Download Inverse Kinematics - Structural Bioinformatics Course 2007

“Inverse Kinematics”
The Loop Closure Problem in Biology
Barak Raveh
Dan Halperin
Course in Structural Bioinformatics
Spring 2006
“Riddle” I
target
“Riddle” II
target
Outline
• Introduction
– The Loop Closure Problem in Proteins
– Direct & Inverse Kinematics
• Brief Linear Algebra of a Kinematic Chain
• Heuristics:
– Cyclic Coordinates Descent Algorithm
(“CCD”)
• Analytical Solution:
– Tripeptide Loop Closure
Loop Closure in Proteins
Want to fill in a
continuous
segment that is
the “loop” that
needs closing
Loop Closure in Proteins
How can we fill
in the gap?
Loop Closure
Loop closure
constraints
Loop Closure
Loop closure
constraints
The Goal of Loop Closure
The ultimate goal of the loop closure problem in
proteins is to find the ensemble of conformations that
can close a fixed gap within the backbone of a protein
using a certain number of amino acids
Loop Closure – When?
•
•
•
•
Protein Loop Design
Flexible Docking & Fold Prediction
Flexible Peptides
And more…
MHC Proteins & Immunology
• MHC (Major Histocompatability Proteins)
– class I
• on the membrane of every cell in our body
– class II
• On memory cells of immune system
• Human MHC = “HLA” (Human Leukocyte
Antigens)
MHC Proteins & Immunology
• MHC class I proteins present
small peptides to the immune
system
– A sample of each protein is digested
in the lysosome to small (8-16)
peptide chunks (“antigen”)
– The “antigen” binds MHC
– The complex transfers to the outer
surface of the cell membrane
– CD8+ T-Cells recognize the MHCpeptide complexes of invader
proteins (viruses, cancer cells, etc.)
MHC I Peptide Binding Domain
is Hyper-Variable
• 1000 possible alleles in
Human MHC (HLA) alone !
• 3-6 different alleles in each
individual
• Each allele binds different
peptides
• Evoloutianary protection of
populations
• Problems in Organ Transplant
MHC-Peptide binding
•~ 1000 MHC alleles
• Huge # of peptides
MHC “Cradle”
Loop Closure for Predicting MHCPeptide Binding
Outline
• Introduction
– The Loop Closure Problem in Proteins
– Direct & Inverse Kinematics
• Brief Linear Algebra of a Kinematic Chain
• Heuristics:
– Cyclic Coordinates Descent Algorithm
(“CCD”)
• Analytical Solution:
– Tripeptide Loop Closure
Kinematic Chains =
Chains of Rigid Links
Protein as Kinematic Chains
Direct Kinematics
• Where will the robot head move when we
change its degrees of freedom?
???
Go right
!!!
Inverse Kinematics
What values of DOFs will bring the robot tool
to the desired position and orientation ?
???
Take the
ball !!!
Inverse Kinematics in Robots
What values of DOFs will bring
the robot tool to the desired
position and orientation ?
Research Questions on
Inverse Kinematics
• Can we find a single solution to an
inverse kinematics problem?
• Can we find all solutions to an inverse
kinematics problem?
• How many solutions exist?
–0?
–1?
– Many ?
– infinite ?
Multiple Solutions
Loop Closure = Inverse
Kinematics
What set of Φ / Ψ angles will
close a certain peptide loop?
Outline
• Introduction
– The Loop Closure Problem in Proteins
– Direct & Inverse Kinematics
• Brief Linear Algebra of a Kinematic Chain
• Heuristics:
– Cyclic Coordinates Descent Algorithm (“CCD”)
• Analytical Solution:
– Tripeptide Loop Closure
Who are the Players?
Rigid Links connected by Joints
(Joints = Degrees of Freedom)
Dihedral angles
Affixing a Coordinate System
(“Frame”) to Each Link
Positions, orientations and
frames
• The position of a point p relative to a coordinate
system A (Ap):
 px 
 
A
p   py 
p 
 z
A
p
ZA
YA
XA
Mapping between Frams
We move from the frame (coordinates system)
of link i to that of link i+1 using a linear
transformation:
Rotation + Translation
A
P  R P  PBorg
A
B
B
A
A
ZB
p
B
p
A
pBorg
ZA
XB
YA
XA
YB
Examples of Rotation Matrices
“Homogenous Transform”:
Translation + Rotation using a single 4x4 Matrix
 A P   BA R


