1. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques117 Jan 2006 Bioinformatics Data Analysis & Tools Molecular simulations & sampling techniques

2. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques217 Jan 2006 Main points Proteins are flexible, crystal structure is an average Peptide folding from simulation –Folding and un-folding in 200 ns –Temperature and Pressure dependence –few relevant non-folded structures: perhaps only 10 9 protein structures needed in stead of 10 90 for simulating folding Phase Space –Motion is a curved line in phase space: trajectory (p(t),q(t)) Molecular Motions: Time & Length-scales Classical (Newton) Mechanics –kinetic energy (K(p), depends on temperature)‏ –potential energy (V(q) depends on interactions)‏ bonded interactions (e.g. bond stretching, angle bending)‏ non-bonded interactions (e.g. van der Waals, electrostatic)‏ –determine for example protein motions.

3. Bioinf. Data Analysis & Tools Main points (2)‏ Calculating Averages through Integration of phase space: –only low energy states are relevant –No analytical solutions -> Numerical integration: by time (Molecular Dynamics)‏ by ensemble (Monte-Carlo)‏ Convergence –Amount of phase-space covered: “Sampling” –You cannot know what you don’t know there might be a “next valley” Apparent Convergence on all timescales! –100 ps – 10 ns Efficiency / Improving Performance Trajectory on Energy Surface

4. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques417 Jan 2006 Trajectory on Energy Surface

5. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques517 Jan 2006 Monte Carlo Sampling Ergodic hypothesis: –Sampling over time (Molecular Dynamics approach); and –Ensemble averaging (Monte Carlo approach) Yield the same result:  (r) = NVE Detailed Balance condition: p(o)  (o  n) = p(n)  (n  o)

6. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques617 Jan 2006 Metropolis Selection Scheme Metropolis acceptance rule that satisfies detailed equilibrium: acc(o  n) = p(n)/p(o) = e -  E/kT if p(n) < p(o) acc(o  n) = 1 if p(n)  p(o)  Metropolis Monte Carlo Ergodic probability density for configurations around r N e -E/kT p(r N ) = ––––––  e -E/kT

7. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques717 Jan 2006 Search Strategies

8. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques817 Jan 2006 Leaps

9. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques917 Jan 2006 Computational Scheme Reduction of the leaps will lead to classical dynamics Control parameter: –RMSD –Angle deviation

10. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1017 Jan 2006 Computational Load: Solvation Most computational time (>95%) spent on calculating (bulk) water-water interactions

11. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1117 Jan 2006 Implicit Solvation

12. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1217 Jan 2006 POPS Solvent accessible area –fast and accurate area calculation –resolution: POPS-A (per atom) POPS-R (per residue)‏ –parametrised on 120000 atoms and 12000 residues –derivable -> MD Free energy of solvation  G solv i = area i ·  i POPS is implemented in GROMOS96 parameters 'sigma' from simulations in water: –amino acids in helix, sheet and extended conformation –peptides in helix and sheet conformation

13. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1317 Jan 2006 POPS server

14. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1417 Jan 2006 Example: Protein & Ligand Dynamics

15. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1517 Jan 2006 Example: Essential Dynamics Analysis Cyt-P450 BM3 7 x 10ns “free” MD simulations

16. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1617 Jan 2006 Example: Minima

17. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1717 Jan 2006 Example: Conformations

18. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1817 Jan 2006 Levinthal’s paradox Protein Folding Problem: –Predict the 3D structure from sequence –Understand the folding process

19. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques1917 Jan 2006 Folding energy Each protein conformation has a certain energy and a certain flexibility (entropy)‏ Corresponds to a point on a multidimensional free energy surface may have higher energy but lower free energy than energy E(x) ‏ coordinate x Three coordinates per atom 3N-6 dimensions possible  G =  H – T  S

20. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques2017 Jan 2006 Folded state Native state = lowest point on the free energy landscape Many possible routes Many possible local minima (misfolded structures)‏

21. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques2117 Jan 2006 Molten globule First step: hydrophobic collapse Molten globule: globular structure, not yet correct folded Local minimum on the free energy surface

22. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques2217 Jan 2006 Force Field “the collection of all forces that we consider to occur in a mechanical atomic system” A generalised description: E total = E bonded + E non-bonded + E crossterm Cross terms: –non-bonded interaction influence the bonded interaction (v.v.). –Most force fields neglect those terms. Note that force fields are (mostly) designed for pairwise atom interactions. –Higher order interactions are implicitly included in the pairwise interaction parameters.

23. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques2317 Jan 2006 Force Field Components: Bonded Interactions

24. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques2417 Jan 2006 Force Field Components: Non-Bonded Interactions

25. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques2517 Jan 2006 All Together…

26. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques2617 Jan 2006 Main points Trajectory on Energy Surface Monte Carlo Sampling –Sampling over time vs. Ensemble averaging –Metropolis Selection Scheme satisfies Detailed Balance condition –Search Strategies Computational Load: Solvation –Implicit Solvation (POPS): Free energy of solvation Examples –Protein & Ligand Dynamics –Minima –Conformations

27. Bioinf. Data Analysis & Tools Main Points (2)‏ Protein Folding Problem –Levinthal’s paradox –Each protein conformation has a certain energy and a certain flexibility (‘entropy’)‏ Corresponds to a point on a multidimensional free energy surface –Folded state (lowest free energy)‏ Many possible routes Many possible local minima (misfolded structures)‏ –Molten globule Force Field: all forces in a system –Bonded Interactions –Non-Bonded Interactions

28. Bioinf. Data Analysis & Tools Molecular Simulations & Sampling Techniques2817 Jan 2006