Molecular Dynamics Molecular dynamics Some random notes on molecular dynamics simulations Seminar based on work by Bert de Groot and many anonymous Googelable colleagues

Molecular Dynamics Most material in this seminar has been produced by Bert de Groot at the MPI in Göttingen.

Molecular Dynamics

Schrödinger equation Born-Oppenheimer approximation Nucleic motion described classically Empirical force field

Molecular Dynamics Inter-atomic interactions

Molecular Dynamics Motions of nuclei are described classically: Potential function E el describes the electronic influence on motions of the nuclei and is approximated empirically  „classical MD“: approximated exact E i bond |R| 0 K B T { Covalent bonds Non-bonded interactions = = R

Molecular Dynamics „Force- Field“ Possible ‘extras’: Planarity Hydrogenbond Weird metal Induced charge Multi-body interaction Pi-Pi stacking and a few more

Molecular Dynamics Non-bonded interactions Lennard-Jones potential Coulomb potential

Molecular Dynamics

Now we need to give all atoms some initial speed, and then, evolve that speed over time using the forces we now know. The average speed of nitrogen in air of 300K is about 520 m/s. The ensemble of speeds is best described by a Maxwell distribution. Back of the enveloppe calculation: 500 m/s = 5.10 Å/s Let’s assume that we can have things fly 0.1 A in a straight line before we calculate forces again, then we need to recalculate forces every 20 femtosecond (one femtosecond is 10 sec. In practice 1 fsec integration steps are being used

Molecular Dynamics Knowing the forces (and some randomized Maxwell distributed initial velocities) we can evolve the forces over time and get a trajectory. Simple Euler integration won’t work as this figure explains. And as the rabbit knows... You can imagine that if you know where you came from, you can over-compensate a bit. These overcompensation algorithms are called Verlet-algorithm, or Leapfrog algorithm. If you take bigger time steps you overshoot your goal. The Shake algorithm can fix that. Shake allows you larger time steps at the cost of little imperfection so that longer simulations can be made in the same (CPU) time.

Molecular Dynamics Molecule: (classical) N-particle system Newtonian equations of motion: Integrate numerically via the „leapfrog“ scheme: (equivalent to the Verlet algorithm) with Δt  1fs!

Molecular Dynamics Solve the Newtonian equations of motion:

Molecular Dynamics Molecular dynamics is very expensive... Example: A one nanosecond Molecular Dynamics simulation of F 1 - ATPase in water (total atoms) needs 10 6 integration steps, which boils down to 8.4 * floating point operations. on a 100 Mflop/s workstation:ca 250 years...but performance has been improved by use of: + multiple time steppingca. 25 years + structure adapted multipole methods*ca. 6 years + FAMUSAMM*ca. 2 years + parallel computers ca. 55 days * Whatever that is

Molecular Dynamics

Role of environment - solvent Explicit or implicit? Box or droplet?

Molecular Dynamics periodic boundary conditions

Molecular Dynamics

Limits of MD-Simulations classical description: chemical reactions not described poor description of H-atoms (proton-transfer) poor description of low-T (quantum) effects simplified electrostatic model simplified force field incomplete force field only small systems accessible ( atoms) only short time spans accessible (ps... μs)

Molecular Dynamics H. Frauenfelder et al., Science 229 (1985) 337

Molecular Dynamics

One example: Thermodynamic Cycle A DC B A -> B -> C -> D -> A ΔG=0!

Molecular Dynamics At Radboud you have seen in ‘Werkcollege 3 Thermodynamica’: Folded 105 C Unfolded 105 C Folded 75 CUnfolded 75 C ? And, for Radboud students only, I type here the answer in Dutch… ΔT kan natuurlijk in Celcius of Kelvin) en is dan of 0 of =30 Cp is heat capacity en kan temepartuuronafhankelijk verondersteld worden. Cp(unfolded)-Cp(folded)=6.28 kJ/molK. Proces 1 is isobaar dus dH1=Cp(folded)*dT Proces 3 is isobaar dus dH3=Cp(unfolded)*dT Proces 2 is isotherm dus ΔH2=ΔH(unfolding;75 C)=509kJ/mol Vul alle getallen in en je krijgt ΔH(unfolding; 105 C)=697.4 kJ/mol.

Molecular Dynamics Thermodynamic Cycle in bioinformatics A DC B ΔG1+ΔG2+ΔG3+ΔG4=0 => ΔG1+ΔG3=-ΔG2-ΔG4 So if you know the difference between ΔG2 and ΔG4, you also know the difference between ΔG1 and ΔG3 (and vice versa). ΔG1 ΔG4 ΔG3 ΔG2 Obviously, all arrows should be bidirectional equilibrium-arrows, but if I draw them that way we are sure to start getting the signs wrong. …

Molecular Dynamics The relations between energy, force and time can be simulated in MD. Obviously you cannot simply put a force on an atom for some time and calculate the Energy from the force, path, and time. But for now, we forget all calibrations, etc, and end up with Energy = Force * time

Molecular Dynamics Stability of a protein is ΔG-folding, which is the ΔG of the process Protein-U Protein-F Wt-U Mut-UMut-F ΔG(fold)wt ΔG(mut)U ΔG(fold)mut ΔG(mut)F Wt-F So we want ΔG(fold)wt- ΔG(fold)mut; which is impossible. But we can calculate ΔG(mut)F-ΔG(mut)U; which gives the same number!

Molecular Dynamics Such cycles can be set up for ligand binding, for membrane insertion, for catalysis, etc. Don’t be surprised if you have to work out a similar cycle in the exam…