A Kinematic View of Loop Closure EVANGELOS A. COUTSIAS, CHAOK SEOK, MATTHEW P. JACOBSON, KEN A. DILL Presented by Keren Lasker

Agenda Problem definition The Tripeptide Loop-Closure Problem Generalization Applications

The Loop Closure problem Finding the ensemble of possible backbone structures of a chain segment of a protein that is geometrically consistent with preceding & following parts of the chain whose structures are given. SER ILE HIS ASP ALA ALA THR SER LEU ASN

R R R Constants Constants : bond lengths, bond angles Variables Variables : backbone torsions

Six free rotation angles The angles form three/four rigid pairs Special case

R R R

Moving to a coarser problem

The Tripeptide Loop-Closure Problem Problem definition : Special case six torsion angles at three Ca atoms located consecutively along a peptide backbone. The atoms are fixed in space 3 variables 3 constrains Output : The exact position of the loop atoms

Notation N C

Finding the bonds length

d

Moving to a polynomial equation

Derivation of a 16th Degree Polynomial for the 6-angle Loop Closure r1r1 r2r2

Find the rotation angles Position the atoms

Noncontiguous Ca atoms The problem characteristic do not depend on the Ca atoms continuity

Additional Dihedral Angle

Rigid sampling coverage of the real protein structure space Dataset : Top500

sampling with perturbation 5 degree perturbation of the NCaC angles 10 degree perturbation of the NCaC angles

Application to Loop Modeling Use PLOP to sample all the torsions except for a three residue gap in the middle of the loop.

Plop (459,0.73) 1.66(236,1.6) 3.25(42,106) 0.27(5000,8.5) 1.04 (5000,6.1) 1.89(5000,23)

Moving to a polynomial equation

C C C N N Ca

Torsion/Dihedral Angle The angle between A-B bonds & C-D bonds when considering four atoms connected in the order A-B-C-D It can also be considered as the angle between two planes defined as A-B-C and B-C-D.

In contrast to the N-C bond, the movements around the N-C  (  angle) and C  -C (  angle) single bonds are restricted only by steric interference between the backbone and the sidechains Thus, the overall conformation of the backbone is almost exclusively determined by the  and  angles The ‘allowed’ values of those angles are shown in the Ramachandran plot Those represent the major types of secondary structures seen in proteins

Defining a peptide chain as a robot Why can we do that How do we do that

questions DOF in tetraheder ? ( do 4 angles determine everything?) Steepest descent iteration Frames Why d is constant?

TODO The exact solution ( all the polynims … )

This extends to the orientation of Cb

A bimodal example

Theta- perturbations are not enough

Biological motivation Homology modeling Monte Carlo simulation

TODO Check that the bond angles are really constant in proteins? Which angles do we try to find in the coarser proble m? Why the consec helps, what is the big problem in non consecutive ? Condtion 3 in the special case?

Special case of the 6R problem Consider all the motions of a chain molecule that involve changes in only six backbone torsions. Input:  molecular chain with inflexible bond lengths & bond angles.  All bond vectors are fixed in space except for a contiguous set.  The changes are made in at most six contiguous dihedral angles. Output:  All the possible arrangements of the molecular chain.

Translation to Robotic problem This is a special case of the 6R problem. [ Define the 6R problem ! ]

N N C C C R R