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\chapter{\label{chapt:lipid}Phospholipid Simulations} |
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\section{\label{lipidSec:Intro}Introduction} |
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In the past 10 years, computer speeds have allowed for the atomistic |
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simulation of phospholipid bilayers. These simulations have ranged |
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from 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 64 to 256 |
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phospholipids.\cite{lindahl00,sum:2003,venable00,gomez:2003,smondyrev:1999,marrink01} |
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This is due to the expense of the computer calculations involved when |
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performing these simulations. 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 the computationally |
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expensive Ewald sum or its slightly faster derivative particle mesh |
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Ewald sum.\cite{nina:2002,norberg:2000,patra:2003} These factors all |
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limit the potential size and time lengths of bilayer simulations. |
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Unfortunately, much of biological interest happens on time and length |
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scales infeasible with current simulation. One such example is the |
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observance of a ripple phase ($P_{\beta^{\prime}}$) between the |
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$L_{\beta}$ and $L_{\alpha}$ phases of certain phospholipid |
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bilayers.\cite{katsaras00,sengupta00} These ripples are shown to |
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have periodicity on the order of 100-200~$\mbox{\AA}$. A simulation on |
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this length scale would have approximately 1,300 lipid molecules with |
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an additional 25 water molecules per lipid to fully solvate the |
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bilayer. A simulation of this size is impractical with current |
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atomistic models. |
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Another class of simulations to consider, are those dealing with the |
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diffusion of molecules through a bilayer. Due to the fluid-like |
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properties of a lipid membrane, not all diffusion across the membrane |
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happens at pores. Some molecules of interest may incorporate |
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themselves directly into the membrane. Once here, they may possess an |
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appreciable waiting time (on the order of 10's to 100's of |
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nanoseconds) within the bilayer. Such long simulation times are |
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difficulty to obtain when integrating the system with atomistic |
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detail. |
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Addressing 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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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 $\frac{1}{r}$), eliminating the |
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need for a costly Ewald sum. |
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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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The following is an outline of this chapter. |
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Sec.~\ref{lipidSec:Methods} is an introduction to the lipid model |
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used in these simulations. As well as clarification about the water |
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model and integration techniques. The various simulation setups |
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explored in this research are outlined in |
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Sec.~\ref{lipidSec:ExpSetup}. Sec.~\ref{lipidSec:Results} and |
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Sec.~\ref{lipidSec:Discussion} give a summary of the results and |
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interpretation of those results respectively. Finally, the |
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conclusions of this chapter are presented in |
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Sec.~\ref{lipidSec:Conclusion}. |
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|
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\section{\label{lipidSec:Methods}Methods} |
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\subsection{\label{lipidSec:lipidModel}The Lipid Model} |
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\begin{figure} |
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\centering |
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\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 |
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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 unified atom beads. They are free of partial charges or |
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dipoles, containing only Lennard-Jones interaction sites at their |
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centers of mass. As with the head groups, their potentials can be |
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scaled by $\sigma_{\text{tail}}$ and $\epsilon_{\text{tail}}$. |
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|
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The long range interactions between lipids are given by the following: |
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\begin{equation} |
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V_{\text{LJ}}(r_{ij}) = |
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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{lipidEq:LJpot} |
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\end{equation} |
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and |
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\begin{equation} |
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V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
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\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
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\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
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- |
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\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
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(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
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{r^{2}_{ij}} \biggr] |
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\label{lipidEq:dipolePot} |
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\end{equation} |
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Where $V_{\text{LJ}}$ is the Lennard-Jones potential and |
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$V_{\text{dipole}}$ is the dipole-dipole potential. As previously |
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stated, $\sigma_{ij}$ and $\epsilon_{ij}$ are the Lennard-Jones |
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parameters which scale the length and depth of the interaction |
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respectively, and $r_{ij}$ is the distance between beads $i$ and $j$. |
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In $V_{\text{dipole}}$, $\mathbf{r}_{ij}$ is the vector starting at |
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bead$i$ and pointing towards bead $j$. Vectors $\mathbf{\Omega}_i$ |
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and $\mathbf{\Omega}_j$ are the orientational degrees of freedom for |
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beads $i$ and $j$. $|\mu_i|$ is the magnitude of the dipole moment of |
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$i$, and $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
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vector of $\boldsymbol{\Omega}_i$. |
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The model also allows for the bonded interactions bends, and torsions. |
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The bond between two beads on a chain is of fixed length, and is |
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maintained according to the {\sc rattle} algorithm.\cite{andersen83} |
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The bends are subject to a harmonic potential: |
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\begin{equation} |
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V_{\text{bend}}(\theta) = k_{\theta}( \theta - \theta_0 )^2 |
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\label{lipidEq:bendPot} |
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\end{equation} |
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where $k_{\theta}$ scales the strength of the harmonic well, and |
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$\theta$ is the angle between bond vectors |
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(Fig.~\ref{lipidFig:lipidModel}). In addition, we have placed a |
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``ghost'' bend on the phospholipid head. The ghost bend adds a |
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potential to keep the dipole pointed along the bilayer surface, where |
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$theta$ is now the angle the dipole makes with respect to the {\sc |
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head}-$\text{{\sc ch}}_2$ bond vector. The torsion potential is given |
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by: |
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\begin{equation} |
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V_{\text{torsion}}(\phi) = |
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k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
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\label{lipidEq:torsionPot} |
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\end{equation} |
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Here, the parameters $k_0$, $k_1$, $k_2$, and $k_3$ fit the cosine |
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power series to the desired torsion potential surface, and $\phi$ is |
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the angle the two end atoms have rotated about the middle bond |
