Graphical Models for Protein Kinetics

1. Graphical Models for Protein Kinetics Nina Singhal CS374 Presentation Nov. 1, 2005

2. Outline • Background material on proteins • Why study protein kinetics • Graphical models for kinetics • Motion planning view (Apaydin et al, 2003) • Molecular dynamics view (Singhal et al, 2004) • Conclusions

3. Alpha Helix Beta Strand and Sheet Beta Barrel Background on Proteins

4. Structure Prediction MTYKLILNGKTLKGETTTEAVDAATAEKVFKQYANDNGVDGEWTYDDATKTFTVTE Given an amino acid sequence, what 3D structure will the protein form? ?

5. Pathways and Kinetics How does a protein actually get from an unfolded configuration to a folded configuration? ?

6. Folding Kinetics • Rate of folding • Uniqueness of pathway • Order of secondary structure formation • Secondary or tertiary structure

7. Applications • Misfolded proteins and diseases • Alzheimer's • Cystic fibrosis • Mad cow disease • Intermediates may be important as drug targets • Protein design

8. Representation of a Protein N1 N1 C Psi omega phi psi Ca N2 N2 R A protein with n amino acids can be represented using 2n phi-psi angles, each in the range [0, 2p)

9. Graphical Models for Protein Kinetics • Protein conformations have different energies • Graphical models discretize the conformation space and connect nearby regions with edges

10. Robotics Motion Planning 2p c1 c1 c2 c2 0 2p Robot with 2 degrees of freedom 2D configuration space Moving the robot arm from c1 to c2 is just finding a path in the configuration space from c1 to c2.

11. Roadmap Method • Randomly sample points in configuration space. Keep feasible ones. • Connect these points to form a graph. • Process path queries using standard graph search techniques. 2p c1 c2 0 2p

12. Protein Folding as a Search Problem • Protein folding can be represented as a search through the protein’s configuration space • Replace collision free constraint with a preference for low energy configurations • Instead of finding any path, want to find all the energetically favorable paths

13. Stochastic Roadmap Simulation (Apaydin et. al. 2003) • Sample protein configuration at random • Add edges between nearby nodes • Take advantage of the many folding pathways contained within a roadmap • Efficiently calculate many properties of the entire landscape

14. Roadmap Construction • Nodes in the graph are sampled uniformly at random • Edges are added between nearest neighbors with probability: if DEij > 0 otherwise

15. Roadmap as a Markov Chain • We can view the molecular motion as a random walk over the roadmap • Roadmap can be regarded as discretely sampled version of Monte Carlo simulation • If fact, in the limit, probability distributions of Monte Carlo simulation and the roadmap converge

16. Transmission Coefficients • Measures “kinetic distance” • Probability that a conformation will fold before unfolding • Can calculate by starting many Monte Carlo simulations from the conformation • Very computationally expensive ? ? Unfolded state Folded state

17. Algebraic Method for Calculating Transmission Coefficients F vi U Pij vj

18. Transmission Coefficients (cont) • System of linear equations • One equation and one unknown for each node • Can be solved iteratively • Low connectivity of the graph results in a sparse matrix

19. Studied a synthetic landscape and a real protein, ROP Protein was represented with 6 degrees of freedom, two vectors connected by a loop Results • Correlation of transmission coefficients calculated by roadmaps and Monte Carlo simulations

20. Benefits and Drawbacks • Extremely efficient at calculating kinetic properties like transmission coefficients • Unclear whether low-dimension representation of protein is adequate • Monte Carlo simulations may not be accurate enough for protein kinetics

21. Molecular dynamics Simulate protein movement using Newton’s laws of motion Bond vibration Isomer- ation Water dynamics Helix forms Fastest folders typical folders slow folders 10-15 femto 10-12 pico 10-9 nano 10-6 micro 10-3 milli 100 seconds MD step long MD run where we need to be where we’d love to be

22. Folding@Home:Worldwide desktop grid computing ~150,000 CPUs over the world (CPU locations from IP address)

23. Markovian Model Method(Singhal et al. JCP 2004) • Generate molecular dynamics trajectories from transition path sampling or independently • Cluster nearby points into macrostates to build roadmap with also include transition time • Calculate the mean first passage time and Pfold using linear algebra

24. Step 1: sampling of paths • Pick a random point from current path • Shoot a path from this point • If path reaches initial or final state by some cutoff time, stop simulation and accept it • Define new current path

25. Step 2: Generation of roadmap • Nodes are accepted points, edges connect successive nodes • Cluster nearby points to make roadmap more connected • Calculate edge weights by counting number of transitions between nodes and normalize

26. Step 3 (opt): Re-weighting of edges • Can analyze roadmap at parameter values other than the simulated ones without need for additional simulations • For temperature, can re-weight edges by the relative probabilities at the two temperatures according to the dynamics • Renormalize edges so outgoing probability sums to one

27. Calculating Pfolds and MFPT • Equation for each node is conditioned on which neighbor it transitions to • One equation and one unknown for each node • Can be solved iteratively

28. Energy landscape and initial pathway • 2-D energy landscape • Initial and final regions defined by circles around the two minima • Initial paths generated by Monte Carlo or Langevin dynamics I F

29. Results - Pfold • Compare Pfold values to those from many direct simulations • Correlation coefficients are 0.99 for both

30. Results - MFPT • Compare MFPT at different temperatures to those from 10,000 direct simulations

31. Results – Trp zipper b-hairpin • Analyzed existing simulation data of a small, 12 residue, protein • 1750 trajectories, each 10 - 450 ns, resolution of 10 ns for non-folding and 250 ps for folding • Combine into roadmap • Depending on clustering cutoffs, MFPT = 2-9 ms • Agrees with experimental results of 2.47 ±0.05 ms and previous analysis of simulation data of 4.5 ms

32. Conclusions • Graphical methods produce a network of possible protein pathways • These networks can be efficiently analyzed to compute kinetic properties • Very fast method for looking at simple protein models or analyzing existing molecular dynamics data