applications of fourier theory: template matchingtemplate matching jochen heyd school of health...
TRANSCRIPT
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Applications of Fourier Theory:Template Matching
Jochen HeydSchool of Health Information Sciences &
Institute of Molecular MedicineUniversity of Texas – Houston
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The Goal
Atomic-levelDescriptionEM Density
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Contents
• Template convolution• Fast translational matching• The correlation landscape and ways to
improve it• Assessment using simulated data• Application to experimental microtubule
data
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Template “Convolution”: An Example
Possible match
Possible match
Scoring function: cross-correlation
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Template “Convolution”: An Example
Better match
Better match
Scoring function: cross-correlation
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g = Gaussian kernel
latticeprojection
low-passfilter
tri-linear interpolation
target density on lattice
Fitting criterion: e.g. linear cross-correlation,evaluate for every rotation R and translation T
rotated probe molecule density projected to the lattice:
em( )ρ r
atomic( )ρ rR
atomic
calc
( )
( )
g ρ
ρ
⊗
≡
rR
r
Template “Convolution”
P. Chacón
r )Tr( )r( T)( 3calcem dC +⋅= ∫ ρρ
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Computational Cost• Three translational degrees of freedom
– N possible locations• Three rotational degrees of freedom
– M possible orientations• Cost for each cross-correlation calculation
– N (number of voxels)
Total cost: N * M * N = M * N2
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FTM (Fast Translational Matching)
r )Tr( )r( T)( 3calcem dC +⋅= ∫ ρρ
Using the Fourier Convolution Theorem, we get
The expression for the cross-correlation is
Needs to be calculated only ONCE
[ ])F( )F(F T)( -1calcemC ρρ ⋅= ∗
This yields ALL possible translations in one step!
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FTM (Fast Translational Matching)
r )Tr( )r( T)( 3calcem dC +⋅= ∫ ρρ
Using the Fourier Convolution Theorem, we get
The expression for the cross-correlation is
[ ])F( )F(F T)( -1calcemC ρρ ⋅= ∗
This yields ALL possible translations in one step!
Need to be calculated for every orientation
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Computational Cost• Three translational degrees of freedom
– N possible locations• Three rotational degrees of freedom
– M possible orientations• Cost for each cross-correlation calculation
– N 2 N log(N)
Total cost: M * 2 N log(N)
For a 503 map this results in a speedup of 4 orders of magnitude!
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FTM: An Example
FFT
FFT
Mul
tiply
IFFT
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LOW-RESOLUTIONEM MAP
ROTATE
FFT
UPDATE LATTICE:SAVE MAX. C VALUE + CORRESPONDING Θ,Φ,Ψ
ATOMICSTRUCTURE
FFT
TARGET PROBE
GAUSSIAN g
em( )ρ r atomic( )ρ r
atomic ( )ρ rR
atomic
calc
( )
( )
g ρ
ρ
⊗
≡
rR
r
ROTATIONAL SPACE3 EULER ANGLES
Θ,Φ,Ψ → R
6D Search with FTM
P. Chacón
[ ])F( )F(F-1calcem ρρ ⋅∗
COMPUTE CORRELATION
∗)F( emρ )F( calcρ
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Correlation Landscape
Translation Function
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P. Chacón
Correlation Landscape
With density cross-correlationwe cannot distinguish between correct and spurious fit
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calcmask 0l,m,n
ρ→ >
Density MaskingRenormalize (mask) the correlation locally:
3em calc
2 3 2 3em calc
( ) ( ) d r
( )( )d r ( ) d r
mask
mask mask
C
ρ ρ
ρ ρ
+
=
×∫
∫ ∫
r r T
Tr r
DOCKEM, A.M. Roseman
• Extends the reliability ofcorrelation based docking (<15Å)
• Cannot be easily FTTaccelerated
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Adding surface/contour information
dockingfilter e
A suitable filter would assign negative values to the interior, positive values to the molecular contour. Both volume and contour matches would provide positive contributions to the correlation criterion:
Density Filtering
P. Chacón
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2
2
2
2
2
22
zyx ∂∂
∂∂
∂∂
++=∇
Laplacian
Contour Filter
JMB (2002) March 29 issueP. Chacón
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∗⋅ ) F( emρ
LOW-RESOLUTIONEM MAP
ROTATE
FFT
UPDATE LATTICE:SAVE MAX. C VALUE + CORRESPONDING Θ,Φ,Ψ
ATOMICSTRUCTURE
FFT
TARGET PROBE
GAUSSIAN g
em( )ρ r atomic( )ρ r
atomic ( )ρ rR
atomic
calc
( )
( )
g ρ
ρ
⊗
≡
rR
r
ROTATIONAL SPACE3 EULER ANGLES
Θ,Φ,Ψ → R
6D Search with FTM and Filtering
P. Chacón
[ ]) F( ) F(F-1calcem ρρ ⋅⋅⋅ ∗
COMPUTE CORRELATION
∗)F( emρ )F( calcρ
Filter
Filter
e ) F( calcρ⋅e
e e
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Effect of Filter on Orientation
No Filter
Laplacian Filter
P. Chacón
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with Laplacian filteringStandard cross-correlation
Example: RecA Translation Function
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Grid size 6ÅResolution 15Å9° steps (30481 rotations)
Only Laplacian filtering successfully restores the initial position
standard cross-correlation
w/ Laplacian filtering
Example: RecA
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Search GranularityTranslationalGranularity
Originates fromvoxel spacing
RotationalGranularity
Originates fromangular sampling
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The exhaustive search is limited to a grid of points in the 6D search space
Off-lattice (6D) local maximizationof the correlation coefficient
Improve the accuracy
Powell’s quadratically convergent maximization method can be used to perform a 6D search around the best fits found on the grid.
Off-Lattice Refinement
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Initial positionIteration 1Iteration 2Iteration 3Iteration 4Iteration 5
Off-Lattice Refinement: Example
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Complete Workflow
6D Search Find Peaks
OutputResults
OptimizePeaks
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Restoring Various Oligomers
Thiolase1AFW
Oxido-reductase
1NIC
RecA2REC
Catalase7CAT
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Grid size 6ÅResolution 15 Å
6D search9º angular sampling
(30481 rotations)
+ + …
Input
ValidationRMSD between
target and docked
structures
Restoration Tests with Simulated Data
P. Chacón
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RecA (2REC), thiolase (1AFW), catalase (7CAT), and oxidoreductase (1NIC).
local mask
Restoring Various Oligomersno filter Laplacian filter
P. Chacón
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β-tubulin
α-tubulin
Resolution 20ÅAngular sampling 9ºGrid size 5Å
Application to Microtubule Data
P. Chacón
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Microtubule Model
P. Chacón
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The Situs Software Package
6D Search Find Peaks
OutputResults
OptimizePeaks
Colores
OptimizeSolutions
ColacorTutorial: Thursday afternoon
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Correlations: SummaryNo filter Local mask
Increasing Fitting Contrast
Summary: Correlation Based Matching
• Rigid Body• Density Filtering• Fast Translational Matching • Fast Rotational Matching (later talk)
Laplacian filter
Situs 6D exhaustive searches:
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Resources and AcknowledgementsWWW:
http://situs.biomachina.orghttp://situs.biomachina.org/tutorial_colores.html
Papers:http://situs.biomachina.org/fref.html
Acknowledgement:Pablo Chacón, Valerio Mariani, Paul Boyle,Willy Wriggers