 1  0 0 0
A
A
B

P B T P
A
B


PBorg
P


1  1 
A
ZB
p
B
p
A
pBorg
ZA
XB
YA
XA
YB
Direct Kinematics
• Where will the robot head move when we
change its degrees of freedom?
???
Go right
!!!
Direct Kinematics =
Linear Algebra
We can move from the frame (coordinates
system) of link i to that of link i+k using
straightforward matrix multiplication
• Each transform can be written as a
combination of a translation and a rotation
• The single transformation that relates frame
{n} to frame {0}:
0
n
T T T
0
1
1
2
n 1
n
T
2D Example with Revolute Joints
Inverse Kinematics in Robots
What values of DOFs will bring
the robot head to correct
orientation?
Analytical Solution to
Inverse Kinematics =
Solving a set of
equations on a matrix
multiplication system
0
n
T 10 T 12T
n 1
n
T
So what is the problem?
Solving the set of equations is
usually unfeasible!
?
0
n
T T T
0
1
1
2
n 1
n
T
Outline
• Introduction
– The Loop Closure Problem in Proteins
– Direct & Inverse Kinematics
• Brief Linear Algebra of a Kinematic Chain
• Heuristics:
– Cyclic Coordinates Descent Algorithm
(“CCD”)
• Analytical Solution:
– Tripeptide Loop Closure
Cyclic Coordinate Descent =
Simple Greedy Heuristics
• Adjusting one link at a time
Tool’s current
position
minimize
Joint to
move
Goal’s
position
Start from Last Link
Cyclic Coordinate Descent
• starts at the last link, adjusting each joint
along the way
repeat until “satisfied”
Summary of CCD algorithm
•
While (“not satisfied”) and
(# of cycles < maximum):
adjust one DOF at a time (iterative) to
minimize tool’s distance to the goal, from
last link backwards
Cyclic Coordinate Descent
Advantages:
–
–
–
–
–
Allow constraints to be placed (at each step)
Free of singularities
Independent of DOFs # (degrees of freedom)
Extremely fast !
Simple to implement
• Disadvantage:
– Heuristics
• Might not find a solution even if one exists
• Does not cover all solutions
CCD for MHC-peptides
interaction
Outline
• Introduction
– The Loop Closure Problem in Proteins
– Direct & Inverse Kinematics
• Brief Linear Algebra of a Kinematic Chain
• Heuristics:
– Cyclic Coordinates Descent Algorithm (“CCD”)
• Analytical Solution:
– Tripeptide Loop Closure
– Generalization
“A Kinematic View of Loop
Closure”
Evangelos A. Coutsias,
Chaok Seok,
Matthew P. Jacobson,
Ken A. Dill
Tripeptide Loop Closure


With the base Cn 1  Cn 1 and
the lengths of the two peptide
virtual bonds fixed, the vertex
C n is constrained to lie on
a circle.
C
N

n
C
C'


Cn 1
Cn 1
Bond vectors
fixed in space
Fixed
distance
Fixed Distance between Cα Atoms
Tripeptide loop closure
• The six-torsion loop closure problem in
simplified representation:
variables: τi (i=1,2,3)
constraints: θi (i=1,2,3)
τ1
θ2
θ1
τ2
θ3
τ3
fixed in space
Constraints & Variables 
Set of Solvable Equations
We omit the details of the analytical solution but bottom line:
•Equations are quite complex
•They are solved using advanced techniques of linear algebra
(“resultants”)
Solving the equations
• We end up with a degree 16 polynomial
• Throretically, there might be up to 16
solutions to this polynomial
 16 = Upper bound on number of
solutions to each tripeptide loop closure
problem
• In practice, at most 10 real solutions has
been found in the article’s research
Summary
• Loop closure of peptides can help in key
challenges of computational biology
• Analytical Solutions exist only for a very
small number of DOFs (Degrees Of
Freedom)
• Efficient heuristics are not guaranteed to
find all solutions, or even a single solution
– But they work well in practice
Thank-You !