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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
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 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
NEdtdm Nela RRR
...,)( ,.
,.
,vdWrepCoul
kwinkelDihedral
dihek
iBindungen
jwinkelBindungs
anglej
bondiel EEEEEEE
Molecular Dynamics„Force-Field“
Molecular DynamicsNon-bonded interactions
Lennard-Jones potential Coulomb potential
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.
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-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.
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
rFrdtdm iii
)()( rVrF ii
)r,...,r(r N
1
Molecular Dynamics
BPTI: Molecular Dynamics (300K)
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
MD-Experiments with Argon Gas
Molecular Dynamics
Role of environment - solvent
Explicit or implicit?
Box or droplet?
Molecular Dynamics
periodic boundary conditions
Molecular Dynamics
H. Frauenfelder et al., Science 229 (1985) 337
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)
