Revision as of 10:15, 3 October 2016

MODELING

ABSTRACT
Bonding of proteins is highly depending on structural properties which in turn are determined by the amino acid sequences. Changing the amino acid sequence of one participating partner could consequently diminish it's binding ability. Therefore it is important to estimate the influence of mutations on the protein structure. This is particularly true for mutations from natural to non-natural amino acids.
To estimate the influence of O-methyl tyrosine on Colicin E2's immunity protein we applied several molecular dynamics simulations leading to 1300 ns in total simulation time. To do this we estimated O-methyl tyrosine parameters for the CHARMm 22 and the GROMOS36a7 force field. We evaluated our simulations by applying several well documented evaluation methods like secondary structure analysis, plotting the solvent accesible surface area, and RMSD and RMSF. Our first simualtion analysis led to the conclusion that O-methyl tyrosine had no influence on the immunity protein.
To estimate possible influences on the thermodynamics of the system we calculated the binding energy between Colicin E2 and it's immunity protein by pulling experiments with following umbrella sampling molecular dynamics simulations. The binding energy was afterwards calculated using the WHAM algorithm showing only minor differences.

THEORETICAL OVERVIEW

Molecular Dynamics Simulation

Introduction

Molecular Dynamics (MD) Simulations is a method to describe atomic and molecular movements. Molecular Dynamics simulations depend on several simplifications that enables the simulation to range from nanoseconds up to severeal milliseconds in systems containg of over one hundred thousand atoms. This enables the possibility to study different biomolecular processes like protein protein binding or enzyme dynamics. Because of the deterministic nature of the system it is possible to calculate thermodynamic properties like free energy or free binding enthalpies.

Assumptions

To describe atomistic or molecular behavior the exact system conditions like positions and energies. The energies of an atomic system are described by the Schr&oumldinger equation (eq. \ref{Schrödinger}) with wavefunction $\Psi$ (eq. \ref{wavefunction}), kinetic energies $\hat{T}_{e}$ and $\hat{T}_{N}$ and potential energies $\hat{V}_{e}$, $\hat{V}_{N}$ and $\hat{V}_{eN}$. Terms with subscript $_{e}$ are terms concerning the electrons and terms with subscript $_{N}$ are terms concerning the nuclei.

$$ \begin{equation} (\hat{T}_{e} + \hat{T}_{N} + \hat{V}_{e} + \hat{V}_{N} + \hat{V}_{eN})\Psi=i \hbar \frac{\partial}{\partial t}\Psi \label{Schrödinger} \end{equation}$$ $$ \begin{equation} \Psi=\Psi(\vec{r}_{1},..., \vec{r}_{N_{e}},\vec{r}_{1},...,\vec{r}_{N_{N}}) \label{wavefunction} \end{equation}$$

Since there is no possibility to solve these equations numerical, it is necessary to simplify the system description. The first assumption depends on the Born-Oppenheimer approximation that the Schr&oumldinger equation can be splitted into two parts, one for the electrons and one for the nuclei respectivly. Since the electrons are far more mobile the dynamic of the system can be defined by the nuclei positions.
Molecular Dynamics simulations depend on several simplifications. First we assume in accord with the Born-Oppenheimer approximation that electronical movement has no influence on the overall atomic momentum because electrons will simply follow the nuclear movements in the simulated time scales. Second we can describe the potential energy function by a sum of simple terms. These terms are described in the so called force field which will be described later on. Third the system potential is evaluated by deriving the forces and applying Newtonian mechanic calculations as shown in equation \ref{newton} and \ref{newton1}.

$$ \begin{equation} M_{K}\frac{d\vec{v}_{1}}{dt} = M_{K}\frac{d^{2}\vec{r}_{1}}{dt^{2}} = \vec{F}_{\vec{r}_{1}} = \frac{\partial V\left(\vec{r}_{1},...,\vec{r}_{N}\right)}{\partial \vec{r}_{1}} \label{newton} \end{equation} $$

$$ \begin{equation} F_{ij}=-\frac{\partial}{\partial r_{ij}}V_{force~field} \label{newton1} \end{equation} $$

To solve these terms numerically we have to discretize the trajectory and therefore use an integrator for the small time steps. Several different integrators were developed today, of which the velocity-Verlet algorithm is the most used (eq. \ref{VV1} & \ref{VV2}).

