Conformational Space of a Flexible Protein Loop

Jean-Claude LatombeComputer Science Department

Stanford University(Joint work with Ankur Dhanik1, Guanfeng Liu2,

Itay Lotan3, Henry van den Bedem4, Jim Milgram5, Nathan Marz6, and Charles Kou6)

1 Graduate student2 Postdoc3 Now a postdoc at U.C. Berkeley4 Joint Center for Structural Genomics, Stanford Linear Accelerator Center5 Department of Mathematics, Stanford University6 Undergraduate CS students

Initial Project“Noise” in electron density maps from X-ray crystallography

4-20 aa fragments unresolved by existing software (RESOLVE, TEXTAL, ARP, MAID)

Fragment Completion Problem

Input:• Electron-density map• Partial structure•Two “anchor” residues•Amino-acid sequence of missing fragment

Output: • Conformations of fragment that

- Respect the closure constraint (IK)- Maximize match with electron-density map

Main part of protein (f olded)

Protein f ragment (f uzzy map)

Anchor 1(3 atoms)

Anchor 2(3 atoms)

Main part of protein (f olded)

Protein f ragment (f uzzy map)

Anchor 1(3 atoms)

Anchor 2(3 atoms)

Two-Stage Method[H. van den Bedem, I. Lotan, J.C. Latombe and A. M. Deacon. Real-space protein-model completion: An inverse-kinematics

approach. Acta Crystallographica, D61:2-13, 2005.]

1. Candidate generations Closed fragments

2. Candidate refinement Optimize fit with EDM

Stage 1: Candidate Generation

Loop:• Generate random conformation of fragment

(only one end is at its “anchor”) • Close fragment – i.e., bring other end to second

anchor – using Cyclic Coordinate Descent (CCD) [A.A. Canutescu and R.L. Dunbrack Jr. Cyclic coordinate descent: A robotics algorithm for protein loop closure. Prot. Sci. 12:963–972, 2003]

Stage 2: Candidate Refinement

Target function T(Q) measuring quality of the fit with the EDM

Minimize T while retaining closure

d3 d2

d1(1,2,3)

Null space

Refinement ProcedureRepeat until minimum is reached: Compute a basis N of the null space at

current Q (using SVD of Jacobian matrix) Compute gradient T of target function at

current Q [Abe et al., Comput. Chem., 1984] Move by small increment along projection

of T into null space (i.e., along dQ = NNT T)+Monte Carlo + simulated annealing protocol to deal with local minima

Tests #1: Artificial Gaps Complete structures (gold standard) resolved

with EDM at 1.6Å resolution Compute EDM at 2, 2.5, and 2.8Å resolution Remove fragments and rebuild

Long Fragments:12: 96% < 1.0Å aaRMSD15: 88% < 1.0Å aaRMSD

Short Fragments: 100% < 1.0Å aaRMSD

Tests #2: True Gaps Structure computed by RESOLVE Gaps completed independently (gold

standard) Example: TM1742 (271 residues) 2.4Å resolution; 5 gaps left by RESOLVE

Length Top scorer Lowest error4 0.22Å 0.22Å5 0.78Å 0.78Å5 0.36Å 0.36Å7 0.72Å 0.66Å10 0.43Å 0.43Å

Produced by H. van den Bedem

TM1621 Green: manually

completed conformation

Blue: conformation computed by stage 1

Pink: conformation computed by stage 2

The aaRMSD improved by 2.4Å to 0.31Å

A323Hist

A316Ser

Two-State Loop

A

B

TM0755: data at 1.8Å 8-residue fragment crystallized in 2 conformations the EDM is difficult to interpret Generate 2 conformations Q1 and Q2 using CCD TH-EDM(Q1,Q2,) = theoretical EDM created by distribution

Q1 + (1-)Q2

Maximize fit of TH-EDM(Q1,Q2,) with experimental EDM by moving in null space N(Q1)N(Q2)[0,1]

