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\usepackage{floatflt} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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\usepackage{subfigure} |
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\usepackage{palatino} |
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\usepackage[ref]{overcite} |
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Simulations of phospholipid bilayers are, by necessity, quite |
| 40 |
|
complex. The lipid molecules are large molecules containing many |
| 41 |
< |
atoms,and the head group of the lipid will typically contain charge |
| 41 |
> |
atoms, and the head group of the lipid will typically contain charge |
| 42 |
|
separated ions which set up a large dipole within the molecule. Adding |
| 43 |
|
to the complexity are the number of water molecules needed to properly |
| 44 |
< |
solvate the lipid bilayer. Because of these factors, many current |
| 45 |
< |
simulations are limited in both length and time scale due to to the |
| 46 |
< |
sheer number of calculations performed at every time step and the |
| 47 |
< |
lifetime of the researcher. A typical |
| 44 |
> |
solvate the lipid bilayer, typically 25 water molecules for every |
| 45 |
> |
lipid molecule. Because of these factors, many current simulations are |
| 46 |
> |
limited in both length and time scale due to to the sheer number of |
| 47 |
> |
calculations performed at every time step and the lifetime of the |
| 48 |
> |
researcher. A typical |
| 49 |
|
simulation\cite{saiz02,lindahl00,venable00,Marrink01} will have around |
| 50 |
|
64 phospholipids forming a bilayer approximately 40~$\mbox{\AA}$ by |
| 51 |
|
50~$\mbox{\AA}$ with roughly 25 waters for every lipid. This means |
| 52 |
|
there are on the order of 8,000 atoms needed to simulate these systems |
| 53 |
< |
and the trajectories in turn are integrated for times up to 10 ns. |
| 53 |
> |
and the trajectories are integrated for times up to 10 ns. |
| 54 |
|
|
| 55 |
|
These limitations make it difficult to study certain biologically |
| 56 |
|
interesting phenomena that don't fit within the short time and length |
| 57 |
< |
scale requirements. One such phenomena is the existence of the ripple |
| 57 |
> |
scale requirements. One such phenomenon is the existence of the ripple |
| 58 |
|
phase ($P_{\beta'}$) of the bilayer between the gel phase |
| 59 |
|
($L_{\beta'}$) and the fluid phase ($L_{\alpha}$). The $P_{\beta'}$ |
| 60 |
|
phase has been shown to have a ripple period of |
| 63 |
|
roughly 25 waters for every lipid to fully solvate the bilayer. With |
| 64 |
|
the large number of atoms involved in a simulation of this magnitude, |
| 65 |
|
steps \emph{must} be taken to simplify the system to the point where |
| 66 |
< |
these numbers are reasonable. |
| 66 |
> |
the numbers of atoms becomes reasonable. |
| 67 |
|
|
| 68 |
|
Another system of interest would be drug molecule diffusion through |
| 69 |
< |
the membrane. Due to the fluid like properties of a lipid membrane, |
| 69 |
> |
the membrane. Due to the fluid-like properties of a lipid membrane, |
| 70 |
|
not all diffusion takes place at membrane channels. It is of interest |
| 71 |
|
to study certain molecules that may incorporate themselves directly |
| 72 |
|
into the membrane. These molecules may then have an appreciable |
| 73 |
|
waiting time (on the order of nanoseconds) within the |
| 74 |
|
bilayer. Simulation of such a long time scale again requires |
| 75 |
|
simplification of the system in order to lower the number of |
| 76 |
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calculations needed at each time step. |
| 76 |
> |
calculations needed at each time step or to increase the length of |
| 77 |
> |
each time step. |
| 78 |
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|
| 79 |
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|
| 80 |
|
\section{Methodology} |
| 83 |
|
|
| 84 |
|
The length scale simplifications are aimed at increasing the number of |
| 85 |
|
molecules that can be simulated without drastically increasing the |
