Conformational Space of a Flexible Protein Loop

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2 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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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?