Ab initio methods: how/why do they work D.Svergun, EMBL-Hamburg - PDF document

Ab initio methods: how/why do they work D.Svergun, EMBL-Hamburg Small- -angle scattering in structural biology angle scattering in structural biology Small Data analysis Detector Resolution, nm: 3 lg I, relative Incident Sample 3.1 1.6 1.0 0.8 beam 2 θ Wave vector 2 Scattering Scattering k, k=2 π / λ Shape curve I(s) curve I(s) determination Scattered Solvent 1 beam, k 1 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 Complementary fragments Complementary Homology Atomic techniques techniques models models Oligomeric MS Distances mixtures EM Additional Additional Orientations Crystallography information information Hierarchical Interfaces NMR systems Bioinformatics Biochemistry 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 than the radius of gyration • Now, an immensely grown number of users are attracted 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 lg I, relative 3.1 1.6 1.0 0.8 [q, Q, s, h, μ , κ …] = 4 π sin( θ )/ λ , 2 I(s)=I 0 *exp(-sR g /3), sR g <1.3 Scattering Scattering curve I(s) curve I(s) 1 [s, S, k, … ] = 2 sin( 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] (3) 2 * sin(theta)/lambda [1/nm] (4)

Scattering from dilute macromolecular Scattering from dilute macromolecular solutions (monodisperse monodisperse systems) systems) solutions ( D sin sr ∫ = π ( ) 4 ( ) I s p r dr sr 0 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. Sample and buffer scattering Sample and buffer scattering

Overall parameters Overall parameters Radius of gyration R g (Guinier, 1939) 1 ≅ − 0 exp( 2 2 I(s) I( ) R s ) 3 g 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 low resolution structure) (or low resolution structure) 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 0.1 0.2 0.3 0.4 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? Trial-and-error 3D search model M parameters 1D scattering data Non-linear search Lack of 3D information Lack of 3D information inevitably leads to inevitably leads to ambiguous interpretation, ambiguous interpretation, and additional information is and additional information is always required always required Ab initio methods methods Ab initio Advanced methods of SAS data analysis employ spherical harmonics (Stuhrmann, 1970) instead of Fourier transformations

The use of spherical harmonics SAS intensity is I(s) = <I( s )> Ω = <{F [ ρ ( r )]} 2 > Ω , where 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 ∞ l ∑ ∑ ρ = ρ ω ( r ) ( ) ( ) r Y lm lm = = − 0 l m l the scattering intensity is expressed as ∞ l 2 2 ∑ ∑ = π 2 I s ( ) A ( ) s lm = =− 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- -EM, 2005 EM, 2005 Cryo SAXS, 1982 SAXS, 1982 Pro- Pro -head head Mature virus Mature virus Agirrezabala, J. M. et al. et al. & Carrascosa J.L. (2005) & Carrascosa J.L. (2005) EMBO J. EMBO J. Svergun, D.I., Feigin, L.A. & Schedrin, B.M. Agirrezabala, J. M. Svergun, D.I., Feigin, L.A. & Schedrin, B.M. 24 , 3820 24 , 3820 (1982) Acta Cryst. (1982) Acta Cryst. A38 A38 , 827 , 827

Shape parameterization by spherical harmonics Homogeneous particle Scattering density in spherical coordinates (r, ω ) = (r, θ , ϕ ) may be described by the ρ envelope function: r ≤ ≤ ω ⎧ 1 , 0 ( ) r F ρ = ( ) r ⎨ > ω 0 , ( ) ⎩ r F F( ω ) is an Shape parameterization by a limited envelope function series of spherical harmonics: l L ∑ ∑ ω ≅ ω = ⋅ ω ( ) ( ) ( ) F F f Y Y lm ( ω ) – orthogonal spherical harmonics, lm lm L = = − l 0 m l f lm – parametrization coefficients, Small-angle scattering intensity from the entire particle is calculated as the sum of scattering from partial harmonics: Stuhrmann, H. B. (1970) Z. l L Physik. Chem. Neue Folge 72, ∑ ∑ 2 2 = π ( ) 2 ( ) 177-198. I s A s lm theor l = m = − l 0 Svergun, D.I. et al . (1996) Acta Crystallogr. A52, 419-426. Shape parameterization by spherical harmonics Homogeneous particle ρ + + r = f 00 + + 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: , R – radius of an equivalent sphere. + ( 1 ) L Number of model parameters f lm is ( L +1) 2 . 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 1 if particle Position ( j ) = x ( j ) = ⎨ ⎩ 0 if solvent Solvent Particle (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 shapes Chacón, P. et al. (1998) Biophys. J. 74, 2r 0 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 Local and global search on the Great Wall Local and global search on the Great Wall Global L o c a l Global Local search always goes to a better Local search always goes to a better point and can thus be trapped in a local point and can thus be trapped in a local minimum minimum Pure Monte- Pure Monte -Carlo search always goes to Carlo search always goes to the closest local minimum (nature: rapid the closest local minimum (nature: rapid quenching and vitreous ice formation) quenching and vitreous ice formation) To get out of local minima, global search To get out of local minima, global search must be able to (sometimes) go to a must be able to (sometimes) go to a worse point worse point Slower annealing allows to search for a Slower annealing allows to search for a global minimum (nature: normal, e.g. global minimum (nature: normal, e.g. slow freezing of water and ice formation) slow freezing of water and ice formation)