| 4 |
|
|
| 5 |
|
\section{\label{lipidSec:Intro}Introduction} |
| 6 |
|
|
| 7 |
< |
\section{\label{lipidSec:Methods}Methods} |
| 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{lindahl00} to the |
| 11 |
> |
spontaneous aggregation of DPPC molecules into fluid phase |
| 12 |
> |
($L_{\alpha}$) bilayers.\cite{marrink01} With the exception of a few |
| 13 |
> |
ambitious |
| 14 |
> |
simulations,\cite{marrink01:undulation,marrink:2002,lindahl00} most |
| 15 |
> |
investigations are limited to 64 to 256 |
| 16 |
> |
phospholipids.\cite{lindahl00,sum:2003,venable00,gomez:2003,smondyrev:1999,marrink01} |
| 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 difficulty 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 lengths of bilayer simulations. |
| 26 |
|
|
| 27 |
< |
\subsection{\label{lipidSec:lipidMedel}The Lipid Model} |
| 27 |
> |
Unfortunately, much of biological interest happens on time and length |
| 28 |
> |
scales infeasible with current simulation. One such example is the |
| 29 |
> |
observance of a ripple phase ($P_{\beta^{\prime}}$) between the |
| 30 |
> |
$L_{\beta}$ and $L_{\alpha}$ phases of certain phospholipid |
| 31 |
> |
bilayers.\cite{katsaras00,sengupta00} 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 |
< |
\begin{figure} |
| 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 |
< |
\caption{Schematic diagram of the single chain phospholipid model} |
| 48 |
> |
Addressing 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 |
< |
\label{lipidFig:lipidModel} |
| 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 $\frac{1}{r}$), eliminating the |
| 72 |
> |
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 |
+ |
|
| 82 |
+ |
The following is an outline of this chapter. |
| 83 |
+ |
Sec.~\ref{lipidSec:Methods} is an introduction to the lipid model |
| 84 |
+ |
used in these simulations. As well as clarification about the water |
| 85 |
+ |
model and integration techniques. The various simulation setups |
| 86 |
+ |
explored in this research are outlined in |
| 87 |
+ |
Sec.~\ref{lipidSec:ExpSetup}. Sec.~\ref{lipidSec:Results} and |
| 88 |
+ |
Sec.~\ref{lipidSec:Discussion} give a summary of the results and |
| 89 |
+ |
interpretation of those results respectively. Finally, the |
| 90 |
+ |
conclusions of this chapter are presented in |
| 91 |
+ |
Sec.~\ref{lipidSec:Conclusion}. |
| 92 |
+ |
|
| 93 |
+ |
\section{\label{lipidSec:Methods}Methods} |
| 94 |
+ |
|
| 95 |
+ |
\subsection{\label{lipidSec:lipidModel}The Lipid Model} |
| 96 |
+ |
|
| 97 |
+ |
\begin{figure} |
| 98 |
+ |
\centering |
| 99 |
+ |
\includegraphics[width=\linewidth]{twoChainFig.eps} |
| 100 |
+ |
\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.} |
| 101 |
+ |
\label{lipidFig:lipidModel} |
| 102 |
|
\end{figure} |
| 103 |
|
|
| 104 |
|
The phospholipid model used in these simulations is based on the |
| 105 |
|
design illustrated in Fig.~\ref{lipidFig:lipidModel}. The head group |
| 106 |
|
of the phospholipid is replaced by a single Lennard-Jones sphere of |
| 107 |
< |
diameter $fix$, with $fix$ scaling the well depth of its van der Walls |
| 108 |
< |
interaction. This sphere also contains a single dipole of magnitude |
| 109 |
< |
$fix$, where $fix$ can be varied to mimic the charge separation of a |
| 107 |
> |
diameter $\sigma_{\text{head}}$, with $\epsilon_{\text{head}}$ scaling |
| 108 |
> |
the well depth of its van der Walls interaction. This sphere also |
| 109 |
> |
contains a single dipole of magnitude $|\boldsymbol{\mu}|$, where |
| 110 |
> |