| 86 |
< |
computational cost of the system. This is done by a combination of |
| 87 |
< |
substituting less expensive interactions for expensive ones and |
| 88 |
< |
decreasing the number of interaction sites per molecule. Namely, |
| 89 |
< |
point charge distributions are replaced with dipoles, and unified atoms are |
| 90 |
< |
used in place of water, phospholipid head groups, and alkyl groups. |
| 86 |
> |
computational cost of the simulation. This is done through a |
| 87 |
> |
combination of substituting less expensive interactions for expensive |
| 88 |
> |
ones and decreasing the number of interaction sites per |
| 89 |
> |
molecule. Namely, groups of point charges are replaced with single |
| 90 |
> |
point-dipoles, and unified atoms are used in place of water, |
| 91 |
> |
phospholipid head groups, and alkyl groups. |
| 92 |
|
|
| 93 |
< |
The replacement of charge distributions with dipoles allows us to |
| 94 |
< |
replace an interaction that has a relatively long range, |
| 95 |
< |
$(\frac{1}{r})$, for the coulomb potential, with that of a relatively |
| 96 |
< |
short range, $(\frac{1}{r^{3}})$, for dipole - dipole |
| 97 |
< |
potentials. Combined with a computational simplification algorithm |
| 98 |
< |
such as a Verlet neighbor list,\cite{allen87:csl} this should give |
| 99 |
< |
computational scaling by $N$. This is in comparison to the Ewald |
| 100 |
< |
sum\cite{leach01:mm} needed to compute the coulomb interactions, which |
| 101 |
< |
scales at best by $N \ln N$. |
| 93 |
> |
The replacement of charges with dipoles allows us to replace an |
| 94 |
> |
interaction that has a relatively long range ($\frac{1}{r}$ for the |
| 95 |
> |
coulomb potential) with that of a relatively short range |
| 96 |
> |
($\frac{1}{r^{3}}$ for dipole - dipole potentials). Combined with |
| 97 |
> |
Verlet neighbor lists,\cite{allen87:csl} this should result in an |
| 98 |
> |
algorithm which scales linearly with increasing system size. This is |
| 99 |
> |
in comparison to the Ewald sum\cite{leach01:mm} needed to compute |
| 100 |
> |
periodic replicas of the coulomb interactions, which scales at |
| 101 |
> |
best\cite{darden93:pme} by $N \ln N$. |
| 102 |
|
|
| 103 |
|
The second step taken to simplify the number of calculations is to |
| 104 |
|
incorporate unified models for groups of atoms. In the case of water, |
| 105 |
|
we use the soft sticky dipole (SSD) model developed by |
| 106 |
< |
Ichiye\cite{Liu96} (Section~\ref{sec:ssdModel}). For the phospholipids, a |
| 107 |
< |
unified head atom with a dipole will replace the atoms in the head |
| 108 |
< |
group, while unified $\text{CH}_2$ and $\text{CH}_3$ atoms will |
| 109 |
< |
replace the alkyl units in the tails (Section~\ref{sec:lipidModel}). |
| 106 |
> |
Ichiye\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md} |
| 107 |
> |
(Section~\ref{sec:ssdModel}). For the phospholipids, a unified head |
| 108 |
> |
atom with a dipole will replace the atoms in the head group, while |
| 109 |
> |
unified $\text{CH}_2$ and $\text{CH}_3$ atoms will replace the alkyl |
| 110 |
> |
units in the tails (Section~\ref{sec:lipidModel}). This model is |
| 111 |
> |
similar in practice to that of Lipowsky and Goetz\cite{goetz98}, where |
| 112 |
> |
the whole system is reduced to attractive and repulsive Lennard-Jones |
| 113 |
> |
spheres. However, our model gives a greater level of detail to each |
| 114 |
> |
unified molecule, namely through dipole, bend, and torsion |
| 115 |
> |
interactions. |
| 116 |
|
|
| 117 |
|
The time scale simplifications are introduced so that we can take |
| 118 |
|
longer time steps. By increasing the size of the time steps taken by |
| 119 |
< |
the simulation, we are able to integrate the simulation trajectory |
| 120 |
< |
with fewer calculations. However, care must be taken that any |
| 121 |
< |
simplifications used, still conserve the total energy of the |
| 119 |
> |
the simulation, we are able to integrate a given length of time using |
| 120 |
> |
fewer calculations. However, care must be taken that any |
| 121 |
> |
simplifications used still conserve the total energy of the |
