| 4 |
|
|
| 5 |
|
\section{\label{lipidSec:Intro}Introduction} |
| 6 |
|
|
| 7 |
< |
\section{\label{lipidSec:Methods}Methods} |
| 7 |
> |
In the past 10 years, increasing computer speeds have allowed for the |
| 8 |
> |
atomistic simulation of phospholipid bilayers for increasingly |
| 9 |
> |
relevant lengths of time. These simulations have ranged from |
| 10 |
> |
simulation of the gel phase ($L_{\beta}$) of |
| 11 |
> |
dipalmitoylphosphatidylcholine (DPPC),\cite{lindahl00} to the |
| 12 |
> |
spontaneous aggregation of DPPC molecules into fluid phase |
| 13 |
> |
($L_{\alpha}$) bilayers.\cite{marrink01} With the exception of a few |
| 14 |
> |
ambitious |
| 15 |
> |
simulations,\cite{marrink01:undulation,marrink:2002,lindahl00} most |
| 16 |
> |
investigations are limited to a range of 64 to 256 |
| 17 |
> |
phospholipids.\cite{lindahl00,sum:2003,venable00,gomez:2003,smondyrev:1999,marrink01} |
| 18 |
> |
The expense of the force calculations involved when performing these |
| 19 |
> |
simulations limits the system size. To properly hydrate a bilayer, one |
| 20 |
> |
typically needs 25 water molecules for every lipid, bringing the total |
| 21 |
> |
number of atoms simulated to roughly 8,000 for a system of 64 DPPC |
| 22 |
> |
molecules. Added to the difficulty is the electrostatic nature of the |
| 23 |
> |
phospholipid head groups and water, requiring either the |
| 24 |
> |
computationally expensive Ewald sum or the faster, particle mesh Ewald |
| 25 |
> |
sum.\cite{nina:2002,norberg:2000,patra:2003} These factors all limit |
| 26 |
> |
the system size and time scales of bilayer simulations. |
| 27 |
|
|
| 28 |
< |
\subsection{\label{lipidSec:lipidMedel}The Lipid Model} |
| 28 |
> |
Unfortunately, much of biological interest happens on time and length |
| 29 |
> |
scales well beyond the range of current simulation technology. One |
| 30 |
> |
such example is the observance of a ripple phase |
| 31 |
> |
($P_{\beta^{\prime}}$) between the $L_{\beta}$ and $L_{\alpha}$ phases |
| 32 |
> |
of certain phospholipid bilayers.\cite{katsaras00,sengupta00} These |
| 33 |
> |
ripples are shown to have periodicity on the order of |
| 34 |
> |
100-200~$\mbox{\AA}$. A simulation on this length scale would have |
| 35 |
> |
approximately 1,300 lipid molecules with an additional 25 water |
| 36 |
> |
molecules per lipid to fully solvate the bilayer. A simulation of this |
| 37 |
> |
size is impractical with current atomistic models. |
| 38 |
|
|
| 39 |
< |
\begin{figure} |
| 39 |
> |
The time and length scale limitations are most striking in transport |
| 40 |
> |
phenomena. Due to the fluid-like properties of a lipid membrane, not |
| 41 |
> |
all diffusion across the membrane happens at pores. Some molecules of |
| 42 |
> |
interest may incorporate themselves directly into the membrane. Once |
| 43 |
> |
here, they may possess an appreciable waiting time (on the order of |
| 44 |
> |
10's to 100's of nanoseconds) within the bilayer. Such long simulation |
| 45 |
> |
times are nearly impossible to obtain when integrating the system with |
| 46 |
> |
atomistic detail. |
| 47 |
|
|
| 48 |
< |
\caption{Schematic diagram of the single chain phospholipid model} |
| 48 |
> |
To address 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 $\frac{1}{r}$), and eliminates |
| 72 |
> |
the 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 |
+ |
molecules. |
| 82 |
+ |
|
| 83 |
+ |
The following is an outline of this chapter. |
| 84 |
+ |
Sec.~\ref{lipidSec:Methods} is an introduction to the lipid model used |
| 85 |
+ |
in these simulations, as well as clarification about the water model |
| 86 |
+ |
and integration techniques. The various simulations explored in this |
| 87 |
+ |
research are outlined in |
| 88 |
+ |
Sec.~\ref{lipidSec:ExpSetup}. Sec.~\ref{lipidSec:resultsDis} gives a |
| 89 |
+ |
summary and interpretation of the results. Finally, the conclusions |
| 90 |
+ |
of this chapter are presented in Sec.~\ref{lipidSec:Conclusion}. |
| 91 |
+ |
|
| 92 |
+ |
\section{\label{lipidSec:Methods}Methods} |
| 93 |
+ |
|
| 94 |
+ |
\subsection{\label{lipidSec:lipidModel}The Lipid Model} |
| 95 |
+ |
|
| 96 |
+ |
\begin{figure} |
| 97 |
+ |
\centering |
| 98 |
+ |
\includegraphics[width=\linewidth]{twoChainFig.eps} |
| 99 |
+ |
\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.} |
| 100 |
+ |
