Computing Protein Structures from Electron Density Maps: The Missing Fragment Problem

Computing Protein Structures from Electron Density Maps: The Missing Fragment Problem Itay Lotan† Henry van den Bedem* Ashley M. Deacon* Jean-Claude Latombe† † Computer Science Dept., Stanford University * Joint Center for Structural Genomics (JCSG) at SSRL

Structure determination X-ray crystallography Bernhard Rupp

Protein Structure Initiative • Reduce cost and time to determine protein structure 152K sequenced genes (30K/year) 25K determined structures (3.6K/year) • Develop software to automatically interpret the electron density map (EDM)

EDM 3-D “image” of atomic structure • High value (electron density) at atom centers • Density falls off exponentially away from center • Limited resolution, sampled on 3D grid

Automated model building • ~90% built at high resolution (2Å) • ~66% built at medium to low resolution (2.5 – 2.8Å) • Gaps left at noisy areas in EDM (blurred density) Gaps need to be resolved manually

The Fragment completion problem • Input • EDM • Partially resolved structure • 2 Anchor residues • Length of missing fragment • Output • A small number of candidate structures for missing fragment A robotics inverse kinematics (IK) problem

Related work Biology/Crystallography • Exact IK solvers • Wedemeyer & Scheraga ’99 • Coutsias et al. ’04 • Optimization IK solvers • Fine et al. ’86 • Canutescu & Dunbrack Jr. ’03 • Ab-initio loop closure • Fiser et al. ’00 • Kolodny et al. ’03 • Database search loop closure • Jones & Thirup ’86 • Van Vlijman & Karplus ’97 • Semi-automatic tools • Jones & Kjeldgaard ’97 • Oldfield ’01 Computer Science • Exact IK solvers • Manocha & Canny ’94 • Manocha et al. ’95 • Optimization IK solvers • Wang & Chen ’91 • Redundant manipulators • Khatib ’87 • Burdick ’89 • Motion planning for closed loops • Han & Amato ’00 • Yakey et al. ’01 • Cortes et al. ’02, ’04

Contributions • Sampling of gap-closing fragments biased by the EDM • Refinement of fit to density without breaking closure • Fully automatic fragment completion software for X-ray Crystallography Novel application of a combination of inverse kinematics techniques

Torsion angle model Protein backbone is a kinematic chain

Two-stage IK method • Candidate generations: Optimize density fit while closing the gap • Refinement: Optimize closed fragments without breaking closure

Stage 1: candidate generation • Generate random conformation • Close using Cyclic Coordinate Descent (CCD) (Wang & Chen ’91, Canutescu & Dunbrack Jr. ’03)

Stage 1: candidate generation • Generate random conformation • Close using Cyclic Coordinate Descent (CCD) (Wang & Chen ’91, Canutescu & Dunbrack ’03)

Stage 1: candidate generation • Generate random conformation • Close using Cyclic Coordinate Descent (CCD) (Wang & Chen ’91, Canutescu & Dunbrack ’03) CCD moves biased toward high-density

Stage 2: refinement • Target function T(goodness of fit to EDM) • Minimize T while retaining closure • Closed conformations lie on Self-motion manifold of lower dimension 1-Dmanifold

Stage 2: null-space minimization Jacobian: linear relation between joint velocities and end-effector linear and angular velocity . Compute minimizing move using: N – orthonormal basis of null space

Stage 2: minimization with closure • Choose sub-fragment with n > 6 DOFs • Compute using SVD • Project onto • Move until minimum is reached or closure is broken Escape from local minima using Monte Carlo with simulated annealing

MC + Minimization (Li & Scheraga ’87) • Suggest large random change • Random move in • Exact IK solution for 3 residues (Coutsias et al. ’04) • Minimize resulting conformation • Accept using Metropolis criterion: • Use simulated annealing

Test: artificial gaps • Completed structure (gold standard) • Good density (1.6Å resolution) • Remove fragment and rebuild Produced by H. van den Bedem

Test: true gaps • Completed structure (gold standard) • OK density (2.4Å resolution) • 6 gaps left by model builder (RESOLVE) Produced by H. van den Bedem

Example: TM0423 PDB: 1KQ3, 376 res. 2.0Å resolution 12 residue gap Best: 0.3Å aaRMSD

Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best: 0.6Å aaRMSD GLU-77 GLY-90

Example: TM0813 PDB: 1J5X, 342 res. 2.8Å resolution 12 residue gap Best 0.6Å aaRMSD GLU-77 GLY-90

Alternative conformations TM0755, 1.8Å res. B A Produced by H. van den Bedem

Conclusion • Sampling of gap-closing fragments biased by the EDM • Refinement of fit to density without breaking closure • Fully automatic fragment completion software for X-ray Crystallography

Thank you

Stage 1: Density-biased CCD • Compute pair that minimizes closure distance • Search square neighborhood for density maximum and move there. • The size of  is reduced with the number of iterations

Stage 2: Target function • EDM - • Computed (model) density - • Least-squares residuals between EDM and model density

Building a missing fragment • Generate 1000 fragments using CCD • Choose top 6 candidates • Refine each candidate 6 times • Save top 2 of each refinement set 12 final candidates are output

Testing: TM1621 • PDB: 1O1Z, SCOP: α/β, 234 res. • 34% helical, 19% strands • Collected at 1.6Å res. • 2mFo-DFc EDMs calculated at 2.0Å, 2.5Å, and 2.8Å • 103 fragments of length 4,8,12 and 15 2Å Res. 2.8Å Res. Produced by H. van den Bedem

Testing: TM1621 - mean - median - %>1Å aaRMSD 2Å Res. 2.8Å Res. Helical fragments (>2/3 helical) account for most misses Produced by H. van den Bedem

Testing: TM1742 • PDB: 1VJR, 271 res. • Collected at 2.4Å • Good quality density • 88% built using RESOLVE • 5 gaps, 1 region built incorrectly Produced by H. van den Bedem

TM1621: running time Times reported in minutes

Computing Protein Structures from Electron Density Maps: The Missing Fragment Problem