| 1 |
%\documentclass[aps,pre,twocolumn,amssymb,showpacs,floatfix]{revtex4} |
| 2 |
\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4} |
| 3 |
\usepackage{graphicx} |
| 4 |
|
| 5 |
\begin{document} |
| 6 |
\renewcommand{\thefootnote}{\fnsymbol{footnote}} |
| 7 |
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} |
| 8 |
|
| 9 |
%\bibliographystyle{aps} |
| 10 |
|
| 11 |
\title{Dipolar Ordering of the Ripple Phase in Lipid Membranes} |
| 12 |
\author{Xiuquan Sun and J. Daniel Gezelter} |
| 13 |
\email[E-mail:]{gezelter@nd.edu} |
| 14 |
\affiliation{Department of Chemistry and Biochemistry,\\ |
| 15 |
University of Notre Dame, \\ |
| 16 |
Notre Dame, Indiana 46556} |
| 17 |
|
| 18 |
\date{\today} |
| 19 |
|
| 20 |
\begin{abstract} |
| 21 |
|
| 22 |
\end{abstract} |
| 23 |
|
| 24 |
\pacs{} |
| 25 |
\maketitle |
| 26 |
|
| 27 |
\section{Introduction} |
| 28 |
\label{sec:Int} |
| 29 |
|
| 30 |
As one of the most important components in the formation of the |
| 31 |
biomembrane, lipid molecules attracted numerous studies in the past |
| 32 |
several decades. Due to their amphiphilic structure, when dispersed in |
| 33 |
water, lipids can self-assemble to construct a bilayer structure. The |
| 34 |
phase behavior of lipid membrane is well understood. The gel-fluid |
| 35 |
phase transition is known as main phase transition. However, there is |
| 36 |
an intermediate phase between gel and fluid phase for some lipid (like |
| 37 |
phosphatidycholine (PC)) membranes. This intermediate phase |
| 38 |
distinguish itself from other phases by its corrugated membrane |
| 39 |
surface, therefore this phase is termed ``ripple'' ($P_{\beta '}$) |
| 40 |
phase. The phase transition between gel-fluid and ripple phase is |
| 41 |
called pretransition. Since the pretransition usually occurs in room |
| 42 |
temperature, there might be some important biofuntions carried by the |
| 43 |
ripple phase for the living organism. |
| 44 |
|
| 45 |
The ripple phase is observed experimentally by x-ray diffraction |
| 46 |
~\cite{Sun96,Katsaras00}, freeze-fracture electron microscopy |
| 47 |
(FFEM)~\cite{Copeland80,Meyer96}, and atomic force microscopy (AFM) |
| 48 |
recently~\cite{Kaasgaard03}. The experimental studies suggest two |
| 49 |
kinds of ripple structures: asymmetric (sawtooth like) and symmetric |
| 50 |
(sinusoidal like) ripple phases. Substantial number of theoretical |
| 51 |
explaination applied on the formation of the ripple |
| 52 |
phases~\cite{Marder84,Carlson87,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02}. |
| 53 |
In contrast, few molecular modelling have been done due to the large |
| 54 |
size of the resulting structures and the time required for the phases |
| 55 |
of interest to develop. One of the interesting molecular simulations |
| 56 |
was carried out by De Vries and Marrink {\it et |
| 57 |
al.}~\cite{deVries05}. According to their dynamic simulation results, |
| 58 |
the ripple consists of two domains, one is gel bilayer, and in the |
| 59 |
other domain, the upper and lower leaves of the bilayer are fully |
| 60 |
interdigitated. The mechanism of the formation of the ripple phase in |
| 61 |
their work suggests the theory that the packing competition between |
| 62 |
head group and tail of lipid molecules is the driving force for the |
| 63 |
formation of the ripple phase~\cite{Carlson87}. Recently, the ripple |
| 64 |
phase is also studied by using monte carlo simulation~\cite{Lenz07}, |
| 65 |
the ripple structure is similar to the results of Marrink except that |
| 66 |
the connection of the upper and lower leaves of the bilayer is an |
| 67 |
interdigitated line instead of the fully interdigitated |
| 68 |
domain. Furthermore, the symmetric ripple phase was also observed in |
| 69 |
