Rigid Body Docking. Willy Wriggers School of Health Information Sciences & Institute of Molecular Medicine University of Texas Houston

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1 Rigid Body Docking Willy Wriggers School of Health Information Sciences & Institute of Molecular Medicine University of Texas Houston

2 14 Years of Multi-Scale Modeling in EM Guoji Wang, Claudine Porta, Zhongguo Chen, Timothy S. Baker, John E. Johnson: Identification of a Fab interaction footprint site on an icosahedral virus by cryoelectron microscopy and X-ray crystallography. Nature, 355:275, Phoebe L. Stewart, Stephen D. Fuller, Roger M. Burnett: Difference imaging of adenovirus: bridging the resolution gap between X-ray crystallography and electron microscopy. EMBO J., 12:2589, At that time, placing an atomic structure into an EM map seemed like a very dangerous idea Phoebe Stewart, 2003

3 1998: Simulated Markers Actin filament: Reconstruction from EM data at 20Å resolution rmsd: 1.1Å Willy Wriggers, Ronald A. Milligan, Klaus Schulten, and J. Andrew McCammon: Self-Organizing Neural Networks Bridge the Biomolecular Resolution Gap. J. Mol. Biol., 284:1247, 1998

4 1999: The First Algorithmic Packages Willy Wriggers, Ronald A. Milligan, and J. Andrew McCammon: Situs: A Package for Docking Crystal Structures into Low-Resolution Maps from Electron Microscopy. J. Structural Biology, 125:185, 1999 Niels Volkmann and Dorit Hanein: Quantitative Fitting of Atomic Models into Observed Densities Derived by Electron Microscopy. J. Structural Biology, 125:176, Present: The Gold Rush era: Many other labs join the effort to develop multi-scale modeling and visualization tools

5 Advantages of a Collaborative Design VMD, Chimera Sculptor

6 Resumé: ~2000 unique, registered users of Situs 6 workshops in Houston, San Diego, New Hampshire, Finland, France ~1000 citations 34 collaborators around the world 25 publications (our lab) ~50 publications using Situs (collaborators)

7 Combining Multi-Resolution Biophys. Data Actin filament (Holmes et al., 1990) Convolution with Gaussian: G () 3r 2 r = exp 2σ 2 2σ=0Å 2σ=10Å 2σ=20Å 2σ=40Å Q: We can lower the resolution of 3D data, but how can one increase it? A: Combine low- with high-resolution data by flexible and rigid-body fitting.

8 6D Search with FTM TARGET PROBE LOW-RESOLUTION EM MAP ATOMIC STRUCTURE ρ r ρatomic( r) em( ) ROTATIONAL SPACE 3 EULER ANGLES Θ,Φ,Ψ R FILTER e FFT ( ) * f e ρ em ( h) g ρ ρ atomic calc ρ atomic ( rr) ( rr) () r GAUSSIAN g FILTER e ROTATE f UPDATE LATTICE: SAVE MAX. C VALUE + CORRESPONDING Θ,Φ,Ψ FFT ( e ρ ) COMPUTE CORRELATION calc ( h) f 1 f f C( T) = ( e ρem ) ( e ρ ) calc *

9 Example: RecA Grid size 6Å Resolution 15Å 9 0 steps (30481 rotations) standard crosscorrelation w/ Laplacian filtering Only Laplacian filtering successfully restores the initial pose

10 Registration of Two EM Maps E. Coli RNA polymerase (Seth Darst) E. Coli RNA polymerase + factor GreB Problem: different helical arrays Need to perform difference mapping to localize GreB (difficult at variable helical symmetry)

11 Registration and Difference Mapping Rigid-body docking: The RNAP jaws are open in presence of GrepB factor, perform flexible map fitting Map fitting new in Situs 2.2.

12 FRM: Fast Rotational Matching Euler angle search is expensive! 9º angular sampling (30481 rotations) requires > 10 minutes on standard workstation for rotations only. Rotations + translations: hours. Our Goal: We seek to FFT-accelerate rotational search in addition to translational search. To do this, we need to do the math in rotational space and take advantage of expressions similar to convolution theorem that are best described by group theory.

