| 101 |
|
\begin{figure} |
| 102 |
|
\centering |
| 103 |
|
\includegraphics[width=\linewidth]{./figures/mdLipidModels.pdf} |
| 104 |
< |
\caption{Three different representations of DPPC lipid molecules, |
| 104 |
> |
\caption[Three different representations of DPPC lipid |
| 105 |
> |
molecules]{Three different representations of DPPC lipid molecules, |
| 106 |
|
including the chemical structure, an atomistic model, and the |
| 107 |
|
head-body ellipsoidal coarse-grained model used in this |
| 108 |
|
work.\label{mdfig:lipidModels}} |
| 244 |
|
\begin{figure} |
| 245 |
|
\centering |
| 246 |
|
\includegraphics[width=\linewidth]{./figures/md2LipidModel.pdf} |
| 247 |
< |
\caption{The parameters defining the behavior of the lipid |
| 248 |
< |
models. $\sigma_h / d$ is the ratio of the head group to body diameter. |
| 249 |
< |
Molecular bodies had a fixed aspect ratio of 3.0. The solvent model |
| 250 |
< |
was a simplified 4-water bead ($\sigma_w \approx d$) that has been |
| 251 |
< |
used in other coarse-grained simulations. The dipolar strength |
| 252 |
< |
(and the temperature and pressure) were the only other parameters that |
| 253 |
< |
were varied systematically.\label{mdfig:lipidModel}} |
| 247 |
> |
\caption[The parameters defining the behavior of the lipid |
| 248 |
> |
models]{The parameters defining the behavior of the lipid |
| 249 |
> |
models. $\sigma_h / d$ is the ratio of the head group to body |
| 250 |
> |
diameter. Molecular bodies had a fixed aspect ratio of 3.0. The |
| 251 |
> |
solvent model was a simplified 4-water bead ($\sigma_w \approx d$) |
| 252 |
> |
that has been used in other coarse-grained simulations. The dipolar |
| 253 |
> |
strength (and the temperature and pressure) were the only other |
| 254 |
> |
parameters that were varied systematically.\label{mdfig:lipidModel}} |
| 255 |
|
\end{figure} |
| 256 |
|
|
| 257 |
|
To take into account the permanent dipolar interactions of the |
| 323 |
|
\begin{table*} |
| 324 |
|
\begin{minipage}{\linewidth} |
| 325 |
|
\begin{center} |
| 326 |
< |
\caption{Potential parameters used for molecular-scale coarse-grained |
| 327 |
< |
lipid simulations} |
| 326 |
> |
\caption{POTENTIAL PARAMETERS USED FOR MOLECULAR SCALE COARSE-GRAINED |
| 327 |
> |
LIPID SIMULATIONS} |
| 328 |
|
\begin{tabular}{llccc} |
| 329 |
|
\hline |
| 330 |
|
& & Head & Chain & Solvent \\ |
| 345 |
|
\end{minipage} |
| 346 |
|
\end{table*} |
| 347 |
|
|
| 348 |
< |
\section{Experimental Methodology} |
| 349 |
< |
\label{mdsec:experiment} |
| 348 |
> |
\section{Simulation Methodology} |
| 349 |
> |
\label{mdsec:simulation} |
| 350 |
|
|
| 351 |
|
The parameters that were systematically varied in this study were the |
| 352 |
|
size of the head group ($\sigma_h$), the strength of the dipole moment |
| 441 |
|
\begin{figure} |
| 442 |
|
\centering |
| 443 |
|
\includegraphics[width=\linewidth]{./figures/mdPhaseCartoon.pdf} |
| 444 |
< |
\caption{The role of the ratio between the head group size and the |
| 445 |
< |
width of the molecular bodies is to increase the local membrane |
| 446 |
< |
curvature. With strong attractive interactions between the head |
| 447 |
< |
groups, this local curvature can be maintained in bilayer structures |
| 448 |
< |
through surface corrugation. Shown above are three phases observed in |
| 449 |
< |
these simulations. With $\sigma_h = 1.20 d$, the bilayer maintains a |
| 450 |
< |
flat topology. For larger heads ($\sigma_h = 1.35 d$) the local |