| 122 |
|
simulation. In practice, this means taking steps small enough to |
| 123 |
|
resolve all motion in the system without accidently moving an object |
| 124 |
|
too far along a repulsive energy surface before it feels the effect of |
| 132 |
|
energy. This is in contrast to the 1 fs time step typically needed to |
| 133 |
|
conserve energy when bonds lengths are allowed to oscillate. |
| 134 |
|
|
| 135 |
< |
\subsection{The Soft Sticky Water Model} |
| 135 |
> |
\subsection{The Soft Sticky Dipole Water Model} |
| 136 |
|
\label{sec:ssdModel} |
| 137 |
|
|
| 138 |
< |
%\begin{floatingfigure}{55mm} |
| 139 |
< |
%\includegraphics[width=45mm]{ssd.epsi} |
| 140 |
< |
%\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference. \vspace{5mm}} |
| 141 |
< |
%\label{fig:ssdModel} |
| 142 |
< |
%\end{floatingfigure} |
| 138 |
> |
\begin{figure} |
| 139 |
> |
\centering |
| 140 |
> |
\includegraphics[width=50mm]{ssd.epsi} |
| 141 |
> |
\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference.} |
| 142 |
> |
\label{fig:ssdModel} |
| 143 |
> |
\end{figure} |
| 144 |
|
|
| 145 |
|
The water model used in our simulations is a modified soft |
| 146 |
|
Stockmayer-sphere model.\cite{stevens95} Like the Stockmayer-sphere, the SSD |
| 147 |
< |
model\cite{Liu96} consists of a Lennard-Jones interaction site and a |
| 147 |
> |
model consists of a Lennard-Jones interaction site and a |
| 148 |
|
dipole both located at the water's center of mass (Figure |
| 149 |
|
\ref{fig:ssdModel}). However, the SSD model extends this by adding a |
| 150 |
|
tetrahedral potential to correct for hydrogen bonding. |
| 151 |
|
|
| 152 |
< |
The SSD water potential is then given by the following equation: |
| 152 |
> |
The SSD water potential for a pair of water molecules is then given by |
| 153 |
> |
the following equation: |
| 154 |
|
\begin{equation} |
| 155 |
|
V_{\text{SSD}} = V_{\text{LJ}}(r_{i\!j}) + V_{\text{dp}}(\mathbf{r}_{i\!j}, |
| 156 |
|
\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 158 |
|
\boldsymbol{\Omega}_{j}) |
| 159 |
|
\label{eq:ssdTotPot} |
| 160 |
|
\end{equation} |
| 161 |
< |
$V_{\text{LJ}}$ is the Lennard-Jones potential: |
| 161 |
> |
where $\mathbf{r}_{ij}$ is the vector between molecules $i$ and $j$, |
| 162 |
> |
and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ are the |
| 163 |
> |
Euler angles of molecule $i$ or $j$ respectively. $V_{\text{LJ}}$ is |
| 164 |
> |
the Lennard-Jones potential: |
| 165 |
|
\begin{equation} |
| 166 |
|
V_{\text{LJ}} = |
| 167 |
|
4\epsilon_{ij} \biggl[ |
| 170 |
|
\biggr] |
| 171 |
|
\label{eq:lennardJonesPot} |
| 172 |
|
\end{equation} |
| 173 |
< |
where $r_{ij}$ is the distance between two $ij$ pairs, $\sigma_{ij}$ |
| 174 |
< |
scales the length of the iteraction, and $\epsilon_{ij}$ scales the |
| 175 |
< |
energy of the potential. For SSD, $\sigma_{\text{SSD}} = 3.051 \mbox{ |
| 173 |
> |
where $\sigma_{ij}$ scales the length of the interaction, and |
| 174 |
> |
$\epsilon_{ij}$ scales the energy of the potential. For SSD, |
| 175 |
> |
$\sigma_{\text{SSD}} = 3.051 \mbox{ |
| 176 |
|
\AA}$ and $\epsilon_{\text{SSD}} = 0.152\text{ kcal/mol}$. |
| 177 |
|
$V_{\text{dp}}$ is the dipole potential: |
| 178 |
|
\begin{equation} |
| 185 |
|
{r^{5}_{ij}} \biggr] |
| 186 |
|
\label{eq:dipolePot} |
| 187 |
|
\end{equation} |
| 188 |
< |
where $\mathbf{r}_{ij}$ is the vector between $i$ and $j$, |
| 189 |
< |
$\boldsymbol{\Omega}$ is the orientation of the species, and |
| 190 |
< |
$\boldsymbol{\mu}$ is the dipole vector. The SSD model specifies a dipole |
| 191 |
< |
magnitude of 2.35~D for water. |
| 188 |
> |
where $\boldsymbol{\mu}_i$ is the dipole vector of molecule $i$, |
| 189 |
> |
$\boldsymbol{\mu}_i$ points along the z-axis in the body fixed |
| 190 |
> |
frame. This frame is then oriented in space by the Euler angles, |
| 191 |
> |
$\boldsymbol{\Omega}_i$. The SSD model specifies a dipole magnitude of |
| 192 |
> |
2.35~D for water. |
| 193 |
|
|
| 194 |
< |
The hydrogen bonding of the model is governed by the $V_{\text{sp}}$ |
| 194 |
> |