13 Expansion in Spherical Harmonics Target map f f ( su) = B 1 l l= 0 m= l fˆ lm ( s) Y lm ( u) Ylm are the spherical harmonic functions Probe map g g( su) = B 1 l l= 0 m= l gˆ lm ( s) Y lm ( u)

14 FRM 3D Method The correlation function is: cr ( ) = f gr ( ) = cα (, βγ, ) α,β,γ are specially chosen Euler angles (origin shift). R 3 Expanding f and g in spherical harmonics as before l π l π l we arrive at: cˆ( p, q, r) = dpq ( ) d ( ) 2 qr I 2 pr where: I l pr = 0 fˆ lp ( s) gˆ l lr ( s) s 2 ds c( α, β, γ) = FFT ( c) 1 ˆ 3D l π The quantities d pq ( 2 ) (which come from rotation group theory) are precomputed using a recursive procedure.

15 Comparison between FRM 3D and Crowther cr ( ) = cα (, βγ, ) π π cpqr ˆ(,, ) = d ( ) d ( ) I 2 2 l Precomputations: FFT setup l π ( 2 ) d pq l l l pq qr pr cr ( ) = cφ (, ψθ ; ) l cˆ( p, r; θ) = dpr ( θ) I Precomputations: FFT setup l l pr FRM Reading maps COMs & sampling f ˆ ( r), gˆ ( r), lp lr cpqr ˆ(,, ) I l pr c( α, β, γ) = FFT ( c) 1 ˆ 3D Crowther k = 0,, B Reading maps COMs & sampling f ˆ ( r), gˆ ( r), lp lr kπ l θ k = d ( θ ), B pr k I l pr cˆ( p, r; θ k ) 1 c( φψθ, ; ) = FFT ( c) k ˆ 2D Postprocessing & Saving Postprocessing & Saving

16 Timings of FRM 3D and Crowther (seconds) angular sampling Crowther FRM

17 FRM 6D (Rigid-Body Matching) 5 angular parameters. The correlation function is now: c( R, R ; δ ) = R 3 ( e ρ ) R ( e ρ ) em ( ) ( R'; δ ). calc δ ρ calc 1 linear parameter remains, distance δ of movement along the z axis. ρ em

18 Rigid-Body Search by 5D FFT The 5D Fourier transform of the correlation function turns out to be: n l l l l ll cnhmh ˆ(,,,, m ; δ ) = ( 1) dnhdhmd nh dh m Imnm ( δ ). ll, ll The quantities Imnm ( δ ) are the so called two-center integrals, corresponding to the spherical harmonic transforms of the two maps, at a distance δ of one another: π ll 1 1 lm l ' m' l 2 l I ( ) ( )( ) ˆ ( ) ˆ mnm δ = l+ l + ( ) 2 2 ρem r ρcalc r dn0( β ) r dr dn0( β)sinβ dβ 0 0

19 Test Cases 1afw (peroxisomal thiolase) 1nic (coppernitrite reductase)

20 Test Cases 7cat (catalase) 1e0j (Gp4D helicase)

21 Efficiency: FRM vs. FTM FTM: 3D FFT + 3D rot. search FRM: 5D FFT + 1D trans. search FTM 1000 FRM 11 sampling, FTM Time (min) 100 standard workstation. FRM 10 1 typical EM map size Number of voxels Gain: 2-4 orders of magnitude for typical EM map

22 Summary: Correlation Based Matching Correlations: Summary Input Situs 6D exhaustive searches: Rigid Body Fast Translational Matching (2.2) Fast Rotational Matching (2.3) Density Filtering No filter Local mask Laplacian filter Increasing Fitting Contrast

23 Questions?

24 Alternative: Simulated Markers Alternative Actin filament: Reconstruction from EM data at 20Å resolution rmsd: 1.1Å

25 Reduced Representations of Biomolecular Reduced Representations Structure of Biomolecular Structure calc w j em w i Feature points (fiducials, landmarks), reduce complexity of search space Useful for: Rigid-body fitting Flexible fitting Interactive fitting / force feedback Building of deformable models

26 Vector Quantization Lloyd (1957) Linde, Buzo, & Gray (1980) Martinetz & Schulten (1993) R d =3 } Digital Signal Processing, Speech and Image Compression. Topology-Representing Network. { } Encode data (in ) using a finite set (j=1,,k) of codebook vectors. R 3 Delaunay triangulation divides into k Voronoi polyhedra ( receptive fields ): V i = w j { } 3 v R v w v w j i j w 4 w 2 w 1 w 3

