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\bibliographystyle{achemso} |
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|
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\title{Dipolar Ordering in Molecular-scale Models of the Ripple Phase |
28 |
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in Lipid Membranes} |
27 |
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\title{Dipolar ordering in the ripple phases of molecular-scale models |
28 |
> |
of lipid membranes} |
29 |
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\author{Xiuquan Sun and J. Daniel Gezelter \\ |
30 |
|
Department of Chemistry and Biochemistry,\\ |
31 |
|
University of Notre Dame, \\ |
38 |
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\maketitle |
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|
40 |
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\begin{abstract} |
41 |
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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} |
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|
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%\maketitle |
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+ |
\newpage |
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|
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\section{Introduction} |
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\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 |
746 |
< |
of the lipid molecules. Many rows of the head-to-tail dipoles are |
747 |
< |
parallel to each other and adopt the antiferroelectric state as a |
748 |
< |
whole. This is the first time the organization of the head groups in |
749 |
< |
ripple phases of the lipid bilayer has been addressed. |
737 |
> |
We have computed translational diffusion constants for lipid molecules |
738 |
> |
from the mean-square displacement, |
739 |
> |
\begin{equation} |
740 |
> |
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
741 |
> |
\end{equation} |
742 |
> |
of the lipid bodies. Translational diffusion constants for the |
743 |
> |
different head-to-tail size ratios (all at 300 K) are shown in table |
744 |
> |
\ref{tab:relaxation}. We have also computed orientational diffusion |
745 |
> |
constants for the head groups from the relaxation of the second-order |
746 |
> |
Legendre polynomial correlation function, |
747 |
> |
\begin{eqnarray} |
748 |
> |
C_{\ell}(t) & = & \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
749 |
> |
\mu}_{i}(0) \right) \rangle \\ \\ |
750 |
> |
& \approx & e^{-\ell(\ell + 1) \theta t}, |
751 |
> |
\end{eqnarray} |
752 |
> |
of the head group dipoles. In this last line, we have assumed a |
753 |
> |
simple Debye-like model for the relaxation of the correlation |
754 |
> |
function, specifically in the case when $\ell = 2$. The computed |
755 |
> |
orientational diffusion constants are given in table |
756 |
> |
\ref{tab:relaxation}. We observe that the head group orientational diffusion |
757 |
> |
constant exhibits an order of magnitude decrease upon entering the |
758 |
> |
rippled phase. |
759 |
|
|
760 |
< |
The most important prediction we can make using the results from this |
761 |
< |
simple model is that if dipolar ordering is driving the surface |
762 |
< |
corrugation, the wave vectors for the ripples should always found to |
763 |
< |
be {\it perpendicular} to the dipole director axis. This prediction |
764 |
< |
should suggest experimental designs which test whether this is really |
765 |
< |
true in the phosphatidylcholine $P_{\beta'}$ phases. The dipole |
766 |
< |
director axis should also be easily computable for the all-atom and |
767 |
< |
coarse-grained simulations that have been published in the literature. |
760 |
> |
Sparrman and Westlund used $T_1$ and $T_2$ measurements in analyzing |
761 |
> |
NMR lineshapes for gel, fluid, and ripple phases and obtained |
762 |
> |
estimates of a correlation time for water translational diffusion |
763 |
> |
($\tau_c$) of 20 ns.\cite{Sparrman2003} This correlation time |
764 |
> |
corresponds to water bound to small regions of the lipid membrane. |
765 |
> |
Sparrman and Westlund further assume that the lipids can explore only |
766 |
> |
a single period of the ripple (essentially moving in an almost |
767 |
> |
one-dimensional path to do so), and the correlation time can therefore |
768 |
> |
be used to estimate a value for the translational diffusion constant |
769 |
> |
of $2.25 \times 10^{-11} m^2 s^{-1}$. The translational diffusion |
770 |
> |
constants we obtain are in reasonable agreement with this |
771 |
> |
experimentally determined value. |
772 |
|
|
773 |
+ |
Our orientational correlation times are substantially in excess of |
774 |
+ |
those provided by. |
775 |
+ |
|
776 |
+ |
|
777 |
+ |
\begin{table*} |
778 |
+ |
\begin{minipage}{\linewidth} |
779 |
+ |
\begin{center} |
780 |
+ |
\caption{Rotational diffusion constants for the head groups |
781 |
+ |
($\theta_h$) and molecular bodies ($\theta_b$) as well as the |
782 |
+ |
translational diffusion coefficients for the molecule as a function of |
783 |
+ |
the head-to-body width ratio. The orientational mobility of the head |
784 |
+ |
groups experiences an {\it order of magnitude decrease} upon entering |
785 |
+ |
the rippled phase, which suggests that the rippling is tied to a |
786 |
+ |
freezing out of head group orientational freedom. Uncertainties in |
787 |
+ |
the last digit are indicated by the values in parentheses.} |
788 |
+ |
\begin{tabular}{lccc} |
789 |
+ |
\hline |
790 |
+ |
$\sigma_h / d$ & $\theta_h (\mu s^{-1})$ & $\theta_b (1/fs)$ & $D ( |
791 |
+ |
\times 10^{-11} m^2 s^{-1})$ \\ |
792 |
+ |
\hline |
793 |
+ |
1.20 & $0.206(1) $ & $0.0175(5) $ & $0.43(1)$ \\ |
794 |
