| 1 |
\chapter{\label{chapt:lipid}LIPID MODELING} |
| 2 |
|
| 3 |
\section{\label{lipidSection:introduction}Introduction} |
| 4 |
|
| 5 |
Under biologically relevant conditions, phospholipids are solvated |
| 6 |
in aqueous solutions at a roughly 25:1 ratio. Solvation can have a |
| 7 |
tremendous impact on transport phenomena in biological membranes |
| 8 |
since it can affect the dynamics of ions and molecules that are |
| 9 |
transferred across membranes. Studies suggest that because of the |
| 10 |
directional hydrogen bonding ability of the lipid headgroups, a |
| 11 |
small number of water molecules are strongly held around the |
| 12 |
different parts of the headgroup and are oriented by them with |
| 13 |
residence times for the first hydration shell being around 0.5 - 1 |
| 14 |
ns. In the second solvation shell, some water molecules are weakly |
| 15 |
bound, but are still essential for determining the properties of the |
| 16 |
system. Transport of various molecular species into living cells is |
| 17 |
one of the major functions of membranes. A thorough understanding of |
| 18 |
the underlying molecular mechanism for solute diffusion is crucial |
| 19 |
to the further studies of other related biological processes. All |
| 20 |
transport across cell membranes takes place by one of two |
| 21 |
fundamental processes: Passive transport is driven by bulk or |
| 22 |
inter-diffusion of the molecules being transported or by membrane |
| 23 |
pores which facilitate crossing. Active transport depends upon the |
| 24 |
expenditure of cellular energy in the form of ATP hydrolysis. As the |
| 25 |
central processes of membrane assembly, translocation of |
| 26 |
phospholipids across membrane bilayers requires the hydrophilic head |
| 27 |
of the phospholipid to pass through the highly hydrophobic interior |
| 28 |
of the membrane, and for the hydrophobic tails to be exposed to the |
| 29 |
aqueous environment. A number of studies indicate that the flipping |
| 30 |
of phospholipids occurs rapidly in the eukaryotic ER and the |
| 31 |
bacterial cytoplasmic membrane via a bi-directional, facilitated |
| 32 |
diffusion process requiring no metabolic energy input. Another |
| 33 |
system of interest would be the distribution of sites occupied by |
| 34 |
inhaled anesthetics in membrane. Although the physiological effects |
| 35 |
of anesthetics have been extensively studied, the controversy over |
| 36 |
their effects on lipid bilayers still continues. Recent deuterium |
| 37 |
NMR measurements on halothane in POPC lipid bilayers suggest the |
| 38 |
anesthetics are primarily located at the hydrocarbon chain region. |
| 39 |
Infrared spectroscopy experiments suggest that halothane in DMPC |
| 40 |
lipid bilayers lives near the membrane/water interface. |
| 41 |
|
| 42 |
Molecular dynamics simulations have proven to be a powerful tool for |
| 43 |
studying the functions of biological systems, providing structural, |
| 44 |
thermodynamic and dynamical information. Unfortunately, much of |
| 45 |
biological interest happens on time and length scales well beyond |
| 46 |
the range of current simulation technologies. Several schemes are |
| 47 |
proposed in this chapter to overcome these difficulties. |
| 48 |
|
| 49 |
\section{\label{lipidSection:model}Model} |
| 50 |
|
| 51 |
\subsection{\label{lipidSection:SSD}The Soft Sticky Dipole Water Model} |
| 52 |
|
| 53 |
