| 311 |
|
|
| 312 |
|
\section{\label{lipidSec:resultsDis}Results and Discussion} |
| 313 |
|
|
| 314 |
< |
\subsection{\label{lipidSec:scd}$\text{S}_{\text{{\sc cd}}}$ order parameters} |
| 314 |
> |
\subsection{\label{lipidSec:diffusion}Lateral Diffusion Constants} |
| 315 |
> |
|
| 316 |
> |
The lateral diffusion constant, $D_L$, is the constant charecterizing |
| 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} |
| 320 |
> |
\begin{equation} |
| 321 |
> |
2tD_L = \frac{1}{2}\langle |\mathbf{r}(t) - \mathbf{r}(0)|^2\rangle |
| 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 |
| 326 |
> |
normal, namely the $xy$-plane. |
| 327 |
> |
|
| 328 |
> |
Fig.~\ref{lipidFig:diffusionFig} shows the lateral diffusion constants |
| 329 |
> |
as a function of temperature. There is a definite increase in the |
| 330 |
> |
lateral diffusion with higher temperatures, which is exactly what one |
| 331 |
> |
would expect with greater fluidity of the chains. However, the |
| 332 |
> |
diffusion constants are all two orders of magnitude smaller than those |
| 333 |
> |
typical of DPPC.\cite{Cevc87} This is counter-intuitive as the DPPC |
| 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 |
| 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. |
| 341 |
> |
|
| 342 |
> |
\begin{figure} |
| 343 |
> |
\centering |
| 344 |
> |
\includegraphics[width=\linewidth]{diffusionFig.eps} |
| 345 |
> |
\caption[The lateral difusion constants versus temperature]{The lateral diffusion constants for the bilayers as a function of temperature.} |
| 346 |
> |
\label{lipidFig:diffusionFig} |
| 347 |
> |
\end{figure} |
| 348 |
> |
|
| 349 |
> |
\subsection{\label{lipidSec:densProf}Density Profile} |
| 350 |
> |
|
| 351 |
> |
Fig.~\ref{lipidFig:densityProfile} illustrates the densities of the |
| 352 |
> |
atoms in the bilayer systems normailzed by the bulk density as a |
| 353 |
> |
function of distance from the center of the box. The profile is taken |
| 354 |
> |
along the bilayer normal, in this case the $z$ axis. The profile at |
| 355 |
> |
270~K shows several structural features that are largerly smoothed out |
| 356 |
> |
by 300~K. The left peak for the {\sc head} atoms is split at 270~K, |
| 357 |
> |
implying that some freezing of the structure might already be occuring |
| 358 |
> |
at this temperature. From the dynamics, the tails at this temperature |
| 359 |
> |
are very much fluid, but the profile could indicate that a phase |
| 360 |
> |
transition may simply be beyond the length scale of the current |
| 361 |
> |
simulation. In all profiles, the water penetrates almost |
| 362 |
> |
5~$\mbox{\AA}$ into the bilayer, completely solvating the {\sc head} |
| 363 |
> |
atoms. The $\text{{\sc ch}}_3$ atoms although mainly centered at the |
| 364 |
> |
middle of the bilayer, show appreciable penetration into the head |
| 365 |
> |
group region. This indicates that the chains have enough mobility to |
| 366 |
> |
bend back upward to allow the ends to explore areas around the {\sc |
| 367 |
> |
head} atoms. It is unlikely that this is penetration from a lipid of |
| 368 |
> |
the opposite face, as the lipids are only 12~$\mbox{\AA}$ in length, |
| 369 |
> |
and the typical leaf spacing as measured from the {\sc head-head} |
| 370 |
> |
spacing in the profile is 17.5~$\mbox{\AA}$. |
| 371 |
|
|
| 372 |
+ |
\begin{figure} |
| 373 |
+ |
\centering |
| 374 |
+ |
\includegraphics[width=\linewidth]{densityProfile.eps} |
| 375 |
+ |
\caption[The density profile of the lipid bilayers]{The density profile of the lipid bilayers along the bilayer normal. The black lines are the {\sc head} atoms, red lines are the {\sc ch} atoms, green lines are the $\text{{\sc ch}}_2$ atoms, blue lines are the $\text{{\sc ch}}_3$ atoms, and the magenta lines are the {\sc ssd} atoms.} |
| 376 |
+ |
\label{lipidFig:densityProfile} |
| 377 |
+ |
\end{figure} |
| 378 |
+ |
|
| 379 |
+ |
|
| 380 |
+ |
\subsection{\label{lipidSec:scd}$\text{S}_{\text{{\sc cd}}}$ Order Parameters} |
| 381 |
+ |
|
| 382 |
|
The $\text{S}_{\text{{\sc cd}}}$ order parameter is often reported in |
| 383 |
|
the experimental charecterizations of phospholipids. It is obtained |