\label{lipidFig:lipidModel} |
| 101 |
|
\end{figure} |
| 102 |
|
|
| 103 |
|
The phospholipid model used in these simulations is based on the |
| 104 |
|
design illustrated in Fig.~\ref{lipidFig:lipidModel}. The head group |
| 105 |
|
of the phospholipid is replaced by a single Lennard-Jones sphere of |
| 106 |
< |
diameter $fix$, with $fix$ scaling the well depth of its van der Walls |
| 107 |
< |
interaction. This sphere also contains a single dipole of magnitude |
| 108 |
< |
$fix$, where $fix$ can be varied to mimic the charge separation of a |
| 106 |
> |
diameter $\sigma_{\text{head}}$, with $\epsilon_{\text{head}}$ scaling |
| 107 |
> |
the well depth of its van der Walls interaction. This sphere also |
| 108 |
> |
contains a single dipole of magnitude $|\boldsymbol{\mu}|$, where |
| 109 |
> |
$|\boldsymbol{\mu}|$ can be varied to mimic the charge separation of a |
| 110 |
|
given phospholipid head group. The atoms of the tail region are |
| 111 |
< |
modeled by unified atom beads. They are free of partial charges or |
| 112 |
< |
dipoles, containing only Lennard-Jones interaction sites at their |
| 113 |
< |
centers of mass. As with the head groups, their potentials can be |
| 114 |
< |
scaled by $fix$ and $fix$. |
| 111 |
> |
modeled by beads representing multiple methyl groups. They are free |
| 112 |
> |
of partial charges or dipoles, and contain only Lennard-Jones |
| 113 |
> |
interaction sites at their centers of mass. As with the head groups, |
| 114 |
> |
their potentials can be scaled by $\sigma_{\text{tail}}$ and |
| 115 |
> |
$\epsilon_{\text{tail}}$. |
| 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 |
> |
3(\boldsymbol{\hat{u}}_i \cdot \mathbf{\hat{r}}_{ij}) % |
| 133 |
> |
(\boldsymbol{\hat{u}}_j \cdot \mathbf{\hat{r}}_{ij} \biggr] |
| 134 |
|
\label{lipidEq:dipolePot} |
| 135 |
|
\end{equation} |
| 136 |
|
Where $V_{\text{LJ}}$ is the Lennard-Jones potential and |
| 139 |
|
parameters which scale the length and depth of the interaction |
| 140 |
|
respectively, and $r_{ij}$ is the distance between beads $i$ and $j$. |
| 141 |
|
In $V_{\text{dipole}}$, $\mathbf{r}_{ij}$ is the vector starting at |
| 142 |
< |
bead$i$ and pointing towards bead $j$. Vectors $\mathbf{\Omega}_i$ |
| 142 |
> |
bead $i$ and pointing towards bead $j$. Vectors $\mathbf{\Omega}_i$ |
| 143 |
|
and $\mathbf{\Omega}_j$ are the orientational degrees of freedom for |
| 144 |
|
beads $i$ and $j$. $|\mu_i|$ is the magnitude of the dipole moment of |
| 145 |
|
$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 146 |
< |
vector of $\boldsymbol{\Omega}_i$. |
| 146 |
> |
vector rotated with Euler angles: $\boldsymbol{\Omega}_i$. |
| 147 |
|
|
| 148 |
< |
The model also allows for the bonded interactions of bonds, bends, and |
| 149 |
< |
torsions. The bonds between two beads on a chain are of fixed length, |
| 150 |
< |
and are maintained according to the {\sc rattle} algorithm. \cite{fix} |
| 148 |
> |
The model also allows for the bonded interactions bends, and torsions. |
| 149 |
> |
The bond between two beads on a chain is of fixed length, and is |
| 150 |
> |
maintained according to the {\sc rattle} algorithm.\cite{andersen83} |
| 151 |
|
The bends are subject to a harmonic potential: |
| 152 |
|
\begin{equation} |
| 153 |
< |
eq here |
| 153 |
> |
V_{\text{bend}}(\theta) = k_{\theta}( \theta - \theta_0 )^2 |
| 154 |
|
\label{lipidEq:bendPot} |
| 155 |
|
\end{equation} |
| 156 |
< |
where $fix$ scales the strength of the harmonic well, and $fix$ is the |
| 157 |
< |
angle between bond vectors $fix$ and $fix$. The torsion potential is |
| 158 |
< |
given by: |
| 156 |
> |
where $k_{\theta}$ scales the strength of the harmonic well, and |
| 157 |
> |
$\theta$ is the angle between bond vectors |
| 158 |
> |
(Fig.~\ref{lipidFig:lipidModel}). In addition, we have placed a |
| 159 |
> |
``ghost'' bend on the phospholipid head. The ghost bend adds a |
| 160 |
> |
potential to keep the dipole pointed along the bilayer surface, where |
| 161 |
> |
$\theta$ is now the angle the dipole makes with respect to the {\sc |
| 162 |
> |
head}-$\text{{\sc ch}}_2$ bond vector |
| 163 |
> |
(Fig.~\ref{lipidFig:ghostBend}). The torsion potential is given by: |
| 164 |
|
\begin{equation} |
| 165 |
< |
eq here |
| 165 |
> |
V_{\text{torsion}}(\phi) = |
| 166 |
> |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 167 |
