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“Inverse Kinematics” The Loop Closure Problem in Biology Barak Raveh Dan Halperin Course in Structural Bioinformatics Spring 2006
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“Riddle” I target
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“Riddle” II target
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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
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Loop Closure in Proteins Want to fill in a continuous segment that is the “loop” that needs closing
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Loop Closure in Proteins How can we fill in the gap?
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Loop Closure Loop closure constraints
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Loop Closure Loop closure constraints
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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
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Loop Closure – When? Protein Loop Design Flexible Docking & Fold Prediction Flexible Peptides And more…
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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)
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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 MHC- peptide complexes of invader proteins (viruses, cancer cells, etc.)
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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
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MHC-peptide binding MHC “Cradle” ~ 1000 MHC alleles Huge # of peptides
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Loop Closure for Predicting MHC- Peptide Binding
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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
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Kinematic Chains = Chains of Rigid Links
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Protein as Kinematic Chains
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Direct Kinematics Where will the robot head move when we change its degrees of freedom? Go right !!! ???
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Inverse Kinematics How can we move the robot head to a certain location at a certain orientation? Take the ball !!! ???
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Inverse Kinematics in Robots What values of DOFs will bring the robot tool to the desired position and orientation?
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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 ?
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Multiple Solutions
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Loop Closure = Inverse Kinematics What set of Φ / Ψ angles will close a certain peptide loop?
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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
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Who are the Players? Rigid Links connected by Joints (Joints = Degrees of Freedom) Dihedral angles
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Affixing a Coordinate System (“Frame”) to Each Link
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Positions, orientations and frames The position of a point p relative to a coordinate system A ( A p):
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We move from the frame (coordinates system) of link i to that of link i+1 using a linear transformation: Rotation + Translation Mapping between Frams
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Examples of Rotation Matrices
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“Homogenous Transform”: Translation + Rotation using a single 4x4 Matrix
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Direct Kinematics Where will the robot head move when we change its degrees of freedom? Go right !!! ???
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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}:
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2D Example with Revolute Joints
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Inverse Kinematics in Robots What values of DOFs will bring the robot tool to the desired position and orientation? Analytical Solution to Inverse Kinematics = Solving a set of equations on a matrix multiplication system
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So what is the problem? Solving the set of equations is usually infeasible! ?
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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
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Cyclic Coordinate Descent = Simple Greedy Heuristics Adjusting one link at a time Tool’s current position Goal’s position minimize Joint to move
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Start from Last Link
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Cyclic Coordinate Descent starts at the last link, adjusting each joint along the way repeat until “satisfied”
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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
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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
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CCD for MHC-peptides interaction
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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
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“A Kinematic View of Loop Closure” Evangelos A. Coutsias, Chaok Seok, Matthew P. Jacobson, Ken A. Dill
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Bond vectors fixed in space Fixed distance Tripeptide Loop Closure With the base and the lengths of the two peptide virtual bonds fixed, the vertex is constrained to lie on a circle.
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Fixed Distance between C α Atoms
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Tripeptide loop closure The six-torsion loop closure problem in simplified representation: fixed in space variables: τ i (i=1,2,3) constraints: θ i (i=1,2,3) τ1τ1 τ2τ2 τ3τ3 θ1θ1 θ2θ2 θ3θ3
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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”)
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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
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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
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Thank-You !
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Formal Definition Finding the ensemble of possible backbone structures of a chain segment of a protein molecule that is geometrically consistent with preceding and following part of the chain whose structures are given
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