| 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 |
| 14 |
> |
ns\cite{Ho1992}. In the second solvation shell, some water molecules |
| 15 |
> |
are weakly bound, but are still essential for determining the |
| 16 |
> |
properties of the system. Transport of various molecular species |
| 17 |
> |
into living cells is one of the major functions of membranes. A |
| 18 |
> |
thorough understanding of the underlying molecular mechanism for |
| 19 |
> |
solute diffusion is crucial to the further studies of other related |
| 20 |
> |
biological processes. All transport across cell membranes takes |
| 21 |
> |
place by one of two fundamental processes: Passive transport is |
| 22 |
> |
driven by bulk or inter-diffusion of the molecules being transported |
| 23 |
> |
or by membrane pores which facilitate crossing. Active transport |
| 24 |
> |
depends upon the expenditure of cellular energy in the form of ATP |
| 25 |
> |
hydrolysis. As the central processes of membrane assembly, |
| 26 |
> |
translocation of phospholipids across membrane bilayers requires the |
| 27 |
> |
hydrophilic head of the phospholipid to pass through the highly |
| 28 |
> |
hydrophobic interior of the membrane, and for the hydrophobic tails |
| 29 |
> |
to be exposed to the aqueous environment\cite{Sasaki2004}. A number |
| 30 |
> |
of studies indicate that the flipping of phospholipids occurs |
| 31 |
> |
rapidly in the eukaryotic endoplasmic reticulum and the bacterial |
| 32 |
> |
cytoplasmic membrane via a bi-directional, facilitated diffusion |
| 33 |
> |
process requiring no metabolic energy input. Another system of |
| 34 |
> |
interest is the distribution of sites occupied by inhaled |
| 35 |
> |
anesthetics in membrane. Although the physiological effects of |
| 36 |
> |
anesthetics have been extensively studied, the controversy over |
| 37 |
|
their effects on lipid bilayers still continues. Recent deuterium |
| 38 |
< |
NMR measurements on halothane in POPC lipid bilayers suggest the |
| 39 |
< |
anesthetics are primarily located at the hydrocarbon chain region. |
| 40 |
< |
Infrared spectroscopy experiments suggest that halothane in DMPC |
| 41 |
< |
lipid bilayers lives near the membrane/water interface. |
| 38 |
> |
NMR measurements on halothane on POPC lipid bilayers suggest the |
| 39 |
> |
anesthetics are primarily located at the hydrocarbon chain |
| 40 |
> |
region\cite{Baber1995}. However, infrared spectroscopy experiments |
| 41 |
> |
suggest that halothane in DMPC lipid bilayers lives near the |
| 42 |
> |
membrane/water interface\cite{Lieb1982}. |
| 43 |
|
|
| 44 |
|
Molecular dynamics simulations have proven to be a powerful tool for |
| 45 |
|
studying the functions of biological systems, providing structural, |
| 48 |
|
the range of current simulation technologies. Several schemes are |
| 49 |
|
proposed in this chapter to overcome these difficulties. |
| 50 |
|
|
| 51 |
< |
\section{\label{lipidSection:model}Model} |
| 51 |
> |
\section{\label{lipidSection:model}Model and Methodology} |
| 52 |
|
|
| 53 |
|
\subsection{\label{lipidSection:SSD}The Soft Sticky Dipole Water Model} |
| 54 |
|
|
| 55 |
|
In a typical bilayer simulation, the dominant portion of the |
| 56 |
|
computation time will be spent calculating water-water interactions. |
| 57 |
|
As an efficient solvent model, the Soft Sticky Dipole (SSD) water |
| 58 |
< |
model is used as the explicit solvent in this project. Unlike other |