|
\label{lipidEq:torsionPot} |
| 168 |
|
\end{equation} |
| 169 |
|
Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine |
| 170 |
|
power series to the desired torsion potential surface, and $\phi$ is |
| 171 |
< |
the angle between bondvectors $fix$ and $fix$ along the vector $fix$ |
| 172 |
< |
(see Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as |
| 173 |
< |
the Lennard-Jones potential are excluded for bead pairs involved in |
| 174 |
< |
the same bond, bend, or torsion. However, internal interactions not |
| 171 |
> |
the angle the two end atoms have rotated about the middle bond |
| 172 |
> |
(Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as the |
| 173 |
> |
Lennard-Jones potential are excluded for atom pairs involved in the |
| 174 |
> |
same bond, bend, or torsion. However, internal interactions not |
| 175 |
|
directly involved in a bonded pair are calculated. |
| 176 |
+ |
|
| 177 |
+ |
\begin{figure} |
| 178 |
+ |
\centering |
| 179 |
+ |
\includegraphics[width=\linewidth]{ghostBendFig.eps} |
| 180 |
+ |
\caption[Depiction of the ``ghost'' bend]{The ``ghost'' bend is a bend potential added to constrain the motion of the dipole on the {\sc head} group. The potential follows Eq.~\ref{lipidEq:bendPot} where $\theta$ is now the angle that the dipole makes with the {\sc head}-$\text{{\sc ch}}_2$ bond vector.} |
| 181 |
+ |
\label{lipidFig:ghostBend} |
| 182 |
+ |
\end{figure} |
| 183 |
+ |
|
| 184 |
+ |
All simulations presented here use a two chained lipid as pictured in |
| 185 |
+ |
Fig.~\ref{lipidFig:lipidModel}. The chains are both eight beads long, |
| 186 |
+ |
and their mass and Lennard Jones parameters are summarized in |
| 187 |
+ |
Table~\ref{lipidTable:tcLJParams}. The magnitude of the dipole moment |
| 188 |
+ |
for the head bead is 10.6~Debye, and the bend and torsion parameters |
| 189 |
+ |
are summarized in Table~\ref{lipidTable:tcBendParams} and |
| 190 |
+ |
\ref{lipidTable:tcTorsionParams}. |
| 191 |
+ |
|
| 192 |
+ |
\begin{table} |
| 193 |
+ |
\caption{The Lennard Jones Parameters for the two chain phospholipids.} |
| 194 |
+ |
\label{lipidTable:tcLJParams} |
| 195 |
+ |
\begin{center} |
| 196 |
+ |
\begin{tabular}{|l|c|c|c|} |
| 197 |
+ |
\hline |
| 198 |
+ |
& mass (amu) & $\sigma$($\mbox{\AA}$) & $\epsilon$ (kcal/mol) \\ \hline |
| 199 |
+ |
{\sc head} & 72 & 4.0 & 0.185 \\ \hline |
| 200 |
+ |
{\sc ch}\cite{Siepmann1998} & 13.02 & 4.0 & 0.0189 \\ \hline |
| 201 |
+ |
$\text{{\sc ch}}_2$\cite{Siepmann1998} & 14.03 & 3.95 & 0.18 \\ \hline |
| 202 |
+ |
$\text{{\sc ch}}_3$\cite{Siepmann1998} & 15.04 & 3.75 & 0.25 \\ \hline |
| 203 |
+ |
{\sc ssd}\cite{liu96:new_model} & 18.03 & 3.051 & 0.152 \\ \hline |
| 204 |
+ |
\end{tabular} |
| 205 |
+ |
\end{center} |
| 206 |
+ |
\end{table} |
| 207 |
+ |
|
| 208 |
+ |
\begin{table} |
| 209 |
+ |
\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}} |
| 210 |
+ |
\label{lipidTable:tcBendParams} |
| 211 |
+ |
\begin{center} |
| 212 |
+ |
\begin{tabular}{|l|c|c|} |
| 213 |
+ |
\hline |
| 214 |
+ |
& $k_{\theta}$ ( kcal/($\text{mol deg}^2$) ) & $\theta_0$ ( deg ) \\ \hline |
| 215 |
+ |
{\sc ghost}-{\sc head}-$\text{{\sc ch}}_2$ & 0.00177 & 129.78 \\ \hline |
| 216 |
+ |
$x$-{\sc ch}-$y$ & 58.84 & 112.0 \\ \hline |
| 217 |
+ |
$x$-$\text{{\sc ch}}_2$-$y$ & 58.84 & 114.0 \\ \hline |
| 218 |
+ |
\end{tabular} |
| 219 |
+ |
\end{center} |
| 220 |
+ |
\end{table} |
| 221 |
+ |
|
| 222 |
+ |
\begin{table} |
| 223 |
+ |
\caption[Torsion Parameters for the two chain phospholipids]{Torsion Parameters for the two chain phospholipids. Alkane parameters based on TraPPE.\cite{Siepmann1998}} |
| 224 |
+ |
\label{lipidTable:tcTorsionParams} |
| 225 |
+ |
\begin{center} |
| 226 |
+ |
\begin{tabular}{|l|c|c|c|c|} |
| 227 |
+ |
\hline |
| 228 |
+ |
All are in kcal/mol $\rightarrow$ & $k_3$ & $k_2$ & $k_1$ & $k_0$ \\ \hline |
| 229 |
+ |
$x$-{\sc ch}-$y$-$z$ & 3.3254 & -0.4215 & -1.686 & 1.1661 \\ \hline |
| 230 |
+ |
$x$-$\text{{\sc ch}}_2$-$\text{{\sc ch}}_2$-$y$ & 5.9602 & -0.568 & -3.802 & 2.1586 \\ \hline |
| 231 |
+ |
\end{tabular} |
| 232 |
+ |
\end{center} |
| 233 |
+ |
\end{table} |
| 234 |
+ |
|
| 235 |
+ |
|
| 236 |
+ |
\section{\label{lipidSec:furtherMethod}Further Methodology} |
| 237 |
+ |
|
| 238 |
+ |
As mentioned previously, the water model used throughout these |
| 239 |
+ |