their work. They claimed the mismatch between the size of the head |
| 70 |
group and tail of the lipid molecules is the driving force for the |
| 71 |
formation of the ripple phase. |
| 72 |
|
| 73 |
Although the organizations of the tails of lipid molecules are |
| 74 |
addressed by these molecular simulations, the ordering of the head |
| 75 |
group in ripple phase is still not settlement. We developed a simple |
| 76 |
``web of dipoles'' spin lattice model which provides some physical |
| 77 |
insight in our previous studies~\cite{Sun2007}, we found the dipoles |
| 78 |
on head groups of the lipid molecules are ordered in an |
| 79 |
antiferroelectric state. The similiar phenomenon is also observed by |
| 80 |
Tsonchev {\it et al.} when they studied the formation of the |
| 81 |
nanotube\cite{Tsonchev04}. |
| 82 |
|
| 83 |
In this paper, we made a more realistic coarse-grained lipid model to |
| 84 |
understand the primary driving force for membrane corrugation and to |
| 85 |
elucidate the organization of the anisotropic interacting head group |
| 86 |
via molecular dynamics simulation. We will talk about our model and |
| 87 |
methodology in section \ref{sec:method}, and details of the simulation |
| 88 |
in section \ref{sec:experiment}. The results are shown in section |
| 89 |
\ref{sec:results}. At last, we will discuss the results in section |
| 90 |
\ref{sec:discussion}. |
| 91 |
|
| 92 |
\section{Methodology and Model} |
| 93 |
\label{sec:method} |
| 94 |
|
| 95 |
Our idea for developing a simple and reasonable lipid model to study |
| 96 |
the ripple phase of lipid bilayers is based on two facts: one is that |
| 97 |
the most essential feature of lipid molecules is their amphiphilic |
| 98 |
structure with polar head groups and non-polar tails. Another fact is |
| 99 |
that dominant numbers of lipid molecules are very rigid in ripple |
| 100 |
phase which allows the details of the lipid molecules neglectable. The |
| 101 |
lipid model is shown in Figure \ref{fig:lipidMM}. Figure |
| 102 |
\ref{fig:lipidMM}a shows the molecular strucure of a DPPC molecule. The |
| 103 |
hydrophilic character of the head group is the effect of the strong |
| 104 |
dipole composed by a positive charge sitting on the nitrogen and a |
| 105 |
negative charge on the phosphate. The hydrophobic tail consists of |
| 106 |
fatty acid chains. In our model shown in Figure \ref{fig:lipidMM}b, |
| 107 |
lipid molecules are represented by rigid bodies made of one head |
| 108 |
sphere with a point dipole sitting on it and one ellipsoid tail, the |
| 109 |
direction of the dipole is fixed to be perpendicular to the tail. The |
| 110 |
breadth and length of tail are $\sigma_0$, $3\sigma_0$. The diameter |
| 111 |
of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$. The model of |
| 112 |
the solvent in our simulations is inspired by the idea of ``DPD'' |
| 113 |
water. Every four water molecules are reprsented by one sphere. |
| 114 |
|
| 115 |
\begin{figure}[htb] |
| 116 |
\centering |
| 117 |
\includegraphics[width=\linewidth]{lipidModels} |
| 118 |
\caption{Three different representations of DPPC lipid molecules, |
| 119 |
including the chemical structure, an atomistic model, and the |
| 120 |
head-body ellipsoidal coarse-grained model used in this |
| 121 |
work.\label{fig:lipidModels}} |
| 122 |
\end{figure} |
| 123 |
|
| 124 |
Spheres interact each other with Lennard-Jones potential |
| 125 |
\begin{eqnarray*} |
| 126 |
V_{ij} = 4\epsilon \left[\left(\frac{\sigma_0}{r_{ij}}\right)^{12} - |
| 127 |
\left(\frac{\sigma_0}{r_{ij}}\right)^6\right] |
| 128 |
\end{eqnarray*} |
| 129 |
here, $\sigma_0$ is diameter of the spherical particle. $r_{ij}$ is |
| 130 |
the distance between two spheres. $\epsilon$ is the well depth. |
| 131 |
Dipoles interact each other with typical dipole potential |