| 59 |
< |
water models which have partial charges distributed throughout the |
| 60 |
< |
whole molecule, the SSD water model consists of a single site which |
| 61 |
< |
is a Lennard-Jones interaction site, as well as a point dipole. A |
| 62 |
< |
tetrahedral potential is added to correct for hydrogen bonding. The |
| 63 |
< |
following equation describes the interaction between two water |
| 64 |
< |
molecules: |
| 65 |
< |
\[ |
| 58 |
> |
model\cite{Chandra1999,Fennell2004} is used as the explicit solvent |
| 59 |
> |
in this project. Unlike other water models which have partial |
| 60 |
> |
charges distributed throughout the whole molecule, the SSD water |
| 61 |
> |
model consists of a single site which is a Lennard-Jones interaction |
| 62 |
> |
site, as well as a point dipole. A tetrahedral potential is added to |
| 63 |
> |
correct for hydrogen bonding. The following equation describes the |
| 64 |
> |
interaction between two water molecules: |
| 65 |
> |
\begin{equation} |
| 66 |
|
V_{SSD} = V_{LJ} (r_{ij} ) + V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) |
| 67 |
|
+ V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) |
| 68 |
< |
\] |
| 69 |
< |
where $r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
| 68 |
> |
\label{lipidSection:ssdEquation} |
| 69 |
> |
\end{equation} |
| 70 |
> |
where$r_{ij}$ is the vector between molecule $i$ and molecule $j$, |
| 71 |
|
$\Omega _i$ and $\Omega _j$ are the orientational degrees of freedom |
| 72 |
< |
for molecule $i$ and molecule $j$ respectively. |
| 73 |
< |
\[ |
| 74 |
< |
V_{LJ} (r_{ij} ) = 4\varepsilon _{ij} \left[ {\left( {\frac{{\sigma |
| 75 |
< |
_{ij} }}{{r_{ij} }}} \right)^{12} - \left( {\frac{{\sigma _{ij} |
| 76 |
< |
}}{{r_{ij} }}} \right)^6 } \right] |
| 77 |
< |
\] |
| 78 |
< |
\[ |
| 79 |
< |
V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) = \frac{1}{{4\pi \varepsilon |
| 80 |
< |
_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }} - |
| 81 |
< |
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
| 82 |
< |
r_{ij} } \right)}}{{r_{ij}^5 }}} \right] |
| 83 |
< |
\] |
| 84 |
< |
\[ |
| 85 |
< |
V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) = v_0 [s(r_{ij} )w(r_{ij} |
| 86 |
< |
,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i ,\Omega _j |
| 84 |
< |
)] |
| 85 |
< |
\] |
| 72 |
> |
for molecule $i$ and molecule $j$ respectively. The potential terms |
| 73 |
> |
in Eq.~\ref{lipidSection:ssdEquation} are given by : |
| 74 |
> |
\begin{eqnarray} |
| 75 |
> |
V_{LJ} (r_{ij} ) &= &4\varepsilon _{ij} \left[ {\left( |
| 76 |
> |
{\frac{{\sigma _{ij} }}{{r_{ij} }}} \right)^{12} - \left( |
| 77 |
> |
{\frac{{\sigma _{ij} |
| 78 |
> |
}}{{r_{ij} }}} \right)^6 } \right], \\ |
| 79 |
> |
V_{dp} (r_{ij} ,\Omega _i ,\Omega _j ) &= & |
| 80 |
> |
\frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 81 |
> |
\hat{u}_{i} \cdot \hat{u}_{j} - 3(\hat{u}_i \cdot \hat{\mathbf{r}}_{ij}) % |
| 82 |
> |
(\hat{u}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr],\\ |
| 83 |
> |
V_{sticky} (r_{ij} ,\Omega _i ,\Omega _j ) &=& v_0 [s(r_{ij} |
| 84 |
> |
)w(r_{ij} ,\Omega _i ,\Omega _j ) + s'(r_{ij} )w'(r_{ij} ,\Omega _i |
| 85 |
> |
,\Omega _j )] |
| 86 |
> |
\end{eqnarray} |
| 87 |
|
where $v_0$ is a strength parameter, $s$ and $s'$ are cubic |
| 88 |
< |
switching functions, while $w$ and $w'$ are responsible for the |
| 88 |
