Lecture 3: Geometric and Signal 3D Processing (and some Visualization)

Transcription

1 Lecture 3: Geometric and Signal 3D Processing (and some Visualization) Chandrajit Bajaj

2 Algorithms & Tools Structure elucidation: filtering, contrast enhancement, segmentation, skeletonization, subunit identification Structure Modeling: finite element meshing, spline representations(aspline,rbf representations) for structural fitting & complementary docking Visualization: multi-dimensional transfer functions, surface and volume texture rendering, wavelet compression, hierarchical representations, cluster based parallelism VolRover TexMol

3 Sub-nanometer Structure Elucidation from 3D Cryo-EM Cryo-EM FFT based 3D Reconstruction Anisotropic and Vector Diffusion Filtering Structure Segmentation Quasi-Atomic Modeling Visualization Rice Dwarf Virus (6.8A) **Sponsored by NSF-ITR, NIH (Collaborators: Wah Chiu,NCMI, Baylor College of Medicine, Andrej Sali, UCSF)

4 A Structure Determination Pipeline for single particle cryo-em

5 Structure Elucidation for Icosahedral Viruses Input map Segmentation of double-layer capsids Find critical points Detect local symmetry Align subunits Compute Sym. score Fast marching method If known Detect global symmetry Segment capsid layer for double layer Segment outer & inner layers For single layer for each layer Detect local symmetry Align subunits Refinement Color subunits colored map Segment subunits Average subunit averaged map Z. Yu and C. Bajaj, Automatic Ultra-structure Segmentation of Reconstructed Cryo-EM Maps of Icosahedral Viruses, IEEE Transactions on Image Processing, Special Issue on Cellular/Molecular Imaging, 4(9), pp , 005. Generate Trans. matrix Trans. matrix

6 Structure Elucidation (A) Adaptive contrast enhancement Bilateral filtering h ( x ξ ) σ d σ r ( x, ξ ) = e e where σ and σ are parameters and f(.) is d r the image intensity value. Anisotropic diffusion filtering φ div( a( φ ) φ) = t where a stands for the diffusion tensor determined by local curvature estimation. Anisotropic gradient vector diffusion ( f ( x) f ( ξ )) C. Bajaj, G. Xu, ACM Transactions on Graphics, (003),(), 4-3. σ Z. Yu & C. Bajaj, Proc. Int l Conf. Image Processing, 00. pp Z. Yu & C. Bajaj, Proc. Int l Conf. Computer Vision and Pattern Recognition, W. Jiang, M. Baker, Q. Wu, C. Bajaj, W. Chiu, Journal of Structural Biology, 44, 5,(003),4-

7 Compute Critical Points Using AGVD : minimum : maximum : saddle (0) (3) (, )

8 Anisotropic Gradient Vector Diffusion (AGVD) + = + = ) )( ( ) )( ( y x y y x x f f f v v t v f f f u u t u µ µ Where: (u(t), v(t)) stands for the evolving vector field; µ is a constant; f is the original image to be diffused; (f x, f y ) = (u(0), v(0)). Isotropic Diffusion (Xu et al., 998) + = + = ) )( ( ) ) ( ( ) )( ( ) ) ( ( y x y y x x f f f v v g t v f f f u u g t u α µ α µ Where (u(t), v(t)) stands for vector field; µ is a constant; (f x, f y ) = (u(0), v(0)). f is the original image to be diffused; g(.) is the angle between two vectors Anisotropic Diffusion (Yu & Bajaj ICPR 0)

9 GVD v.s. AGVD Isotropic diffusion Anisotropic diffusion

10 Multi-seed Fast Marching Method Structure Elucidation (B) Classify map critical points as seeds based on local symmetry. Each seed initializes a contour, with its group s membership. Contours march simultaneously. Contours with same membership are merged, while contours with different membership stop each other. RDV Φ9 Z. Yu, and C. Bajaj, IEEE Trans.on Image Process, (-), pp

11 Global and Local Symmetry Automatic structure unit identification in a 3D Map R T P Q S Two-fold vertices Three-fold vertices Five-fold vertices Example: RDV

12 Symmetry Detection: Correlation Search C Algorithm: detect 5-fold rotation symmetry Compute the scoring function For every angular bin B j, compute θ, { For every critical point C i { r k ( Ci, B j ) = R( θ, ϕ,kπ /5) Ci, k = 0,,,3,4 Dev( C SF( B, B Locate the symmetry axes The peaks j ) = Refine the symmetry axes i ) = ( θ, ϕ) = f ( r ) f ( R( θ, ϕ,π /5) r V p j p i= 0 In order to locate a perfect icosahedron 5 r j 4 (rotate the axes by 0 0, , , 80 0 ) j k = 0 r ( f ( i j k Dev( C, B ) ) } f ) j ϕ j } r ) Inverted and normalized SF(Bj)