The hydrogen bonding is modeled by the $V_{\text{sp}}$ |
| 195 |
|
term of the potential. Its form is as follows: |
| 196 |
|
\begin{equation} |
| 197 |
|
V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
| 225 |
|
\label{eq:spCorrection} |
| 226 |
|
\end{equation} |
| 227 |
|
The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical |
| 228 |
< |
polar coordinates of the position of sphere $j$ in the reference frame |
| 229 |
< |
fixed on sphere $i$ with the z-axis aligned with the dipole moment. |
| 228 |
> |
coordinates of the position of molecule $j$ in the reference frame |
| 229 |
> |
fixed on molecule $i$ with the z-axis aligned with the dipole moment. |
| 230 |
|
The correction |
| 231 |
|
$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
| 232 |
|
is needed because |
| 233 |
|
$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
| 234 |
|
vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. |
| 235 |
|
|
| 236 |
< |
Finally, the sticky potential is scaled by a cutoff function, |
| 237 |
< |
$s(r_{ij})$ that scales smoothly between 0 and 1. It is represented |
| 236 |
> |
Finally, the sticky potential is scaled by a switching function, |
| 237 |
> |
$s(r_{ij})$, that scales smoothly between 0 and 1. It is represented |
| 238 |
|
by: |
| 239 |
|
\begin{equation} |
| 240 |
|
s(r_{ij}) = |
| 249 |
|
\end{equation} |
| 250 |
|
|
| 251 |
|
Despite the apparent complexity of Equation \ref{eq:spPot}, the SSD |
| 252 |
< |
model is still computationaly inexpensive. This is due to Equation |
| 253 |
< |
\ref{eq:spCutoff}. With $r_{L}$ being 2.75~$\mbox\AA}$ and $r_{U}$ |
| 252 |
> |
model is still computationally inexpensive. This is due to Equation |
| 253 |
> |
\ref{eq:spCutoff}. With $r_{L}$ being 2.75~$\mbox{\AA}$ and $r_{U}$ |
| 254 |
|
being equal to either 3.35~$\mbox{\AA}$ for $s(r_{ij})$ or |
| 255 |
|
4.0~$\mbox{\AA}$ for $s'(r_{ij})$, the sticky potential is only active |
| 256 |
< |
over an extremly short range, and then only with other SSD |
| 257 |
< |
molecules. Therefore, it's predominant interaction is through it's |
| 258 |
< |
point dipole and Lennard-Jones sphere. |
| 256 |
> |
over an extremely short range, and then only with other SSD |
| 257 |
> |
molecules. Therefore, it's predominant interaction is through the |
| 258 |
> |
point dipole and the Lennard-Jones sphere. Of these, only the dipole |
| 259 |
> |
interaction can be considered ``long-range''. |
| 260 |
|
|
| 261 |
|
\subsection{The Phospholipid Model} |
| 262 |
|
\label{sec:lipidModel} |
| 263 |
|
|
| 264 |
< |
\begin{floatingfigure}{90mm} |
| 264 |
> |
\begin{figure} |
| 265 |
> |
\centering |
| 266 |
|
\includegraphics[angle=-90,width=80mm]{lipidModel.epsi} |
| 267 |
< |
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length. \vspace{5mm}} |
| 267 |
> |
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
| 268 |
|
\label{fig:lipidModel} |
| 269 |
< |
\end{floatingfigure} |
| 269 |
> |
\end{figure} |
| 270 |
|
|
| 271 |
|
The lipid molecules in our simulations are unified atom models. Figure |
| 272 |
< |
\ref{fig:lipidModel} shows a template drawing for one of our |
| 273 |
< |
lipids. The Head group of the phospholipid is replaced by a single |
| 272 |
> |
\ref{fig:lipidModel} shows a schematic for one of our |
| 273 |
> |
lipids. The head group of the phospholipid is replaced by a single |
| 274 |
|
Lennard-Jones sphere with a freely oriented dipole placed at it's |
| 275 |
< |
center. The magnitude of it's dipole moment is 20.6 D. The tail atoms |
| 276 |
< |
are unified $\text{CH}_2$ and $\text{CH}_3$ atoms and are also modeled |
| 277 |
< |
as Lennard-Jones spheres. The total potential for the lipid is |
| 278 |
< |
represented by Equation \ref{eq:lipidModelPot}. |
| 275 |
> |
center. The magnitude of the dipole moment is 20.6 D, chosen to match |
| 276 |
> |
that of DPPC\cite{Cevc87}. The tail atoms are unified $\text{CH}_2$ |
| 277 |
> |
and $\text{CH}_3$ atoms and are also modeled as Lennard-Jones |
| 278 |
> |
spheres. The total potential for the lipid is represented by Equation |
| 279 |
> |
\ref{eq:lipidModelPot}. |
| 280 |
|
|
| 281 |
|
\begin{equation} |
