Inverse Kinematics
Inverse Kinematics (IK)
Given a kinematic chain (serial linkage),
the position/orientation of one
q4
end relative to the other
(closed chain), find the values
of the joint parameters
q2
q5
q3
q1
rigid groups
of atoms
T
Why is IK useful for proteins?
 Filling gaps in structure determination by Xray 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
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
1Badger
(2003) Acta Cryst. D59
The Completion Problem
 Input:
Anchor 1
(3 atoms)
• Electron-density map
• Partial structure
• Two anchor residues
• Amino-acid sequence of
missing fragment
(typically 4 – 15 residues long)
Anchor 2
(3 atoms)
Protein fragment (fuzzy map)
Main part of protein (folded)
 Output:
• Few candidate conformation(s) of fragment that
- Respect the closure constraint (IK)
- Maximize match with electron-density map
Example: TM0813
PDB: 1J5X, 342 res.
2.8Å resolution
12 residue gap
Best: 0.6Å aaRMSD
GLU-77
GLY-90
Example: TM0813
PDB: 1J5X, 342 res.
2.8Å resolution
12 residue gap
Best 0.6Å aaRMSD
GLU-77
GLY-90
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 a helices and b 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 a helices and b 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
IK Problem
 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
T
Relation to Robotics
Some Bibliographical References
Biology/Crystallography
Robotics/Computer Science
•
–
–
Manocha & Canny ’94
Manocha et al. ’95
–
Wang & Chen ’91
–
–
Khatib ’87
Burdick ’89
–
–
–
Han & Amato ’00
Yakey et al. ’01
Cortes et al. ’02, ’04
Optimization IK solvers
•
Redundant manipulators
Motion planning for closed loops
Exact IK solvers
–
–
Exact IK solvers
•
•
•
•
Optimization IK solvers
–
–
•
Fiser et al. ’00
Kolodny et al. ’03
Database search loop closure
–
–
•
Fine et al. ’86
Canutescu & Dunbrack Jr. ’03
Ab-initio loop closure
–
–
•
Wedemeyer & Scheraga ’99
Coutsias et al. ’04
Jones & Thirup ’86
Van Vlijman & Karplus ’97
Semi-automatic tools
–
–
Jones & Kjeldgaard ’97
Oldfield ’01
Forward Kinematics
q2
d2
d1
q1
(x,y)
x = d1 cos q1 + d2 cos(q1+q2)
y = d1 sin q1 + d2 sin(q1+q2)
Inverse Kinematics
q2
d2
d1
(x,y)
q2 = cos-1
q1
q1 =
x2 + y2 – d12 – d22
2d1d2
-x(d2sinq2) + y(d1 + d2cosq2)
y(d2sinq2) + x(d1 + d2cosq2)
Inverse Kinematics
d2
d1
(x,y)
q2 = cos-1
q1 =
Two solutions
x2 + y2 – d12 – d22
2d1d2
-x(d2sinq2) + y(d1 + d2cosq2)
y(d2sinq2) + x(d1 + d2cosq2)
More Complicated Example
q2
d2
(x,y)
d3
q3
d1
q1
 Redundant linkage
 Infinite number of solutions
 Self-motion space
More Complicated Example
q2
d2
(x,y)
d3
q3
d1
q1
dq3
(q1,q2,q3)
dq2
dq1
1-D space
(self-motion
space)
More Complicated Example
q2
d2
(x,y,f)
d3
q3
d1
q1
dq3
(q1,q2,q3)
 No redundancy
 Finite number of solutions
dq2
dq1
General Results from Kinematics
 Number of DOFs of a linkage (dimensionality of
velocity space):
NDOF = k(Nlink – 1) – (k–1)Njoint
where k = 3 if the linkage is planar and k = 6 if it
is in 3-D space (Grübler formula, 1883).
 Examples:
- Open chain: Njoint = Nlink – 1  NDOF = Njoint
- Closed chain: Njoint = Nlink  NDOF = Njoint – k
Nlink = 4
Njoint = 3
NDOF = 3(4-1)-(3-1)3 = 3
Nlink = 4
Njoint = 4
NDOF = 1
General Results from Kinematics
 Number of DOFs of a linkage (dimension of
velocity space):
NDOF = k(Nlink – 1) – (k–1)Njoint
where k = 3 if the linkage is planar and k = 6 if it
is in 3-D space (Grübler formula, 1883).
 Examples:
- Open chain: Njoint = Nlink – 1  NDOF = Njoint
- Closed chain: Njoint = Nlink  NDOF = Njoint – k
Nlink = 4
Njoint = 3
NDOF = 3(4-1)-(3-1)3 = 3
Nlink =
Njoint =
NDOF =
General Results from Kinematics
 Number of DOFs of a linkage (dimension of
velocity space):
NDOF = k(Nlink – 1) – (k–1)Njoint
where k = 3 if the linkage is planar and k = 6 if it
is in 3-D space (Grübler formula, 1883).
 Examples:
- Open chain: Njoint = Nlink – 1  NDOF = Njoint
- Closed chain: Njoint = Nlink  NDOF = Njoint – k
Nlink = 4
Njoint = 3
NDOF = 3(4-1)-(3-1)3 = 3
Nlink = 3
Njoint = 3
NDOF = 0
General Results from Kinematics
 Number of DOFs of a linkage (dimension of
velocity space):
NDOF = k(Nlink – 1) – (k–1)Njoint
where k = 3 if the linkage is planar and k = 6 if it
is in 3-D space (Grübler formula, 1883).
 Examples:
- Open chain: Njoint = Nlink – 1  NDOF = Njoint
- Closed chain: Njoint = Nlink  NDOF = Njoint – k
5 amino-acids
10 f-y joints
10 links
NDOF = 4
General Results from Kinematics
6-joint chain in 3-D space:
 NDOF=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(q/2)
 Simplify, e.g., using the fact that two
consecutive torsional angles f and y 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, 25:510-528, 2004
Decomposition Method for Randomly
Sampling Conformations of Closed Chains
Decompose closed chain into:

