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 Bernhard Rupp X-ray crystallography

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

Electron Density Map (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 Electron Density Map (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 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 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

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 1.Candidate generations: Optimize density fit while closing the gap 2.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)

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 1-D manifold  Target function T (goodness of fit to EDM)  Minimize T while retaining closure  Closed conformations lie on Self-motion manifold of lower dimension

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 1.Choose sub-fragment with n > 6 DOFs 2.Compute using SVD 3.Project onto 4.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 LengthHigh - 2.0ÅMedium - 2.5ÅLow - 2.8Å 4100% (0.14Å)100% (0.19Å)100% (0.32Å) 8100% (0.18Å)100% (0.23Å)100% (0.36Å) 1291% (0.51Å)96% (0.41Å)91% (0.52Å) 1591% (0.53Å)88% (0.63Å)83% (0.76Å) 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) LengthError 40.40Å 40.22Å 50.78Å 50.36Å 70.66Å Å Produced by H. van den Bedem

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

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

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

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

Alternative conformations A B TM0755, 1.8Å res. 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

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 1.Generate 1000 fragments using CCD 2.Choose top 6 candidates 3.Refine each candidate 6 times 4.Save top 2 of each refinement set 12 final candidates are output

Testing: TM1621 2Å Res.2.8Å Res. 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 Produced by H. van den Bedem

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

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 LengthHigh (2.0)Medium (2.5Å)Low (2.8Å) Times reported in minutes