| 1 |
|
%\documentclass[aps,pre,twocolumn,amssymb,showpacs,floatfix]{revtex4} |
| 2 |
< |
\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4} |
| 2 |
> |
%\documentclass[aps,pre,preprint,amssymb]{revtex4} |
| 3 |
> |
\documentclass[12pt]{article} |
| 4 |
> |
\usepackage{times} |
| 5 |
> |
\usepackage{mathptm} |
| 6 |
> |
\usepackage{tabularx} |
| 7 |
> |
\usepackage{setspace} |
| 8 |
|
\usepackage{amsmath} |
| 9 |
|
\usepackage{amssymb} |
| 10 |
|
\usepackage{graphicx} |
| 11 |
+ |
\usepackage[ref]{overcite} |
| 12 |
+ |
\pagestyle{plain} |
| 13 |
+ |
\pagenumbering{arabic} |
| 14 |
+ |
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
| 15 |
+ |
\topmargin -21pt \headsep 10pt |
| 16 |
+ |
\textheight 9.0in \textwidth 6.5in |
| 17 |
+ |
\brokenpenalty=10000 |
| 18 |
+ |
\renewcommand{\baselinestretch}{1.2} |
| 19 |
+ |
\renewcommand\citemid{\ } % no comma in optional reference note |
| 20 |
|
|
| 21 |
|
\begin{document} |
| 22 |
< |
\renewcommand{\thefootnote}{\fnsymbol{footnote}} |
| 23 |
< |
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} |
| 22 |
> |
%\renewcommand{\thefootnote}{\fnsymbol{footnote}} |
| 23 |
> |
%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} |
| 24 |
|
|
| 25 |
< |
%\bibliographystyle{aps} |
| 25 |
> |
\bibliographystyle{achemso} |
| 26 |
|
|
| 27 |
|
\title{Dipolar Ordering in Molecular-scale Models of the Ripple Phase |
| 28 |
|
in Lipid Membranes} |
| 29 |
< |
\author{Xiuquan Sun and J. Daniel Gezelter} |
| 30 |
< |
\email[E-mail:]{gezelter@nd.edu} |
| 17 |
< |
\affiliation{Department of Chemistry and Biochemistry,\\ |
| 29 |
> |
\author{Xiuquan Sun and J. Daniel Gezelter \\ |
| 30 |
> |
Department of Chemistry and Biochemistry,\\ |
| 31 |
|
University of Notre Dame, \\ |
| 32 |
|
Notre Dame, Indiana 46556} |
| 33 |
|
|
| 34 |
+ |
%\email[E-mail:]{gezelter@nd.edu} |
| 35 |
+ |
|
| 36 |
|
\date{\today} |
| 37 |
|
|
| 38 |
+ |
\maketitle |
| 39 |
+ |
|
| 40 |
|
\begin{abstract} |
| 41 |
|
The ripple phase in phosphatidylcholine (PC) bilayers has never been |
| 42 |
|
completely explained. |
| 43 |
|
\end{abstract} |
| 44 |
|
|
| 45 |
< |
\pacs{} |
| 29 |
< |
\maketitle |
| 45 |
> |
%\maketitle |
| 46 |
|
|
| 47 |
|
\section{Introduction} |
| 48 |
|
\label{sec:Int} |
| 194 |
|
Pechukas.\cite{Berne72} The potential is constructed in the familiar |
| 195 |
|
form of the Lennard-Jones function using orientation-dependent |
| 196 |
|
$\sigma$ and $\epsilon$ parameters, |
| 197 |
< |
\begin{eqnarray*} |
| 197 |
> |
\begin{equation*} |
| 198 |
|
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 199 |
|
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 200 |
|
{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 202 |
|
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 203 |
|
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 204 |
|
\label{eq:gb} |
| 205 |
< |
\end{eqnarray*} |
| 205 |
> |
\end{equation*} |
| 206 |
|
|
| 207 |
|
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 208 |
|
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
| 211 |
|
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
| 212 |
|
intermolecular separation (${\bf \hat{r}}_{ij}$). $\sigma$ and |
| 213 |
|
$\sigma_0$ are also governed by shape mixing and anisotropy variables, |
| 214 |
< |
\begin {equation} |
| 199 |
< |
\begin{array}{rcl} |
| 214 |
> |
\begin {eqnarray*} |
| 215 |
|
\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
| 216 |
|
\chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 - |
| 217 |
|
d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^2 + |
| 219 |
|
\alpha^2 & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 + |
