| 271 |
|
\epsilon^{r} & < & 1 |
| 272 |
|
\end{array} |
| 273 |
|
\end{equation} |
| 274 |
+ |
A sketch of the various structural elements of our molecular-scale |
| 275 |
+ |
lipid / solvent model is shown in figure \ref{fig:lipidModel}. The |
| 276 |
+ |
actual parameters used in our simulations are given in table |
| 277 |
+ |
\ref{tab:parameters}. |
| 278 |
|
|
| 279 |
|
\begin{figure}[htb] |
| 280 |
|
\centering |
| 296 |
|
varied between $1.20 d$ and $1.41 d$. The head groups interact with |
| 297 |
|
each other using a combination of Lennard-Jones, |
| 298 |
|
\begin{eqnarray*} |
| 299 |
< |
V_{ij} = 4\epsilon \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
| 299 |
> |
V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
| 300 |
|
\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
| 301 |
|
\end{eqnarray*} |
| 302 |
|
and dipole-dipole, |
| 303 |
|
\begin{eqnarray*} |
| 304 |
< |
V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
| 304 |
> |
V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 305 |
> |
\hat{r}}_{ij})) = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
| 306 |
|
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
| 307 |
|
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
| 308 |
|
\end{eqnarray*} |
| 314 |
|
For the interaction between nonequivalent uniaxial ellipsoids (in this |
| 315 |
|
case, between spheres and ellipsoids), the spheres are treated as |
| 316 |
|
ellipsoids with an aspect ratio of 1 ($d = l$) and with an well depth |
| 317 |
< |
ratio of 1 ($\epsilon^e = \epsilon^s$). The form of the Gay-Berne |
| 318 |
< |
potential we are using was generalized by Cleaver {\it et al.} and is |
| 319 |
< |
appropriate for dissimilar uniaxial ellipsoids.\cite{Cleaver96} |
| 317 |
> |
ratio ($\epsilon^r$) of 1 ($\epsilon^e = \epsilon^s$). The form of |
| 318 |
> |
the Gay-Berne potential we are using was generalized by Cleaver {\it |
| 319 |
> |
et al.} and is appropriate for dissimilar uniaxial |
| 320 |
> |
ellipsoids.\cite{Cleaver96} |
| 321 |
|
|
| 322 |
|
The solvent model in our simulations is identical to one used by |
| 323 |
|
Marrink {\it et al.} in their dissipative particle dynamics (DPD) |
| 335 |
|
\hline |
| 336 |
|
& & Head & Chain & Solvent \\ |
| 337 |
|
\hline |
| 338 |
< |
$d$ (\AA) & & varied & 4.6 & 4.7 \\ |
| 339 |
< |
$l$ (\AA) & & 1 & 3 & 1 \\ |
| 338 |
> |
$d$ (\AA) & & varied & 4.6 & 4.7 \\ |
| 339 |
> |
$l$ (\AA) & & $= d$ & 13.8 & 4.7 \\ |
| 340 |
|
$\epsilon^s$ (kcal/mol) & & 0.185 & 1.0 & 0.8 \\ |
| 341 |
|
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 \\ |
| 342 |
< |
$m$ (amu) & & 196 & 760 & 72.06112 \\ |
| 342 |
> |
$m$ (amu) & & 196 & 760 & 72.06 \\ |
| 343 |
|
$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
| 344 |
|
\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
| 345 |
|
\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
| 353 |
|
|
| 354 |
|
A switching function has been applied to all potentials to smoothly |
| 355 |
|
turn off the interactions between a range of $22$ and $25$ \AA. |
| 356 |
+ |
|
| 357 |
+ |
The parameters that were systematically varied in this study were the |
| 358 |
+ |
size of the head group ($\sigma_h$), the strength of the dipole moment |
| 359 |
+ |
($\mu$), and the temperature of the system. Values for $\sigma_h$ |
| 360 |
+ |
ranged from 5.5 \AA\ to 6.5 \AA\ . If the width of the tails is |
| 361 |
+ |
taken to be the unit of length, these head groups correspond to a |
| 362 |
+ |