simulations was the {\sc ssd} model of |
| 240 |
+ |
Ichiye.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md} A |
| 241 |
+ |
discussion of the model can be found in Sec.~\ref{oopseSec:SSD}. As |
| 242 |
+ |
for the integration of the equations of motion, all simulations were |
| 243 |
+ |
performed in an orthorhombic periodic box with a thermostat on |
| 244 |
+ |
velocities, and an independent barostat on each Cartesian axis $x$, |
| 245 |
+ |
$y$, and $z$. This is the $\text{NPT}_{xyz}$. ensemble described in |
| 246 |
+ |
Sec.~\ref{oopseSec:integrate}. |
| 247 |
+ |
|
| 248 |
+ |
|
| 249 |
+ |
\subsection{\label{lipidSec:ExpSetup}Experimental Setup} |
| 250 |
+ |
|
| 251 |
+ |
Two main starting configuration classes were used in this research: |
| 252 |
+ |
random and ordered bilayers. The ordered bilayer starting |
| 253 |
+ |
configurations were all started from an equilibrated bilayer at |
| 254 |
+ |
300~K. The original configuration for the first 300~K run was |
| 255 |
+ |
assembled by placing the phospholipids centers of mass on a planar |
| 256 |
+ |
hexagonal lattice. The lipids were oriented with their long axis |
| 257 |
+ |
perpendicular to the plane. The second leaf simply mirrored the first |
| 258 |
+ |
leaf, and the appropriate number of waters were then added above and |
| 259 |
+ |
below the bilayer. |
| 260 |
+ |
|
| 261 |
+ |
The random configurations took more work to generate. To begin, a |
| 262 |
+ |
test lipid was placed in a simulation box already containing water at |
| 263 |
+ |
the intended density. The waters were then tested for overlap with |
| 264 |
+ |
the lipid using a 5.0~$\mbox{\AA}$ buffer distance. This gave an |
| 265 |
+ |
estimate for the number of waters each lipid would displace in a |
| 266 |
+ |
simulation box. A target number of waters was then defined which |
| 267 |
+ |
included the number of waters each lipid would displace, the number of |
| 268 |
+ |
waters desired to solvate each lipid, and a factor to pad the |
| 269 |
+ |
initial box with a few extra water molecules. |
| 270 |
+ |
|
| 271 |
+ |
Next, a cubic simulation box was created that contained at least the |
| 272 |
+ |
target number of waters in an FCC lattice (the lattice was for ease of |
| 273 |
+ |
placement). What followed was a RSA simulation similar to those of |
| 274 |
+ |
Chapt.~\ref{chapt:RSA}. The lipids were sequentially given a random |
| 275 |
+ |
position and orientation within the box. If a lipid's position caused |
| 276 |
+ |
atomic overlap with any previously placed lipid, its position and |
| 277 |
+ |
orientation were rejected, and a new random placement site was |
| 278 |
+ |
attempted. The RSA simulation proceeded until all phospholipids had |
| 279 |
+ |
been adsorbed. After placement of all lipid molecules, water |
| 280 |
+ |
molecules with locations that overlapped with the atomic coordinates |
| 281 |
+ |
of the lipids were removed. |
| 282 |
+ |
|
| 283 |
+ |
Finally, water molecules were removed at random until the desired |
| 284 |
+ |
number of waters per lipid was reached. The typical low final density |
| 285 |
+ |
for these initial configurations was not a problem, as the box shrinks |
| 286 |
+ |
to an appropriate size within the first 50~ps of a simulation in the |
| 287 |
+ |
$\text{NPT}_{xyz}$ ensemble. |
| 288 |
+ |
|
| 289 |
+ |
\subsection{\label{lipidSec:Configs}Configurations} |
| 290 |
+ |
|
| 291 |
+ |
The first class of simulations were started from ordered |
| 292 |
+ |
bilayers. They were all configurations consisting of 60 lipid |
| 293 |
+ |
molecules with 30 lipids on each leaf, and were hydrated with 1620 |
| 294 |
+ |
{\sc ssd} molecules. The original configuration was assembled |
| 295 |
+ |
according to Sec.~\ref{lipidSec:ExpSetup} and simulated for a length |
| 296 |
+ |
of 10~ns at 300~K. The other temperature runs were started from a |
| 297 |
+ |
frame 7~ns into the 300~K simulation. Their temperatures were reset |
| 298 |
+ |
with the thermostating algorithm in the $\text{NPT}_{xyz}$ |
| 299 |
+ |
integrator. All of the temperature variants were also run for 10~ns, |
| 300 |
+ |
with only the last 5~ns being used for accumulation of statistics. |
| 301 |
|
|
| 302 |
+ |
The second class of simulations were two configurations started from |
| 303 |
+ |
randomly dispersed lipids in a ``gas'' of water. The first |
| 304 |
+ |