+ |
1.28 & $0.179(2) $ & $0.055(2) $ & $5.91(3)$ \\ |
795 |
+ |
1.35 & $0.025(1) $ & $0.195(3) $ & $3.42(1)$ \\ |
796 |
+ |
1.41 & $0.023(1) $ & $0.024(3) $ & $7.16(1)$ \\ |
797 |
+ |
\end{tabular} |
798 |
+ |
\label{tab:relaxation} |
799 |
+ |
\end{center} |
800 |
+ |
\end{minipage} |
801 |
+ |
\end{table*} |
802 |
+ |
|
803 |
+ |
\section{Discussion} |
804 |
+ |
\label{sec:discussion} |
805 |
+ |
|
806 |
+ |
Symmetric and asymmetric ripple phases have been observed to form in |
807 |
+ |
our molecular dynamics simulations of a simple molecular-scale lipid |
808 |
+ |
model. The lipid model consists of an dipolar head group and an |
809 |
+ |
ellipsoidal tail. Within the limits of this model, an explanation for |
810 |
+ |
generalized membrane curvature is a simple mismatch in the size of the |
811 |
+ |
heads with the width of the molecular bodies. With heads |
812 |
+ |
substantially larger than the bodies of the molecule, this curvature |
813 |
+ |
should be convex nearly everywhere, a requirement which could be |
814 |
+ |
resolved either with micellar or cylindrical phases. |
815 |
+ |
|
816 |
+ |
The persistence of a {\it bilayer} structure therefore requires either |
817 |
+ |
strong attractive forces between the head groups or exclusionary |
818 |
+ |
forces from the solvent phase. To have a persistent bilayer structure |
819 |
+ |
with the added requirement of convex membrane curvature appears to |
820 |
+ |
result in corrugated structures like the ones pictured in |
821 |
+ |
Fig. \ref{fig:phaseCartoon}. In each of the sections of these |
822 |
+ |
corrugated phases, the local curvature near a most of the head groups |
823 |
+ |
is convex. These structures are held together by the extremely strong |
824 |
+ |
and directional interactions between the head groups. |
825 |
+ |
|
826 |
+ |
Dipolar head groups are key for the maintaining the bilayer structures |
827 |
+ |
exhibited by this model. The dipoles are likely to form head-to-tail |
828 |
+ |
configurations even in flat configurations, but the temperatures are |
829 |
+ |
high enough that vortex defects become prevalent in the flat phase. |
830 |
+ |
The flat phase we observed therefore appears to be substantially above |
831 |
+ |
the Kosterlitz-Thouless transition temperature for a planar system of |
832 |
+ |
dipoles with this set of parameters. For this reason, it would be |
833 |
+ |
interesting to observe the thermal behavior of the flat phase at |
834 |
+ |
substantially lower temperatures. |
835 |
+ |
|
836 |
+ |
One feature of this model is that an energetically favorable |
837 |
+ |
orientational ordering of the dipoles can be achieved by forming |
838 |
+ |
ripples. The corrugation of the surface breaks the symmetry of the |
839 |
+ |
plane, making vortex defects somewhat more expensive, and stabilizing |
840 |
+ |
the long range orientational ordering for the dipoles in the head |
841 |
+ |
groups. Most of the rows of the head-to-tail dipoles are parallel to |
842 |
+ |
each other and the system adopts a bulk anti-ferroelectric state. We |
843 |
+ |
believe that this is the first time the organization of the head |
844 |
+ |
groups in ripple phases has been addressed. |
845 |
+ |
|
846 |
+ |
Although the size-mismatch between the heads and molecular bodies |
847 |
+ |
appears to be the primary driving force for surface convexity, the |
848 |
+ |
persistence of the bilayer through the use of rippled structures is a |
849 |
+ |
function of the strong, attractive interactions between the heads. |
850 |
+ |
One important prediction we can make using the results from this |
851 |
+ |
simple model is that if the dipole-dipole interaction is the leading |
852 |
+ |
contributor to the head group attractions, the wave vectors for the |
853 |
+ |
ripples should always be found {\it perpendicular} to the dipole |
854 |
+ |
director axis. This echoes the prediction we made earlier for simple |
855 |
+ |
elastic dipolar membranes, and may suggest experimental designs which |
856 |
+ |
will test whether this is really the case in the phosphatidylcholine |
857 |
+ |
$P_{\beta'}$ phases. The dipole director axis should also be easily |
858 |
+ |
computable for the all-atom and coarse-grained simulations that have |
859 |
+ |
been published in the literature.\cite{deVries05} |
860 |
+ |
|
861 |
|
Although our model is simple, it exhibits some rich and unexpected |
862 |
< |
behaviors. It would clearly be a closer approximation to the reality |
863 |
< |
if we allowed greater translational freedom to the dipoles and |
864 |
< |
replaced the somewhat artificial lattice packing and the harmonic |
865 |
< |
elastic tension with more realistic molecular modeling potentials. |
866 |
< |
What we have done is to present a simple model which exhibits bulk |
867 |
< |
non-thermal corrugation, and our explanation of this rippling |
862 |
> |
behaviors. It would clearly be a closer approximation to reality if |
863 |
> |
we allowed bending motions between the dipoles and the molecular |
864 |
> |
bodies, and if we replaced the rigid ellipsoids with ball-and-chain |
865 |
> |
tails. However, the advantages of this simple model (large system |
866 |
> |
sizes, 50 fs time steps) allow us to rapidly explore the phase diagram |
867 |
> |
for a wide range of parameters. Our explanation of this rippling |
868 |
|
phenomenon will help us design more accurate molecular models for |
869 |
< |
corrugated membranes and experiments to test whether rippling is |
870 |
< |
dipole-driven or not. |
720 |
< |
|
869 |
> |
corrugated membranes and experiments to test whether or not |
870 |
> |
dipole-dipole interactions exert an influence on membrane rippling. |
871 |
|
\newpage |
872 |
|
\bibliography{mdripple} |
873 |
|
\end{document} |