range from $1.2 d$ to $1.41 d$. Since the solvent beads are nearly |
| 363 |
+ |
identical in diameter to the tail ellipsoids, all distances that |
| 364 |
+ |
follow will be measured relative to this unit of distance. |
| 365 |
|
|
| 366 |
|
\section{Experimental Methodology} |
| 367 |
|
\label{sec:experiment} |
| 368 |
|
|
| 369 |
|
To create unbiased bilayers, all simulations were started from two |
| 370 |
< |
perfectly flat monolayers separated by a 20 \AA\ gap between the |
| 370 |
> |
perfectly flat monolayers separated by a 26 \AA\ gap between the |
| 371 |
|
molecular bodies of the upper and lower leaves. The separated |
| 372 |
|
monolayers were evolved in a vaccum with $x-y$ anisotropic pressure |
| 373 |
|
coupling. The length of $z$ axis of the simulations was fixed and a |
| 374 |
|
constant surface tension was applied to enable real fluctuations of |
| 375 |
< |
the bilayer. Periodic boundaries were used, and $480-720$ lipid |
| 376 |
< |
molecules were present in the simulations depending on the size of the |
| 377 |
< |
head beads. The two monolayers spontaneously collapse into bilayer |
| 378 |
< |
structures within 100 ps, and following this collapse, all systems |
| 379 |
< |
were equlibrated for $100$ ns at $300$ K. |
| 375 |
> |
the bilayer. Periodic boundary conditions were used, and $480-720$ |
| 376 |
> |
lipid molecules were present in the simulations, depending on the size |
| 377 |
> |
of the head beads. In all cases, the two monolayers spontaneously |
| 378 |
> |
collapsed into bilayer structures within 100 ps. Following this |
| 379 |
> |
collapse, all systems were equlibrated for $100$ ns at $300$ K. |
| 380 |
|
|
| 381 |
< |
The resulting structures were then solvated at a ratio of $6$ DPD |
| 381 |
> |
The resulting bilayer structures were then solvated at a ratio of $6$ |
| 382 |
|
solvent beads (24 water molecules) per lipid. These configurations |
| 383 |
< |
were then equilibrated for another $30$ ns. All simulations with |
| 384 |
< |
solvent were carried out at constant pressure ($P=1$ atm) by $3$D |
| 385 |
< |
anisotropic coupling, and constant surface tension ($\gamma=0.015$ |
| 386 |
< |
UNIT). Given the absence of fast degrees of freedom in this model, a |
| 387 |
< |
timestep of $50$ fs was utilized. Data collection for structural |
| 388 |
< |
properties of the bilayers was carried out during a final 5 ns run |
| 389 |
< |
following the solvent equilibration. All simulations were performed |
| 390 |
< |
using the OOPSE molecular modeling program.\cite{Meineke05} |
| 383 |
> |
were then equilibrated for another $30$ ns. All simulations utilizing |
| 384 |
> |
the solvent were carried out at constant pressure ($P=1$ atm) with |
| 385 |
> |
$3$D anisotropic coupling, and constant surface tension |
| 386 |
> |
($\gamma=0.015$ UNIT). Given the absence of fast degrees of freedom in |
| 387 |
> |
this model, a timestep of $50$ fs was utilized with excellent energy |
| 388 |
> |
conservation. Data collection for structural properties of the |
| 389 |
> |
bilayers was carried out during a final 5 ns run following the solvent |
| 390 |
> |
equilibration. All simulations were performed using the OOPSE |
| 391 |
> |
molecular modeling program.\cite{Meineke05} |
| 392 |
|
|
| 393 |
|
\section{Results} |
| 394 |
|
\label{sec:results} |
| 395 |
|
|
| 396 |
|
Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
| 397 |
< |
more corrugated increasing size of the head groups. The surface is |
| 398 |
< |
nearly flat when $\sigma_h=1.20\sigma_0$. With |
