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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