Inverse Kinematics (IK)

Transcription

1 Inverse Kinematics

2 Inverse Kinematics (IK) Given a kinematic chain (serial linkage), the position/orientation of one end relative to the other q 4 q (closed chain), find the values 5 of the joint parameters q 2 q 3 q 1 rigid groups of atoms T

3 Why is IK useful for proteins? Filling gaps in structure determination by X- ray crystallography

4 Structure Determination X-Ray Crystallography

5 Automated Model Building Software systems: RESOLVE, TEXTAL, ARP/wARP, MAID 1.0Å < d < 2.3Å ~ 90% completeness 2.3Å d < 3.0Å ~ 67% completeness (varies widely) 1 1.0Å 3.0Å JCSG: 43% of data sets 2.3Å Manually completing a model: Labor intensive, time consuming Existing tools are highly interactive Model completion is high-throughput bottleneck 1 Badger (2003) Acta Cryst. D59

6 The Completion Problem Input: Electron-density map Partial structure Two anchor residues Amino-acid sequence of missing fragment (typically 4 15 residues long) Anchor 1 (3 atoms) Protein fragment (fuzzy map) Main part of protein (folded) Anchor 2 (3 atoms) Output: Few candidate conformation(s) of fragment that - Respect the closure constraint (IK) - Maximize match with electron-density map

7 Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best: 0.6Å aarmsd GLU-77 GLY-90

8 Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best 0.6Å aarmsd GLU-77 GLY-90

9 Why is IK useful for proteins? Filling gaps in structure determination by X-ray crystallography Studying the motion space of loops (secondary structure elements connecting α helices and β strands), which often play a key role in: enzyme catalysis, ligand binding (induced fit), protein protein interactions

10 Loop motion in Amylosucrase 17-residue loop that plays important role in protein s activity

11 Loop 7 of 1G5A Conformations obtained by deformation sampling

12 1K96

13 Why is IK useful for proteins? Filling gaps in structure determination by X-ray crystallography Studying the motion space of loops (secondary structure elements connecting α helices and β strands), which often play a key role in: enzyme catalysis, ligand binding (induced fit), protein protein interactions Sampling conformations using homology modeling Chain tweaking for better prediction of folded state R. [Singh and B. Berger. ChainTweak: Sampling from the Neighbourhood of a Protein Conformation. Proc. Pacific Symposium on Biocomputing, 10:52-63, 2005.]

14 Generic Problem Definition Inputs: Protein structure with missing fragment(s) (typically 4 15 residues long, each) Amino-acid sequence of each missing fragment Outputs: Conformation of fragment or distribution of conformations that Respect the closure constraint (IK) Avoid atomic clashes Satisfy other constraints, e.g., maximize match with electron density map, minimize energy function, etc

15 Inputs: Closed kinematic chain with n degrees of freedom Relative positions/orientations X of end frames Target function T(Q) Ρ Outputs: Conformation(s) that Achieve closure OptimizeT IK Problem T

16 Relation to Robotics

17 Some Bibliographical References Robotics/Computer Science Exact IK solvers Manocha & Canny 94 Manocha et al. 95 Optimization IK solvers Wang & Chen 91 Redundant manipulators Khatib 87 Burdick 89 Motion planning for closed loops Han & Amato 00 Yakey et al. 01 Cortes et al. 02, 04 Biology/Crystallography Exact IK solvers Wedemeyer & Scheraga 99 Coutsias et al. 04 Optimization IK solvers Fine et al. 86 Canutescu & Dunbrack Jr. 03 Ab-initio loop closure Fiser et al. 00 Kolodny et al. 03 Database search loop closure Jones & Thirup 86 Van Vlijman & Karplus 97 Semi-automatic tools Jones & Kjeldgaard 97 Oldfield 01

18 Forward Kinematics θ 2 d 1 d 2 (x,y) θ 1 x = d 1 cos θ 1 + d 2 cos(θ 1 +θ 2 ) y = d 1 sin θ 1 + d 2 sin(θ 1 +θ 2 )

19 Inverse Kinematics θ 2 d 1 θ 1 d 2 (x,y) θ 2 = cos -1 x 2 + y 2 d 12 d 2 2 2d 1 d 2 θ 1 = -x(d 2 sinθ 2 ) + y(d 1 + d 2 cosθ 2 ) y(d 2 sinθ 2 ) + x(d 1 + d 2 cosθ 2 )

20 Inverse Kinematics d 1 d 2 (x,y) θ 2 = cos -1 x 2 + y 2 d 12 d 2 2 2d 1 d 2 θ 1 = -x(d 2 sinθ 2 ) + y(d 1 + d 2 cosθ 2 ) y(d 2 sinθ 2 ) + x(d 1 + d 2 cosθ 2 ) Two solutions

