Geometric Search and Crystal Structure Determination

1. Geometric Search and Crystal Structure Determination Andy Wilson Geometric Search and Crystal Structure Determination

2. Overview • Driving Problem • Crystallographic Methods • Relation to computational geometry • Solving for phases • Complications and Future Work • Conclusions Geometric Search and Crystal Structure Determination

3. Driving Problem • What is a protein’s shape? • Shape determines function • DNA sequencing gives amino acid sequence • AA sequence determines primary structure • Unknown mapping between AA sequence and secondary structure Geometric Search and Crystal Structure Determination

4. Two Molecules Geometric Search and Crystal Structure Determination

5. More Molecules Geometric Search and Crystal Structure Determination

6. Driving Problem 2 • Mechanical optimization alone won’t work • Problem space has several thousand dimensions • Local minima are everywhere • Can refine a “close” guess • Humans are good at fitting models • Need something to fit to • Can hand off to automated methods Geometric Search and Crystal Structure Determination

7. Objective • Generate an approximate electron density map. • Have a scientist fit a model to this map. • Refine using other methods. Geometric Search and Crystal Structure Determination

8. Observing Proteins • X-ray crystallography to the rescue! • Crystallize protein, exploit repetitive structure • Observe electron density of molecule • XRC alone isn’t good enough • Output is (roughly) the Fourier transform of the electron density map • BUT… the phases are lost irretrievably • Magnitudes aren’t enough to reconstruct input Geometric Search and Crystal Structure Determination

9. Recovering Phases • Random phases • doesn’t work - not enough structure in magnitudes • Similar structure • Guess that trial molecule is like a known one • Direct methods • Exploit relationships between reflections Geometric Search and Crystal Structure Determination

10. Input • Cloud of reflections • each corresponds to a beam of X-rays • has position, magnitude, (unknown) phase • arranged on regular lattice • Symmetry group • If molecule has symmetry, only need solve part of it Geometric Search and Crystal Structure Determination

11. Output • System of linear constraints on phases • Phase and magnitude for reflections • Approximate electron density map Geometric Search and Crystal Structure Determination

12. Direct Methods • Solve for sums of phases of a group of reflections • Probabilistic, symbolic method • Relates phases of 3 or 4 reflections • Objective: solve for enough phases to synthesize a rough electron map • Method: search for phase invariants Geometric Search and Crystal Structure Determination

13. Phase Invariants • Groups of 3 or 4 reflections with a certain geometric relationship • Sum of phases remains constant • probably! Probability increases with strength of reflections • Invariant to rotation of molecule Geometric Search and Crystal Structure Determination

14. Finding Invariants • Compute strength of reflections • Choose triplets • Test strength • Incorporate into constraints Geometric Search and Crystal Structure Determination

15. Reflection Strength • For each reflection: • Find average magnitude of nearby reflections • “Strength” of a reflection is its magnitude divided by average over neighborhood • An especially strong reflection has magnitude greater than 2x average Geometric Search and Crystal Structure Determination

16. Searching for Invariants • Choose three reflections h, k, -(h-k) • If vector sum of positions is zero, sum of phases is (probably) zero Geometric Search and Crystal Structure Determination

17. Searching for invariants • Naïve search is at least N choose 2 (or 3) • O(n2) or O(n3), which is expensive with >20000 reflections • Accelerate search with a spatial data structure • k-D tree is well suited to this task • Has to support nearest-neighbor queries • Could probably fake it with range queries Geometric Search and Crystal Structure Determination

18. Let the computer search • Idea 1: search for third reflection • Pick the first two with for-loops • Search nearest neighbors to look for the third • Accept or reject based on distance, strength • Idea 2: search small chunks • Subdivide space with a regular grid • Choose 3 chunks in “invariant pattern” Geometric Search and Crystal Structure Determination

19. Using the results • Invariants specify constraints on phases • Fix one phase, then solve for others • Use phases and magnitudes to construct electron map • Let scientist try to fit model to map • Generate more constraints if necessary Geometric Search and Crystal Structure Determination

20. Complications • Strength of reflections • As problem size increases, strength goes down • Do certain structures make certain patterns? • Disulfide bonds • Alpha helices • Beta sheets/barrels • Memory locality • Nested loops in search are harmful Geometric Search and Crystal Structure Determination

21. Future Work • Implement invariant search. • CORWIN already has groundwork • Consider substructure invariants. • Finding them is hard • Searching for them is even harder Geometric Search and Crystal Structure Determination

22. For More Information • GRIP library or team members • Dickerson and Geis, Protein Structure and Action. • Glusker and Trueblood, Crystal Structure Analysis: A Primer. • Schenk, Introduction to Structure Invariants and Seminvariants. • See Andy or Darlene Freedman to get these. Geometric Search and Crystal Structure Determination