($\text{R}_{\text{I}}$) was a simulation containing 72 lipids with |
| 305 |
+ |
1800 {\sc ssd} molecules simulated at 300~K. The second |
| 306 |
+ |
($\text{R}_{\text{II}}$) was 90 lipids with 1350 {\sc ssd} molecules |
| 307 |
+ |
simulated at 350~K. Both simulations were integrated for more than |
| 308 |
+ |
20~ns, and illustrate the spontaneous aggregation of the lipid model |
| 309 |
+ |
into phospholipid macro-structures: $\text{R}_{\text{I}}$ into a |
| 310 |
+ |
bilayer, and $\text{R}_{\text{II}}$ into a inverted rod. |
| 311 |
|
|
| 312 |
+ |
\section{\label{lipidSec:resultsDis}Results and Discussion} |
| 313 |
+ |
|
| 314 |
+ |
\subsection{\label{lipidSec:diffusion}Lateral Diffusion Constants} |
| 315 |
+ |
|
| 316 |
+ |
The lateral diffusion constant, $D_L$, is the constant characterizing |
| 317 |
+ |
the diffusive motion of the lipid within the plane of the bilayer. It |
| 318 |
+ |
is given by the following Einstein relation valid at long |
| 319 |
+ |
times:\cite{allen87:csl} |
| 320 |
+ |
\begin{equation} |
| 321 |
+ |
2tD_L = \frac{1}{2}\langle |\mathbf{r}(t) - \mathbf{r}(0)|^2\rangle |
| 322 |
+ |
\end{equation} |
| 323 |
+ |
Where $\mathbf{r}(t)$ is the position of the lipid at time $t$, and is |
| 324 |
+ |
constrained to lie within a plane. For the bilayer simulations the |
| 325 |
+ |
plane of constrained motion was that perpendicular to the bilayer |
| 326 |
+ |
normal, namely the $xy$-plane. |
| 327 |
+ |
|
| 328 |
+ |
Fig.~\ref{lipidFig:diffusionFig} shows the lateral diffusion constants |
| 329 |
+ |
as a function of temperature. There is a definite increase in the |
| 330 |
+ |
lateral diffusion with higher temperatures, which is exactly what one |
| 331 |
+ |
would expect with greater fluidity of the chains. However, the |
| 332 |
+ |
diffusion constants are all two orders of magnitude smaller than those |
| 333 |
+ |
typical of DPPC.\cite{Cevc87} This is counter-intuitive as the DPPC |
| 334 |
+ |
molecule is sterically larger and heavier than our model. This could |
| 335 |
+ |
be an indication that our model's chains are too interwoven and hinder |
| 336 |
+ |
the motion of the lipid, or that a simplification in the model's |
| 337 |
+ |
forces has led to a slowing of diffusive behavior within the |
| 338 |
+ |
bilayer. In contrast, the diffusion constant of the {\sc ssd} water, |
| 339 |
+ |
$9.84\times 10^{-6}\,\text{cm}^2/\text{s}$, compares favorably with |
| 340 |
+ |
that of bulk water ($2.2999\times |
| 341 |
+ |
10^{-5}\,\text{cm}^2/\text{s}$\cite{Holz00}). |
| 342 |
+ |
|
| 343 |
+ |
\begin{figure} |
| 344 |
+ |
\centering |
| 345 |
+ |
\includegraphics[width=\linewidth]{diffusionFig.eps} |
| 346 |
+ |
\caption[The lateral diffusion constants versus temperature]{The lateral diffusion constants for the bilayers as a function of temperature.} |
| 347 |
+ |
\label{lipidFig:diffusionFig} |
| 348 |
+ |
\end{figure} |
| 349 |
+ |
|
| 350 |
+ |
\subsection{\label{lipidSec:densProf}Density Profile} |
| 351 |
+ |
|
| 352 |
+ |
Fig.~\ref{lipidFig:densityProfile} illustrates the densities of the |
| 353 |
+ |
atoms in the bilayer systems normalized by the bulk density as a |
| 354 |
+ |
function of distance from the center of the box. The profile is taken |
| 355 |
+ |
along the bilayer normal, in this case the $z$ axis. The profile at |
| 356 |
+ |
270~K shows several structural features that are largely smoothed out |
| 357 |
+ |
by 300~K. The left peak for the {\sc head} atoms is split at 270~K, |
| 358 |
+ |
implying that some freezing of the structure might already be occurring |
| 359 |
+ |
at this temperature. From the dynamics, the tails at this temperature |
| 360 |
+ |
are very much fluid, but the profile could indicate that a phase |
| 361 |
+ |
transition may simply be beyond the length scale of the current |
| 362 |
+ |
simulation. In all profiles, the water penetrates almost |
| 363 |
+ |
5~$\mbox{\AA}$ into the bilayer, completely solvating the {\sc head} |
| 364 |
+ |
atoms. The $\text{{\sc ch}}_3$ atoms although mainly centered at the |
| 365 |
+ |
middle of the bilayer, show appreciable penetration into the head |
| 366 |
+ |
group region. This indicates that the chains have enough mobility to |
| 367 |
+ |
bend back upward to allow the ends to explore areas around the {\sc |
| 368 |
+ |
head} atoms. It is unlikely that this is penetration from a lipid of |
