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Robotics Algorithms for the Study of
Protein Structure and Motion
Jean-Claude LatombeComputer Science Department
Stanford University
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ProteinLong sequence of amino-acids (dozens to thousands), from a dictionary of 20 distinct amino-acids
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Central Dogma of Molecular Biology
Physiological conditions: aqueous solution, 37°C, pH 7,atmospheric pressure
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Why Proteins? They are the workhorses of living organisms
• They perform many vital functions, e.g.:- catalysis of reactions - storage of energy- transmission of signals - building blocks of muscles
They raise challenging computational issues• Large molecules (100s to several 1000s of atoms)• Made of building blocks drawn from a small “dictionary”• Unusual kinematic structure
They are associated with many critical problems• Folded structure determination• Global and local structural similarities • Prediction of folding and binding motions
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Kinematic Linkage Model
peptide group
side-chain group
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Molecule and Robot
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Two problems Structure determination from
electron density maps• Inverse kinematics techniques
[Itay Lotan, Henry van den Bedem, Ashley Deacon (Joint Center for Structural Genomics)]
Energy maintenance during Monte Carlo simulation• Collision detection techniques
[Itay Lotan, Fabian Schwarzer, and Danny Halperin (Tel Aviv University)]
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Structure Determination/Prediction
Experimental tools
Computational tools• Homology, threading• Molecular dynamics
NMR spectrometryX-ray crystallography
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Protein Data Bank
1990 250 new structures1999 2500 new structures2000 >20,000 structures total2004 ~30,000 structures total
Only about 10% of structures have been determined for known protein sequences
Protein Structure Initiative (PSI)
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X-Ray Crystallography
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Automated Model Building Software systems: RESOLVE, TEXTAL, ARP/wARP, MAID
• 1.0Å < d < 2.3Å ~ 90% completeness• 2.3Å ≤ d < 3.0Å ~ 67% completeness (varies widely)1
Manually completing a model:
• Labor intensive, time consuming• Existing tools are highly
interactive
JCSG: 43% of data sets 2.3Å
1Badger (2003) Acta Cryst. D59
Model completion is high-throughput bottleneck
1.0Å 3.0Å
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The Completion Problem
Input:• Electron-density map• Partial structure•Two anchor residues•Amino-acid sequence of missing fragment (typically 4 – 15 residues long)
Output: • Few candidate conformation(s) 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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Input:• Closed kinematic chain with n > 6 degrees of freedom
• Relative positions/orientations X of end frames• Target function T(Q) →
Output:• Joint angles Q that
- Achieve closure- Optimize T
IK Problem
T
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Related Work
Robotics/Computer 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
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Two-Stage IK Method
1. Candidate generations Closed fragments
2. Candidate refinement Optimize fit with EDM
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Stage 1: Candidate Generation
1. Generate random conformation of fragment (only one end attached to anchor)
2. Close fragment (i.e., bring other end to second anchor) using Cyclic Coordinate Descent (CCD) (Wang & Chen ’91, Canutescu & Dunbrack ’03)
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fixed end
moving end
Closure Distance
Closure Distance: 2 22
S N N C C C C
Compute
+ bias toward EDM+ avoid steric clashes
s.t. 0ii
Sq
q
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
1-D manifold
Target function T (Q) measuring quality of the fit with the EDM
Minimize T while retaining closure Closed conformations lie on a self-motion
manifold of lower dimension
d3d2
d1(1,2,3)
Null space
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Closure and Null Space
dX = J dQ, where J is the 6n Jacobian matrix (n > 6)
Null space {dQ | J dQ = 0} has dim = n – 6 N: orthonormal basis of null space Pseudo-inverse J+ such that JJ+ = I dQ = J+dX + NNTy
y = T(Q)
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dX U66 VT6n dQ66
=
Computation of J+ and NSVD of J
12
6
J+ = V + UT where +=diag[1/i]
Gram-Schmidt orthogonalization
0
(n-6) basis N of null space
NT
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Refinement Procedure
Repeat until minimum is reached: Compute J, J+ and N at current Q• Compute T at current Q
(analytical expression of T + linear-time recursive computation [Abe et al., Comput. Chem., 1984])
• Move along dQ = J+dX + NNT T until minimum is reached or closure is broken
+Monte Carlo + simulated annealing protocol to deal with local minima
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Monte Carlo OptimizationRepeat:1. Perform a random move of the
fragment:– either by picking a random direction in
null space– or by using an exact IK solver over 6
dofs [Coutsias et al, 2004] ( big jumps)
2. Minimize T(Q)3. Accept move with Metropolis-
criterion probability ~exp(-T/Temp)
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Tests #1: Artificial Gaps
TM1621 (234 residues) and TM0423 (376 residues), SCOP classification a/b
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
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TM1621 103 Fragments from TM1621 at 2.5Å
Produced by H. van den Bedem
Long Fragments:
12: 96% < 1.0Å aaRMSD15: 88% < 1.0Å aaRMSD
Short Fragments:
100% < 1.0Å aaRMSD
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Comparison Across Resolutions
Resolution = 2.0Å Resolution = 2.8ÅResolution = 2.5Å
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Example: TM0423PDB: 1KQ3, 376 res.2.0Å resolution12 residue gapBest: 0.3Å 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 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
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TM0813
GLU-83
GLY-96
PDB: 1J5X, 342 res.2.8Å resolution12 residue gap
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TM0813
GLU-83
GLY-96
PDB: 1J5X, 342 res.2.8Å resolution12 residue gapBest 0.6Å aaRMSD
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TM1621
Green: manually completed conformation
Cyan: conformation computed by stage 1
Magenta: conformation computed by stage 2
The aaRMSD improved by 2.4Å to 0.31Å
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resolution: 2.0Åinitial model: ARP/wARPcontour: 1.0sPDB: 1VJGaaRMSD: 0.33Å
Alr1529D72-D78
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TM0542
• Top-scoring fragment in cyan• Manually completed fragment in green• Residues A259 and A260 are flipped
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Current/Future Work
A
B
Software actively being used at the JCSG
What about multi-modal loops?
