Inverse Kinematics

Inverse Kinematics (IK) T q1q1 q2q2 q3q3 q4q4 q5q5 Given a kinematic chain (serial linkage), the position/orientation of one end relative to the other (closed chain), find the values of the joint parameters rigid groups of atoms

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

Structure Determination X-Ray Crystallography

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  Manually completing a model: Labor intensive, time consuming Existing tools are highly interactive JCSG: 43% of data sets  2.3Å 1 Badger (2003) Acta Cryst. D59  Model completion is high-throughput bottleneck 1.0Å3.0Å

The Completion Problem  Input: Electron-density map Partial structure Two anchor residues Amino-acid sequence of missing fragment (typically 4 – 15 residues long)  Output: Few candidate conformation(s) of fragment that - Respect the closure constraint (IK) - Maximize match with electron-density map

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

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

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  helices and  strands), which often play a key role in: enzyme catalysis, ligand binding (induced fit), protein – protein interactions

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

Loop 7 of 1G5A Conformations obtained by deformation sampling

1K96

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  helices and  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.]

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

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

Relation to Robotics

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

Forward Kinematics d1d1 d2d2 11 22 (x,y) x = d 1 cos  1 + d 2 cos(  1 +  2 ) y = d 1 sin  1 + d 2 sin(  1 +  2 )

Inverse Kinematics d1d1 d2d2 11 22 (x,y)  2 = cos -1 x 2 + y 2 – d 1 2 – d 2 2 2d 1 d 2 -x(d 2 sin  2 ) + y(d 1 + d 2 cos  2 ) y(d 2 sin  2 ) + x(d 1 + d 2 cos  2 )  1 =

Inverse Kinematics d1d1 d2d2 (x,y)  2 = cos -1 x 2 + y 2 – d 1 2 – d 2 2 2d 1 d 2 -x(d 2 sin  2 ) + y(d 1 + d 2 cos  2 ) y(d 2 sin  2 ) + x(d 1 + d 2 cos  2 )  1 = Two solutions

More Complicated Example d1d1 d2d2 11 22 d3d3 (x,y) 33  Redundant linkage  Infinite number of solutions  Self-motion space

More Complicated Example d1d1 d2d2 11 22 d3d3 (x,y) 33 1-D space (self-motion space) d3d3 d2d2 d1d1 (1,2,3)(1,2,3)

More Complicated Example d1d1 d2d2 11 22 d3d3 33  No redundancy  Finite number of solutions (x,y,  ) d3d3 d2d2 d1d1 (1,2,3)(1,2,3)

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

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 =

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

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  -  joints 10 links N DOF = 4

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

IK Methods  Analytical (exact) techniques (only for 6 joints)  Write forward kinematics in the form of polynomial equations (use t = tan(  /2)  Simplify, e.g., using the fact that two consecutive torsional angles  and  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

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

 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 Decomposition Method for Randomly Sampling Conformations of Closed Chains 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

Application of Decomposition Method Amylosucrase

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

CCD (Cyclic Coordinate Descent) Method  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 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.

fixed end moving end Application of CCD to Proteins Closure Distance: Compute and move A.A. Canutescu and R.L. Dunbrack Jr. Cyclic coordinate descent: A robotics algorithm for protein loop closure. Prot. Sci. 12:963–972, 2003.

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

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

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

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

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 )

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

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 ] d q 1 +…+ [∂f i (Q)/∂ q n ] d q 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.

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

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 )

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

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

Computation of J + 1.SVD decomposition  J = U  V T where: - U in an 6  6 square orthonormal matrix - V is an n  6 square orthonormal matrix -  is of the form diag[  i ]: 2.J + = V  + U T where  + =diag[1/  i ] 11 22 66 0

dXU66U66 VT6nVT6n dQ 6666 = Getting Null space J

dXU66U66 VTnnVTnn dQ 6n6n 0 = Getting Null space J Gram-Schmidt orthogonalization

dXU66U66 VTnnVTnn dQ 6n6n 0 (n-6) basis N of null space = Getting Null space J NTNT

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)Computing Protein Structures from Electron Density Maps: The Missing Loop Problem.

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

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

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

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

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