21 More Complicated Example θ 2 d 2 d 3 (x,y) d 1 θ 3 θ 1 Redundant linkage Infinite number of solutions Self-motion space

22 More Complicated Example θ 2 d 2 d 3 (x,y) d 1 θ 3 θ 1 (θ 1,θ 2,θ 3 ) dθ 3 dθ 2 dθ 1 1-D space (self-motion space)

23 More Complicated Example θ 2 d 2 d 3 (x,y,φ) d 1 θ3 θ 1 No redundancy Finite number of solutions dθ 3 dθ 2 (θ 1,θ 2,θ 3 ) dθ 1

24 General Results from Kinematics Number of DOFs of a linkage (dimensionality of velocity space): N DOF = k (N link 1) (k 1) N joint where k = 3 if the linkage is planar and k = 6 if it is in 3-D space (Grübler formula, 1883). Examples: - Open chain: N joint = N link 1 N DOF = N joint - Closed chain: N joint = N link N DOF = N joint k N link = 4 N joint = 3 N DOF = 3(4-1)-(3-1)3 = 3 N link = 4 N joint = 4 N DOF = 1

25 General Results from Kinematics Number of DOFs of a linkage (dimension of velocity space): N DOF = k (N link 1) (k 1) N joint where k = 3 if the linkage is planar and k = 6 if it is in 3-D space (Grübler formula, 1883). Examples: - Open chain: N joint = N link 1 N DOF = N joint - Closed chain: N joint = N link N DOF = N joint k N link = 4 N joint = 3 N DOF = 3(4-1)-(3-1)3 = 3 N link = N joint = N DOF =

26 General Results from Kinematics Number of DOFs of a linkage (dimension of velocity space): N DOF = k (N link 1) (k 1) N joint where k = 3 if the linkage is planar and k = 6 if it is in 3-D space (Grübler formula, 1883). Examples: - Open chain: N joint = N link 1 N DOF = N joint - Closed chain: N joint = N link N DOF = N joint k N link = 4 N joint = 3 N DOF = 3(4-1)-(3-1)3 = 3 N link = 3 N joint = 3 N DOF = 0

27 General Results from Kinematics Number of DOFs of a linkage (dimension of velocity space): N DOF = k (N link 1) (k 1) N joint where k = 3 if the linkage is planar and k = 6 if it is in 3-D space (Grübler formula, 1883). Examples: - Open chain: N joint = N link 1 N DOF = N joint - Closed chain: N joint = N link N DOF = N joint k 5 amino-acids 10 φ-ψ joints 10 links N DOF = 4

28 General Results from Kinematics 6-joint chain in 3-D space: N DOF =0 At most 16 distinct IK solutions

29 IK Methods Analytical (exact) techniques (only for 6 joints) Write forward kinematics in the form of polynomial equations (use t = tan(θ/2) Simplify, e.g., using the fact that two consecutive torsional angles φ and ψ have intersecting axes [Coutsias, Seck, Jacobson, Dill, 2004] Solve E.A. Coutsias, C. Seok, M.P. Jacobson, and K.A. Dill. A Kinematic View of Loop Closure. J. Comp. Chemistry, 25: , 2004

30 Decomposition Method for Randomly Sampling Conformations of Closed Chains Decompose closed chain into: 6 passive joints n-6 active joints

31 Decomposition Method for Randomly Sampling Conformations of Closed Chains Decompose closed chain into: 6 passive joints n-6 active joints Sample the active joint parameters Compute the passive joint parameters using exact IK solver J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. Geometric Algorithms for the Conformational Analysis of Long Protein Loops. J. Comp. Chemistry, 25: , 2004

32 Application of Decomposition Method Amylosucrase

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34 IK Methods Analytical (exact) techniques (only for 6 joints) Write forward kinematics in the form of polynomial equations (use t = tan(θ/2) Simplify, e.g., using the fact that two consecutive torsional angles φ and ψ have intersecting axes [Coutsias, Seck, Jacobson, Dill, 2004] Solve Iterative (approximate) techniques

35 CCD (Cyclic Coordinate Descent) Generate random conformation with one end of chain at required position/orientation Repeat until other end is at required position/orientation or algorithm is stuck at local minimum Pick one DOF Change to minimize closure distance Method L.T. Wang and C.C. Chen. A Combined Optimization Method for Solving the Inverse Kinematics Problem of Mechanical Manipulators. IEEE Tr. On Robotics and Automation, 7: , 1991.

