| 24 |
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|
| 25 |
|
\bibliographystyle{achemso} |
| 26 |
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|
| 27 |
< |
\title{Dipolar Ordering in Molecular-scale Models of the Ripple Phase |
| 28 |
< |
in Lipid Membranes} |
| 27 |
> |
\title{Dipolar ordering in the ripple phases of molecular-scale models |
| 28 |
> |
of lipid membranes} |
| 29 |
|
\author{Xiuquan Sun and J. Daniel Gezelter \\ |
| 30 |
|
Department of Chemistry and Biochemistry,\\ |
| 31 |
|
University of Notre Dame, \\ |
| 38 |
|
\maketitle |
| 39 |
|
|
| 40 |
|
\begin{abstract} |
| 41 |
< |
The ripple phase in phosphatidylcholine (PC) bilayers has never been |
| 42 |
< |
completely explained. |
| 41 |
> |
Symmetric and asymmetric ripple phases have been observed to form in |
| 42 |
> |
molecular dynamics simulations of a simple molecular-scale lipid |
| 43 |
> |
model. The lipid model consists of an dipolar head group and an |
| 44 |
> |
ellipsoidal tail. Within the limits of this model, an explanation for |
| 45 |
> |
generalized membrane curvature is a simple mismatch in the size of the |
| 46 |
> |
heads with the width of the molecular bodies. The persistence of a |
| 47 |
> |
{\it bilayer} structure requires strong attractive forces between the |
| 48 |
> |
head groups. One feature of this model is that an energetically |
| 49 |
> |
favorable orientational ordering of the dipoles can be achieved by |
| 50 |
> |
out-of-plane membrane corrugation. The corrugation of the surface |
| 51 |
> |
stabilizes the long range orientational ordering for the dipoles in the |
| 52 |
> |
head groups which then adopt a bulk anti-ferroelectric state. We |
| 53 |
> |
observe a common feature of the corrugated dipolar membranes: the wave |
| 54 |
> |
vectors for the surface ripples are always found to be perpendicular |
| 55 |
> |
to the dipole director axis. |
| 56 |
|
\end{abstract} |
| 57 |
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|
| 58 |
|
%\maketitle |
| 59 |
+ |
\newpage |
| 60 |
|
|
| 61 |
|
\section{Introduction} |
| 62 |
|
\label{sec:Int} |
| 75 |
|
experimental results provide strong support for a 2-dimensional |
| 76 |
|
hexagonal packing lattice of the lipid molecules within the ripple |
| 77 |
|
phase. This is a notable change from the observed lipid packing |
| 78 |
< |
within the gel phase.~\cite{Cevc87} |
| 78 |
> |
within the gel phase.~\cite{Cevc87} The X-ray diffraction work by |
| 79 |
> |
Katsaras {\it et al.} showed that a rich phase diagram exhibiting both |
| 80 |
> |
{\it asymmetric} and {\it symmetric} ripples is possible for lecithin |
| 81 |
> |
bilayers.\cite{Katsaras00} |
| 82 |
|
|
| 83 |
|
A number of theoretical models have been presented to explain the |
| 84 |
|
formation of the ripple phase. Marder {\it et al.} used a |
| 85 |
< |
curvature-dependent Landau-de Gennes free-energy functional to predict |
| 85 |
> |
curvature-dependent Landau-de~Gennes free-energy functional to predict |
| 86 |
|
a rippled phase.~\cite{Marder84} This model and other related continuum |
| 87 |
|
models predict higher fluidity in convex regions and that concave |
| 88 |
|
portions of the membrane correspond to more solid-like regions. |
| 108 |
|
regions.~\cite{Heimburg00} Kubica has suggested that a lattice model of |
| 109 |
|
polar head groups could be valuable in trying to understand bilayer |
| 110 |
|
phase formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations |
| 111 |
< |
of lamellar stacks of hexagonal lattices to show that large headgroups |
| 111 |
> |
of lamellar stacks of hexagonal lattices to show that large head groups |
| 112 |
|
and molecular tilt with respect to the membrane normal vector can |
| 113 |
|
cause bulk rippling.~\cite{Bannerjee02} |
| 114 |
|
|
| 115 |
< |
In contrast, few large-scale molecular modelling studies have been |
| 115 |
> |
In contrast, few large-scale molecular modeling studies have been |
| 116 |
|
done due to the large size of the resulting structures and the time |
| 117 |
|
required for the phases of interest to develop. With all-atom (and |
| 118 |
|
even unified-atom) simulations, only one period of the ripple can be |
| 119 |
< |
observed and only for timescales in the range of 10-100 ns. One of |
| 120 |
< |
the most interesting molecular simulations was carried out by De Vries |
| 119 |
> |
observed and only for time scales in the range of 10-100 ns. One of |
| 120 |
> |
the most interesting molecular simulations was carried out by de~Vries |
| 121 |
|
{\it et al.}~\cite{deVries05}. According to their simulation results, |
| 122 |
|
the ripple consists of two domains, one resembling the gel bilayer, |
| 123 |
|
while in the other, the two leaves of the bilayer are fully |
| 139 |
|
|
| 140 |
|
Although the organization of the tails of lipid molecules are |
| 141 |
|
addressed by these molecular simulations and the packing competition |
| 142 |
< |
between headgroups and tails is strongly implicated as the primary |
| 142 |
> |
between head groups and tails is strongly implicated as the primary |
| 143 |
|
driving force for ripple formation, questions about the ordering of |
| 144 |
< |
the head groups in ripple phase has not been settled. |
| 144 |
> |
the head groups in ripple phase have not been settled. |
