1 |
|
2 |
|
3 |
\chapter{\label{chapt:lipid}Phospholipid Simulations} |
4 |
|
5 |
\section{\label{lipidSec:Intro}Introduction} |
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 |
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 |
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} |
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 |
21 |
number of atoms simulated to roughly 8,000 for a system of 64 DPPC |
22 |
molecules. Added to the difficulty is the electrostatic nature of the |
23 |
phospholipid head groups and water, requiring either the |
24 |
computationally expensive Ewald sum or the faster, particle mesh Ewald |
25 |
sum.\cite{nina:2002,norberg:2000,patra:2003} These factors all limit |
26 |
the system size and time scales of bilayer simulations. |
27 |
|
28 |
Unfortunately, much of biological interest happens on time and length |
29 |
scales well beyond the range of current simulation technology. One |
30 |
such example is the observance of a ripple phase |
31 |
($P_{\beta^{\prime}}$) between the $L_{\beta}$ and $L_{\alpha}$ phases |
32 |
of certain phospholipid bilayers.\cite{katsaras00,sengupta00} These |
33 |
ripples are shown to have periodicity on the order of |
34 |
100-200~$\mbox{\AA}$. A simulation on this length scale would have |
35 |
approximately 1,300 lipid molecules with an additional 25 water |
36 |
molecules per lipid to fully solvate the bilayer. A simulation of this |
37 |
size is impractical with current atomistic models. |
38 |
|
39 |
The time and length scale limitations are most striking in transport |
40 |
phenomena. Due to the fluid-like properties of a lipid membrane, not |
41 |
all diffusion across the membrane happens at pores. Some molecules of |
42 |
interest may incorporate themselves directly into the membrane. Once |
43 |
here, they may possess an appreciable waiting time (on the order of |
44 |
10's to 100's of nanoseconds) within the bilayer. Such long simulation |
45 |
times are nearly impossible to obtain when integrating the system with |
46 |
atomistic detail. |
47 |
|
48 |
To address these issues, several schemes have been proposed. One |
49 |
approach by Goetz and Liposky\cite{goetz98} is to model the entire |
50 |
system as Lennard-Jones spheres. Phospholipids are represented by |
51 |
chains of beads with the top most beads identified as the head |
52 |
atoms. Polar and non-polar interactions are mimicked through |
53 |
attractive and soft-repulsive potentials respectively. A similar |
54 |
model proposed by Marrinck \emph{et. al}.\cite{marrink04}~uses a |
55 |
similar technique for modeling polar and non-polar interactions with |
56 |
Lennard-Jones spheres. However, they also include charges on the head |
57 |
group spheres to mimic the electrostatic interactions of the |
58 |
bilayer. While the solvent spheres are kept charge-neutral and |
59 |
interact with the bilayer solely through an attractive Lennard-Jones |
60 |
potential. |
61 |
|
62 |
The model used in this investigation adds more information to the |
63 |
interactions than the previous two models, while still balancing the |
64 |
need for simplifications over atomistic detail. The model uses |
65 |
Lennard-Jones spheres for the head and tail groups of the |
66 |
phospholipids, allowing for the ability to scale the parameters to |
67 |
reflect various sized chain configurations while keeping the number of |
68 |
interactions small. What sets this model apart, however, is the use |
69 |
of dipoles to represent the electrostatic nature of the |
70 |
phospholipids. The dipole electrostatic interaction is shorter range |
71 |
than Coulombic ($\frac{1}{r^3}$ versus $\frac{1}{r}$), and eliminates |
72 |
the need for a costly Ewald sum. |