> |
switching functions, while $w$ and $w'$ are responsible for the |
| 89 |
|
tetrahedral potential and the short-range correction to the dipolar |
| 90 |
< |
interaction respectively. |
| 91 |
< |
\[ |
| 92 |
< |
\begin{array}{l} |
| 93 |
< |
w(r_{ij} ,\Omega _i ,\Omega _j ) = \sin \theta _{ij} \sin 2\theta _{ij} \cos 2\phi _{ij} \\ |
| 94 |
< |
w'(r_{ij} ,\Omega _i ,\Omega _j ) = (\cos \theta _{ij} - 0.6)^2 (\cos \theta _{ij} + 0.8)^2 - w_0 \\ |
| 95 |
< |
\end{array} |
| 96 |
< |
\] |
| 97 |
< |
Although dipole-dipole and sticky interactions are more |
| 90 |
> |
interaction respectively: |
| 91 |
> |
\begin{eqnarray} |
| 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 - |
| 94 |
> |
w_0. |
| 95 |
> |
\end{eqnarray} |
| 96 |
> |
Here $\theta _{ij}$ and $\phi _{ij}$ are the spherical polar angles |
| 97 |
> |
representing relative orientations between molecule $i$ and molecule |
| 98 |
> |
$j$. Although the dipole-dipole and sticky interactions are more |
| 99 |
|
mathematically complicated than Coulomb interactions, the number of |
| 100 |
|
pair interactions is reduced dramatically both because the model |
| 101 |
|
only contains a single-point as well as "short range" nature of the |
| 102 |
< |
higher order interaction. |
| 102 |
> |
more expensive interaction. |
| 103 |
|
|
| 104 |
|
\subsection{\label{lipidSection:coarseGrained}The Coarse-Grained Phospholipid Model} |
| 105 |
|
|
| 106 |
< |
Figure 1 shows a schematic for our coarse-grained phospholipid |
| 107 |
< |
model. The lipid head group is modeled by a linear rigid body which |
| 108 |
< |
consists of three Lennard-Jones spheres and a centrally located |
| 109 |
< |
point-dipole. The backbone atoms in the glycerol motif are modeled |
| 110 |
< |
by Lennard-Jones spheres with dipoles. Alkyl groups in hydrocarbon |
| 111 |
< |
chains are replaced with unified CH2 or CH3 atoms. |
| 106 |
> |
Fig.~\ref{lipidFigure:coarseGrained} shows a schematic for our |
| 107 |
> |
coarse-grained phospholipid model. The lipid head group is modeled |
| 108 |
> |
by a linear rigid body which consists of three Lennard-Jones spheres |
| 109 |
> |
and a centrally located point-dipole. The backbone atoms in the |
| 110 |
> |
glycerol motif are modeled by Lennard-Jones spheres with dipoles. |
| 111 |
> |
Alkyl groups in hydrocarbon chains are replaced with unified |
| 112 |
> |
$\text{{\sc CH}}_2$ or $\text{{\sc CH}}_3$ atoms. |
| 113 |
|
|
| 114 |
+ |
\begin{figure} |
| 115 |
+ |
\centering |
| 116 |
+ |
\includegraphics[width=3in]{coarse_grained.eps} |
| 117 |
+ |
\caption[A representation of coarse-grained phospholipid model]{A |
| 118 |
+ |
representation of coarse-grained phospholipid model. The lipid |
| 119 |
+ |
headgroup consists of $\text{{\sc PO}}_4$ group (yellow), |
| 120 |
+ |
$\text{{\sc NC}}_4$ group (blue) and a united C atom (gray) with a |
| 121 |
+ |
dipole, while the glycerol backbone includes dipolar $\text{{\sc |
| 122 |
+ |
CE}}$ (red) and $\text{{\sc CK}}$ (pink) groups. Alkyl groups in |
| 123 |
+ |
hydrocarbon chains are simply represented by gray united atoms.} |
| 124 |
+ |
\label{lipidFigure:coarseGrained} |
| 125 |
+ |
\end{figure} |
| 126 |
+ |
|
| 127 |
|
Accurate and efficient computation of electrostatics is one of the |
| 128 |
|
most difficult tasks in molecular modeling. Traditionally, the |
| 129 |
|
electrostatic interaction between two molecular species is |
| 135 |
|
\] |