| 220 |
|
d_i^2 \right)}{\left( l_j^2 - d_j^2 \right) \left(l_i^2 + |
| 221 |
|
d_j^2 \right)}\right]^{1/2}, |
| 222 |
< |
\end{array} |
| 208 |
< |
\end{equation} |
| 222 |
> |
\end{eqnarray*} |
| 223 |
|
where $l$ and $d$ describe the length and width of each uniaxial |
| 224 |
|
ellipsoid. These shape anisotropy parameters can then be used to |
| 225 |
|
calculate the range function, |
| 226 |
< |
\begin {equation} |
| 226 |
> |
\begin{equation*} |
| 227 |
|
\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \sigma_{0} |
| 228 |
|
\left[ 1- \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
| 229 |
|
\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
| 232 |
|
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
| 233 |
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
| 234 |
|
\right]^{-1/2} |
| 235 |
< |
\end{equation} |
| 235 |
> |
\end{equation*} |
| 236 |
|
|
| 237 |
|
Gay-Berne ellipsoids also have an energy scaling parameter, |
| 238 |
|
$\epsilon^s$, which describes the well depth for two identical |
| 268 |
|
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
| 269 |
|
\end {eqnarray*} |
| 270 |
|
although many of the quantities and derivatives are identical with |
| 271 |
< |
those obtained for the range parameter. Ref. \onlinecite{Luckhurst90} |
| 271 |
> |
those obtained for the range parameter. Ref. \citen{Luckhurst90} |
| 272 |
|
has a particularly good explanation of the choice of the Gay-Berne |
| 273 |
|
parameters $\mu$ and $\nu$ for modeling liquid crystal molecules. An |
| 274 |
|
excellent overview of the computational methods that can be used to |
| 275 |
|
efficiently compute forces and torques for this potential can be found |
| 276 |
< |
in Ref. \onlinecite{Golubkov06} |
| 276 |
> |
in Ref. \citen{Golubkov06} |
| 277 |
|
|
| 278 |
|
The choices of parameters we have used in this study correspond to a |
| 279 |
|
shape anisotropy of 3 for the chain portion of the molecule. In |
| 309 |
|
are protected by a head ``bead'' with a range parameter which we have |
| 310 |
|
varied between $1.20 d$ and $1.41 d$. The head groups interact with |
| 311 |
|
each other using a combination of Lennard-Jones, |
| 312 |
< |
\begin{eqnarray*} |
| 312 |
> |
\begin{equation} |
| 313 |
|
V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
| 314 |
|
\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
| 315 |
< |
\end{eqnarray*} |
| 315 |
> |
\end{equation} |
| 316 |
|
and dipole-dipole, |
| 317 |
< |
\begin{eqnarray*} |
| 317 |
> |
\begin{equation} |
| 318 |
|
V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 319 |
|
\hat{r}}_{ij})) = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
| 320 |
|
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
| 321 |
|
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
| 322 |
< |
\end{eqnarray*} |
| 322 |
> |
\end{equation} |
| 323 |
|
potentials. |
| 324 |
|
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 325 |
|
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
| 487 |
|
different direction from the upper leaf.\label{fig:topView}} |
| 488 |
|
\end{figure} |
| 489 |
|
|
| 490 |
+ |
The principal method for observing orientational ordering in dipolar |
| 491 |
+ |
or liquid crystalline systems is the $P_2$ order parameter (defined |
| 492 |
+ |
as $1.5 \times \lambda_{max}$, where $\lambda_{max}$ is the largest |
| 493 |
+ |
eigenvalue of the matrix, |
| 494 |
+ |
\begin{equation} |
| 495 |
+ |
{\mathsf{S}} = \frac{1}{N} \sum_i \left( |
| 496 |
+ |
\begin{array}{ccc} |
| 497 |
+ |
u^{x}_i u^{x}_i-\frac{1}{3} & u^{x}_i u^{y}_i & u^{x}_i u^{z}_i \\ |
