Conformational Space of a
Flexible Protein Loop
Jean

Claude Latombe
Computer Science Department
Stanford University
(Joint work with Ankur Dhanik
1
, Guanfeng Liu
2
,
Itay Lotan
3
, Henry van den Bedem
4
, Jim Milgram
5
,
Nathan Marz
6
, and Charles Kou
6
)
1 Graduate student
2 Postdoc
3 Now a postdoc at U.C. Berkeley
4 Joint Center for Structural Genomics, Stanford Linear Accelerator Center
5 Department of Mathematics, Stanford University
6 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)
Model completion is high

throughput bottleneck
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
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
O灴業楺攠晩f w楴h 䕄E
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
d
q
3
d
q
2
d
q
1
(
q
1
,
q
2
,
q
3
)
Null space
Refinement Procedure
Repeat 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 = NN
T
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 aaRMSD
15: 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 error
4
0.22Å
0.22Å
5
0.78Å
0.78Å
5
0.36Å
0.36Å
7
0.72Å
0.66Å
10
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
A323
Hist
A316
Ser
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 Q
1
and Q
2
using CCD
TH

EDM(Q
1
,Q
2
,
a
) = theoretical EDM created by distribution
a
Q
1
+ (1

a
)Q
2
Maximize fit of TH

EDM(Q
1
,Q
2
,
a
) with experimental EDM by
moving in null space N(Q
1
)
N(Q
2
)
[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 joint
l
1
l
2
l
3
l
4
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]
l
1
l
2
l
3
l
4
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 the
singularities of the
red linkage’s endpoint
map: C
2
l
1
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]
l
1
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
I
S
1
I
(S
1
S
1
)
S
1

S
1
S
1

S
1
Images of the
singularities of the
red linkage’s
endpoint
map: C
2
C
a
C
N
N
How does it apply to a protein loop?
C
a
C
N
N
How does it apply to a protein loop?
C
a
C
N
N
How does it apply to a protein loop?
C
a
C
N
N
Images of the
singularities of the
red linkage map:
C
3
SO(3)
2D surface
in
3
SO(3)
C
a
C
N
Kinematic Model
~60dg
Singularities of Map C
R
3
Rank 1 singularities: Planar linkage
Rank 2 singularities:
•
Type 1
•
Type 2
Singularities of Map C
R
3
Rank 1 singularities: Planar linkage
Rank 2 singularities:
•
Type 1
•
Type 2
Planar sub

linkages
P
0
Line contained in P
0
Singularities of Map C
R
3
Rank 1 singularities: Planar linkage
Rank 2 singularities:
•
Type 1
•
Type 2
P
0
P
1
P
2
There is a line L
contained in P
2
to
which P
0
and P
1
are //
L
Must be // to
each other and
// to last plane
Endpoint is
contained in all
planes P
0
, P
1
, and P
2
Images of Singularities
Singularities are on the periphery of the endpoint’s reachable space
rank 1
singularity
Impact on Flexible Loops?
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