Molecular Dynamics – Advance Applications

Why Molecular Dynamics? 1. Scale: Large collections of interacting particles that cannot (and should not) be studied by quantum mechanics Classical limit? ~115 nm ~2,000,000 atoms ~25 nm ~500,000 atoms Quantum Still far from bulk material… Lattice fluids Continuum models 2. Dynamics: time dependent behavior and non-equilibrium processes. Monte Carlo methods can predict many of the same things, but do not provide info on time dependent properties

A Few Theoretical Concepts Statistical Ensembles     microcanonical grand canonical Isothermal-isobaric In Equilibrium MD, we want to sample the ensemble as best as possible!

A Few Theoretical Concepts Classical configuration Integral    

The van Der Waals Equation Potentials modeling pairwise interactions. Lennard-Jones Potential Hard sphere Potential

Molecular Dynamics: Nuts and Bolts Particles (atoms and molecules): non-reactive, stable species (generally). trajectories determined by solving Newtonian equations of motion Forces on particles due to molecular mechanics force fields. Basic Force Field components + other possibilities (implicit solvent, external potentials, multiple center non-bonded interactions, dipole-dipole interactions….)

Common types of force fields United Atom All Atom Includes explicit H atoms Partial charges on most atoms OPLS-AA, CHARMM, AMBER and many others OPLS-UA, TraPPE-UA, Beads include H atoms in CH2 CH3 etc. Most atoms uncharged, Exceptions for O-H groups etc. Tip4p Coarse Grain (graphic: sklog Wiki) Beads include large functional groups. SPC σ/2 1 (nonpolar) Martini water bead represents 4 water molecules…  http://dx.doi.org/10.1063/1.4863329 (http://espressomd.org)

Force Field Parametrization Force Fields are largely empirical You only get so many parameters: Make them fit experiment! Ab initio calculations can be used How does one get charges? OPLS obtained dihedrals from QM simulations

Practical stipulations: AMBER—all residues have integer charge TraPPE—Carbon hybridization dictates parameters CHARMM22—parameterized for tip3P water Transferability Want to model systems containing mixtures of particles which were parameterized (want parameters to “transfer” to new systems). Possible if all parameters derived in same way…

How do the dynamics happen? Forces on each particle are calculated at time t. The forces provide trajectories, which are propagated for a small duration of time, Δt, producing new particle positions at time t+ Δt. Forces due to new positions are then calculated and the process continues: The **basic** idea…

How do the dynamics happen? What is a suitably short time step? Must be significantly shorter than the fastest motion in your simulation: What is frequency of C-H stretch. O-H stretch? Huge restoring force: simulation crashes Normal restoring force Time step Too long Adequately Short Time step Minimum time step depends on what you are monitoring. At least, simulation must be stable. Constraint algorithms: Shake, Rattle, LINCS

Cuttoffs and Boundaries Do we use the entire configuration integral? Graphic credit: http://beam.acclab.helsinki.fi/~koehenri How do you keep your particles from driftingout of the cell? What’s the maximum cutoff? Graphic credit: http://server.ccl.net/cca/documents/molecular-modeling/node9.html

Cutoff (modified cutoff) radii For L-J, 2.5σ PERIODIC BOUNDARY CONDITIONS Cutoff (modified cutoff) radii For L-J, 2.5σ Long-range electrostatics Ewald Sums Graphic credit: http://server.ccl.net/cca/documents/molecular-modeling/node9.html

Temperature control The natural ensemble for the most basic MD simulations is the microcanonical (N,V,E) ensemble. Often we want to use the canonical (NVT) isothermal, isobaric (NPT). Stochastic thermostats: Introduce random perturbations to velocities—can be viewed as random reassignment rather than rescaling.

Thermostats Andersen: Randomly select particles and reassigns velocities from a distribution determined by the desired temperature. Langevin or Browninan dynamics: include a stochastic term in dynamics equations that perturbs normal velocities by assigning random forces from within a distribution determined by the temperature. ***These cause drag and slower dynamics. Weak temperature coupling is advised. Non-Equilibrium MD simulations: Does strict temperature control make sense? Is the center of mass moving? (Probably) Must you apply a thermostat to the whole system? Weak temperature coupling is advised!

Pressure Control If we want to perform a simulation at constant pressure we generally have to let the volume fluctuate (NPT ensemble). The simplest barostat is the Berendsen barostat, which is similar to the Berendsen thermostat such that the volume of the simulation cell is corrected at each (specified) time step by a scaling factor:

rem- barostats Where μ is the scaling factor for of one side of the (cubic) simulation cell (and other lengths in the system) and τ is a time coupling constant. Again the Berendsen method does not generate any known statistical ensemble, especially for small systems, but it is stable. For production runs, use the Parinello-Rahman method, which is somewhat analagous to the Nose-Hoover thermostat method. Pressure coupling only works at equilibrium!

MD Condition controls All types of temperature control alter velocities and therefore disturb dynamics . In Non-stochastic methods, the “flying ice cube” artifact can be circumvented by removing translational and rotational center of mass motion. The Nose-Hoover method produces the correct velocity distributions for a canonical ensemble and maintains smooth trajectories; consequently it is a method of choice for equilibrium NVT and NPT simulations.

Practical Remarks Stochastic temperature control typically disturbs dynamics more than non-stochastic methods. Velocity reassignment causes significant drag on dynamics and produces discontinuous trajectories. Non-equilibrium simulations should use stochastic temperature control in order to avoid build up of kinetic energy in translational and/or rotational modes. Issues with thermostats may be circumvented by selectively thermostatting different groups (i.e. solvent or an specific area in space). The Rahman-Parrinello and MTTK pressure coupling schemes generate the correct ensemble, but they are sensitive and unstable far from equilibrium, epsecially R-P. A more stable Berendsen scheme can be used initially to reach the correct volume. Pressure coupling only makes sense for equilibrium simulations.

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