73 |
|
74 |
Another key feature of this model, is the use of a dipolar water model |
75 |
to represent the solvent. The soft sticky dipole ({\sc ssd}) water |
76 |
\cite{liu96:new_model} relies on the dipole for long range electrostatic |
77 |
effects, but also contains a short range correction for hydrogen |
78 |
bonding. In this way the systems in this research mimic the entropic |
79 |
contribution to the hydrophobic effect due to hydrogen-bond network |
80 |
deformation around a non-polar entity, \emph{i.e.}~the phospholipid |
81 |
molecules. |
82 |
|
83 |
The following is an outline of this chapter. |
84 |
Sec.~\ref{lipidSec:Methods} is an introduction to the lipid model used |
85 |
in these simulations, as well as clarification about the water model |
86 |
and integration techniques. The various simulations explored in this |
87 |
research are outlined in |
88 |
Sec.~\ref{lipidSec:ExpSetup}. Sec.~\ref{lipidSec:resultsDis} gives a |
89 |
summary and interpretation of the results. Finally, the conclusions |
90 |
of this chapter are presented in Sec.~\ref{lipidSec:Conclusion}. |
91 |
|
92 |
\section{\label{lipidSec:Methods}Methods} |
93 |
|
94 |
\subsection{\label{lipidSec:lipidModel}The Lipid Model} |
95 |
|
96 |
\begin{figure} |
97 |
\centering |
98 |
\includegraphics[width=\linewidth]{twoChainFig.eps} |
99 |
\caption[The two chained lipid model]{Schematic diagram of the double chain phospholipid model. The head group (in red) has a point dipole, $\boldsymbol{\mu}$, located at its center of mass. The two chains are eight methylene groups in length.} |
100 |
\label{lipidFig:lipidModel} |
101 |
\end{figure} |
102 |
|
103 |
The phospholipid model used in these simulations is based on the |
104 |
design illustrated in Fig.~\ref{lipidFig:lipidModel}. The head group |
105 |
of the phospholipid is replaced by a single Lennard-Jones sphere of |
106 |
diameter $\sigma_{\text{head}}$, with $\epsilon_{\text{head}}$ scaling |
107 |
the well depth of its van der Walls interaction. This sphere also |
108 |
contains a single dipole of magnitude $|\boldsymbol{\mu}|$, where |
109 |
$|\boldsymbol{\mu}|$ can be varied to mimic the charge separation of a |
110 |
given phospholipid head group. The atoms of the tail region are |
111 |
modeled by beads representing multiple methyl groups. They are free |
112 |
of partial charges or dipoles, and contain only Lennard-Jones |
113 |
interaction sites at their centers of mass. As with the head groups, |
114 |
their potentials can be scaled by $\sigma_{\text{tail}}$ and |
115 |
$\epsilon_{\text{tail}}$. |
116 |
|
117 |
The long range interactions between lipids are given by the following: |
118 |
\begin{equation} |
119 |
V_{\text{LJ}}(r_{ij}) = |
120 |
4\epsilon_{ij} \biggl[ |
121 |
\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
122 |
- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
123 |
\biggr] |
124 |
\label{lipidEq:LJpot} |
125 |
\end{equation} |
126 |
and |
127 |
\begin{equation} |
128 |
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
129 |
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
130 |
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
131 |
- |
132 |
3(\boldsymbol{\hat{u}}_i \cdot \mathbf{\hat{r}}_{ij}) % |
133 |
(\boldsymbol{\hat{u}}_j \cdot \mathbf{\hat{r}}_{ij} \biggr] |
134 |
\label{lipidEq:dipolePot} |
135 |
\end{equation} |
136 |
Where $V_{\text{LJ}}$ is the Lennard-Jones potential and |
137 |
$V_{\text{dipole}}$ is the dipole-dipole potential. As previously |
138 |
stated, $\sigma_{ij}$ and $\epsilon_{ij}$ are the Lennard-Jones |
139 |
parameters which scale the length and depth of the interaction |
140 |
respectively, and $r_{ij}$ is the distance between beads $i$ and $j$. |