| 136 |
|
where $N_A$ and $N_B$ are the number of point charges in the two |
| 137 |
|
molecular species. Originally developed to study ionic crystals, the |
| 138 |
< |
Ewald summation method mathematically transforms this |
| 139 |
< |
straightforward but conditionally convergent summation into two more |
| 140 |
< |
complicated but rapidly convergent sums. One summation is carried |
| 141 |
< |
out in reciprocal space while the other is carried out in real |
| 142 |
< |
space. An alternative approach is a multipole expansion, which is |
| 143 |
< |
based on electrostatic moments, such as charge (monopole), dipole, |
| 128 |
< |
quadruple etc. |
| 138 |
> |
Ewald sum method mathematically transforms this straightforward but |
| 139 |
> |
conditionally convergent summation into two more complicated but |
| 140 |
> |
rapidly convergent sums. One summation is carried out in reciprocal |
| 141 |
> |
space while the other is carried out in real space. An alternative |
| 142 |
> |
approach is the multipole expansion, which is based on electrostatic |
| 143 |
> |
moments, such as charge (monopole), dipole, quadrupole etc. |
| 144 |
|
|
| 145 |
|
Here we consider a linear molecule which consists of two point |
| 146 |
|
charges $\pm q$ located at $ ( \pm \frac{d}{2},0)$. The |
| 153 |
|
\theta } }}} \right) |
| 154 |
|
\] |
| 155 |
|
|
| 156 |
+ |
\begin{figure} |
| 157 |
+ |
\centering |
| 158 |
+ |
\includegraphics[width=3in]{charge_dipole.eps} |
| 159 |
+ |
\caption[An illustration of split-dipole |
| 160 |
+ |
approximation]{Electrostatic potential due to a linear molecule |
| 161 |
+ |
comprising two point charges with opposite charges. } |
| 162 |
+ |
\label{lipidFigure:chargeDipole} |
| 163 |
+ |
\end{figure} |
| 164 |
+ |
|
| 165 |
|
The basic assumption of the multipole expansion is $r \gg d$ , thus, |
| 166 |
|
$\frac{{d^2 }}{4}$ inside the square root of the denominator is |
| 167 |
|
neglected. This is a reasonable approximation in most cases. |
| 168 |
|
Unfortunately, in our headgroup model, the distance of charge |
| 169 |
< |
separation $d$ (4.63 $\AA$ in PC headgroup) may be comparable to |
| 170 |
< |
$r$. Nevertheless, we could still assume $ \cos \theta \approx 0$ |
| 171 |
< |
in the central region of the headgroup. Using Taylor expansion and |
| 169 |
> |
separation $d$ (4.63 \AA in PC headgroup) may be comparable to $r$. |
| 170 |
> |
Nevertheless, we could still assume $ \cos \theta \approx 0$ in |
| 171 |
> |
the central region of the headgroup. Using Taylor expansion and |
| 172 |
|
associating appropriate terms with electric moments will result in a |
| 173 |
|
"split-dipole" approximation: |
| 174 |
|
\[ |
| 186 |
|
\] |
| 187 |
|
where $\mu _i$ and $\mu _j$ are the dipole moment of molecule $i$ |
| 188 |
|
and molecule $j$ respectively, $r_{ij}$ is vector between molecule |
| 189 |
< |
$i$ and molecule $j$, and $R_{ij{$ is given by, |
| 189 |
> |
$i$ and molecule $j$, and $R_{ij}$ is given by, |
| 190 |
|
\[ |
| 191 |
|
R_{ij} = \sqrt {r_{ij}^2 + \frac{{d_i^2 }}{4} + \frac{{d_j^2 |
| 192 |
|
}}{4}} |
| 195 |
|
and respectively. This approximation to the multipole expansion |
| 196 |
|
maintains the fast fall-off of the multipole potentials but lacks |
| 197 |
|
the normal divergences when two polar groups get close to one |
| 198 |
< |
another. |
| 199 |
< |
|
| 198 |
> |
another. The comparision between different electrostatic |
| 199 |