27 Linde, Buzo, Gray (LBG) Algorithm Encoding Distortion Error: w 4 w 3 w 2 w 1 E = i (atoms, voxels) v i w j( i) 2 m i ({ w () t }) E j r : ε E ( t) wr r rj( i) 2 w Lower iteratively: Gradient descent w r () t w ( t 1) = = ε δ ( v w ) m. Inline (Monte Carlo) approach for a sequence selected at random according to weights w r ( t) = ~ ε δrj i) m i : ( v ( t) w ). ( i r How do we avoid getting trapped in the many local minima of E? r v i ( t) i i r i

28 r ( v (){ t w }) s, i ~ E j v Soft-Max Adaptation Avoid local minima by smoothing of energy function (here: TRN method): r : ( ) ~ λ r wr t = ε e i r i s w r j0 = 0 v i w s k r j1 = 1 v i s w r j( k 1) = k 1 ({ } ) λ w j, λ = e v w mi. r= 1 s r s Where is the closeness rank: λ 0 : i ( v () t w ), Note: LBG algorithm. λ 0 : not only winner w j () i, also second, third,... closest are updated. Can show that this corresponds to stochastic gradient descent on ~ λ 0 : E E. : E ~ Note: LBG algorithm. parabolic (single minimum). i j( i) } ( ) λ λ t 2

29 Convergence and Variability Q: How do we know that we have found the global minimum of E? A: We don t (in general). { } QuestionAnswer w j But we can compute the statistical variability of the by repeating the calculation with different seeds for random number generator. Codebook vector variability arises due to: statistical uncertainty, spread of local minima. A small variability indicates good convergence behavior. Optimum choice of # of vectors k: variability is minimal.

30 Single-Molecule Rigid-Body Docking (h) w j (l ) w j Xtal structure EM low res. data Estimate optimum k with variability criterion. Index map I: m n (m, n = 1,,k). k! = k (k-1) 2 possible combinations. ( h) (l ) For each index map I perform a least squares fit of the w I ( j) to the w j. Quality of I: residual rms deviation I = 1 k k j= 1 w ( h) I( j) w ( l) j 2 Find optimal I by direct enumeration of the k! cases (minimum of ). I

31 Application Example ncd monomer and dimer-decorated microtubules (Milligan et al., 1997) ncd monomer crystal structure (Fletterick et al., 1996,1998)

32 Search for Conformations Two possible ranking criteria: Codebook vector rms deviation ( ). Overlap between both data sets: Correlation coefficient: 2 1,, 2,, 2 1,, 2,,,,,,,, = z y x z y x z y x z y x z y x z y x z y x hl l h l h C I ncd motor (white, shown with ATP nucleotide) docked to EM map (black) using k=7 codebook vectors

33 Reduced Search Features Top 20, 7!=5040 possible pairs of codebook vectors. I C hl I (permutation) For a fixed k, codebook rmsd is more stringent criterion than correlation coefficient!

34 Performance (I) Dependence on resolution (simulated EM map, automatic assignment of k from 3 k 9): Deviation from start structure (PDB: 1TOP) used to generate simulated EM map. Catastrophic misalignment Accurate matching up to ~30Å

35 Performance (II) Is minimum vector variability a suitable choice for optimum k? Wriggers & Birmanns, J. Struct. Biol 133, (2001) 10 test systems, 3 k 9 simulated EM densities from 2-100Å. 2-20Å (reliable fitting) 22-50Å (borderline) Å (mismatches) Reasonable correlation with actual deviation No false positives for resolution values < 20Å and variability < 1Å.

36 Summary: Classic Situs (Versions 1.x) Advantages of vector quantization: Fast (seconds of compute time). Reduced search is robust. Limitations: Original k->k algorithm works best for single molecules, not for matching subunits to larger densities. Largely superseded by FTM and FRM in Situs 2.x But newer application allows k->l matching (Sculptor: Stefan Birmanns)

37 Resources and Further Reading WWW: Tools: colores (FTM) Contact me for beta version of FRM qvol, qpdb, qdock, qrange (vector quantization) WWW: Acknowledgement: Yao Cong, Stefan Birmanns, Julio Kovacs, Pablo Chacon (