•
•
6 “passive” joints
n-6 “active” joints
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,
25:956-967, 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(q/2)
 Simplify, e.g., using the fact that two
consecutive torsional angles f and y 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, 7:489-498, 1991.
Application of CCD to Proteins
Closure Distance: S  N  N  Ca  Ca  C  C
2
moving end
2
2
A.A. Canutescu and R.L. Dunbrack Jr.
Cyclic coordinate descent: A robotics
algorithm for protein loop closure.
Prot. Sci. 12:963–972, 2003.
fixed end
S
 0 and move
Compute qi s.t.
qi
Example: TM0813
PDB: 1J5X, 342 res.
2.8Å resolution
12 residue gap
Best: 0.6Å aaRMSD
GLU-77
GLY-90
Example: TM0813
PDB: 1J5X, 342 res.
2.8Å resolution
12 residue gap
Best: 0.6Å aaRMSD
GLU-77
GLY-90
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 ψ
φ
CCD with Ramachandran Maps
 Ramachandran maps assign probabilities
to φ-ψ pairs
 Change a pair (φi,ψi) at each iteration:
 Compute change to φi
 Compute change to ψi based on change to φi
 Accept with probability min(1,Pnew/Pold)
IK Methods
 Analytical (exact) techniques
(only for 6 joints)
 Write forward kinematics in the form of
polynomial equations (use t = tan(q/2)
 Simplify, e.g., using the fact that two
consecutive torsional angles f and y 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)
 dxi = [∂fi(Q)/∂q1] dq1 +…+ [∂fi(Q)/∂qn] dqn
 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. 850-856, Sand Francisco, April 2000.
Jacobian Matrix J
∂f1(Q)/∂q1 ∂f1(Q)/∂q2
∂f2(Q)/∂q1 ∂f2(Q)/∂q2
…
…
∂f6(Q)/∂q1 ∂f6(Q)/∂q2
…
…
∂f1(Q)/∂qn
∂f2(Q)/∂qn
…
∂f6(Q)/∂qn
Case where n = 6




J is a square 6x6 matrix.
Problem: Given X, find Q such that X= F(Q)
Start at any X0 = F(Q0)
Method:
1. Interpolate linearly between X0 and X
 sequence X1, X2, …, Xp = X
2. For i = 1,…,p do
a) Qi = Qi-1 + J-1(Qi-1)(Xi-Xi-1)
b) Reset Xi to F(Qi)
Case where n > 6



dX = J dQ
J is an 6n matrix. Assume rank(J) = 6.
Null space { dQ0 | J dQ0 = 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 { dQ0 | J dQ0 = 0} has dim = n - 6
dQ = J+dX + dQ0
arbitrarily chosen
in null space
Computation of
1.
+
J
SVD decomposition  J = U S VT where:
- U in an 66 square orthonormal matrix
- V is an n6 square orthonormal matrix
- S is of the form diag[si]:
s1
s2
0
s6
2. J+ = V S+ UT where S+=diag[1/si]
Getting Null space
J
dX
U66
=
S66
VT6n
dQ
Getting Null space
J
dX
U66
VTnn
S6n
=
Gram-Schmidt orthogonalization
0
dQ
Getting Null space
J
dX
U66
=
VTnn
S6n
0
NT
(n-6) basis N of null space
dQ
Minimization of Target Function
T with Closure when n > 6
Input: Chain with both ends at goal positions
and orientations
Repeat
1.
2.
3.
4.
Compute Jacobian matrix J at current q
Compute null-space basis N using SVD of J
Compute gradient T(q)
Move along projection NNTy 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)
Example: TM0813
PDB: 1J5X, 342 res.
2.8Å resolution
12 residue gap
Best: 0.6Å aaRMSD
GLU-77
GLY-90
Example: TM0813
PDB: 1J5X, 342 res.
2.8Å resolution
12 residue gap
Best: 0.6Å aaRMSD
GLU-77
GLY-90
Example: TM0813
PDB: 1J5X, 342 res.
2.8Å resolution
12 residue gap
Best 0.6Å aaRMSD
GLU-77
GLY-90
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
A323
Hist
A316
Ser
Produced by H. van den Bedem