> |
approximation is shown in \ref{lipidFigure:splitDipole}. Due to the |
| 200 |
> |
divergence at the central region of the headgroup introduced by |
| 201 |
> |
dipole-dipole approximation, we discover that water molecules are |
| 202 |
> |
locked into the central region in the simulation. This artifact can |
| 203 |
> |
be corrected using split-dipole approximation or other accurate |
| 204 |
> |
methods. |
| 205 |
|
\begin{figure} |
| 206 |
|
\centering |
| 207 |
|
\includegraphics[width=\linewidth]{split_dipole.eps} |
| 208 |
< |
\caption[Comparison between electrostatic approximation]{Electron |
| 209 |
< |
density profile along the bilayer normal.} |
| 210 |
< |
\label{lipidFigure:splitDipole} |
| 208 |
> |
\caption[Comparison between electrostatic |
| 209 |
> |
approximation]{Electrostatic potential map for two pairs of charges |
| 210 |
> |
with different alignments: (a) illustration of different alignments; |
| 211 |
> |
(b) charge-charge interaction; (c) dipole-dipole approximation; (d) |
| 212 |
> |
split-dipole approximation.} \label{lipidFigure:splitDipole} |
| 213 |
|
\end{figure} |
| 214 |
|
|
| 215 |
|
%\section{\label{lipidSection:methods}Methods} |
| 218 |
|
|
| 219 |
|
\subsection{One Lipid in Sea of Water Molecules} |
| 220 |
|
|
| 221 |
< |
To exclude the inter-headgroup interaction, atomistic models of one |
| 222 |
< |
lipid (DMPC or DLPE) in sea of water molecules (TIP3P) were built |
| 223 |
< |
and studied using atomistic molecular dynamics. The simulation was |
| 224 |
< |
analyzed using a set of radial distribution functions, which give |
| 225 |
< |
the probability of finding a pair of molecular species a distance |
| 226 |
< |
apart, relative to the probability expected for a completely random |
| 227 |
< |
distribution function at the same density. |
| 221 |
> |
To tune our parameters without the inter-headgroup interactions, |
| 222 |
> |
atomistic models of one lipid (DMPC or DLPE) in sea of water |
| 223 |
> |
molecules (TIP3P) were built and studied using atomistic molecular |
| 224 |
> |
dynamics. The simulation was analyzed using a set of radial |
| 225 |
> |
distribution functions, which give the probability of finding a pair |
| 226 |
> |
of molecular species a distance apart, relative to the probability |
| 227 |
> |
expected for a completely random distribution function at the same |
| 228 |
> |
density. |
| 229 |
|
|
| 230 |
|
\begin{equation} |
| 231 |
|
g_{AB} (r) = \frac{1}{{\rho _B }}\frac{1}{{N_A }} < \sum\limits_{i |
| 237 |
|
} \delta (\cos \theta _{ij} - \cos \theta ) > |
| 238 |
|
\end{equation} |
| 239 |
|
|
| 240 |
< |
From figure 4(a), we can identify the first solvation shell (3.5 |
| 241 |
< |
$\AA$) and the second solvation shell (5.0 $\AA$) from both plots. |
| 242 |
< |
However, the corresponding orientations are different. In DLPE, |
| 243 |
< |
water molecules prefer to sit around -NH3 group due to the hydrogen |
| 244 |
< |
bonding. In contrast, because of the hydrophobic effect of the |
| 245 |
< |
-N(CH3)3 group, the preferred position of water molecules in DMPC is |
| 246 |
< |
around the -PO4 Group. When the water molecules are far from the |
| 247 |
< |
headgroup, the distribution of the two angles should be uniform. The |
| 248 |
< |
correlation close to center of the headgroup dipole (< 5 $\AA$) in |
| 249 |
< |
both plots tell us that in the closely-bound region, the dipoles of |
| 250 |
