6 |
|
|
7 |
|
In the past 10 years, increasing computer speeds have allowed for the |
8 |
|
atomistic simulation of phospholipid bilayers for increasingly |
9 |
< |
relevant lenghths of time. These simulations have ranged from |
9 |
> |
relevant lengths of time. These simulations have ranged from |
10 |
|
simulation of the gel phase ($L_{\beta}$) of |
11 |
|
dipalmitoylphosphatidylcholine (DPPC),\cite{lindahl00} to the |
12 |
|
spontaneous aggregation of DPPC molecules into fluid phase |
143 |
|
and $\mathbf{\Omega}_j$ are the orientational degrees of freedom for |
144 |
|
beads $i$ and $j$. $|\mu_i|$ is the magnitude of the dipole moment of |
145 |
|
$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
146 |
< |
vector rotated with euler angles: $\boldsymbol{\Omega}_i$. |
146 |
> |
vector rotated with Euler angles: $\boldsymbol{\Omega}_i$. |
147 |
|
|
148 |
|
The model also allows for the bonded interactions bends, and torsions. |
149 |
|
The bond between two beads on a chain is of fixed length, and is |
243 |
|
performed in an orthorhombic periodic box with a thermostat on |
244 |
|
velocities, and an independent barostat on each Cartesian axis $x$, |
245 |
|
$y$, and $z$. This is the $\text{NPT}_{xyz}$. ensemble described in |
246 |
< |
Sec.~\ref{oopseSec:Ensembles}. |
246 |
> |
Sec.~\ref{oopseSec:integrate}. |
247 |
|
|
248 |
|
|
249 |
|
\subsection{\label{lipidSec:ExpSetup}Experimental Setup} |
306 |
|
($\text{R}_{\text{II}}$) was 90 lipids with 1350 {\sc ssd} molecules |
307 |
|
simulated at 350~K. Both simulations were integrated for more than |
308 |
|
20~ns, and illustrate the spontaneous aggregation of the lipid model |
309 |
< |
into phospholipid macrostructures: $\text{R}_{\text{I}}$ into a |
309 |
> |
into phospholipid macro-structures: $\text{R}_{\text{I}}$ into a |
310 |
|
bilayer, and $\text{R}_{\text{II}}$ into a inverted rod. |
311 |
|
|
312 |
|
\section{\label{lipidSec:resultsDis}Results and Discussion} |
313 |
|
|
314 |
|
\subsection{\label{lipidSec:diffusion}Lateral Diffusion Constants} |
315 |
|
|
316 |
< |
The lateral diffusion constant, $D_L$, is the constant charecterizing |
316 |
> |
The lateral diffusion constant, $D_L$, is the constant characterizing |
317 |
|
the diffusive motion of the lipid within the plane of the bilayer. It |
318 |
|
is given by the following Einstein relation valid at long |
319 |
|
times:\cite{allen87:csl} |
322 |
|
\end{equation} |
323 |
|
Where $\mathbf{r}(t)$ is the position of the lipid at time $t$, and is |
324 |
|
constrained to lie within a plane. For the bilayer simulations the |
325 |
< |
plane of constrained motion was that perpindicular to the bilayer |
325 |
> |
plane of constrained motion was that perpendicular to the bilayer |
326 |
|
normal, namely the $xy$-plane. |
327 |
|
|
328 |
|
Fig.~\ref{lipidFig:diffusionFig} shows the lateral diffusion constants |
334 |
|
molecule is sterically larger and heavier than our model. This could |
335 |
|
be an indication that our model's chains are too interwoven and hinder |
336 |
|
the motion of the lipid, or that a simplification in the model's |
337 |
< |
forces has led to a slowing of diffusive behaviour within the |
337 |
> |
forces has led to a slowing of diffusive behavior within the |
338 |
|
bilayer. In contrast, the diffusion constant of the {\sc ssd} water, |
339 |
|
$9.84\times 10^{-6}\,\text{cm}^2/\text{s}$, compares favorably with |
340 |
< |
that of bulk water. |
340 |
> |
that of bulk water ($2.2999\times |
341 |
> |
10^{-5}\,\text{cm}^2/\text{s}$\cite{Holz00}). |
342 |
|
|
343 |
|
\begin{figure} |
344 |
|
\centering |
345 |
|
\includegraphics[width=\linewidth]{diffusionFig.eps} |
346 |
< |
\caption[The lateral difusion constants versus temperature]{The lateral diffusion constants for the bilayers as a function of temperature.} |
346 |
> |