| 498 |
+ |
u^{y}_i u^{x}_i & u^{y}_i u^{y}_i -\frac{1}{3} & u^{y}_i u^{z}_i \\ |
| 499 |
+ |
u^{z}_i u^{x}_i & u^{z}_i u^{y}_i & u^{z}_i u^{z}_i -\frac{1}{3} |
| 500 |
+ |
\end{array} \right). |
| 501 |
+ |
\label{eq:opmatrix} |
| 502 |
+ |
\end{equation} |
| 503 |
+ |
Here $u^{\alpha}_i$ is the $\alpha=x,y,z$ component of the unit vector |
| 504 |
+ |
for molecule $i$. (Here, $\hat{\bf u}_i$ can refer either to the |
| 505 |
+ |
principal axis of the molecular body or to the dipole on the head |
| 506 |
+ |
group of the molecule.) $P_2$ will be $1.0$ for a perfectly-ordered |
| 507 |
+ |
system and near $0$ for a randomized system. Note that this order |
| 508 |
+ |
parameter is {\em not} equal to the polarization of the system. For |
| 509 |
+ |
example, the polarization of a perfect anti-ferroelectric arrangement |
| 510 |
+ |
of point dipoles is $0$, but $P_2$ for the same system is $1$. The |
| 511 |
+ |
eigenvector of $\mathsf{S}$ corresponding to the largest eigenvalue is |
| 512 |
+ |
familiar as the director axis, which can be used to determine a |
| 513 |
+ |
privileged axis for an orientationally-ordered system. Since the |
| 514 |
+ |
molecular bodies are perpendicular to the head group dipoles, it is |
| 515 |
+ |
possible for the director axes for the molecular bodies and the head |
| 516 |
+ |
groups to be completely decoupled from each other. |
| 517 |
+ |
|
| 518 |
|
Figure \ref{fig:topView} shows snapshots of bird's-eye views of the |
| 519 |
|
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.41, 1.35 d$) |
| 520 |
|
bilayers. The directions of the dipoles on the head groups are |
| 521 |
|
represented with two colored half spheres: blue (phosphate) and yellow |
| 522 |
|
(amino). For flat bilayers, the system exhibits signs of |
| 523 |
< |
orientational frustration; some disorder in the dipolar chains is |
| 524 |
< |
evident with kinks visible at the edges between different ordered |
| 525 |
< |
domains. The lipids can also move independently of lipids in the |
| 526 |
< |
opposing leaf, so the ordering of the dipoles on one leaf is not |
| 527 |
< |
necessarily consistant with the ordering on the other. These two |
| 523 |
> |
orientational frustration; some disorder in the dipolar head-to-tail |
| 524 |
> |
chains is evident with kinks visible at the edges between differently |
| 525 |
> |
ordered domains. The lipids can also move independently of lipids in |
| 526 |
> |
the opposing leaf, so the ordering of the dipoles on one leaf is not |
| 527 |
> |
necessarily consistent with the ordering on the other. These two |
| 528 |
|
factors keep the total dipolar order parameter relatively low for the |
| 529 |
|
flat phases. |
| 530 |
|
|
| 531 |
|
With increasing head group size, the surface becomes corrugated, and |
| 532 |
|
the dipoles cannot move as freely on the surface. Therefore, the |
| 533 |
|
translational freedom of lipids in one layer is dependent upon the |
| 534 |
< |
position of lipids in the other layer. As a result, the ordering of |
| 534 |
> |
position of the lipids in the other layer. As a result, the ordering of |
| 535 |
|
the dipoles on head groups in one leaf is correlated with the ordering |
| 536 |
|
in the other leaf. Furthermore, as the membrane deforms due to the |
| 537 |
|
corrugation, the symmetry of the allowed dipolar ordering on each leaf |
| 539 |
|
configuration, and the dipolar order parameter increases dramatically. |
| 540 |
|
However, the total polarization of the system is still close to zero. |
| 541 |
|
This is strong evidence that the corrugated structure is an |
| 542 |
< |