| 145 |
|
|
| 146 |
|
In a recent paper, we presented a simple ``web of dipoles'' spin |
| 147 |
|
lattice model which provides some physical insight into relationship |
| 148 |
|
between dipolar ordering and membrane buckling.\cite{Sun2007} We found |
| 149 |
|
that dipolar elastic membranes can spontaneously buckle, forming |
| 150 |
< |
ripple-like topologies. The driving force for the buckling in dipolar |
| 151 |
< |
elastic membranes the antiferroelectric ordering of the dipoles, and |
| 152 |
< |
this was evident in the ordering of the dipole director axis |
| 153 |
< |
perpendicular to the wave vector of the surface ripples. A similiar |
| 150 |
> |
ripple-like topologies. The driving force for the buckling of dipolar |
| 151 |
> |
elastic membranes is the anti-ferroelectric ordering of the dipoles. |
| 152 |
> |
This was evident in the ordering of the dipole director axis |
| 153 |
> |
perpendicular to the wave vector of the surface ripples. A similar |
| 154 |
|
phenomenon has also been observed by Tsonchev {\it et al.} in their |
| 155 |
|
work on the spontaneous formation of dipolar peptide chains into |
| 156 |
|
curved nano-structures.\cite{Tsonchev04,Tsonchev04II} |
| 184 |
|
non-polar tails. Another fact is that the majority of lipid molecules |
| 185 |
|
in the ripple phase are relatively rigid (i.e. gel-like) which makes |
| 186 |
|
some fraction of the details of the chain dynamics negligible. Figure |
| 187 |
< |
\ref{fig:lipidModels} shows the molecular strucure of a DPPC |
| 187 |
> |
\ref{fig:lipidModels} shows the molecular structure of a DPPC |
| 188 |
|
molecule, as well as atomistic and molecular-scale representations of |
| 189 |
|
a DPPC molecule. The hydrophilic character of the head group is |
| 190 |
|
largely due to the separation of charge between the nitrogen and |
| 203 |
|
The ellipsoidal portions of the model interact via the Gay-Berne |
| 204 |
|
potential which has seen widespread use in the liquid crystal |
| 205 |
|
community. Ayton and Voth have also used Gay-Berne ellipsoids for |
| 206 |
< |
modelling large length-scale properties of lipid |
| 206 |
> |
modeling large length-scale properties of lipid |
| 207 |
|
bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
| 208 |
|
was a single site model for the interactions of rigid ellipsoidal |
| 209 |
|
molecules.\cite{Gay81} It can be thought of as a modification of the |
| 253 |
|
|
| 254 |
|
Gay-Berne ellipsoids also have an energy scaling parameter, |
| 255 |
|
$\epsilon^s$, which describes the well depth for two identical |
| 256 |
< |
ellipsoids in a {\it side-by-side} configuration. Additionaly, a well |
| 256 |
> |
ellipsoids in a {\it side-by-side} configuration. Additionally, a well |
| 257 |
|
depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes |
| 258 |
|
the ratio between the well depths in the {\it end-to-end} and |
| 259 |
|
side-by-side configurations. As in the range parameter, a set of |
| 320 |
|
\end{figure} |
| 321 |
|
|
| 322 |
|
To take into account the permanent dipolar interactions of the |
| 323 |
< |
zwitterionic head groups, we place fixed dipole moments $\mu_{i}$ at |
| 323 |
> |
zwitterionic head groups, we have placed fixed dipole moments $\mu_{i}$ at |
| 324 |
|
one end of the Gay-Berne particles. The dipoles are oriented at an |
| 325 |
|
angle $\theta = \pi / 2$ relative to the major axis. These dipoles |
| 326 |
< |
are protected by a head ``bead'' with a range parameter which we have |
| 326 |
> |
are protected by a head ``bead'' with a range parameter ($\sigma_h$) which we have |
| 327 |
|
varied between $1.20 d$ and $1.41 d$. The head groups interact with |
| 328 |
|
each other using a combination of Lennard-Jones, |
| 329 |
|
\begin{equation} |
| 352 |
|
|
| 353 |
|
The solvent model in our simulations is identical to one used by |
| 354 |
|
Marrink {\it et al.} in their dissipative particle dynamics (DPD) |
| 355 |
< |
simulation of lipid bilayers.\cite{Marrink04} This solvent bead is a single |
| 356 |
< |
site that represents four water molecules (m = 72 amu) and has |
| 355 |
> |
simulation of lipid bilayers.\cite{Marrink04} This solvent bead is a |
| 356 |
> |
single site that represents four water molecules (m = 72 amu) and has |
| 357 |
|
comparable density and diffusive behavior to liquid water. However, |
| 358 |
|
since there are no electrostatic sites on these beads, this solvent |
| 359 |
< |
model cannot replicate the dielectric properties of water. |
| 359 |
> |
model cannot replicate the dielectric properties of water. |
| 360 |
> |
|
| 361 |
|
\begin{table*} |
| 362 |
|
\begin{minipage}{\linewidth} |
| 363 |
|
\begin{center} |
| 383 |
|
\end{minipage} |
| 384 |
|
\end{table*} |
| 385 |
|
|
| 386 |
< |
A switching function has been applied to all potentials to smoothly |
| 387 |
< |
turn off the interactions between a range of $22$ and $25$ \AA. |
| 386 |
> |
\section{Experimental Methodology} |
| 387 |
> |
\label{sec:experiment} |
| 388 |
|
|
| 389 |
|
The parameters that were systematically varied in this study were the |
| 390 |
|
size of the head group ($\sigma_h$), the strength of the dipole moment |
| 391 |
|
($\mu$), and the temperature of the system. Values for $\sigma_h$ |
| 392 |
< |