\caption[The lateral diffusion constants versus temperature]{The lateral diffusion constants for the bilayers as a function of temperature.} |
347 |
|
\label{lipidFig:diffusionFig} |
348 |
|
\end{figure} |
349 |
|
|
350 |
|
\subsection{\label{lipidSec:densProf}Density Profile} |
351 |
|
|
352 |
|
Fig.~\ref{lipidFig:densityProfile} illustrates the densities of the |
353 |
< |
atoms in the bilayer systems normailzed by the bulk density as a |
353 |
> |
atoms in the bilayer systems normalized by the bulk density as a |
354 |
|
function of distance from the center of the box. The profile is taken |
355 |
|
along the bilayer normal, in this case the $z$ axis. The profile at |
356 |
< |
270~K shows several structural features that are largerly smoothed out |
356 |
> |
270~K shows several structural features that are largely smoothed out |
357 |
|
by 300~K. The left peak for the {\sc head} atoms is split at 270~K, |
358 |
< |
implying that some freezing of the structure might already be occuring |
358 |
> |
implying that some freezing of the structure might already be occurring |
359 |
|
at this temperature. From the dynamics, the tails at this temperature |
360 |
|
are very much fluid, but the profile could indicate that a phase |
361 |
|
transition may simply be beyond the length scale of the current |
381 |
|
\subsection{\label{lipidSec:scd}$\text{S}_{\text{{\sc cd}}}$ Order Parameters} |
382 |
|
|
383 |
|
The $\text{S}_{\text{{\sc cd}}}$ order parameter is often reported in |
384 |
< |
the experimental charecterizations of phospholipids. It is obtained |
384 |
> |
the experimental characterizations of phospholipids. It is obtained |
385 |
|
through deuterium NMR, and measures the ordering of the carbon |
386 |
|
deuterium bond in relation to the bilayer normal at various points |
387 |
|
along the chains. In our model, there are no explicit hydrogens, but |
401 |
|
the carbon atom makes with the bilayer normal. The brackets denote an |
402 |
|
average over time and molecules. The carbon atom axes are defined: |
403 |
|
$\mathbf{\hat{z}}\rightarrow$ vector from $C_{n-1}$ to $C_{n+1}$; |
404 |
< |
$\mathbf{\hat{y}}\rightarrow$ vector that is perpindicular to $z$ and |
404 |
> |
$\mathbf{\hat{y}}\rightarrow$ vector that is perpendicular to $z$ and |
405 |
|
in the plane through $C_{n-1}$, $C_{n}$, and $C_{n+1}$; |
406 |
< |
$\mathbf{\hat{x}}\rightarrow$ vector perpindicular to |
406 |
> |
$\mathbf{\hat{x}}\rightarrow$ vector perpendicular to |
407 |
|
$\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$. |
408 |
|
|
409 |
|
The order parameter has a range of $[1,-\frac{1}{2}]$. A value of 1 |
410 |
|
implies full order aligned to the bilayer axis, 0 implies full |
411 |
< |
disorder, and $-\frac{1}{2}$ implies full order perpindicular to the |
411 |
> |
disorder, and $-\frac{1}{2}$ implies full order perpendicular to the |
412 |
|
bilayer axis. The {\sc cd} bond vector for carbons near the head group |
413 |
< |
are usually ordered perpindicular to the bilayer normal, with tails |
414 |
< |
farther away tending toward disorder. This makes the order paramter |
413 |
> |
are usually ordered perpendicular to the bilayer normal, with tails |
414 |
> |
farther away tending toward disorder. This makes the order parameter |
415 |
|
negative for most carbons, and as such $|S_{\text{{\sc cd}}}|$ is more |
416 |
|
commonly reported than $S_{\text{{\sc cd}}}$. |
417 |
|
|
420 |
|
difference in the plots for the various temperatures, however, there |
421 |
|
is a larger difference between our models ordering, and that of |
422 |
|
DMPC. As our values are closer to $-\frac{1}{2}$, this implies more |
423 |
< |
ordering perpindicular to the normal than in a real system. This is |
423 |
> |
ordering perpendicular to the normal than in a real system. This is |
424 |
|
due to the model having only one carbon group separating the chains |
425 |
|
from the top of the lipid. In DMPC, with the flexibility inherent in a |