< |
the water molecules are preferentially anti-aligned with the dipole |
| 251 |
< |
of headgroup. |
| 240 |
> |
From Fig.~\ref{lipidFigure:PCFAtom}, we can identify the first |
| 241 |
> |
solvation shell (3.5 \AA) and the second solvation shell (5.0 \AA) |
| 242 |
> |
from both plots. However, the corresponding orientations are |
| 243 |
> |
different. In DLPE, water molecules prefer to sit around $\text{{\sc |
| 244 |
> |
NH}}_3$ group due to the hydrogen bonding. In contrast, because of |
| 245 |
> |
the hydrophobic effect of the $ {\rm{N(CH}}_{\rm{3}} |
| 246 |
> |
{\rm{)}}_{\rm{3}} $ group, the preferred position of water molecules |
| 247 |
> |
in DMPC is around the $\text{{\sc PO}}_4$ Group. When the water |
| 248 |
> |
molecules are far from the headgroup, the distribution of the two |
| 249 |
> |
angles should be uniform. The correlation close to center of the |
| 250 |
> |
headgroup dipole in both plots tells us that in the closely-bound |
| 251 |
> |
region, the dipoles of the water molecules are preferentially |
| 252 |
> |
anti-aligned with the dipole of headgroup. When the water molecules |
| 253 |
> |
are far from the headgroup, the distribution of the two angles |
| 254 |
> |
should be uniform. The correlation close to center of the headgroup |
| 255 |
> |
dipole in both plots tell us that in the closely-bound region, the |
| 256 |
> |
dipoles of the water molecules are preferentially aligned with the |
| 257 |
> |
dipole of headgroup. |
| 258 |
|
|
| 259 |
|
\begin{figure} |
| 260 |
|
\centering |
| 261 |
|
\includegraphics[width=\linewidth]{g_atom.eps} |
| 262 |
< |
\caption[The pair correlation functions for atomistic models]{} |
| 262 |
> |
\caption[The pair correlation functions for atomistic models]{The |
| 263 |
> |
pair correlation functions for atomistic models: (a)$g(r,\cos \theta |
| 264 |
> |
)$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE; (c)$g(r,\cos \omega |
| 265 |
> |
)$ for DMPC; (d)$g(r,\cos \omega )$ for DLPE; (e)$g(\cos \theta,\cos |
| 266 |
> |
\omega)$ for DMPC; (f)$g(\cos \theta,\cos \omega)$ for DMLPE.} |
| 267 |
|
\label{lipidFigure:PCFAtom} |
| 268 |
|
\end{figure} |
| 269 |
|
|
| 282 |
|
\begin{figure} |
| 283 |
|
\centering |
| 284 |
|
\includegraphics[width=\linewidth]{g_coarse.eps} |
| 285 |
< |
\caption[The pair correlation functions for coarse-grained models]{} |
| 285 |
> |
\caption[The pair correlation functions for coarse-grained |
| 286 |
> |
models]{The pair correlation functions for coarse-grained models: |
| 287 |
> |
(a)$g(r,\cos \theta )$ for DMPC; (b) $g(r,\cos \theta )$ for DLPE.} |
| 288 |
|
\label{lipidFigure:PCFCoarse} |
| 289 |
|
\end{figure} |
| 290 |
|
|
| 291 |
|
\begin{figure} |
| 292 |
|
\centering |
| 293 |
|
\includegraphics[width=\linewidth]{EWD_coarse.eps} |
| 294 |
< |
\caption[Excess water density of coarse-grained phospholipids]{ } |
| 295 |
< |
\label{lipidFigure:EWDCoarse} |
| 294 |
> |
\caption[Excess water density of coarse-grained |
| 295 |
> |
phospholipids]{Excess water density of coarse-grained |
| 296 |
> |
phospholipids.} \label{lipidFigure:EWDCoarse} |
| 297 |
|
\end{figure} |
| 298 |
|
|
| 299 |
|
\begin{table} |
| 304 |
|
\hline |
| 305 |
|
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... |
| 306 |
|
Atom type & Mass & $\sigma$ & $\epsilon$ & charge & Dipole \\ |
| 262 |
– |
|
| 307 |
|
$\text{{\sc CH}}_2$ & 14.03 & 3.95 & 0.0914 & 0 & 0 \\ |
| 308 |
|