141 |
In $V_{\text{dipole}}$, $\mathbf{r}_{ij}$ is the vector starting at |
142 |
bead $i$ and pointing towards bead $j$. Vectors $\mathbf{\Omega}_i$ |
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: $\boldsymbol{\Omega}_i$. |
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 |
150 |
maintained according to the {\sc rattle} algorithm.\cite{andersen83} |
151 |
The bends are subject to a harmonic potential: |
152 |
\begin{equation} |
153 |
V_{\text{bend}}(\theta) = k_{\theta}( \theta - \theta_0 )^2 |
154 |
\label{lipidEq:bendPot} |
155 |
\end{equation} |
156 |
where $k_{\theta}$ scales the strength of the harmonic well, and |
157 |
$\theta$ is the angle between bond vectors |
158 |
(Fig.~\ref{lipidFig:lipidModel}). In addition, we have placed a |
159 |
``ghost'' bend on the phospholipid head. The ghost bend adds a |
160 |
potential to keep the dipole pointed along the bilayer surface, where |
161 |
$\theta$ is now the angle the dipole makes with respect to the {\sc |
162 |
head}-$\text{{\sc ch}}_2$ bond vector |
163 |
(Fig.~\ref{lipidFig:ghostBend}). The torsion potential is given by: |
164 |
\begin{equation} |
165 |
V_{\text{torsion}}(\phi) = |
166 |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
167 |
\label{lipidEq:torsionPot} |
168 |
\end{equation} |
169 |
Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine |
170 |
power series to the desired torsion potential surface, and $\phi$ is |
171 |
the angle the two end atoms have rotated about the middle bond |
172 |
(Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as the |
173 |
Lennard-Jones potential are excluded for atom pairs involved in the |
174 |
same bond, bend, or torsion. However, internal interactions not |
175 |
directly involved in a bonded pair are calculated. |
176 |
|
177 |
\begin{figure} |
178 |
\centering |
179 |
\includegraphics[width=\linewidth]{ghostBendFig.eps} |
180 |
\caption[Depiction of the ``ghost'' bend]{The ``ghost'' bend is a bend potential added to constrain the motion of the dipole on the {\sc head} group. The potential follows Eq.~\ref{lipidEq:bendPot} where $\theta$ is now the angle that the dipole makes with the {\sc head}-$\text{{\sc ch}}_2$ bond vector.} |
181 |
\label{lipidFig:ghostBend} |
182 |
\end{figure} |
183 |
|
184 |
All simulations presented here use a two chained lipid as pictured in |
185 |
Fig.~\ref{lipidFig:lipidModel}. The chains are both eight beads long, |
186 |
and their mass and Lennard Jones parameters are summarized in |
187 |
Table~\ref{lipidTable:tcLJParams}. The magnitude of the dipole moment |
188 |
for the head bead is 10.6~Debye, and the bend and torsion parameters |
189 |
are summarized in Table~\ref{lipidTable:tcBendParams} and |
190 |
\ref{lipidTable:tcTorsionParams}. |
191 |
|
192 |
\begin{table} |
193 |
\caption{The Lennard Jones Parameters for the two chain phospholipids.} |
194 |
\label{lipidTable:tcLJParams} |
195 |
\begin{center} |
196 |
\begin{tabular}{|l|c|c|c|} |
197 |
\hline |
198 |
& mass (amu) & $\sigma$($\mbox{\AA}$) & $\epsilon$ (kcal/mol) \\ \hline |
199 |
{\sc head} & 72 & 4.0 & 0.185 \\ \hline |
200 |
{\sc ch}\cite{Siepmann1998} & 13.02 & 4.0 & 0.0189 \\ \hline |
201 |
$\text{{\sc ch}}_2$\cite{Siepmann1998} & 14.03 & 3.95 & 0.18 \\ \hline |
202 |
$\text{{\sc ch}}_3$\cite{Siepmann1998} & 15.04 & 3.75 & 0.25 \\ \hline |
203 |
{\sc ssd}\cite{liu96:new_model} & 18.03 & 3.051 & 0.152 \\ \hline |
204 |
\end{tabular} |
205 |
\end{center} |
206 |
\end{table} |
207 |
|
208 |
\begin{table} |
209 |
\caption[Bend Parameters for the two chain phospholipids]{Bend Parameters for the two chain phospholipids. All alkane parameters are based off of those from TraPPE.\cite{Siepmann1998}} |
210 |
\label{lipidTable:tcBendParams} |