| 132 |
\begin{eqnarray*} |
| 133 |
V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
| 134 |
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
| 135 |
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
| 136 |
\end{eqnarray*} |
| 137 |
In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 138 |
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
| 139 |
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. Identical |
| 140 |
ellipsoids interact each other with Gay-Berne potential. |
| 141 |
\begin{eqnarray*} |
| 142 |
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 143 |
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 144 |
{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 145 |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
| 146 |
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 147 |
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 148 |
\end{eqnarray*} |
| 149 |
where $\sigma_0 = \sqrt 2\sigma_s$, and orietation-dependent range |
| 150 |
parameter is given by |
| 151 |
\begin{eqnarray*} |
| 152 |
\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 153 |
{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{({\mathbf{\hat r}_{ij}} |
| 154 |
\cdot {\mathbf{\hat u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 155 |
u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
| 156 |
\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}} |
| 157 |
\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
| 158 |
{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
| 159 |
\end{eqnarray*} |
| 160 |
and the strength anisotropy function is, |
| 161 |
\begin{eqnarray*} |
| 162 |
\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 163 |
{\epsilon^\nu}({\mathbf{\hat u}_i}, {\mathbf{\hat |
| 164 |
u}_j}){\epsilon'^\mu}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 165 |
{\mathbf{\hat r}_{ij}}) |
| 166 |
\end{eqnarray*} |
| 167 |
with $\nu$ and $\mu$ being adjustable exponent, and |
| 168 |
$\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j})$, |
| 169 |
$\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 170 |
r}_{ij}})$ defined as |
| 171 |
\begin{eqnarray*} |
| 172 |
\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}) = |
| 173 |
\epsilon_0\left[1-\chi^2({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
| 174 |
u}_j})^2\right]^{-\frac{1}{2}} |
| 175 |
\end{eqnarray*} |
| 176 |
\begin{eqnarray*} |
| 177 |
\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 178 |
1-\frac{1}{2}\chi'\left[\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 179 |
u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 180 |
u}_j})^2}{1+\chi'({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
| 181 |
\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}} |
| 182 |
\cdot {\mathbf{\hat u}_j})^2}{1-\chi'({\mathbf{\hat u}_i} \cdot |
| 183 |
{\mathbf{\hat u}_j})} \right] |
| 184 |
\end{eqnarray*} |
| 185 |
the diameter dependent parameter $\chi$ is given by |
| 186 |
\begin{eqnarray*} |
| 187 |
\chi = \frac{({\sigma_s}^2 - |
| 188 |
{\sigma_e}^2)}{({\sigma_s}^2 + {\sigma_e}^2)} |
| 189 |
\end{eqnarray*} |
| 190 |
and the well depth dependent parameter $\chi'$ is given by |
| 191 |
\begin{eqnarray*} |
| 192 |
\chi' = \frac{({\epsilon_s}^{\frac{1}{\mu}} - |
| 193 |
{\epsilon_e}^{\frac{1}{\mu}})}{({\epsilon_s}^{\frac{1}{\mu}} + |
| 194 |
{\epsilon_e}^{\frac{1}{\mu}})} |
| 195 |
\end{eqnarray*} |
| 196 |
$\sigma_s$ is the side-by-side width, and $\sigma_e$ is the end-to-end |
| 197 |
length. $\epsilon_s$ is the side-by-side well depth, and $\epsilon_e$ |
| 198 |
is the end-to-end well depth. For the interaction between |
| 199 |
nonequivalent uniaxial ellipsoids (in this case, between spheres and |
| 200 |
ellipsoids), the range parameter is generalized as\cite{Cleaver96} |
| 201 |