| 451 |
< |
curvature resolves into a symmetrically rippled phase with little or |
| 452 |
< |
no interdigitation between the upper and lower leaves of the membrane. |
| 453 |
< |
The largest heads studied ($\sigma_h = 1.41 d$) resolve into an |
| 454 |
< |
asymmetric rippled phases with interdigitation between the two |
| 455 |
< |
leaves.\label{mdfig:phaseCartoon}} |
| 444 |
> |
\caption[ three phases observed in the simulations]{The role of the |
| 445 |
> |
ratio between the head group size and the width of the molecular |
| 446 |
> |
bodies is to increase the local membrane curvature. With strong |
| 447 |
> |
attractive interactions between the head groups, this local curvature |
| 448 |
> |
can be maintained in bilayer structures through surface corrugation. |
| 449 |
> |
Shown above are three phases observed in these simulations. With |
| 450 |
> |
$\sigma_h = 1.20 d$, the bilayer maintains a flat topology. For |
| 451 |
> |
larger heads ($\sigma_h = 1.35 d$) the local curvature resolves into a |
| 452 |
> |
symmetrically rippled phase with little or no interdigitation between |
| 453 |
> |
the upper and lower leaves of the membrane. The largest heads studied |
| 454 |
> |
($\sigma_h = 1.41 d$) resolve into an asymmetric rippled phases with |
| 455 |
> |
interdigitation between the two leaves.\label{mdfig:phaseCartoon}} |
| 456 |
|
\end{figure} |
| 457 |
|
|
| 458 |
|
Sample structures for the flat ($\sigma_h = 1.20 d$), symmetric |
| 478 |
|
for PE head groups. |
| 479 |
|
|
| 480 |
|
\begin{table*} |
| 479 |
– |
\begin{minipage}{\linewidth} |
| 481 |
|
\begin{center} |
| 482 |
< |
\caption{Phase, bilayer spacing, area per lipid, ripple wavelength |
| 483 |
< |
and amplitude observed as a function of the ratio between the head |
| 484 |
< |
beads and the diameters of the tails. Ripple wavelengths and |
| 484 |
< |
amplitudes are normalized to the diameter of the tail ellipsoids.} |
| 482 |
> |
\caption{PHASE, BILAYER SPACING, AREA PER LIPID, RIPPLE WAVELENGTH AND |
| 483 |
> |
AMPLITUDE OBSERVED AS A FUNCTION OF THE RATIO BETWEEN THE HEAD BEADS |
| 484 |
> |
AND THE DIAMETERS OF THE TAILS} |
| 485 |
|
\begin{tabular}{lccccc} |
| 486 |
|
\hline |
| 487 |
|
$\sigma_h / d$ & type of phase & bilayer spacing (\AA) & area per |
| 492 |
|
1.35 & symmetric ripple & 42.9 & 51.7 & 17.2 & 2.2 \\ |
| 493 |
|
1.41 & asymmetric ripple & 37.1 & 63.1 & 15.4 & 1.5 \\ |
| 494 |
|
\end{tabular} |
| 495 |
+ |
\begin{minipage}{\linewidth} |
| 496 |
+ |
%\centering |
| 497 |
+ |
\vspace{2mm} |
| 498 |
+ |
Ripple wavelengths and amplitudes are normalized to the diameter of |
| 499 |
+ |
the tail ellipsoids. |
| 500 |
|
\label{mdtab:property} |
| 496 |
– |
\end{center} |
| 501 |
|
\end{minipage} |
| 502 |
+ |
\end{center} |
| 503 |
|
\end{table*} |
| 504 |
|
|
| 505 |
|
The membrane structures and the reduced wavelength $\lambda / d$, |
| 515 |
|
\begin{figure} |
| 516 |
|
\centering |
| 517 |
|
\includegraphics[width=\linewidth]{./figures/mdTopDown.pdf} |
| 518 |
< |
\caption{Top views of the flat (upper), symmetric ripple (middle), |
| 519 |
< |
and asymmetric ripple (lower) phases. Note that the head-group |
| 520 |
< |
dipoles have formed head-to-tail chains in all three of these phases, |
| 521 |
< |
but in the two rippled phases, the dipolar chains are all aligned {\it |
| 522 |
< |
perpendicular} to the direction of the ripple. Note that the flat |
| 523 |
< |
membrane has multiple vortex defects in the dipolar ordering, and the |