211 |
\begin{center} |
212 |
\begin{tabular}{|l|c|c|} |
213 |
\hline |
214 |
& $k_{\theta}$ ( kcal/($\text{mol deg}^2$) ) & $\theta_0$ ( deg ) \\ \hline |
215 |
{\sc ghost}-{\sc head}-$\text{{\sc ch}}_2$ & 0.00177 & 129.78 \\ \hline |
216 |
$x$-{\sc ch}-$y$ & 58.84 & 112.0 \\ \hline |
217 |
$x$-$\text{{\sc ch}}_2$-$y$ & 58.84 & 114.0 \\ \hline |
218 |
\end{tabular} |
219 |
\end{center} |
220 |
\end{table} |
221 |
|
222 |
\begin{table} |
223 |
\caption[Torsion Parameters for the two chain phospholipids]{Torsion Parameters for the two chain phospholipids. Alkane parameters based on TraPPE.\cite{Siepmann1998}} |
224 |
\label{lipidTable:tcTorsionParams} |
225 |
\begin{center} |
226 |
\begin{tabular}{|l|c|c|c|c|} |
227 |
\hline |
228 |
All are in kcal/mol $\rightarrow$ & $k_3$ & $k_2$ & $k_1$ & $k_0$ \\ \hline |
229 |
$x$-{\sc ch}-$y$-$z$ & 3.3254 & -0.4215 & -1.686 & 1.1661 \\ \hline |
230 |
$x$-$\text{{\sc ch}}_2$-$\text{{\sc ch}}_2$-$y$ & 5.9602 & -0.568 & -3.802 & 2.1586 \\ \hline |
231 |
\end{tabular} |
232 |
\end{center} |
233 |
\end{table} |
234 |
|
235 |
|
236 |
\section{\label{lipidSec:furtherMethod}Further Methodology} |
237 |
|
238 |
As mentioned previously, the water model used throughout these |
239 |
simulations was the {\sc ssd} model of |
240 |
Ichiye.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md} A |
241 |
discussion of the model can be found in Sec.~\ref{oopseSec:SSD}. As |
242 |
for the integration of the equations of motion, all simulations were |
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}. |
247 |
|
248 |
|
249 |
\subsection{\label{lipidSec:ExpSetup}Experimental Setup} |
250 |
|
251 |
Two main starting configuration classes were used in this research: |
252 |
random and ordered bilayers. The ordered bilayer starting |
253 |
configurations were all started from an equilibrated bilayer at |
254 |
300~K. The original configuration for the first 300~K run was |
255 |
assembled by placing the phospholipids centers of mass on a planar |
256 |
hexagonal lattice. The lipids were oriented with their long axis |
257 |
perpendicular to the plane. The second leaf simply mirrored the first |
258 |
leaf, and the appropriate number of waters were then added above and |
259 |
below the bilayer. |
260 |
|
261 |
The random configurations took more work to generate. To begin, a |
262 |
test lipid was placed in a simulation box already containing water at |
263 |
the intended density. The waters were then tested for overlap with |
264 |
the lipid using a 5.0~$\mbox{\AA}$ buffer distance. This gave an |
265 |
estimate for the number of waters each lipid would displace in a |
266 |
simulation box. A target number of waters was then defined which |
267 |
included the number of waters each lipid would displace, the number of |
268 |
waters desired to solvate each lipid, and a factor to pad the |
269 |
initial box with a few extra water molecules. |
270 |
|
271 |
Next, a cubic simulation box was created that contained at least the |
272 |
target number of waters in an FCC lattice (the lattice was for ease of |
273 |
placement). What followed was a RSA simulation similar to those of |
274 |
Chapt.~\ref{chapt:RSA}. The lipids were sequentially given a random |
275 |
position and orientation within the box. If a lipid's position caused |
276 |
atomic overlap with any previously placed lipid, its position and |
277 |
orientation were rejected, and a new random placement site was |
278 |
attempted. The RSA simulation proceeded until all phospholipids had |
279 |
been adsorbed. After placement of all lipid molecules, water |
280 |
molecules with locations that overlapped with the atomic coordinates |
281 |
of the lipids were removed. |
282 |
|
283 |
