Taking a Numeric Path

1. Taking a Numeric Path Idan Szpektor

2. The Input • A partial description of a molecule: • The atoms • The bonds • The bonds lengths and angles • Spatial constraints on some of the atoms of a molecule.

3. The Questions • Is there a conformation of the molecule that answers the spatial constraints? • Provide explicit description of such conformations. • How many “really” different conformations are there? Which one is the best?

4. Hard life… Torsion angles are (usually) unconstrained  a large number of degrees of freedom  The answers to the questions are not easy to find analytically. They are also resource consuming (running time).

5. A Numeric Path • Two algorithms take different approaches to limit the space of possible conformations to search in: • Numeric randomized algorithm • Semi numeric algorithm on a simpler problem

6. A Randomized Kinematics-Based Approach to Pharmacophore-Constrained Conformational Search and Database Screening S. M. Lavalle, P. W. Finn, L. E. Kavraki, J. C. Latombe (2000)

7. The Pharmacophore Problem • The problem: to find a molecule conformation that satisfies a set of constraints out of a database of flexible molecules (e.g. for ligand docking). • Constraints: • Different atoms and their features in the molecule (types, charges etc.) • The bond length and angles between bonds • Static bond torsion angles • Relative locations of some atoms from an anchor atom

8. The Molecule Model – no rings

9. The Molecule Model (Cont…) • An atom ai carries standard information • A bond bi carries the following information: • li – the bond length • αi – the angle from the previous bond • The set of possible torsion angles θi [0, 2π) • All info besides the torsion angles is fixed. • Θis the m dimensional vector of variables that defines the conformation

10. The Pharmacophore Model • A finite set of features corresponding to a subset of atoms • Constraints on the relative positions between features (the atoms), when one of the features is designated as aanch, the origin of a global xyz coordinate

11. The Kinematic Model • The bond length, angles and the torsion angles Θ can be used to give the positions of all of the atoms, relative to aanch • We look at each atom center as a local coordinate frame. We would like to use the transformation from one coordinate frame to the another.

12. The Kinematic Model (Cont…)

13. The Kinematic Model (Cont…) • The homogeneous transformation is:

14. The Kinematic Model (Cont…) • The xyz position of atom ai is given by:

15. The Kinematic Model (Cont…) The coordinate frame of the molecule could be rotated with respect to the global coordinate frame of the pharmacophore feature positions  Another global coordinate frame transformation is needed

16. The Kinematic Model (Cont…) • Global rotation transformation based on Euler angles γ,φ,ψ:

17. The Kinematic Model (Cont…) • The complete xyz position of atom ai is given by:

18. The Kinematic Error Function • Given Θ, γ,φ and ψ, the total amount of error between the requested feature positions G and the actual feature positions g can be measured as:

19. The Energy Function • In a sense, the energy function measures the likelihood that the molecule will achieve a conformation in nature. • An example for an energy function:

20. Two Questions • For a given molecule from a database and a pharmacophore: • Can the molecule achieve a low-energy conformation that satisfies the given pharmacophore? • What are the “distinct” low-energy conformations that satisfy the pharmacophore?

21. Randomized Conformation Search with Constraints – The Motivation • For the problem, the system of equations is generally under constrained, which leads to a complicated multidimensional solution set. • This consideration, and the need for efficiency, led to the choice of a numerical randomized technique.

22. Randomized Conformation Search with Constraints – The Approach

23. The Search – Gradient Descent • Randomly sample the neighborhood of Pi • Search for a point Pi+1 such that f(Pi+1) < f(Pi) • If such Pi+1 is found, move to Pi+1 and repeat the search

24. Distance Minimization

25. Energy Minimization

26. Integrating into a Database Search • A small set (~several hundreds) of candidate molecules (configurations) are chosen from a database (using 2D information) • Kinematics-based conformational search is performed to further reduce the set of candidates

27. Search in a Set of Candidates • Each time a sample conformation fails to match, the likelihood that the molecule will ever succeed decrease • On the other hand, after any number of ‘fail’ iterations, it is impossible to conclude that the molecule will never succeed

28. Search in a Set of Candidates (cont…) • Perform one attempt (random sample + match search) per molecule • Go through all the molecules in the set and repeat • Stop when a requested number of matches was found or a maximum number of iterations was reached

29. Conformation Clustering • In general, having alternative low-energy conformations is useful because in many cases it is not the lowest-energy conformation that results in docking.

