Molecular Motion Pathways: Computation of Ensemble Properties with Probabilistic Roadmaps 1)A.P. Singh, J.C. Latombe, and D.L. Brutlag. A Motion Planning Approach to Flexible Ligand Binding. Proc. 7th Int. Conf. on Intelligent Systems for Molecular Biology (ISMB), AAAI Press, Menlo Park, CA, pp , )N.M. Amato, K.A. Dill, and G. Song. Using Motion Planning to Map Protein Folding Landscapes and Analyze Folding Kinetics of Known Native Structures. J. Comp. Biology, 10(2): , )M.S. Apaydin, D.L. Brutlag, C. Guestrin, D. Hsu, J.C. Latombe, and C. Varma. Stochastic Roadmap Simulation: An Efficient Representation and Algorithm for Analyzing Molecular Motion. J. Comp. Biology, 10(3-4): , )N. Singhal, C.D. Snow, and V.S. Pande. Using Path Sampling to Build Better Markovian State Models: Predicting the Folding Rate and Mechanism of a Tryptophan Zipper Beta Hairpin, J. Chemical Physics, 121(1): , )J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. Geometric Algorithms for the Conformational Analysis of Long Protein Loops. J. Comp. Chemistry, 25: , 2004.

Mad cow disease is caused by misfolding Drug molecules act by binding to proteins Molecular motion is an essential process of life

So, studying molecular motion is of critical importance in molecular biology Stanford BioX cluster NMR spectrometer However, few tools are available Computer simulation: - Monte Carlo simulation - Molecular Dynamics

Two Major Drawbacks of MD and MC Simulation 1)Each simulation run yields a single pathway, while molecules tend to move along many different pathways  Interest in ensemble properties

Example of Ensemble Property: Probability of Folding p fold Unfolded stateFolded state p fold 1- p fold Measure kinetic distance to folded state Du, Pande, Grosberg, Tanaka, and Shakhnovich. On the Transition Coordinate for Protein Folding. Journal of Chemical Physics (1998).

Other Examples of Ensemble Properties  Folding: Order of formation of SSE’s Folding rate / Mean first passage time Key intermediates  Binding: Average time to escape from active site Average energy barrier

Two Major Drawbacks of MD and MC Simulation 1)Each simulation run yields a single pathway, while molecules tend to move along many different pathways 2)Each simulation run tends to waste much time in local minima

 Roadmap-Based Representation  Compact representation of many motion pathways  Coarse resolution relative to MC and MD simulation  Efficient algorithms for analyzing multiple pathways

Roadmaps for Robot Motion Planning free space [Kavraki, Svetska, Latombe,Overmars, 96]

Initial Work A.P. Singh, J.C. Latombe, and D.L. Brutlag. A Motion Planning Approach to Flexible Ligand Binding. Proc. 7th ISMB, pp , 1999  Study of ligand-protein binding  The ligand is a small flexible molecule, but the protein is assumed rigid  A fixed coordinate system P is attached to the protein and a moving coordinate system L is defined using three bonded atoms in the ligand  A conformation of the ligand is defined by the position and orientation of L relative to P and the torsional angles of the ligand

Roadmap Construction (Node Generation)  The nodes of the roadmap are generated by sampling conformations of the ligand uniformly at random in the parameter space (around the protein)  The energy E at each sampled conformation is computed: E = E interaction + E internal E interaction = electrostatic + van der Waals potential E internal =  non-bonded pairs of atoms electrostatic + van der Waals

Roadmap Construction (Node Generation)  The nodes of the roadmap are generated by sampling conformations of the ligand uniformly at random in the parameter space (around the protein)  The energy E at each sampled conformation is computed: E = E interaction + E internal E interaction = electrostatic + van der Waals potential E internal =  non-bonded pairs of atoms electrostatic + van der Waals  A sampled conformation is retained as a node of the roadmap with probability: 0if E > E max E max -E E max -E min 1if E < E min  Denser distribution of nodes in low-energy regions of conformational space P = if E min  E  E max

