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|
| 5 |
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\section{\label{lipidSec:Intro}Introduction} |
| 6 |
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|
| 7 |
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In the past 10 years, computer speeds have allowed for the atomistic |
| 8 |
+ |
simulation of phospholipid bilayers. These simulations have ranged |
| 9 |
+ |
from simulation of the gel phase ($L_{\beta}$) of |
| 10 |
+ |
dipalmitoylphosphatidylcholine (DPPC), \cite{Lindahl:2000} to the |
| 11 |
+ |
spontaneous aggregation of DPPC molecules into fluid phase |
| 12 |
+ |
($L_{\alpha}$ bilayers. \cite{Marrinck:2001} With the exception of a |
| 13 |
+ |
few ambitious |
| 14 |
+ |
simulations,\cite{Marrinch:2001b,Marrinck:2002,Lindahl:2000} most |
| 15 |
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investigations are limited to 64 to 256 |
| 16 |
+ |
phospholipids.\cite{Lindal:2000,Sum:2003,Venable:2000,Gomez:2003,Smondyrev:1999,Marrinck:2001a} |
| 17 |
+ |
This is due to the expense of the computer calculations involved when |
| 18 |
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performing these simulations. To properly hydrate a bilayer, one |
| 19 |
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typically needs 25 water molecules for every lipid, bringing the total |
| 20 |
+ |
number of atoms simulated to roughly 8,000 for a system of 64 DPPC |
| 21 |
+ |
molecules. Added to the difficluty is the electrostatic nature of the |
| 22 |
+ |
phospholipid head groups and water, requiring the computationally |
| 23 |
+ |
expensive Ewald sum or its slightly faster derivative particle mesh |
| 24 |
+ |
Ewald sum.\cite{Nina:2002,Norberg:2000,Patra:2003} These factors all |
| 25 |
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limit the potential size and time lenghts of bilayer simulations. |
| 26 |
+ |
|
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+ |
Unfortunately, much of biological interest happens on time and length |
| 28 |
+ |
scales unfeasible with current simulation. One such example is the |
| 29 |
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observance of a ripple phase ($P_{\beta'}$) between the $L_{\beta}$ |
| 30 |
+ |
and $L_{\alpha}$ phases of certain phospholipid |
| 31 |
+ |
bilayers.\cite{Katsaras:2000,Sengupta:2000} These ripples are shown to |
| 32 |
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have periodicity on the order of 100-200~$\mbox{\AA}$. A simulation on |
| 33 |
+ |
this length scale would have approximately 1,300 lipid molecules with |
| 34 |
+ |
an additional 25 water molecules per lipid to fully solvate the |
| 35 |
+ |
bilayer. A simulation of this size is impractical with current |
| 36 |
+ |
atomistic models. |
| 37 |
+ |
|
| 38 |
+ |
Another class of simulations to consider, are those dealing with the |
| 39 |
+ |
diffusion of molecules through a bilayer. Due to the fluid-like |
| 40 |
+ |
properties of a lipid membrane, not all diffusion across the membrane |
| 41 |
+ |
happens at pores. Some molecules of interest may incorporate |
| 42 |
+ |
themselves directly into the membrane. Once here, they may possess an |
| 43 |
+ |
appreciable waiting time (on the order of 10's to 100's of |
| 44 |
+ |
nanoseconds) within the bilayer. Such long simulation times are |
| 45 |
+ |
difficulty to obtain when integrating the system with atomistic |
| 46 |
+ |
detail. |
| 47 |
+ |
|
| 48 |
+ |
Addressing these issues, several schemes have been proposed. One |
| 49 |
+ |
approach by Goetz and Liposky\cite{Goetz:1998} 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{Marrinck:2004}~ 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, |
| 64 |
+ |
|
| 65 |
|
\section{\label{lipidSec:Methods}Methods} |
| 66 |
|
|
| 67 |
|
\subsection{\label{lipidSec:lipidMedel}The Lipid Model} |