13 Structure Elucidation Results: RDV (Bakeoff) surface rendering (outside) volume rendering (inside) volume rendering (asymmetric unit) averaged trimer (side) averaged trimer (bottom) segmented monomers

14 Structure Elucidation (C): Secondary Structure Identification Beta-sheet G σ I I I x x x I I y z I I I x y y I I y z I I x I I y z z I z Alpha-helix The eigenvectors of the local structure tensor give the principal directions of the local features: λ λ λ λ 3 λ λ 3 Line structure (alpha-helix) plane structure (beta-sheet) λ λ >> λ 0 λ >> λ λ3 0 3

15 Monomeric Unit of Outer Capsid of RDV Beta-sheet Alpha-helix

16 Monomeric Unit of Inner Capsid of RDV Alpha-helix Beta-sheet

17 Structure Elucidation Results: GroEL 6Å Å Data courtesy: Dr. Wah Chiu

18 Segmentation Results: Ribosome (Bakeoff) 70S ribosome from E. coli complex. 70S-tRNAfMet-MF-tRNAPhe. Data courtesy: EBI & J.Frank

19 Structure Elucidation for Symmetric Capsid Viruses Input map Segmentation of double-layer capsids Find critical points Detect local symmetry Align subunits Compute Sym. score Fast marching method If known Detect global symmetry Segment capsid layer for double layer Segment outer & inner layers For single layer for each layer Detect local symmetry Align subunits Refinement Color subunits colored map Segment subunits Average subunit averaged map Z. Yu and C. Bajaj, Automatic Ultra-structure Segmentation of Reconstructed Cryo-EM Maps of Icosahedral Viruses, IEEE Transactions on Image Processing, Special Issue on Cellular/Molecular Imaging, 4(9), pp , 005. Generate Trans. matrix Trans. matrix

20 Subunit alignment (): averaging Φ9 The above two pictures (left: outer; right: inner) show the averaged capsid layer, calculated from one 5-fold subunit (orange) and one 6-fold subunit (green). The tail structure (blue) is augmented after the averaging. Data courtesy: Tim Baker

21 Structure Elucidation (C): Subunit Alignment Cross-correlation #0 # # #3 Φ9. Symmetry score #0 # # # auto-correlation #4 #5 #6 # # # # # # # #0 # # #3 #4 #5 #6 #7 #8 #9 segmented subunit #8 #

22 Structure Elucidation Results: Φ9 Data courtesy: Tim Baker

23 Subunit alignment (): Fitting The PDB structure of one monomer is matched & fit into the cryo- EM map (as shaded in green in the left figure). Then all the quasisymmetric 5-fold subunits are computationally fit with the PDB structure using the transform matrices obtained in subunit alignment. Similar procedure can be applied to all 6-fold subunits.

24 Interactive Fitting Models 3D maps Blurring Synthetic map Critical point evaluation Viewing Transformations Correlation Modules The GroEL map docked with a blur map of OEL.PDB Correlation Department of Computer Sciences University of Texas at Austin September 007

25 Gro-EL: X-ray structures docked in Cryo-EM

26 Interactive Correlation Analysis C = 0.35 C = 0.69 C = A GroEL map and OEL.pdb C = 0.08 C = C = A RDV map and UF.pdb

27 Approximate Correlation Analysis score i= N i= = N f ( c ) g( c ) Where f is the normalized density function of the blurred crystal structure; g is the normalized density function of the cryo-em map; c i, i=,, N, are the critical points of the blurred crystal structure; c i, i=,, N, are the transformations of the critical points. (T, R) i i ' i max( f ( c ), g( c )) ' i c i c i Cryo-EM map (density function: g) Blurred model (density function: f )

28 Blurring I For a molecule with M atoms, we can define a 3D electron density map as M _ dens ( x) = i ( x) i= f elec G For quadratic decay kernels, A i =e d : x R 3 f elec_dens (x) = PM i= A i e à r d x î(c i ) JJE For linear decay kernels, A i =e dri : f elec_dens (x) = PM i= A i e àd x î(c i ) JJ

29 Atomic Shape Parameters Isotropic Quadratic Kernel Isotropic Linear Kernel where The decay d controls the shape of the Gaussian function. The van der Waals radius is r i The center of the atom is x c. à d r G i (x) =e ((xàx ci ) àr ) i i G i (x) =e àd( xàx ci àr i ) Different atom representations quadratic decay linear decay hard sphere d =.3 r =.8A d values suggested in (Boys 50), (Grant Pickup 99) Anisotropic Kernels angstroms

30 Blurring II For quadratic decay kernels, A i =e d : f elec_dens (x) = PM i= A i e à r d x î(c i ) For linear decay kernels, A i =e dri : f elec_dens (x) = PM i= A i e àd x î(c i ) For above kernels G: f elec_dens (x) =G ê PM i= A i î(c i ) PM Ai î( à x à à x i) i= G