Roadmap Construction (Edge Generation) qq’  Each node is connected to its closest neighbors by straight edges  Each edge is discretized so that between q i and q i+1 no atom moves by more than some ε (= 1Å)  If any E(q i ) > E max, then the edge is rejected qiqi q i+1 E E max

Heuristic measure of energetic difficulty or moving from q to q’ Roadmap Construction (Edge Generation) qq’  Any two nodes closer apart than some threshold distance are connected by a straight edge  Each edge is discretized so that between q i and q i+1 no atom moves by more than some ε (= 1Å)  If all E(q i )  E max, then the edge is retained and is assigned two weights w(q  q’) and w(q’  q) where: (probability that the ligand moves from q i to q i+1 when it is constrained to move along the edge) qiqi q i+1

 For a given goal node q g (e.g., binding conformation), the Dijkstra’s single-source algorithm computes the lowest-weight paths from q g to each node (in either direction) in O(N logN) time, where N = number of nodes  Various quantities can then be easily computed in O(N) time, e.g., average weights of all paths entering q g and of all paths leaving q g (~ binding and dissociation rates K on and K off ) Querying the Roadmap Protein: Lactate dehydrogenase Ligand: Oxamate (7 degrees of freedom)

Experiments on 3 Complexes 1)PDB ID: 1ldm Receptor: Lactate Dehydrogenase (2386 atoms, 309 residues) Ligand: Oxamate (6 atoms, 7 dofs) 2)PDB ID: 4ts1 Receptor: Mutant of tyrosyl-transfer-RNA synthetase (2423 atoms, 319 residues) Ligand: L- leucyl-hydroxylamine (13 atoms, 9 dofs) 3)PDB ID: 1stp Receptor: Streptavidin (901 atoms, 121 residues) Ligand: Biotin (16 atoms, 11 dofs)

Computation of Potential Binding Conformations 1)Sample many (several 1000’s) ligand’s conformations at random around protein 2)Repeat several times:  Select lowest-energy conformations that are close to protein surface  Resample around them 3)Retain k (~10) lowest-energy conformations whose centers of mass are at least 5Å apart lactate dehydrogenase active site

Results for 1ldm  Some potential binding sites have slightly lower energy than the active site  Energy is not a discriminating factor  Average path weights (energetic difficulty) to enter and leave binding site are significantly greater for the active site  Indicates that the active site is surrounded by an energy barrier that “traps” the ligand

Energy Conformation Potential binding site Active site

 Known native state  Degrees of freedom: φ-ψ angles  Energy: van der Waals, hydrogen bonds, hydrophobic effect  New idea: Sampling strategy  Application: Finding order of SSE formation Application of Roadmaps to Protein Folding N.M. Amato, K.A. Dill, and G. Song. Using Motion Planning to Map Protein Folding Landscapes and Analyze Folding Kinetics of Known Native Structures. J. Comp. Biology, 10(2): , 2003

 High dimensionality  non-uniform sampling  Conformations are sampled using Gaussian distribution around native state  Conformations are sorted into bins by number of native contacts (pairs of C  atoms that are close apart in native structure)  Sampling ends when all bins have minimum number of conformations  “good” coverage of conformational space Sampling Strategy (Node Generation)

 The lowest-weight path is extracted from each denatured conformation to the folded one  The order of formation of SSE’s is computed along each path  The formation order that appears the most often over all paths is considered the SSE formation order of the protein Application: Order of Formation of Secondary Structures

1)The contact matrix showing the time step when each native contact appears is built Method

Protein CI2 (1  + 4  )

Protein CI2 (1  + 4  ) 60 5 The native contact between residues 5 and 60 appears at step 216

1)The contact matrix showing the time step when each native contact appears is built 2)The time step at which a structure appears is approximated as the average of the appearance time steps of its contacts Method