In a typical bilayer simulation, the dominant portion of the |
| 54 |
computation time will be spent calculating water-water interactions. |
| 55 |
As an efficient solvent model, the Soft Sticky Dipole (SSD) water |
| 56 |
model is used as the explicit solvent in this project. Unlike other |
| 57 |
water models which have partial charges distributed throughout the |
| 58 |
whole molecule, the SSD water model consists of a single site which |
| 59 |
is a Lennard-Jones interaction site, as well as a point dipole. A |
| 60 |
tetrahedral potential is added to correct for hydrogen bonding. The |
| 61 |
following equation describes the interaction between two water |
| 62 |
molecules: |
| 63 |
\[ |
| 64 |
V_{SSD} = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) |
| 65 |
+ V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) |
| 66 |
\] |
| 67 |
where $r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
| 68 |
$\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom |
| 69 |
for molecule $i$ and molecule $j$ respectively. |
| 70 |
\[ |
| 71 |
V_{LJ} (r_{ij} ) = 4\varepsilon _{ij} \left[ {\left( {\frac{{\sigma |
| 72 |
_{ij} }}{{r_{ij} }}} \right)^{12} - \left( {\frac{{\sigma _{ij} |
| 73 |
}}{{r_{ij} }}} \right)^6 } \right] |
| 74 |
\] |
| 75 |
\[ |
| 76 |
V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
| 77 |
_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }} - |
| 78 |
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
| 79 |
r_{ij} } \right)}}{{r_{ij}^5 }}} \right] |
| 80 |
\] |
| 81 |
\[ |
| 82 |
V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) = v_0 [s(r_{ij} )w(r_{ij} |
| 83 |
,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i ,\Omega _j |
| 84 |
)] |
| 85 |
\] |
| 86 |
where $v_0$ is a strength parameter, $s$ and $s'$ are cubic |
| 87 |
switching functions, while $w$ and $w'$ are responsible for the |
| 88 |
tetrahedral potential and the short-range correction to the dipolar |
| 89 |
interaction respectively. |
| 90 |
\[ |
| 91 |
\begin{array}{l} |
| 92 |
w(r_{ij} ,\Omega _i ,\Omega _j ) = \sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij} \\ |
| 93 |
w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - w_0 \\ |
| 94 |
\end{array} |
| 95 |
\] |
| 96 |
Although dipole-dipole and sticky interactions are more |
| 97 |
mathematically complicated than Coulomb interactions, the number of |
| 98 |
pair interactions is reduced dramatically both because the model |
| 99 |
only contains a single-point as well as "short range" nature of the |
| 100 |
higher order interaction. |
| 101 |
|
| 102 |
\subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model} |
| 103 |
|
| 104 |
Figure 1 shows a schematic for our coarse-grained phospholipid |
| 105 |
model. The lipid head group is modeled by a linear rigid body which |
| 106 |
consists of three Lennard-Jones spheres and a centrally located |
| 107 |
point-dipole. The backbone atoms in the glycerol motif are modeled |
| 108 |
by Lennard-Jones spheres with dipoles. Alkyl groups in hydrocarbon |
| 109 |
chains are replaced with unified CH2 or CH3 atoms. |
| 110 |
|
| 111 |
Accurate and efficient computation of electrostatics is one of the |
| 112 |
most difficult tasks in molecular modeling. Traditionally, the |
| 113 |
electrostatic interaction between two molecular species is |
| 114 |
calculated as a sum of interactions between pairs of point charges, |
| 115 |
using Coulomb's law: |