| 399 |
< |
$\sigma_h=1.28\sigma_0$, although the surface is still flat, the |
| 400 |
< |
bilayer starts to splay inward; the upper leaf of the bilayer is |
| 401 |
< |
connected to the lower leaf with an interdigitated line defect. Two |
| 402 |
< |
periodicities with $100$ \AA\ width were observed in the |
| 403 |
< |
simulation. This structure is very similiar to the structure observed |
| 404 |
< |
by de Vries and Lenz {\it et al.}. The same basic structure is also |
| 405 |
< |
observed when $\sigma_h=1.41\sigma_0$, but the wavelength of the |
| 406 |
< |
surface corrugations depends sensitively on the size of the ``head'' |
| 407 |
< |
beads. From the undulation spectrum, the corrugation is clearly |
| 392 |
< |
non-thermal. |
| 397 |
> |
more corrugated with increasing size of the head groups. The surface |
| 398 |
> |
is nearly flat when $\sigma_h=1.20 d$. With $\sigma_h=1.28 d$, |
| 399 |
> |
although the surface is still flat, the bilayer starts to splay |
| 400 |
> |
inward; the upper leaf of the bilayer is connected to the lower leaf |
| 401 |
> |
with an interdigitated line defect. Two periodicities with $100$ \AA\ |
| 402 |
> |
wavelengths were observed in the simulation. This structure is very |
| 403 |
> |
similiar to the structure observed by de Vries and Lenz {\it et |
| 404 |
> |
al.}. The same basic structure is also observed when $\sigma_h=1.41 |
| 405 |
> |
d$, but the wavelength of the surface corrugations depends sensitively |
| 406 |
> |
on the size of the ``head'' beads. From the undulation spectrum, the |
| 407 |
> |
corrugation is clearly non-thermal. |
| 408 |
|
\begin{figure}[htb] |
| 409 |
|
\centering |
| 410 |
|
\includegraphics[width=4in]{phaseCartoon} |
| 412 |
|
our simulations.\label{fig:phaseCartoon}} |
| 413 |
|
\end{figure} |
| 414 |
|
|
| 415 |
< |
When $\sigma_h=1.35\sigma_0$, we observed another corrugated surface |
| 416 |
< |
morphology. This structure is different from the asymmetric rippled |
| 415 |
> |
When $\sigma_h=1.35 d$, we observed another corrugated surface |
| 416 |
> |
morphology. This structure is different from the asymmetric rippled |
| 417 |
|
surface; there is no interdigitation between the upper and lower |
| 418 |
|
leaves of the bilayer. Each leaf of the bilayer is broken into several |
| 419 |
|
hemicylinderical sections, and opposite leaves are fitted together |
| 420 |
|
much like roof tiles. Unlike the surface in which the upper |
| 421 |
|
hemicylinder is always interdigitated on the leading or trailing edge |
| 422 |
< |
of lower hemicylinder, the symmetric ripple has no prefered direction. |
| 423 |
< |
The corresponding cartoons are shown in Figure |
| 422 |
> |
of lower hemicylinder, this ``symmetric'' ripple has no prefered |
| 423 |
> |
direction. The corresponding structures are shown in Figure |
| 424 |
|
\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
| 425 |
< |
different phases. Figure \ref{fig:phaseCartoon} (a) is the flat phase, |
| 426 |
< |
(b) is the asymmetric ripple phase corresponding to the lipid |
| 427 |
< |
organization when $\sigma_h=1.28\sigma_0$ and $\sigma_h=1.41\sigma_0$, |
| 428 |
< |
and (c) is the symmetric ripple phase observed when |
| 429 |
< |
$\sigma_h=1.35\sigma_0$. In symmetric ripple phase, the bilayer is |
| 430 |
< |
continuous everywhere on the whole membrane, however, in asymmetric |
| 431 |
< |
ripple phase, the bilayer is intermittent domains connected by thin |
| 432 |
< |
interdigitated monolayer which consists of upper and lower leaves of |
