| 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 |
< |
stablizes the long range orientational ordering for the dipoles in the |
| 52 |
< |
head groups which then adopt a bulk antiferroelectric state. We |
| 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. |
| 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 have not been settled. |
| 145 |
|
|
| 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 of dipolar |
| 151 |
< |
elastic membranes is the antiferroelectric ordering of the dipoles. |
| 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 similiar |
| 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 |
| 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 headgroups. |
| 399 |
> |
Debye dipole moment of the PC head groups. |
| 400 |
|
|
| 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 |
| 416 |
|
the solvent were carried out at constant pressure ($P=1$ atm) with |
| 417 |
|
$3$D anisotropic coupling, and constant surface tension |
| 418 |
|
($\gamma=0.015$ N/m). Given the absence of fast degrees of freedom in |
| 419 |
< |
this model, a timestep of $50$ fs was utilized with excellent energy |
| 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 |
| 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. The |
| 433 |
< |
mean spacing between the head groups is XXX \AA, and the mean |
| 434 |
< |
area per lipid in this phase is \AA$^2$. This corresponds |
| 435 |
< |
reasonably well to a bilayer of DPPC.\cite{XXX} |
| 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 |
| 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 similiar to the structures observed by both de Vries {\it et al.} |
| 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 $\sigma_h = 1.35 d$. |
| 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 & flat & N/A & 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} |
| 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). |
| 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 $P_2$ |
| 770 |
|
One feature of this model is that an energetically favorable |
| 771 |
|
orientational ordering of the dipoles can be achieved by forming |
| 772 |
|
ripples. The corrugation of the surface breaks the symmetry of the |
| 773 |
< |
plane, making vortex defects somewhat more expensive, and stablizing |
| 773 |
> |
plane, making vortex defects somewhat more expensive, and stabilizing |
| 774 |
|
the long range orientational ordering for the dipoles in the head |
| 775 |
|
groups. Most of the rows of the head-to-tail dipoles are parallel to |
| 776 |
< |
each other and the system adopts a bulk antiferroelectric state. We |
| 776 |
> |
each other and the system adopts a bulk anti-ferroelectric state. We |
| 777 |
|
believe that this is the first time the organization of the head |
| 778 |
|
groups in ripple phases has been addressed. |
| 779 |
|
|
| 797 |
|
we allowed bending motions between the dipoles and the molecular |
| 798 |
|
bodies, and if we replaced the rigid ellipsoids with ball-and-chain |
| 799 |
|
tails. However, the advantages of this simple model (large system |
| 800 |
< |
sizes, 50 fs timesteps) allow us to rapidly explore the phase diagram |
| 800 |
> |
sizes, 50 fs time steps) allow us to rapidly explore the phase diagram |
| 801 |
|
for a wide range of parameters. Our explanation of this rippling |
| 802 |
|
phenomenon will help us design more accurate molecular models for |
| 803 |
|
corrugated membranes and experiments to test whether or not |
