Algorithmic Robotics and Molecular Modeling

Dan HalperinSchool of Computer Science

Tel Aviv University

June 2007

Robotics

RAS field of interest (ICRA, Rome, April 2007) :

Robotics focuses on sensor and actuator systems that operateautonomously or semi-autonomously (in cooperation with humans) inunpredictable environments.  Robot systems emphasize intelligence andadaptability,  may be networked, and are being developed for manyapplications  such as service and personal assistants; surgery andrehabilitation; haptics; space, underwater, and remote exploration andteleoperation; education, entertainment; search  and rescue; defense;agriculture; and intelligent vehicles.

Algorithmic Robotics and Motion Planning

[Latombe et al[

Proteins as Robots Long sequence of amino-acids (dozens to thousands), also

called residues from a dictionary of 20 amino-acids

Robots with Many Dofs

NN

NN

C’

C’

C’

C’

O

O O

O

C

C

C

C

C

C C

C

Resi Resi+1 Resi+2 Resi+3

http://www.youtube.com/watch?v=k-VgI4wNyTo

Simulation and Predicition of Molecular Motion

[Enosh-Raveh 2007][Enosh,Fleishman,Ben-Tal,H 2007]

Exploiting the Kinematic Structure of Molecules

[Lotan et al 2004]

Krebs et al. (2003) J. Biol. Chem. 278, 50217.

[Enosh et al 2004]

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1CTF 1JB01HTB1LE2

ChainTree

Grid

Tim

e (

in m

Se

c.)

Speeding up MCS

The ChainTree [Lotan,Schwarzer,H,Latombe 2004]

TAB A

TBC B

TCD C

TDE D

TEF E

TFG F

TGH G

THI H

TAC AB TCE CD TEG EF TGI GH

TAE AD TEI EH

TAI AH

A BC D

E FG H

I

Molecular SimulationsMonte Carlo Simulation (MCS)

Popular method for sampling the conformation space of proteins:

Estimate thermodynamic quantities

Search for low-energy conformations and the folded structure

MCS: How it works

2. Compute energy E of new conformation3. Accept with probability:

Requires >>106 steps to sample adequately

/( ) min 1, bE k TP accept e

1. Propose random change in conformation

Energy function

Bonded terms: Bond lengths: Bond angles: Dihedral angles:

Non-bonded terms: Van der Waals: Electrostatic: Heuristic

Pair-wise interactions

Cutoff distance (6 - 12Å) Linear number of interactions

contribute to energy (H-Overmars ’98)

Challenge: Find all interacting pairs without enumerating all pairs

Related work Computer Science Bounding volume

hierarchies for collision detection

Gotschalk et al. ’96 Larsen et al. ’00 Guibas et al. ’02

Space partition methods for collision detection

Faverjon ’84 Halperin & Overmars ’98

Collisions detection for chains

Halperin et al. ’97 Guibas et al. ’02

Biology Neighbor lists

Verlet ’67 Brooks et al. ’83

Grid Quentrec & Brot ’73 Hockney et al. ’74 Van Gunsteren et al. ’84

Neighbor lists + grid Yip & Elber ’89 Petrella ’02

Grid method

d: Cutoff distance

ddd

Linear complexity Optimal in worst case

Contributions

Efficient maintenance and self-collision detection for kinematic chains

Efficient computation of pair-wise interactions in MCS of proteins

Scheme for caching and reusing partial energy sums during MCS

MCS software*

Much faster than existing algorithm (grid method)

*Download at: http://robotics.stanford.edu/~itayl/mcs

Properties of kinematic chains

Small changes large effects

Properties of kinematic chains

Small changes large effects

Properties of kinematic chains

Small changes large effects Local changes global effects

Properties of kinematic chains

Small changes large effects Local changes global effects Few DoF changes long rigid sub-

chains

Properties of kinematic chains

Small changes large effects Local changes global effects Few DoF changes long rigid sub-

chains

ChainTree: A tale of two hierarchies

Transform hierarchy: approximates kinematics of protein backbone at successive resolutions