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(Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as the |
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Lennard-Jones potential are excluded for atom pairs involved in the |
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same bond, bend, or torsion. However, internal interactions not |
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directly involved in a bonded pair are calculated. |
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All simulations presented here use a two chained lipid as pictured in |
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Fig.~\ref{lipidFig:lipidModel}. The chains are both eight beads long, |
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and their mass and Lennard Jones parameters are summarized in |
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Table~\ref{lipidTable:tcLJParams}. The magnitude of the dipole moment |
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for the head bead is 10.6~Debye, and the bend and torsion parameters |
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are summarized in Table~\ref{lipidTable:tcBendParams} and |
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\ref{lipidTable:tcTorsionParams}. |
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|
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\begin{table} |
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\caption{The Lennard Jones Parameters for the two chain phospholipids.} |
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\label{lipidTable:tcLJParams} |
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\begin{center} |
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\begin{tabular}{|l|c|c|c|} |
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\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 |
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\end{tabular} |
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\end{center} |
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\end{table} |
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|
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\begin{table} |
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\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}} |
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\label{lipidTable:tcBendParams} |
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\begin{center} |
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\begin{tabular}{|l|c|c|} |
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\hline |
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& $k_{\theta}$ ( kcal/($\text{mol deg}^2$) ) & $\theta_0$ ( deg ) \\ \hline |
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{\sc ghost}-{\sc head}-$\text{{\sc ch}}_2$ & 0.00177 & 129.78 \\ \hline |
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$x$-{\sc ch}-$y$ & 58.84 & 112.0 \\ \hline |
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$x$-$\text{{\sc ch}}_2$-$y$ & 58.84 & 114.0 \\ \hline |
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\end{tabular} |
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\end{center} |
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\end{table} |
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|
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\begin{table} |
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\caption[Torsion Parameters for the two chain phospholipids]{Torsion Parameters for the two chain phospholipids. Alkane parameters based on TraPPE.\cite{Siepmann1998}} |
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\label{lipidTable:tcTorsionParams} |
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\begin{center} |
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\begin{tabular}{|l|c|c|c|c|} |
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\hline |
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All are in kcal/mol $\rightarrow$ & $k_3$ & $k_2$ & $k_1$ & $k_0$ \\ \hline |
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$x$-{\sc ch}-$y$-$z$ & 3.3254 & -0.4215 & -1.686 & 1.1661 \\ \hline |
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$x$-$\text{{\sc ch}}_2$-$\text{{\sc ch}}_2$-$y$ & 5.9602 & -0.568 & -3.802 & 2.1586 \\ \hline |
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\end{tabular} |
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\end{center} |
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\end{table} |
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|
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|
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\section{\label{lipidSec:furtherMethod}Further Methodology} |
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|
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As mentioned previously, the water model used throughout these |
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simulations was the {\sc ssd} model of |
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Ichiye.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md} A |
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discussion of the model can be found in Sec.~\ref{oopseSec:SSD}. As |
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for the integration of the equations of motion, all simulations were |
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performed in an orthorhombic periodic box with a thermostat on |
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velocities, and an independent barostat on each Cartesian axis $x$, |
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$y$, and $z$. This is the $\text{NPT}_{xyz}$. ensemble described in |
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Sec.~\ref{oopseSec:Ensembles}. |
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|
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|
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\subsection{\label{lipidSec:ExpSetup}Experimental Setup} |
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|
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Two main starting configuration classes were used in this research: |
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random and ordered bilayers. The ordered bilayer starting |
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configurations were all started from an equilibrated bilayer at |
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300~K. The original configuration for the first 300~K run was |
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assembled by placing the phospholipids centers of mass on a planar |
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hexagonal lattice. The lipids were oriented with their long axis |
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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 |
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below the bilayer. |
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The random configurations took more work to generate. To begin, a |
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test lipid was placed in a simulation box already containing water at |
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the intended density. The waters were then tested for overlap with |
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the lipid using a 5.0~$\mbox{\AA}$ buffer distance. This gave an |
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estimate for the number of waters each lipid would displace in a |
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simulation box. A target number of waters was then defined which |
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included the number of waters each lipid would displace, the number of |
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waters desired to solvate each lipid, and a fudge factor to pad the |
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initialization. |
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|
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Next, a cubic simulation box was created that contained at least the |
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target number of waters in an FCC lattice (the lattice was for ease of |
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placement). What followed was a RSA simulation similar to those of |
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Chapt.~\ref{chapt:RSA}. The lipids were sequentially given a random |
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position and orientation within the box. If a lipid's position caused |
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atomic overlap with any previously adsorbed lipid, its position and |
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orientation were rejected, and a new random adsorption site was |
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attempted. The RSA simulation proceeded until all phospholipids had |
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been adsorbed. After adsorption, all water molecules with locations |
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that overlapped with the atomic coordinates of the lipids were |
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removed. |
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|
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Finally, water molecules were removed one by one at random until the |
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desired number of waters per lipid was reached. The typical low final |
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density for these initial configurations was not a problem, as the box |
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would shrink to an appropriate size within the first 50~ps of a |
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simulation in the $\text{NPT}_{xyz}$ ensemble. |
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|
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\subsection{\label{lipidSec:Configs}The simulation configurations} |
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|
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Table ~\ref{lipidTable:simNames} summarizes the names and important |
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details of the simulations. The B set of simulations were all started |
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in an ordered bilayer and observed over a period of 10~ns. Simulation |
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RL was integrated for approximately 20~ns starting from a random |
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configuration as an example of spontaneous bilayer aggregation. |
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Lastly, simulation RH was also started from a random configuration, |
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but with a lesser water content and higher temperature to show the |
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spontaneous aggregation of an inverted hexagonal lamellar phase. |
