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Revision 971 by mmeineke, Wed Jan 21 02:25:18 2004 UTC vs.
Revision 1026 by mmeineke, Thu Feb 5 00:37:36 2004 UTC

+\section{\label{lipidSec:Intro}Introduction}
-+
+In the past 10 years, computer speeds have allowed for the atomistic
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+simulation of phospholipid bilayers.  These simulations have ranged
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+from simulation of the gel phase ($L_{\beta}$) of
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+dipalmitoylphosphatidylcholine (DPPC), \cite{Lindahl:2000} to the
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+spontaneous aggregation of DPPC molecules into fluid phase
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+($L_{\alpha}$ bilayers. \cite{Marrinck:2001} With the exception of a
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+few ambitious
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+simulations,\cite{Marrinch:2001b,Marrinck:2002,Lindahl:2000} most
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+investigations are limited to 64 to 256
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+phospholipids.\cite{Lindal:2000,Sum:2003,Venable:2000,Gomez:2003,Smondyrev:1999,Marrinck:2001a}
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+This is due to the expense of the computer calculations involved when
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+performing these simulations.  To properly hydrate a bilayer, one
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+typically needs 25 water molecules for every lipid, bringing the total
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+number of atoms simulated to roughly 8,000 for a system of 64 DPPC
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+molecules. Added to the difficluty is the electrostatic nature of the
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+phospholipid head groups and water, requiring the computationally
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+expensive Ewald sum or its slightly faster derivative particle mesh
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+Ewald sum.\cite{Nina:2002,Norberg:2000,Patra:2003} These factors all
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+limit the potential size and time lenghts of bilayer simulations.
-+
-+
+Unfortunately, much of biological interest happens on time and length
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+scales unfeasible with current simulation. One such example is the
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+observance of a ripple phase ($P_{\beta'}$) between the $L_{\beta}$
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+and $L_{\alpha}$ phases of certain phospholipid
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+bilayers.\cite{Katsaras:2000,Sengupta:2000} These ripples are shown to
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+have periodicity on the order of 100-200~$\mbox{\AA}$. A simulation on
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+this length scale would have approximately 1,300 lipid molecules with
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+an additional 25 water molecules per lipid to fully solvate the
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+bilayer. A simulation of this size is impractical with current
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+atomistic models.
-+
-+
+Another class of simulations to consider, are those dealing with the
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+diffusion of molecules through a bilayer.  Due to the fluid-like
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+properties of a lipid membrane, not all diffusion across the membrane
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+happens at pores.  Some molecules of interest may incorporate
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+themselves directly into the membrane.  Once here, they may possess an
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+appreciable waiting time (on the order of 10's to 100's of
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+nanoseconds) within the bilayer.  Such long simulation times are
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+difficulty to obtain when integrating the system with atomistic
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+detail.
-+
-+
+Addressing these issues, several schemes have been proposed.  One
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+approach by Goetz and Liposky\cite{Goetz:1998} is to model the entire
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+system as Lennard-Jones spheres. Phospholipids are represented by
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+chains of beads with the top most beads identified as the head
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+atoms. Polar and non-polar interactions are mimicked through
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+attractive and soft-repulsive potentials respectively.  A similar
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+model proposed by Marrinck \emph{et. al.}\cite{Marrinck:2004}~ uses a
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+similar technique for modeling polar and non-polar interactions with
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+Lennard-Jones spheres. However, they also include charges on the head
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+group spheres to mimic the electrostatic interactions of the
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+bilayer. While the solvent spheres are kept charge-neutral and
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+interact with the bilayer solely through an attractive Lennard-Jones
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+potential.
-+
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+The model used in this investigation adds more information to the
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+interactions than the previous two models, while still balancing the
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+need for simplifications over atomistic detail.  The model uses
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+Lennard-Jones spheres for the head and tail groups of the
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+phopholipids, allowing for the ability to scale the parameters to
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+reflect various sized chain configurations while keeping the number of
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+interactions small.  What sets this model apart, however, is the use
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+of dipoles to represent the electrosttaic nature of the
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+phospholipids. The dipole electrostatic interaction is shorter range
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+than coulombic ($\frac{1}{r^3}$ versus $\frac{1}{r}$), eliminating the
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+need for a costly Ewald sum.
-+
-+
+Another key feature of this model, is the use of a dipolar water model
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+to represent the solvent. The soft sticky dipole ({\scssd})
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+water \cite{Liu:1996a} relies on the dipole for long range
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+electrostatic effects, butalso contains a short range correction for
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+hydrogen bonding. In this way the systems in this research mimic the
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+entropic contribution to the hydrophobic effect due to hydrogen-bond
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+network deformation around a non-polar entity, \emph{i.e.}~ the
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+phospholipid.
-+
