Ab initio methods: how/why do they work Ab i iti th d h / h d th - PowerPoint PPT Presentation

Ab initio methods: how/why do they work Ab i iti th d h / h d th k D.Svergun

Small Small- -angle scattering in structural biology angle scattering in structural biology Data analysis Detector Resolution, nm: R l ti 3 Incident Sample g I, relative 3.1 1.6 1.0 0.8 beam Wave vector 2 θ 2 Scattering k, k=2 π / λ k=2 π / λ k Shape Sh l curve I(s) I( ) determination Scattered Solvent 1 beam, k 1 Rigid body Rigid body Radiation sources: modelling 0 2 4 6 8 X-ray tube ( λ = 0.1 - 0.2 nm) s, nm -1 Synchrotron ( λ = 0.05 - 0.5 nm) Thermal neutrons ( λ = 0.1 - 1 nm) Missing g fragments Complementary Complementary Homology Atomic techniques techniques models models Oligomeric MS Distances mixtures EM Additional Additional Orientations Crystallography information information Hierarchical Interfaces NMR systems Bioinformatics Biochemistry h Flexible AUC systems EPR FRET

Major problem for biologists using SAS Major problem for biologists using SAS • In the past, many biologists did not believe that SAS yields more not believe that SAS yields more than the radius of gyration • Now, an immensely grown number of users are attracted by u b o u a a a d by new possibilities of SAS and they want rapid answers to more and more complicated Questions • The users often have to perform numerous cumbersome actions during the experiment and data analysis, to become each of the Answers Now we shall go through the major steps required on the way

Step 1: know, which units are used Step 1: know, which units are used The momentum transfer Resolution, nm: 3 relative 3.1 1.6 1.0 0.8 [q, Q, s, h, μ , κ …] = 4 π sin( θ )/ λ , [q μ ] ( ) lg I, I(s)=I 0 *exp(-sR g /3), sR g <1.3 2 Scattering curve I(s) 1 [ [s, S, k, … ] = 2 sin( θ )/ λ S k ] 2 i ( θ )/ λ I(s)=I 0 *exp(-2 π sR g /3), sR g <1.3*2 π 0 2 4 6 8 s, nm -1 GNOM or CRYSOL input: Angular units in the input file: 4*pi*sin(theta)/lambda [1/angstrom] (1) 4*pi*sin(theta)/lambda [1/nm] (2) 2 * sin(theta)/lambda [1/angstrom] 2 sin(theta)/lambda [1/angstrom] (3) (3) 2 * sin(theta)/lambda [1/nm] (4)

Scattering from dilute macromolecular Scattering from dilute macromolecular solutions (monodisperse systems) solutions (monodisperse systems) D sin sr ∫ ∫ = π ( ) 4 ( ) I s p r dr sr 0 The scattering is proportional to that The scattering is proportional to that of a single particle averaged over all orientations, which allows one to determine size, shape and internal structure of the particle at low ( 1-10 nm ) resolution. ) l ti

Sample and buffer scattering Sample and buffer scattering

Overall parameters Overall parameters Radius of gyration R (Guinier 1939) Radius of gyration R g (Guinier, 1939) 1 ≅ − 2 2 0 exp( I(s) I( ) R s ) g 3 Molecular mass (from I(0)) Maximum size D max : p(r)=0 for r> Dmax Excluded particle volume (Porod, 1952) ∞ ∫ = π 2 = 2 V 2 I(0)/Q; ( ) Q s I s ds 0

The scattering is related to the shape The scattering is related to the shape (or lo (or lo (or low resolution structure) (or low resolution structure) resol tion str ct re) resol tion str ct re) Solid sphere lg I(s), relative lg I(s), relative lg I(s), relative lg I(s), relative lg I(s), relative 0 0 0 0 0 -1 -1 -1 -1 -1 Hollow sphere -2 -2 -2 -2 -2 -3 -3 -3 -3 -3 -4 -4 -4 -4 -4 -5 -5 -5 -5 -5 -6 -6 -6 -6 -6 0.0 0 0 0 1 0.1 0.2 0 2 0 3 0.3 0.4 0 4 0 5 0.5 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 Dumbbell s, nm -1 s, nm -1 s, nm -1 s, nm -1 s, nm -1 Flat disc Long rod

Shape determination: how? Shape determination: how? p Trial-and-error 3D search model M parameters 1D scattering g data Non linear Non-linear search Lack of 3D information Lack of 3D information inevitably leads to inevitably leads to bl bl l l d d ambiguous interpretation, ambiguous interpretation, and additional information is and additional information is and additional information is and additional information is always required always required

Ab initio methods Ab initio methods Advanced methods of SAS data analysis employ spherical harmonics l i l h i l h i (Stuhrmann, 1970) instead of Fourier t transformations f ti