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TM0755: data at 1.8Å 8-residue fragment crystallized in 2 conformations Overlapping density: Difficult to interpret
manually
Algorithm successfully identified and built both conformations
A323Hist
A316Ser
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Current/Future Work
A
B
Software actively being used at the JCSG
What about multi-modal loops?
Fuzziness in EDM can then be exploited
Use EDM to infer probability measure over the conformation space of the loop
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Amylosucrase
J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. J. Comp. Chemistry, 25:956-967, 2004
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Energy maintenance during Monte Carlo simulation
joint work with Itay Lotan, Fabian Schwarzer, and Dan Halperin1
1 Computer Science Department, Tel Aviv University
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Random walk through conformation space At each attempted step:
• Perturb current conformation at random• Accept step with probability:
The conformations generated by an arbitrarily long MCS are Boltzman distributed, i.e.,
#conformations in V ~
/( ) min 1, bE k TP accept e
Monte Carlo Simulation (MCS)
E
-kT
Ve dV
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Used to:• sample meaningful distributions of conformations • generate energetically plausible motion pathways
A simulation run may consist of millions of steps
energy must be evaluated frequently
Problem: How to maintain energy efficiently?
Monte Carlo Simulation (MCS)
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Energy Function E = bonded terms
+ non-bonded terms + solvation terms
Bonded terms - O(n)
Non-bonded terms - E.g., e.g. Van der Waals and electrostatic- Depend on distances between pairs of atoms - O(n2) Expensive to compute
Solvation terms- May require computing molecular surface
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Non-Bonded Terms Energy terms go to 0 when distance
increases Cutoff distance (6 - 12Å)
vdW forces prevent atoms from bunching up Only O(n) interacting pairs [Halperin&Overmars 98]
Problem: How to find interacting pairswithout enumerating all atom pairs?
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Grid Method
dcutoff
Subdivide 3-space into cubic cells
Compute cell that contains each atom center
Represent grid as hashtable
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Grid Method
dcutoff Θ(n) time to build grid O(1) time to find
interactive pairs for each atom
Θ(n) to find all interactive pairs of atoms [Halperin&Overmars, 98]
Asymptotically optimal in worst-case
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Can we do better on average?
Few DOFs are changed at each MC step
Number kof DOF changes
0 10 20 305
simulationof 100,000attempted steps
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Can we do better on average?
Few DOFs are changed at each MC step Proteins are long chain kinematics
Long sub-chains stay rigid at each step Many partial energy sums remain constant
Problem: How to retrieve the unchanged partial sums?
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Hierarchical Collision Checking
Widely used technique in robotics/graphics to approximate distances between objects
Pre-computation of bounding-volume hierarchy
How to update this hierarchy if the objects deform
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Two New Data Structures
1. ChainTree Fast detection of interacting atom pairs
2. EnergyTree Retrieval of unchanged partial energy sums
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ChainTree(Twofold Hierarchy: BVs +
Transforms)
links
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TNO
TJK
TAB
joints
ChainTree(Twofold Hierarchy: BVs +
Transforms)
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Updating the ChainTree
Update path to root:– Recompute transforms that “shortcut” the DOF change– Recompute BVs that contain the DOF change– O(k log(n/k)) work for k changes
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Finding Interacting Pairs
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Finding Interacting Pairs
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Finding Interacting Pairs
Do not search inside rigid sub-chains (unmarked nodes)
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Finding Interacting Pairs
Do not search inside rigid sub-chains (unmarked nodes)
Do not test two nodes with no marked node between them
New interacting pairs
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EnergyTree
E(N,N)
E(J,L)
E(K.L)
E(L,L)
E(M,M)
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EnergyTree
E(N,N)
E(J,L)
E(K.L)
E(L,L)
E(M,M)
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Complexity
n : total number of DOFs k : number of DOF changes at each MCS step k << n
Complexity of: updating ChainTree: O(k log(n/k)) finding interacting pairs: O(n4/3)
but performs much better in practice!!!
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Experimental Setup
Energy function: Van der Waals Electrostatic Attraction between native contacts Cutoff at 12Å
300,000 steps MCS with Grid and ChainTree
Steps are the same with both methods Early rejection for large vdW terms
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Results: 1-DOF change
(68) (144) (374) (755)# amino acids
3.5
12.5
5.8
7.8
speedup
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Results: 5-DOF change
(68) (144) (374) (755)
2.2
3.4
4.5
5.9
speedup
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Two-Pass ChainTree (ChainTree+)
1st pass: small cutoff distance to detect steric clashes2nd pass: normal cutoff distance
>5Tests around native state
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Interaction with Solvent
Explicit solvent models: 100s or 1000s of discrete solvent molecules
Implicit solvent models: solvent as continuous medium, interface is solvent-accessible surface
E. Eyal, D. Halperin. Dynamic Maintenance of Molecular Surfaces underConformational Changes. http://www.give.nl/movie/publications/telaviv/EH04.pdf
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Summary
Inverse kinematics techniques Improve structure determination from fuzzy electron density maps
Collision detection techniques Speedup energy maintenance during Monte Carlo simulation
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About Computational Biology
Computational Biology is more than using computers to biological problems or mimicking nature (e.g., performing MD simulation)
One of its goals is to achieve algorithmic efficiency by exploiting properties of molecules, e.g.: • Proteins are long kinematic chains• Atoms cannot bunch up together• Forces have relatively short ranges