Protein CI2 (1  + 4  )  forms at time step 122 (II)  3 and  4 come together at 187 (V)  2 and  3 come together at 210 (IV)  1 and  4 come together at 214 (I)  and  4 come together at 214 (III)

1)The contact matrix showing the time step when each native contact appears is built 2)The time step at which a structure appears is approximated as the average of the appearance time steps of its contacts Method

Comparison with Experimental Data CI2 1  +5  33 1  +4  5126, 70k 5471, 104k 7975, 104k 8357, 119k roadmap sizeSSE’s

Stochastic Roadmaps M.S. Apaydin, D.L. Brutlag, C. Guestrin, D. Hsu, J.C. Latombe and C. Varma. Stochastic Roadmap Simulation: An Efficient Representation and Algorithm for Analyzing Molecular Motion. J. Comp. Biol., 10(3-4): , 2003 New Idea: Capture the stochastic nature of molecular motion by assigning probabilities to edges vivi vjvj P ij

Edge probabilities Follow Metropolis criteria: Self-transition probability: vjvj vivi P ij P ii [Roadmap nodes are sampled uniformly at random and energy profile along edges is not considered]

V Stochastic Roadmap Simulation P ij Stochastic roadmap simulation and Monte Carlo simulation converge to the Boltzmann distribution, i.e., the number of times SRS is at a node in V converges toward when the number of nodes grows (and they are uniformly distributed)

Roadmap as Markov Chain  Transition probability P ij depends only on i and j P ij i j

Example #1: Probability of Folding p fold Unfolded stateFolded state p fold 1- p fold

P ii F: Folded stateU: Unfolded state First-Step Analysis P ij i k j l m P ik P il P im Let f i = p fold (i) After one step: f i = P ii f i + P ij f j + P ik f k + P il f l + P im f m =1  One linear equation per node  Solution gives p fold for all nodes  No explicit simulation run  All pathways are taken into account  Sparse linear system

Number of Self-Avoiding Walks on a 2D Grid 1, 2, 12, 184, 8512, , , , , (10x10) , (11x11) , (12x12) >

In contrast … Computing p fold with MC simulation requires: For every conformation q of interest  Perform many MC simulation runs from q  Count number of times F is attained first

Computational Tests 1ROP (repressor of primer) 2  helices 6 DOF 1HDD (Engrailed homeodomain) 3  helices 12 DOF H-P energy model with steric clash exclusion [Sun et al., 95]

1ROP Correlation with MC Approach

p fold for ß hairpin Immunoglobin binding protein (Protein G) Last 16 amino acids Cα based representation Go model energy function 42 DOFs [Zhou and Karplus, `99]

Comparison between SRS and MC for ß hairpin for ~100 conformations

Computation Times (ß hairpin) Monte Carlo (30 simulations): 1 conformation ~10 hours of computer time Over 10 7 energy computations Roadmap: 2000 conformations 23 seconds of computer time ~50,000 energy computations ~6 orders of magnitude speedup!

Example #2: Ligand-Protein Interaction Computation of escape time from funnels of attraction around potential binding sites Funnel of attraction = ball of 10Å rmsd around bound state [Camacho and Vajda, 01]

Computation Through Simulation [Sept, Elcock and McCammon `99] 10K to 30K independent simulations

Computing Escape Time with Roadmap  i = 1 + P ii  i + P ij  j + P ik  k + P il  l + P im  m (escape time is measured as number of steps of stochastic simulation) = 0 Funnel of Attraction i j k l m P ii P im P il P ik P ij

Distinguishing Active Site Given several potential binding sites, which one is the active one? Energy: electrostatic + van der Waals + solvation free energy terms

Complexes Studied ligandprotein# random nodes # DOFs oxamate1ldm80007 Streptavidin1stp Hydroxylamine4ts COT1cjw THK1aid IPM1ao PTI3tpi800013