| 524 |
< |
ordering on the lower leaf of the bilayer can be in an entirely |
| 525 |
< |
different direction from the upper leaf.\label{mdfig:topView}} |
| 518 |
> |
\caption[Top views of the flat, symmetric ripple, and asymmetric |
| 519 |
> |
ripple phases]{Top views of the flat (upper), symmetric ripple |
| 520 |
> |
(middle), and asymmetric ripple (lower) phases. Note that the |
| 521 |
> |
head-group dipoles have formed head-to-tail chains in all three of |
| 522 |
> |
these phases, but in the two rippled phases, the dipolar chains are |
| 523 |
> |
all aligned {\it perpendicular} to the direction of the ripple. Note |
| 524 |
> |
that the flat membrane has multiple vortex defects in the dipolar |
| 525 |
> |
ordering, and the ordering on the lower leaf of the bilayer can be in |
| 526 |
> |
an entirely different direction from the upper |
| 527 |
> |
leaf.\label{mdfig:topView}} |
| 528 |
|
\end{figure} |
| 529 |
|
|
| 530 |
|
The orientational ordering in the system is observed by $P_2$ order |
| 589 |
|
\begin{figure} |
| 590 |
|
\centering |
| 591 |
|
\includegraphics[width=\linewidth]{./figures/mdRP2.pdf} |
| 592 |
< |
\caption{The $P_2$ order parameters for head groups (circles) and |
| 593 |
< |
molecular bodies (squares) as a function of the ratio of head group |
| 594 |
< |
size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{mdfig:rP2}} |
| 592 |
> |
\caption[The $P_2$ order parameters as a function of the ratio of head group |
| 593 |
> |
size to the width of the molecular bodies]{The $P_2$ order parameters |
| 594 |
> |
for head groups (circles) and molecular bodies (squares) as a function |
| 595 |
> |
of the ratio of head group size ($\sigma_h$) to the width of the |
| 596 |
> |
molecular bodies ($d$). \label{mdfig:rP2}} |
| 597 |
|
\end{figure} |
| 598 |
|
|
| 599 |
|
In addition to varying the size of the head groups, we studied the |
| 642 |
|
\begin{figure} |
| 643 |
|
\centering |
| 644 |
|
\includegraphics[width=\linewidth]{./figures/mdSP2.pdf} |
| 645 |
< |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
| 646 |
< |
molecular bodies (b) as a function of the strength of the dipoles. |
| 645 |
> |
\caption[The $P_2$ order parameters as a function of the strength of |
| 646 |
> |
the dipoles.]{The $P_2$ order parameters for head group dipoles (a) |
| 647 |
> |
and molecular bodies (b) as a function of the strength of the dipoles. |
| 648 |
|
These order parameters are shown for four values of the head group / |
| 649 |
|
molecular width ratio ($\sigma_h / d$). \label{mdfig:sP2}} |
| 650 |
|
\end{figure} |
| 675 |
|
\begin{figure} |
| 676 |
|
\centering |
| 677 |
|
\includegraphics[width=\linewidth]{./figures/mdTP2.pdf} |
| 678 |
< |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
| 679 |
< |
molecular bodies (b) as a function of temperature. |
| 680 |
< |
These order parameters are shown for four values of the head group / |
| 681 |
< |
molecular width ratio ($\sigma_h / d$).\label{mdfig:tP2}} |
| 678 |
> |
\caption[The $P_2$ order parameters as a function of temperature]{The |
| 679 |
> |
$P_2$ order parameters for head group dipoles (a) and molecular bodies |
| 680 |
> |
(b) as a function of temperature. These order parameters are shown |
| 681 |
> |
for four values of the head group / molecular width ratio ($\sigma_h / |
| 682 |
> |
d$).\label{mdfig:tP2}} |
| 683 |
|
\end{figure} |
| 684 |
|
|
| 685 |
|
Fig. \ref{mdfig:phaseDiagram} shows a phase diagram for the model as a |