ranged from 5.5 \AA\ to 6.5 \AA\ . If the width of the tails is |
| 393 |
< |
taken to be the unit of length, these head groups correspond to a |
| 394 |
< |
range from $1.2 d$ to $1.41 d$. Since the solvent beads are nearly |
| 395 |
< |
identical in diameter to the tail ellipsoids, all distances that |
| 396 |
< |
follow will be measured relative to this unit of distance. |
| 392 |
> |
ranged from 5.5 \AA\ to 6.5 \AA\ . If the width of the tails is taken |
| 393 |
> |
to be the unit of length, these head groups correspond to a range from |
| 394 |
> |
$1.2 d$ to $1.41 d$. Since the solvent beads are nearly identical in |
| 395 |
> |
diameter to the tail ellipsoids, all distances that follow will be |
| 396 |
> |
measured relative to this unit of distance. Because the solvent we |
| 397 |
> |
are using is non-polar and has a dielectric constant of 1, values for |
| 398 |
> |
$\mu$ are sampled from a range that is somewhat smaller than the 20.6 |
| 399 |
> |
Debye dipole moment of the PC head groups. |
| 400 |
|
|
| 380 |
– |
\section{Experimental Methodology} |
| 381 |
– |
\label{sec:experiment} |
| 382 |
– |
|
| 401 |
|
To create unbiased bilayers, all simulations were started from two |
| 402 |
|
perfectly flat monolayers separated by a 26 \AA\ gap between the |
| 403 |
|
molecular bodies of the upper and lower leaves. The separated |
| 404 |
< |
monolayers were evolved in a vaccum with $x-y$ anisotropic pressure |
| 404 |
> |
monolayers were evolved in a vacuum with $x-y$ anisotropic pressure |
| 405 |
|
coupling. The length of $z$ axis of the simulations was fixed and a |
| 406 |
|
constant surface tension was applied to enable real fluctuations of |
| 407 |
|
the bilayer. Periodic boundary conditions were used, and $480-720$ |
| 408 |
|
lipid molecules were present in the simulations, depending on the size |
| 409 |
|
of the head beads. In all cases, the two monolayers spontaneously |
| 410 |
|
collapsed into bilayer structures within 100 ps. Following this |
| 411 |
< |
collapse, all systems were equlibrated for $100$ ns at $300$ K. |
| 411 |
> |
collapse, all systems were equilibrated for $100$ ns at $300$ K. |
| 412 |
|
|
| 413 |
|
The resulting bilayer structures were then solvated at a ratio of $6$ |
| 414 |
|
solvent beads (24 water molecules) per lipid. These configurations |
| 415 |
|
were then equilibrated for another $30$ ns. All simulations utilizing |
| 416 |
|
the solvent were carried out at constant pressure ($P=1$ atm) with |
| 417 |
|
$3$D anisotropic coupling, and constant surface tension |
| 418 |
< |
($\gamma=0.015$ UNIT). Given the absence of fast degrees of freedom in |
| 419 |
< |
this model, a timestep of $50$ fs was utilized with excellent energy |
| 418 |
> |
($\gamma=0.015$ N/m). Given the absence of fast degrees of freedom in |
| 419 |
> |
this model, a time step of $50$ fs was utilized with excellent energy |
| 420 |
|
conservation. Data collection for structural properties of the |
| 421 |
|
bilayers was carried out during a final 5 ns run following the solvent |
| 422 |
|
equilibration. All simulations were performed using the OOPSE |
| 423 |
|
molecular modeling program.\cite{Meineke05} |
| 424 |
|
|
| 425 |
+ |
A switching function was applied to all potentials to smoothly turn |
| 426 |
+ |
off the interactions between a range of $22$ and $25$ \AA. |
| 427 |
+ |
|
| 428 |
|
\section{Results} |
| 429 |
|
\label{sec:results} |
| 430 |
|
|
| 431 |
< |
Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
| 432 |
< |
more corrugated with increasing size of the head groups. The surface |
| 433 |
< |
is nearly flat when $\sigma_h=1.20 d$. With $\sigma_h=1.28 d$, |
| 434 |
< |
although the surface is still flat, the bilayer starts to splay |
| 435 |
< |
inward; the upper leaf of the bilayer is connected to the lower leaf |
| 436 |
< |
with an interdigitated line defect. Two periodicities with $100$ \AA\ |
| 437 |
< |
wavelengths were observed in the simulation. This structure is very |
| 438 |
< |
similiar to the structure observed by de Vries and Lenz {\it et |
| 439 |
< |
al.}. The same basic structure is also observed when $\sigma_h=1.41 |
| 440 |
< |
d$, but the wavelength of the surface corrugations depends sensitively |
| 441 |
< |
on the size of the ``head'' beads. From the undulation spectrum, the |
| 442 |
< |
corrugation is clearly non-thermal. |
| 431 |
> |
The membranes in our simulations exhibit a number of interesting |
| 432 |
> |
bilayer phases. The surface topology of these phases depends most |
| 433 |
> |
sensitively on the ratio of the size of the head groups to the width |
| 434 |
> |
of the molecular bodies. With heads only slightly larger than the |
| 435 |
> |
bodies ($\sigma_h=1.20 d$) the membrane exhibits a flat bilayer. |
| 436 |
> |
|
| 437 |
> |
Increasing the head / body size ratio increases the local membrane |
| 438 |
> |
curvature around each of the lipids. With $\sigma_h=1.28 d$, the |
| 439 |
> |
surface is still essentially flat, but the bilayer starts to exhibit |
| 440 |
> |
signs of instability. We have observed occasional defects where a |
| 441 |
> |
line of lipid molecules on one leaf of the bilayer will dip down to |
| 442 |
> |
interdigitate with the other leaf. This gives each of the two bilayer |