\begin{eqnarray*} |
| 202 |
\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 203 |
{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{(\alpha{\mathbf{\hat r}_{ij}} |
| 204 |
\cdot {\mathbf{\hat u}_i} + \alpha^{-1}{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 205 |
u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
| 206 |
\frac{(\alpha{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - \alpha^{-1}{\mathbf{\hat r}_{ij}} |
| 207 |
\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
| 208 |
{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
| 209 |
\end{eqnarray*} |
| 210 |
where $\alpha$ is given by |
| 211 |
\begin{eqnarray*} |
| 212 |
\alpha^2 = |
| 213 |
\left[\frac{\left({\sigma_e}_i^2-{\sigma_s}_i^2\right)\left({\sigma_e}_j^2+{\sigma_s}_i^2\right)}{\left({\sigma_e}_j^2-{\sigma_s}_j^2\right)\left({\sigma_e}_i^2+{\sigma_s}_j^2\right)} |
| 214 |
\right]^{\frac{1}{2}} |
| 215 |
\end{eqnarray*} |
| 216 |
the strength parameter is adjusted by the suggestion of |
| 217 |
\cite{Cleaver96}. All potentials are truncated at $25$ \AA\ and |
| 218 |
shifted at $22$ \AA. |
| 219 |
|
| 220 |
\begin{figure}[htb] |
| 221 |
\centering |
| 222 |
\includegraphics[height=4in]{lipidModel} |
| 223 |
\caption{The parameters defining the behavior of the lipid |
| 224 |
models. $\sigma_h / \sigma_0$ is the ratio of the head group to body |
| 225 |
diameter. Molecular bodies all had an aspect ratio of 3.0. The |
| 226 |
dipolar strength (and the temperature and pressure) wer the only other |
| 227 |
parameters that wer varied systematically.\label{fig:lipidModel}} |
| 228 |
\end{figure} |
| 229 |
|
| 230 |
\section{Experiment} |
| 231 |
\label{sec:experiment} |
| 232 |
|
| 233 |
To make the simulations less expensive and to observe long-time |
| 234 |
behavior of the lipid membranes, all simulations were started from two |
| 235 |
separate monolayers in the vaccum with $x-y$ anisotropic pressure |
| 236 |
coupling. The length of $z$ axis of the simulations was fixed and a |
| 237 |
constant surface tension was applied to enable real fluctuations of |
| 238 |
the bilayer. Periodic boundaries were used. There were $480-720$ lipid |
| 239 |
molecules in the simulations depending on the size of the head |
| 240 |
beads. All the simulations were equlibrated for $100$ ns at $300$ |
| 241 |
K. The resulting structures were solvated in water ($6$ DPD |
| 242 |
water/lipid molecule). These configurations were relaxed for another |
| 243 |
$30$ ns relaxation. All simulations with water were carried out at |
| 244 |
constant pressure ($P=1$ atm) by $3$D anisotropic coupling, and |
| 245 |
constant surface tension ($\gamma=0.015$). Given the absence of fast |
| 246 |
degrees of freedom in this model, a timestep of $50$ fs was |
| 247 |
utilized. Simulations were performed by using OOPSE |
| 248 |
package\cite{Meineke05}. |
| 249 |
|
| 250 |
\section{Results and Analysis} |
| 251 |
\label{sec:results} |
| 252 |
|
| 253 |
Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
| 254 |
more corrugated increasing size of the head groups. The surface is |
| 255 |
nearly flat when $\sigma_h=1.20\sigma_0$. With |
| 256 |
$\sigma_h=1.28\sigma_0$, although the surface is still flat, the |
| 257 |
bilayer starts to splay inward; the upper leaf of the bilayer is |
| 258 |
connected to the lower leaf with an interdigitated line defect. Two |
| 259 |
periodicities with $100$ \AA\ width were observed in the |
| 260 |
simulation. This structure is very similiar to the structure observed |
| 261 |
by de Vries and Lenz {\it et al.}. The same basic structure is also |
| 262 |
observed when $\sigma_h=1.41\sigma_0$, but the wavelength of the |
| 263 |
surface corrugations depends sensitively on the size of the ``head'' |
| 264 |
beads. From the undulation spectrum, the corrugation is clearly |