| 116 |
\[ |
| 117 |
V = \sum\limits_{i = 1}^{N_A } {\sum\limits_{j = 1}^{N_B } |
| 118 |
{\frac{{q_i q_j }}{{4\pi \varepsilon _0 r_{ij} }}} } |
| 119 |
\] |
| 120 |
where $N_A$ and $N_B$ are the number of point charges in the two |
| 121 |
molecular species. Originally developed to study ionic crystals, the |
| 122 |
Ewald summation method mathematically transforms this |
| 123 |
straightforward but conditionally convergent summation into two more |
| 124 |
complicated but rapidly convergent sums. One summation is carried |
| 125 |
out in reciprocal space while the other is carried out in real |
| 126 |
space. An alternative approach is a multipole expansion, which is |
| 127 |
based on electrostatic moments, such as charge (monopole), dipole, |
| 128 |
quadruple etc. |
| 129 |
|
| 130 |
Here we consider a linear molecule which consists of two point |
| 131 |
charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The |
| 132 |
electrostatic potential at point $P$ is given by: |
| 133 |
\[ |
| 134 |
\frac{1}{{4\pi \varepsilon _0 }}\left( {\frac{{ - q}}{{r_ - }} + |
| 135 |
\frac{{ + q}}{{r_ + }}} \right) = \frac{1}{{4\pi \varepsilon _0 |
| 136 |
}}\left( {\frac{{ - q}}{{\sqrt {r^2 + \frac{{d^2 }}{4} + rd\cos |
| 137 |
\theta } }} + \frac{q}{{\sqrt {r^2 + \frac{{d^2 }}{4} - rd\cos |
| 138 |
\theta } }}} \right) |
| 139 |
\] |
| 140 |
|
| 141 |
The basic assumption of the multipole expansion is $r \gg d$ , thus, |
| 142 |
$\frac{{d^2 }}{4}$ inside the square root of the denominator is |
| 143 |
neglected. This is a reasonable approximation in most cases. |
| 144 |
Unfortunately, in our headgroup model, the distance of charge |
| 145 |
separation $d$ (4.63 $\AA$ in PC headgroup) may be comparable to |
| 146 |
$r$. Nevertheless, we could still assume $ \cos \theta \approx 0$ |
| 147 |
in the central region of the headgroup. Using Taylor expansion and |
| 148 |
associating appropriate terms with electric moments will result in a |
| 149 |
"split-dipole" approximation: |
| 150 |
\[ |
| 151 |
V(r) = \frac{1}{{4\pi \varepsilon _0 }}\frac{{r\mu \cos \theta |
| 152 |
}}{{R^3 }} |
| 153 |
\] |
| 154 |
where$R = \sqrt {r^2 + \frac{{d^2 }}{4}}$ Placing a dipole at point |
| 155 |
$P$ and applying the same strategy, the interaction between two |
| 156 |
split-dipoles is then given by: |
| 157 |
\[ |
| 158 |
V_{sd} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
| 159 |
_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{R_{ij}^3 }} - |
| 160 |
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
| 161 |
r_{ij} } \right)}}{{R_{ij}^5 }}} \right] |
| 162 |
\] |
| 163 |
where $\mu _i$ and $\mu _j$ are the dipole moment of molecule $i$ |
| 164 |
and molecule $j$ respectively, $r_{ij}$ is vector between molecule |
| 165 |
$i$ and molecule $j$, and $R_{ij{$ is given by, |
| 166 |
\[ |
| 167 |
R_{ij} = \sqrt {r_{ij}^2 + \frac{{d_i^2 }}{4} + \frac{{d_j^2 |
| 168 |
}}{4}} |
| 169 |
\] |
| 170 |
where $d_i$ and $d_j$ are the charge separation distance of dipole |
| 171 |
and respectively. This approximation to the multipole expansion |
| 172 |
maintains the fast fall-off of the multipole potentials but lacks |
| 173 |
the normal divergences when two polar groups get close to one |
| 174 |
another. |
| 175 |
|
| 176 |
\begin{figure} |
| 177 |
\centering |
| 178 |
\includegraphics[width=\linewidth]{split_dipole.eps} |
| 179 |
\caption[Comparison between electrostatic approximation]{Electron |