| 369 |
+ |
the opposite face, as the lipids are only 12~$\mbox{\AA}$ in length, |
| 370 |
+ |
and the typical leaf spacing as measured from the {\sc head-head} |
| 371 |
+ |
spacing in the profile is 17.5~$\mbox{\AA}$. |
| 372 |
+ |
|
| 373 |
+ |
\begin{figure} |
| 374 |
+ |
\centering |
| 375 |
+ |
\includegraphics[width=\linewidth]{densityProfile.eps} |
| 376 |
+ |
\caption[The density profile of the lipid bilayers]{The density profile of the lipid bilayers along the bilayer normal. The black lines are the {\sc head} atoms, red lines are the {\sc ch} atoms, green lines are the $\text{{\sc ch}}_2$ atoms, blue lines are the $\text{{\sc ch}}_3$ atoms, and the magenta lines are the {\sc ssd} atoms.} |
| 377 |
+ |
\label{lipidFig:densityProfile} |
| 378 |
+ |
\end{figure} |
| 379 |
+ |
|
| 380 |
+ |
|
| 381 |
+ |
\subsection{\label{lipidSec:scd}$\text{S}_{\text{{\sc cd}}}$ Order Parameters} |
| 382 |
+ |
|
| 383 |
+ |
The $\text{S}_{\text{{\sc cd}}}$ order parameter is often reported in |
| 384 |
+ |
the experimental characterizations of phospholipids. It is obtained |
| 385 |
+ |
through deuterium NMR, and measures the ordering of the carbon |
| 386 |
+ |
deuterium bond in relation to the bilayer normal at various points |
| 387 |
+ |
along the chains. In our model, there are no explicit hydrogens, but |
| 388 |
+ |
the order parameter can be written in terms of the carbon ordering at |
| 389 |
+ |
each point in the chain:\cite{egberts88} |
| 390 |
+ |
\begin{equation} |
| 391 |
+ |
S_{\text{{\sc cd}}} = \frac{2}{3}S_{xx} + \frac{1}{3}S_{yy} |
| 392 |
+ |
\label{lipidEq:scd1} |
| 393 |
+ |
\end{equation} |
| 394 |
+ |
Where $S_{ij}$ is given by: |
| 395 |
+ |
\begin{equation} |
| 396 |
+ |
S_{ij} = \frac{1}{2}\Bigl\langle(3\cos\Theta_i\cos\Theta_j |
| 397 |
+ |
- \delta_{ij})\Bigr\rangle |
| 398 |
+ |
\label{lipidEq:scd2} |
| 399 |
+ |
\end{equation} |
| 400 |
+ |
Here, $\Theta_i$ is the angle the $i$th axis in the reference frame of |
| 401 |
+ |
the carbon atom makes with the bilayer normal. The brackets denote an |
| 402 |
+ |
average over time and molecules. The carbon atom axes are defined: |
| 403 |
+ |
$\mathbf{\hat{z}}\rightarrow$ vector from $C_{n-1}$ to $C_{n+1}$; |
| 404 |
+ |
$\mathbf{\hat{y}}\rightarrow$ vector that is perpendicular to $z$ and |
| 405 |
+ |
in the plane through $C_{n-1}$, $C_{n}$, and $C_{n+1}$; |
| 406 |
+ |
$\mathbf{\hat{x}}\rightarrow$ vector perpendicular to |
| 407 |
+ |
$\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$. |
| 408 |
+ |
|
| 409 |
+ |
The order parameter has a range of $[1,-\frac{1}{2}]$. A value of 1 |
| 410 |
+ |
implies full order aligned to the bilayer axis, 0 implies full |
| 411 |
+ |
disorder, and $-\frac{1}{2}$ implies full order perpendicular to the |
| 412 |
+ |
bilayer axis. The {\sc cd} bond vector for carbons near the head group |
| 413 |
+ |
are usually ordered perpendicular to the bilayer normal, with tails |
| 414 |
+ |
farther away tending toward disorder. This makes the order parameter |
| 415 |
+ |
negative for most carbons, and as such $|S_{\text{{\sc cd}}}|$ is more |
| 416 |
+ |
commonly reported than $S_{\text{{\sc cd}}}$. |
| 417 |
+ |
|
| 418 |
+ |
Fig.~\ref{lipidFig:scdFig} shows the $S_{\text{{\sc cd}}}$ order |
| 419 |
+ |
parameters for the bilayer system at 300~K. There is no appreciable |
| 420 |
+ |
difference in the plots for the various temperatures, however, there |
| 421 |
+ |
is a larger difference between our models ordering, and that of |
| 422 |
+ |
DMPC. As our values are closer to $-\frac{1}{2}$, this implies more |
| 423 |
+ |
ordering perpendicular to the normal than in a real system. This is |
| 424 |
+ |
due to the model having only one carbon group separating the chains |
| 425 |
+ |
from the top of the lipid. In DMPC, with the flexibility inherent in a |
| 426 |
+ |
multiple atom head group, as well as a glycerol linkage between the |
| 427 |
+ |
head group and the acyl chains, there is more loss of ordering by the |
| 428 |
+ |
point when the chains start. |
| 429 |
+ |
|
| 430 |
+ |
\begin{figure} |
| 431 |
+ |
\centering |
| 432 |
+ |
\includegraphics[width=\linewidth]{scdFig.eps} |
| 433 |
+ |
\caption[$\text{S}_{\text{{\sc cd}}}$ order parameter for our model]{A comparison of $|\text{S}_{\text{{\sc cd}}}|$ between our model (blue) and DMPC\cite{petrache00} (black) near 300~K.} |