426 |
|
multiple atom head group, as well as a glycerol linkage between the |
434 |
|
\label{lipidFig:scdFig} |
435 |
|
\end{figure} |
436 |
|
|
437 |
+ |
\subsection{\label{lipidSec:p2Order}$P_2$ Order Parameter} |
438 |
|
|
439 |
+ |
The $P_2$ order parameter allows us to measure the amount of |
440 |
+ |
directional ordering that exists in the bilayer. Each lipid molecule |
441 |
+ |
can be thought of as a cylindrical tube with the head group at the |
442 |
+ |
top. If all of the cylinders are perfectly aligned, the $P_2$ order |
443 |
+ |
parameter will be $1.0$. If the cylinders are completely dispersed, |
444 |
+ |
the $P_2$ order parameter will be 0. For a collection of unit vectors, |
445 |
+ |
the $P_2$ order parameter can be solved via the following |
446 |
+ |
method.\cite{zannoni94} |
447 |
|
|
448 |
+ |
Define an ordering matrix $\mathbf{Q}$, such that, |
449 |
+ |
\begin{equation} |
450 |
+ |
\mathbf{Q} = \frac{1}{N}\sum_i^N % |
451 |
+ |
\begin{pmatrix} % |
452 |
+ |
u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
453 |
+ |
u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
454 |
+ |
u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
455 |
+ |
\end{pmatrix} |
456 |
+ |
\label{lipidEq:po1} |
457 |
+ |
\end{equation} |
458 |
+ |
Where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
459 |
+ |
$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
460 |
+ |
collection of unit vectors. This allows the matrix element |
461 |
+ |
$Q_{\alpha\beta}$ to be written: |
462 |
+ |
\begin{equation} |
463 |
+ |
Q_{\alpha\beta} = \langle u_{\alpha}u_{\beta} - |
464 |
+ |
\frac{1}{3}\delta_{\alpha\beta} \rangle |
465 |
+ |
\label{lipidEq:po2} |
466 |
+ |
\end{equation} |
467 |
|
|
468 |
+ |
Having constructed the matrix, diagonalizing $\mathbf{Q}$ yields three |
469 |
+ |
eigenvalues and eigenvectors. The eigenvector associated with the |
470 |
+ |
largest eigenvalue, $\lambda_{\text{max}}$, is the director for the |
471 |
+ |
system of unit vectors. The director is the average direction all of |
472 |
+ |
the unit vectors are pointing. The $P_2$ order parameter is then |
473 |
+ |
simply |
474 |
+ |
\begin{equation} |
475 |
+ |
\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}} |
476 |
+ |
\label{lipidEq:po3} |
477 |
+ |
\end{equation} |
478 |
|
|
479 |
+ |
Table~\ref{lipidTab:blSummary} summarizes the $P_2$ values for the |
480 |
+ |
bilayers, as well as the dipole orientations. The unit vector for the |
481 |
+ |
lipid molecules was defined by finding the moment of inertia for each |
482 |
+ |
lipid, then setting $\mathbf{\hat{u}}$ to point along the axis of |
483 |
+ |
minimum inertia. For the {\sc head} atoms, the unit vector simply |
484 |
+ |
pointed in the same direction as the dipole moment. For the lipid |
485 |
+ |
molecules, the ordering was consistent across all temperatures, with |
486 |
+ |
the director pointed along the $z$ axis of the box. More |
487 |
+ |
interestingly, is the high degree of ordering the dipoles impose on |
488 |
+ |
the {\sc head} atoms. The directors for the dipoles consistently |
489 |
+ |
pointed along the plane of the bilayer, with the directors |
490 |
+ |
anti-aligned on the top and bottom leaf. |
491 |
+ |
|
492 |
|
\begin{table} |
493 |
|
\caption[Structural properties of the bilayers]{Bilayer Structural properties as a function of temperature.} |
494 |
+ |
\label{lipidTab:blSummary} |
495 |
|
\begin{center} |
496 |
|
\begin{tabular}{|c|c|c|c|c|} |
497 |
|
\hline |
510 |
|
\end{tabular} |
511 |
|
\end{center} |
512 |
|
\end{table} |
513 |
+ |
|
514 |
+ |
\subsection{\label{lipidSec:miscData}Further Bilayer Data} |
515 |
+ |
|
516 |
+ |
Also summarized in Table~\ref{lipidTab:blSummary}, are the bilayer |
517 |
+ |
thickness and area per lipid. The bilayer thickness was measured from |
518 |
+ |
the peak to peak {\sc head} atom distance in the density profiles. The |