| 443 |
> |
leaves some local convexity near the line defect. These structures, |
| 444 |
> |
once developed in a simulation, are very stable and are spaced |
| 445 |
> |
approximately 100 \AA\ away from each other. |
| 446 |
> |
|
| 447 |
> |
With larger heads ($\sigma_h = 1.35 d$) the membrane curvature |
| 448 |
> |
resolves into a ``symmetric'' ripple phase. Each leaf of the bilayer |
| 449 |
> |
is broken into several convex, hemicylinderical sections, and opposite |
| 450 |
> |
leaves are fitted together much like roof tiles. There is no |
| 451 |
> |
interdigitation between the upper and lower leaves of the bilayer. |
| 452 |
> |
|
| 453 |
> |
For the largest head / body ratios studied ($\sigma_h = 1.41 d$) the |
| 454 |
> |
local curvature is substantially larger, and the resulting bilayer |
| 455 |
> |
structure resolves into an asymmetric ripple phase. This structure is |
| 456 |
> |
very similar to the structures observed by both de~Vries {\it et al.} |
| 457 |
> |
and Lenz {\it et al.}. For a given ripple wave vector, there are two |
| 458 |
> |
possible asymmetric ripples, which is not the case for the symmetric |
| 459 |
> |
phase observed when $\sigma_h = 1.35 d$. |
| 460 |
> |
|
| 461 |
|
\begin{figure}[htb] |
| 462 |
|
\centering |
| 463 |
|
\includegraphics[width=4in]{phaseCartoon} |
| 464 |
< |
\caption{A sketch to discribe the structure of the phases observed in |
| 465 |
< |
our simulations.\label{fig:phaseCartoon}} |
| 464 |
> |
\caption{The role of the ratio between the head group size and the |
| 465 |
> |
width of the molecular bodies is to increase the local membrane |
| 466 |
> |
curvature. With strong attractive interactions between the head |
| 467 |
> |
groups, this local curvature can be maintained in bilayer structures |
| 468 |
> |
through surface corrugation. Shown above are three phases observed in |
| 469 |
> |
these simulations. With $\sigma_h = 1.20 d$, the bilayer maintains a |
| 470 |
> |
flat topology. For larger heads ($\sigma_h = 1.35 d$) the local |
| 471 |
> |
curvature resolves into a symmetrically rippled phase with little or |
| 472 |
> |
no interdigitation between the upper and lower leaves of the membrane. |
| 473 |
> |
The largest heads studied ($\sigma_h = 1.41 d$) resolve into an |
| 474 |
> |
asymmetric rippled phases with interdigitation between the two |
| 475 |
> |
leaves.\label{fig:phaseCartoon}} |
| 476 |
|
\end{figure} |
| 477 |
|
|
| 478 |
< |
When $\sigma_h=1.35 d$, we observed another corrugated surface |
| 479 |
< |
morphology. This structure is different from the asymmetric rippled |
| 480 |
< |
surface; there is no interdigitation between the upper and lower |
| 481 |
< |
leaves of the bilayer. Each leaf of the bilayer is broken into several |
| 482 |
< |
hemicylinderical sections, and opposite leaves are fitted together |
| 483 |
< |
much like roof tiles. Unlike the surface in which the upper |
| 484 |
< |
hemicylinder is always interdigitated on the leading or trailing edge |
| 485 |
< |
of lower hemicylinder, this ``symmetric'' ripple has no prefered |
| 486 |
< |
direction. The corresponding structures are shown in Figure |
| 487 |
< |
\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
| 488 |
< |
different phases. The top panel in figure \ref{fig:phaseCartoon} is |
| 489 |
< |
the flat phase, the middle panel shows the asymmetric ripple phase |
| 490 |
< |
corresponding to $\sigma_h = 1.41 d$ and the lower panel shows the |
| 491 |
< |
symmetric ripple phase observed when $\sigma_h=1.35 d$. In the |
| 492 |
< |
symmetric ripple, the bilayer is continuous over the whole membrane, |
| 493 |
< |
however, in asymmetric ripple phase, the bilayer domains are connected |
| 494 |
< |
by thin interdigitated monolayers that share molecules between the |
| 495 |
< |
upper and lower leaves. |
| 478 |
> |
Sample structures for the flat ($\sigma_h = 1.20 d$), symmetric |
| 479 |
> |
($\sigma_h = 1.35 d$, and asymmetric ($\sigma_h = 1.41 d$) ripple |
| 480 |
> |
phases are shown in Figure \ref{fig:phaseCartoon}. |
| 481 |
> |
|
| 482 |
> |
It is reasonable to ask how well the parameters we used can produce |
| 483 |
> |
bilayer properties that match experimentally known values for real |
| 484 |
> |
lipid bilayers. Using a value of $l = 13.8$ \AA for the ellipsoidal |
| 485 |
> |
tails and the fixed ellipsoidal aspect ratio of 3, our values for the |
| 486 |
> |
area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend |
| 487 |
> |
entirely on the size of the head bead relative to the molecular body. |
| 488 |
> |
These values are tabulated in table \ref{tab:property}. Kucera {\it |
| 489 |
> |
et al.} have measured values for the head group spacings for a number |
| 490 |
> |
of PC lipid bilayers that range from 30.8 \AA (DLPC) to 37.8 (DPPC). |
| 491 |
> |
They have also measured values for the area per lipid that range from |
| 492 |
> |
60.6 |
| 493 |
> |
\AA$^2$ (DMPC) to 64.2 \AA$^2$ |
| 494 |
> |
(DPPC).\cite{NorbertKucerka04012005,NorbertKucerka06012006} Only the |
| 495 |
> |
largest of the head groups we modeled ($\sigma_h = 1.41 d$) produces |
| 496 |
> |
bilayers (specifically the area per lipid) that resemble real PC |
| 497 |
> |
bilayers. The smaller head beads we used are perhaps better models |
| 498 |