| 698 |
|
\begin{figure} |
| 699 |
|
\centering |
| 700 |
|
\includegraphics[width=\linewidth]{./figures/mdPhaseDiagram.pdf} |
| 701 |
< |
\caption{Phase diagram for the simple molecular model as a function |
| 702 |
< |
of the head group / molecular width ratio ($\sigma_h / d$) and the |
| 703 |
< |
strength of the head group dipole moment |
| 704 |
< |
($\mu$).\label{mdfig:phaseDiagram}} |
| 701 |
> |
\caption[Phase diagram for the simple molecular model]{Phase diagram |
| 702 |
> |
for the simple molecular model as a function of the head group / |
| 703 |
> |
molecular width ratio ($\sigma_h / d$) and the strength of the head |
| 704 |
> |
group dipole moment ($\mu$).\label{mdfig:phaseDiagram}} |
| 705 |
|
\end{figure} |
| 706 |
|
|
| 707 |
|
We have computed translational diffusion constants for lipid molecules |
| 750 |
|
times that are too fast when compared with experimental measurements. |
| 751 |
|
|
| 752 |
|
\begin{table*} |
| 742 |
– |
\begin{minipage}{\linewidth} |
| 753 |
|
\begin{center} |
| 754 |
< |
\caption{Fit values for the rotational correlation times for the head |
| 755 |
< |
groups ($\tau^h$) and molecular bodies ($\tau^b$) as well as the |
| 756 |
< |
translational diffusion constants for the molecule as a function of |
| 757 |
< |
the head-to-body width ratio. All correlation functions and transport |
| 748 |
< |
coefficients were computed from microcanonical simulations with an |
| 749 |
< |
average temperture of 300 K. In all of the phases, the head group |
| 750 |
< |
correlation functions decay with an fast librational contribution ($12 |
| 751 |
< |
\pm 1$ ps). There are additional moderate ($\tau^h_{\rm mid}$) and |
| 752 |
< |
slow $\tau^h_{\rm slow}$ contributions to orientational decay that |
| 753 |
< |
depend strongly on the phase exhibited by the lipids. The symmetric |
| 754 |
< |
ripple phase ($\sigma_h / d = 1.35$) appears to exhibit the slowest |
| 755 |
< |
molecular reorientation.} |
| 754 |
> |
\caption{FIT VALUES FOR THE ROTATIONAL CORRELATION TIMES FOR THE HEAD |
| 755 |
> |
GROUPS ($\tau^h$) AND MOLECULAR BODIES ($\tau^b$) AS WELL AS THE |
| 756 |
> |
TRANSLATIONAL DIFFUSION CONSTANTS FOR THE MOL\-E\-CULE AS A FUNCTION |
| 757 |
> |
OF THE HEAD-TO-BODY WIDTH RATIO} |
| 758 |
|
\begin{tabular}{lcccc} |
| 759 |
|
\hline |
| 760 |
|
$\sigma_h / d$ & $\tau^h_{\rm mid} (ns)$ & $\tau^h_{\rm |
| 765 |
|
1.35 & $3.2$ & $4.0$ & $0.9$ & $3.42(1)$ \\ |
| 766 |
|
1.41 & $0.3$ & $23.8$ & $6.9$ & $7.16(1)$ \\ |
| 767 |
|
\end{tabular} |
| 768 |
+ |
\begin{minipage}{\linewidth} |
| 769 |
+ |
%\centering |
| 770 |
+ |
\vspace{2mm} |
| 771 |
+ |
All correlation functions and transport coefficients were computed |
| 772 |
+ |
from microcanonical simulations with an average temperture of 300 K. |
| 773 |
+ |
In all of the phases, the head group correlation functions decay with |
| 774 |
+ |
an fast librational contribution ($12 \pm 1$ ps). There are |
| 775 |
+ |
additional moderate ($\tau^h_{\rm mid}$) and slow $\tau^h_{\rm slow}$ |
| 776 |
+ |
contributions to orientational decay that depend strongly on the phase |
| 777 |
+ |
exhibited by the lipids. The symmetric ripple phase ($\sigma_h / d = |
| 778 |
+ |
1.35$) appears to exhibit the slowest molecular reorientation. |
| 779 |
|
\label{mdtab:relaxation} |
| 767 |
– |
\end{center} |
| 780 |
|
\end{minipage} |
| 781 |
+ |
\end{center} |
| 782 |
|
\end{table*} |
| 783 |
|
|
| 784 |
|
\section{Discussion} |
