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mmeineke |
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\chapter{\label{chapt:lipid}Phospholipid Simulations} |
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\section{\label{lipidSec:Intro}Introduction} |
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mmeineke |
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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{Lindahl:2000} to the |
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spontaneous aggregation of DPPC molecules into fluid phase |
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($L_{\alpha}$ bilayers. \cite{Marrinck:2001} With the exception of a |
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few ambitious |
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simulations,\cite{Marrinch:2001b,Marrinck:2002,Lindahl:2000} most |
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investigations are limited to 64 to 256 |
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phospholipids.\cite{Lindal:2000,Sum:2003,Venable:2000,Gomez:2003,Smondyrev:1999,Marrinck:2001a} |
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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 difficluty 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 lenghts of bilayer simulations. |
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Unfortunately, much of biological interest happens on time and length |
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scales unfeasible with current simulation. One such example is the |
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observance of a ripple phase ($P_{\beta'}$) between the $L_{\beta}$ |
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and $L_{\alpha}$ phases of certain phospholipid |
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bilayers.\cite{Katsaras:2000,Sengupta:2000} 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{Goetz:1998} 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{Marrinck:2004}~ 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, |
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mmeineke |
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\section{\label{lipidSec:Methods}Methods} |
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\subsection{\label{lipidSec:lipidMedel}The Lipid Model} |
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\begin{figure} |
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\caption{Schematic diagram of the single chain phospholipid model} |
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\label{lipidFig:lipidModel} |
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\end{figure} |
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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 $fix$, with $fix$ scaling the well depth of its van der Walls |
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interaction. This sphere also contains a single dipole of magnitude |
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$fix$, where $fix$ 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 $fix$ and $fix$. |
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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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EQ Here |
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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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EQ Here |
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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 of bonds, bends, and |
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torsions. The bonds between two beads on a chain are of fixed length, |
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and are maintained according to the {\sc rattle} algorithm. \cite{fix} |
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The bends are subject to a harmonic potential: |
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\begin{equation} |
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eq here |
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\label{lipidEq:bendPot} |
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\end{equation} |
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where $fix$ scales the strength of the harmonic well, and $fix$ is the |
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angle between bond vectors $fix$ and $fix$. The torsion potential is |
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given by: |
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\begin{equation} |
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eq here |
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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 between bondvectors $fix$ and $fix$ along the vector $fix$ |
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(see Fig.:\ref{lipidFig:lipidModel}). Long range interactions such as |
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the Lennard-Jones potential are excluded for bead pairs involved in |
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the 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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