| 180 |
density profile along the bilayer normal.} |
| 181 |
\label{lipidFigure:splitDipole} |
| 182 |
\end{figure} |
| 183 |
|
| 184 |
%\section{\label{lipidSection:methods}Methods} |
| 185 |
|
| 186 |
\section{\label{lipidSection:resultDiscussion}Results and Discussion} |
| 187 |
|
| 188 |
\subsection{One Lipid in Sea of Water Molecules} |
| 189 |
|
| 190 |
To exclude the inter-headgroup interaction, atomistic models of one |
| 191 |
lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built |
| 192 |
and studied using atomistic molecular dynamics. The simulation was |
| 193 |
analyzed using a set of radial distribution functions, which give |
| 194 |
the probability of finding a pair of molecular species a distance |
| 195 |
apart, relative to the probability expected for a completely random |
| 196 |
distribution function at the same density. |
| 197 |
|
| 198 |
\begin{equation} |
| 199 |
g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i |
| 200 |
\in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} } > |
| 201 |
\end{equation} |
| 202 |
\begin{equation} |
| 203 |
g_{AB} (r,\cos \theta ) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < |
| 204 |
\sum\limits_{i \in A} {\sum\limits_{j \in B} {\delta (r - r_{ij} )} |
| 205 |
} \delta (\cos \theta _{ij} - \cos \theta ) > |
| 206 |
\end{equation} |
| 207 |
|
| 208 |
From figure 4(a), we can identify the first solvation shell (3.5 |
| 209 |
$\AA$) and the second solvation shell (5.0 $\AA$) from both plots. |
| 210 |
However, the corresponding orientations are different. In DLPE, |
| 211 |
water molecules prefer to sit around -NH3 group due to the hydrogen |
| 212 |
bonding. In contrast, because of the hydrophobic effect of the |
| 213 |
-N(CH3)3 group, the preferred position of water molecules in DMPC is |
| 214 |
around the -PO4 Group. When the water molecules are far from the |
| 215 |
headgroup, the distribution of the two angles should be uniform. The |
| 216 |
correlation close to center of the headgroup dipole (< 5 $\AA$) in |
| 217 |
both plots tell us that in the closely-bound region, the dipoles of |
| 218 |
the water molecules are preferentially anti-aligned with the dipole |
| 219 |
of headgroup. |
| 220 |
|
| 221 |
\begin{figure} |
| 222 |
\centering |
| 223 |
\includegraphics[width=\linewidth]{g_atom.eps} |
| 224 |
\caption[The pair correlation functions for atomistic models]{} |
| 225 |
\label{lipidFigure:PCFAtom} |
| 226 |
\end{figure} |
| 227 |
|
| 228 |
The initial configurations of coarse-grained systems are constructed |
| 229 |
from the previous atomistic ones. The parameters for the |
| 230 |
coarse-grained model in Table~\ref{lipidTable:parameter} are |
| 231 |
estimated and tuned using isothermal-isobaric molecular dynamics. |
| 232 |
Pair distribution functions calculated from coarse-grained models |
| 233 |
preserve the basic characteristics of the atomistic simulations. The |
| 234 |
water density, measured in a head-group-fixed reference frame, |
| 235 |
surrounding two phospholipid headgroups is shown in |
| 236 |
Fig.~\ref{lipidFigure:PCFCoarse}. It is clear that the phosphate end |
| 237 |
in DMPC and the amine end in DMPE are the two most heavily solvated |
| 238 |
atoms. |
| 239 |
|
| 240 |
\begin{figure} |
| 241 |
\centering |
| 242 |
\includegraphics[width=\linewidth]{g_coarse.eps} |
| 243 |
\caption[The pair correlation functions for coarse-grained models]{} |
