| 1 |
|
| 2 |
|
| 3 |
\chapter{\label{chapt:lipid}Phospholipid Simulations} |
| 4 |
|
| 5 |
\section{\label{lipidSec:Intro}Introduction} |
| 6 |
|
| 7 |
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 |
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 |
performing these simulations. To properly hydrate a bilayer, one |
| 19 |
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 |
limit the potential size and time lenghts of bilayer simulations. |
| 26 |
|
| 27 |
Unfortunately, much of biological interest happens on time and length |
| 28 |
scales unfeasible with current simulation. One such example is the |
| 29 |
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 |
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} |
| 68 |
|
| 69 |
\begin{figure} |
| 70 |
|
| 71 |
\caption{Schematic diagram of the single chain phospholipid model} |
| 72 |
|
| 73 |
\label{lipidFig:lipidModel} |
| 74 |
|
| 75 |
\end{figure} |
| 76 |
|
| 77 |
The phospholipid model used in these simulations is based on the |
| 78 |
design illustrated in Fig.~\ref{lipidFig:lipidModel}. The head group |
| 79 |
of the phospholipid is replaced by a single Lennard-Jones sphere of |
| 80 |
diameter $fix$, with $fix$ scaling the well depth of its van der Walls |
| 81 |
interaction. This sphere also contains a single dipole of magnitude |
| 82 |
$fix$, where $fix$ can be varied to mimic the charge separation of a |
| 83 |
given phospholipid head group. The atoms of the tail region are |
| 84 |
modeled by unified atom beads. They are free of partial charges or |
| 85 |
dipoles, containing only Lennard-Jones interaction sites at their |
| 86 |
centers of mass. As with the head groups, their potentials can be |
| 87 |
scaled by $fix$ and $fix$. |
| 88 |
|
| 89 |
The long range interactions between lipids are given by the following: |
| 90 |
\begin{equation} |
| 91 |
EQ Here |
| 92 |
\label{lipidEq:LJpot} |
| 93 |
\end{equation} |
| 94 |
and |
| 95 |
\begin{equation} |
| 96 |
EQ Here |
| 97 |
\label{lipidEq:dipolePot} |
| 98 |
\end{equation} |
| 99 |
Where $V_{\text{LJ}}$ is the Lennard-Jones potential and |
| 100 |
$V_{\text{dipole}}$ is the dipole-dipole potential. As previously |
| 101 |
stated, $\sigma_{ij}$ and $\epsilon_{ij}$ are the Lennard-Jones |
| 102 |
parameters which scale the length and depth of the interaction |
| 103 |
respectively, and $r_{ij}$ is the distance between beads $i$ and $j$. |
| 104 |
In $V_{\text{dipole}}$, $\mathbf{r}_{ij}$ is the vector starting at |
| 105 |
bead$i$ and pointing towards bead $j$. Vectors $\mathbf{\Omega}_i$ |
| 106 |
and $\mathbf{\Omega}_j$ are the orientational degrees of freedom for |
| 107 |
beads $i$ and $j$. $|\mu_i|$ is the magnitude of the dipole moment of |
| 108 |
$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 109 |
vector of $\boldsymbol{\Omega}_i$. |
| 110 |
|
| 111 |
The model also allows for the bonded interactions of bonds, bends, and |
| 112 |
torsions. The bonds between two beads on a chain are of fixed length, |
| 113 |
and are maintained according to the {\sc rattle} algorithm. \cite{fix} |
| 114 |
The bends are subject to a harmonic potential: |
| 115 |
\begin{equation} |
| 116 |
eq here |
| 117 |
\label{lipidEq:bendPot} |
| 118 |
\end{equation} |
| 119 |
where $fix$ scales the strength of the harmonic well, and $fix$ is the |
| 120 |
angle between bond vectors $fix$ and $fix$. The torsion potential is |
| 121 |
given by: |
| 122 |
\begin{equation} |
| 123 |
eq here |
| 124 |
\label{lipidEq:torsionPot} |
| 125 |
\end{equation} |
| 126 |
Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine |
| 127 |
power series to the desired torsion potential surface, and $\phi$ is |
| 128 |
the angle between bondvectors $fix$ and $fix$ along the vector $fix$ |
| 129 |
(see Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as |
| 130 |
the Lennard-Jones potential are excluded for bead pairs involved in |
| 131 |
the same bond, bend, or torsion. However, internal interactions not |
| 132 |
directly involved in a bonded pair are calculated. |
| 133 |
|
| 134 |
|