Status Software running with Xsolve, JCSG’s

structure-solution software suite Used by crystallographers at JCSG for

structure determination Contributed to determining several

structures recently deposited in PDB

Lesson “Fuzziness” in EDM due to loop

motion is not “noise”

Instead, it may be exploited to extract information on loop mobility

New 4-year NSF project (DMS-0443939, Bio-Math program)

Goal: Create a representation (probabilistic roadmap) of the conformation space of a protein loop, with a probabilistic distribution over this representation

Applications:• Motion from X-ray crystallography• Improvement of homology methods• Predicting loop motion for drug design• Conformation tweaking (MC optimization, decoy

generation)

Predicting Loop Motion

[J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. J. Comp. Chemistry, 25:956-967, 2004]

Ongoing Work

1. Develop software tools to create and manipulate loop conformations

2. Study the topological structure of a loop conformational space

Software tools implemented

CCD Exact IK for 3 residues (non-necessarily

contiguous) Creation of loop conformations

Exact IK for 3 Residues[E.A. Coutsias, C. Seok, M.J. Jacobson, K.A. Dill. A Kinematic View of

Loop Closure, J. Comp. Chemistry, 25(4):510 – 528, 2004]

Maximal number of solutions: 10, 12?

Closing loops using CCD + Exact IK

Closing loops using CCD + Exact IK

Software tools implemented

CCD Exact IK for 3 residues (non-necessarily

contiguous) Creation of loop conformations Computation of pseudo-inverse of Jacobian

and null-space basis Loop deformation in null space Conformation sampling

Moving an atom along a line

Interpolating between two conformations

Sampling many conformations

Software tools implemented

CCD Exact IK for 3 residues (non-necessarily

contiguous) Creation of loop conformations Computation of pseudo-inverse of

Jacobian and null-space basis Loop deformation in null space Conformation sampling Detection of steric clashes (grid

method)

Topological Structure of Conformational Space

Inspired by work of Trinkle and Milgram on closed-loop kinematic chains

Leads to studying singularities of open protein chains and of their images

Configuration Space of a 4R Closed-Loop Chain

[J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002]

Rigid link

Revolute jointl1

l2

l3

l4

Configuration Space of a 4R Closed-Loop Chain

[J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002]

l1l2

l3

l4

Configuration Space of a 4R Closed-Loop Chain

[J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002]

Images of thesingularities of the red linkage’s endpoint map: C 2

l1

Configuration Space of a 4R Closed-Loop Chain

[J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002]

l1

Configuration Space of a 4R Closed-Loop Chain

[J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002]

[J.C. Trinkle and R.J. Milgram, Complete Path Planning for Closed Kinematic Chains with Spherical Joints, Int. J. of Robotics Research, 21(9):773-789, 2002]

Configuration Space of a 5R Closed-Loop Chain

IS1

I(S1 S1)

S1|S1

S1|S1

Images of thesingularities of the red linkage’s endpoint map: C 2

C

C

N

N

How does it apply to a protein loop?

C

C

N

N

How does it apply to a protein loop?

C

C

N

N

How does it apply to a protein loop?

C

C

N

N

Images of thesingularities of the red linkage map: C 3SO(3)

2D surfacein 3SO(3)

C C

N

Kinematic Model

~60dg

Singularities of Map C R3

Rank 1 singularities: Planar linkage Rank 2 singularities:

• Type 1• Type 2

Singularities of Map C R3

Rank 1 singularities: Planar linkage Rank 2 singularities:

• Type 1• Type 2

Planar sub-linkages

P0

Line contained in P0

Singularities of Map C R3

Rank 1 singularities: Planar linkage Rank 2 singularities:

• Type 1• Type 2

P0

P1

P2 There is a line L

contained in P2 to which P0 and P1 are //

L

Must be // to each other and // to last plane

Endpoint iscontained in all planes P0, P1, and P2

Images of Singularities

Singularities are on the periphery of the endpoint’s reachable space

rank 1 singularity

Impact on Flexible Loops?