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\begin{document} |
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\title{A Mesoscale Model for Phospholipid Simulations} |
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\author{Matthew A. Meineke\\ |
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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\section{Background and Research Goals} |
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Simulations of phospholipid bilayers are, by necessity, quite |
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complex. The lipid molecules are large molecules containing many |
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atoms,and the head group of the lipid will typically contain charge |
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separated ions which set up a large dipole within the molecule. Adding |
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to the complexity are the number of water molecules needed to properly |
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solvate the lipid bilayer. Because of these factors, many current |
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simulations are limited in both length and time scale due to to the |
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sheer number of calculations performed at every time step and the |
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lifetime of the researcher. A typical |
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simulation\cite{saiz02,lindahl00,venable00,Marrink01} will have around |
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64 phospholipids forming a bilayer approximately 40~$\mbox{\AA}$ by |
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50~$\mbox{\AA}$ with roughly 25 waters for every lipid. This means |
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there are on the order of 8,000 atoms needed to simulate these systems |
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and the trajectories in turn are integrated for times up to 10 ns. |
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These limitations make it difficult to study certain biologically |
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interesting phenomena that don't fit within the short time and length |
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scale requirements. One such phenomena is the existence of the ripple |
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phase ($P_{\beta'}$) of the bilayer between the gel phase |
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($L_{\beta'}$) and the fluid phase ($L_{\alpha}$). The $P_{\beta'}$ |
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phase has been shown to have a ripple period of |
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100-200~$\mbox{\AA}$.\cite{katsaras00,sengupta00} A simulation of this |
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length scale would require approximately 1,300 lipid molecules and |
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roughly 25 waters for every lipid to fully solvate the bilayer. With |
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the large number of atoms involved in a simulation of this magnitude, |
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steps \emph{must} be taken to simplify the system to the point where |
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these numbers are reasonable. |
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Another system of interest would be drug molecule diffusion through |
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the membrane. Due to the fluid like properties of a lipid membrane, |
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not all diffusion takes place at membrane channels. It is of interest |
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to study certain molecules that may incorporate themselves directly |
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into the membrane. These molecules may then have an appreciable |
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waiting time (on the order of nanoseconds) within the |
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bilayer. Simulation of such a long time scale again requires |
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simplification of the system in order to lower the number of |
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calculations needed at each time step. |
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\section{Methodology} |
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\subsection{Length and Time Scale Simplifications} |
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The length scale simplifications are aimed at increasing the number of |
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molecules that can be simulated without drastically increasing the |
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computational cost of the system. This is done by a combination of |
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substituting less expensive interactions for expensive ones and |
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decreasing the number of interaction sites per molecule. Namely, |
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point charge distributions are replaced with dipoles, and unified atoms are |
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used in place of water, phospholipid head groups, and alkyl groups. |
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The replacement of charge distributions with dipoles allows us to |
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replace an interaction that has a relatively long range, |
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$(\frac{1}{r})$, for the coulomb potential, with that of a relatively |
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short range, $(\frac{1}{r^{3}})$, for dipole - dipole |
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potentials. Combined with a computational simplification algorithm |
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such as a Verlet neighbor list,\cite{allen87:csl} this should give |
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computational scaling by $N$. This is in comparison to the Ewald |
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sum\cite{leach01:mm} needed to compute the coulomb interactions, which |
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scales at best by $N \ln N$. |
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The second step taken to simplify the number of calculations is to |
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incorporate unified models for groups of atoms. In the case of water, |
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we use the soft sticky dipole (SSD) model developed by |
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Ichiye\cite{Liu96} (Section~\ref{sec:ssdModel}). For the phospholipids, a |
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unified head atom with a dipole will replace the atoms in the head |
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group, while unified $\text{CH}_2$ and $\text{CH}_3$ atoms will |
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replace the alkyl units in the tails (Section~\ref{sec:lipidModel}). |
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The time scale simplifications are introduced so that we can take |
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longer time steps. By increasing the size of the time steps taken by |
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the simulation, we are able to integrate the simulation trajectory |