| 282 |
< |
V_{\mbox{lipid}} = \overbrace{% |
| 283 |
< |
V_{\text{bend}}(\theta_{ijk}) +% |
| 284 |
< |
V_{\text{tors.}}(\phi_{ijkl})}^{bonded} |
| 285 |
< |
+ \overbrace{% |
| 286 |
< |
V_{\text{LJ}}(r_{i\!j}) + |
| 287 |
< |
V_{\text{dp}}(r_{i\!j},\Omega_{i},\Omega_{j})% |
| 268 |
< |
}^{non-bonded} |
| 282 |
> |
V_{\text{lipid}} = |
| 283 |
> |
\sum_{i}V_{i}^{\text{internal}} |
| 284 |
> |
+ \sum_i \sum_{j>i} \sum_{\alpha_i} |
| 285 |
> |
\sum_{\beta_j} |
| 286 |
> |
V_{\text{LJ}}(r_{\alpha_{i}\beta_{j}}) |
| 287 |
> |
+\sum_i\sum_{j>i}V_{\text{dp}}(r_{1_i,1_j},\Omega_{1_i},\Omega_{1_j}) |
| 288 |
|
\label{eq:lipidModelPot} |
| 289 |
|
\end{equation} |
| 290 |
+ |
where, |
| 291 |
+ |
\begin{equation} |
| 292 |
+ |
V_{i}^{\text{internal}} = |
| 293 |
+ |
\sum_{\text{bends}}V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
| 294 |
+ |
+ \sum_{\text{torsions}}V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
| 295 |
+ |
+ \sum_{\alpha_i} \sum_{\beta_i > \alpha_i}V_{\text{LJ}} |
| 296 |
+ |
(r_{\alpha_i \beta_i}) |
| 297 |
+ |
\label{eq:lipidModelPotInternal} |
| 298 |
+ |
\end{equation} |
| 299 |
|
|
| 300 |
|
The non-bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are |
| 301 |
|
the Lennard-Jones and dipole-dipole interactions respectively. For the |
| 302 |
< |
non-bonded potentials, only the bend and the torsional potentials are |
| 302 |
> |
bonded potentials, only the bend and the torsional potentials are |
| 303 |
|
calculated. The bond potential is not calculated, and the bond lengths |
| 304 |
|
are constrained via RATTLE.\cite{leach01:mm} The bend potential is of |
| 305 |
|
the form: |
| 306 |
|
\begin{equation} |
| 307 |
< |
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}\frac{(\theta_{ijk} - \theta_0)^2}{2} |
| 307 |
> |
V_{\text{bend}}(\theta_{\alpha\beta\gamma}) |
| 308 |
> |
= k_{\theta}\frac{(\theta_{\alpha\beta\gamma} - \theta_0)^2}{2} |
| 309 |
|
\label{eq:bendPot} |
| 310 |
|
\end{equation} |
| 311 |
|
Where $k_{\theta}$ sets the stiffness of the bend potential, and $\theta_0$ |
| 312 |
|
sets the equilibrium bend angle. The torsion potential was given by: |
| 313 |
|
\begin{equation} |
| 314 |
< |
V_{\text{tors.}}(\phi_{ijkl})= c_1[1+\cos\phi_{ijkl}] |
| 315 |
< |
+ c_2 [1 - \cos(2\phi_{ijkl})] + c_3[1 + \cos(3\phi_{ijkl})] |
| 314 |
> |
V_{\text{tors.}}(\phi_{\alpha\beta\gamma\zeta}) |
| 315 |
> |
= c_1 [1+\cos\phi_{\alpha\beta\gamma\zeta}] |
| 316 |
> |
+ c_2 [1 - \cos(2\phi_{\alpha\beta\gamma\zeta})] |
| 317 |
> |
+ c_3 [1 + \cos(3\phi_{\alpha\beta\gamma\zeta})] |
| 318 |
|
\label{eq:torsPot} |
| 319 |
|
\end{equation} |
| 320 |
|
All parameters for bonded and non-bonded potentials in the tail atoms |
| 331 |
|
\subsection{Starting Configuration and Parameters} |
| 332 |
|
\label{sec:5x5Start} |
| 333 |
|
|
| 334 |
< |
Our first simulation was an array of 25 single chained lipids in a sea |
| 334 |
> |
\begin{figure} |
| 335 |
> |
\centering |
| 336 |
> |
\mbox{ |
| 337 |
> |
\subfigure[The starting configuration of the 25 lipid system.]{% |
| 338 |
> |
\label{fig:5x5Start}% |
| 339 |
> |
\includegraphics[width=70mm]{5x5-initial.eps}}\quad |
| 340 |
> |
\subfigure[The 25 lipid system at 6.27~ns.]{% |
| 341 |
> |
\label{fig:5x5Final}% |
| 342 |
> |
\includegraphics[width=70mm]{5x5-6.27ns.eps}} |
| 343 |
> |
} |
| 344 |
> |
\caption{Snapshots of the 25 lipid system. Carbon tail atoms are drawn in gray, the phospholipid head groups are colored blue, and the waters are scaled down for visibility. A box has been drawn around the periodic image.} |
| 345 |
> |
\end{figure} |
| 346 |
> |
|
| 347 |
> |
Our first simulation is an array of 25 single chain lipids in a sea |
| 348 |
|
of water (Figure \ref{fig:5x5Start}). The total number of water |
| 349 |
< |
molecules was 1386, giving a final of water concentration of 70\% |
| 350 |
< |
wt. The simulation box measured 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
| 351 |
< |
x 39.4~$\mbox{\AA}$ with periodic boundary conditions invoked. The |
| 352 |
< |
system was simulated in the micro-canonical (NVE) ensemble with an |
| 349 |
> |
