An improved hybrid Monte Carlo method for conformational sampling of proteins

1. An improved hybrid Monte Carlo method for conformational sampling of proteins Jesús A. Izaguirre and Scott Hampton Department of Computer Science and EngineeringUniversity of Notre Dame March 5, 2003 This work is partially supported by two NSF grants (CAREER and BIOCOMPLEXITY) and two grants from University of Notre Dame

2. Overview 2. Methods for sampling (MD, HMC) 1. Motivation: sampling conformational space of proteins 3. Evaluation of new Shadow HMC 4. Future applications

3. Protein: The Machinery of Life NH2-Val-His-Leu-Thr-Pro-Glu-Glu- Lys-Ser-Ala-Val-Thr-Ala-Leu-Trp- Gly-Lys-Val-Asn-Val-Asp-Glu-Val- Gly-Gly-Glu-…..

4. Protein Structure

5. Why protein folding? • Huge gap: sequence data and 3D structure data • EMBL/GENBANK, DNA (nucleotide) sequences 15 million sequence, 15,000 million base pairs • SWISSPROT, protein sequences120,000 entries • PDB, 3D protein structures20,000 entries • Bridging the gap through prediction • Aim of structural genomics: • “Structurally characterize most of the protein sequences by an efficient combination of experiment and prediction,” Baker and Sali (2001) • Thermodynamics hypothesis: Native state is at the global free energy minimumAnfinsen (1973)

6. Questions related to folding I • Long time kinetics: dynamics of folding • only statistical correctness possible • ensemble dynamics • e.g., folding@home • Short time kinetics • strong correctness possible • e.g., transport properties, diffusion coefficients

7. Questions related to folding II • Sampling • Compute equilibrium averages by visiting all (most) of “important” conformations • Examples: • Equilibrium distribution of solvent molecules in vacancies • Free energies • Characteristic conformations (misfolded and folded states)

8. Overview 2. Methods for sampling (MD, HMC) 1. Motivation: sampling conformational space of proteins 3. Evaluation of new Shadow HMC 4. Future applications

9. Classical molecular dynamics • Newton’s equations of motion: • Atoms • Molecules • CHARMM force field(Chemistry at Harvard MolecularMechanics) Bonds, angles and torsions

10. What is a Forcefield? In molecular dynamics a molecule is described as a series of charged points (atoms) linked by springs (bonds). The forcefield is a collection of equations and associated constants designed to reproduce molecular geometry and selected properties of tested structures.

11. Energy Terms Described in the CHARMm forcefield Bond Angle Dihedral Improper

12. Energy Functions Ubond = oscillations about the equilibrium bond length Uangle = oscillations of 3 atoms about an equilibrium angle Udihedral = torsional rotation of 4 atoms about a central bond Unonbond = non-bonded energy terms (electrostatics and Lennard-Jones)

13. Molecular Dynamics –what does it mean? MD = change in conformation over time using a forcefield Energy Energy supplied to the minimized system at the start of the simulation Conformation impossible to access through MD Conformational change

14. MD, MC, and HMC in sampling • Molecular Dynamics takes long steps in phase space, but it may get trapped • Monte Carlo makes a random walk (short steps), it may escape minima due to randomness • Can we combine these two methods? MD MC HMC

15. Hybrid Monte Carlo • We can sample from a distribution with density p(x) by simulating a Markov chain with the following transitions: • From the current state, x, a candidate state x’ is drawn from a proposal distribution S(x,x’). The proposed state is accepted with prob.min[1,(p(x’) S(x’,x)) / (p(x) S(x,x’))] • If the proposal distribution is symmetric, S(x’,x)) = S(x,x’)), then the acceptance prob. only depends on p(x’) / p(x)

16. Hybrid Monte Carlo II • Proposal functions must be reversible: if x’ = s(x), then x = s(x’) • Proposal functions must preserve volume Jacobian must have absolute value one • Valid proposal: x’ = -x • Invalid proposals: • x’ = 1 / x (Jacobian not 1) • x’ = x + 5 (not reversible)

17. Hybrid Monte Carlo III • Hamiltonian dynamics preserve volume in phase space • Hamiltonian dynamics conserve the Hamiltonian H(q,p) • Reversible symplectic integrators for Hamiltonian systems preserve volume in phase space • Conservation of the Hamiltonian depends on the accuracy of the integrator • Hybrid Monte Carlo: Use reversible symplectic integrator for MD to generate the next proposal in MC

18. HMC Algorithm Perform the following steps: 1. Draw random values for the momenta p from normal distribution; use given positions q 2. Perform cyclelength steps of MD, using a symplectic reversible integrator with timestep t, generating (q’,p’) 3. Compute change in total energy • H = H(q’,p’) - H(q,p) 4. Accept new state based on exp(- H )