| 434 |
+ |
\label{lipidFig:scdFig} |
| 435 |
+ |
\end{figure} |
| 436 |
+ |
|
| 437 |
+ |
\subsection{\label{lipidSec:p2Order}$P_2$ Order Parameter} |
| 438 |
+ |
|
| 439 |
+ |
The $P_2$ order parameter allows us to measure the amount of |
| 440 |
+ |
directional ordering that exists in the bilayer. Each lipid molecule |
| 441 |
+ |
can be thought of as a cylindrical tube with the head group at the |
| 442 |
+ |
top. If all of the cylinders are perfectly aligned, the $P_2$ order |
| 443 |
+ |
parameter will be $1.0$. If the cylinders are completely dispersed, |
| 444 |
+ |
the $P_2$ order parameter will be 0. For a collection of unit vectors, |
| 445 |
+ |
the $P_2$ order parameter can be solved via the following |
| 446 |
+ |
method.\cite{zannoni94} |
| 447 |
+ |
|
| 448 |
+ |
Define an ordering matrix $\mathbf{Q}$, such that, |
| 449 |
+ |
\begin{equation} |
| 450 |
+ |
\mathbf{Q} = \frac{1}{N}\sum_i^N % |
| 451 |
+ |
\begin{pmatrix} % |
| 452 |
+ |
u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
| 453 |
+ |
u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
| 454 |
+ |
u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
| 455 |
+ |
\end{pmatrix} |
| 456 |
+ |
\label{lipidEq:po1} |
| 457 |
+ |
\end{equation} |
| 458 |
+ |
Where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
| 459 |
+ |
$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
| 460 |
+ |
collection of unit vectors. This allows the matrix element |
| 461 |
+ |
$Q_{\alpha\beta}$ to be written: |
| 462 |
+ |
\begin{equation} |
| 463 |
+ |
Q_{\alpha\beta} = \langle u_{\alpha}u_{\beta} - |
| 464 |
+ |
\frac{1}{3}\delta_{\alpha\beta} \rangle |
| 465 |
+ |
\label{lipidEq:po2} |
| 466 |
+ |
\end{equation} |
| 467 |
+ |
|
| 468 |
+ |
Having constructed the matrix, diagonalizing $\mathbf{Q}$ yields three |
| 469 |
+ |
eigenvalues and eigenvectors. The eigenvector associated with the |
| 470 |
+ |
largest eigenvalue, $\lambda_{\text{max}}$, is the director for the |
| 471 |
+ |
system of unit vectors. The director is the average direction all of |
| 472 |
+ |
the unit vectors are pointing. The $P_2$ order parameter is then |
| 473 |
+ |
simply |
| 474 |
+ |
\begin{equation} |
| 475 |
+ |
\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}} |
| 476 |
+ |
\label{lipidEq:po3} |
| 477 |
+ |
\end{equation} |
| 478 |
+ |
|
| 479 |
+ |
Table~\ref{lipidTab:blSummary} summarizes the $P_2$ values for the |
| 480 |
+ |
bilayers, as well as the dipole orientations. The unit vector for the |
| 481 |
+ |
lipid molecules was defined by finding the moment of inertia for each |
| 482 |
+ |
lipid, then setting $\mathbf{\hat{u}}$ to point along the axis of |
| 483 |
+ |
minimum inertia. For the {\sc head} atoms, the unit vector simply |
| 484 |
+ |
pointed in the same direction as the dipole moment. For the lipid |
| 485 |
+ |
molecules, the ordering was consistent across all temperatures, with |
| 486 |
+ |
the director pointed along the $z$ axis of the box. More |
| 487 |
+ |
interestingly, is the high degree of ordering the dipoles impose on |
| 488 |
+ |
the {\sc head} atoms. The directors for the dipoles consistently |
| 489 |
+ |
pointed along the plane of the bilayer, with the directors |
| 490 |
+ |
anti-aligned on the top and bottom leaf. |
| 491 |
+ |
|
| 492 |
+ |
\begin{table} |
| 493 |
+ |
\caption[Structural properties of the bilayers]{Bilayer Structural properties as a function of temperature.} |
| 494 |
+ |
\label{lipidTab:blSummary} |
| 495 |
+ |
\begin{center} |
| 496 |
+ |
\begin{tabular}{|c|c|c|c|c|} |
| 497 |
+ |
\hline |
| 498 |
+ |
Temperature (K) & $\langle L_{\perp}\rangle$ ($\mbox{\AA}$) & % |
| 499 |
+ |
$\langle A_{\parallel}\rangle$ ($\mbox{\AA}^2$) & % |
| 500 |
+ |
$\langle P_2\rangle_{\text{Lipid}}$ & % |
| 501 |
+ |
$\langle P_2\rangle_{\text{{\sc head}}}$ \\ \hline |
| 502 |
+ |
270 & 18.1 & 58.1 & 0.253 & 0.494 \\ \hline |
| 503 |
+ |
275 & 17.2 & 56.7 & 0.295 & 0.514 \\ \hline |
| 504 |
+ |
277 & 16.9 & 58.0 & 0.301 & 0.541 \\ \hline |
| 505 |
+ |
280 & 17.4 & 58.0 & 0.274 & 0.488 \\ \hline |
| 506 |
+ |
285 & 16.9 & 57.6 & 0.270 & 0.616 \\ \hline |
| 507 |
+ |
290 & 17.0 & 57.6 & 0.263 & 0.534 \\ \hline |
| 508 |
+ |
293 & 17.5 & 58.0 & 0.227 & 0.643 \\ \hline |
| 509 |
+ |
300 & 16.9 & 57.6 & 0.315 & 0.536 \\ \hline |
| 510 |
+ |
\end{tabular} |
| 511 |
+ |
\end{center} |
| 512 |
