Algorithmic Robotics and Molecular Modeling Dan Halperin School 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 operate autonomously or semi-autonomously (in cooperation with humans) in unpredictable environments. Robot systems emphasize intelligence and adaptability, may be networked, and are being developed for many applications such as service and personal assistants; surgery and rehabilitation; haptics; space, underwater, and remote exploration and teleoperation; 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

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, [Enosh et al 2004] Speeding up MCS

The ChainTree [Lotan,Schwarzer,H,Latombe 2004] T AB B A T BC B B T CD B C T DE B D T EF B E T FG B F T GH B G T HI B H T AC B AB T CE B CD T EG B EF T GI B GH T AE B AD T EI B EH T AI B AH A B C D E F G H I

Molecular Simulations Monte 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 conformation 3. Accept with probability: Requires >>10 6 steps to sample adequately 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 ( Å )  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  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:

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

A B C D E F G H I T AB T BC T AC T HI T CD T DE T EF T FG T GH T CE T EG T GI T AE T EI T AI

Hierarchy of bounding volumesB BABA BHBH BGBG BFBF BEBE BDBD BCBC B CD B EF B GH B AB B AD B EH B AH

The ChainTree T AB B A T BC B B T CD B C T DE B D T EF B E T FG B F T GH B G T HI B H T AC B AB T CE B CD T EG B EF T GI B GH T AE B AD T EI B EH T AI B AH A B C D E F G H I

Updating the ChainTree T AB B A T BC B B T CD B C T DE B D T EF B E T FG B F T GH B G T HI B H T AC B AB T CE B CD T EG B EF T GI B GH T AE B AD T EI B EH T AI B AH A B C D E F G H I

Computing the energy ABCDEF GH JKLM NO P Pruning rules: 1.Prune search when distance between bounding volumes is more than cutoff distance 2.Do not search inside rigid sub-chains Recursively search ChainTree for interactions

ABCDEF GH JKLM NO P Computing the energy [ P ]

ABCDEF GH JKLM NO P [ N ] [ P ]

ABCDEF GH JKLM NO P [ N ][ O ] [ P ]

ABCDEF GH JKLM NO P [ N-O ][ N ][ O ] [ P ]

Computing the energy [ N-O ] [ J-K ] [ A-C ] [ B-C ] [ A-D ] [ B-D ] ABCDEF GH JKLM NO 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 ] ABCDEF GH JKLM NO P

Computing the energy E(O) ABCDEF GH JKLM NO 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 worst case bound Much faster in practice

Test [68 res.][144 res.][374 res.][755 res.] [68 res.][144 res.][374 res.][755 res.] 1-DoF change5-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)

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