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 crystallographic structure Structure obtained with out hydrogens Some bad clashes result after hydrogens are added Red spikes = bad clashes Blue dots = favorable interactions

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 –Given positions of first and last two atoms –Determine all possible positions of the n-4 intervening atoms a2a2 anan a1a1 a n-1 aiai

zizi yiyi xixi atom i-1 atom i atom i+1 b i-1 bibi b i+1 θiθi ωiωi Local frame, F i = {X i, Y i, Z i }, at atom i Let R i = R Xi (ω i )* R Zi (θ i )*T Zi (d i ), where d i = |b i | Then, F i = R i * F i-1 Denavit-Hartenberg local frames

Loop closure: three residues –9 atoms –Assume peptide bonds are planar –Fix position and orientation of N 1 and C 3 –Assume ideal bond geometry C β1 C β2 C β3 N1N1 C3C3

Loop closure: three residues –9 atoms –Assume peptide bonds are planar –Fix position and orientation of N 1 and C 3 –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 7 th DoF for continuous movement –variable bond angle Derivative –direction and magnitude of movement –with respect to the variable angle

7 th variable angle N-C α -C bond angle (Tau) Derivative with respect to Tau angle 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 C β C β1 C β2 C β3 φ1φ1 φ2φ2 φ3φ3 ψ1ψ1 ψ2ψ2 ψ3ψ3 N1N1 C3C3 Three residue segment –fix outermost atoms N 1 and C 3 –6 free dihedrals –modify center tau Tau

One swinging C β

Two swinging C β ‘s Four residue segment –fix outermost C α ‘s –6 free dihedrals –modify one intermediate tau ψ3ψ3 ψ2ψ2 ψ1ψ1 C β2 C β1 C β3 C β4 φ1φ1 φ2φ2 φ3φ3 C α1 C α4 Tau

Two swinging C β ‘s

Ongoing work Extend analytic solution –to handle variable geometry Determine closed form solution for derivative Extend to several geometric modifications

The End