| 10 |
|
simulation of the gel ($L_{\beta}$) phase of |
| 11 |
|
dipalmitoylphosphatidylcholine (DPPC),\cite{lindahl00} to the |
| 12 |
|
spontaneous aggregation of DPPC molecules into fluid phase |
| 13 |
< |
($L_{\alpha}$) bilayers.\cite{marrink01} With the exception of a few |
| 13 |
> |
($L_{\alpha}$) bilayers.\cite{Marrink01} With the exception of a few |
| 14 |
|
ambitious |
| 15 |
|
simulations, \cite{marrink01:undulation,marrink:2002,lindahl00} most |
| 16 |
|
investigations are limited to a range of 64 to 256 |
| 17 |
< |
phospholipids.\cite{lindahl00,sum:2003,venable00,gomez:2003,smondyrev:1999,marrink01} |
| 17 |
> |
phospholipids.\cite{lindahl00,sum:2003,venable00,gomez:2003,smondyrev:1999,Marrink01} |
| 18 |
|
The expense of the force calculations involved when performing these |
| 19 |
|
simulations limits the system size. To properly hydrate a bilayer, one |
| 20 |
|
typically needs 25 water molecules for every lipid, bringing the total |
| 207 |
|
\ref{lipidTable:tcTorsionParams}. |
| 208 |
|
|
| 209 |
|
\begin{table} |
| 210 |
< |
\caption[Lennard-Jones parameters for the two chain phospholipids]{THE LENNARD JONES PARAMETERS FOR THE TWO CHAIN PHOSPHOLIPIDS} |
| 210 |
> |
\caption{THE LENNARD JONES PARAMETERS FOR THE TWO CHAIN PHOSPHOLIPIDS} |
| 211 |
|
\label{lipidTable:tcLJParams} |
| 212 |
|
\begin{center} |
| 213 |
|
\begin{tabular}{|l|c|c|c|c|} |
| 224 |
|
\end{table} |
| 225 |
|
|
| 226 |
|
\begin{table} |
| 227 |
< |
\caption[Bend Parameters for the two chain phospholipids]{BEND PARAMETERS FOR THE TWO CHAIN PHOSPHOLIPIDS} |
| 227 |
> |
\caption{BEND PARAMETERS FOR THE TWO CHAIN PHOSPHOLIPIDS} |
| 228 |
|
\label{lipidTable:tcBendParams} |
| 229 |
|
\begin{center} |
| 230 |
|
\begin{tabular}{|l|c|c|} |
| 244 |
|
\end{table} |
| 245 |
|
|
| 246 |
|
\begin{table} |
| 247 |
< |
\caption[Torsion Parameters for the two chain phospholipids]{TORSION PARAMETERS FOR THE TWO CHAIN PHOSPHOLIPIDS} |
| 247 |
> |
\caption{TORSION PARAMETERS FOR THE TWO CHAIN PHOSPHOLIPIDS} |
| 248 |
|
\label{lipidTable:tcTorsionParams} |
| 249 |
|
\begin{center} |
| 250 |
|
\begin{tabular}{|l|c|c|c|c|} |
| 354 |
|
Fig.~\ref{lipidFig:densityProfile} illustrates the densities of the |
| 355 |
|
atoms in the bilayer systems normalized by the bulk density as a |
| 356 |
|
function of distance from the center of the box. The profile is taken |
| 357 |
< |
along the bilayer normal (in this case the $z$ axis). The first interesting point to note, is the penetration of water into the membrane. Water is present about 5~$\mbox{\AA}$ into the bilayer, completely solvating the head groups. This is common in atomistic and some coarse grain simulations of phospholipid bilayers.\cite{Marrink01,marrink04,klein01} It is an indication that the water molecules are very attracted to the head region, yet there is still enough of a hydrophobic effect to exclude water from the inside of the bilayer. |
| 357 |
> |
along the bilayer normal (in this case the $z$ axis). The first interesting point to note is the penetration of water into the membrane. Water penetrates about 5~$\mbox{\AA}$ into the bilayer, completely solvating the head groups. This is common in atomistic and some coarse grain simulations of phospholipid bilayers.\cite{Marrink01,marrink04,klein01} It is an indication that the water molecules are very attracted to the polar head region, yet there is still enough of a hydrophobic effect to exclude water from the inside of the bilayer. |
| 358 |
|
|
| 359 |
|
Another interesting point is the fluidity of the chains. Although the ends of the tails, the $\text{{\sc ch}}_3$ atoms, are mostly concentrated at the centers of the bilayers, they have a significant density around the head regions. This indicates that there is much freedom of movement in the chains of our model. Typical atomistic simulations of DPPC show the terminal groups concentrated at the center of the bilayer.\cite{marrink03:vesicles} This is most likely an indication that our chain lengths are too small, and given longer chains, the tail groups would stay more deeply buried in the bilayer. |
| 360 |
|
|
| 361 |
< |