$\text{{\sc CH}}_3$ & 15.04 & 3.75 & 0.195 & 0 & 0 \\ |
| 309 |
< |
$\text{{\sc CE}}$ & 28.01 & 3.427& 0.294 & 0 & 1.693 |
| 309 |
> |
$\text{{\sc CE}}$ & 28.01 & 3.427& 0.294 & 0 & 1.693 \\ |
| 310 |
|
$\text{{\sc CK}}$ & 28.01 & 3.592& 0.311 & 0 & 2.478 \\ |
| 311 |
|
$\text{{\sc PO}}_4$ & 108.995& 3.9 & 1.88 & -1& 0 \\ |
| 312 |
|
$\text{{\sc HDP}}$ & 14.03 & 4.0 & 0.13 & 0 & 0 \\ |
| 320 |
|
\subsection{Bilayer Simulations Using Coarse-grained Model} |
| 321 |
|
|
| 322 |
|
A bilayer system consisting of 128 DMPC lipids and 3655 water |
| 323 |
< |
molecules has been constructed from an atomistic coordinate |
| 324 |
< |
file.[15] The MD simulation is performed at constant temperature, T |
| 325 |
< |
= 300K, and constant pressure, p = 1 atm, and consisted of an |
| 326 |
< |
equilibration period of 2 ns. During the equilibration period, the |
| 327 |
< |
system was initially simulated at constant volume for 1ns. Once the |
| 328 |
< |
system was equilibrated at constant volume, the cell dimensions of |
| 329 |
< |
the system was relaxed by performing under NPT conditions using |
| 330 |
< |
Nos¨¦-Hoover extended system isothermal-isobaric dynamics. After |
| 331 |
< |
equilibration, different properties were evaluated over a production |
| 288 |
< |
run of 5 ns. |
| 323 |
> |
molecules has been constructed from an atomistic coordinate file. |
| 324 |
> |
The MD simulation is performed at constant temperature, T = 300K, |
| 325 |
> |
and constant pressure, p = 1 atm, and consisted of an equilibration |
| 326 |
> |
period of 2 ns. During the equilibration period, the system was |
| 327 |
> |
initially simulated at constant volume for 1 ns. Once the system was |
| 328 |
> |
equilibrated at constant volume, the cell dimensions of the system |
| 329 |
> |
was relaxed by performing under NPT conditions using Nos\'{e}-Hoover |
| 330 |
> |
extended system isothermal-isobaric dynamics. After equilibration, |
| 331 |
> |
different properties were evaluated over a production run of 5 ns. |
| 332 |
|
|
| 333 |
|
\begin{figure} |
| 334 |
|
\centering |
| 339 |
|
\label{lipidFigure:bilayer} |
| 340 |
|
\end{figure} |
| 341 |
|
|
| 342 |
< |
\subsubsection{Electron Density Profile (EDP)} |
| 342 |
> |
\subsubsection{\textbf{Electron Density Profile (EDP)}} |
| 343 |
|
|
| 344 |
|
Assuming a gaussian distribution of electrons on each atomic center |
| 345 |
|
with a variance estimated from the size of the van der Waals radius, |
| 346 |
|
the EDPs which are proportional to the density profiles measured |
| 347 |
|
along the bilayer normal obtained by x-ray scattering experiment, |
| 348 |
< |
can be expressed by |
| 348 |
> |
can be expressed by\cite{Tu1995} |
| 349 |
|
\begin{equation} |
| 350 |
|
\rho _{x - ray} (z)dz \propto \sum\limits_{i = 1}^N {\frac{{n_i |
| 351 |
|
}}{V}\frac{1}{{\sqrt {2\pi \sigma ^2 } }}e^{ - (z - z_i )^2 /2\sigma |
| 360 |
|
electron density is in the hydrocarbon region. As a good |
| 361 |
|
approximation to the thickness of the bilayer, the headgroup spacing |
| 362 |
|
, is defined as the distance between two peaks in the electron |
| 363 |
< |
density profile, calculated from our simulations to be 34.1 $\AA$. |
| 363 |
> |
density profile, calculated from our simulations to be 34.1 \AA. |
| 364 |
|
This value is close to the x-ray diffraction experimental value 34.4 |
| 365 |
< |
$\AA$. |
| 365 |
> |
\AA\cite{Petrache1998}. |
| 366 |
|
|