> |
for PE head groups. |
| 499 |
> |
|
| 500 |
|
\begin{table*} |
| 501 |
|
\begin{minipage}{\linewidth} |
| 502 |
|
\begin{center} |
| 503 |
< |
\caption{Phases, ripple wavelengths and amplitudes observed as a |
| 504 |
< |
function of the ratio between the head beads and the diameters of the |
| 505 |
< |
tails. All lengths are normalized to the diameter of the tail |
| 506 |
< |
ellipsoids.} |
| 507 |
< |
\begin{tabular}{lccc} |
| 503 |
> |
\caption{Phase, bilayer spacing, area per lipid, ripple wavelength |
| 504 |
> |
and amplitude observed as a function of the ratio between the head |
| 505 |
> |
beads and the diameters of the tails. Ripple wavelengths and |
| 506 |
> |
amplitudes are normalized to the diameter of the tail ellipsoids.} |
| 507 |
> |
\begin{tabular}{lccccc} |
| 508 |
|
\hline |
| 509 |
< |
$\sigma_h / d$ & type of phase & $\lambda / d$ & $A / d$\\ |
| 509 |
> |
$\sigma_h / d$ & type of phase & bilayer spacing (\AA) & area per |
| 510 |
> |
lipid (\AA$^2$) & $\lambda / d$ & $A / d$\\ |
| 511 |
|
\hline |
| 512 |
< |
1.20 & flat & N/A & N/A \\ |
| 513 |
< |
1.28 & asymmetric ripple or flat & 21.7 & N/A \\ |
| 514 |
< |
1.35 & symmetric ripple & 17.2 & 2.2 \\ |
| 515 |
< |
1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
| 512 |
> |
1.20 & flat & 33.4 & 49.6 & N/A & N/A \\ |
| 513 |
> |
1.28 & flat & 33.7 & 54.7 & N/A & N/A \\ |
| 514 |
> |
1.35 & symmetric ripple & 42.9 & 51.7 & 17.2 & 2.2 \\ |
| 515 |
> |
1.41 & asymmetric ripple & 37.1 & 63.1 & 15.4 & 1.5 \\ |
| 516 |
|
\end{tabular} |
| 517 |
|
\label{tab:property} |
| 518 |
|
\end{center} |
| 521 |
|
|
| 522 |
|
The membrane structures and the reduced wavelength $\lambda / d$, |
| 523 |
|
reduced amplitude $A / d$ of the ripples are summarized in Table |
| 524 |
< |
\ref{tab:property}. The wavelength range is $15~21$ molecular bodies |
| 524 |
> |
\ref{tab:property}. The wavelength range is $15 - 17$ molecular bodies |
| 525 |
|
and the amplitude is $1.5$ molecular bodies for asymmetric ripple and |
| 526 |
< |
$2.2$ for symmetric ripple. These values are consistent to the |
| 527 |
< |
experimental results. Note, that given the lack of structural freedom |
| 528 |
< |
in the tails of our model lipids, the amplitudes observed from these |
| 529 |
< |
simulations are likely to underestimate of the true amplitudes. |
| 526 |
> |
$2.2$ for symmetric ripple. These values are reasonably consistent |
| 527 |
> |
with experimental measurements.\cite{Sun96,Katsaras00,Kaasgaard03} |
| 528 |
> |
Note, that given the lack of structural freedom in the tails of our |
| 529 |
> |
model lipids, the amplitudes observed from these simulations are |
| 530 |
> |
likely to underestimate of the true amplitudes. |
| 531 |
|
|
| 532 |
|
\begin{figure}[htb] |
| 533 |
|
\centering |
| 534 |
|
\includegraphics[width=4in]{topDown} |
| 535 |
< |
\caption{Top views of the flat (upper), asymmetric ripple (middle), |
| 536 |
< |
and symmetric ripple (lower) phases. Note that the head-group dipoles |
| 537 |
< |
have formed head-to-tail chains in all three of these phases, but in |
| 538 |
< |
the two rippled phases, the dipolar chains are all aligned |
| 539 |
< |
{\it perpendicular} to the direction of the ripple. The flat membrane |
| 540 |
< |
has multiple point defects in the dipolar orientational ordering, and |
| 541 |
< |
the dipolar ordering on the lower leaf of the bilayer can be in a |
| 542 |
< |
different direction from the upper leaf.\label{fig:topView}} |
| 535 |
> |
\caption{Top views of the flat (upper), symmetric ripple (middle), |
| 536 |
> |
and asymmetric ripple (lower) phases. Note that the head-group |
| 537 |
> |
dipoles have formed head-to-tail chains in all three of these phases, |
| 538 |
> |
but in the two rippled phases, the dipolar chains are all aligned {\it |
| 539 |
> |
perpendicular} to the direction of the ripple. Note that the flat |
| 540 |
> |
membrane has multiple vortex defects in the dipolar ordering, and the |
| 541 |
> |
ordering on the lower leaf of the bilayer can be in an entirely |
| 542 |
> |
different direction from the upper leaf.\label{fig:topView}} |
| 543 |
|
\end{figure} |
| 544 |
|
|
| 545 |
|
The principal method for observing orientational ordering in dipolar |
| 571 |
|
groups to be completely decoupled from each other. |
| 572 |
|
|
| 573 |
|
Figure \ref{fig:topView} shows snapshots of bird's-eye views of the |
| 574 |
< |
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.41, 1.35 d$) |
| 574 |
> |
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$) |
| 575 |
|
bilayers. The directions of the dipoles on the head groups are |
| 576 |
|
represented with two colored half spheres: blue (phosphate) and yellow |
| 577 |
|
(amino). For flat bilayers, the system exhibits signs of |
| 594 |
|
configuration, and the dipolar order parameter increases dramatically. |
| 595 |
|
However, the total polarization of the system is still close to zero. |
| 596 |
|
This is strong evidence that the corrugated structure is an |
| 597 |
< |
antiferroelectric state. It is also notable that the head-to-tail |
| 597 |
> |
anti-ferroelectric state. It is also notable that the head-to-tail |
| 598 |
|
arrangement of the dipoles is always observed in a direction |
| 599 |
|