antiferroelectric state. It is also notable that the head-to-tail |
| 543 |
< |
arrangement of the dipoles is in a direction perpendicular to the wave |
| 544 |
< |
vector for the surface corrugation. This is a similar finding to what |
| 545 |
< |
we observed in our earlier work on the elastic dipolar |
| 546 |
< |
membranes.\cite{Sun07} |
| 542 |
> |
antiferroelectric state. It is also notable that the head-to-tail |
| 543 |
> |
arrangement of the dipoles is always observed in a direction |
| 544 |
> |
perpendicular to the wave vector for the surface corrugation. This is |
| 545 |
> |
a similar finding to what we observed in our earlier work on the |
| 546 |
> |
elastic dipolar membranes.\cite{Sun2007} |
| 547 |
|
|
| 548 |
|
The $P_2$ order parameters (for both the molecular bodies and the head |
| 549 |
|
group dipoles) have been calculated to quantify the ordering in these |
| 550 |
< |
phases. $P_2 = 1$ implies a perfectly ordered structure, and $P_2 = 0$ |
| 551 |
< |
implies complete orientational randomization. Figure \ref{fig:rP2} |
| 552 |
< |
shows the $P_2$ order parameter for the head-group dipoles increasing |
| 553 |
< |
with increasing head group size. When the heads of the lipid molecules |
| 554 |
< |
are small, the membrane is nearly flat. The dipolar ordering exhibits |
| 555 |
< |
frustrated orientational ordering in this circumstance. |
| 550 |
> |
phases. Figure \ref{fig:rP2} shows that the $P_2$ order parameter for |
| 551 |
> |
the head-group dipoles increases with increasing head group size. When |
| 552 |
> |
the heads of the lipid molecules are small, the membrane is nearly |
| 553 |
> |
flat. Since the in-plane packing is essentially a close packing of the |
| 554 |
> |
head groups, the head dipoles exhibit frustration in their |
| 555 |
> |
orientational ordering. |
| 556 |
|
|
| 557 |
< |
The ordering of the tails is essentially opposite to the ordering of |
| 558 |
< |
the dipoles on head group. The $P_2$ order parameter {\it decreases} |
| 559 |
< |
with increasing head size. This indicates that the surface is more |
| 560 |
< |
curved with larger head / tail size ratios. When the surface is flat, |
| 561 |
< |
all tails are pointing in the same direction (parallel to the normal |
| 562 |
< |
of the surface). This simplified model appears to be exhibiting a |
| 563 |
< |
smectic A fluid phase, similar to the real $L_{\beta}$ phase. We have |
| 564 |
< |
not observed a smectic C gel phase ($L_{\beta'}$) for this model |
| 565 |
< |
system. Increasing the size of the heads, results in rapidly |
| 566 |
< |
decreasing $P_2$ ordering for the molecular bodies. |
| 557 |
> |
The ordering trends for the tails are essentially opposite to the |
| 558 |
> |
ordering of the head group dipoles. The tail $P_2$ order parameter |
| 559 |
> |
{\it decreases} with increasing head size. This indicates that the |
| 560 |
> |
surface is more curved with larger head / tail size ratios. When the |
| 561 |
> |
surface is flat, all tails are pointing in the same direction (normal |
| 562 |
> |
to the bilayer surface). This simplified model appears to be |
| 563 |
> |
exhibiting a smectic A fluid phase, similar to the real $L_{\beta}$ |
| 564 |
> |
phase. We have not observed a smectic C gel phase ($L_{\beta'}$) for |
| 565 |
> |
this model system. Increasing the size of the heads results in |
| 566 |
> |
rapidly decreasing $P_2$ ordering for the molecular bodies. |
| 567 |
|
|
| 568 |
|
\begin{figure}[htb] |
| 569 |
|
\centering |
| 570 |
|
\includegraphics[width=\linewidth]{rP2} |
| 571 |
< |
\caption{The $P_2$ order parameter as a function of the ratio of |
| 572 |
< |
