Multi-scale computations of simplified polypeptide models

1. Multi-scale computations of simplified polypeptide models Dov Bai Cornell University

2. Objectives • Application of the same algorithm developed for simple polymers (Bai & Brandt). • Starting point for development of multi-scale algorithms for complex protein systems (e.g protein folding). • Backbone structure is similar enough to polymer so that it can bridge between polymers and proteins.

3. Models • Backbone model with major energy terms, (H. Scheraga and coworkers, Brant & Flory) but no side chains (polyglycine). • BLN model – chain with 46 beads of three types: hydrophobic(B), hydrophilic(L), neutral(N). Forces depend on bead properties (work in progress).

4. Multi-scale algorithms Provides a systematic algorithm to produce coarse level Hamiltonians (energy functionals, force fields) with reduced number of degrees of freedom and computationally efficient potentials. Coarse Hamiltonians are used for simulations on coarse levels. Reduce computational work by 1-2 orders of magnitude per level. Coarse simulations should provide highly accurate statistical representation of fine level. General coarsening –Eliminating degrees of freedom, e.g. solvent degrees of freedom.

5. Backbone model • Backbone consists of repeating units of • Each unit corresponds to a single amino acid. • Units are connected by an amide bond between • Torsion angle is planar.

6. Strong length and angle bonding, as in polymers. • Torsion angles: are the main structural degrees of freedom: • Torsion barriers are smaller than in polymers.

7. Backbone/no-dipole Bonding only (21 units)

8. Dipole-dipole interaction (representing electrostatic interaction): Nearest neighbors in coil, longer in helix.

9. Bonding & dipole-dipole (21 units)

10. Central (Fine) Central fine dihedral with / without dipole-dipole

11. Van-der-Waals forces (WCA, long range, repulsive part of LJ (6-12)):

12. Hydrogen Bonds … Between O(C) and H(N) Operates only in very short distances. Can cause major structural changes in the protein: a-helix (i & i+4 amino acids) or b-sheet.

13. General multiscale approach • Derive coarse level Hamiltonian (energy functional) on small pieces of polypeptide (local set). • Local interactions only – local set atoms do not interact with the rest of polypeptide. • Require that on local set, coarse Hamiltonian produces correctly larger-scale quantities ( e.g. end-to-end, radius of gyration distributions). • Gradually add important interactions to local set if needed, e.g. • solvent degrees of freedom.

14. Coarsening – backbone model Coarsening ratio of 1:3 : Geometric centers of groups of 3 atoms. CMC validation: MC on fine for fixed coarse configurations. No transitions between torsion minima. Natural coarsening: Each coarse point represents a single amino-acid

15. Central (Fine) Central fine dihedral with / without dipole-dipole

16. Central coarse dihedral distribution No dipole-dipole /with dipole-dipole

17. Central coarse dihedral distribution Dipole-dipole & WCA

18. Derivation of Coarse Hamiltonian Express coarse Hamiltonian as: Observables: , Observables measured and compared on fine and coarse level simulations.

19. Iterative Process Update : By solving: Swendsen, 1979 (Ising)

20. i+3 i+1 i i+2 i Li - Distance between atom i, i+1 Ai- Angle formed by atoms i, i+1, i+1 Di- Dihedral angle formed by i, i+1, i+2, i+3 Ui - Any internal coordinate starting at point i

21. Coarse Hamiltonian terms: Internal coordinates terms: i-internal coordinate j-interval (bin) Probability of i –th internal coordinate to be in the j-th Interval Observable

22. Correlation terms: neighboring internal coordinates e.g.neighboring lengths term: Hamiltonian term Observable

24. - Dihedral angle - Transformation of dihedral angle due to periodicity

25. With bonding & dipole-dipole interactions: With bonding, dipole-dipole, WCA interactions:

26. Coarse WCA terms Similar in form to fine WCA terms Different parameter values

27. Correlations between internal Coordinates

28. End-to-end distribution with bonding & dipole-dipole

29. End-to-end distribution with bonding, dipole-dipole & WCA potential (long range)

30. Convergence of Newtonian iterations

31. Convergence of correlations

32. BLN model • Polymer-like bonding forces. • Hydrophobic(B),hydrophilic(L),neutral(N) non-bonding interactions. Chain of atoms: B-hydrophobic L-hydrophilic N-neutral J. D. Honeycutt, D. Thirummalai, PNAS, 1990

33. B-hydrophobic (black) L-hydrophilic (light grey) N-neutral (white) M. A. Miller, D. J. Wales, JCP, 1999

34. BLN model force field single N multiple N

35. Coarsening - CMC • For a given coarse configuration, check for a unique local minimum (SA, random compatible configurations). • Tested for T=0.5: center-of-mass, 1:3 – nearly unique, 1:2 – totally unique.

36. Coarse level correlations • 1:2 coarsening ratio. • Relative correlations between internal coordinates > 0.15 : about 35. • Examples of strong correlations:

37. ANGLE 9 - ANGLE 10-0.789 ANGLE 15 - ANGLE 16-0.782 ANGLE 3 - ANGLE 4-0.732 DIHED 3 - DIHED 4-0.520 Always neighboring coordinates!