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with fewer calculations. However, care must be taken that any |
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simplifications used, still conserve the total energy of the |
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simulation. In practice, this means taking steps small enough to |
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resolve all motion in the system without accidently moving an object |
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too far along a repulsive energy surface before it feels the effect of |
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the surface. |
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In our simulation we have chosen to constrain all bonds to be of fixed |
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length. This means the bonds are no longer allowed to vibrate about |
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their equilibrium positions. Bond vibrations are typically the fastest |
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periodic motion in a dynamics simulation. By taking this action, we |
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are able to take time steps of 3 fs while still maintaining constant |
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energy. This is in contrast to the 1 fs time step typically needed to |
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conserve energy when bonds lengths are allowed to oscillate. |
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\subsection{The Soft Sticky Water Model} |
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\label{sec:ssdModel} |
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%\begin{floatingfigure}{55mm} |
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%\includegraphics[width=45mm]{ssd.epsi} |
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%\caption{The SSD model with the oxygen and hydrogen atoms drawn in for reference. \vspace{5mm}} |
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%\label{fig:ssdModel} |
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%\end{floatingfigure} |
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The water model used in our simulations is a modified soft |
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Stockmayer-sphere model.\cite{stevens95} Like the Stockmayer-sphere, the SSD |
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model\cite{Liu96} consists of a Lennard-Jones interaction site and a |
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dipole both located at the water's center of mass (Figure |
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\ref{fig:ssdModel}). However, the SSD model extends this by adding a |
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tetrahedral potential to correct for hydrogen bonding. |
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The SSD water potential is then given by the following equation: |
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\begin{equation} |
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V_{\text{SSD}} = V_{\text{LJ}}(r_{i\!j}) + V_{\text{dp}}(\mathbf{r}_{i\!j}, |
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\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
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+ V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) |
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\label{eq:ssdTotPot} |
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\end{equation} |
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$V_{\text{LJ}}$ is the Lennard-Jones potential: |
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\begin{equation} |
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V_{\text{LJ}} = |
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4\epsilon_{ij} \biggl[ |
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\biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12} |
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- \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6} |
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\biggr] |
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\label{eq:lennardJonesPot} |
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\end{equation} |
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where $r_{ij}$ is the distance between two $ij$ pairs, $\sigma_{ij}$ |
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scales the length of the iteraction, and $\epsilon_{ij}$ scales the |
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energy of the potential. For SSD, $\sigma_{\text{SSD}} = 3.051 \mbox{ |
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\AA}$ and $\epsilon_{\text{SSD}} = 0.152\text{ kcal/mol}$. |
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$V_{\text{dp}}$ is the dipole potential: |
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\begin{equation} |
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V_{\text{dp}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
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\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
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- |
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\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
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(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
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{r^{5}_{ij}} \biggr] |
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\label{eq:dipolePot} |
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\end{equation} |
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where $\mathbf{r}_{ij}$ is the vector between $i$ and $j$, |
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$\boldsymbol{\Omega}$ is the orientation of the species, and |
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$\boldsymbol{\mu}$ is the dipole vector. The SSD model specifies a dipole |
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magnitude of 2.35~D for water. |
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The hydrogen bonding of the model is governed by the $V_{\text{sp}}$ |
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term of the potential. Its form is as follows: |
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\begin{equation} |
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V_{\text{sp}}(\mathbf{r}_{i\!j},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = |
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v^{\circ}[s(r_{ij})w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) |
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+ |
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s'(r_{ij})w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j})] |
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\label{eq:spPot} |
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\end{equation} |
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Where $v^\circ$ scales strength of the interaction. |
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$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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and |
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$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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are responsible for the tetrahedral potential and a correction to the |
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tetrahedral potential respectively. They are, |
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\begin{equation} |
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w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
