |
3268 |
estimates of a correlation time for water translational diffusion |
813 |
|
|
($\tau_c$) of 20 ns.\cite{Sparrman2003} This correlation time |
814 |
|
|
corresponds to water bound to small regions of the lipid membrane. |
815 |
gezelter |
3269 |
They further assume that the lipids can explore only a single period |
816 |
|
|
of the ripple (essentially moving in a nearly one-dimensional path to |
817 |
|
|
do so), and the correlation time can therefore be used to estimate a |
818 |
|
|
value for the translational diffusion constant of $2.25 \times |
819 |
|
|
10^{-11} m^2 s^{-1}$. The translational diffusion constants we obtain |
820 |
|
|
are in reasonable agreement with this experimentally determined |
821 |
|
|
value. However, the $T_2$ relaxation times obtained by Sparrman and |
822 |
|
|
Westlund are consistent with P-N vector reorientation timescales of |
823 |
|
|
2-5 ms. This is substantially slower than even the slowest component |
824 |
|
|
we observe in the decay of our orientational correlation functions. |
825 |
|
|
Other than the dipole-dipole interactions, our head groups have no |
826 |
|
|
shape anisotropy which would force them to move as a unit with |
827 |
|
|
neighboring molecules. This would naturally lead to P-N reorientation |
828 |
|
|
times that are too fast when compared with experimental measurements. |
829 |
gezelter |
3264 |
|
830 |
xsun |
3262 |
\begin{table*} |
831 |
|
|
\begin{minipage}{\linewidth} |
832 |
|
|
\begin{center} |
833 |
gezelter |
3269 |
\caption{Fit values for the rotational correlation times for the head |
834 |
|
|
groups ($\tau^h$) and molecular bodies ($\tau^b$) as well as the |
835 |
|
|
translational diffusion constants for the molecule as a function of |
836 |
gezelter |
3351 |
the head-to-body width ratio. All correlation functions and transport |
837 |
|
|
coefficients were computed from microcanonical simulations with an |
838 |
|
|
average temperture of 300 K. In all of the phases, the head group |
839 |
|
|
correlation functions decay with an fast librational contribution ($12 |
840 |
|
|
\pm 1$ ps). There are additional moderate ($\tau^h_{\rm mid}$) and |
841 |
|
|
slow $\tau^h_{\rm slow}$ contributions to orientational decay that |
842 |
|
|
depend strongly on the phase exhibited by the lipids. The symmetric |
843 |
|
|
ripple phase ($\sigma_h / d = 1.35$) appears to exhibit the slowest |
844 |
|
|
molecular reorientation.} |
845 |
gezelter |
3269 |
\begin{tabular}{lcccc} |
846 |
xsun |
3262 |
\hline |
847 |
gezelter |
3269 |
$\sigma_h / d$ & $\tau^h_{\rm mid} (ns)$ & $\tau^h_{\rm |
848 |
gezelter |
3351 |
slow} (\mu s)$ & $\tau^b (\mu s)$ & $D (\times 10^{-11} m^2 s^{-1})$ \\ |
849 |
xsun |
3262 |
\hline |
850 |
gezelter |
3269 |
1.20 & $0.4$ & $9.6$ & $9.5$ & $0.43(1)$ \\ |
851 |
|
|
1.28 & $2.0$ & $13.5$ & $3.0$ & $5.91(3)$ \\ |
852 |
|
|
1.35 & $3.2$ & $4.0$ & $0.9$ & $3.42(1)$ \\ |
853 |
|
|
1.41 & $0.3$ & $23.8$ & $6.9$ & $7.16(1)$ \\ |
854 |
xsun |
3262 |
\end{tabular} |
855 |
|
|
\label{tab:relaxation} |
856 |
|
|
\end{center} |
857 |
|
|
\end{minipage} |
858 |
|
|
\end{table*} |
859 |
|
|
|
860 |
xsun |
3174 |
\section{Discussion} |
861 |
|
|
\label{sec:discussion} |
862 |
xsun |
3147 |
|
863 |
gezelter |
3203 |
Symmetric and asymmetric ripple phases have been observed to form in |
864 |
|
|
our molecular dynamics simulations of a simple molecular-scale lipid |
865 |
|
|
model. The lipid model consists of an dipolar head group and an |
866 |
|
|
ellipsoidal tail. Within the limits of this model, an explanation for |
867 |
|
|
generalized membrane curvature is a simple mismatch in the size of the |
868 |
|
|
heads with the width of the molecular bodies. With heads |
869 |
|
|
substantially larger than the bodies of the molecule, this curvature |
870 |
|
|
should be convex nearly everywhere, a requirement which could be |
871 |
|
|
resolved either with micellar or cylindrical phases. |
872 |
xsun |
3201 |
|
873 |
gezelter |
3203 |
The persistence of a {\it bilayer} structure therefore requires either |
874 |
|
|
strong attractive forces between the head groups or exclusionary |
875 |
|
|
forces from the solvent phase. To have a persistent bilayer structure |
876 |
|
|
with the added requirement of convex membrane curvature appears to |
877 |
|
|
result in corrugated structures like the ones pictured in |
878 |
|
|
Fig. \ref{fig:phaseCartoon}. In each of the sections of these |
879 |
|
|
corrugated phases, the local curvature near a most of the head groups |
880 |
|
|
is convex. These structures are held together by the extremely strong |
881 |
|
|
and directional interactions between the head groups. |
882 |
xsun |
3201 |
|
883 |
gezelter |
3269 |
The attractive forces holding the bilayer together could either be |
884 |
|
|
non-directional (as in the work of Kranenburg and |
885 |
|
|
Smit),\cite{Kranenburg2005} or directional (as we have utilized in |
886 |
|
|
these simulations). The dipolar head groups are key for the |
887 |
|
|
maintaining the bilayer structures exhibited by this particular model; |
