“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 MHC- peptide 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 MHC “Cradle” ~ 1000 MHC alleles Huge # of peptides

Loop Closure for Predicting MHC- Peptide 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 How can we move the robot head to a certain location at a certain 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 ( A p):

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

Examples of Rotation Matrices

“Homogenous Transform”: Translation + Rotation using a single 4x4 Matrix

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}:

2D Example with Revolute Joints

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

So what is the problem? Solving the set of equations is usually infeasible! ?

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 Goal’s position minimize Joint to move

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

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.

Fixed Distance between C α Atoms

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

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 !

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