36 |
|
|
37 |
|
Simulations of phospholipid bilayers are, by necessity, quite |
38 |
|
complex. The lipid molecules are large molecules containing many |
39 |
< |
atoms,and the head group of the lipid will typically contain charge |
39 |
> |
atoms, and the head group of the lipid will typically contain charge |
40 |
|
separated ions which set up a large dipole within the molecule. Adding |
41 |
|
to the complexity are the number of water molecules needed to properly |
42 |
< |
solvate the lipid bilayer. Because of these factors, many current |
43 |
< |
simulations are limited in both length and time scale due to to the |
44 |
< |
sheer number of calculations performed at every time step and the |
45 |
< |
lifetime of the researcher. A typical |
42 |
> |
solvate the lipid bilayer, typically 25 water molecules for every |
43 |
> |
lipid molecule. Because of these factors, many current simulations are |
44 |
> |
limited in both length and time scale due to to the sheer number of |
45 |
> |
calculations performed at every time step and the lifetime of the |
46 |
> |
researcher. A typical |
47 |
|
simulation\cite{saiz02,lindahl00,venable00,Marrink01} will have around |
48 |
|
64 phospholipids forming a bilayer approximately 40~$\mbox{\AA}$ by |
49 |
|
50~$\mbox{\AA}$ with roughly 25 waters for every lipid. This means |
50 |
|
there are on the order of 8,000 atoms needed to simulate these systems |
51 |
< |
and the trajectories in turn are integrated for times up to 10 ns. |
51 |
> |
and the trajectories are integrated for times up to 10 ns. |
52 |
|
|
53 |
|
These limitations make it difficult to study certain biologically |
54 |
|
interesting phenomena that don't fit within the short time and length |
61 |
|
roughly 25 waters for every lipid to fully solvate the bilayer. With |
62 |
|
the large number of atoms involved in a simulation of this magnitude, |
63 |
|
steps \emph{must} be taken to simplify the system to the point where |
64 |
< |
these numbers are reasonable. |
64 |
> |
the numbers of atoms becomes reasonable. |
65 |
|
|
66 |
|
Another system of interest would be drug molecule diffusion through |
67 |
< |
the membrane. Due to the fluid like properties of a lipid membrane, |
67 |
> |
the membrane. Due to the fluid-like properties of a lipid membrane, |
68 |
|
not all diffusion takes place at membrane channels. It is of interest |
69 |
|
to study certain molecules that may incorporate themselves directly |
70 |
|
into the membrane. These molecules may then have an appreciable |
71 |
|
waiting time (on the order of nanoseconds) within the |
72 |
|
bilayer. Simulation of such a long time scale again requires |
73 |
|
simplification of the system in order to lower the number of |
74 |
< |
calculations needed at each time step. |
74 |
> |
calculations needed at each time step or to increase the length of |
75 |
> |
each time step. |
76 |
|
|
77 |
|
|
78 |
|
\section{Methodology} |
81 |
|
|
82 |
|
The length scale simplifications are aimed at increasing the number of |
83 |
|
molecules that can be simulated without drastically increasing the |
84 |
< |
computational cost of the system. This is done by a combination of |
85 |
< |
substituting less expensive interactions for expensive ones and |
86 |
< |
decreasing the number of interaction sites per molecule. Namely, |
87 |
< |
point charge distributions are replaced with dipoles, and unified atoms are |
88 |
< |
used in place of water, phospholipid head groups, and alkyl groups. |
84 |
> |
computational cost of the simulation. This is done through a |
85 |
> |
combination of substituting less expensive interactions for expensive |
86 |
> |
ones and decreasing the number of interaction sites per |
87 |
> |
molecule. Namely, point charge distributions are replaced with |
88 |
> |
dipoles, and unified atoms are used in place of water, phospholipid |
89 |
> |
head groups, and alkyl groups. |
90 |
|
|
91 |
|
The replacement of charge distributions with dipoles allows us to |
92 |
< |
replace an interaction that has a relatively long range, |
93 |
< |
$(\frac{1}{r})$, for the coulomb potential, with that of a relatively |
94 |
< |
short range, $(\frac{1}{r^{3}})$, for dipole - dipole |
95 |
< |
potentials. Combined with a computational simplification algorithm |
96 |
< |
such as a Verlet neighbor list,\cite{allen87:csl} this should give |
97 |
< |
computational scaling by $N$. This is in comparison to the Ewald |
