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

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)

EDM3-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 fragmentA robotics inverse kinematics (IK) problem

Related workComputer 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

NN

NN

C’C’

C’C’

O

O O

O

C

C

C

C

C

C C

C

Resi Resi+1 Resi+2 Resi+3

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)


(CCD) (Wang & Chen ’91, Canutescu & Dunbrack ’03)


(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 minimizationJacobian: linear relation between joint velocities and end-effector linear and angular velocity .

(6 matrix)x J q q n

Compute minimizing move using:

† T T qq J q x N N

q

null | 0J q J q

qx

N – orthonormal basis of null space

Stage 2: minimization with closure1. Choose sub-fragment with n > 6 DOFs2. Compute using SVD3. Project onto 4. Move until minimum is reached or

closure is broken

( )T q q null( )Jnull( )J

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

exp prev newT q T q

P acceptTemp

null( )J

Test: artificial gaps Completed structure (gold standard) Good density (1.6Å resolution) Remove fragment and rebuild

Length High - 2.0Å Medium - 2.5Å Low - 2.8Å4 100% (0.14Å) 100% (0.19Å) 100% (0.32Å)8 100% (0.18Å) 100% (0.23Å) 100% (0.36Å)12 91% (0.51Å) 96% (0.41Å) 91% (0.52Å)15 91% (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)

Length Error4 0.40Å4 0.22Å5 0.78Å5 0.36Å7 0.66Å10 0.43Å


Example: TM0423PDB: 1KQ3, 376 res.2.0Å resolution12 residue gapBest: 0.3Å aaRMSD

Example: TM0813

GLU-77

GLY-90

PDB: 1J5X, 342 res.2.8Å resolution12 residue gapBest: 0.6Å aaRMSD

Example: TM0813

GLU-77

GLY-90

PDB: 1J5X, 342 res.2.8Å resolution12 residue gapBest 0.6Å aaRMSD

Alternative conformations

AB

TM0755, 1.8Å res.


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

,i i

, ,t t t ti i i i

Stage 2: Target function EDM - Computed (model) density -

5 2

1

expci

ii

ra b

Least-squares residuals between EDM and model density

2

i

o ci i

g V

T q S g k g

oc

Building a missing fragment1. Generate 1000 fragments using CCD2. Choose top 6 candidates3. Refine each candidate 6 times4. 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


Testing: TM1621

2Å Res. 2.8Å Res.


Helical fragments (>2/3 helical) account for most misses

- mean- median- %>1Å aaRMSD

xxp

Testing: TM1742• PDB: 1VJR, 271 res. • Collected at 2.4Å• Good quality density

• 88% built using RESOLVE • 5 gaps, 1 region built incorrectly


TM1621: running timeLength High (2.0) Medium (2.5Å) Low (2.8Å)

4 40 29 28

8 92 63 58

12 134 82 73

15 178 105 95

Times reported in minutes

computing protein structures from electron density maps: the missing fragment problem

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