Download Geometric Search and Crystal Structure Determination

Geometric Search and Crystal
Structure Determination
Andy Wilson
Geometric Search and Crystal
Structure Determination
Overview
Driving Problem
 Crystallographic Methods
 Relation to computational geometry
 Solving for phases
 Complications and Future Work
 Conclusions

Geometric Search and Crystal
Structure Determination
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
Two Molecules
Geometric Search and Crystal
Structure Determination
More Molecules
Geometric Search and Crystal
Structure Determination
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
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
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
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
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
Output

System of linear constraints on phases

Phase and magnitude for reflections

Approximate electron density map
Geometric Search and Crystal
Structure Determination
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
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Geometric Search and Crystal
Structure Determination
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
Finding Invariants
Compute strength of reflections
 Choose triplets
 Test strength
 Incorporate into constraints

Geometric Search and Crystal
Structure Determination
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
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
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
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
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
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
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
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
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Seminvariants.
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See Andy or Darlene Freedman to get these.
Geometric Search and Crystal
Structure Determination