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\sin\theta_{ij} \sin 2\theta_{ij} \cos 2\phi_{ij} |
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+ \sin \theta_{ji} \sin 2\theta_{ji} \cos 2\phi_{ji} |
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\label{eq:spPot2} |
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\end{equation} |
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and |
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\begin{equation} |
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\begin{split} |
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w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) = |
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&(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij} + 0.8)^2 \\ |
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&+ (\cos\theta_{ji}-0.6)^2(\cos\theta_{ji} + 0.8)^2 - 2w^{\circ} |
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\end{split} |
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\label{eq:spCorrection} |
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\end{equation} |
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The angles $\theta_{ij}$ and $\phi_{ij}$ are defined by the spherical |
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polar coordinates of the position of sphere $j$ in the reference frame |
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fixed on sphere $i$ with the z-axis aligned with the dipole moment. |
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The correction |
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$w^{x}_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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is needed because |
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$w_{ij}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})$ |
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vanishes when $\theta_{ij}$ is $0^\circ$ or $180^\circ$. |
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Finally, the sticky potential is scaled by a cutoff function, |
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$s(r_{ij})$ that scales smoothly between 0 and 1. It is represented |
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by: |
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\begin{equation} |
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s(r_{ij}) = |
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\begin{cases} |
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1& \text{if $r_{ij} < r_{L}$}, \\ |
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\frac{(r_{U} - r_{ij})^2 (r_{U} + 2r_{ij} |
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- 3r_{L})}{(r_{U}-r_{L})^3}& |
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\text{if $r_{L} \leq r_{ij} \leq r_{U}$},\\ |
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0& \text{if $r_{ij} \geq r_{U}$}. |
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\end{cases} |
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\label{eq:spCutoff} |
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\end{equation} |
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Despite the apparent complexity of Equation \ref{eq:spPot}, the SSD |
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model is still computationaly inexpensive. This is due to Equation |
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\ref{eq:spCutoff}. With $r_{L}$ being 2.75~$\mbox\AA}$ and $r_{U}$ |
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being equal to either 3.35~$\mbox{\AA}$ for $s(r_{ij})$ or |
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4.0~$\mbox{\AA}$ for $s'(r_{ij})$, the sticky potential is only active |
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over an extremly short range, and then only with other SSD |
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molecules. Therefore, it's predominant interaction is through it's |
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point dipole and Lennard-Jones sphere. |
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\subsection{The Phospholipid Model} |
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\label{sec:lipidModel} |
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\begin{floatingfigure}{90mm} |
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\includegraphics[angle=-90,width=80mm]{lipidModel.epsi} |
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\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}} |
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\label{fig:lipidModel} |
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\end{floatingfigure} |
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The lipid molecules in our simulations are unified atom models. Figure |
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\ref{fig:lipidModel} shows a template drawing for one of our |
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lipids. The Head group of the phospholipid is replaced by a single |
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Lennard-Jones sphere with a freely oriented dipole placed at it's |
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center. The magnitude of it's dipole moment is 20.6 D. The tail atoms |
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are unified $\text{CH}_2$ and $\text{CH}_3$ atoms and are also modeled |
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as Lennard-Jones spheres. The total potential for the lipid is |
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represented by Equation \ref{eq:lipidModelPot}. |
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\begin{equation} |
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V_{\mbox{lipid}} = \overbrace{% |
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V_{\text{bend}}(\theta_{ijk}) +% |
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V_{\text{tors.}}(\phi_{ijkl})}^{bonded} |
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+ \overbrace{% |
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V_{\text{LJ}}(r_{i\!j}) + |
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V_{\text{dp}}(r_{i\!j},\Omega_{i},\Omega_{j})% |
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}^{non-bonded} |
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\label{eq:lipidModelPot} |
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\end{equation} |
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The non-bonded interactions, $V_{\text{LJ}}$ and $V_{\text{dp}}$, are |
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the Lennard-Jones and dipole-dipole interactions respectively. For the |
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non-bonded potentials, only the bend and the torsional potentials are |
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calculated. The bond potential is not calculated, and the bond lengths |
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are constrained via RATTLE.\cite{leach01:mm} The bend potential is of |
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the form: |
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\begin{equation} |
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V_{\text{bend}}(\theta_{ijk}) = k_{\theta}\frac{(\theta_{ijk} - \theta_0)^2}{2} |
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\label{eq:bendPot} |
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\end{equation} |
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Where $k_{\theta}$ sets the stiffness of the bend potential, and $\theta_0$ |
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sets the equilibrium bend angle. The torsion potential was given by: |