molecules is 1386, giving a final of water concentration of 70\% |
| 350 |
> |
wt. The simulation box measures 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
| 351 |
> |
x 39.4~$\mbox{\AA}$ with periodic boundary conditions imposed. The |
| 352 |
> |
system is simulated in the micro-canonical (NVE) ensemble with an |
| 353 |
|
average temperature of 300~K. |
| 354 |
|
|
| 355 |
|
\subsection{Results} |
| 356 |
|
\label{sec:5x5Results} |
| 357 |
|
|
| 358 |
+ |
\begin{figure} |
| 359 |
+ |
\centering |
| 360 |
+ |
\subfigure[The self correlation of the phospholipid head groups. $g(r)$ is on the top, the bottom chart is the $g_\gamma(r)$.]{% |
| 361 |
+ |
\label{fig:5x5HHCorr}% |
| 362 |
+ |
\includegraphics[angle=-90,width=80mm]{all5x5-HEAD-HEAD.epsi}% |
| 363 |
+ |
} |
| 364 |
+ |
\subfigure[The $g(r)$ for the $\text{CH}_2$ molecules in the chain tails]{% |
| 365 |
+ |
\label{fig:5x5CCg}% |
| 366 |
+ |
\includegraphics[angle=-90,width=80mm]{all5x5-CH2-CH2.epsi}} |
| 367 |
+ |
\subfigure% |
| 368 |
+ |
[The pair correlations between the head groups and the water]{% |
| 369 |
+ |
\label{fig:5x5HXCorr}% |
| 370 |
+ |
\includegraphics[angle=-90,width=80mm]{all5x5-HEAD-X.epsi}} |
| 371 |
+ |
\caption{The pair correlation functions for the 25 lipid system} |
| 372 |
+ |
\end{figure} |
| 373 |
+ |
|
| 374 |
+ |
|
| 375 |
|
Figure \ref{fig:5x5Final} shows a snapshot of the system at |
| 376 |
< |
3.6~ns. Here it is exciting to note that the system has spontaneously |
| 377 |
< |
self assembled into a bilayer. Discussion of the length scales of the |
| 378 |
< |
bilayer will follow in this section. However, it is interesting to |
| 379 |
< |
note several qualitative properties of the system revealed by this |
| 380 |
< |
snapshot. First, is how the tail chains are tilted to the bilayer |
| 381 |
< |
normal. This is usually indicative of the gel ($L_{\beta'}$) |
| 382 |
< |
phase. Likely, the box size is too small for the bilayer to relax to |
| 383 |
< |
the fluid ($P_{\alpha}$ phase. Showing the need for a constant |
| 384 |
< |
pressure simulation versus the constant volume of the current |
| 324 |
< |
ensemble. |
| 376 |
> |
6.27~ns. Note that the system has spontaneously self assembled into a |
| 377 |
> |
bilayer. Discussion of the length scales of the bilayer will follow in |
| 378 |
> |
this section. However, it is interesting to note a key qualitative |
| 379 |
> |
property of the system revealed by this snapshot, the tail chains are |
| 380 |
> |
tilted to the bilayer normal. This is usually indicative of the gel |
| 381 |
> |
($L_{\beta'}$) phase. In this system, the box size is probably too |
| 382 |
> |
small for the bilayer to relax to the fluid ($P_{\alpha}$) phase. This |
| 383 |
> |
demonstrates a need for an isobaric-isothermal ensemble where the box |
| 384 |
> |
size may relax or expand to keep the system at 1~atm. |
| 385 |
|
|
| 386 |
< |
The trajectory of the simulation was analyzed using the pair-wise |
| 387 |
< |
radial distribution function, $g(r)$, which has the form: |
| 386 |
> |
The simulation was analyzed using the radial distribution function, |
| 387 |
> |
$g(r)$, which has the form: |
| 388 |
|
\begin{equation} |
| 389 |
< |
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j \neq i} |
| 389 |
> |
g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j > i} |
| 390 |
|
\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 391 |
|
\label{eq:gofr} |
| 392 |
|
\end{equation} |
| 399 |
|
fluid contains no long range structure to contribute peaks in the |
| 400 |
|
number density. |
| 401 |
|
|
| 402 |
< |
For the species containing dipoles, a second pair wise distribution |
| 402 |
> |
For the species containing dipoles, a second pair-wise distribution |
| 403 |
|
function was used, $g_{\gamma}(r)$. It is of the form: |
| 404 |
|
\begin{equation} |
| 405 |
< |
g_{\gamma}(r) = foobar |
| 405 |
> |
g_{\gamma}(r) = \langle \sum_i \sum_{j>i} |
| 406 |
> |
(\cos \gamma_{ij}) \delta(| \mathbf{r} - \mathbf{r}_{ij}|) \rangle |
| 407 |
|
\label{eq:gammaofr} |
| 408 |
|
\end{equation} |
| 409 |
|
Where $\gamma_{ij}$ is the angle between the dipole of atom $j$ with |