Bounding volume hierarchy: approximates geometry of protein at successive resolutions

Hierarchy of transforms

Hierarchy of transforms

A BC D

E FG H

I

TAB TBC

TAC

THITCD TDE TEF TFG TGH

TCE TEG TGI

TAE TEI

TAI

Hierarchy of bounding volumes

BA HGFEDC

CD EF GHAB

AD EH

AH

The ChainTree

TAB A

TBC B

TCD C

TDE D

TEF E

TFG F

TGH G

THI H

TAC AB TCE CD TEG EF TGI GH

TAE AD TEI EH

TAI AH

A BC D

E FG H

I

Updating the ChainTree

TAB A

TBC B

TCD C

TDE D

TEF E

TFG F

TGH G

THI H

TAC AB TCE CD TEG EF TGI GH

TAE AD TEI EH

TAI AH

A BC D

E FG H

I

Computing the energy

A B C D E F G H

J K L M

N O

P

Pruning rules:1. Prune search when distance between bounding volumes

is more than cutoff distance2. Do not search inside rigid sub-chains

Recursively search ChainTree for interactions

A B C D E F G H

J K L M

N O

P

Computing the energy

[P]

A B C D E F G H

J K L M

N O

P

Computing the energy

[N]

[P]

A B C D E F G H

J K L M

N O

P

Computing the energy

[N] [O]

[P]

A B C D E F G H

J K L M

N O

P

Computing the energy

[N-O][N] [O]

[P]

Computing the energy

[N-O]

[J-K]

[A-C]

[B-C][A-D]

[B-D]

A B C D E F G H

J K L M

N O

P

[J]

[N]

[K]

[C]

[D][C-D]

[O]

[P]

Computing the energy

[P]

[N] [N-O]

[J-K] [K] [K-L][J-M][J-L] [K-M]

[A-G]

[B-G][A-H]

[B-H]

[A-C]

[B-C][A-D]

[B-D]

[C]

[D][C-D]

[A-E]

[B-E][A-F]

[B-F]

[C-E][C-F]

[C-G][C-H][D-G][D-H]

[J]

[A]

[B][A-B]

[D-E][D-F]

[O]

[L] [L-M] [M]

[E]

[F][E-F]

[E-G]

[F-G][E-H]

[F-H]

[H]

[G][H-G]

A B C D E F G H

J K L M

N O

P

Computing the energy

E(O)

A B C D E F G H

J K L M

N O

P

[P]

[N] [N-O]

[J-K] [K] [K-L][J-M][J-L] [K-M]

[A-G]

[B-G][A-H]

[B-H]

[A-C]

[B-C][A-D]

[B-D]

[C]

[D][C-D]

[A-E]

[B-E][A-F]

[B-F]

[C-E][C-F]

[C-G][C-H][D-G][D-H]

[J]

[A]

[B][A-B]

[D-E][D-F]

[O]

[L] [L-M] [M]

[E]

[F][E-F]

[E-G]

[F-G][E-H]

[F-H]

[H]

[G][H-G]

Computing the energy

Only changed interactions are found

Reuse unaffected partial sums

Better performance for

Longer proteins

Fewer simultaneous changes

Updating:

Searching:

Computational complexity

log nO k k

43n worst case bound

Much faster in practice

Test

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1CTF 1JB01HTB1LE2

ChainTree

Grid

Tim

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in m

Se

c.)

[68 res.] [144 res.] [374 res.] [755 res.]

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1CTF 1JB01HTB1LE2

ChainTree

Grid

Tim

e (

in m

Se

c.)

[68 res.] [144 res.] [374 res.] [755 res.]

1-DoF change 5-DoF change

Dynamic Maintenance of Molecular Surfaces [Eyal-H 2005]

Major Goals

Dynamic maintenance of molecular properties in MD-type simulations

Simulation and prediction of motion with more dofs

Fast and accurate IK (loop closure)

THE END