| 265 |
non-thermal. |
| 266 |
\begin{figure}[htb] |
| 267 |
\centering |
| 268 |
\includegraphics[width=\linewidth]{phaseCartoon} |
| 269 |
\caption{A sketch to discribe the structure of the phases observed in |
| 270 |
our simulations.\label{fig:phaseCartoon}} |
| 271 |
\end{figure} |
| 272 |
|
| 273 |
When $\sigma_h=1.35\sigma_0$, we observed another corrugated surface |
| 274 |
morphology. This structure is different from the asymmetric rippled |
| 275 |
surface; there is no interdigitation between the upper and lower |
| 276 |
leaves of the bilayer. Each leaf of the bilayer is broken into several |
| 277 |
hemicylinderical sections, and opposite leaves are fitted together |
| 278 |
much like roof tiles. Unlike the surface in which the upper |
| 279 |
hemicylinder is always interdigitated on the leading or trailing edge |
| 280 |
of lower hemicylinder, the symmetric ripple has no prefered direction. |
| 281 |
The corresponding cartoons are shown in Figure |
| 282 |
\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
| 283 |
different phases. Figure \ref{fig:phaseCartoon} (a) is the flat phase, |
| 284 |
(b) is the asymmetric ripple phase corresponding to the lipid |
| 285 |
organization when $\sigma_h=1.28\sigma_0$ and $\sigma_h=1.41\sigma_0$, |
| 286 |
and (c) is the symmetric ripple phase observed when |
| 287 |
$\sigma_h=1.35\sigma_0$. In symmetric ripple phase, the bilayer is |
| 288 |
continuous everywhere on the whole membrane, however, in asymmetric |
| 289 |
ripple phase, the bilayer is intermittent domains connected by thin |
| 290 |
interdigitated monolayer which consists of upper and lower leaves of |
| 291 |
the bilayer. |
| 292 |
\begin{table*} |
| 293 |
\begin{minipage}{\linewidth} |
| 294 |
\begin{center} |
| 295 |
\caption{} |
| 296 |
\begin{tabular}{lccc} |
| 297 |
\hline |
| 298 |
$\sigma_h/\sigma_0$ & type of phase & $\lambda/\sigma_0$ & $A/\sigma_0$\\ |
| 299 |
\hline |
| 300 |
1.20 & flat & N/A & N/A \\ |
| 301 |
1.28 & asymmetric flat & 21.7 & N/A \\ |
| 302 |
1.35 & symmetric ripple & 17.2 & 2.2 \\ |
| 303 |
1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
| 304 |
\end{tabular} |
| 305 |
\label{tab:property} |
| 306 |
\end{center} |
| 307 |
\end{minipage} |
| 308 |
\end{table*} |
| 309 |
|
| 310 |
The membrane structures and the reduced wavelength $\lambda/\sigma_0$, |
| 311 |
reduced amplitude $A/\sigma_0$ of the ripples are summerized in Table |
| 312 |
\ref{tab:property}. The wavelength range is $15~21$ and the amplitude |
| 313 |
is $1.5$ for asymmetric ripple and $2.2$ for symmetric ripple. These |
| 314 |
values are consistent to the experimental results. Note, the |
| 315 |
amplitudes are underestimated without the melted tails in our |
| 316 |
simulations. |
| 317 |
|
| 318 |
The $P_2$ order paramters (for molecular bodies and head group |
| 319 |
dipoles) have been calculated to clarify the ordering in these phases |
| 320 |
quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$ |
| 321 |
implies orientational randomization. Figure \ref{fig:rP2} shows the |
| 322 |
$P_2$ order paramter of the dipoles on head group rising with |
| 323 |
increasing head group size. When the heads of the lipid molecules are |
| 324 |
small, the membrane is flat. The dipolar ordering is essentially |
| 325 |
frustrated on orientational ordering in this circumstance. Figure |
| 326 |
\ref{} shows the snapshots of the top view for the flat system |
| 327 |
($\sigma_h=1.20\sigma$) and rippled system |
| 328 |
($\sigma_h=1.41\sigma$). The pointing direction of the dipoles on the |
| 329 |
head groups are represented by two colored half spheres from blue to |
| 330 |
yellow. For flat surfaces, the system obviously shows frustration on |
| 331 |
the dipolar ordering, there are kinks on the edge of defferent |