Distinction Using Escape Time ProteinBound state Best potential binding site 1stp3.4E+91.1E+7 4ts13.8E+101.8E+6 3tpi1.3E+115.9E+5 1ldm8.1E+53.4E+6 1cjw5.4E+84.2E+6 1aid9.7E+51.6E+8 1ao56.6E+75.7E+6 (# steps) Able to distinguish catalytic site Not able

Using Path Sampling to Construct Roadmaps N. Singhal, C.D. Snow, and V.S. Pande. Using Path Sampling to Build Better Markovian State Models: Predicting the Folding Rate and Mechanism of a Tryptophan Zipper Beta Hairpin, J. Chemical Physics, 121(1): , 2004 New idea: Paths computed with Molecular Dynamics simulation techniques are used to create the nodes of the roadmap  More pertinent/better distributed nodes  Edges are labeled with the time needed to traverse them

 t U F Sampling Nodes from Computed Paths (Path Shooting)

U F i j t ij p ij Example: Langevin dynamics equation of motion is where R is a Gaussian random force

Node Merging If two nodes are closer apart than some , they are merged into one and merging rules are applied to update edge probabilities and times P 12, t 12 P 14, t ’ P 12’, t 12’ P 12’ = P 12 + P 14 t 12’ = P 12 x t 12 + P 14 x t 14

Node Merging If two nodes are closer apart than some , they are merged into one and merging rules are applied to update edge probabilities and times P 12, t 12 P 14, t ’ P 12’, t 12’ P 12’ = P 12 + P 14 t 12’ = P 12 x t 12 + P 14 x t 14  Approximately uniform distribution of nodes over the reachable subset of conformational space

Application: Computation of MFPT  Mean First Passage Time: the average time when a protein first reaches its folded state  First-Step Analysis yields:  MPFT(i) =  j P ij x (t ij + MPFT(j))  MPFT(i) = 0 if i  F  Assuming first-order kinetics, the probability that a protein folds at time t is: where r is the folding rate  MFPT = =1/r

Computational Test  12-residue tryptophan zipper beta hairpin (TZ2)  used to generate trajectories (fully atomistic simulation) ranging from 10 to 450 ns  1750 trajectories (14 reaching folded state)  22,400-node roadmap  MFPT ~ 2-9  s, which is similar to experimental measurements (from fluorescence and IR)

Conformational Analysis of Protein Loops J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. Geometric Algorithms for the Conformational Analysis of Long Protein Loops. J. Comp. Chemistry, 25: , 2004 New idea: Explore the clash-free subset of the conformational space of a loop, by building a tree-shaped roadmap Kinematic model:  -  angles on the backbone +  i torsional angles in side-chains

Amylosucrase (AS) - Only enzyme in its family that acts on sucrose substrate -The 17-residue loop (named loop 7) between Gly433 and Gly449 is believed to play a pivotal role

Roadmap Construction  A tree-shaped roadmap is created from a start conformation q start  At each step of the roadmap construction, a conformation q rand of the loop is picked at random, and a new roadmap node is created by iteratively pulling toward it the existing node that is closest to q rand

Roadmap Construction C C free C closed q start q rand

Roadmap Construction C C free C closed q start q rand

Roadmap Construction C C free C closed q start q rand

Roadmap Construction C C free C closed q start q rand Stops when one can’t get closer to q rand or a clash is detected

Computational Results  Surprisingly, loop 7 can’t move much  Main bottleneck is residue Asp231 Positions of the C  atom of middle residue (Ser441)

Computational Results  Surprisingly, loop 7 can’t move much  Main bottleneck is residue Asp231

Computational Results  If residue Asp231 is “removed”, then loop 7’s mobility increases dramatically. The C  atom of Ser441 can be displaced by more than 9Å from its crystallographic position

Conclusion  Probabilistic roadmaps are a recent, but promising tool for exploring conformational space and computing ensemble properties of molecular pathways  Current/future research: Better sampling strategies able to handle more complex molecular models (protein-protein binding) More work to include time information in roadmaps More thorough experimental validation to compare computed and measured quantitative properties