| 418 |
< |
the bilayer. |
| 425 |
> |
different phases. The top panel in figure \ref{fig:phaseCartoon} is |
| 426 |
> |
the flat phase, the middle panel shows the asymmetric ripple phase |
| 427 |
> |
corresponding to $\sigma_h = 1.41 d$ and the lower panel shows the |
| 428 |
> |
symmetric ripple phase observed when $\sigma_h=1.35 d$. In the |
| 429 |
> |
symmetric ripple, the bilayer is continuous over the whole membrane, |
| 430 |
> |
however, in asymmetric ripple phase, the bilayer domains are connected |
| 431 |
> |
by thin interdigitated monolayers that share molecules between the |
| 432 |
> |
upper and lower leaves. |
| 433 |
|
\begin{table*} |
| 434 |
|
\begin{minipage}{\linewidth} |
| 435 |
|
\begin{center} |
| 436 |
< |
\caption{} |
| 436 |
> |
\caption{Phases, ripple wavelengths and amplitudes observed as a |
| 437 |
> |
function of the ratio between the head beads and the diameters of the |
| 438 |
> |
tails. All lengths are normalized to the diameter of the tail |
| 439 |
> |
ellipsoids.} |
| 440 |
|
\begin{tabular}{lccc} |
| 441 |
|
\hline |
| 442 |
< |
$\sigma_h/\sigma_0$ & type of phase & $\lambda/\sigma_0$ & $A/\sigma_0$\\ |
| 442 |
> |
$\sigma_h / d$ & type of phase & $\lambda / d$ & $A / d$\\ |
| 443 |
|
\hline |
| 444 |
|
1.20 & flat & N/A & N/A \\ |
| 445 |
< |
1.28 & asymmetric flat & 21.7 & N/A \\ |
| 445 |
> |
1.28 & asymmetric ripple or flat & 21.7 & N/A \\ |
| 446 |
|
1.35 & symmetric ripple & 17.2 & 2.2 \\ |
| 447 |
|
1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
| 448 |
|
\end{tabular} |
| 451 |
|
\end{minipage} |
| 452 |
|
\end{table*} |
| 453 |
|
|
| 454 |
< |
The membrane structures and the reduced wavelength $\lambda/\sigma_0$, |
| 455 |
< |
reduced amplitude $A/\sigma_0$ of the ripples are summerized in Table |
| 456 |
< |
\ref{tab:property}. The wavelength range is $15~21$ and the amplitude |
| 457 |
< |
is $1.5$ for asymmetric ripple and $2.2$ for symmetric ripple. These |
| 458 |
< |
values are consistent to the experimental results. Note, the |
| 459 |
< |
amplitudes are underestimated without the melted tails in our |
| 460 |
< |
simulations. |
| 454 |
> |
The membrane structures and the reduced wavelength $\lambda / d$, |
| 455 |
> |
reduced amplitude $A / d$ of the ripples are summarized in Table |
| 456 |
> |
\ref{tab:property}. The wavelength range is $15~21$ molecular bodies |
| 457 |
> |
and the amplitude is $1.5$ molecular bodies for asymmetric ripple and |
| 458 |
> |
$2.2$ for symmetric ripple. These values are consistent to the |
| 459 |
> |
experimental results. Note, that given the lack of structural freedom |
| 460 |
> |
in the tails of our model lipids, the amplitudes observed from these |
| 461 |
> |
simulations are likely to underestimate of the true amplitudes. |
| 462 |
|
|
| 463 |
|
\begin{figure}[htb] |
| 464 |
|
\centering |
| 473 |
|
different direction from the upper leaf.\label{fig:topView}} |
| 474 |
|
\end{figure} |
| 475 |
|
|
| 476 |
< |
The $P_2$ order paramters (for molecular bodies and head group |
| 477 |
< |
dipoles) have been calculated to clarify the ordering in these phases |
| 478 |
< |
quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$ |
| 479 |
< |
implies orientational randomization. Figure \ref{fig:rP2} shows the |
| 480 |
< |
$P_2$ order paramter of the dipoles on head group rising with |
| 481 |
< |
increasing head group size. When the heads of the lipid molecules are |
| 482 |
< |
small, the membrane is flat. The dipolar ordering is essentially |