$|\boldsymbol{\mu}|$ can be varied to mimic the charge separation of a |
| 111 |
|
given phospholipid head group. The atoms of the tail region are |
| 112 |
|
modeled by unified atom beads. They are free of partial charges or |
| 113 |
|
dipoles, containing only Lennard-Jones interaction sites at their |
| 114 |
|
centers of mass. As with the head groups, their potentials can be |
| 115 |
< |
scaled by $fix$ and $fix$. |
| 115 |
> |
scaled by $\sigma_{\text{tail}}$ and $\epsilon_{\text{tail}}$. |
| 116 |
|
|
| 117 |
|
The long range interactions between lipids are given by the following: |
| 118 |
|
\begin{equation} |
| 119 |
< |
EQ Here |
| 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 |
< |
EQ Here |
| 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 |
> |
\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
| 133 |
> |
(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
| 134 |
> |
{r^{2}_{ij}} \biggr] |
| 135 |
|
\label{lipidEq:dipolePot} |
| 136 |
|
\end{equation} |
| 137 |
|
Where $V_{\text{LJ}}$ is the Lennard-Jones potential and |
| 146 |
|
$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 147 |
|
vector of $\boldsymbol{\Omega}_i$. |
| 148 |
|
|
| 149 |
< |
The model also allows for the bonded interactions of bonds, bends, and |
| 150 |
< |
torsions. The bonds between two beads on a chain are of fixed length, |
| 151 |
< |
and are maintained according to the {\sc rattle} algorithm. \cite{fix} |
| 149 |
> |
The model also allows for the bonded interactions bends, and torsions. |
| 150 |
> |
The bond between two beads on a chain is of fixed length, and is |
| 151 |
> |
maintained according to the {\sc rattle} algorithm.\cite{andersen83} |
| 152 |
|
The bends are subject to a harmonic potential: |
| 153 |
|
\begin{equation} |
| 154 |
< |
eq here |
| 154 |
> |
V_{\text{bend}}(\theta) = k_{\theta}( \theta - \theta_0 )^2 |
| 155 |
|
\label{lipidEq:bendPot} |
| 156 |
|
\end{equation} |
| 157 |
< |
where $fix$ scales the strength of the harmonic well, and $fix$ is the |
| 158 |
< |
angle between bond vectors $fix$ and $fix$. The torsion potential is |
| 159 |
< |
given by: |
| 157 |
> |
where $k_{\theta}$ scales the strength of the harmonic well, and |
| 158 |
> |
$\theta$ is the angle between bond vectors |
| 159 |
> |
(Fig.~\ref{lipidFig:lipidModel}). In addition, we have placed a |
| 160 |
> |
``ghost'' bend on the phospholipid head. The ghost bend adds a |
| 161 |
> |
potential to keep the dipole pointed along the bilayer surface, where |
| 162 |
> |
$theta$ is now the angle the dipole makes with respect to the {\sc |
| 163 |
> |
head}-$\text{{\sc ch}}_2$ bond vector. The torsion potential is given |
| 164 |
> |
by: |
| 165 |
|
\begin{equation} |
| 166 |
< |
eq here |
| 166 |
> |
V_{\text{torsion}}(\phi) = |
| 167 |
> |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 168 |
|
\label{lipidEq:torsionPot} |
| 169 |
|
\end{equation} |
| 170 |
|
Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine |
| 171 |
|
power series to the desired torsion potential surface, and $\phi$ is |
| 172 |
< |
the angle between bondvectors $fix$ and $fix$ along the vector $fix$ |
| 173 |
< |
(see Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as |
| 174 |
< |
the Lennard-Jones potential are excluded for bead pairs involved in |
| 175 |
< |
the same bond, bend, or torsion. However, internal interactions not |
| 172 |
> |
the angle the two end atoms have rotated about the middle bond |
| 173 |
> |
(Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as the |
| 174 |
> |
Lennard-Jones potential are excluded for atom pairs involved in the |