| 367 |
|
\begin{figure} |
| 368 |
|
\centering |
| 375 |
|
\label{lipidFigure:electronDensity} |
| 376 |
|
\end{figure} |
| 377 |
|
|
| 378 |
< |
\subsubsection{$\text{S}_{\text{{\sc cd}}}$ Order Parameter} |
| 378 |
> |
\subsubsection{\textbf{$\text{S}_{\text{{\sc cd}}}$ Order Parameter}} |
| 379 |
|
|
| 380 |
|
Measuring deuterium order parameters by NMR is a useful technique to |
| 381 |
|
study the orientation of hydrocarbon chains in phospholipids. The |
| 396 |
|
\end{itemize} |
| 397 |
|
In coarse-grained model, although there are no explicit hydrogens, |
| 398 |
|
the order parameter can still be written in terms of carbon ordering |
| 399 |
< |
at each point of the chain |
| 399 |
> |
at each point of the chain\cite{Egberts1988} |
| 400 |
|
\begin{equation} |
| 401 |
|
S_{ij} = \frac{1}{2} < 3\cos \theta _i \cos \theta _j - \delta |
| 402 |
|
_{ij} >. |
| 404 |
|
|
| 405 |
|
Fig.~\ref{lipidFigure:Scd} shows the order parameter profile |
| 406 |
|
calculated for our coarse-grained DMPC bilayer system at 300K. Also |
| 407 |
< |
shown are the experimental data of Tiburu. The fact that |
| 407 |
> |
shown are the experimental data of Tu\cite{Tu1995}. The fact that |
| 408 |
|
$\text{S}_{\text{{\sc cd}}}$ order parameters calculated from |
| 409 |
|
simulation are higher than the experimental ones is ascribed to the |
| 410 |
|
assumption of the locations of implicit hydrogen atoms which are |
| 416 |
|
\includegraphics[width=\linewidth]{scd.eps} |
| 417 |
|
\caption[$\text{S}_{\text{{\sc cd}}}$ order parameter]{A comparison |
| 418 |
|
of $|\text{S}_{\text{{\sc cd}}}|$ between coarse-grained model |
| 419 |
< |
(blue) and DMPC\cite{petrache00} (black) near 300~K.} |
| 419 |
> |
(blue) and DMPC\cite{Petrache2000} (black) near 300~K.} |
| 420 |
|
\label{lipidFigure:Scd} |
| 421 |
|
\end{figure} |
| 422 |
+ |
|
| 423 |
+ |
%\subsection{Bilayer Simulations Using Gay-Berne Ellipsoid Model} |
| 424 |
+ |
|
| 425 |
+ |
\section{\label{lipidSection:Conclusion}Conclusion} |
| 426 |
+ |
|
| 427 |
+ |
Atomistic simulations have been used in this study to determine the |
| 428 |
+ |
preferred orientation and location of water molecules relative to |
| 429 |
+ |
the location and orientation of the PC and PE lipid headgroups. |
| 430 |
+ |
Based on the results from our all-atom simulations, we developed a |
| 431 |
+ |
simple coarse-grained model which captures the essential features of |
| 432 |
+ |
the headgroup solvation which is crucial to transport process in |
| 433 |
+ |
membrane system. In addition, the model has been explored in a |
| 434 |
+ |
bilayer system was shown to have reasonable electron density |
| 435 |
+ |
profiles, $\text{S}_{\text{{\sc cd}}}$ order parameter and other |
| 436 |
+ |
structural properties. The accuracy of this model is achieved by |
| 437 |
+ |
matching atomistic result. It is also easy to represent different |
| 438 |
+ |
phospholipids by changing a few parameters of the model. Another |
| 439 |
+ |
important characteristic of this model distinguishing itself from |
| 440 |
+ |
other models\cite{Goetz1998,Marrink2004} is the computational speed |
| 441 |
+ |
gained by introducing a short range electrostatic approximation. |
| 442 |
+ |
Further studies of this system using z-constraint method could |
| 443 |
+ |
extend the time length of the simulations to study transport |
| 444 |
+ |
phenomena in large-scale membrane systems. |