98 |
< |
sum\cite{leach01:mm} needed to compute the coulomb interactions, which |
99 |
< |
scales at best by $N \ln N$. |
92 |
> |
replace an interaction that has a relatively long range ($\frac{1}{r}$ |
93 |
> |
for the coulomb potential) with that of a relatively short range |
94 |
> |
($\frac{1}{r^{3}}$ for dipole - dipole potentials). Combined with |
95 |
> |
Verlet neighbor lists,\cite{allen87:csl} this should result in an |
96 |
> |
algorithm wich scales linearly with increasing system size. This is in |
97 |
> |
comparison to the Ewald sum\cite{leach01:mm} needed to compute |
98 |
> |
periodic replicas of the coulomb interactions, which scales at best by |
99 |
> |
$N \ln N$. |
100 |
|
|
101 |
|
The second step taken to simplify the number of calculations is to |
102 |
|
incorporate unified models for groups of atoms. In the case of water, |
103 |
|
we use the soft sticky dipole (SSD) model developed by |
104 |
< |
Ichiye\cite{Liu96} (Section~\ref{sec:ssdModel}). For the phospholipids, a |
105 |
< |
unified head atom with a dipole will replace the atoms in the head |
106 |
< |
group, while unified $\text{CH}_2$ and $\text{CH}_3$ atoms will |
107 |
< |
replace the alkyl units in the tails (Section~\ref{sec:lipidModel}). |
104 |
> |
Ichiye\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md} |
105 |
> |
(Section~\ref{sec:ssdModel}). For the phospholipids, a unified head |
106 |
> |
atom with a dipole will replace the atoms in the head group, while |
107 |
> |
unified $\text{CH}_2$ and $\text{CH}_3$ atoms will replace the alkyl |
108 |
> |
units in the tails (Section~\ref{sec:lipidModel}). |
109 |
|
|
110 |
|
The time scale simplifications are introduced so that we can take |
111 |
|
longer time steps. By increasing the size of the time steps taken by |
112 |
< |
the simulation, we are able to integrate the simulation trajectory |
113 |
< |
with fewer calculations. However, care must be taken that any |
112 |
> |
the simulation, we are able to integrate a given length of time using |
113 |
> |
fewer calculations. However, care must be taken that any |
114 |
|
simplifications used, still conserve the total energy of the |
115 |
|
simulation. In practice, this means taking steps small enough to |
116 |
|
resolve all motion in the system without accidently moving an object |
128 |
|
\subsection{The Soft Sticky Water Model} |
129 |
|
\label{sec:ssdModel} |
130 |
|
|
131 |
< |
%\begin{floatingfigure}{55mm} |
132 |
< |
%\includegraphics[width=45mm]{ssd.epsi} |
133 |
< |
%\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference. \vspace{5mm}} |
134 |
< |
%\label{fig:ssdModel} |
135 |
< |
%\end{floatingfigure} |
131 |
> |
\begin{figure} |
132 |
> |
\begin{center} |
133 |
> |
\includegraphics[width=50mm]{ssd.epsi} |
134 |
> |
\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference.} |
135 |
> |
\end{center} |
136 |
> |
\label{fig:ssdModel} |
137 |
> |
\end{figure} |
138 |
|
|
139 |
|
The water model used in our simulations is a modified soft |
140 |
|
Stockmayer-sphere model.\cite{stevens95} Like the Stockmayer-sphere, the SSD |
141 |
< |
model\cite{Liu96} consists of a Lennard-Jones interaction site and a |
141 |
> |
model consists of a Lennard-Jones interaction site and a |
142 |
|
dipole both located at the water's center of mass (Figure |
143 |
|
\ref{fig:ssdModel}). However, the SSD model extends this by adding a |
144 |
|
tetrahedral potential to correct for hydrogen bonding. |
145 |
|
|
146 |
< |
The SSD water potential is then given by the following equation: |
146 |
> |
The SSD water potential for a pair of water molecules is then given by |
147 |
> |
the following equation: |
148 |
|
\begin{equation} |
149 |
|
V_{\text{SSD}} = V_{\text{LJ}}(r_{i\!j}) + V_{\text{dp}}(\mathbf{r}_{i\!j}, |
150 |
|
\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
152 |
|
\boldsymbol{\Omega}_{j}) |
153 |
|
\label{eq:ssdTotPot} |
154 |
|
\end{equation} |
155 |
< |
$V_{\text{LJ}}$ is the Lennard-Jones potential: |
155 |
> |
where $\mathbf{r}_{ij}$ is the vector between molecules $i$ and $j$, |
156 |
> |
and $\boldsymbol{\Omega}$ is the orientation of molecule $i$ or $j$ |
157 |
> |
respectively. $V_{\text{LJ}}$ is the Lennard-Jones potential: |
158 |
|
\begin{equation} |
159 |
|
V_{\text{LJ}} = |
160 |
|
4\epsilon_{ij} \biggl[ |
163 |
|
\biggr] |
164 |
|
\label{eq:lennardJonesPot} |
165 |
|
