3D and 6D Fast Rotational Matching

Transcription

1 3D and 6D Fast Rotationa Matching Juio Kovacs, Ph.D. Department of Moecuar Bioogy The Scripps Research Institute N. Torrey Pines Road, Mai TPC6 La Joa, Caifornia, 9037 Situs Modeing Workshop, San Diego, CA, Feb. 3-5, 003

2 Highights Computationay efficient method to perform rigid body docking. Combines spherica harmonics with new set of parameters for rotations, aowing us to dea with the rotationa correation function by means of Fourier techniques. Appications to: Docking of atomic structures into EM density maps. The Moecuar Repacement Method in crystaography. Mutipe moecue ( interior ) docking. Versions of FRM FRMr: radia version the starting point. FRM3D: voumetric version. Suitabe for compicated maps such as Patterson maps in crystaography. FRM6D: for mutipe moecue docking. FRMD: anaog of FRM6D for -D images. Has appications to reconstructions of EM maps (Yao Cong s ecture).

3 Radia FRM Description of the method origina map radiaized map Radiaization procedure -dimensiona sice of the density map. (Eectron microscopy map or burred atomic structure)

4 Radiaization procedure First, the center of mass of each density map is computed Radiaization procedure Then the desired eve surface is determined on an equianguar grid, giving the radiaized map

5 What do we do with the radiaized maps? Radiaized map f f = B 1 = 0 m= f ˆ Y m m Ym are the spherica harmonic functions Radiaized map g g = B 1 = 0 m= gˆ m Y m f and g represent the same object There is a scaing factor λ and a rotation R such that λ g( R) f, where g(r) denotes rotation of g by R. Criterion is to maximize the rotationa correation function: c( R) = f g( R) S

6 Euer anges φ, θ, ψ change to ξ = φ π / η = π θ ω = ψ π / c ( R) = T ( ξ, η, ω) The Fourier transform of T turns out to be: Tˆ( m, h, m ) = π π d ( ) d mh hm ( ) fˆ mgˆ m π The quantities d mn ( ) (which come from rotation group theory) are precomputed using a recursive procedure, since the expicit formua is quite invoved T ˆ( m, h, m ) inverse FFT T ( ξ, η, ω)

7 Timing test: Performance of FRMr Timings shown correspond to a 1GHz Pentium III Linux PC ncd (69 atoms) RNA poymerase (4837 atoms) Times for FRMr and FTM (6 anguar samping) ncd RNAp main processes FRMr FTM FRMr FTM precomputations centers of mass samping spherica coefficients Tˆ inverse FFT subtota 0.39 s 13 m 0.56 s side processes h FTM = Fast Transationa Matching

8 Robustness test of FRMr LYSOZYME (194 atoms) BARNASE (64 atoms) RIBONUCLEASE INHIBITOR (3410 atoms) HUMAN SERUM ALBUMIN (4475 atoms) LETHAL FACTOR PRECURSOR (5947 atoms) HALOPEROXIDASE (4567 atoms) UREASE γ SUBUNIT (6040 atoms) CHOLERA TOXIN (4098 atoms) RNA poymerase (4837 atoms) rmsd vs. resoution (6 anguar samping) 6 5 rmsd 4 3 average minimum maximum 1 trendine for the average resoution

9 Voumetric FRM (FRM3D) Here we hande the origina 3D density functions instead of their radiaized versions. The correation function is: c ( R) = f g( R) = T ( ξ, η, ω) f ( ru) = B 1 fˆ m = 0 m= 3 R Expanding in spherica harmonics: ( r) Y m ( u) g( ru) = B 1 = 0 m= π π We arrive at: T ˆ( m, h, m ) = d ( ) d ( ) I where: mm = 0 I fˆ m ( r) gˆ m mh ( r) r dr hm gˆ mm m ( r) Y m ( u) Crowther s Fast Rotation Function This cassica method (197) uses the reguar Euer anges as parameters of the correation function: c ( R) = f g( R) = T ( φ, ψ ; θ ) 3 R By doing this, ony φ and ψ can be Fourier-transformed: ˆ = ) T ( m, m ; θ ) d m m ( θ I mm whie θ remains as a parameter that needs to be scanned sequentiay.