| 414 |
|
distance. |
| 415 |
|
|
| 416 |
|
Figure \ref{fig:5x5HHCorr} shows the two self correlation functions |
| 417 |
< |
for the Head groups of the lipids. The first peak at 4.03~$\mbox{\AA}$ is the |
| 418 |
< |
nearest neighbor separation of the heads of two lipids. |
| 417 |
> |
for the Head groups of the lipids. The first peak in the $g(r)$ at |
| 418 |
> |
4.03~$\mbox{\AA}$ is the nearest neighbor separation of the heads of |
| 419 |
> |
two lipids. This corresponds to a maximum in the $g_{\gamma}(r)$ which |
| 420 |
> |
means that the two neighbors on the same leaf have their dipoles |
| 421 |
> |
aligned. The broad peak at 6.5~$\mbox{\AA}$ is the inter-surface |
| 422 |
> |
spacing. Here, there is a corresponding anti-alignment in the angular |
| 423 |
> |
correlation. This means that although the dipoles are aligned on the |
| 424 |
> |
same monolayer, the dipoles will orient themselves to be anti-aligned |
| 425 |
> |
on a opposite facing monolayer. With this information, the two peaks |
| 426 |
> |
at 7.0~$\mbox{\AA}$ and 7.4~$\mbox{\AA}$ are head groups on the same |
| 427 |
> |
monolayer, and they are the second nearest neighbors to the head |
| 428 |
> |
group. The peak is likely a split peak because of the small statistics |
| 429 |
> |
of this system. Finally, the peak at 8.0~$\mbox{\AA}$ is likely the |
| 430 |
> |
second nearest neighbor on the opposite monolayer because of the |
| 431 |
> |
anti-alignment evident in the angular correlation. |
| 432 |
|
|
| 433 |
+ |
Figure \ref{fig:5x5CCg} shows the radial distribution function for the |
| 434 |
+ |
$\text{CH}_2$ unified atoms. The spacing of the atoms along the tail |
| 435 |
+ |
chains accounts for the regularly spaced sharp peaks, but the broad |
| 436 |
+ |
underlying peak with its maximum at 4.6~$\mbox{\AA}$ is the |
| 437 |
+ |
distribution of chain-chain distances between two lipids. The final |
| 438 |
+ |
Figure, Figure \ref{fig:5x5HXCorr}, includes the correlation functions |
| 439 |
+ |
between the Head group and the SSD atoms. The peak in $g(r)$ at |
| 440 |
+ |
3.6~$\mbox{\AA}$ is the distance of closest approach between the two, |
| 441 |
+ |
and $g_{\gamma}(r)$ shows that the SSD atoms will align their dipoles |
| 442 |
+ |
with the head groups at close distance. However, as one increases the |
| 443 |
+ |
distance, the SSD atoms are no longer aligned. |
| 444 |
|
|
| 445 |
+ |
\section{Second Simulation: 50 randomly oriented lipids in water} |
| 446 |
+ |
\label{sec:r50} |
| 447 |
|
|
| 448 |
+ |
\subsection{Starting Configuration and Parameters} |
| 449 |
+ |
\label{sec:r50Start} |
| 450 |
|
|
| 451 |
< |
\section{Second Simulation: 50 randomly oriented lipids in water} |
| 451 |
> |
\begin{figure} |
| 452 |
> |
\centering |
| 453 |
> |
\mbox{ |
| 454 |
> |
\subfigure[The starting configuration of the 50 lipid system.]{% |
| 455 |
> |
\label{fig:r50Start}% |
| 456 |
> |
\includegraphics[width=70mm]{r50-initial.eps}}\quad |
| 457 |
> |
\subfigure[The 50 lipid system at 2.21~ns]{% |
| 458 |
> |
\label{fig:r50Final}% |
| 459 |
> |
\includegraphics[width=70mm]{r50-2.21ns.eps}} |
| 460 |
> |
} |
| 461 |
> |
\caption{Snapshots of the 50 lipid system} |
| 462 |
> |
\end{figure} |
| 463 |
|
|
| 464 |
< |
the second simulation |
| 464 |
> |
The second simulation consists of 50 single chained lipid molecules |
| 465 |
> |
embedded in a sea of 1384 SSD waters (54\% wt.). The lipids in this |
| 466 |
> |
simulation were started with random orientation and location (Figure |
| 467 |
> |
\ref{fig:r50Start} ) The simulation box measured 26.6~$\mbox{\AA}$ x |
| 468 |
> |
26.6~$\mbox{\AA}$ x 108.4~$\mbox{\AA}$ with periodic boundary conditions |
| 469 |
> |
imposed. The simulation was run in the NVE ensemble with an average |
| 470 |
> |
temperature of 300~K. |
| 471 |
|
|
| 472 |
+ |
\subsection{Results} |
| 473 |
+ |
\label{sec:r50Results} |
| 474 |
+ |
|
| 475 |
+ |
\begin{figure} |
| 476 |
+ |
\centering |
| 477 |
+ |
\subfigure[The self correlation of the phospholipid head groups.]{% |
| 478 |
+ |
\label{fig:r50HHCorr}% |
| 479 |
+ |
