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![Page 1: Conformational Space of a Flexible Protein Loop Jean-Claude Latombe Computer Science Department Stanford University (Joint work with Ankur Dhanik 1, Guanfeng.](https://reader036.fdocuments.us/reader036/viewer/2022062523/5a4d1b0c7f8b9ab05998be40/html5/thumbnails/1.jpg)
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
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Initial Project“Noise” in electron density maps from X-ray crystallography
4-20 aa fragments unresolved by existing software (RESOLVE, TEXTAL, ARP, MAID)
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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)
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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
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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]
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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
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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
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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
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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
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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
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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]
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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
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Lesson “Fuzziness” in EDM due to loop
motion is not “noise”
Instead, it may be exploited to extract information on loop mobility
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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)
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Predicting Loop Motion
[J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. J. Comp. Chemistry, 25:956-967, 2004]
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Ongoing Work
1. Develop software tools to create and manipulate loop conformations
2. Study the topological structure of a loop conformational space
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Software tools implemented
CCD Exact IK for 3 residues (non-necessarily
contiguous) Creation of loop conformations
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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?
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Closing loops using CCD + Exact IK
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Closing loops using CCD + Exact IK
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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
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Moving an atom along a line
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Interpolating between two conformations
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Sampling many conformations
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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)
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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
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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
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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
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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
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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]
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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]
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[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
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C
C
N
N
How does it apply to a protein loop?
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C
C
N
N
How does it apply to a protein loop?
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C
C
N
N
How does it apply to a protein loop?
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C
C
N
N
Images of thesingularities of the red linkage map: C 3SO(3)
2D surfacein 3SO(3)
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C C
N
Kinematic Model
~60dg
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Singularities of Map C R3
Rank 1 singularities: Planar linkage Rank 2 singularities:
• Type 1• Type 2
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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
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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
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Images of Singularities
Singularities are on the periphery of the endpoint’s reachable space
rank 1 singularity
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Impact on Flexible Loops?