Molecular Dynamics Molecular dynamics Some random notes on

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

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: Non-bonded interactions Covalent bonds Eibond approximated exact = = R KBT { 0 |R| Potential function Eel describes the electronic influence on motions of the nuclei and is approximated empirically „classical MD“:

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

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

Molecular Dynamics

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 300 K 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 12Å/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 -15 sec. In practice 1 fsec integration steps are being used. http: //en. wikipedia. org/wiki/Verlet_integration http: //en. wikipedia. org/wiki/Maxwell_speed_distribution

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. http: //en. wikipedia. org/wiki/Verlet_integration

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

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 183 674 atoms) needs 106 integration steps, which boils down to 8. 4 * 1017 floating point operations. on a 100 Mflop/s workstation: ca 250 years . . . but performance has been improved by use of: + multiple time stepping ca. 25 years + structure adapted multipole methods* ca. 6 years + FAMUSAMM* ca. 2 years + parallel computers ca. 55 days * Whatever that is

Molecular Dynamics

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

Molecular Dynamics periodic boundary conditions

Molecular Dynamics

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 (104. . . 106 atoms) only short time spans accessible (ps. . . μs)

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

Molecular Dynamics

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

Molecular Dynamics At Radboud you have seen in ‘Werkcollege 3 Thermodynamica’: Folded 105 C ? 1 Unfolded 105 C 3 Folded 75 C Unfolded 75 C 2 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 105 -75=30 Cp is heat capacity en kan temepartuuronafhankelijk verondersteld worden. Cp(unfolded)-Cp(folded)=6. 28 k. J/mol. K. Proces 1 is isobaar dus d. H 1=Cp(folded)*d. T Proces 3 is isobaar dus d. H 3=Cp(unfolded)*d. T Proces 2 is isotherm dus ΔH 2=ΔH(unfolding; 75 C)=509 k. J/mol Vul alle getallen in en je krijgt ΔH(unfolding; 105 C)=697. 4 k. J/mol.

Molecular Dynamics Thermodynamic Cycle in bioinformatics ΔG 1 A B ΔG 2 ΔG 4 D C ΔG 3 ΔG 1+ΔG 2+ΔG 3+ΔG 4=0 => ΔG 1+ΔG 3=-ΔG 2 -ΔG 4 So if you know the difference between ΔG 2 and ΔG 4, you also know the difference between ΔG 1 and ΔG 3 (and vice versa). 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 ΔG(fold)wt Wt-U Wt-F ΔG(mut)U Mut-U ΔG(fold)mut Mut-F So we want ΔG(fold)wtΔG(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…