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

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

3. Molecular Dynamics Schrödinger equation Born-Oppenheimer approximation Nucleic motion described classically Empirical force field

4. Molecular Dynamics Inter-atomic interactions

5. 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

6. Molecular Dynamics „Force- Field“

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

8. Molecular Dynamics http://en.wikipedia.org/wiki/Verlet_integration http://en.wikipedia.org/wiki/Maxwell_speed_distribution 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. 12 -15

9. Molecular Dynamics http://en.wikipedia.org/wiki/Verlet_integration 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. 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.

10. 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!

11. Molecular Dynamics BPTI: Molecular Dynamics (300K)

12. Molecular Dynamics Solve the Newtonian equations of motion:

13. Molecular Dynamics Molecular dynamics is very expensive... Example: A one nanosecond Molecular Dynamics simulation of F 1 - ATPase in water (total 183 674 atoms) needs 10 6 integration steps, which boils down to 8.4 * 10 17 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

14. Molecular Dynamics MD-Experiments with Argon Gas

15. Molecular Dynamics Role of environment - solvent Explicit or implicit? Box or droplet?

16. Molecular Dynamics periodic boundary conditions

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

18. 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 (10 4... 10 6 atoms) only short time spans accessible (ps... μs)