30. Conformation Clustering (cont...) • Use a metric m(θ1, θ2), such as RMS of the displacements of the atoms between two conformations • A threshold mmax is the maximum distance that still regards two conformations as identical (in the same cluster)

31. Clustering Algorithm – Idea • Always keep one representative for each cluster • The representative is the conformation with the lowest energy

32. Clustering Algorithm – Detailed • for a new conformation θi: • if exists another conformation θk such that m(θk, θi) ≤ mmax and e(θk) ≤ e(θi), discard θi • otherwise: • add θi as a new cluster • remove all θk such that m(θk, θi) ≤ mmax

33. Experiments • 2 different pharmacophores, for ACE and Thermolysin inhibitors • 6 different molecules for each pharmacophore, kept in the database in randomly picked conformations • The docked conformations are known

34. Experiments (Cont…) • Cluster distance mmax=1.5 Ǻ • 20 iterations for the different candidates

35. Experiments (Cont…) • In general, conformations’ energies are within 2–7 kcal/mol of the energy of the known docked conformation. • The RMS distance of the conformations from the known docked conformations: • Thermolysin inhibitors – 0.50, 2.96, 0.59, 0.81, 2.40, and 2.56 Å. • ACE inhibitors – 1.26, 1.79, 0.94, 2.03, 1.87, and 1.98 Å.

36. Experiments (Cont…) • A sufficient clustering record for a single molecule required about 5–20 min. • A claim: “Our previous work with randomized techniques has shown that if we continue iterating our algorithm we increase our chances of covering the conformational space of the molecule, and hence, our chances of providing exhaustive information about the constrained conformations of the molecule.”

37. A Cluster record

38. Cyclic coordinate descent: A robotics algorithm for protein loop closure A. A. Canutescu and R. L. Dunbrack Jr. (2003)

39. The Loop Closure Problem • The problem: matching a given loop to a given backbone (e.g. for Homology modeling). • Constraint: connecting the two protein segments on either end of the loop, termed N and C-terminal anchors.

40. Loop Closure Problem (Cont…)

41. Previous Solutions • Number of available conformation is enormous. • Analytical: for 6 degrees of freedom. • Numerical: changing all torsion angles at once to the next “best” position. • Numerical methods are computationally expensive and sometimes unstable.

42. Cyclic Coordinate Descent (CCD) • Originally developed for robotics. • An iterative relaxation algorithm. • Adjust only one degree of freedom at a time.

43. CCD Algorithm • Proceed in an iterative fashion along the chain of degrees of freedom. • Modify each torsion angle so that the end of the loop gets as close as possible to the desired position.

44. CCD Simplicity • One equation in one unknown for each degree of freedom. • The equation provides: • Optimum setting for the variable • First and second derivatives

45. The Main Equation

46. The Main Equation (Cont...)

47. The Main Equation (Cont...) • Multiplying the last two terms by: • Defining: • We get:

48. The Main Equation (Cont...)

49. CCD Benefits • Computationally Fast. • Analytically simple - no singularities. • Constraints can be placed on any degree of freedom. • Using derivatives, small increments in change can be done in preference to of large changes.

50. Test 1 – Success Percentage • 2752 different loops. • 100 randomly different starting conformations for each loop. • A match (closed loop) is when distance from the terminals is less than 0.08 Ǻ. • Maximum 5000 iterative cycles (through all torsion angles) per search.