\end{equation} |
166 |
< |
where $r_{ij}$ is the distance between two $ij$ pairs, $\sigma_{ij}$ |
166 |
> |
here $\sigma_{ij}$ |
167 |
|
scales the length of the interaction, and $\epsilon_{ij}$ scales the |
168 |
|
energy of the potential. For SSD, $\sigma_{\text{SSD}} = 3.051 \mbox{ |
169 |
|
\AA}$ and $\epsilon_{\text{SSD}} = 0.152\text{ kcal/mol}$. |
178 |
|
{r^{5}_{ij}} \biggr] |
179 |
|
\label{eq:dipolePot} |
180 |
|
\end{equation} |
181 |
< |
where $\mathbf{r}_{ij}$ is the vector between $i$ and $j$, |
182 |
< |
$\boldsymbol{\Omega}$ is the orientation of the species, and |
183 |
< |
$\boldsymbol{\mu}$ is the dipole vector. The SSD model specifies a dipole |
184 |
< |
magnitude of 2.35~D for water. |
181 |
> |
where $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$, |
182 |
> |
$\boldsymbol{\mu}_i$ takes its orientation from |
183 |
> |
$\boldsymbol{\Omega}_i$. The SSD model specifies a dipole magnitude of |
184 |
> |
2.35~D for water. |
185 |
|
|
186 |
< |
The hydrogen bonding of the model is governed by the $V_{\text{sp}}$ |
186 |
> |
The hydrogen bonding is modeled by the $V_{\text{sp}}$ |
187 |
|
term of the potential. Its form is as follows: |
188 |
|
\begin{equation} |
189 |
|
V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
217 |
|
\label{eq:spCorrection} |
218 |
|
\end{equation} |
219 |
|
The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical |
220 |
< |
polar coordinates of the position of sphere $j$ in the reference frame |
221 |
< |
fixed on sphere $i$ with the z-axis aligned with the dipole moment. |
220 |
> |
coordinates of the position of molecule $j$ in the reference frame |
221 |
> |
fixed on molecule $i$ with the z-axis aligned with the dipole moment. |
222 |
|
The correction |
223 |
|
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
224 |
|
is needed because |
226 |
|
vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. |
227 |
|
|
228 |
|
Finally, the sticky potential is scaled by a cutoff function, |
229 |
< |
$s(r_{ij})$ that scales smoothly between 0 and 1. It is represented |
229 |
> |
$s(r_{ij})$, that scales smoothly between 0 and 1. It is represented |
230 |
|
by: |
231 |
|
\begin{equation} |
232 |
|
s(r_{ij}) = |
246 |
|
being equal to either 3.35~$\mbox{\AA}$ for $s(r_{ij})$ or |
247 |
|
4.0~$\mbox{\AA}$ for $s'(r_{ij})$, the sticky potential is only active |
248 |
|
over an extremely short range, and then only with other SSD |
249 |
< |
molecules. Therefore, it's predominant interaction is through it's |
250 |
< |
point dipole and Lennard-Jones sphere. |
249 |
> |
molecules. Therefore, it's predominant interaction is through the |
250 |
> |
point dipole and the Lennard-Jones sphere. Of these, only the dipole |
251 |
> |
interaction can be considered ``long-range''. |
252 |
|
|
253 |
|
\subsection{The Phospholipid Model} |
254 |
|
\label{sec:lipidModel} |
255 |
|
|
256 |
< |
%\begin{floatingfigure}{90mm} |
257 |
< |
%\includegraphics[angle=-90,width=80mm]{lipidModel.epsi} |
258 |
< |
%\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length. \vspace{5mm}} |
259 |
< |
%\label{fig:lipidModel} |
260 |
< |
%\end{floatingfigure} |
256 |
> |
\begin{figure} |
257 |
> |
\begin{center} |
258 |
> |
\includegraphics[angle=-90,width=80mm]{lipidModel.epsi} |
259 |
> |
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length. \vspace{5mm}} |
260 |
> |
\end{center} |
261 |
> |
\label{fig:lipidModel} |
262 |
> |
\end{figure} |
263 |
|
|
264 |
|
The lipid molecules in our simulations are unified atom models. Figure |
265 |
< |
\ref{fig:lipidModel} shows a template drawing for one of our |
265 |
> |
\ref{fig:lipidModel} shows a schematic for one of our |
266 |
|
lipids. The Head group of the phospholipid is replaced by a single |
267 |
|
Lennard-Jones sphere with a freely oriented dipole placed at it's |
268 |
< |
center. The magnitude of it's dipole moment is 20.6 D. The tail atoms |
269 |
< |
are unified $\text{CH}_2$ and $\text{CH}_3$ atoms and are also modeled |
270 |
< |
as Lennard-Jones spheres. The total potential for the lipid is |
271 |
< |
represented by Equation \ref{eq:lipidModelPot}. |
268 |
> |
center. The magnitude of the dipole moment is 20.6 D, chosen to match |
269 |
> |
that of DPPC\cite{Cevc87}. The tail atoms are unified $\text{CH}_2$ |
270 |
> |
and $\text{CH}_3$ atoms and are also modeled as Lennard-Jones |
271 |
> |
spheres. The total potential for the lipid is represented by Equation |
