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|
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|
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\chapter{\label{chapt:lipid}Phospholipid Simulations} |
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|
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\section{\label{lipidSec:Intro}Introduction} |
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|
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In the past 10 years, increasing computer speeds have allowed for the |
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atomistic simulation of phospholipid bilayers for increasingly |
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relevant lengths of time. These simulations have ranged from |
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simulation of the gel phase ($L_{\beta}$) of |
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dipalmitoylphosphatidylcholine (DPPC),\cite{lindahl00} to the |
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spontaneous aggregation of DPPC molecules into fluid phase |
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($L_{\alpha}$) bilayers.\cite{marrink01} With the exception of a few |
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ambitious |
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simulations,\cite{marrink01:undulation,marrink:2002,lindahl00} most |
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investigations are limited to a range of 64 to 256 |
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phospholipids.\cite{lindahl00,sum:2003,venable00,gomez:2003,smondyrev:1999,marrink01} |
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The expense of the force calculations involved when performing these |
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simulations limits the system size. To properly hydrate a bilayer, one |
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typically needs 25 water molecules for every lipid, bringing the total |
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number of atoms simulated to roughly 8,000 for a system of 64 DPPC |
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molecules. Added to the difficulty is the electrostatic nature of the |
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phospholipid head groups and water, requiring either the |
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computationally expensive Ewald sum or the faster, particle mesh Ewald |
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sum.\cite{nina:2002,norberg:2000,patra:2003} These factors all limit |
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the system size and time scales of bilayer simulations. |
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|
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Unfortunately, much of biological interest happens on time and length |
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scales well beyond the range of current simulation technology. One |
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such example is the observance of a ripple phase |
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($P_{\beta^{\prime}}$) between the $L_{\beta}$ and $L_{\alpha}$ phases |
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of certain phospholipid bilayers.\cite{katsaras00,sengupta00} These |
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ripples are shown to have periodicity on the order of |
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100-200~$\mbox{\AA}$. A simulation on this length scale would have |
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approximately 1,300 lipid molecules with an additional 25 water |
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molecules per lipid to fully solvate the bilayer. A simulation of this |
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size is impractical with current atomistic models. |
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|
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The time and length scale limitations are most striking in transport |
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phenomena. Due to the fluid-like properties of a lipid membrane, not |
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all diffusion across the membrane happens at pores. Some molecules of |
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interest may incorporate themselves directly into the membrane. Once |
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here, they may possess an appreciable waiting time (on the order of |
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10's to 100's of nanoseconds) within the bilayer. Such long simulation |
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times are nearly impossible to obtain when integrating the system with |
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atomistic detail. |
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|
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To address these issues, several schemes have been proposed. One |
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approach by Goetz and Liposky\cite{goetz98} is to model the entire |
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system as Lennard-Jones spheres. Phospholipids are represented by |
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chains of beads with the top most beads identified as the head |
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atoms. Polar and non-polar interactions are mimicked through |
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attractive and soft-repulsive potentials respectively. A similar |
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model proposed by Marrinck \emph{et. al}.\cite{marrink04}~uses a |
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similar technique for modeling polar and non-polar interactions with |
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Lennard-Jones spheres. However, they also include charges on the head |
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group spheres to mimic the electrostatic interactions of the |
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bilayer. While the solvent spheres are kept charge-neutral and |
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interact with the bilayer solely through an attractive Lennard-Jones |
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potential. |
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|
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The model used in this investigation adds more information to the |
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interactions than the previous two models, while still balancing the |
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need for simplifications over atomistic detail. The model uses |
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Lennard-Jones spheres for the head and tail groups of the |
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phospholipids, allowing for the ability to scale the parameters to |
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reflect various sized chain configurations while keeping the number of |
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interactions small. What sets this model apart, however, is the use |
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of dipoles to represent the electrostatic nature of the |
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phospholipids. The dipole electrostatic interaction is shorter range |
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than Coulombic ($\frac{1}{r^3}$ versus $\frac{1}{r}$), and eliminates |
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the need for a costly Ewald sum. |
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|
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Another key feature of this model, is the use of a dipolar water model |
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to represent the solvent. The soft sticky dipole ({\sc ssd}) water |
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\cite{liu96:new_model} relies on the dipole for long range electrostatic |
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effects, but also contains a short range correction for hydrogen |
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bonding. In this way the systems in this research mimic the entropic |
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contribution to the hydrophobic effect due to hydrogen-bond network |
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deformation around a non-polar entity, \emph{i.e.}~the phospholipid |
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molecules. |
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|
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The following is an outline of this chapter. |
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Sec.~\ref{lipidSec:Methods} is an introduction to the lipid model used |
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in these simulations, as well as clarification about the water model |
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and integration techniques. The various simulations explored in this |
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research are outlined in |
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Sec.~\ref{lipidSec:ExpSetup}. Sec.~\ref{lipidSec:resultsDis} gives a |
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summary and interpretation of the results. Finally, the conclusions |
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of this chapter are presented in Sec.~\ref{lipidSec:Conclusion}. |
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|
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\section{\label{lipidSec:Methods}Methods} |
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|
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\subsection{\label{lipidSec:lipidModel}The Lipid Model} |
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|
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\begin{figure} |
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\centering |
98 |
\includegraphics[width=\linewidth]{twoChainFig.eps} |
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\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.} |
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\label{lipidFig:lipidModel} |
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\end{figure} |
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|
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The phospholipid model used in these simulations is based on the |
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design illustrated in Fig.~\ref{lipidFig:lipidModel}. The head group |
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of the phospholipid is replaced by a single Lennard-Jones sphere of |
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diameter $\sigma_{\text{head}}$, with $\epsilon_{\text{head}}$ scaling |
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the well depth of its van der Walls interaction. This sphere also |
108 |
contains a single dipole of magnitude $|\boldsymbol{\mu}|$, where |
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$|\boldsymbol{\mu}|$ can be varied to mimic the charge separation of a |
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given phospholipid head group. The atoms of the tail region are |
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modeled by beads representing multiple methyl groups. They are free |
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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 |
$\epsilon_{\text{tail}}$. |
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|
117 |
The long range interactions between lipids are given by the following: |
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\begin{equation} |
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 |
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] |
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\label{lipidEq:dipolePot} |
135 |
\end{equation} |
136 |
Where $V_{\text{LJ}}$ is the Lennard-Jones potential and |
137 |
$V_{\text{dipole}}$ is the dipole-dipole potential. As previously |
138 |
stated, $\sigma_{ij}$ and $\epsilon_{ij}$ are the Lennard-Jones |
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 $\mathbf{\Omega}_i$ |
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 rotated with Euler angles: $\boldsymbol{\Omega}_i$. |
147 |
|
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 |
V_{\text{bend}}(\theta) = k_{\theta}( \theta - \theta_0 )^2 |
154 |
\label{lipidEq:bendPot} |
155 |
\end{equation} |
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 |
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 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 |
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& mass (amu) & $\sigma$($\mbox{\AA}$) & $\epsilon$ (kcal/mol) \\ \hline |
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{\sc head} & 72 & 4.0 & 0.185 \\ \hline |
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{\sc ch}\cite{Siepmann1998} & 13.02 & 4.0 & 0.0189 \\ \hline |
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$\text{{\sc ch}}_2$\cite{Siepmann1998} & 14.03 & 3.95 & 0.18 \\ \hline |
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$\text{{\sc ch}}_3$\cite{Siepmann1998} & 15.04 & 3.75 & 0.25 \\ \hline |
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{\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 |
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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 |
|
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The first class of simulations were started from ordered |
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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. |