| 175 |
> |
same bond, bend, or torsion. However, internal interactions not |
| 176 |
|
directly involved in a bonded pair are calculated. |
| 177 |
|
|
| 178 |
+ |
All simulations presented here use a two chained lipid as pictured in |
| 179 |
+ |
Fig.~\ref{lipidFig:lipidModel}. The chains are both eight beads long, |
| 180 |
+ |
and their mass and Lennard Jones parameters are summarized in |
| 181 |
+ |
Table~\ref{lipidTable:tcLJParams}. The magnitude of the dipole moment |
| 182 |
+ |
for the head bead is 10.6~Debye, and the bend and torsion parameters |
| 183 |
+ |
are summarized in Table~\ref{lipidTable:tcBendParams} and |
| 184 |
+ |
\ref{lipidTable:tcTorsionParams}. |
| 185 |
|
|
| 186 |
+ |
\begin{table} |
| 187 |
+ |
\caption{The Lennard Jones Parameters for the two chain phospholipids.} |
| 188 |
+ |
\label{lipidTable:tcLJParams} |
| 189 |
+ |
\begin{center} |
| 190 |
+ |
\begin{tabular}{|l|c|c|c|} |
| 191 |
+ |
\hline |
| 192 |
+ |
& mass (amu) & $\sigma$($\mbox{\AA}$) & $\epsilon$ (kcal/mol) \\ \hline |
| 193 |
+ |
{\sc head} & 72 & 4.0 & 0.185 \\ \hline |
| 194 |
+ |
{\sc ch}\cite{Siepmann1998} & 13.02 & 4.0 & 0.0189 \\ \hline |
| 195 |
+ |
$\text{{\sc ch}}_2$\cite{Siepmann1998} & 14.03 & 3.95 & 0.18 \\ \hline |
| 196 |
+ |
$\text{{\sc ch}}_3$\cite{Siepmann1998} & 15.04 & 3.75 & 0.25 \\ \hline |
| 197 |
+ |
{\sc ssd}\cite{liu96:new_model} & 18.03 & 3.051 & 0.152 \\ \hline |
| 198 |
+ |
\end{tabular} |
| 199 |
+ |
\end{center} |
| 200 |
+ |
\end{table} |
| 201 |
+ |
|
| 202 |
+ |
\begin{table} |
| 203 |
+ |
\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}} |
| 204 |
+ |
\label{lipidTable:tcBendParams} |
| 205 |
+ |
\begin{center} |
| 206 |
+ |
\begin{tabular}{|l|c|c|} |
| 207 |
+ |
\hline |
| 208 |
+ |
& $k_{\theta}$ ( kcal/($\text{mol deg}^2$) ) & $\theta_0$ ( deg ) \\ \hline |
| 209 |
+ |
{\sc ghost}-{\sc head}-$\text{{\sc ch}}_2$ & 0.00177 & 129.78 \\ \hline |
| 210 |
+ |
$x$-{\sc ch}-$y$ & 58.84 & 112.0 \\ \hline |
| 211 |
+ |
$x$-$\text{{\sc ch}}_2$-$y$ & 58.84 & 114.0 \\ \hline |
| 212 |
+ |
\end{tabular} |
| 213 |
+ |
\end{center} |
| 214 |
+ |
\end{table} |
| 215 |
+ |
|
| 216 |
+ |
\begin{table} |
| 217 |
+ |
\caption[Torsion Parameters for the two chain phospholipids]{Torsion Parameters for the two chain phospholipids. Alkane parameters based on TraPPE.\cite{Siepmann1998}} |
| 218 |
+ |
\label{lipidTable:tcTorsionParams} |
| 219 |
+ |
\begin{center} |
| 220 |
+ |
\begin{tabular}{|l|c|c|c|c|} |
| 221 |
+ |
\hline |
| 222 |
+ |
All are in kcal/mol $\rightarrow$ & $k_3$ & $k_2$ & $k_1$ & $k_0$ \\ \hline |
| 223 |
+ |
$x$-{\sc ch}-$y$-$z$ & 3.3254 & -0.4215 & -1.686 & 1.1661 \\ \hline |
| 224 |
+ |
$x$-$\text{{\sc ch}}_2$-$\text{{\sc ch}}_2$-$y$ & 5.9602 & -0.568 & -3.802 & 2.1586 \\ \hline |
| 225 |
+ |
\end{tabular} |
| 226 |
+ |
\end{center} |
| 227 |
+ |
\end{table} |
| 228 |
+ |
|
| 229 |
+ |
|
| 230 |
+ |
\section{\label{lipidSec:furtherMethod}Further Methodology} |
| 231 |
+ |
|
| 232 |
+ |
As mentioned previously, the water model used throughout these |
| 233 |
+ |
simulations was the {\sc ssd} model of |
| 234 |
+ |
Ichiye.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md} A |
| 235 |
+ |
discussion of the model can be found in Sec.~\ref{oopseSec:SSD}. As |
| 236 |
+ |
for the integration of the equations of motion, all simulations were |
| 237 |
+ |
performed in an orthorhombic periodic box with a thermostat on |
| 238 |
+ |
velocities, and an independent barostat on each Cartesian axis $x$, |
| 239 |
+ |
