1. Conformational Space of a Flexible Protein Loop Jean-Claude Latombe Computer Science Department Stanford University (Joint work with Ankur Dhanik 1, Guanfeng Liu 2, Itay Lotan 3, Henry van den Bedem 4, Jim Milgram 5, Nathan Marz 6, and Charles Kou 6 ) 1 Graduate student 2 Postdoc 3 Now a postdoc at U.C. Berkeley 4 Joint Center for Structural Genomics, Stanford Linear Accelerator Center 5 Department of Mathematics, Stanford University 6 Undergraduate CS students

2. Initial Project “Noise” in electron density maps from X-ray crystallography  4-20 aa fragments unresolved by existing software (RESOLVE, TEXTAL, ARP, MAID)  Model completion is high-throughput bottleneck

3. Fragment Completion Problem  Input: Electron-density map Partial structure Two “anchor” residues Amino-acid sequence of missing fragment  Output: Conformations of fragment that - Respect the closure constraint (IK) - Maximize match with electron-density map

4. Two-Stage Method [H. van den Bedem, I. Lotan, J.C. Latombe and A. M. Deacon. Real-space protein-model completion: An inverse-kinematics approach. Acta Crystallographica, D61:2-13, 2005.] 1.Candidate generations  Closed fragments 2.Candidate refinement  Optimize fit with EDM

5. Stage 1: Candidate Generation Loop: Generate random conformation of fragment (only one end is at its “anchor”) Close fragment – i.e., bring other end to second anchor – using Cyclic Coordinate Descent (CCD) [ A.A. Canutescu and R.L. Dunbrack Jr. Cyclic coordinate descent: A robotics algorithm for protein loop closure. Prot. Sci. 12:963–972, 2003]

6. Stage 2: Candidate Refinement  Target function T(Q) measuring quality of the fit with the EDM  Minimize T while retaining closure d3d3 d2d2 d1d1 (1,2,3)(1,2,3) Null space

7. Refinement Procedure Repeat until minimum is reached:  Compute a basis N of the null space at current Q (using SVD of Jacobian matrix)  Compute gradient  T of target function at current Q [Abe et al., Comput. Chem., 1984]  Move by small increment along projection of  T into null space (i.e., along dQ = NN T  T) + Monte Carlo + simulated annealing protocol to deal with local minima

8. Tests #1: Artificial Gaps  Complete structures (gold standard) resolved with EDM at 1.6Å resolution  Compute EDM at 2, 2.5, and 2.8Å resolution  Remove fragments and rebuild Long Fragments: 12: 96% < 1.0Å aaRMSD 15: 88% < 1.0Å aaRMSD Short Fragments: 100% < 1.0Å aaRMSD

9. Tests #2: True Gaps  Structure computed by RESOLVE  Gaps completed independently (gold standard)  Example: TM1742 (271 residues)  2.4Å resolution; 5 gaps left by RESOLVE LengthTop scorerLowest error 40.22Å 50.78Å 50.36Å 70.72Å0.66Å 100.43Å Produced by H. van den Bedem

10. TM1621  Green: manually completed conformation  Blue: conformation computed by stage 1  Pink: conformation computed by stage 2  The aaRMSD improved by 2.4Å to 0.31Å

11. A323 Hist A316 Ser Two-State Loop A B  TM0755: data at 1.8Å  8-residue fragment crystallized in 2 conformations  the EDM is difficult to interpret  Generate 2 conformations Q 1 and Q 2 using CCD  TH-EDM(Q 1,Q 2,  ) = theoretical EDM created by distribution  Q 1 + (1-  )Q 2  Maximize fit of TH-EDM(Q 1,Q 2,  ) with experimental EDM by moving in null space N(Q 1 )  N(Q 2 )  [0,1]

12. Status  Software running with Xsolve, JCSG’s structure-solution software suite  Used by crystallographers at JCSG for structure determination  Contributed to determining several structures recently deposited in PDB

13. Lesson  “Fuzziness” in EDM due to loop motion is not “noise”  Instead, it may be exploited to extract information on loop mobility

14. New 4-year NSF project (DMS-0443939, Bio-Math program)  Goal: Create a representation (probabilistic roadmap) of the conformation space of a protein loop, with a probabilistic distribution over this representation  Applications: Motion from X-ray crystallography Improvement of homology methods Predicting loop motion for drug design Conformation tweaking (MC optimization, decoy generation)

15. Predicting Loop Motion [J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. J. Comp. Chemistry, 25:956-967, 2004]

16. Ongoing Work 1.Develop software tools to create and manipulate loop conformations 2.Study the topological structure of a loop conformational space

17. Software tools implemented  CCD  Exact IK for 3 residues (non-necessarily contiguous)  Creation of loop conformations

18. Exact IK for 3 Residues [E.A. Coutsias, C. Seok, M.J. Jacobson, K.A. Dill. A Kinematic View of Loop Closure, J. Comp. Chemistry, 25(4):510 – 528, 2004] Maximal number of solutions: 10, 12?

19. Closing loops using CCD + Exact IK

21. Software tools implemented  CCD  Exact IK for 3 residues (non-necessarily contiguous)  Creation of loop conformations  Computation of pseudo-inverse of Jacobian and null-space basis  Loop deformation in null space  Conformation sampling

22. Moving an atom along a line

23. Interpolating between two conformations

24. Sampling many conformations

25. Software tools implemented  CCD  Exact IK for 3 residues (non-necessarily contiguous)  Creation of loop conformations  Computation of pseudo-inverse of Jacobian and null-space basis  Loop deformation in null space  Conformation sampling  Detection of steric clashes (grid method)

26. Topological Structure of Conformational Space  Inspired by work of Trinkle and Milgram on closed-loop kinematic chains  Leads to studying singularities of open protein chains and of their images

27. Configuration Space of a 4R Closed-Loop Chain [J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002] Rigid link Revolute joint l1l1 l2l2 l3l3 l4l4

28. Configuration Space of a 4R Closed-Loop Chain [J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002] l1l1 l2l2 l3l3 l4l4

29. Configuration Space of a 4R Closed-Loop Chain [J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002] Images of the singularities of the red linkage’s endpoint map: C   2

30. l1l1 Configuration Space of a 4R Closed-Loop Chain [J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002]

31. l1l1 Configuration Space of a 4R Closed-Loop Chain [J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002]

32. Configuration Space of a 5R Closed-Loop Chain IS1IS1 I  (S 1  S 1 ) S1|S1S1|S1 S1|S1S1|S1 Images of the singularities of the red linkage’s endpoint map: C   2

33. CC C N N How does it apply to a protein loop?

34. CC C N N

35. CC C N N

36. CC C N N Images of the singularities of the red linkage map: C   3  SO(3) 2D surface in  3  SO(3)

37. CC C N Kinematic Model ~60dg

38. Singularities of Map C  R 3  Rank 1 singularities: Planar linkage  Rank 2 singularities: Type 1 Type 2

39. Singularities of Map C  R 3  Rank 1 singularities: Planar linkage  Rank 2 singularities: Type 1 Type 2 Planar sub-linkages P0P0 Line contained in P 0

40. Singularities of Map C  R 3  Rank 1 singularities: Planar linkage  Rank 2 singularities: Type 1 Type 2 P0P0 P1P1 P2P2 There is a line L contained in P 2 to which P 0 and P 1 are // L Must be // to each other and // to last plane Endpoint is contained in all planes P 0, P 1, and P 2

41. Images of Singularities Singularities are on the periphery of the endpoint’s reachable space rank 1 singularity

42. Impact on Flexible Loops?