$\sigma_h$ to $d$. \label{fig:rP2}} |
| 571 |
> |
\caption{The $P_2$ order parameters for head groups (circles) and |
| 572 |
> |
molecular bodies (squares) as a function of the ratio of head group |
| 573 |
> |
size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{fig:rP2}} |
| 574 |
|
\end{figure} |
| 575 |
|
|
| 576 |
< |
We studied the effects of the interactions between head groups on the |
| 577 |
< |
structure of lipid bilayer by changing the strength of the dipole. |
| 578 |
< |
Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with |
| 579 |
< |
increasing strength of the dipole. Generally the dipoles on the head |
| 580 |
< |
group are more ordered by increase in the strength of the interaction |
| 581 |
< |
between heads and are more disordered by decreasing the interaction |
| 582 |
< |
stength. When the interaction between the heads is weak enough, the |
| 583 |
< |
bilayer structure does not persist; all lipid molecules are solvated |
| 584 |
< |
directly in the water. The critial value of the strength of the dipole |
| 585 |
< |
depends on the head size. The perfectly flat surface melts at $5$ |
| 586 |
< |
$0.03$ debye, the asymmetric rippled surfaces melt at $8$ $0.04$ |
| 587 |
< |
$0.03$ debye, the symmetric rippled surfaces melt at $10$ $0.04$ |
| 588 |
< |
debye. The ordering of the tails is the same as the ordering of the |
| 589 |
< |
dipoles except for the flat phase. Since the surface is already |
| 590 |
< |
perfect flat, the order parameter does not change much until the |
| 591 |
< |
strength of the dipole is $15$ debye. However, the order parameter |
| 592 |
< |
decreases quickly when the strength of the dipole is further |
| 593 |
< |
increased. The head groups of the lipid molecules are brought closer |
| 594 |
< |
by stronger interactions between them. For a flat surface, a large |
| 595 |
< |
amount of free volume between the head groups is available, but when |
| 596 |
< |
the head groups are brought closer, the tails will splay outward, |
| 597 |
< |
forming an inverse micelle. When $\sigma_h=1.28\sigma_0$, the $P_2$ |
| 598 |
< |
order parameter decreases slightly after the strength of the dipole is |
| 599 |
< |
increased to $16$ debye. For rippled surfaces, there is less free |
| 600 |
< |
volume available between the head groups. Therefore there is little |
| 601 |
< |
effect on the structure of the membrane due to increasing dipolar |
| 602 |
< |
strength. However, the increase of the $P_2$ order parameter implies |
| 603 |
< |
the membranes are flatten by the increase of the strength of the |
| 604 |
< |
dipole. Unlike other systems that melt directly when the interaction |
| 605 |
< |
is weak enough, for $\sigma_h=1.41\sigma_0$, part of the membrane |
| 606 |
< |
melts into itself first. The upper leaf of the bilayer becomes totally |
| 607 |
< |
interdigitated with the lower leaf. This is different behavior than |
| 608 |
< |
what is exhibited with the interdigitated lines in the rippled phase |
| 609 |
< |
where only one interdigitated line connects the two leaves of bilayer. |
| 576 |
> |
In addition to varying the size of the head groups, we studied the |
| 577 |
> |
effects of the interactions between head groups on the structure of |
| 578 |
> |
lipid bilayer by changing the strength of the dipoles. Figure |
| 579 |
> |
\ref{fig:sP2} shows how the $P_2$ order parameter changes with |
| 580 |
> |
increasing strength of the dipole. Generally, the dipoles on the head |
| 581 |
> |
groups become more ordered as the strength of the interaction between |
| 582 |
> |
heads is increased and become more disordered by decreasing the |