888 |
|
|
reducing the strength of the dipole has the tendency to make the |
889 |
|
|
rippled phase disappear. The dipoles are likely to form attractive |
890 |
|
|
head-to-tail configurations even in flat configurations, but the |
891 |
|
|
temperatures are high enough that vortex defects become prevalent in |
892 |
|
|
the flat phase. The flat phase we observed therefore appears to be |
893 |
|
|
substantially above the Kosterlitz-Thouless transition temperature for |
894 |
|
|
a planar system of dipoles with this set of parameters. For this |
895 |
|
|
reason, it would be interesting to observe the thermal behavior of the |
896 |
|
|
flat phase at substantially lower temperatures. |
897 |
xsun |
3201 |
|
898 |
gezelter |
3203 |
One feature of this model is that an energetically favorable |
899 |
|
|
orientational ordering of the dipoles can be achieved by forming |
900 |
|
|
ripples. The corrugation of the surface breaks the symmetry of the |
901 |
gezelter |
3204 |
plane, making vortex defects somewhat more expensive, and stabilizing |
902 |
gezelter |
3203 |
the long range orientational ordering for the dipoles in the head |
903 |
|
|
groups. Most of the rows of the head-to-tail dipoles are parallel to |
904 |
gezelter |
3204 |
each other and the system adopts a bulk anti-ferroelectric state. We |
905 |
gezelter |
3203 |
believe that this is the first time the organization of the head |
906 |
|
|
groups in ripple phases has been addressed. |
907 |
|
|
|
908 |
|
|
Although the size-mismatch between the heads and molecular bodies |
909 |
|
|
appears to be the primary driving force for surface convexity, the |
910 |
|
|
persistence of the bilayer through the use of rippled structures is a |
911 |
|
|
function of the strong, attractive interactions between the heads. |
912 |
|
|
One important prediction we can make using the results from this |
913 |
|
|
simple model is that if the dipole-dipole interaction is the leading |
914 |
|
|
contributor to the head group attractions, the wave vectors for the |
915 |
|
|
ripples should always be found {\it perpendicular} to the dipole |
916 |
|
|
director axis. This echoes the prediction we made earlier for simple |
917 |
|
|
elastic dipolar membranes, and may suggest experimental designs which |
918 |
|
|
will test whether this is really the case in the phosphatidylcholine |
919 |
|
|
$P_{\beta'}$ phases. The dipole director axis should also be easily |
920 |
|
|
computable for the all-atom and coarse-grained simulations that have |
921 |
|
|
been published in the literature.\cite{deVries05} |
922 |
|
|
|
923 |
gezelter |
3351 |
Experimental verification of our predictions of dipolar orientation |
924 |
|
|
correlating with the ripple direction would require knowing both the |
925 |
|
|
local orientation of a rippled region of the membrane (available via |
926 |
|
|
AFM studies of supported bilayers) as well as the local ordering of |
927 |
|
|
the membrane dipoles. Obtaining information about the local |
928 |
|
|
orientations of the membrane dipoles may be available from |
929 |
|
|
fluorescence detected linear dichroism (LD). Benninger {\it et al.} |
930 |
|
|
have recently used axially-specific chromophores |
931 |
|
|
2-(4,4-difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-pentanoyl)-1-hexadecanoyl-sn-glycero-3-phospocholine |
932 |
|
|
($\beta$-BODIPY FL C5-HPC or BODIPY-PC) and 3,3' |
933 |
|
|
dioctadecyloxacarbocyanine perchlorate (DiO) in their |
934 |
|
|
fluorescence-detected linear dichroism (LD) studies of plasma |
935 |
|
|
membranes of living cells.\cite{Benninger:2005qy} The DiO dye aligns |
936 |
|
|
its transition moment perpendicular to the membrane normal, while the |
937 |
|
|
BODIPY-PC transition dipole is parallel with the membrane normal. |
938 |
|
|
Without a doubt, using fluorescence detection of linear dichroism in |
939 |
|
|
concert with AFM surface scanning would be difficult experiments to |
940 |
|
|
carry out. However, there is some hope of performing experiments to |
941 |
|
|
either verify or falsify the predictions of our simulations. |
942 |
|
|
|
943 |
xsun |
3201 |
Although our model is simple, it exhibits some rich and unexpected |
944 |
gezelter |
3203 |
behaviors. It would clearly be a closer approximation to reality if |
945 |
|
|
we allowed bending motions between the dipoles and the molecular |
946 |
|
|
bodies, and if we replaced the rigid ellipsoids with ball-and-chain |
947 |
|
|
tails. However, the advantages of this simple model (large system |
948 |
gezelter |
3204 |
sizes, 50 fs time steps) allow us to rapidly explore the phase diagram |
949 |
gezelter |
3203 |
for a wide range of parameters. Our explanation of this rippling |
950 |
xsun |
3201 |
phenomenon will help us design more accurate molecular models for |
951 |
gezelter |
3203 |
corrugated membranes and experiments to test whether or not |
952 |
|
|
dipole-dipole interactions exert an influence on membrane rippling. |
953 |
gezelter |
3199 |
\newpage |
954 |
xsun |
3147 |
\bibliography{mdripple} |
955 |
|
|
\end{document} |