Geometric Search and Crystal Structure Determination

Geometric Search and Crystal Geometric Search and Crystal Structure DeterminationStructure Determination

Andy WilsonAndy Wilson

Geometric Search and Crystal Structure Determination

OverviewOverview

Driving ProblemDriving Problem Crystallographic MethodsCrystallographic Methods Relation to computational geometryRelation to computational geometry Solving for phasesSolving for phases Complications and Future WorkComplications and Future Work ConclusionsConclusions

Geometric Search and Crystal Structure Determination

Driving ProblemDriving Problem

What is a protein’s shape?What is a protein’s shape?– Shape determines functionShape determines function– DNA sequencing gives amino acid sequenceDNA sequencing gives amino acid sequence– AA sequence determines primary structureAA sequence determines primary structure– Unknown mapping between AA sequence and Unknown mapping between AA sequence and

secondary structuresecondary structure

Geometric Search and Crystal Structure Determination

Two MoleculesTwo Molecules

Geometric Search and Crystal Structure Determination

More MoleculesMore Molecules

Geometric Search and Crystal Structure Determination

Driving Problem 2Driving Problem 2

Mechanical optimization alone won’t workMechanical optimization alone won’t work– Problem space has several thousand dimensionsProblem space has several thousand dimensions– Local minima are Local minima are everywhereeverywhere– Can Can refine a “close” guessrefine a “close” guess

Humans are good at fitting modelsHumans are good at fitting models– Need something to fit toNeed something to fit to– Can hand off to automated methodsCan hand off to automated methods

Geometric Search and Crystal Structure Determination

ObjectiveObjective

Generate an approximate electron density Generate an approximate electron density map.map.

Have a scientist fit a model to this map.Have a scientist fit a model to this map. Refine using other methods.Refine using other methods.

Geometric Search and Crystal Structure Determination

Observing ProteinsObserving Proteins

X-ray crystallography to the rescue!X-ray crystallography to the rescue!– Crystallize protein, exploit repetitive structureCrystallize protein, exploit repetitive structure– Observe electron density of moleculeObserve electron density of molecule

XRC alone isn’t good enoughXRC alone isn’t good enough– Output is (roughly) the Fourier transform of the Output is (roughly) the Fourier transform of the

electron density mapelectron density map– BUT… the phases are lost irretrievablyBUT… the phases are lost irretrievably– Magnitudes aren’t enough to reconstruct inputMagnitudes aren’t enough to reconstruct input

Geometric Search and Crystal Structure Determination

Recovering PhasesRecovering Phases

Random phasesRandom phases– doesn’t work - not enough structure in doesn’t work - not enough structure in

magnitudesmagnitudes Similar structureSimilar structure

– Guess that trial molecule is like a known oneGuess that trial molecule is like a known one Direct methodsDirect methods

– Exploit relationships between reflectionsExploit relationships between reflections

Geometric Search and Crystal Structure Determination

InputInput Cloud of reflectionsCloud of reflections

– each corresponds to a beam each corresponds to a beam of X-raysof X-rays

– has position, magnitude, has position, magnitude, (unknown) phase(unknown) phase

– arranged on regular latticearranged on regular lattice Symmetry groupSymmetry group

– If molecule has symmetry, If molecule has symmetry, only need solve part of itonly need solve part of it

Geometric Search and Crystal Structure Determination

OutputOutput

System of linear constraints on phasesSystem of linear constraints on phases

Phase and magnitude for reflectionsPhase and magnitude for reflections

Approximate electron density mapApproximate electron density map

Geometric Search and Crystal Structure Determination

Direct MethodsDirect Methods

Solve for sums of phases of a group of Solve for sums of phases of a group of reflectionsreflections

Probabilistic, symbolic methodProbabilistic, symbolic method Relates phases of 3 or 4 reflectionsRelates phases of 3 or 4 reflections Objective: solve for enough phases to Objective: solve for enough phases to

synthesize a rough electron mapsynthesize a rough electron map Method: search for Method: search for phase invariantsphase invariants

Geometric Search and Crystal Structure Determination

Phase InvariantsPhase Invariants

Groups of 3 or 4 reflections with a certain Groups of 3 or 4 reflections with a certain geometric relationshipgeometric relationship

