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:

Potential function Eel describes the electronic influence on motions of the nuclei and is approximated empirically „classical MD“:

approximated

exact

Eibond

|R|0

KBT {

Covalent bonds Non-bonded interactions

==R

.,...,1),,...,()( 12

2

NEdt

dm Nela RRR

...,)( ,.

,.

,vdWrepCoul

kwinkelDihedral

dihek

iBindungen

jwinkelBindungs

anglej

bondiel EEEEEEE

Molecular Dynamics„Force-Field“

Possible ‘extras’:PlanarityHydrogenbondWeird metalInduced chargeMulti-body interactionPi-Pi stackingand a few more

Molecular DynamicsNon-bonded interactions

Lennard-Jones potential Coulomb potential

Molecular Dynamics

Molecular Dynamics

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

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. 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!

)(2

2

rFrdt

dm iii

)()( rVrF ii

)r,...,r(r N

1

Molecular Dynamics

Solve the Newtonian equations of motion:

Molecular Dynamics

Molecular dynamics is very expensive ... Example: A one nanosecond Molecular Dynamics simulation of F1-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 describedpoor description of H-atoms (proton-transfer)poor description of low-T (quantum) effectssimplified electrostatic modelsimplified force fieldincomplete 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

D C

BA -> B -> C -> D -> AΔG=0!

Molecular Dynamics

At Radboud you have seen in

‘Werkcollege 3 Thermodynamica’:

Folded 105 C Unfolded 105 C

Folded 75 C Unfolded 75 C

?

1

2

3

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 kJ/molK.Proces 1 is isobaar dus dH1=Cp(folded)*dTProces 3 is isobaar dus dH3=Cp(unfolded)*dTProces 2 is isotherm dus ΔH2=ΔH(unfolding;75 C)=509kJ/molVul alle getallen in en je krijgt ΔH(unfolding; 105 C)=697.4 kJ/mol.

Molecular Dynamics

Thermodynamic Cycle in bioinformatics

A

D C

BΔG1+ΔG2+ΔG3+ΔG4=0 =>ΔG1+ΔG3=-ΔG2-ΔG4So 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-U Mut-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…