36 Application of CCD to Proteins Closure Distance: fixed end moving end S = N N + Cα Cα + C C A.A. Canutescu and R.L. Dunbrack Jr. Cyclic coordinate descent: A robotics algorithm for protein loop closure. Prot. Sci. 12: , S Compute qi s.t. = 0 and move q i

37 Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best: 0.6Å aarmsd GLU-77 GLY-90

38 Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best: 0.6Å aarmsd GLU-77 GLY-90

39 Advantages of CCD Simplicity No singularity problem Possibility to constrain each joint independent of all others But may get stuck at local minima!

40 CCD with Ramachandran Maps Ramachandran maps assign probabilities to φ-ψ pairs ψ φ

41 CCD with Ramachandran Maps Ramachandran maps assign probabilities to φ-ψ pairs Change a pair (φ i,ψ i ) at each iteration: Compute change to φ i Compute change to ψ i based on change to φ i Accept with probability min(1,p new /P old )

42 IK Methods Analytical (exact) techniques (only for 6 joints) Write forward kinematics in the form of polynomial equations (use t = tan(θ/2) Simplify, e.g., using the fact that two consecutive torsional angles φ and ψ have intersecting axes [Coutsias, Seck, Jacobson, Dill, 2004] Solve Iterative (approximate) techniques

43 Jacobian Matrix Q: n-vector of internal coordinates X: 6-vector defining endpoint s position/orientation n 6 Forward kinematics: X = F(Q) dx i = [ f i (Q)/ q 1 ] dq [ f i (Q)/ q n ] dq n dx = J dq Efficient algorithm to compute Jacobian: K.S. Chang and O. Khatib. Operational Space Dynamics: Efficient Algorithms for Modeling and Control of Branching Mechanisms. IEEE Int. Conf. on Robotics and Automation (ICRA),pp , Sand Francisco, April 2000.

44 Jacobian Matrix J f 1 (Q)/ q 1 f 1 (Q)/ q 2 f 1 (Q)/ q n f 2 (Q)/ q 1 f 2 (Q)/ q 2 f 2 (Q)/ q n f 6 (Q)/ q 1 f 6 (Q)/ q 2 f 6 (Q)/ q n

45 Case where n = 6 J is a square 6x6 matrix. Problem: Given X, find Q such that X= F(Q) Start at any X 0 = F(Q 0 ) Method: 1. Interpolate linearly between X 0 and X sequence X 1, X 2,, X p = X 2. For i = 1,,p do a) Q i = Q i-1 + J -1 (Q i-1 )(X i -X i-1 ) b) Reset Xi to F(Q i )

46 Case where n > 6 dx = J dq J is an 6 n matrix. Assume rank(j) = 6. Null space { dq 0 J dq 0 = 0} has dim = n - 6

47 Case where n > 6 dx = J dq J is an 6 n matrix. Assume rank(j) = 6 Find J + (pseudo-inverse) such that JJ + = I dq = J + dx Null space { dq 0 J dq 0 = 0} has dim = n - 6 dq = J + dx + dq 0 arbitrarily chosen in null space

48 Computation of J + 1. SVD decomposition J = U Σ V T where: - U in an 6 6 square orthonormal matrix - V is an n 6 square orthonormal matrix -Σis of the form diag[σ i ]: σ 1σ2 σ J + = V Σ + U T where Σ + =diag[1/σ i ]

49 Getting Null space J dx U 6 6 Σ V T n dq =

50 Getting Null space J dx U 6 6 Σ V T 6 n n n dq = 0 Gram-Schmidt orthogonalization

51 Getting Null space J dx U 6 6 Σ V T 6 n n n dq = 0 N T (n-6) basis N of null space

52 Minimization of Target Function T with Closure when n > 6 Input: Chain with both ends at goal positions and orientations Repeat 1. Compute Jacobian matrix J at current q 2. Compute null-space basis N using SVD of J 3. Compute gradient T(q) 4. Move along projection NN T y of y=- T(q) onto N until minimum is reached or closure is broken New q I. Lotan, H. van den Bedem, A.M. Deacon and J.-C Latombe. Computing Protein Structures from Electron Density Maps: The Missing Loop Problem. Proc. 6th Workshop on Algorithmic Foundations of Robotics (WAFR `04)

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54 Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best: 0.6Å aarmsd GLU-77 GLY-90

55 Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best: 0.6Å aarmsd GLU-77 GLY-90

56 Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best 0.6Å aarmsd GLU-77 GLY-90

57 TM1621 Green: manually completed conformation Cyan: conformation computed by stage 1 Magenta: conformation computed by stage 2 The aarmsd improved by 2.4Å to 0.31Å Produced by H. van den Bedem

58 Multi-Modal Loop A323 Hist A316 Ser Produced by H. van den Bedem