Application of Probabilistic Roadmaps to the Study of Protein Motion

Proteins  Proteins are the workhorses of all living organisms  They perform many vital functions, e.g: Catalysis of reactions Transport of molecules Building blocks of muscles Storage of energy Transmission of signals Defense against intruders  They are large molecules (few 100s to several 1000s of atoms)  They are made of building blocks (amino acids) drawn from a small “library” of 20 amino-acids  They have an unusual kinematic structure: long serial linkage (backbone) with short side-chains

Protein Sequence O N N N N OO O  Long sequence of amino-acids (dozens to thousands), also called residues  Dictionary of 20 amino-acids (several billion years old) (residue i-1)

Central Dogma of Molecular Biology Physiological conditions: aqueous solution, 37°C, pH 7, atmospheric pressure

Mad cow disease is caused by mis-folding 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

Motion occurs at very different frequencies HIV-1 protease Low-frequency motions (diffusive motions) are more directly related to protein functions

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

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

Kinematic Models  Atomistic model: The position of each atom is defined by its coordinates in 3-D space (x 4,y 4,z 4 ) (x 2,y 2,z 2 ) (x 3,y 3,z 3 ) (x 5,y 5,z 5 ) (x 6,y 6,z 6 ) (x 8,y 8,z 8 ) (x 7,y 7,z 7 ) (x 1,y 1,z 1 ) p atoms  3p parameters Drawback: The bond structure is not taken into account

Kinematic Models  Linkage model: The protein consists of atoms connected by rotatable bonds

Roadmap-Based Representation  Compact representation of many motion pathways  Coarse resolution relative to MC and MD simulation (  only low-frequency motions are represented)  Efficient algorithms for analyzing multiple pathways

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)

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

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)

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!

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)

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