1. Exploring Folding Landscapes with Motion Planning Techniques Bonnie Kirkpatrick Montana State University Dr. Nancy Amato Guang Song Xinyu Tang Texas A&M University Parallel Architectures, Algorithms, and Optimizations Laboratory

2. Outline Motivation: Biopolymers Goal: Folding Landscapes Method: Motion Planning Application 1: Protein Folding Application 2: RNA Folding

3. Motivation: Biopolymers

4. Protein and RNA Molecules Protein and RNA molecules are a complex 3-dimensional folding of a sequence of bases. Primary Structure – Sequence of bases – Each base is represented by a letter of the alphabet – i.e. ACGUGCCAUCG – Obtained from experiment Tertiary Structure – The sequence loops back on itself and folds in 3-dimensions. Tertiary representation of an RNA molecule.

5. Tertiary Structure Chemical bonds (or contacts) form between complementary bases in close proximity. There are many possible conformations of the primary sequence. – Example sequence: CACAGAGUGU – Two possible conformations are shown. Potential energy calculations based on number and types of bonds are used to classify conformations. – The lowest energy conformation is known as the native structure. – Conformations with few bonds and high energy are referred to as unfolded. Two possible conformations of the sequence. Bonds are blue.

6. Goal: Folding Landscapes

7. The Folding Process (The Black Box) AGGCUACUGGGAGCCUUCUCCCC Physical Laws cause folding Unfolded Conformation (high energy) Native Conformation (low energy)

8. Folding Landscapes Description of the “black box” A space in which every point corresponds to a conformation (or set of conformations) and its associated potential energy value (C-space). A complete folding landscape contains a point for every possible conformation of a given sequence. Tetrahymena Ribozyme Landscape [Russell, Zhuang, Babcock, Millett, Doniach, Chu, and Herschlag, 2002] Native State Conformation Space Potential

9. Folding Landscapes (cont.) Conformational changes describe how a molecule changes physically to fold from one conformation to another – Continuous Protein Folding Model Bond angles change with continuous rotations – Discrete RNA Folding Model Bonds either exist or do not exist

10. Native State Features of Folding Landscapes Folding pathways consist of the set of conformational changes a molecule is likely to fold though when moving from one conformation to another. – N to X to Y Energy barriers are areas of the landscape with high energy that separate groups of conformations. – Y is separated from X and N Intermediate states are conformations lying on the folding pathway represent local minimums of potential. – Y and X Mutant α mRNA fragment [Chen and Dill, 2000]

11. A Protein Folding Pathway unfoldedfolded

12. A RNA Folding Pathway Phenylalanine tRNA [Hofacker, 1998] Energy Barrier Native State Unfolded

13. Mapping Folding Landscapes Existing techniques for mapping landscapes are limited to relatively short sequences (~200 nucleotides). A robotics motion planning technique called PRM has successfully been applied to protein folding.

14. Method: Motion Planning

15. Motion Planning start goalobstacles (Basic) Motion Planning (in a nutshell): Given a movable object, find a sequence of valid configurations that moves the object from the start to the goal. Motion Planning for Foldable Objects: Given a foldable object, find a valid folding sequence that transforms the object from one folded state to another.

16. Native state Construct the roadmap: 1. Generate nodes. 2. Connect to form roadmap The Roadmap is like a net being laid down on protein’s potential landscape. A conformation Conformation space Potential Now the roadmap can be used: 1.To find a path 2.To extract multiple paths Probabilistic Roadmap Method (PRM) [Kavraki, Svestka, and Latombe, 1996]

17. Application 1: Protein Folding

18. Outline Probabilistic Roadmap Methods (PRMs) for Protein Folding – the native fold is known – bias sampling around native fold Results – Protein folding landscapes – Secondary structure formation order and validation timed contact map – Folding kinetics

19. Model of a protein [Song and Amato, 2001] amino acid: pair of phi/psi angles protein: a sequence of amino acids. – conformation node is :

20. PRM: Node Generation N Take advantage of the known native state. – map the potential landscape/funnel leading to it. – sample around it and gradually grow out. – generate conformations by randomly selecting phi/psi angles Criterion for accepting a node: Compute potential energy E of each node and retain it with probability P(E):