+ |
\end{table} |
| 513 |
+ |
|
| 514 |
+ |
\subsection{\label{lipidSec:miscData}Further Bilayer Data} |
| 515 |
+ |
|
| 516 |
+ |
Also summarized in Table~\ref{lipidTab:blSummary}, are the bilayer |
| 517 |
+ |
thickness and area per lipid. The bilayer thickness was measured from |
| 518 |
+ |
the peak to peak {\sc head} atom distance in the density profiles. The |
| 519 |
+ |
area per lipid data compares favorably with values typically seen for |
| 520 |
+ |
DMPC (60.0~$\mbox{\AA}^2$ at 303~K)\cite{petrache00}. Although are |
| 521 |
+ |
values are lower this is most likely due to the shorter chain length |
| 522 |
+ |
of our model (8 versus 14 for DMPC). |
| 523 |
+ |
|
| 524 |
+ |
\subsection{\label{lipidSec:randBilayer}Bilayer Aggregation} |
| 525 |
+ |
|
| 526 |
+ |
A very important accomplishment for our model is its ability to |
| 527 |
+ |
spontaneously form bilayers from a randomly dispersed starting |
| 528 |
+ |
configuration. Fig.~\ref{lipidFig:blImage} shows an image sequence for |
| 529 |
+ |
the bilayer aggregation. After 3.0~ns, the basic form of the bilayer |
| 530 |
+ |
can already be seen. By 7.0~ns, the bilayer has a lipid bridge |
| 531 |
+ |
stretched across the simulation box to itself that will turn out to be |
| 532 |
+ |
very long lived ($\sim$20~ns), as well as a water pore, that will |
| 533 |
+ |
persist for the length of the current simulation. At 24~ns, the lipid |
| 534 |
+ |
bridge is dispersed, and the bilayer is still integrating the lipid |
| 535 |
+ |
molecules from the bridge into itself, and has still been unable to |
| 536 |
+ |
disperse the water pore. |
| 537 |
+ |
|
| 538 |
+ |
\begin{figure} |
| 539 |
+ |
\centering |
| 540 |
+ |
\includegraphics[width=\linewidth]{bLayerImage.eps} |
| 541 |
+ |
\caption[Image sequence of the bilayer aggregation]{Image sequence of the bilayer aggregation. The blue beads are the {\sc head} atoms the grey beads are the chains, and the red and white bead are the water molecules. A box has been drawn around the periodic image.} |
| 542 |
+ |
\label{lipidFig:blImage} |
| 543 |
+ |
\end{figure} |
| 544 |
+ |
|
| 545 |
+ |
\subsection{\label{lipidSec:randIrod}Inverted Rod Aggregation} |
| 546 |
+ |
|
| 547 |
+ |
Fig.~\ref{lipidFig:iRimage} shows a second aggregation sequence |
| 548 |
+ |
simulated in this research. Here the fraction of water had been |
| 549 |
+ |
significantly decreased to observe how the model would respond. After |
| 550 |
+ |
1.5~ns, The main body of water in the system has already collected |
| 551 |
+ |
into a central water channel. By 10.0~ns, the channel has widened |
| 552 |
+ |
slightly, but there are still many sub channels permeating the lipid |
| 553 |
+ |
macro-structure. At 23.0~ns, the central water channel has stabilized |
| 554 |
+ |
and several smaller water channels have been absorbed to main |
| 555 |
+ |
one. However, there are still several other channels that persist |
| 556 |
+ |
through the lipid structure. |
| 557 |
+ |
|
| 558 |
+ |
\begin{figure} |
| 559 |
+ |
\centering |
| 560 |
+ |
\includegraphics[width=\linewidth]{iRodImage.eps} |
| 561 |
+ |
\caption[Image sequence of the inverted rod aggregation]{Image sequence of the inverted rod aggregation. color scheme is the same as in Fig.~\ref{lipidFig:blImage}.} |
| 562 |
+ |
\label{lipidFig:iRimage} |
| 563 |
+ |
\end{figure} |
| 564 |
+ |
|
| 565 |
+ |
\section{\label{lipidSec:Conclusion}Conclusion} |
| 566 |
+ |
|
| 567 |
+ |
We have presented a phospholipid model capable of spontaneous |
| 568 |
+ |
aggregation into a bilayer and an inverted rod structure. The time |
| 569 |
+ |
scales of the macro-molecular aggregations are in excess of 24~ns. In |
| 570 |
+ |
addition the model's bilayer properties have been explored over a |
| 571 |
+ |
range of temperatures through prefabricated bilayers. No freezing |
| 572 |
+ |
transition is seen in the temperature range of our current |
| 573 |
+ |
simulations. However, structural information from the lowest |
| 574 |
+ |
temperature may imply that a freezing event is on a much longer time |
| 575 |
+ |
scale than that explored in this current research. Further studies of |
| 576 |
+ |
this system could extend the time length of the simulations at the low |
| 577 |
+ |
temperatures to observe whether lipid crystallization occurs within the |
| 578 |
+ |
framework of this model. |