| 483 |
< |
frustrated on orientational ordering in this circumstance. Figure |
| 484 |
< |
\ref{fig:topView} shows the snapshots of the top view for the flat system |
| 485 |
< |
($\sigma_h=1.20\sigma$) and rippled system |
| 486 |
< |
($\sigma_h=1.41\sigma$). The pointing direction of the dipoles on the |
| 487 |
< |
head groups are represented by two colored half spheres from blue to |
| 470 |
< |
yellow. For flat surfaces, the system obviously shows frustration on |
| 471 |
< |
the dipolar ordering, there are kinks on the edge of defferent |
| 472 |
< |
domains. Another reason is that the lipids can move independently in |
| 473 |
< |
each monolayer, it is not nessasory for the direction of dipoles on |
| 474 |
< |
one leaf is consistant to another layer, which makes total order |
| 475 |
< |
parameter is relatively low. With increasing head group size, the |
| 476 |
< |
surface is corrugated, and dipoles do not move as freely on the |
| 477 |
< |
surface. Therefore, the translational freedom of lipids in one layer |
| 478 |
< |
is dependent upon the position of lipids in another layer, as a |
| 479 |
< |
result, the symmetry of the dipoles on head group in one layer is tied |
| 480 |
< |
to the symmetry in the other layer. Furthermore, as the membrane |
| 481 |
< |
deforms from two to three dimensions due to the corrugation, the |
| 482 |
< |
symmetry of the ordering for the dipoles embedded on each leaf is |
| 483 |
< |
broken. The dipoles then self-assemble in a head-tail configuration, |
| 484 |
< |
and the order parameter increases dramaticaly. However, the total |
| 485 |
< |
polarization of the system is still close to zero. This is strong |
| 486 |
< |
evidence that the corrugated structure is an antiferroelectric |
| 487 |
< |
state. From the snapshot in Figure \ref{}, the dipoles arrange as |
| 488 |
< |
arrays along $Y$ axis and fall into head-to-tail configuration in each |
| 489 |
< |
line, but every $3$ or $4$ lines of dipoles change their direction |
| 490 |
< |
from neighbour lines. The system shows antiferroelectric |
| 491 |
< |
charactoristic as a whole. The orientation of the dipolar is always |
| 492 |
< |
perpendicular to the ripple wave vector. These results are consistent |
| 493 |
< |
with our previous study on dipolar membranes. |
| 476 |
> |
Figure \ref{fig:topView} shows snapshots of bird's-eye views of the |
| 477 |
> |
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.41, 1.35 d$) |
| 478 |
> |
bilayers. The directions of the dipoles on the head groups are |
| 479 |
> |
represented with two colored half spheres: blue (phosphate) and yellow |
| 480 |
> |
(amino). For flat bilayers, the system exhibits signs of |
| 481 |
> |
orientational frustration; some disorder in the dipolar chains is |
| 482 |
> |
evident with kinks visible at the edges between different ordered |
| 483 |
> |
domains. The lipids can also move independently of lipids in the |
| 484 |
> |
opposing leaf, so the ordering of the dipoles on one leaf is not |
| 485 |
> |
necessarily consistant with the ordering on the other. These two |
| 486 |
> |
factors keep the total dipolar order parameter relatively low for the |
| 487 |
> |
flat phases. |
| 488 |
|
|
| 489 |
+ |
With increasing head group size, the surface becomes corrugated, and |
| 490 |
+ |
the dipoles cannot move as freely on the surface. Therefore, the |
| 491 |
+ |
translational freedom of lipids in one layer is dependent upon the |
| 492 |
+ |
