Robotics Algorithms for the Study of Protein Structure and Motion Based on Itay Lotan’s PhD Jean-Claude Latombe Computer Science Department Stanford University

Unfolded (denatured) state Folded (native) state Many pathways

Loops connect  helices and  strands Folded State

amino-acid (residue) peptide bonds Protein Sequence Structure

 Kinematic Linkage Model  Conformational space

Molecule  Robot

Why Studying Proteins?  They perform many vital functions, e.g.: catalysis of reactions storage of energy transmission of signals building blocks of muscles  They are linked to key biological problems that raise major computational challenges mostly due to their large sizes (100s to several 1000s of atoms), many degrees of kinematic freedom, and their huge number (millions)

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 Distance computation techniques [Itay Lotan, Fabian Schwarzer, and Danny Halperin (Tel Aviv University)]

Structure Determination: X-Ray Crystallography

Software 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 1 Badger (2003) Acta Cryst. D59  Model completion is high-throughput bottleneck 1.0Å3.0Å

The Completion Problem  Input: Electron-density map Partial structure Two anchor residues Amino-acid sequence of missing fragment (typically 4 – 15 residues long)  Output: Ranked conformations Q of fragment that - Respect the closure constraint - Maximize target function T(Q) measuring fit with electron-density map - No atomic clashes Partial structure (folded) (Inverse Kinematics)

Two-Stage IK Method 1.Candidate generations  Closed fragments 2.Candidate refinement  Optimize fit with EDM

Stage 1: Candidate Generation 1.Generate a 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)

fixed end moving end Closure Distance Closure Distance: Compute + bias toward avoiding steric clashes A.A. Canutescu and R.L. Dunbrack Jr. Cyclic coordinate descent: A robotics algorithm for protein loop closure. Prot. Sci. 12:963–972, 2003.

Exact Inverse Kinematics Repeat for each conformation of a closed fragment: 1.Pick 3 amino-acids at random (3 pairs of  -  angles) 2.Apply exact IK solver to generate all IK solutions [Coutsias et al, 2004]

TM0813 GLU-83 GLY-96

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 d3d3 d2d2 d1d1 (1,2,3)(1,2,3) Null space

Closure and Null Space  dX = J dQ, where J is the 6  n Jacobian matrix (n > 6)  Null space {dQ | J dQ = 0} has dim = n – 6  N: orthonormal basis of null space  dQ = NN T  T(Q) X

dXU66U66 VT6nVT6n dQ 6666 = Computation of N SVD of J 11 22 66 Gram-Schmidt orthogonalization 0 (n-6) basis N of null space NTNT

Refinement Procedure Repeat until minimum of T is reached: 1.Compute J and N at current Q 2.Compute  T at current Q (analytical expression of  T + linear-time recursive computation [Abe et al., Comput. Chem., 1984]) 3.Move by small increment along dQ = NN T  T (+ Monte Carlo / simulated annealing protocol to deal with local minima)

TM0813 GLU-83 GLY-96

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

TM Fragments from TM1621 at 2.5Å Produced by H. van den Bedem Long Fragments: 12: 96% < 1.0Å aaRMSD 15: 88% < 1.0Å aaRMSD Short Fragments: 100% < 1.0Å aaRMSD

Example: TM0423 PDB: 1KQ3, 376 res. 2.0Å resolution 12 residue gap Best: 0.3Å 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 LengthTop scorer 40.22Å 50.78Å 50.36Å 70.72Å Å Produced by H. van den Bedem

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

Current/Future Work A B  Software actively being used at the JCSG  What about multi-modal loops?

 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 A323 Hist A316 Ser

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

Amylosucrase J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. J. Comp. Chemistry, 25: , 2004

Energy maintenance during Monte Carlo simulation joint work with Itay Lotan, Fabian Schwarzer, and Dan Halperin 1 1 Computer Science Department, Tel Aviv University

 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 ~ Monte Carlo Simulation (MCS)

 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 a large number of times Problem: How to maintain energy efficiently? Monte Carlo Simulation (MCS)

Energy Function  E =  bonded terms +  non-bonded terms +  solvation terms  Bonded terms - O(n)  Non-bonded terms - E.g., Van der Waals and electrostatic - Depend on distances between pairs of atoms - O(n 2 )  Expensive to compute  Solvation terms - May require computing molecular surface

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 pairs without enumerating all atom pairs?

Grid Method d cutoff  Subdivide 3-space into cubic cells  Compute cell that contains each atom center  Represent grid as hashtable

Grid Method d cutoff  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

Can we do better on average?  Few DOFs are changed at each MC step Number k of DOF changes simulation of 100,000 attempted steps

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 interacting pairs of atoms are unchanged  Many partial energy sums remain constant Problem: How to find new interacting pairs and retrieve unchanged partial sums?

Two New Data Structures 1.ChainTree  Fast detection of interacting atom pairs 2.EnergyTree  Retrieval of unchanged partial energy sums

ChainTree (Twofold Hierarchy: BVs + Transforms) links

T NO T JK T AB joints ChainTree (Twofold Hierarchy: BVs + Transforms)

Updating the ChainTree Update path to root: –Recompute transforms that “shortcut” the DOF change –Recompute BVs that contain the DOF change –O(k log 2 (2n/k)) work for k changes

Finding Interacting Pairs 

Finding Interacting Pairs

 Do not search inside rigid sub-chains (unmarked nodes)

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

EnergyTree E(N,N) E(J,L) E(K.L) E(L,L) E(M,M)

EnergyTree E(N,N) E(J,L) E(K.L) E(L,L) E(M,M)

Complexity  n : total number of DOFs  k : number of DOF changes at each MCS step  k << n  Complexity of:  updating ChainTree: O(k log 2 (2n/k))  finding interacting pairs: O(n 4/3 ) but p erforms much better in practice!!!

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

Results: 1-DOF change (68)(144)(374) (755) # amino acids speedup

Results: 5-DOF change (68)(144)(374)(755) speedup

Two-Pass ChainTree (ChainTree+) 1 st pass: small cutoff distance to detect steric clashes 2 nd pass: normal cutoff distance >5 Tests around native state

Interaction with Solvent  Implicit solvent model: solvent as continuous medium, interface is solvent-accessible surface E. Eyal, D. Halperin. Dynamic Maintenance of Molecular Surfaces under Conformational Changes.

Summary  Inverse kinematics techniques  Improve structure determination from fuzzy electron density maps  Collision detection techniques  Speedup energy maintenance during Monte Carlo simulation

About Computational Biology  Computational Biology is more than mimicking nature (e.g., performing Molecular Dynamic simulation)  One of its goals is to achieve algorithmic efficiency by exploiting properties of molecules, e.g.: Atoms cannot bunch up together Forces have relatively short ranges Proteins are long kinematic chains