$y$, and $z$. This is the $\text{NPT}_{xyz}$. ensemble described in |
| 240 |
+ |
Sec.~\ref{oopseSec:Ensembles}. |
| 241 |
+ |
|
| 242 |
+ |
|
| 243 |
+ |
\subsection{\label{lipidSec:ExpSetup}Experimental Setup} |
| 244 |
+ |
|
| 245 |
+ |
Two main starting configuration classes were used in this research: |
| 246 |
+ |
random and ordered bilayers. The ordered bilayer starting |
| 247 |
+ |
configurations were all started from an equilibrated bilayer at |
| 248 |
+ |
300~K. The original configuration for the first 300~K run was |
| 249 |
+ |
assembled by placing the phospholipids centers of mass on a planar |
| 250 |
+ |
hexagonal lattice. The lipids were oriented with their long axis |
| 251 |
+ |
perpendicular to the plane. The second leaf simply mirrored the first |
| 252 |
+ |
leaf, and the appropriate number of waters were then added above and |
| 253 |
+ |
below the bilayer. |
| 254 |
+ |
|
| 255 |
+ |
The random configurations took more work to generate. To begin, a |
| 256 |
+ |
test lipid was placed in a simulation box already containing water at |
| 257 |
+ |
the intended density. The waters were then tested for overlap with |
| 258 |
+ |
the lipid using a 5.0~$\mbox{\AA}$ buffer distance. This gave an |
| 259 |
+ |
estimate for the number of waters each lipid would displace in a |
| 260 |
+ |
simulation box. A target number of waters was then defined which |
| 261 |
+ |
included the number of waters each lipid would displace, the number of |
| 262 |
+ |
waters desired to solvate each lipid, and a fudge factor to pad the |
| 263 |
+ |
initialization. |
| 264 |
+ |
|
| 265 |
+ |
Next, a cubic simulation box was created that contained at least the |
| 266 |
+ |
target number of waters in an FCC lattice (the lattice was for ease of |
| 267 |
+ |
placement). What followed was a RSA simulation similar to those of |
| 268 |
+ |
Chapt.~\ref{chapt:RSA}. The lipids were sequentially given a random |
| 269 |
+ |
position and orientation within the box. If a lipid's position caused |
| 270 |
+ |
atomic overlap with any previously adsorbed lipid, its position and |
| 271 |
+ |
orientation were rejected, and a new random adsorption site was |
| 272 |
+ |
attempted. The RSA simulation proceeded until all phospholipids had |
| 273 |
+ |
been adsorbed. After adsorption, all water molecules with locations |
| 274 |
+ |
that overlapped with the atomic coordinates of the lipids were |
| 275 |
+ |
removed. |
| 276 |
+ |
|
| 277 |
+ |
Finally, water molecules were removed one by one at random until the |
| 278 |
+ |
desired number of waters per lipid was reached. The typical low final |
| 279 |
+ |
density for these initial configurations was not a problem, as the box |
| 280 |
+ |
would shrink to an appropriate size within the first 50~ps of a |
| 281 |
+ |
simulation in the $\text{NPT}_{xyz}$ ensemble. |
| 282 |
+ |
|
| 283 |
+ |
\subsection{\label{lipidSec:Configs}The simulation configurations} |
| 284 |
+ |
|
| 285 |
+ |
Table ~\ref{lipidTable:simNames} summarizes the names and important |
| 286 |
+ |
details of the simulations. The B set of simulations were all started |
| 287 |
+ |
in an ordered bilayer and observed over a period of 10~ns. Simulation |
| 288 |
+ |
RL was integrated for approximately 20~ns starting from a random |
| 289 |
+ |
configuration as an example of spontaneous bilayer aggregation. |
| 290 |
+ |
Lastly, simulation RH was also started from a random configuration, |
| 291 |
+ |
but with a lesser water content and higher temperature to show the |
| 292 |
+ |
spontaneous aggregation of an inverted hexagonal lamellar phase. |