21. PRM: Node Generation Start with native structure. Gradually grow out. Denser distribution around native state Native state

22. PRM: Roadmap Connection 1.Find k closest nodes for each roadmap node 2.Assign edge weight to reflect energetic feasibility: lower weight  more feasible [Singh, Latombe, Brutlag, 1999] Native state

23. Energy Computation where Potential ( ref. Levitt’83 ) – van der Waals + hydrogen bonds + disulphide bonds + hydrophobic effect – All-atom model Free Energy ( ref. Fiebig & Dill ’93, Munoz & Eaton’99)

24. PRMs for Protein Folding: Key Issues Validation – In RECOMB ‘01 (Song & Amato), our results validated with hydrogen exchange experiments. [Li & Woodward 1999] Energy Functions – The degree to which the roadmap accurately reflects folding landscape depends on the quality of energy calculation.

25. Analysis of Landscape Folding Potential Landscape Secondary structure formation order – timed contact map – experimental validation Studying Folding Kinetics – 2-state folding kinetics – calculation of folding rates – identifying 2-state, 3-state, … k-state kinetics

26. Distributions for different types: Potential Energy vs. RMSD for roadmap nodes all alphaalpha + betaall beta

27. Timed Contact Map: formation order for a Path protein G (domain B1) (IV:  1-4 ) 140 143 140 143 140 141 142 144 139 143 143 114 142 135 131   1-4  3-4 Average T = 142 Formation order: ,  3-4,  1-2,  1-4 residue # 1-2 

28. Validating Folding Pathways Protein GB1 (56 amino acids) — 1 alpha helix & 4 beta-strands Hydrogen Exchange Results first helix, and beta 3-4 Our Paths 80%: helix, beta 3-4, beta 1-2, beta 1-4 20%: helix, beta 1-2, beta 3-4, beta 1-4 [Li & Woodward 1999] our paths are: from all the nodes with little structure to the native state secondary structure formation order checked on each path w/ timed contact map

29. Secondary structure formation order and validation Proteins primarily from [Munoz & Eaton PNAS’99] for comparison purposes Contact us if you want us to analyze your proteins! PDB nameNum of Residues 2 nd structuresComparison w/ Exp. [Li & Woodward ’99] 1GB1561 alpha + 4 betaAgreed 1BDD603 alphaAgreed 1SHG625 betaN/a 1COA641 alpha + 4betaAgreed 1SRL645 betaN/a 1CSP677 betaN/a 1NYF673 betaN/a 1MJC697 betaN/a 2AIT747 betaN/a 1UBQ761alpha + 5 betaAgreed 1PKS791 alpha + 5 betaN/a 1PBA813 alpha + 3 betaN/a 2ABD865 alphaN/a 1BRN1103 alpha + 7 betaNot sure

30. Folding kinetics: statistical mechanical model Proteins treated as statistical system Define interactions => partition function Free energy as a function of reaction coordinate (R) Then decide folding kinetics and folding rate [Munoz & Eaton, Alm & Baker. PNAS’99] Assumption and limitation of statistical model – Very limited interactions to simplify partition function calculation – Assume selected reaction coordinate good (monotonically increasing) – Cannot provide folding trajectories Strength: as a theoretical model, it is good for analysis R F U N

31. Free Energy Landscape: 2-state folding kinetics Our method can produce similar results (plus more). – 2-state folding kinetics indicated Both plots ‘imply’ nativelikeness should always increase Plots like these lose info because of averaging effect statistical model [Munoz & Eaton PNAS’99] Native Contacts Free Energy our roadmap model Blue line: free energy average Blue, red and green lines are three levels of approximation

32. Studying folding kinetics at pathway level A B cluster paths into several groups extract 2-state,3-state, …, k- state kinetics from same roadmap not possible with statistical mechanical models AB Average 2-state 3-state 2-state

33. Studying folding kinetics at pathway level Native contacts Free energy trajectories not available from statistical model native contact not monotonically increasing Diverse free energy profiles Protein G

34. Protein Folding: Conclusion & Future Work PRM roadmaps approximate folding landscapes Efficiently produce multiple folding pathways –Secondary structure formation order –better than trajectory-based simulation methods, such as Monte Carlo, molecular dynamics Provide a good way to study folding kinetics –multiple folding kinetics in same landscape (roadmap) –natural way to study the statistical behavior of folding –more realistic than statistical models (e.g. Lattice models, Baker’s model PNAS’99, Munoz’s model, PNAS’99 )