The last point to consider, is the splitting in the density peak of the {\sc head} atom at 270~K. This implies that there is some locking in of structure at this temperature. By 280~K, this feature is smoothed out, demonstrating a more fluid phase in the bilayer. Within the time scale of the simulation, there is no gelling of the chains within the bilayer at 270~K, so this splitting in the peak is likely a glassy transition in the head groups, and could possibly indicate that we are simulating in a super cooled region of our phospholipid model. |
| 361 |
> |
The last point to consider is the splitting in the density peak of the {\sc head} atom at 270~K. This implies that there is some freezing of structure at this temperature. By 280~K, this feature is smoothed out, demonstrating a more fluid phase in the bilayer. Within the time scale of the simulation, the gel phase has not formed at 270~K, so this splitting in the peak is likely a glassy transition in the head groups, and could possibly indicate that we are simulating in a super cooled region of our phospholipid model. |
| 362 |
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|
| 363 |
|
\begin{figure} |
| 364 |
|
\centering |
| 419 |
|
group separating the chains from the top of the lipid. In DMPC, with |
| 420 |
|
the flexibility inherent in a multiple atom head group, as well as a |
| 421 |
|
glycerol linkage between the head group and the acyl chains, there is |
| 422 |
< |
more loss of ordering by the point when the chains start. Also, there is more ordering in the model due to the our assumptions about the locations of the hydrogen atoms. Are method assumes a rigid location for each hydrogen atom, based on the carbon positions. This does not allow for any small fluctuations in their positions that would be inherent in a real system or even an atomistic simulation. These small fluctuations would serve to lower the the ordering measured in the $S_{\text{{\sc cd}}}$. |
| 422 |
> |
more loss of ordering by the point when the chains start. Also, there is more ordering in the model due to the our assumptions about the locations of the hydrogen atoms. Our method assumes a rigid location for each hydrogen atom based on the carbon positions. This does not allow for any small fluctuations in their positions that would be inherent in an atomistic simulation or in experiments. These small fluctuations would serve to lower the ordering measured in the $S_{\text{{\sc cd}}}$. |
| 423 |
|
|
| 424 |
|
\begin{figure} |
| 425 |
|
\centering |
| 472 |
|
\end{equation} |
| 473 |
|
|
| 474 |
|
Table~\ref{lipidTab:blSummary} summarizes the $P_2$ values for the |
| 475 |
< |
bilayers, as well as the dipole orientations. The unit vector for the |
| 475 |
> |
bilayers, as well as for the dipole orientations. The unit vector for the |
| 476 |
|
lipid molecules was defined by finding the moment of inertia for each |
| 477 |
|
lipid, then setting $\mathbf{\hat{u}}$ to point along the axis of |
| 478 |
|
minimum inertia (the long axis). For the {\sc head} atoms, the unit vector simply |
| 481 |
|
the director pointed along the $z$ axis of the box. More |
| 482 |
|
interestingly, is the high degree of ordering the dipoles impose on |
| 483 |
|
the {\sc head} atoms. The directors for the dipoles themselves |
| 484 |
< |
consistently pointed along the plane of the bilayer, with head groups lining up in rows of alternating alignment. These lines are an unphysical situation for the phospholipids, and implies that the dipole interaction is a little too strong, or that perhaps the dipoles are allowed to approach each other a little too closely. A possible change in future models would alter the size or shape of the molecule to discourage too rigid ordering of the dipoles. |
| 484 |
> |
consistently pointed along the plane of the bilayer, with head groups lining up in rows of alternating alignment. The ordering implies that the dipole interaction is a little too strong, or that perhaps the dipoles are allowed to approach each other a bit too closely. A possible change in future models would alter the size or shape of the head group to discourage too rigid ordering of the dipoles. |
| 485 |
|
|
| 486 |
|
\begin{table} |
| 487 |
< |