The use of spherical harmonics SAS intensity is I(s) = <I( s )> Ω = <{F [ ρ ( r )]} 2 > Ω , where F SAS i t it i I( ) <I( )> <{F [ ( )]} 2 > h F denotes the Fourier transform, <> Ω stands for the spherical average, and s=( s , Ω ) is the scattering vector. Expanding ρ ( r ) in spherical harmonics g ρ ( ) p p ∞ l ∑ ∑ ρ = ρ ω ( ) ( ) ( ) r r Y lm lm = = − 0 l m l the scattering intensity is expressed as ∞ l 2 2 ∑ ∑ ∑ ∑ = = π π 2 2 I s I s ( ) ( ) A A ( ) ( ) s s lm l = =− 0 l m l where the partial amplitudes A lm (s) are the Hankel transforms from the radial functions ∞ 2 2 ∫ l = ρ A ( ) s i ( ) ( r j sr r dr ) π lm lm l 0 and j l (sr) are the spherical Bessel functions. Stuhrmann, H.B. Acta Cryst., A26 (1970) 297.

Structure of bacterial virus T7 Structure of bacterial virus T7 Cryo Cryo- -EM, 2005 EM, 2005 SAXS, 1982 SAXS, 1982 Pro Pro- -head head Mature virus Mature virus Agirrezabala, J. M. et al. Agirrezabala, J. M. et al. & Carrascosa J.L. (2005) & Carrascosa J.L. (2005) EMBO J. EMBO J. Svergun, D.I., Feigin, L.A. & Schedrin, B.M. Svergun, D.I., Feigin, L.A. & Schedrin, B.M. 24 24 , 3820 , 3820 (1982) Acta Cryst. (1982) Acta Cryst. A38 A38 , 827 , 827

Shape parameterization by spherical harmonics Homogeneous particle Homogeneous particle Scattering density in spherical coordinates Scattering density in spherical coordinates (r, ω ) = (r, θ , ϕ ) may be described by the envelope function: ρ r ≤ r ≤ ω ⎧ ⎧ 1 , , 0 ( ( ) ) F ρ = ( ( r ) ) ⎨ ⎨ r > ω 0 , ( ) ⎩ F Shape parameterization by a limited F( ω ) is an envelope function envelope function series of spherical harmonics: series of spherical harmonics: l L ∑ ∑ ∑ ∑ ω ω ≅ ≅ ω ω = ⋅ ω ω ( ( ) ) ( ( ) ) ( ( ) ) F F F F f f Y Y l lm l lm Y lm ( ω ) – orthogonal spherical harmonics, L L = = − l 0 m l f lm – parametrization coefficients, Small-angle g scattering g intensity y from the entire p particle is calculated as the sum of scattering from partial harmonics: Stuhrmann, H. B. (1970) Z. l Physik. Chem. Neue Folge 72, L ∑ ∑ ∑ ∑ 2 2 2 2 = π ( ) 2 ( ) 177-198. 177 198 I s A s lm theor = = − l 0 m l Svergun, D.I. et al . (1996) Acta Crystallogr. A52, 419-426.

Shape parameterization by spherical harmonics H Homogeneous particle ti l ρ + + r r + + = = + + f 00 f 00 - f f 11 A 00 (s) A 11 (s) F( ω ) is an ( ) + envelope function + + - - + +… + f 20 f 22 - + A 20 (s) A 22 (s) π δ = R Spatial resolution: p , R – radius of an equivalent sphere. , q p + + ( ( 1 1 ) ) L L Number of model parameters f lm is ( L +1) 2 . One can easily impose symmetry by selecting appropriate harmonics in the sum. One can easily impose symmetry by selecting appropriate harmonics in the sum. This significantly reduces the number of parameters describing F( ω ) for a given L .

Program SASHA Program SASHA

Bead (dummy atoms) model Bead (dummy atoms) model A sphere of radius D max is filled by Vector of model parameters: densely packed beads of radius ⎧ ⎧ r 0 << D max r 0 D max 1 1 if if particle particle Position ( j ) = x ( j ) = ⎨ ⎩ 0 if solvent Particle Solvent (phase assignments) Number of model parameters M ≈ (D max / r 0 ) 3 ≈ 10 3 is too big for conventional minimization methods – Monte-Carlo like approaches are to be used But: This model is able to describe rather complex describe rather complex shapes Chacón, P. et al. (1998) Biophys. J. 74, 2r 0 2r 0 2760 2775 2760-2775. D max Svergun, D.I. (1999) Biophys. J. 76, 2879-2886

Finding a global minimum Finding a global minimum Pure Monte Carlo runs in a danger to be trapped into a Pure Monte Carlo runs in a danger to be trapped into a local minimum local minimum Solution: use a global minimization method like Solution: use a global minimization method like simulated annealing or genetic algorithm simulated annealing or genetic algorithm