perpendicular to the wave vector for the surface corrugation. This is |
| 600 |
|
a similar finding to what we observed in our earlier work on the |
| 635 |
|
increasing strength of the dipole. Generally, the dipoles on the head |
| 636 |
|
groups become more ordered as the strength of the interaction between |
| 637 |
|
heads is increased and become more disordered by decreasing the |
| 638 |
< |
interaction stength. When the interaction between the heads becomes |
| 638 |
> |
interaction strength. When the interaction between the heads becomes |
| 639 |
|
too weak, the bilayer structure does not persist; all lipid molecules |
| 640 |
|
become dispersed in the solvent (which is non-polar in this |
| 641 |
< |
molecular-scale model). The critial value of the strength of the |
| 641 |
> |
molecular-scale model). The critical value of the strength of the |
| 642 |
|
dipole depends on the size of the head groups. The perfectly flat |
| 643 |
|
surface becomes unstable below $5$ Debye, while the rippled |
| 644 |
|
surfaces melt at $8$ Debye (asymmetric) or $10$ Debye (symmetric). |
| 651 |
|
close to each other and distort the bilayer structure. For a flat |
| 652 |
|
surface, a substantial amount of free volume between the head groups |
| 653 |
|
is normally available. When the head groups are brought closer by |
| 654 |
< |
dipolar interactions, the tails are forced to splay outward, forming |
| 655 |
< |
first curved bilayers, and then inverted micelles. |
| 654 |
> |
dipolar interactions, the tails are forced to splay outward, first forming |
| 655 |
> |
curved bilayers, and then inverted micelles. |
| 656 |
|
|
| 657 |
|
When $\sigma_h=1.28 d$, the $P_2$ order parameter decreases slightly |
| 658 |
< |
when the strength of the dipole is increased above $16$ debye. For |
| 658 |
> |
when the strength of the dipole is increased above $16$ Debye. For |
| 659 |
|
rippled bilayers, there is less free volume available between the head |
| 660 |
|
groups. Therefore increasing dipolar strength weakly influences the |
| 661 |
|
structure of the membrane. However, the increase in the body $P_2$ |
| 712 |
|
molecular width ratio ($\sigma_h / d$).\label{fig:tP2}} |
| 713 |
|
\end{figure} |
| 714 |
|
|
| 715 |
< |
\section{Discussion} |
| 716 |
< |
\label{sec:discussion} |
| 715 |
> |
Fig. \ref{fig:phaseDiagram} shows a phase diagram for the model as a |
| 716 |
> |
function of the head group / molecular width ratio ($\sigma_h / d$) |
| 717 |
> |
and the strength of the head group dipole moment ($\mu$). Note that |
| 718 |
> |
the specific form of the bilayer phase is governed almost entirely by |
| 719 |
> |
the head group / molecular width ratio, while the strength of the |
| 720 |
> |
dipolar interactions between the head groups governs the stability of |
| 721 |
> |
the bilayer phase. Weaker dipoles result in unstable bilayer phases, |
| 722 |
> |
while extremely strong dipoles can shift the equilibrium to an |
| 723 |
> |
inverted micelle phase when the head groups are small. Temperature |
| 724 |
> |
has little effect on the actual bilayer phase observed, although higher |
| 725 |
> |
temperatures can cause the unstable region to grow into the higher |
| 726 |
> |
dipole region of this diagram. |
| 727 |
|
|
| 728 |
< |
The ripple phases have been observed in our molecular dynamic |
| 729 |
< |
simulations using a simple molecular lipid model. The lipid model |
| 730 |
< |
consists of an anisotropic interacting dipolar head group and an |
| 731 |
< |
ellipsoid shape tail. According to our simulations, the explanation of |
| 732 |
< |
the formation for the ripples are originated in the size mismatch |
| 733 |
< |
between the head groups and the tails. The ripple phases are only |
| 734 |
< |
observed in the studies using larger head group lipid models. However, |
| 735 |
< |
there is a mismatch betweent the size of the head groups and the size |
| 671 |
< |
of the tails in the simulations of the flat surface. This indicates |
| 672 |
< |
the competition between the anisotropic dipolar interaction and the |
| 673 |
< |
packing of the tails also plays a major role for formation of the |
| 674 |
< |
ripple phase. The larger head groups provide more free volume for the |
| 675 |
< |
tails, while these hydrophobic ellipsoids trying to be close to each |
| 676 |
< |
other, this gives the origin of the spontanous curvature of the |
| 677 |
< |
surface, which is believed as the beginning of the ripple phases. The |
| 678 |
< |
lager head groups cause the spontanous curvature inward for both of |
| 679 |
< |
leaves of the bilayer. This results in a steric strain when the tails |
| 680 |
< |
of two leaves too close to each other. The membrane has to be broken |
| 681 |
< |
to release this strain. There are two ways to arrange these broken |
| 682 |
< |
curvatures: symmetric and asymmetric ripples. Both of the ripple |
| 683 |
< |
phases have been observed in our studies. The difference between these |
| 684 |
< |
two ripples is that the bilayer is continuum in the symmetric ripple |
| 685 |
< |
phase and is disrupt in the asymmetric ripple phase. |
| 728 |
> |
\begin{figure}[htb] |
| 729 |
> |
\centering |
| 730 |
> |
\includegraphics[width=\linewidth]{phaseDiagram} |
| 731 |
> |