35. Application 2: RNA Folding

36. Outline RNA Model using secondary structure Conformation Space Node Generation Node Evaluation Node Connection Edge Weights

37. Ribonucleic Acid (RNA) Bases in RNA are: – Adenine (A) – Cytosine (C) – Guanine (G) – Uracil (U) Base pair interactions (a.k.a. contact pairs) – Watson-Crick Pairs: A, U G, C – Wobble Pair G, U

38. RNA Secondary Structure Two-dimensional representation of the tertiary structure Planar representation Sufficient structural information Pseudo knots are considered a tertiary structure, rather than a secondary structure

39. Violates criteria (2) Violates criteria (1) Secondary Structure Formalized A secondary structure conformation is specified by a set of intra-chain contacts (bonds between base pairs) that follow certain rules. Given any two intra-chain contacts [i, j] with i < j and [k, l] with k < l, then: 1) If i = k, then j = l Each base can appear in only one contact pair 2) If k < j, then i < k < l < j No pseudo-knots

40. Representations of RNA Planar Graph – M: Multi-loop – I: Internal-loop – B: Bulge-loop – H: hairpin-loop –: W-C pairs – -: GU pairs

41. Representations (cont.) Hydrogen bonds between intra-chain pairs are represented by circular arcs All representations are equivalent

42. Representations (cont.) Contact Map A dot is placed in the i th row and jth column of a triangular array to represent the intra- chain contact [i, j]

43. PRM: Conformation Space Let U be the set of every possible combination of contact pairs. Let C-space (the conformation space), C, be the sub-set of U containing only valid secondary structures. C-space is smaller than U, but is still very large. – Sequence: (ACGU) 10 – Length: 40 nucleotides – C-Space: 1.6x10 8 structures Purpose: generate nodes in C-space that describe the space without covering it C-space Where n is the number of possible contact pairs

44. PRM: Node Generation Random Node Generation Algorithm – Starting with an empty configuration, c, random contacts are added to c one at a time. – Each step preserves the condition that c contains a valid set of base pair contacts. – Contacts are added until there are no more contacts that do not conflict with the contact set of c. Every node generated has valid secondary structure and is a member of C-space. Since every generated node has the maximal number of contacts, the sampling is biased toward the area of C-space near the native state. C-space

45. PRM: Node Evaluation Evaluation of Nodes – Potential energy determines how good a node is. – Only add a node to the roadmap if it has a low energy. – Probability of a node q being added to the roadmap:

46. PRM: Node Connection Given two nodes in C-Space, C 1 and C 2, find a path between them consisting of a sequence of nodes: { C 1 = S 1, S 2, …, S n-1, S n = C 2 } The path must have the property that for each i, 1 < i < n, the set of contact pair of S i differs from that of S i-1 by the application of one transformation operation: (1) open or (2) close a single contact pair. C 1 = S 1 S2S2 S n-1 S n = C 2 …

47. Node Connection (cont.) There exists a path between any two nodes in C-Space. Not just any path will do; we want a good one. Bad paths have high energy nodes in them. How do we find the lowest energy path?

48. Node Connection (cont.) more contacts  less potential energy Heuristic: if a contact is opened by the transition from one node to another, try to close a contact in the next transition c1 = s1:..(.((..))).. open s2:..(.(....)).. close s3:..(.((.).)).. open s4:..(..(.)..).. close C2 = s5:..(.((.)).)..

49. Edge Weight Depends on the nodes generated in the node connection phase. Difference in potential energy ΔE i = E(s i+1 ) – E(s i )

50. Future Work Analysis of the roadmap – Finding the low energy folding pathways – Shortest path algorithm Validation – How do we know if our results agree with experimental results? – Proposal Compare a fully enumerated roadmap to experimental folding rates Compare a more sparse roadmap with a fully enumerated roadmap – Proposal Solve the master equation using stacking pairs (they are representative of all the dynamics) and our model Compare our results with results from the Zhang and Chen’s statistical mechanical model [2002]

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