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.

Simulation Analysis

Because of the vast amount of data that is produced by Molecular Dynamics simulations it is essential to process the data into more accesible formats. To perform this task we applied several approaches like comparison of the solvent accesible surface area (SASA) over time.

Root Mean Square Deviation

The Root Mean Square Deviation (RMSD) describes the sum of distances of all selected atoms \(n\) between themselves in a selceted timestep \(\tau\) and a reference timestep \(r\) (eq. \ref{RMSD}). Plotted over time it is possible to detect fluctuations in the whole molecular configuration and therefore it is possible to conclude structures of high stability from plateaus in the RMSD.

$$ \begin{equation} RMSD_{\tau}=\sqrt{\sum_{n=1}^{N}{\left((x_{\tau,n}-x_{r,n})^{2}+(y_{\tau}-y_{r,n})^{2}+(z_{\tau,n}-z_{r,n})^{2}\right)}} \label{RMSD} \end{equation} $$

By chosing the right atom selection it is possible to evaluate different behaviour of protein subgroups. For example if the RMSD between Cα atoms is calculated it is possible to plot the backbone movement over time and hence detect configurations that differ from the starting structure. This is important if the one wants to search for different thermodynamic stable ensembles of the protein or molecul of interest.

Root Mean Square Fluctuation

Similar to the RMSD the Root Mean Square Fluctuation describes the sum of distances of all selected atoms. In this case the distance per atom between all selected configuartions is calculated and summed over time. Therefore it is possible to spot out residues with strong mobility and consequntly residues that are part of fluctuating and disordered protein subunits.

$$ \begin{equation} RMSF_{n}=\sqrt{\sum_{\tau=1}^{T}{\left((x_{n}-x_{0})^{2}+(y_{n}-y_{0})^{2}+(z_{n}-z_{0})^{2}\right)}} \label{RMSF} \end{equation} $$

DSSP

Define Secondary Structures of Proteins (DSSP) by Wolfgang Kabsch and Christian Sander is a standard program to analyse secondary structure properties of proteins. The main idea to discriminate between different secondary structures is to based on the presence of H bonds because this can be represented by one energy value. This definition enables the algorithm to distinguish different types of α helices, β sheets, and turns.
The electrostatic interations between two groups are calculated by assigning partial charges to each C (\(+q_{1}\)), O (\(-q_{1}\)), N (\(-q_{2}\)), and H (\(+q_{2}\)), with \(q_{1} = 0.42~e\) and \( q_{2} = 0.20~e\) and r(AB) being the distance between two atoms A and B in Angstr&oumlm. An H bond is defined by \( E < -0.5~\frac{kcal}{mol}\).

$$ \begin{equation} E = q_{1} q_{2} \left( \frac{1}{r(ON)} + \frac{1}{r(CH)} + \frac{1}{r(OH)} + \frac{1}{r(CN)} \right) 332~\frac{kcal}{mol} \label{DSSP} \end{equation} $$

Binding Energy Calculations

Several methods to calculate binding energies from molecular dynamics simulations have been developed. Each of these methods rely on a stepwise cancelling of the binding state by either decoupling the binding interactions often called alchemistry perturbation mehtods or by pulling the molecules apart by an additional biasing potential. In each of these methods several molecular dynamics simulations have to be run with in case of umbrella sampeling different distances between the molecules of interest. This is done to construct a thermodynamic cycle to calculate the free energy differences between the bound and unbound state and hence gain the binding energy. In case of the perturbation methods a coupling parameter λ is introduced that ranges from 0 to 1 over the simulations and regulates the strength of the intermolecule interactions.
In every energy calculation concept the simualtions have to be similar enough so that a part of the ensembles is present in the neighbouring simulations and consequntly a sufficient phase space overlap is reached. Unfortunately it is difficult to guess how the simulations have to be set up to gain this overlap. Therefore a careful evaluation has to be applied that results in a trial-and-error procedure. This is especially true for pulling experiments that have four major adjustment parameters: The pulling direction, the pulling speed, the pulling force, and the number of and spacing between the conformations that are subsequently used. To evaluate these simulations two algorithms are often in applied today: The Bennet Acceptence Ration (BAR) and the Weighted Histogram Analysis Method (WHAM).
To calculate the experimentally determinable parameter of the binding constant \(K_{d}\) one can use the relation in equation \ref{gibbs}.

$$ \begin{equation} \Delta G = R T ln \left( \frac{K_{d}}{mol/L} \right) \label{gibbs} \end{equation} $$

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