Finally, water molecules were removed at random until the desired |
284 |
number of waters per lipid was reached. The typical low final density |
285 |
for these initial configurations was not a problem, as the box shrinks |
286 |
to an appropriate size within the first 50~ps of a simulation in the |
287 |
$\text{NPT}_{xyz}$ ensemble. |
288 |
|
289 |
\subsection{\label{lipidSec:Configs}Configurations} |
290 |
|
291 |
The first class of simulations were started from ordered |
292 |
bilayers. They were all configurations consisting of 60 lipid |
293 |
molecules with 30 lipids on each leaf, and were hydrated with 1620 |
294 |
{\sc ssd} molecules. The original configuration was assembled |
295 |
according to Sec.~\ref{lipidSec:ExpSetup} and simulated for a length |
296 |
of 10~ns at 300~K. The other temperature runs were started from a |
297 |
frame 7~ns into the 300~K simulation. Their temperatures were reset |
298 |
with the thermostating algorithm in the $\text{NPT}_{xyz}$ |
299 |
integrator. All of the temperature variants were also run for 10~ns, |
300 |
with only the last 5~ns being used for accumulation of statistics. |
301 |
|
302 |
The second class of simulations were two configurations started from |
303 |
randomly dispersed lipids in a ``gas'' of water. The first |
304 |
($\text{R}_{\text{I}}$) was a simulation containing 72 lipids with |
305 |
1800 {\sc ssd} molecules simulated at 300~K. The second |
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 |
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 |
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 |
385 |
deuterium bond in relation to the bilayer normal at various points |
386 |
along the chains. In our model, there are no explicit hydrogens, but |
387 |
the order parameter can be written in terms of the carbon ordering at |
388 |
each point in the chain:\cite{egberts88} |
389 |
\begin{equation} |
390 |
S_{\text{{\sc cd}}} = \frac{2}{3}S_{xx} + \frac{1}{3}S_{yy} |
391 |
\label{lipidEq:scd1} |
392 |
\end{equation} |
393 |
Where $S_{ij}$ is given by: |
394 |
\begin{equation} |
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 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 $C_{n+1}$; |
405 |
$\mathbf{\hat{x}}\rightarrow$ vector perpindicular to |
406 |
$\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$. |
407 |
|
408 |
The order parameter has a range of $[1,-\frac{1}{2}]$. A value of 1 |
409 |
implies full order aligned to the bilayer axis, 0 implies full |
410 |
disorder, and $-\frac{1}{2}$ implies full order perpindicular to the |
411 |
bilayer axis. The {\sc cd} bond vector for carbons near the head group |
412 |
are usually ordered perpindicular to the bilayer normal, with tails |
413 |
farther away tending toward disorder. This makes the order paramter |
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 |
|
429 |
\begin{figure} |
430 |
\centering |
431 |
\includegraphics[width=\linewidth]{scdFig.eps} |
432 |
\caption[$\text{S}_{\text{{\sc cd}}}$ order parameter for our model]{A comparison of $|\text{S}_{\text{{\sc cd}}}|$ between our model (blue) and DMPC\cite{petrache00} (black) near 300~K.} |
433 |
\label{lipidFig:scdFig} |
434 |
\end{figure} |
435 |
|
436 |
|
437 |
|
438 |
|
439 |
|
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) & $\langle L_{\perp}\rangle$ ($\mbox{\AA}$) & % |
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 |
452 |
280 & 17.4 & 58.0 & 0.274 & 0.488 \\ \hline |
453 |
285 & 16.9 & 57.6 & 0.270 & 0.616 \\ \hline |
454 |
290 & 17.0 & 57.6 & 0.263 & 0.534 \\ \hline |
455 |
293 & 17.5 & 58.0 & 0.227 & 0.643 \\ \hline |
456 |
300 & 16.9 & 57.6 & 0.315 & 0.536 \\ \hline |
457 |
\end{tabular} |
458 |
\end{center} |
459 |
\end{table} |