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\begin{equation} |
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V_{\text{tors.}}(\phi_{ijkl})= c_1[1+\cos\phi_{ijkl}] |
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+ c_2 [1 - \cos(2\phi_{ijkl})] + c_3[1 + \cos(3\phi_{ijkl})] |
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\label{eq:torsPot} |
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\end{equation} |
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All parameters for bonded and non-bonded potentials in the tail atoms |
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were taken from TraPPE.\cite{Siepmann1998} The bonded interactions for |
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the head atom were also taken from TraPPE, however it's dipole moment |
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and mass were based on the properties of the phosphatidylcholine head |
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group. The Lennard-Jones parameter for the head group was chosen such |
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that it was roughly twice the size of a $\text{CH}_3$ atom, and it's |
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well depth was set to be approximately equal to that of $\text{CH}_3$. |
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\section{Initial Simulation: 25 lipids in water} |
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\label{sec:5x5} |
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\subsection{Starting Configuration and Parameters} |
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\label{sec:5x5Start} |
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Our first simulation was an array of 25 single chained lipids in a sea |
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of water (Figure \ref{fig:5x5Start}). The total number of water |
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molecules was 1386, giving a final of water concentration of 70\% |
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wt. The simulation box measured 34.5~$\mbox{\AA}$ x 39.4~$\mbox{\AA}$ |
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x 39.4~$\mbox{\AA}$ with periodic boundary conditions invoked. The |
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system was simulated in the micro-canonical (NVE) ensemble with an |
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average temperature of 300~K. |
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\subsection{Results} |
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\label{sec:5x5Results} |
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Figure \ref{fig:5x5Final} shows a snapshot of the system at |
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3.6~ns. Here it is exciting to note that the system has spontaneously |
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self assembled into a bilayer. Discussion of the length scales of the |
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bilayer will follow in this section. However, it is interesting to |
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note several qualitative properties of the system revealed by this |
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snapshot. First, is how the tail chains are tilted to the bilayer |
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normal. This is usually indicative of the gel ($L_{\beta'}$) |
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phase. Likely, the box size is too small for the bilayer to relax to |
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the fluid ($P_{\alpha}$ phase. Showing the need for a constant |
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pressure simulation versus the constant volume of the current |
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ensemble. |
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|
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The trajectory of the simulation was analyzed using the pair-wise |
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radial distribution function, $g(r)$, which has the form: |
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\begin{equation} |
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g(r) = \frac{V}{N_{\text{pairs}}}\langle \sum_{i} \sum_{j \neq i} |
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\delta(|\mathbf{r} - \mathbf{r}_{ij}|) \rangle |
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\label{eq:gofr} |
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\end{equation} |
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Equation \ref{eq:gofr} gives us information about the spacing of two |
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species as a function of radius. Essentially, if the observer were |
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located at atom $i$ and were looking out in all directions, $g(r)$ |
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shows the relative density of atom $j$ at any given radius, $r$, |
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normalized by the expected density of atom $j$ at $r$. In a |
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homogeneously mixed fluid, $g(r)$ will approach 1 at large $r$, as a |
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fluid contains no long range structure to contribute peaks in the |
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number density. |
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|
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For the species containing dipoles, a second pair wise distribution |
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function was used, $g_{\gamma}(r)$. It is of the form: |
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\begin{equation} |
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g_{\gamma}(r) = foobar |
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\label{eq:gammaofr} |
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\end{equation} |
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Where $\gamma_{ij}$ is the angle between the dipole of atom $j$ with |
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respect to the dipole of atom $i$. This correlation will vary between |
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$+1$ and $-1$ when the two dipoles are perfectly aligned and |
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anti-aligned respectively. This then gives us information about how |
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directional species are aligned with each other as a function of |
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distance. |
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|
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Figure \ref{fig:5x5HHCorr} shows the two self correlation functions |
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for the Head groups of the lipids. The first peak at 4.03~$\mbox{\AA}$ is the |
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nearest neighbor separation of the heads of two lipids. |
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|
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\section{Second Simulation: 50 randomly oriented lipids in water} |
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|
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the second simulation |
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\section{Future Directions} |
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\pagebreak |
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\bibliographystyle{achemso} |
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\bibliography{canidacy_paper} \end{document} |