\caption{Phase diagram for the simple molecular model as a function |
| 732 |
> |
of the head group / molecular width ratio ($\sigma_h / d$) and the |
| 733 |
> |
strength of the head group dipole moment |
| 734 |
> |
($\mu$).\label{fig:phaseDiagram}} |
| 735 |
> |
\end{figure} |
| 736 |
|
|
| 737 |
< |
Dipolar head groups are the key elements for the maintaining of the |
| 738 |
< |
bilayer structure. The lipids are solvated in water when lowering the |
| 739 |
< |
the strength of the dipole on the head groups. The long range |
| 740 |
< |
orientational ordering of the dipoles can be achieved by forming the |
| 741 |
< |
ripples, although the dipoles are likely to form head-to-tail |
| 742 |
< |
configurations even in flat surface, the frustration prevents the |
| 743 |
< |
formation of the long range orientational ordering for dipoles. The |
| 744 |
< |
corrugation of the surface breaks the frustration and stablizes the |
| 745 |
< |
long range oreintational ordering for the dipoles in the head groups |
| 696 |
< |
of the lipid molecules. Many rows of the head-to-tail dipoles are |
| 697 |
< |
parallel to each other and adopt the antiferroelectric state as a |
| 698 |
< |
whole. This is the first time the organization of the head groups in |
| 699 |
< |
ripple phases of the lipid bilayer has been addressed. |
| 737 |
> |
We have computed translational diffusion coefficients for lipid |
| 738 |
> |
molecules from the mean square displacement, |
| 739 |
> |
\begin{eqnarray} |
| 740 |
> |
\langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle \\ \\ |
| 741 |
> |
& = & 6Dt |
| 742 |
> |
\end{eqnarray} |
| 743 |
> |
of the lipid bodies. The values of the translational diffusion |
| 744 |
> |
coefficient for different head-to-tail size ratio are shown in table |
| 745 |
> |
\ref{tab:relaxation}. |
| 746 |
|
|
| 747 |
< |
The most important prediction we can make using the results from this |
| 748 |
< |
simple model is that if dipolar ordering is driving the surface |
| 749 |
< |
corrugation, the wave vectors for the ripples should always found to |
| 750 |
< |
be {\it perpendicular} to the dipole director axis. This prediction |
| 751 |
< |
should suggest experimental designs which test whether this is really |
| 752 |
< |
true in the phosphatidylcholine $P_{\beta'}$ phases. The dipole |
| 753 |
< |
director axis should also be easily computable for the all-atom and |
| 754 |
< |
coarse-grained simulations that have been published in the literature. |
| 747 |
> |
We have also computed orientational diffusion constants for the head |
| 748 |
> |
groups from the relaxation of the second-order Legendre polynomial |
| 749 |
> |
correlation function, |
| 750 |
> |
\begin{eqnarray} |
| 751 |
> |
C_{\ell}(t) & = & \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 752 |
> |
\mu}_{i}(0) \right) \rangle \\ \\ |
| 753 |
> |
& \approx & e^{-\ell(\ell + 1) \theta t}, |
| 754 |
> |
\end{eqnarray} |
| 755 |
> |
of the head group dipoles. In this last line, we have used a simple |
| 756 |
> |
``Debye''-like model for the relaxation of the correlation function, |
| 757 |
> |
specifically in the case when $\ell = 2$. The computed orientational |
| 758 |
> |
diffusion constants are given in table \ref{tab:relaxation}. The |
| 759 |
> |
notable feature we observe is that the orientational diffusion |
| 760 |
> |
constant for the head group exhibits an order of magnitude decrease |
| 761 |
> |
upon entering the rippled phase. Our orientational correlation times |
| 762 |
> |
are substantially in excess of those provided by... |
| 763 |
|
|
| 764 |
+ |
|
| 765 |
+ |
\begin{table*} |
| 766 |
+ |
\begin{minipage}{\linewidth} |
| 767 |
+ |
\begin{center} |
| 768 |
+ |
\caption{Rotational diffusion constants for the head groups |
| 769 |
+ |
($\theta_h$) and molecular bodies ($\theta_b$) as well as the |
| 770 |
+ |
translational diffusion coefficients for the molecule as a function of |
| 771 |
+ |
the head-to-body width ratio. The orientational mobility of the head |
| 772 |
+ |
groups experiences an {\it order of magnitude decrease} upon entering |
| 773 |
+ |
the rippled phase, which suggests that the rippling is tied to a |
| 774 |
+ |
freezing out of head group orientational freedom. Uncertainties in |
| 775 |
+ |
the last digit are indicated by the values in parentheses.} |
| 776 |
+ |
\begin{tabular}{lccc} |
| 777 |
+ |
\hline |
| 778 |
+ |
$\sigma_h / d$ & $\theta_h (\mu s^{-1})$ & $\theta_b (1/fs)$ & $D ( |
| 779 |
+ |
\times 10^{-11} m^2 s^{-1}) \\ |
| 780 |
+ |
\hline |
| 781 |
+ |
1.20 & $0.206(1) $ & $0.0175(5) $ & $0.43(1)$ \\ |
| 782 |
+ |
1.28 & $0.179(2) $ & $0.055(2) $ & $5.91(3)$ \\ |
| 783 |
+ |
1.35 & $0.025(1) $ & $0.195(3) $ & $3.42(1)$ \\ |
| 784 |
+ |
1.41 & $0.023(1) $ & $0.024(3) $ & $7.16(1)$ \\ |
| 785 |
+ |
\end{tabular} |
| 786 |
+ |
\label{tab:relaxation} |
| 787 |
+ |
\end{center} |
| 788 |
+ |
\end{minipage} |
| 789 |
+ |
\end{table*} |
| 790 |
+ |
|
| 791 |
+ |
\section{Discussion} |
| 792 |
+ |
\label{sec:discussion} |
| 793 |
+ |
|
| 794 |
+ |
Symmetric and asymmetric ripple phases have been observed to form in |
| 795 |
+ |
our molecular dynamics simulations of a simple molecular-scale lipid |
| 796 |
+ |
model. The lipid model consists of an dipolar head group and an |
| 797 |
