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 problemFinding 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

ConstantsConstants : bond lengths, bond angles

VariablesVariables : backbone torsions

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

Special case

nC

z

y

1nC

x

R

R

R

nC

1nC

x

y

z

xy

z

iii

Moving to a coarser problem

The Tripeptide Loop-Closure ProblemC

CC

NN

Ca

CaC

CC

NN

Ca

Ca

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

3311 ,,, CCCN

Output :

The exact position of

the loop atoms

Notation

N

C

Finding the bonds length

x

cosxn

sinxm

sincos mnx

d

r

r

ix

y

1ix y

iz

1iz

Moving to a polynomial equation

Derivation of a 16th Degree Polynomial for the 6-angle Loop Closure

iii rr cos1 ki

ji

kj

ijkiii uupuuP 1

2

0,

)(1 :),(

),,,,( 1 iiiiiip

r1

r2

0)( 3

16

0316

j

jjkuruR

0)( 3

16

0316

j

jjkuruR

iii

uuu

123 ,

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

resolved

unresolved

Dataset : Top500

sampling with perturbation

reso lved

unresolved

resolved

unresolved

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 -

0.29(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

Moving to a polynomial equation

1

a2

b2

3 4

a4

b4

5

a5 b5a1 b1

1

2

4

51

2

4

5

'3

o90

This extends to the orientation of Cb1

2

o111

A bimodal example

oo

o

10110

111

Theta-perturbations are not enough

oo

o

519

111

oo

o

519

111

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?

N

N C

C

C

R

R