272 |
> |
\ref{eq:lipidModelPot}. |
273 |
|
|
274 |
|
\begin{equation} |
275 |
< |
V_{\mbox{lipid}} = \overbrace{% |
276 |
< |
V_{\text{bend}}(\theta_{ijk}) +% |
277 |
< |
V_{\text{tors.}}(\phi_{ijkl})}^{bonded} |
278 |
< |
+ \overbrace{% |
279 |
< |
V_{\text{LJ}}(r_{i\!j}) + |
280 |
< |
V_{\text{dp}}(r_{i\!j},\Omega_{i},\Omega_{j})% |
268 |
< |
}^{non-bonded} |
275 |
> |
V_{\text{lipid}} = |
276 |
> |
\sum_{i}V_{i}^{\text{internal}} |
277 |
> |
+ \sum_i \sum_{j>i} \sum_{\text{$\alpha$ in $i$}} |
278 |
> |
\sum_{\text{$\beta$ in $j$}} |
279 |
> |
V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}}) |
280 |
> |
+\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j}) |
281 |
|
\label{eq:lipidModelPot} |
282 |
|
\end{equation} |
283 |
+ |
where, |
284 |
+ |
\begin{equation} |
285 |
+ |
V_{i}^{\text{internal}} = |
286 |
+ |
\sum_{\text{bends}}V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
287 |
+ |
+ \sum_{\text{torsions}}V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
288 |
+ |
+ \sum_{\alpha} \sum_{\beta>\alpha}V_{\text{LJ}}(r_{\alpha \beta}) |
289 |
+ |
\label{eq:lipidModelPotInternal} |
290 |
+ |
\end{equation} |
291 |
|
|
292 |
|
The non-bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are |
293 |
|
the Lennard-Jones and dipole-dipole interactions respectively. For the |
294 |
< |
non-bonded potentials, only the bend and the torsional potentials are |
294 |
> |
bonded potentials, only the bend and the torsional potentials are |
295 |
|
calculated. The bond potential is not calculated, and the bond lengths |
296 |
|
are constrained via RATTLE.\cite{leach01:mm} The bend potential is of |
297 |
|
the form: |
298 |
|
\begin{equation} |
299 |
< |
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}\frac{(\theta_{ijk} - \theta_0)^2}{2} |
299 |
> |
V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
300 |
> |
= k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \theta_0)^2}{2} |
301 |
|
\label{eq:bendPot} |
302 |
|
\end{equation} |
303 |
|
Where $k_{\theta}$ sets the stiffness of the bend potential, and $\theta_0$ |
304 |
|
sets the equilibrium bend angle. The torsion potential was given by: |
305 |
|
\begin{equation} |
306 |
< |
V_{\text{tors.}}(\phi_{ijkl})= c_1[1+\cos\phi_{ijkl}] |
307 |
< |
+ c_2 [1 - \cos(2\phi_{ijkl})] + c_3[1 + \cos(3\phi_{ijkl})] |
306 |
> |
V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
307 |
> |
= c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}] |
308 |
> |
+ c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] |
309 |
> |
+ c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})] |
310 |
|
\label{eq:torsPot} |
311 |
|
\end{equation} |
312 |
|
All parameters for bonded and non-bonded potentials in the tail atoms |
323 |
|
\subsection{Starting Configuration and Parameters} |
324 |
|
\label{sec:5x5Start} |
325 |
|
|
326 |
< |
Our first simulation is an array of 25 single chained lipids in a sea |
326 |
> |
\begin{figure} |
327 |
> |
\begin{center} |
328 |
> |
\includegraphics[width=70mm]{5x5-initial.eps} |
329 |
> |
\caption{The starting configuration of the 25 lipid system. A box is drawn around the periodic image.} |
330 |
> |
\end{center} |
331 |
> |
\label{fig:5x5Start} |
332 |
> |
\end{figure} |
333 |
> |
|
334 |
> |
\begin{figure} |
335 |
> |
\begin{center} |
336 |
> |
\includegraphics[width=70mm]{5x5-6.27ns.eps} |
337 |
> |
\caption{The 25 lipid system at 6.27~ns} |
338 |
> |
\end{center} |
339 |
> |
\label{fig:5x5Final} |
340 |
> |
\end{figure} |
341 |
> |
|
342 |
> |
Our first simulation is an array of 25 single chain lipids in a sea |
343 |
|
of water (Figure \ref{fig:5x5Start}). The total number of water |
344 |
|
molecules is 1386, giving a final of water concentration of 70\% |
345 |
|
wt. The simulation box measures 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
351 |
|
\label{sec:5x5Results} |
352 |
|
|
353 |
|
Figure \ref{fig:5x5Final} shows a snapshot of the system at |
354 |
< |
3.6~ns. Note that the system has spontaneously self assembled into a |
354 |
> |
6.27~ns. Note that the system has spontaneously self assembled into a |
355 |
|
bilayer. Discussion of the length scales of the bilayer will follow in |
356 |
|
this section. However, it is interesting to note a key qualitative |
357 |
|
property of the system revealed by this snapshot, the tail chains are |
380 |
|
For the species containing dipoles, a second pair wise distribution |