+ |
ellipsoidal tail. Within the limits of this model, an explanation for |
| 798 |
+ |
generalized membrane curvature is a simple mismatch in the size of the |
| 799 |
+ |
heads with the width of the molecular bodies. With heads |
| 800 |
+ |
substantially larger than the bodies of the molecule, this curvature |
| 801 |
+ |
should be convex nearly everywhere, a requirement which could be |
| 802 |
+ |
resolved either with micellar or cylindrical phases. |
| 803 |
+ |
|
| 804 |
+ |
The persistence of a {\it bilayer} structure therefore requires either |
| 805 |
+ |
strong attractive forces between the head groups or exclusionary |
| 806 |
+ |
forces from the solvent phase. To have a persistent bilayer structure |
| 807 |
+ |
with the added requirement of convex membrane curvature appears to |
| 808 |
+ |
result in corrugated structures like the ones pictured in |
| 809 |
+ |
Fig. \ref{fig:phaseCartoon}. In each of the sections of these |
| 810 |
+ |
corrugated phases, the local curvature near a most of the head groups |
| 811 |
+ |
is convex. These structures are held together by the extremely strong |
| 812 |
+ |
and directional interactions between the head groups. |
| 813 |
+ |
|
| 814 |
+ |
Dipolar head groups are key for the maintaining the bilayer structures |
| 815 |
+ |
exhibited by this model. The dipoles are likely to form head-to-tail |
| 816 |
+ |
configurations even in flat configurations, but the temperatures are |
| 817 |
+ |
high enough that vortex defects become prevalent in the flat phase. |
| 818 |
+ |
The flat phase we observed therefore appears to be substantially above |
| 819 |
+ |
the Kosterlitz-Thouless transition temperature for a planar system of |
| 820 |
+ |
dipoles with this set of parameters. For this reason, it would be |
| 821 |
+ |
interesting to observe the thermal behavior of the flat phase at |
| 822 |
+ |
substantially lower temperatures. |
| 823 |
+ |
|
| 824 |
+ |
One feature of this model is that an energetically favorable |
| 825 |
+ |
orientational ordering of the dipoles can be achieved by forming |
| 826 |
+ |
ripples. The corrugation of the surface breaks the symmetry of the |
| 827 |
+ |
plane, making vortex defects somewhat more expensive, and stabilizing |
| 828 |
+ |
the long range orientational ordering for the dipoles in the head |
| 829 |
+ |
groups. Most of the rows of the head-to-tail dipoles are parallel to |
| 830 |
+ |
each other and the system adopts a bulk anti-ferroelectric state. We |
| 831 |
+ |
believe that this is the first time the organization of the head |
| 832 |
+ |
groups in ripple phases has been addressed. |
| 833 |
+ |
|
| 834 |
+ |
Although the size-mismatch between the heads and molecular bodies |
| 835 |
+ |
appears to be the primary driving force for surface convexity, the |
| 836 |
+ |
persistence of the bilayer through the use of rippled structures is a |
| 837 |
+ |
function of the strong, attractive interactions between the heads. |
| 838 |
+ |
One important prediction we can make using the results from this |
| 839 |
+ |
simple model is that if the dipole-dipole interaction is the leading |
| 840 |
+ |
contributor to the head group attractions, the wave vectors for the |
| 841 |
+ |
ripples should always be found {\it perpendicular} to the dipole |
| 842 |
+ |
director axis. This echoes the prediction we made earlier for simple |
| 843 |
+ |
elastic dipolar membranes, and may suggest experimental designs which |
| 844 |
+ |
will test whether this is really the case in the phosphatidylcholine |
| 845 |
+ |
$P_{\beta'}$ phases. The dipole director axis should also be easily |
| 846 |
+ |
computable for the all-atom and coarse-grained simulations that have |
| 847 |
+ |
been published in the literature.\cite{deVries05} |
| 848 |
+ |
|
| 849 |
|
Although our model is simple, it exhibits some rich and unexpected |
| 850 |
< |
behaviors. It would clearly be a closer approximation to the reality |
| 851 |
< |
if we allowed greater translational freedom to the dipoles and |
| 852 |
< |
replaced the somewhat artificial lattice packing and the harmonic |
| 853 |
< |
elastic tension with more realistic molecular modeling potentials. |
| 854 |
< |
What we have done is to present a simple model which exhibits bulk |
| 855 |
< |
non-thermal corrugation, and our explanation of this rippling |
| 850 |
> |
behaviors. It would clearly be a closer approximation to reality if |
| 851 |
> |
we allowed bending motions between the dipoles and the molecular |
| 852 |
> |
bodies, and if we replaced the rigid ellipsoids with ball-and-chain |
| 853 |
> |
tails. However, the advantages of this simple model (large system |
| 854 |
> |
sizes, 50 fs time steps) allow us to rapidly explore the phase diagram |
| 855 |
> |
for a wide range of parameters. Our explanation of this rippling |
| 856 |
|
phenomenon will help us design more accurate molecular models for |
| 857 |
< |
corrugated membranes and experiments to test whether rippling is |
| 858 |
< |
dipole-driven or not. |
| 720 |
< |
|
| 857 |
> |
corrugated membranes and experiments to test whether or not |
| 858 |
> |
dipole-dipole interactions exert an influence on membrane rippling. |
| 859 |
|
\newpage |
| 860 |
|
\bibliography{mdripple} |
| 861 |
|
\end{document} |
