Download C bond angle - BioGeometry

Derivatives of Backbone
Motion
Kimberly Noonan, Jack Snoeyink
UNC Chapel Hill
Computer Science
Outline
• Protein design
• Related work
• Local backbone motion
• Derivative algorithm
• Ongoing work
Protein design
• Operations
– Visualize structure
• Mage, Chime
– Modify structure
• Dezymer
• Example [Hellinga]
– RBP (Ribose Binding Protein)
• bind zinc
• bind TNT
Dezymer software
• H. Hellinga, L. Looger …
• Input: fixed backbone and ligand
• Output: top-ranked receptor designs
• Method:
– Identifies molten zone
– Freezes side chains outside zone
– Frees side chains inside zone by mutation to
Alanine.
– Ranks all possible mutation configurations and
ligand orientations using energy functions
Binding site design
RBP binding TNT
[Hellinga]
Dezymer decorated
wild type backbone
Binding site design improved?
RBP binding TNT
[Hellinga]
Dezymer decorated
wild type backbone
vs.
Dezymer’s redesign
of rubbed backbone
Crystallographic refinement
Structure obtained with out
hydrogens
Some bad clashes result after
hydrogens are added
Red spikes = bad clashes
Blue dots = favorable interactions
crystallographic
structure
Crystallographic refinement
crystallographic
structure
best choice of
rotamer?
Crystallographic refinement
crystallographic
structure
best choice of
rotamer?
rubbed backbone
with same rotamer
Protein modification
• Operations
– Side chain mutation
– Rotamer selection
– Backbone movement
• CAD for local backbone motion?
– Modify segment of backbone,
leave remainder of chain fixed
Geometry for proteins
• Loop Closure Problem
– Given n-atom chain
linked by fixed bond
lengths and angles
ai
an-1
– Given positions of first
and last two atoms
– Determine all possible
positions of the n-4
intervening atoms
an
a2
a1
Denavit-Hartenberg
local frames
xi
atom i
bi-1
atom i-1
zi
ωi b
i
yi
θi
atom i+1
bi+1
Local frame, Fi = {Xi , Yi , Zi }, at atom i
Let Ri = RXi(ωi)* RZi(θi)*TZi(di), where di = |bi|
Then, Fi = Ri * Fi-1
Loop closure: three residues
Cβ2
Cβ3
C3
Cβ1
N1
– 9 atoms
– Assume peptide bonds
are planar
– Fix position and
orientation of
N1 and C3
– Assume ideal bond
geometry
Loop closure: three residues
Cβ2
Cβ3
φ2
ψ2
φ3
ψ3
C3
Cβ1
ψ1
φ1
N1
– 9 atoms
– Assume peptide bonds
are planar
– Fix position and
orientation of
N1 and C3
– Assume ideal bond
geometry
– Free dihedral angles
• (φ, ψ)
– 6 degrees of freedom
Related work:
• Computational tool
– Manocha, Canny, 95
– Eigenvalue problem
– Returns set of feasible solutions
• Exact analytical solution
– Wedemeyer, Sheraga, 99
– spherical geometry
– 16 degree polynomial
• empirically at most 8 feasible solutions
Local backbone motion
• 6 degrees of freedom
– yields discrete solutions
• Need 7th DoF for continuous movement
– variable bond angle
• Derivative
– direction and magnitude
of movement
– with respect to the
variable angle
7th variable angle
N-Cα-C bond angle (Tau)
Cβ2
Cβ3
φ2
Tau
ψ2
Derivative with respect
to Tau angle
φ3
ψ3
C3
Cβ1
ψ1
φ1
N1
• Closed form solution
(adapt exact analytic)
• Estimate derivative with
algorithm
Derivative algorithm
• Input:
– Chain length and geometry
– Desired bond angle to be varied
• Output:
– Derivative estimate
• Method:
– Fixes local frames of outermost atoms
– Frees all intermediate φ, ψ angles
– Matlab optimization technique to solve for
resulting atom positions
One swinging Cβ
Cβ2
Cβ3
φ2
Tau
ψ2
φ3
Three residue segment
ψ3
C3
Cβ1
ψ1
φ1
N1
– fix outermost atoms
N1 and C3
– 6 free dihedrals
– modify center tau
One swinging Cβ
Two swinging Cβ ‘s
Cβ2
φ2
ψ2
φ1
Tau
ψ3
Cβ3
φ3
Cα4
ψ1
Cβ1
Cα1
Cβ4
Four residue segment
– fix outermost Cα ‘s
– 6 free dihedrals
– modify one
intermediate tau
Two swinging Cβ ‘s
Ongoing work
• Extend analytic solution
– to handle variable geometry
• Determine closed form solution for
derivative
• Extend to several geometric
modifications
The End