| 583 |
> |
interaction stength. When the interaction between the heads becomes |
| 584 |
> |
too weak, the bilayer structure does not persist; all lipid molecules |
| 585 |
> |
become dispersed in the solvent (which is non-polar in this |
| 586 |
> |
molecular-scale model). The critial value of the strength of the |
| 587 |
> |
dipole depends on the size of the head groups. The perfectly flat |
| 588 |
> |
surface becomes unstable below $5$ Debye, while the rippled |
| 589 |
> |
surfaces melt at $8$ Debye (asymmetric) or $10$ Debye (symmetric). |
| 590 |
> |
|
| 591 |
> |
The ordering of the tails mirrors the ordering of the dipoles {\it |
| 592 |
> |
except for the flat phase}. Since the surface is nearly flat in this |
| 593 |
> |
phase, the order parameters are only weakly dependent on dipolar |
| 594 |
> |
strength until it reaches $15$ Debye. Once it reaches this value, the |
| 595 |
> |
head group interactions are strong enough to pull the head groups |
| 596 |
> |
close to each other and distort the bilayer structure. For a flat |
| 597 |
> |
surface, a substantial amount of free volume between the head groups |
| 598 |
> |
is normally available. When the head groups are brought closer by |
| 599 |
> |
dipolar interactions, the tails are forced to splay outward, forming |
| 600 |
> |
first curved bilayers, and then inverted micelles. |
| 601 |
> |
|
| 602 |
> |
When $\sigma_h=1.28 d$, the $P_2$ order parameter decreases slightly |
| 603 |
> |
when the strength of the dipole is increased above $16$ debye. For |
| 604 |
> |
rippled bilayers, there is less free volume available between the head |
| 605 |
> |
groups. Therefore increasing dipolar strength weakly influences the |
| 606 |
> |
structure of the membrane. However, the increase in the body $P_2$ |
| 607 |
> |
order parameters implies that the membranes are being slightly |
| 608 |
> |
flattened due to the effects of increasing head-group attraction. |
| 609 |
> |
|
| 610 |
> |
A very interesting behavior takes place when the head groups are very |
| 611 |
> |
large relative to the molecular bodies ($\sigma_h = 1.41 d$) and the |
| 612 |
> |
dipolar strength is relatively weak ($\mu < 9$ Debye). In this case, |
| 613 |
> |
the two leaves of the bilayer become totally interdigitated with each |
| 614 |
> |
other in large patches of the membrane. With higher dipolar |
| 615 |
> |
strength, the interdigitation is limited to single lines that run |
| 616 |
> |
through the bilayer in a direction perpendicular to the ripple wave |
| 617 |
> |
vector. |
| 618 |
> |
|
| 619 |
|
\begin{figure}[htb] |
| 620 |
|
\centering |
| 621 |
|
\includegraphics[width=\linewidth]{sP2} |
| 622 |
< |
\caption{The $P_2$ order parameter as a funtion of the strength of the |
| 623 |
< |
dipole.\label{fig:sP2}} |
| 622 |
> |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
| 623 |
> |
molecular bodies (b) as a function of the strength of the dipoles. |
| 624 |
> |
These order parameters are shown for four values of the head group / |
| 625 |
> |
molecular width ratio ($\sigma_h / d$). \label{fig:sP2}} |
| 626 |
|
\end{figure} |
| 627 |
|
|
| 628 |
< |
Figure \ref{fig:tP2} shows the dependence of the order parameter on |
| 629 |
< |
temperature. The behavior of the $P_2$ order paramter is |
| 630 |
< |
straightforward. Systems are more ordered at low temperature, and more |
| 631 |
< |
disordered at high temperatures. When the temperature is high enough, |
| 632 |
< |
the membranes are instable. For flat surfaces ($\sigma_h=1.20\sigma_0$ |
| 633 |
< |
and $\sigma_h=1.28\sigma_0$), when the temperature is increased to |