| 384 |
|
through deuterium NMR, and measures the ordering of the carbon |
| 392 |
|
\end{equation} |
| 393 |
|
Where $S_{ij}$ is given by: |
| 394 |
|
\begin{equation} |
| 395 |
< |
S_{ij} = \frac{1}{2}\Bigl<(3\cos\Theta_i\cos\Theta_j - \delta_{ij})\Bigr> |
| 395 |
> |
S_{ij} = \frac{1}{2}\Bigl\langle(3\cos\Theta_i\cos\Theta_j |
| 396 |
> |
- \delta_{ij})\Bigr\rangle |
| 397 |
|
\label{lipidEq:scd2} |
| 398 |
|
\end{equation} |
| 399 |
< |
Here, $\Theta_i$ is the angle the $i$th carbon atom frame axis makes |
| 400 |
< |
with the bilayer normal. The brackets denote an average over time and |
| 401 |
< |
molecules. The carbon atom axes are defined: |
| 399 |
> |
Here, $\Theta_i$ is the angle the $i$th axis in the reference frame of |
| 400 |
> |
the carbon atom makes with the bilayer normal. The brackets denote an |
| 401 |
> |
average over time and molecules. The carbon atom axes are defined: |
| 402 |
|
$\mathbf{\hat{z}}\rightarrow$ vector from $C_{n-1}$ to $C_{n+1}$; |
| 403 |
|
$\mathbf{\hat{y}}\rightarrow$ vector that is perpindicular to $z$ and |
| 404 |
|
in the plane through $C_{n-1}$, $C_{n}$, and $C_{n+1}$; |
| 414 |
|
negative for most carbons, and as such $|S_{\text{{\sc cd}}}|$ is more |
| 415 |
|
commonly reported than $S_{\text{{\sc cd}}}$. |
| 416 |
|
|
| 417 |
+ |
Fig.~\ref{lipidFig:scdFig} shows the $S_{\text{{\sc cd}}}$ order |
| 418 |
+ |
parameters for the bilayer system at 300~K. There is no appreciable |
| 419 |
+ |
difference in the plots for the various temperatures, however, there |
| 420 |
+ |
is a larger difference between our models ordering, and that of |
| 421 |
+ |
DMPC. As our values are closer to $-\frac{1}{2}$, this implies more |
| 422 |
+ |
ordering perpindicular to the normal than in a real system. This is |
| 423 |
+ |
due to the model having only one carbon group separating the chains |
| 424 |
+ |
from the top of the lipid. In DMPC, with the flexibility inherent in a |
| 425 |
+ |
multiple atom head group, as well as a glycerol linkage between the |
| 426 |
+ |
head group and the acyl chains, there is more loss of ordering by the |
| 427 |
+ |
point when the chains start. |
| 428 |
|
|
| 351 |
– |
|
| 352 |
– |
|
| 429 |
|
\begin{figure} |
| 430 |
|
\centering |
| 431 |
|
\includegraphics[width=\linewidth]{scdFig.eps} |
| 434 |
|
\end{figure} |
| 435 |
|
|
| 436 |
|
|
| 361 |
– |
\begin{figure} |
| 362 |
– |
\centering |
| 363 |
– |
\includegraphics[width=\linewidth]{densityProfile.eps} |
| 364 |
– |
\caption[The density profile of the lipid bilayers]{The density profile of the lipid bilayers along the bilayer normal. The black lines are the {\sc head} atoms, red lines are the {\sc ch} atoms, green lines are the $\text{{\sc ch}}_2$ atoms, blue lines are the $\text{{\sc ch}}_3$ atoms, and the magenta lines are the {\sc ssd} atoms.} |
| 365 |
– |
\label{lipidFig:densityProfile} |
| 366 |
– |
\end{figure} |
| 437 |
|
|
| 438 |
|
|
| 439 |
|
|
| 370 |
– |
\begin{figure} |
| 371 |
– |
\centering |
| 372 |
– |
\includegraphics[width=\linewidth]{diffusionFig.eps} |
| 373 |
– |
\caption[The lateral difusion constants versus temperature]{The lateral diffusion constants for the bilayers as a function of temperature.} |
| 374 |
– |
\label{lipidFig:diffusionFig} |
| 375 |
– |
\end{figure} |
| 376 |
– |
|
| 440 |
|
\begin{table} |
| 441 |
|
\caption[Structural properties of the bilayers]{Bilayer Structural properties as a function of temperature.} |
| 442 |
|
\begin{center} |
| 443 |
|
\begin{tabular}{|c|c|c|c|c|} |
| 444 |
|
\hline |
| 445 |
< |
Temperature (K) & $<L_{\perp}>$ ($\mbox{\AA}$) & % |
| 446 |
< |
$<A_{\parallel}>$ ($\mbox{\AA}^2$) & $<P_2>_{\text{Lipid}}$ & % |
| 447 |
< |
$<P_2>_{\text{{\sc head}}}$ \\ \hline |
| 445 |
> |
Temperature (K) & $\langle L_{\perp}\rangle$ ($\mbox{\AA}$) & % |
| 446 |
> |
$\langle A_{\parallel}\rangle$ ($\mbox{\AA}^2$) & % |
| 447 |
> |
$\langle P_2\rangle_{\text{Lipid}}$ & % |
| 448 |
> |
$\langle P_2\rangle_{\text{{\sc head}}}$ \\ \hline |
| 449 |
|
270 & 18.1 & 58.1 & 0.253 & 0.494 \\ \hline |
| 450 |
|
275 & 17.2 & 56.7 & 0.295 & 0.514 \\ \hline |
| 451 |
|
277 & 16.9 & 58.0 & 0.301 & 0.541 \\ \hline |