\caption[Structural properties of the bilayers]{BILAYER STRUCTURAL PROPERTIES AS A FUNCTION OF TEMPERATURE} |
| 487 |
> |
\caption{BILAYER STRUCTURAL PROPERTIES AS A FUNCTION OF TEMPERATURE} |
| 488 |
|
\label{lipidTab:blSummary} |
| 489 |
|
\begin{center} |
| 490 |
|
\begin{tabular}{|c|c|c|c|c|} |
| 526 |
|
- \mathbf{r}(0)|^2\rangle, |
| 527 |
|
\end{equation} |
| 528 |
|
where $\mathbf{r}(t)$ is the $xy$ position of the lipid at time $t$ |
| 529 |
< |
(assuming the $z$-axis is parallel to the bilayer normal). Calculating the $D_L$ involves first plotting the mean square displacement, $\langle |\mathbf{r}(t) - \mathbf{r}(0)|^2\rangle$, finding the slope at long times, and dividing the slope by 4 to give the diffusion constant (Fig.~\ref{lipidFig:msdFig}). When finding the slope only the long time region is considered, in addition points at the longest time are discarded due to the lack of good statistics at long time intervals. |
| 529 |
> |
(assuming the $z$-axis is parallel to the bilayer normal). Calculating the $D_L$ involves first plotting the mean square displacement, $\langle |\mathbf{r}(t) - \mathbf{r}(0)|^2\rangle$, finding the slope at long times, and dividing the slope by 4 to give the diffusion constant (Fig.~\ref{lipidFig:msdFig}). When finding the slope only the 1~ns to 3~ns times are considered. Points at the longer times are not included due to the lack of good statistics at long time intervals. |
| 530 |
|
|
| 531 |
|
Fig.~\ref{lipidFig:diffusionFig} shows the lateral diffusion constants |
| 532 |
|
as a function of temperature. There is a definite increase in the |
| 542 |
|
\begin{figure} |
| 543 |
|
\centering |
| 544 |
|
\includegraphics[width=\linewidth]{msdFig.eps} |
| 545 |
< |
\caption[Lateral mean square displacement for the phospholipid at 300~K]{This is a representative lateral mean square displacement for the center of mass motion of the phospholipid model. This particular example is from the 300~K run. The box is drawn about the region used in the calculation of the diffusion constant.} |
| 545 |
> |
\caption[Lateral mean square displacement for the phospholipid at 300~K]{This is a representative lateral mean square displacement for the center of mass motion of the phospholipid model. This particular example is from the 300~K simulation. The box is drawn about the region used in the calculation of the diffusion constant.} |
| 546 |
|
\label{lipidFig:msdFig} |
| 547 |
|
\end{figure} |
| 548 |
|
|
| 558 |
|
A very important accomplishment for our model is its ability to |
| 559 |
|
spontaneously form bilayers from a randomly dispersed starting |
| 560 |
|
configuration. Fig.~\ref{lipidFig:blImage} shows an image sequence for |
| 561 |
< |
the bilayer aggregation. After 1.0~ns, bulk aggregation has occured. By 5.0~ns, the basic bilayer aggregation can be seen, however there is a vertical lipid bridge connecting the periodic image of the bilayer to itself. At 15.0~ns, the lipid bridge has finally broken up, and the lipid molecules are starting to re-incorporate themselves into the bilayer. The water pore is still present through the membrane. In the last frame at 42.0~ns, the water pore is still present, although does show some signs of breaking up. These behaviors are typical for coarse grain model simulations, which can have lipid bridge lifetimes of up to 20~ns, and water pores typically lasting 3 to 25~ns.\cite{marrink04} |
| 561 |
> |
the bilayer aggregation. After 1.0~ns, bulk aggregation has occured. By 5.0~ns, the basic bilayer aggregation can be seen, however there is a vertical lipid bridge connecting the periodic image of the bilayer to itself. At 15.0~ns, the lipid bridge has finally broken up, and the lipid molecules are starting to re-incorporate themselves into the bilayer. A water pore is still present through the membrane. In the last frame at 42.0~ns, the water pore is still present, although does show some signs of rejoining the bulk water section. These behaviors are typical for coarse grain model simulations, which can have lipid bridge lifetimes of up to 20~ns, and water pores typically lasting 3 to 25~ns.\cite{marrink04} |
| 562 |
|
|
| 563 |
|
\begin{figure} |
| 564 |
|
\centering |