381 |
|
function was used, $g_{\gamma}(r)$. It is of the form: |
382 |
|
\begin{equation} |
383 |
< |
g_{\gamma}(r) = foobar |
383 |
> |
g_{\gamma}(r) = \langle \sum_i \sum_{j>i} |
384 |
> |
(\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle |
385 |
|
\label{eq:gammaofr} |
386 |
|
\end{equation} |
387 |
|
Where $\gamma_{ij}$ is the angle between the dipole of atom $j$ with |
395 |
|
for the Head groups of the lipids. The first peak in the $g(r)$ at |
396 |
|
4.03~$\mbox{\AA}$ is the nearest neighbor separation of the heads of |
397 |
|
two lipids. This corresponds to a maximum in the $g_{\gamma}(r)$ which |
398 |
< |
means that the two neighbors on the same monolayer have their dipoles |
399 |
< |
aligned. The broad peak at 6.5~$\mbox{\AA}$ is the inter-bilayer |
398 |
> |
means that the two neighbors on the same leaf have their dipoles |
399 |
> |
aligned. The broad peak at 6.5~$\mbox{\AA}$ is the inter-surface |
400 |
|
spacing. Here, there is a corresponding anti-alignment in the angular |
401 |
|
correlation. This means that although the dipoles are aligned on the |
402 |
|
same monolayer, the dipoles will orient themselves to be anti-aligned |
426 |
|
\subsection{Starting Configuration and Parameters} |
427 |
|
\label{sec:r50Start} |
428 |
|
|
429 |
+ |
\begin{figure} |
430 |
+ |
\begin{center} |
431 |
+ |
\includegraphics[width=70mm]{r50-initial.eps} |
432 |
+ |
\caption{The starting configuration of the 50 lipid system.} |
433 |
+ |
\end{center} |
434 |
+ |
\label{fig:r50Start} |
435 |
+ |
\end{figure} |
436 |
+ |
|
437 |
+ |
\begin{figure} |
438 |
+ |
\begin{center} |
439 |
+ |
\includegraphics[width=70mm]{r50-2.21ns.eps} |
440 |
+ |
\caption{The 50 lipid system at 2.21~ns} |
441 |
+ |
\end{center} |
442 |
+ |
\label{fig:r50Final} |
443 |
+ |
\end{figure} |
444 |
+ |
|
445 |
|
The second simulation consists of 50 single chained lipid molecules |
446 |
|
embedded in a sea of 1384 SSD waters (54\% wt.). The lipids in this |
447 |
|
simulation were started with random orientation and location (Figure |
448 |
< |
\ref{fig:r50Start} ) The simulation box measured 34.5~$\mbox{\AA}$ x |
449 |
< |
39.4~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ with periodic boundary conditions |
448 |
> |
\ref{fig:r50Start} ) The simulation box measured 26.6~$\mbox{\AA}$ x |
449 |
> |
26.6~$\mbox{\AA}$ x 108.4~$\mbox{\AA}$ with periodic boundary conditions |
450 |
|
imposed. The simulation was run in the NVE ensemble with an average |
451 |
|
temperature of 300~K. |
452 |
|
|
462 |
|
Figures \ref{fig:r50HHCorr}, \ref{fig:r50CCg}, and \ref{fig:r50} are |
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|
the same correlation functions for the random 50 simulation as for the |
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|
previous simulation of 25 lipids. What is most interesting to note, is |
465 |
< |
the high degree of similarity between the correlation functions for |
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< |
each system. Even though the 25 lipid simulation formed a bilayer and |
467 |
< |
the random 50 simulation is still in the micelle stage, both have a |
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< |
inter surface spacing of 6.5~$\mbox{\AA}$ with the same characteristic |
469 |
< |
anti-alignment between surfaces. Not as surprising, is the consistency |
470 |
< |
of the closest packing statistics between systems. Namely, a head-head |
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< |
closest approach distance of 4~$\mbox{\AA}$, and similar findings for |
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< |
the chain-chain and head-water distributions as in the 25 lipid |
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< |
system. |
465 |
> |
the high degree of similarity between the correlation functions |
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> |
between systems. Even though the 25 lipid simulation formed a bilayer |
467 |
> |
and the random 50 simulation is still in the micelle stage, both have |
468 |
> |
an inter-surface spacing of 6.5~$\mbox{\AA}$ with the same |
469 |
> |
characteristic anti-alignment between surfaces. Not as surprising, is |
470 |
> |
the consistency of the closest packing statistics between |
471 |
> |
systems. Namely, a head-head closest approach distance of |
472 |
> |