10 FRM3D c ( R) = T ( ξ, η, ω) π π ˆ( m, h, m ) = dmh( ) dhm ( ) I T mm Precomputations: FST π d mn ( ) FFTW pan Reading maps COMs & samping f ˆ m( r), gˆ m ( r), Imm T ˆ( m, h, m ) 1 T ( ξ, η, ω) = FFT 3 ( Tˆ) FRM3D vs. Crowther D Crowther c ( R) = T ( φ, ψ ; θ ) ˆ ( m, m ; θ ) = d m m ( θ ) I T mm k=0,,b Precomputations: FST FFTW pan Reading maps COMs & samping f ˆ m( r), gˆ m ( r), Imm kπ θ k = d ( θ ), B mm k Tˆ( m, m ; θk ) 1 T ( φ, ψ ; θ ) = FFT ( Tˆ) k D postprocessing &saving postprocessing &saving Timings (in seconds) of Crowther and FRM3D B Anguar samping 6 3 1½ Crowther FRM3D

11 FRM6D Description of the method The given density objects are expanded into spherica harmonics: f ( ru) = g( ru) = B 1 fˆ m = 0 m= B 1 = 0 m= gˆ m ( r) Y ( r) Y m m ( u) ( u) Theideaistorotateboth objects whie transating one of them aong the positive z axis ony. In this way, the 6D search is performed over 5 anguar and 1 inear parameters. 6 DOF matching setup The correation is therefore a function of rotations and a distance: crr (, ; ρ) = f( R) gr ( ; ρ). 3 R Euer anges R( φ, θ, ψ ) R ( φ, θ, ψ ) ξ = φ π / η = π θ ω = ψ π / ξ = φ π / η = π θ ω = ψ π /

12 I Using these parameters the correation function is written as: Tˆ( n, h, m, h, m ; ρ) = ( 1) n, d nh d hm d nh d h m I mnm ( ρ). The quantities I mnm (ρ) are the so caed two-center integras, corresponding to the spherica harmonic transforms of the two maps, at a distance ρ of one another: mnm c( R, R ; ρ ) = T ( ξ ξ, η, ω, η, ω ; ρ), and the Fourier transform of T turns out to be: π 1 1 ρ = + + ˆ ( ) ( )( ) f ( r) gˆ ( r ) d β 0( ) r dr d 0( β )sin β dβ m m n n 0 0 To compute these integras more efficienty, we make the foowing change of variabes: z = r cos whereby the integras become: I β s = r sin β ( s ds 1 1 = + + ˆ mnm ρ) ( )( ) f ( ) ˆ ( ) m r g m r dn0( β ) dn0( β ) dz 0 In order to do the numerica evauation, we discretize z and s: z = zh s = sh

13 where the barred variabes are integers and h is the step size. Introducing these we get: I mnm ( ρ) h 3 B ( + z s = 1 z= z1 1 )( + d n0 B ]] d This is the actua expression used in the code. 1 [[ ) ˆ s fm[[ h s z r + z ]] gˆ x n0 [[ m [[ h z ρ r B ]] x s + z ρz + ρ ]] Test cases 1afw (peroxisoma thioase) 1nic (coppernitrite reductase)

14 Test cases 7cat (cataase) 1e0j (Gp4D heicase) Comparison timings and trends afw 1nic 7cat pi 1e0j 1der 1aw5 FTM FRM 1000 FTM 1000 Time (min) 100 Time (min) FRM Voxe size (Å) Number of voxes Times for various moecues versus the voxe size of the ow resoution maps. Timing trends versus the number of voxes of the ow resoution maps. Notes: the figure on the eft shows average times for 3 vaues of resoutions (10, 15 and 0Å). In a cases, B=16, or 11 samping. A tests were done appying the Lapacian fiter prior to processing.

15 Concusions Times of few seconds make FRMr and FRM3D suitabe for interactive docking sessions. FRMr is scae-independent and surface-based. FRM3D and FRM6D can be made surface-based by Lapace-fitering the origina maps. Surface fiters, such as Lapacian, are needed in order to avoid spurious fitting resuts when using FRM6D. Wi be incuded in Situs.