-+
+The following is an outline of this chapter.
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+Sec.~\ref{lipoidSec:Methods} is an introduction to the lipid model
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+used in these simulations.  As well as clarification about the water
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+model and integration techniques.  The various simulation setups
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+explored in this research are outlined in
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+Sec.~\ref{lipidSec:ExpSetup}. Sec.~\ref{lipidSec:Results} and
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+Sec.~\ref{lipidSec:Discussion} give a summary of the results and
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+interpretation of those results respectively.  Finally, the
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+conclusions of this chapter are presented in
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+Sec.~\ref{lipidSec:Conclusion}.
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+\section{\label{lipidSec:Methods}Methods}
-–
+\subsection{\label{lipidSec:lipidMedel}The Lipid Model}
-+
-+
+\subsection{\label{lipidSec:lipidModel}The Lipid Model}
-+
+\begin{figure}
+\caption{Schematic diagram of the single chain phospholipid model}
+the same bond, bend, or torsion.  However, internal interactions not
+directly involved in a bonded pair are calculated.
-+
+All simulations presented here use a two chained lipid as pictured in
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+Fig.~\ref{lipidFig:twochain}.  The chains are both eight beads long,
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+and their mass and Lennard Jones parameters are summarized in
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+Table~\ref{lipidTable:tcLJParams}. The magnitude of the dipole moment
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+for the head bead is 10.6~Debye, and the bend and torsion parameters
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+are summarized in Table~\ref{lipidTable:teBTParams}.
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+\section{label{lipidSec:furtherMethod}Further Methodology}
-+
-+
+As mentioned previously, the water model used throughout these
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+simulations was the {\scssd} model of
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+Ichiye.\cite{liu:1996a,Liu:1996b,Chandra:1999} A discussion of the
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+model can be found in Sec.~\ref{oopseSec:SSD}. As for the integration
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+of the equations of motion, all simulations were performed in an
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+orthorhombic periodic box with a thermostat on velocities, and an
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+independent barostat on each cartesian axis $x$, $y$, and $z$.  This
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+is the $\text{NPT}_{xyz}$. ensemble described in Sec.~\ref{oopseSec:Ensembles}.
-+
-+
-+
+\subsection{\label{lipidSec:ExpSetup}Experimental Setup}
-+
-+
+Two main starting configuration classes were used in this research:
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+random and ordered bilayers.  The ordered bilayer starting
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+configurations were all started from an equilibrated bilayer at
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+~K. The original configuration for the first 300~K run was
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+assembled by placing the phospholipids centers of mass on a planar
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+hexagonal lattice.  The lipids were oriented with their long axis
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+perpendicular to the plane.  The second leaf simply mirrored the first
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+leaf, and the appropriate number of waters were then added above and
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+below the bilayer.
-+
-+
+The random configurations took more work to generate.  To begin, a
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+test lipid was placed in a simulation box already containing water at
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+the intended density.  The waters were then tested for overlap with
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+the lipid using a 5.0~$\mbox{\AA}$ buffer distance.  This gave an
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+estimate for the number of waters each lipid would displace in a
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+simulation box. A target number of waters was then defined which
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+included the number of waters each lipid would displace, the number of
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+waters desired to solvate each lipid, and a fudge factor to pad the
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+initialization.
-+
-+
+Next, a cubic simulation box was created that contained at least the
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+target number of waters in an FCC lattice (the lattice was for ease of
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+placement).  What followed was a RSA simulation similar to those of
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+Chapt.~\ref{chapt:RSA}. The lipids were sequentially given a random
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+position and orientation within the box.  If a lipid's position caused
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+atomic overlap with any previously adsorbed lipid, its position and
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+orientation were rejected, and a new random adsorption site was
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+attempted. The RSA simulation proceeded until all phospholipids had
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+been adsorbed.  After adsorption, all water molecules with locations
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+that overlapped with the atomic coordinates of the lipids were
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+removed.
-+
-+
+Finally, water molecules were removed one by one at random until the
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+desired number of waters per lipid was reached.  The typical low final
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+density for these initial configurations was not a problem, as the box
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+would shrink to an appropriate size within the first 50~ps of a
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+simulation in the $\text{NPT}_{xyz}$ ensemble.
-+
-+
+\subsection{\label{lipidSec:Configs}The simulation configurations}
-+
-+
+Table ~\ref{lipidTable:simNames} summarizes the names and important
-+
+details of the simulations.  The B set of simulations were all started
-+
+in an ordered bilayer and observed over a period of 10~ns. Simulution
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+RL was integrated for approximately 20~ns starting from a random
-+
+configuration as an example of spontaneous bilayer aggregation.
-+
+Lastly, simulation RH was also started from a random configuration,
-+
+but with a lesser water content and higher temperature to show the
-+
+spontaneous aggregation of an inverted hexagonal lamellar phase.

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Comparing trunk/mattDisertation/lipid.tex (file contents): Revision 971 by mmeineke, Wed Jan 21 02:25:18 2004 UTC vs. Revision 1026 by mmeineke, Thu Feb 5 00:37:36 2004 UTC

Diff Legend

Comparing trunk/mattDisertation/lipid.tex (file contents):
Revision 971 by mmeineke, Wed Jan 21 02:25:18 2004 UTC vs.
Revision 1026 by mmeineke, Thu Feb 5 00:37:36 2004 UTC