$$ \begin{equation} r_{i}(t_{0} + \Delta t) = r_{i}(t_{0}) + v_{i}(t_{0})\Delta t + \frac{1}{2}a_{i}(t_{0})\Delta t^{2} \label{VV1} \end{equation} $$

$$ \begin{equation} v_{i}(t_{o}+\Delta t) = v_{i}(t_{0}) + \frac{1}{2}[a_{i}(t_{o} + \Delta t)]\Delta t \label{VV2} \end{equation} $$

The temperature of the system is directly correlated to the distribution of kinetic energies. Therefore the temperature of the system can be controlled by manipulating the atom velocities. A possible way to do this was proposed by Berendsen by coupling the system to a heat bath resulting in a NVT ensemble (eq. \ref{Berendsen}).

$$ \begin{equation} a_{i}=\frac{F_{i}}{m_{i}} + \frac{1}{2 \tau_{T}} \left( \frac{T_{B}}{T_{t}} -1 \right) v_{i} \label{Berendsen} \end{equation} $$

Empirical Force Fields

Empirical Force Fields are the backbone of every Molecular Dynamics simulation. Typically the force fields are diveded into two parts, bonded and nonbonded interactions. Bonded interactions consist of chemical bond stretching, angle bending, and rotation of dihedrals and impropers. Nonbonded interactions are approximated by Coulomb interactions (ionic) and Lennard-Jones potentials. The overall CHARMm (Chemistry at HARvard Macromolekular mechanics) potential is calculated by summing up these main potentials ( $ V_{CHARMm} = V_{bonded} + V_{nonbonded} $ ).
In equation \ref{CHARMM_bonded} and \ref{CHARMM_nonbonded} the bonded and nonbonded Potentials of the CHARMM force field are displayed. All terms consist of an equilibrium value marked with $0$ and a force constant $K$.

$$ \begin{equation} V_{bonded} = \sum_{bonds}{K_{b}(b-b_{0})^{2}} + \sum_{angels}{K_{\theta}(\theta-\theta_{0})^{2}} + \sum_{torsions}{K_{\phi}(1+cos(n\phi-\delta))} + \\ \sum_{impropers}{K_{\psi}(\psi-\psi_{0})^{2}} + \sum_{Urey-Bradley}{K_{UB}(r_{1,3}-r_{1,3,o})^{2}} + \sum_{\phi\psi}{V_{CMAP}} \label{CHARMM_bonded} \end{equation} $$ $$ \begin{equation} V_{nonbonded}=\sum_{nonbonded}{\frac{q_{i}q_{j}}{4\pi D r_{ij}}}+ \sum_{nonbonded}{\epsilon_{ij}\left[\left(\frac{R_{min,ij}}{r_{ij}}\right)^{12}-2\left(\frac{R_{min,ij}}{r_{ij}}\right)^{6}\right] } \label{CHARMM_nonbonded} \end{equation} $$

The additional terms CMAP and Urey-Bradley are correctional terms for backbone atoms and 1, 3 interactions respectively.

METHODS

RESULTS

"- yes, their son, Harry -"Mr. Dursley was the director of a firm called Grunnings, which made drills. He was a big, beefy man with hardly any neck, although he did have a very large mustache. Mrs. Dursley was thin and blonde and had nearly twice the usual amount of neck, which came in very useful as she spent so much of her time craning over garden fences, spying on the neighbors. The Dursleys had a small son called Dudley and in their opinion there was no finer boy anywhere.
The Dursleys had everything they wanted, but they also had a secret, and their greatest fear was that somebody would discover it. They didn't think they could bear it if anyone found out about the Potters. Mrs. Potter was Mrs. Dursley's sister, but they hadn't met for several years; in fact, Mrs. Dursley pretended she didn't have a sister, because her sister and her good-for-nothing husband were as unDursleyish as it was possible to be. The Dursleys shuddered to think what the neighbors would say if the Potters arrived in the street. The Dursleys knew that the Potters had a small son, too, but they had never even seen him. This boy was another good reason for keeping the Potters away; they didn't want Dudley mixing with a child like that.
When Mr. and Mrs. Dursley woke up on the dull, gray Tuesday our story starts, there was nothing about the cloudy sky outside to suggest that strange and mysterious things would soon be happening all over the country. Mr. Dursley hummed as he picked out his most boring tie for work, and Mrs. Dursley gossiped away happily as she wrestled a screaming Dudley into his high chair.
None of them noticed a large, tawny owl flutter past the window.
[....]

Theoretical Overview

Methods

Results

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