| 332 |
domains. Another reason is that the lipids can move independently in |
| 333 |
each monolayer, it is not nessasory for the direction of dipoles on |
| 334 |
one leaf is consistant to another layer, which makes total order |
| 335 |
parameter is relatively low. With increasing head group size, the |
| 336 |
surface is corrugated, and dipoles do not move as freely on the |
| 337 |
surface. Therefore, the translational freedom of lipids in one layer |
| 338 |
is dependent upon the position of lipids in another layer, as a |
| 339 |
result, the symmetry of the dipoles on head group in one layer is tied |
| 340 |
to the symmetry in the other layer. Furthermore, as the membrane |
| 341 |
deforms from two to three dimensions due to the corrugation, the |
| 342 |
symmetry of the ordering for the dipoles embedded on each leaf is |
| 343 |
broken. The dipoles then self-assemble in a head-tail configuration, |
| 344 |
and the order parameter increases dramaticaly. However, the total |
| 345 |
polarization of the system is still close to zero. This is strong |
| 346 |
evidence that the corrugated structure is an antiferroelectric |
| 347 |
state. From the snapshot in Figure \ref{}, the dipoles arrange as |
| 348 |
arrays along $Y$ axis and fall into head-to-tail configuration in each |
| 349 |
line, but every $3$ or $4$ lines of dipoles change their direction |
| 350 |
from neighbour lines. The system shows antiferroelectric |
| 351 |
charactoristic as a whole. The orientation of the dipolar is always |
| 352 |
perpendicular to the ripple wave vector. These results are consistent |
| 353 |
with our previous study on dipolar membranes. |
| 354 |
|
| 355 |
The ordering of the tails is essentially opposite to the ordering of |
| 356 |
the dipoles on head group. The $P_2$ order parameter decreases with |
| 357 |
increasing head size. This indicates the surface is more curved with |
| 358 |
larger head groups. When the surface is flat, all tails are pointing |
| 359 |
in the same direction; in this case, all tails are parallel to the |
| 360 |
normal of the surface,(making this structure remindcent of the |
| 361 |
$L_{\beta}$ phase. Increasing the size of the heads, results in |
| 362 |
rapidly decreasing $P_2$ ordering for the molecular bodies. |
| 363 |
\begin{figure}[htb] |
| 364 |
\centering |
| 365 |
\includegraphics[width=\linewidth]{rP2} |
| 366 |
\caption{The $P_2$ order parameter as a funtion of the ratio of |
| 367 |
$\sigma_h$ to $\sigma_0$.\label{fig:rP2}} |
| 368 |
\end{figure} |
| 369 |
|
| 370 |
We studied the effects of the interactions between head groups on the |
| 371 |
structure of lipid bilayer by changing the strength of the dipole. |
| 372 |
Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with |
| 373 |
increasing strength of the dipole. Generally the dipoles on the head |
| 374 |
group are more ordered by increase in the strength of the interaction |
| 375 |
between heads and are more disordered by decreasing the interaction |
| 376 |
stength. When the interaction between the heads is weak enough, the |
| 377 |
bilayer structure does not persist; all lipid molecules are solvated |
| 378 |
directly in the water. The critial value of the strength of the dipole |
| 379 |
depends on the head size. The perfectly flat surface melts at $5$ |
| 380 |
$0.03$ debye, the asymmetric rippled surfaces melt at $8$ $0.04$ |
| 381 |
$0.03$ debye, the symmetric rippled surfaces melt at $10$ $0.04$ |
| 382 |
debye. The ordering of the tails is the same as the ordering of the |
| 383 |
dipoles except for the flat phase. Since the surface is already |
| 384 |
perfect flat, the order parameter does not change much until the |
| 385 |
strength of the dipole is $15$ debye. However, the order parameter |
| 386 |