Sum of phases remains constantSum of phases remains constant– probably! Probability increases with strength of probably! Probability increases with strength of

reflectionsreflections Invariant to rotation of moleculeInvariant to rotation of molecule

Geometric Search and Crystal Structure Determination

Finding InvariantsFinding Invariants

Compute strength of reflectionsCompute strength of reflections Choose tripletsChoose triplets Test strengthTest strength Incorporate into constraintsIncorporate into constraints

Geometric Search and Crystal Structure Determination

Reflection StrengthReflection Strength For each reflection:For each reflection:

– Find average magnitude of Find average magnitude of nearby reflectionsnearby reflections

– ““Strength” of a reflection is its Strength” of a reflection is its magnitude divided by average magnitude divided by average over neighborhoodover neighborhood

– An especially strong reflection An especially strong reflection has magnitude greater than 2x has magnitude greater than 2x averageaverage

Geometric Search and Crystal Structure Determination

Searching for InvariantsSearching for Invariants

Choose three reflections Choose three reflections hh, , kk, , -(h-k)-(h-k)

If vector sum of positions is zero, sum of If vector sum of positions is zero, sum of phases is (probably) zerophases is (probably) zero

Geometric Search and Crystal Structure Determination

Searching for invariantsSearching for invariants

Naïve search is at least N choose 2 (or 3)Naïve search is at least N choose 2 (or 3)– O(nO(n22) or O(n) or O(n33), which is expensive with >20000 ), which is expensive with >20000

reflectionsreflections Accelerate search with a spatial data Accelerate search with a spatial data

structurestructure– k-D tree is well suited to this taskk-D tree is well suited to this task– Has to support nearest-neighbor queriesHas to support nearest-neighbor queries– Could probably fake it with range queriesCould probably fake it with range queries

Geometric Search and Crystal Structure Determination

Let the computer searchLet the computer search

Idea 1: search for third reflectionIdea 1: search for third reflection– Pick the first two with for-loopsPick the first two with for-loops– Search nearest neighbors to look for the thirdSearch nearest neighbors to look for the third– Accept or reject based on distance, strengthAccept or reject based on distance, strength

Idea 2: search small chunksIdea 2: search small chunks– Subdivide space with a regular gridSubdivide space with a regular grid– Choose 3 chunks in “invariant pattern”Choose 3 chunks in “invariant pattern”

Geometric Search and Crystal Structure Determination

Using the resultsUsing the results

Invariants specify constraints on phasesInvariants specify constraints on phases Fix one phase, then solve for othersFix one phase, then solve for others Use phases and magnitudes to construct Use phases and magnitudes to construct

electron mapelectron map Let scientist try to fit model to mapLet scientist try to fit model to map Generate more constraints if necessaryGenerate more constraints if necessary

Geometric Search and Crystal Structure Determination

ComplicationsComplications Strength of reflectionsStrength of reflections

– As problem size increases, strength goes downAs problem size increases, strength goes down Do certain structures make certain patterns?Do certain structures make certain patterns?

– Disulfide bondsDisulfide bonds– Alpha helicesAlpha helices– Beta sheets/barrelsBeta sheets/barrels

Memory localityMemory locality– Nested loops in search are harmfulNested loops in search are harmful

Geometric Search and Crystal Structure Determination

Future WorkFuture Work

Implement invariant search.Implement invariant search.– CORWIN already has groundworkCORWIN already has groundwork

Consider substructure invariants.Consider substructure invariants.– Finding them is hardFinding them is hard– Searching for them is even harderSearching for them is even harder

Geometric Search and Crystal Structure Determination

For More InformationFor More Information

GRIP library or team membersGRIP library or team members Dickerson and Geis, Dickerson and Geis, Protein Structure and Action.Protein Structure and Action. Glusker and Trueblood,Glusker and Trueblood, Crystal Structure Analysis: A Crystal Structure Analysis: A

Primer.Primer. Schenk, Schenk, Introduction to Structure Invariants and Introduction to Structure Invariants and

SeminvariantsSeminvariants..

See Andy or Darlene Freedman to get these.See Andy or Darlene Freedman to get these.