\includegraphics[angle=-90,width=80mm]{r50-HEAD-HEAD.epsi}% |
| 480 |
+ |
} |
| 481 |
+ |
\subfigure% |
| 482 |
+ |
[The pair correlations between the head groups and the water]{% |
| 483 |
+ |
\label{fig:r50HXCorr}% |
| 484 |
+ |
\includegraphics[angle=-90,width=80mm]{r50-HEAD-X.epsi}% |
| 485 |
+ |
} |
| 486 |
+ |
\subfigure[The $g(r)$ for the $\text{CH}_2$ molecules in the chain tails]{% |
| 487 |
+ |
\label{fig:r50CCg}% |
| 488 |
+ |
\includegraphics[angle=-90,width=80mm]{r50-CH2-CH2.epsi}} |
| 489 |
+ |
|
| 490 |
+ |
\caption{The pair correlation functions for the 50 lipid system} |
| 491 |
+ |
\end{figure} |
| 492 |
+ |
|
| 493 |
+ |
Figure \ref{fig:r50Final} is a snapshot of the system at 2.21~ns. Here |
| 494 |
+ |
we see that the system has already aggregated into several micelles |
| 495 |
+ |
and two are even starting to merge. It will be interesting to watch as |
| 496 |
+ |
this simulation continues what the total time scale for the micelle |
| 497 |
+ |
aggregation and bilayer formation will be, in Marrink's\cite{Marrink01} |
| 498 |
+ |
simulation, bilayer aggregation is predicted to occur around 10~ns. |
| 499 |
+ |
|
| 500 |
+ |
Figures \ref{fig:r50HHCorr}, \ref{fig:r50HXCorr}, and \ref{fig:r50CCg} are |
| 501 |
+ |
the same correlation functions for the random 50 simulation as for the |
| 502 |
+ |
previous simulation of 25 lipids. What is most interesting to note, is |
| 503 |
+ |
the high degree of similarity between the correlation functions |
| 504 |
+ |
between systems. Even though the 25 lipid simulation formed a bilayer |
| 505 |
+ |
and the random 50 simulation is still in the micelle stage, both have |
| 506 |
+ |
an inter-surface spacing of 6.5~$\mbox{\AA}$ with the same |
| 507 |
+ |
characteristic anti-alignment between surfaces. Not as surprising, is |
| 508 |
+ |
the consistency of the closest packing statistics between |
| 509 |
+ |
systems. Namely, a head-head closest approach distance of |
| 510 |
+ |
4~$\mbox{\AA}$, and similar findings for the chain-chain and |
| 511 |
+ |
head-water distributions as in the 25 lipid system. |
| 512 |
+ |
|
| 513 |
|
\section{Future Directions} |
| 514 |
|
|
| 515 |
+ |
Current simulations indicate that our model is a feasible one, however |
| 516 |
+ |
improvements will need to be made to allow the system to be simulated |
| 517 |
+ |
in the isobaric-isothermal ensemble. This will relax the system to an |
| 518 |
+ |
equilibrium configuration at room temperature and pressure allowing us |
| 519 |
+ |
to compare our model to experimental results. Also, we are in the |
| 520 |
+ |
process of parallelizing the code for an even greater speedup. This |
| 521 |
+ |
will allow us to simulate large enough systems to examine phenomena |
| 522 |
+ |
such as the ripple phase and drug molecule diffusion |
| 523 |
|
|
| 524 |
< |
\pagebreak |
| 525 |
< |
\bibliographystyle{achemso} |
| 526 |
< |
\bibliography{canidacy_paper} \end{document} |
| 524 |
> |
Once the work has been completed on the simulation engine, we will |
| 525 |
> |
then use it to explore the phase diagram for our model. By |
| 526 |
> |
characterizing how our model parameters affect the bilayer properties, |
| 527 |
> |
we will tailor our model to more closely match real biological |
| 528 |
> |
molecules. With this information, we will then incorporate |
| 529 |
> |
biologically relevant molecules into the system and observe their |
| 530 |
> |
transport properties across the membrane. |
| 531 |
> |
|
| 532 |
> |
\section{Acknowledgments} |
| 533 |
> |
|
| 534 |
> |
I would like to thank Dr.~J.~Daniel Gezelter for his guidance on this |
| 535 |
> |
project. I would also like to acknowledge the following for their help |
| 536 |
> |
and discussions during this project: Christopher Fennell, Charles |
| 537 |
> |
Vardeman, Teng Lin, Megan Sprague, Patrick Conforti, and Dan |
| 538 |
> |
Combest. Funding for this project came from the National Science |
| 539 |
> |
Foundation. |
| 540 |
> |
|
| 541 |
> |
\pagebreak |
| 542 |
> |
\bibliographystyle{achemso} |
| 543 |
> |
\bibliography{canidacy_paper} |
| 544 |
> |
\end{document} |