| 634 |
< |
$310$, the $P_2$ order parameter increases slightly instead of |
| 635 |
< |
decreases like ripple surface. This is an evidence of the frustration |
| 636 |
< |
of the dipolar ordering in each leaf of the lipid bilayer, at low |
| 637 |
< |
temperature, the systems are locked in a local minimum energy state, |
| 638 |
< |
with increase of the temperature, the system can jump out the local |
| 639 |
< |
energy well to find the lower energy state which is the longer range |
| 640 |
< |
orientational ordering. Like the dipolar ordering of the flat |
| 641 |
< |
surfaces, the ordering of the tails of the lipid molecules for ripple |
| 642 |
< |
membranes ($\sigma_h=1.35\sigma_0$ and $\sigma_h=1.41\sigma_0$) also |
| 643 |
< |
show some nonthermal characteristic. With increase of the temperature, |
| 644 |
< |
the $P_2$ order parameter decreases firstly, and increases afterward |
| 645 |
< |
when the temperature is greater than $290 K$. The increase of the |
| 646 |
< |
$P_2$ order parameter indicates a more ordered structure for the tails |
| 647 |
< |
of the lipid molecules which corresponds to a more flat surface. Since |
| 648 |
< |
our model lacks the detailed information on lipid tails, we can not |
| 649 |
< |
simulate the fluid phase with melted fatty acid chains. Moreover, the |
| 650 |
< |
formation of the tilted $L_{\beta'}$ phase also depends on the |
| 597 |
< |
organization of fatty groups on tails. |
| 628 |
> |
Figure \ref{fig:tP2} shows the dependence of the order parameters on |
| 629 |
> |
temperature. As expected, systems are more ordered at low |
| 630 |
> |
temperatures, and more disordered at high temperatures. All of the |
| 631 |
> |
bilayers we studied can become unstable if the temperature becomes |
| 632 |
> |
high enough. The only interesting feature of the temperature |
| 633 |
> |
dependence is in the flat surfaces ($\sigma_h=1.20 d$ and |
| 634 |
> |
$\sigma_h=1.28 d$). Here, when the temperature is increased above |
| 635 |
> |
$310$K, there is enough jostling of the head groups to allow the |
| 636 |
> |
dipolar frustration to resolve into more ordered states. This results |
| 637 |
> |
in a slight increase in the $P_2$ order parameter above this |
| 638 |
> |
temperature. |
| 639 |
> |
|
| 640 |
> |
For the rippled surfaces ($\sigma_h=1.35 d$ and $\sigma_h=1.41 d$), |
| 641 |
> |
there is a slightly increased orientational ordering in the molecular |
| 642 |
> |
bodies above $290$K. Since our model lacks the detailed information |
| 643 |
> |
about the behavior of the lipid tails, this is the closest the model |
| 644 |
> |
can come to depicting the ripple ($P_{\beta'}$) to fluid |
| 645 |
> |
($L_{\alpha}$) phase transition. What we are observing is a |
| 646 |
> |
flattening of the rippled structures made possible by thermal |
| 647 |
> |
expansion of the tightly-packed head groups. The lack of detailed |
| 648 |
> |
chain configurations also makes it impossible for this model to depict |
| 649 |
> |
the ripple to gel ($L_{\beta'}$) phase transition. |
| 650 |
> |
|
| 651 |
|
\begin{figure}[htb] |
| 652 |
|
\centering |
| 653 |
|
\includegraphics[width=\linewidth]{tP2} |
| 654 |
< |
\caption{The $P_2$ order parameter as a function of |
| 655 |
< |
temperature.\label{fig:tP2}} |
| 654 |
> |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
| 655 |
> |
molecular bodies (b) as a function of temperature. |
| 656 |
> |
These order parameters are shown for four values of the head group / |
| 657 |
> |
molecular width ratio ($\sigma_h / d$).\label{fig:tP2}} |
| 658 |
|
\end{figure} |
| 659 |
|
|
| 660 |
|
\section{Discussion} |