519 |
+ |
area per lipid data compares favorably with values typically seen for |
520 |
+ |
DMPC (60.0~$\mbox{\AA}^2$ at 303~K)\cite{petrache00}. Although are |
521 |
+ |
values are lower this is most likely due to the shorter chain length |
522 |
+ |
of our model (8 versus 14 for DMPC). |
523 |
+ |
|
524 |
+ |
\subsection{\label{lipidSec:randBilayer}Bilayer Aggregation} |
525 |
+ |
|
526 |
+ |
A very important accomplishment for our model is its ability to |
527 |
+ |
spontaneously form bilayers from a randomly dispersed starting |
528 |
+ |
configuration. Fig.~\ref{lipidFig:blImage} shows an image sequence for |
529 |
+ |
the bilayer aggregation. After 3.0~ns, the basic form of the bilayer |
530 |
+ |
can already be seen. By 7.0~ns, the bilayer has a lipid bridge |
531 |
+ |
stretched across the simulation box to itself that will turn out to be |
532 |
+ |
very long lived ($\sim$20~ns), as well as a water pore, that will |
533 |
+ |
persist for the length of the current simulation. At 24~ns, the lipid |
534 |
+ |
bridge is dispersed, and the bilayer is still integrating the lipid |
535 |
+ |
molecules from the bridge into itself, and has still been unable to |
536 |
+ |
disperse the water pore. |
537 |
+ |
|
538 |
+ |
\begin{figure} |
539 |
+ |
\centering |
540 |
+ |
\includegraphics[width=\linewidth]{bLayerImage.eps} |
541 |
+ |
\caption[Image sequence of the bilayer aggregation]{Image sequence of the bilayer aggregation. The blue beads are the {\sc head} atoms the grey beads are the chains, and the red and white bead are the water molecules. A box has been drawn around the periodic image.} |
542 |
+ |
\label{lipidFig:blImage} |
543 |
+ |
\end{figure} |
544 |
+ |
|
545 |
+ |
\subsection{\label{lipidSec:randIrod}Inverted Rod Aggregation} |
546 |
+ |
|
547 |
+ |
Fig.~\ref{lipidFig:iRimage} shows a second aggregation sequence |
548 |
+ |
simulated in this research. Here the fraction of water had been |
549 |
+ |
significantly decreased to observe how the model would respond. After |
550 |
+ |
1.5~ns, The main body of water in the system has already collected |
551 |
+ |
into a central water channel. By 10.0~ns, the channel has widened |
552 |
+ |
slightly, but there are still many sub channels permeating the lipid |
553 |
+ |
macro-structure. At 23.0~ns, the central water channel has stabilized |
554 |
+ |
and several smaller water channels have been absorbed to main |
555 |
+ |
one. However, there are still several other channels that persist |
556 |
+ |
through the lipid structure. |
557 |
+ |
|
558 |
+ |
\begin{figure} |
559 |
+ |
\centering |
560 |
+ |
\includegraphics[width=\linewidth]{iRodImage.eps} |
561 |
+ |
\caption[Image sequence of the inverted rod aggregation]{Image sequence of the inverted rod aggregation. color scheme is the same as in Fig.~\ref{lipidFig:blImage}.} |
562 |
+ |
\label{lipidFig:iRimage} |
563 |
+ |
\end{figure} |
564 |
+ |
|
565 |
+ |
\section{\label{lipidSec:Conclusion}Conclusion} |
566 |
+ |
|
567 |
+ |
We have presented a phospholipid model capable of spontaneous |
568 |
+ |
aggregation into a bilayer and an inverted rod structure. The time |
569 |
+ |
scales of the macro-molecular aggregations are in excess of 24~ns. In |
570 |
+ |
addition the model's bilayer properties have been explored over a |
571 |
+ |
range of temperatures through prefabricated bilayers. No freezing |
572 |
+ |
transition is seen in the temperature range of our current |
573 |
+ |
simulations. However, structural information from the lowest |
574 |
+ |
temperature may imply that a freezing event is on a much longer time |
575 |
+ |
scale than that explored in this current research. Further studies of |
576 |
+ |
this system could extend the time length of the simulations at the low |
577 |
+ |
temperatures to observe whether lipid crystallization occurs within the |
578 |
+ |
framework of this model. |