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Revision 1086 by mmeineke, Fri Mar 5 03:09:26 2004 UTC vs.
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# Line 6 | Line 6 | atomistic simulation of phospholipid bilayers for incr
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
# Line 143 | Line 143 | $i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit
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
# Line 243 | Line 243 | $y$, and $z$.  This is the $\text{NPT}_{xyz}$. ensembl
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}
# Line 306 | Line 306 | simulated at 350~K. Both simulations were integrated f
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}
# Line 322 | Line 322 | constrained to lie within a plane. For the bilayer sim
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
# Line 334 | Line 334 | the motion of the lipid, or that a simplification in t
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
# Line 380 | Line 381 | The $\text{S}_{\text{{\sc cd}}}$ order parameter is of
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
# Line 400 | Line 401 | $\mathbf{\hat{z}}\rightarrow$ vector from $C_{n-1}$ to
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  
# Line 419 | Line 420 | DMPC. As our values are closer to $-\frac{1}{2}$, this
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
# Line 433 | Line 434 | point when the chains start.
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
# Line 457 | Line 510 | Temperature (K) & $\langle L_{\perp}\rangle$ ($\mbox{\
510   \end{tabular}
511   \end{center}
512   \end{table}
513 +
514 + \subsection{\label{lipidSec:miscData}Further Bilayer Data}
515 +
516 + Also summarized in Table~\ref{lipidTab:blSummary}, are the bilayer
517 + thickness and area per lipid. The bilayer thickness was measured from
518 + the peak to peak {\sc head} atom distance in the density profiles. The
519 + area per lipid data compares favorably with values typically seen for
520 + DMPC (60.0~$\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.

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