4~$\mbox{\AA}$, and similar findings for the chain-chain and |
473 |
> |
head-water distributions as in the 25 lipid system. |
474 |
|
|
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|
\section{Future Directions} |
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|
|
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|
Current simulations indicate that our model is a feasible one, however |
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< |
improvements will need to be made to allow the system to simulate an |
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< |
isobaric-isothermal ensemble. This will allow the system to relax to |
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< |
an equilibrium configuration at room temperature and pressure allowing |
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< |
us to compare our model to experimental results. Also, we plan to |
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< |
parallelize the code for an even greater speedup. This will allow us |
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< |
to simulate the size systems needed to examine phenomena such as the |
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< |
ripple phase and drug molecule diffusion |
478 |
> |
improvements will need to be made to allow the system to be simulated |
479 |
> |
in the isobaric-isothermal ensemble. This will relax the system to an |
480 |
> |
equilibrium configuration at room temperature and pressure allowing us |
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> |
to compare our model to experimental results. Also, we are in the |
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> |
process of parallelizeing the code for an even greater speedup. This |
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> |
will allow us to simulate the size systems needed to examine phenomena |
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> |
such as the ripple phase and drug molecule diffusion |
485 |
|
|
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< |
Once the work has completed on the simulation engine, we would then |
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< |
like to use it to explore phase diagram for our model. By |
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> |
Once the work has been completed on the simulation engine, we will |
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> |
then use it to explore the phase diagram for our model. By |
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|
characterizing how our model parameters affect the bilayer properties, |
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< |
we hope to tailor our model to more closely match real biological |
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< |
molecules. With this information, we then hope to incorporate |
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> |
we will tailor our model to more closely match real biological |
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> |
molecules. With this information, we will then incorporate |
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|
biologically relevant molecules into the system and observe their |
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|
transport properties across the membrane. |
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|
|
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|
I would like to thank Dr. J.Daniel Gezelter for his guidance on this |
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|
project. I would also like to acknowledge the following for their help |
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|
and discussions during this project: Christopher Fennell, Charles |
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< |
Vardeman, Teng Lin, Megan Sprague, Patrick Conforti, and Dan Combest. |
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> |
Vardeman, Teng Lin, Megan Sprague, Patrick Conforti, and Dan |
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> |
Combest. Funding for this project came from the National Science |
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> |
Foundation. |
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|
|
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|
\pagebreak |
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\bibliographystyle{achemso} |