Intro 2 Md Simulation

Molecular Dynamics simulations

Bert de Groot Max Planck institute for biophysical chemistry Göttingen, Germany

Molecular Dynamics Simulations Schrödinger equation

Born-Oppenheimer approximation

Nucleic motion described classically

Empirical force field

Molecular Dynamics Simulations

Interatomic interactions

Molecular dynamics-(MD) simulations of Biopolymers d2 • Motions of nuclei are described classically, ( ∗ ) ma 2 Rα = −∇α Eel ( R1 ,..., RN ), α = 1,..., N . dt • Potential function Eel describes the electronic influence on motions of the nuclei and is approximated empirically  „classical MD“:

E el ≈

∑

E ibond +

Bindungen i

∑E

Bindungs − winkel j

angle j

+

∑E

Dihedral − winkel k

Covalent bonds Eibond

exact ν0

. rep . vdW + ∑ ( EαCoul ,β + Eα ,β + Eα ,β ) + ...,

Non-bonded interactions

approximated

KBT {

dihe k

= R= |R|

„ForceField“

Molecular Dynamics Simulation Molecule: (classical) N-particle system Newtonian equations of motion:

with

d2   mi 2 ri = Fi ( r ) dt  Fi (r ) = −∇ iV (r )

  r = ( r1 ,..., rN )

Integrate numerically via the „leapfrog“ scheme: with Δt ≈ 1fs!

(equivalent to the Verlet algorithm)

BPTI: Molecular Dynamics (300K)

Computational task: Solve the Newtonian equations of motion:

Non-bonded interactions

Lennard-Jones potential

Coulomb potential

Use of constraints to increase the integration step

The „SHAKE“ algorithm Δt = 1fs --> 2 fs

Molecular dynamics is very expensive ... Example: F1-ATPase in water (183 674 atoms), 1 nanosecond: 106 integration steps 8.4 * 1011 flop per step [n(n-1)/2 interactions]

total:

8.4 * 1017 flop

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

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

MD-Experiments with Argon Gas

Role of environment - solvent explicit or implicit?

box or droplet?

Surface (tension) effects? periodic boundary conditions and the minimum image convention

Proteins jump between many, hierarchically ordered „conformational substates“

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

Reversible Folding Dynamics of a β-Peptide

X. Daura, B. Jaun, D. Seebach, W.F. van Gunsteren, A.E. Mark, J. Mol. Biol. 280 (1998) 925

MD Simulations • external coupling: temperature (potential truncation, integration errors) pressure (density equilibration) system translation/rotation • analysis energies (individual terms, pressure, temperature) coordinates (numerical analysis, visual inspection!)  mechanisms