19. Hybrid Monte Carlo IV Advantages of HMC: • HMC can propose and accept distant points in phase space, provided the accuracy of the MD integrator is high enough • HMC can move in a biased way, rather than in a random walk (distance k vs sqrt(k)) • HMC can quickly change the probability density

20. Hybrid Monte Carlo V • As the number of atoms increases, the total error in the H(q,p) increases. The error is related to the time step used in MD • Analysis of N replicas of multivariate Gaussian distributions shows that HMC takes O(N5/4)withtime step t = O(N-1/4) Kennedy & Pendleton, 91

21. Hybrid Monte Carlo VI • The key problem in scaling is the accuracy of the MD integrator • More accurate methods could help scaling • Creutz and Gocksch89 proposed higher order symplectic methods for HMC • In MD, however, these methods are more expensive than the scaling gain. They need more force evaluations per step

22. Overview 2. Methods for sampling (MD, HMC) 1. Motivation: sampling conformational space of proteins 3. Evaluation of new Shadow HMC 4. Future applications

23. Improved HMC • Symplectic integrators conserve exactly (within roundoff error) a modified Hamiltonian that for short MD simulations (such as in HMC) stays close to the true Hamiltonian Sanz-Serna & Calvo 94 • Our idea is to use highly accurate approximations to the modified Hamiltonian in order to improve the scaling of HMC

24. Shadow Hamiltonian Work by Skeel and Hardy, 2001, shows how to compute an arbitrarily accurate approximation to the modified Hamiltonian, called the Shadow Hamiltonian • Hamiltonian: H=1/2pTM-1p + U(q) • Modified Hamiltonian: HM = H + O(t p) • Shadow Hamiltonian: SH2p = HM + O(t 2p) • Arbitrary accuracy • Easy to compute • Stable energy graph • Example, SH4 = H – f( qn-1, qn-2, pn-1, pn-2 ,βn-1 ,βn-2)

25. See comparison of SHADOW and ENERGY

26. Shadow HMC • Replace total energy H with shadow energy • SH2m = SH2m (q’,p’) – SH2m (q,p) • Nearly linear scalability of sampling rate Computational cost SHMC, N(1+1/2m),where m is accuracy order of integrator • Extra storage (m copies of q and p) • Moderate overhead (25% for small proteins)

27. Example Shadow Hamiltonian

28. Front-end libfrontend Middle layer libintegrators back-end libbase, libtopology libparallel, libforces ProtoMol: a framework for MD Matthey, et al, ACM Tran. Math. Software (TOMS), submitted Modular design of ProtoMol (Prototyping Molecular dynamics). Available at http://www.cse.nd.edu/~lcls/protomol

29. SHMC implementation • Shadow Hamiltonian requires propagation of β • Can work for any integrator

30. Systems tested

31. Sampling Metric 1 • Generate a plot of dihedral angle vs. energy for each angle • Find local maxima • Label ‘bins’ between maxima • For each dihedral angle, print the label of the energy bin that it is currently in

32. Sampling Metric 2 • Round each dihedral angle to the nearest degree • Print label according to degree

33. Acceptance Rates

34. More Acceptance Rates

35. Sampling rate for decalanine (dt = 2 fs)

36. Sampling rate for 2mlt

37. Sampling rate comparison • Cost per conformation is total simulation time divided by number of new conformations discovered (2mlt, dt = 0.5 fs) • HMC 122 s/conformation • SHMC 16 s/conformation • HMC discovered 270 conformations in 33000 seconds • SHMC discovered 2340 conformations in 38000 seconds

38. Conclusions • SHMC has a much higher acceptance rate, particularly as system size and timestep increase • SHMC discovers new conformations more quickly • SHMC requires extra storage and moderate overhead. • SHMC works best at relatively large timesteps

39. Future work • Multiscale problems for rugged energy surface • Multiple time stepping algorithms plus constraining • Temperature tempering and multicanonical ensemble • Potential smoothing • System size • Parallel Multigrid O(N) electrostatics • Applications • Free energy estimation for drug design • Folding and metastable conformations • Average estimation

40. Acknowledgments • Dr. Thierry Matthey, co-developer of ProtoMol, University of Bergen, Norway • Graduate students: Qun Ma, Alice Ko, Yao Wang, Trevor Cickovski • Students in CSE 598K, “Computational Biology,” Spring 2002 • Dr. Robert Skeel, Dr. Ruhong Zhou, and Dr. Christoph Schutte for valuable discussions • Dr. Radford Neal’s presentation “Markov Chain Sampling Using Hamiltonian Dynamics” (http://www.cs.utoronto.ca ) • Dr. Klaus Schulten’s presentation “An introduction to molecular dynamics simulations” (http://www.ks.uiuc.edu )