| 244 |
\label{lipidFigure:PCFCoarse} |
| 245 |
\end{figure} |
| 246 |
|
| 247 |
\begin{figure} |
| 248 |
\centering |
| 249 |
\includegraphics[width=\linewidth]{EWD_coarse.eps} |
| 250 |
\caption[Excess water density of coarse-grained phospholipids]{ } |
| 251 |
\label{lipidFigure:EWDCoarse} |
| 252 |
\end{figure} |
| 253 |
|
| 254 |
\begin{table} |
| 255 |
\caption{The Parameters For Coarse-grained Phospholipids} |
| 256 |
\label{lipidTable:parameter} |
| 257 |
\begin{center} |
| 258 |
\begin{tabular}{|l|c|c|c|c|c|} |
| 259 |
\hline |
| 260 |
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
| 261 |
Atom type & Mass & $\sigma$ & $\epsilon$ & charge & Dipole \\ |
| 262 |
|
| 263 |
$\text{{\sc CH}}_2$ & 14.03 & 3.95 & 0.0914 & 0 & 0 \\ |
| 264 |
$\text{{\sc CH}}_3$ & 15.04 & 3.75 & 0.195 & 0 & 0 \\ |
| 265 |
$\text{{\sc CE}}$ & 28.01 & 3.427& 0.294 & 0 & 1.693 |
| 266 |
$\text{{\sc CK}}$ & 28.01 & 3.592& 0.311 & 0 & 2.478 \\ |
| 267 |
$\text{{\sc PO}}_4$ & 108.995& 3.9 & 1.88 & -1& 0 \\ |
| 268 |
$\text{{\sc HDP}}$ & 14.03 & 4.0 & 0.13 & 0 & 0 \\ |
| 269 |
$\text{{\sc NC}}_4$ & 73.137 & 4.9 & 0.88 & +1& 0 \\ |
| 270 |
$\text{{\sc NH}}_3$ & 17.03 & 3.25 & 0.17 & +1& 0\\ |
| 271 |
\hline |
| 272 |
\end{tabular} |
| 273 |
\end{center} |
| 274 |
\end{table} |
| 275 |
|
| 276 |
\subsection{Bilayer Simulations Using Coarse-grained Model} |
| 277 |
|
| 278 |
A bilayer system consisting of 128 DMPC lipids and 3655 water |
| 279 |
molecules has been constructed from an atomistic coordinate |
| 280 |
file.[15] The MD simulation is performed at constant temperature, T |
| 281 |
= 300K, and constant pressure, p = 1 atm, and consisted of an |
| 282 |
equilibration period of 2 ns. During the equilibration period, the |
| 283 |
system was initially simulated at constant volume for 1ns. Once the |
| 284 |
system was equilibrated at constant volume, the cell dimensions of |
| 285 |
the system was relaxed by performing under NPT conditions using |
| 286 |
Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After |
| 287 |
equilibration, different properties were evaluated over a production |
| 288 |
run of 5 ns. |
| 289 |
|
| 290 |
\begin{figure} |
| 291 |
\centering |
| 292 |
\includegraphics[width=\linewidth]{bilayer.eps} |
| 293 |
\caption[Image of a coarse-grained bilayer system]{A coarse-grained |
| 294 |
bilayer system consisting of 128 DMPC lipids and 3655 SSD water |
| 295 |
molecules.} |
| 296 |
\label{lipidFigure:bilayer} |
| 297 |
\end{figure} |
| 298 |
|
| 299 |
\subsubsection{Electron Density Profile (EDP)} |
| 300 |
|
| 301 |
Assuming a gaussian distribution of electrons on each atomic center |
| 302 |
with a variance estimated from the size of the van der Waals radius, |
| 303 |
the EDPs which are proportional to the density profiles measured |
| 304 |
along the bilayer normal obtained by x-ray scattering experiment, |
| 305 |
can be expressed by |
| 306 |
\begin{equation} |
| 307 |
\rho _{x - ray} (z)dz \propto \sum\limits_{i = 1}^N {\frac{{n_i |
| 308 |
}}{V}\frac{1}{{\sqrt {2\pi \sigma ^2 } }}e^{ - (z - z_i )^2 /2\sigma |
| 309 |
^2 } dz}, |
| 310 |
\end{equation} |
| 311 |
where $\sigma$ is the variance equal to the van der Waals radius, |
| 312 |
$n_i$ is the atomic number of site $i$ and $V$ is the volume of the |
| 313 |
slab between $z$ and $z+dz$ . The highest density of total EDP |
| 314 |
appears at the position of lipid-water interface corresponding to |
