1. Stochastic roadmap simulation for the study of ligand-protein interactions Mehmet Serkan Apaydin, Carlos E. Guestrin, Chris Varma, Douglas L. Brutlag and Jean- Claude Latombe (Stanford Departments of Computer Science and Biochemistry) Presented by Max Shneider

2. Definitions Stochastic - Random, probabilistic Roadmap - Compact graph structure Torsional – twisting or turning Putative binding sites – different cavities on a protein where a ligand could potentially bind Funnel of Attraction – all ligand conformations within 10 Å RMSD of a binding site conformation

3. Monte Carlo Simulation (MC) Generate paths corresponding to potential motions of the ligand and protein: 1. Select initial conformation of interest 2. Sample new conformation according to move set 3. Accept or reject new conformation based on energy difference with original Drawbacks: Only generates one simulation path at a time Can get stuck in local minima of the energy function (repeatedly sampling many similar conformations)

4. Stochastic Roadmap Simulation (SRS) Example conformation representation - ligand and protein parameters specified as vector (  1,  2, …,  d ) Ligand parameters – 3D coordinates of one atom, torsional angles of remaining atoms Protein parameters – backbone torsional angles Conformational parameters determine interaction between atoms of molecule and between molecules and the medium (ie. van der Waals, electrostatic) Assumes that interactions are described by an energy function that depends only on the conformation of the molecules

5. SRS Roadmap Encode many pathways as a directed graph Node = conformation with each  i sampled randomly from allowed range according to some distribution Find nearest neighbors using some metric (ie. RMS, Euclidean Distance) Edge = probability of the molecules transitioning from a node i to one of its neighbors j: Roadmap contains many simultaneous MC paths Can get individual MC path by starting with top node, choosing successor node at random according to edge probabilities (note: you never need to do this with SRS)

6. SRS Roadmap (cont.) AA BCBC C B A D DD E F G EEFFGG P AB P AC P BD P BE P CF P CG P BB P AA P CC

7. SRS - Properties Implicitly defines a Markov chain that captures stochastic nature of molecular motion Markov property: probability of where the system will go next depends only on its current states, not where it has been Doesn’t suffer from MC’s drawbacks No local minimization problems Orders of magnitude speed-up (can process paths simultaneously in closed form using linear algebra methods)

8. Escape Time Measure of binding affinity (expected number of MC simulation steps for ligand to escape the “funnel of attraction” of the protein’s binding site) Longer escape time = high energy barriers around catalytic site Averaged over many molecular motion pathways Naïve approach – perform many simulation runs on roadmap (start from potential bound conformation, end when ligand escapes from funnel), average number of steps taken in each run Slow, and only provides estimate of escape time!

9. Escape Time (cont.) Better solution – use first-step analysis (from Markov chain theory) Each of the neighboring nodes is either: In the funnel (expected number of further steps = that node’s escape time) Not in the funnel (stop) Can define a linear system with one equation for each roadmap node, solve all escape times simultaneously Very fast, and computes escape times exactly! I J1J1 J2J2 F P IJ2 P IJ1

10. Ligand-Protein Modeling Proteins rigid, ligand flexible One atom in ligand designated as the base with 5 DOF, each additional atom had 1 torsional DOF Bonds in a ring were rigid, no DOF Bond angles and lengths assumed constant Potential function used to calculate free energy incorporated electrostatic, van der Walls, and solvation free energies (resolution of 1 Å or 0.5 Å) Modeled solvent with dielectric of 80, solute with dielectric of 1 Repeated experiments with 6 Å and 8 Å funnels, obtained similar results

11. Study 1 - Effects of Mutations Site-directed mutagenesis – biological method in which a few amino acids are deleted, replaced by other amino acids, or have their side chains altered Computational mutagenesis – same as above, but using computers (faster/easier, but less accurate) Lactate dehydrogenase (LDH) – catalyzes reduction of pyruvate to lactate when bound to NADH Mutated residues near LDH’s catalytic site (computational mutagenesis), observed effects on binding affinity (via escape time)

12. Study 1 – Mutations His193  Ala, Arg106  Ala, both of these together Cause large reduction in energetic structure of active site Show sensitivity of SRS to coarse changes in system Asp195  Asn, Gln101  Arg, Thr245  Gly Cause small or no reduction in energetic structure of active site Show sensitivity of SRS to fine changes in system Generated roadmaps contained 4,000 nodes sampled over whole conformation space, 100 extra nodes sampled around bound conformation Other sampling schemes gave similar results

13. Study 1 – Results = 

14. Study 2 – Distinguishing Catalytic Site Shape and electrostatic complementarity between catalytic site and ligand  tight bond Singh et al. (1999) – Studied 3 different ligand-protein complexes: Bound state energy not good at discriminating catalytic site from other putative sites Instead compared average path weight of most energetically stable paths entering and leaving the sites  energy barrier around catalytic site Study expands on this idea: SRS/first-step analysis measures whole energy barrier, not just small part corresponding to most feasible paths Escape time more precise than average path weight

15. Study 2 - Methods Each test conducted with true bound conformation and 4 other putative conformations with: Lowest energies, close to protein surface ( 10 Å) 20 roadmaps/complex, each with set of random conformations and 100 extra conformations around each putative binding site conformation Took < 4 mins. to generate roadmaps, and < 4.5 mins to compute escape times on desktop computer

16. Study 2 - Results

17. Summary Can compute escape time efficiently with SRS to analyze ligand-protein interactions Study 1 – showed high sensitivity of SRS by performing computational mutagenesis on catalytic site of protein In all 6 cases, SRS simulation results agreed with expected biological interpretation of mutation Study 2 – used escape time as metric to distinguish catalytic site from 4 other putative sites In 5/7 cases, escape time distinguished the catalytic site by over 2 orders of magnitude difference from other putative binding sites

18. Future SRS is a general tool, and could be used to efficiently compute other interesting metrics in addition to escape time (binding time, total energy difference along binding paths, etc.) Combine SRS with other techniques to model simultaneously interactions of many molecules (current representation only models one ligand-protein complex)

19. Discussion Questions What are some explanations for why the escape times of the putative sites were higher than the catalytic site in study 2’s failed cases? The paper showed that escape time could be useful in distinguishing the catalytic site. What are other possible applications of escape time? Did the way in which they modeled ligands and proteins affect the results of the studies?