position of lipids in the other layer. As a result, the ordering of |
| 493 |
+ |
the dipoles on head groups in one leaf is correlated with the ordering |
| 494 |
+ |
in the other leaf. Furthermore, as the membrane deforms due to the |
| 495 |
+ |
corrugation, the symmetry of the allowed dipolar ordering on each leaf |
| 496 |
+ |
is broken. The dipoles then self-assemble in a head-to-tail |
| 497 |
+ |
configuration, and the dipolar order parameter increases dramatically. |
| 498 |
+ |
However, the total polarization of the system is still close to zero. |
| 499 |
+ |
This is strong evidence that the corrugated structure is an |
| 500 |
+ |
antiferroelectric state. It is also notable that the head-to-tail |
| 501 |
+ |
arrangement of the dipoles is in a direction perpendicular to the wave |
| 502 |
+ |
vector for the surface corrugation. This is a similar finding to what |
| 503 |
+ |
we observed in our earlier work on the elastic dipolar |
| 504 |
+ |
membranes.\cite{Sun07} |
| 505 |
+ |
|
| 506 |
+ |
The $P_2$ order parameters (for both the molecular bodies and the head |
| 507 |
+ |
group dipoles) have been calculated to quantify the ordering in these |
| 508 |
+ |
phases. $P_2 = 1$ implies a perfectly ordered structure, and $P_2 = 0$ |
| 509 |
+ |
implies complete orientational randomization. Figure \ref{fig:rP2} |
| 510 |
+ |
shows the $P_2$ order parameter for the head-group dipoles increasing |
| 511 |
+ |
with increasing head group size. When the heads of the lipid molecules |
| 512 |
+ |
are small, the membrane is nearly flat. The dipolar ordering exhibits |
| 513 |
+ |
frustrated orientational ordering in this circumstance. |
| 514 |
+ |
|
| 515 |
|
The ordering of the tails is essentially opposite to the ordering of |
| 516 |
< |
the dipoles on head group. The $P_2$ order parameter decreases with |
| 517 |
< |
increasing head size. This indicates the surface is more curved with |
| 518 |
< |
larger head groups. When the surface is flat, all tails are pointing |
| 519 |
< |
in the same direction; in this case, all tails are parallel to the |
| 520 |
< |
normal of the surface,(making this structure remindcent of the |
| 521 |
< |
$L_{\beta}$ phase. Increasing the size of the heads, results in |
| 522 |
< |
rapidly decreasing $P_2$ ordering for the molecular bodies. |
| 516 |
> |
the dipoles on head group. The $P_2$ order parameter {\it decreases} |
| 517 |
> |
with increasing head size. This indicates that the surface is more |
| 518 |
> |
curved with larger head / tail size ratios. When the surface is flat, |
| 519 |
> |
all tails are pointing in the same direction (parallel to the normal |
| 520 |
> |
of the surface). This simplified model appears to be exhibiting a |
| 521 |
> |
smectic A fluid phase, similar to the real $L_{\beta}$ phase. We have |
| 522 |
> |
not observed a smectic C gel phase ($L_{\beta'}$) for this model |
| 523 |
> |
system. Increasing the size of the heads, results in rapidly |
| 524 |
> |
decreasing $P_2$ ordering for the molecular bodies. |
| 525 |
|
|
| 526 |
|
\begin{figure}[htb] |
| 527 |
|
\centering |
| 528 |
|
\includegraphics[width=\linewidth]{rP2} |
| 529 |
< |
\caption{The $P_2$ order parameter as a funtion of the ratio of |
| 530 |
< |
$\sigma_h$ to $\sigma_0$.\label{fig:rP2}} |
| 529 |
> |
\caption{The $P_2$ order parameter as a function of the ratio of |
| 530 |
> |
$\sigma_h$ to $d$. \label{fig:rP2}} |
| 531 |
|
\end{figure} |
| 532 |
|
|
| 533 |
|
We studied the effects of the interactions between head groups on the |