| 315 |
headgroup, glycerol, and carbonyl groups of the lipids and the |
| 316 |
distribution of water locked near the head groups, while the lowest |
| 317 |
electron density is in the hydrocarbon region. As a good |
| 318 |
approximation to the thickness of the bilayer, the headgroup spacing |
| 319 |
, is defined as the distance between two peaks in the electron |
| 320 |
density profile, calculated from our simulations to be 34.1 $\AA$. |
| 321 |
This value is close to the x-ray diffraction experimental value 34.4 |
| 322 |
$\AA$. |
| 323 |
|
| 324 |
\begin{figure} |
| 325 |
\centering |
| 326 |
\includegraphics[width=\linewidth]{electron_density.eps} |
| 327 |
\caption[The density profile of the lipid bilayers]{Electron density |
| 328 |
profile along the bilayer normal. The water density is shown in red, |
| 329 |
the density due to the headgroups in green, the glycerol backbone in |
| 330 |
brown, $\text{{\sc CH}}_2$ in yellow, $\text{{\sc CH}}_3$ in cyan, |
| 331 |
and total density due to DMPC in blue.} |
| 332 |
\label{lipidFigure:electronDensity} |
| 333 |
\end{figure} |
| 334 |
|
| 335 |
\subsubsection{$\text{S}_{\text{{\sc cd}}}$ Order Parameter} |
| 336 |
|
| 337 |
Measuring deuterium order parameters by NMR is a useful technique to |
| 338 |
study the orientation of hydrocarbon chains in phospholipids. The |
| 339 |
order parameter tensor $S$ is defined by: |
| 340 |
\begin{equation} |
| 341 |
S_{ij} = \frac{1}{2} < 3\cos \theta _i \cos \theta _j - \delta |
| 342 |
_{ij} > |
| 343 |
\end{equation} |
| 344 |
where $\theta$ is the angle between the $i$th molecular axis and |
| 345 |
the bilayer normal ($z$ axis). The brackets denote an average over |
| 346 |
time and molecules. The molecular axes are defined: |
| 347 |
\begin{itemize} |
| 348 |
\item $\mathbf{\hat{z}}$ is the vector from $C_{n-1}$ to $C_{n+1}$. |
| 349 |
\item $\mathbf{\hat{y}}$ is the vector that is perpendicular to $z$ and |
| 350 |
in the plane through $C_{n-1}$, $C_{n}$, and $C_{n+1}$. |
| 351 |
\item $\mathbf{\hat{x}}$ is the vector perpendicular to |
| 352 |
$\mathbf{\hat{y}}$ and $\mathbf{\hat{z}}$. |
| 353 |
\end{itemize} |
| 354 |
In coarse-grained model, although there are no explicit hydrogens, |
| 355 |
the order parameter can still be written in terms of carbon ordering |
| 356 |
at each point of the chain |
| 357 |
\begin{equation} |
| 358 |
S_{ij} = \frac{1}{2} < 3\cos \theta _i \cos \theta _j - \delta |
| 359 |
_{ij} >. |
| 360 |
\end{equation} |
| 361 |
|
| 362 |
Fig.~\ref{lipidFigure:Scd} shows the order parameter profile |
| 363 |
calculated for our coarse-grained DMPC bilayer system at 300K. Also |
| 364 |
shown are the experimental data of Tiburu. The fact that |
| 365 |
$\text{S}_{\text{{\sc cd}}}$ order parameters calculated from |
| 366 |
simulation are higher than the experimental ones is ascribed to the |
| 367 |
assumption of the locations of implicit hydrogen atoms which are |
| 368 |
fixed in coarse-grained models at positions relative to the CC |
| 369 |
vector. |
| 370 |
|
| 371 |
\begin{figure} |
| 372 |
\centering |
| 373 |
\includegraphics[width=\linewidth]{scd.eps} |
| 374 |
\caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison |
| 375 |
of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model |
| 376 |
(blue) and DMPC\cite{petrache00} (black) near 300~K.} |
| 377 |
\label{lipidFigure:Scd} |
| 378 |
\end{figure} |