decreases quickly when the strength of the dipole is further |
| 387 |
increased. The head groups of the lipid molecules are brought closer |
| 388 |
by stronger interactions between them. For a flat surface, a large |
| 389 |
amount of free volume between the head groups is available, but when |
| 390 |
the head groups are brought closer, the tails will splay outward, |
| 391 |
forming an inverse micelle. When $\sigma_h=1.28\sigma_0$, the $P_2$ |
| 392 |
order parameter decreases slightly after the strength of the dipole is |
| 393 |
increased to $16$ debye. For rippled surfaces, there is less free |
| 394 |
volume available between the head groups. Therefore there is little |
| 395 |
effect on the structure of the membrane due to increasing dipolar |
| 396 |
strength. However, the increase of the $P_2$ order parameter implies |
| 397 |
the membranes are flatten by the increase of the strength of the |
| 398 |
dipole. Unlike other systems that melt directly when the interaction |
| 399 |
is weak enough, for $\sigma_h=1.41\sigma_0$, part of the membrane |
| 400 |
melts into itself first. The upper leaf of the bilayer becomes totally |
| 401 |
interdigitated with the lower leaf. This is different behavior than |
| 402 |
what is exhibited with the interdigitated lines in the rippled phase |
| 403 |
where only one interdigitated line connects the two leaves of bilayer. |
| 404 |
\begin{figure}[htb] |
| 405 |
\centering |
| 406 |
\includegraphics[width=\linewidth]{sP2} |
| 407 |
\caption{The $P_2$ order parameter as a funtion of the strength of the |
| 408 |
dipole.\label{fig:sP2}} |
| 409 |
\end{figure} |
| 410 |
|
| 411 |
Figure \ref{fig:tP2} shows the dependence of the order parameter on |
| 412 |
temperature. The behavior of the $P_2$ order paramter is |
| 413 |
straightforward. Systems are more ordered at low temperature, and more |
| 414 |
disordered at high temperatures. When the temperature is high enough, |
| 415 |
the membranes are instable. For flat surfaces ($\sigma_h=1.20\sigma_0$ |
| 416 |
and $\sigma_h=1.28\sigma_0$), when the temperature is increased to |
| 417 |
$310$, the $P_2$ order parameter increases slightly instead of |
| 418 |
decreases like ripple surface. This is an evidence of the frustration |
| 419 |
of the dipolar ordering in each leaf of the lipid bilayer, at low |
| 420 |
temperature, the systems are locked in a local minimum energy state, |
| 421 |
with increase of the temperature, the system can jump out the local |
| 422 |
energy well to find the lower energy state which is the longer range |
| 423 |
orientational ordering. Like the dipolar ordering of the flat |
| 424 |
surfaces, the ordering of the tails of the lipid molecules for ripple |
| 425 |
membranes ($\sigma_h=1.35\sigma_0$ and $\sigma_h=1.41\sigma_0$) also |
| 426 |
show some nonthermal characteristic. With increase of the temperature, |
| 427 |
the $P_2$ order parameter decreases firstly, and increases afterward |
| 428 |
when the temperature is greater than $290 K$. The increase of the |
| 429 |
$P_2$ order parameter indicates a more ordered structure for the tails |
| 430 |
of the lipid molecules which corresponds to a more flat surface. Since |
| 431 |
our model lacks the detailed information on lipid tails, we can not |
| 432 |
simulate the fluid phase with melted fatty acid chains. Moreover, the |
| 433 |
formation of the tilted $L_{\beta'}$ phase also depends on the |
| 434 |
organization of fatty groups on tails. |
| 435 |
\begin{figure}[htb] |
| 436 |
\centering |
| 437 |
\includegraphics[width=\linewidth]{tP2} |
| 438 |
\caption{The $P_2$ order parameter as a funtion of |
| 439 |
temperature.\label{fig:tP2}} |
| 440 |
\end{figure} |
| 441 |
|
| 442 |
\section{Discussion} |
| 443 |
\label{sec:discussion} |
| 444 |
|
| 445 |
\bibliography{mdripple} |
| 446 |
\end{document} |
