Major problem for biologists using SAS Major problem for biologists - PDF document

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

Small Small- -angle scattering in structural biology angle scattering in structural biology Data analysis Detector Resolution, nm: 3 Incident Sample lg I, relative 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 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 Flexible AUC systems EPR FRET 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 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 Sample and buffer scattering Sample and buffer scattering

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 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 Ab initio program DAMMIN program DAMMIN Ab initio Using simulated annealing, finds a compact dummy Using simulated annealing, finds a compact dummy atoms configuration X that fits the scattering data by atoms configuration X that fits the scattering data by minimizing minimizing = χ 2 + α ( ) [ ( ), ( , )] ( ) f X I s I s X P X exp where χ χ is the discrepancy between the experimental where is the discrepancy between the experimental and calculated curves, and calculated curves, P(X) P(X) is the penalty to ensure is the penalty to ensure compactness and connectivity, α α > 0 > 0 its weight. its weight. compactness and connectivity, compact compact loose loose disconnected disconnected

Why/how do ab initio ab initio methods work methods work Why/how do The 3D model is required not only to fit the data but also to fulfill (often stringent) physical and/or biochemical constrains Why/how do ab initio ab initio methods work methods work Why/how do The 3D model is required not only to fit the data but also to fulfill (often stringent) physical and/or biochemical constrains

A test ab initio ab initio shape determination run shape determination run A test Program Slow mode DAMMIN Bovine serum albumin, molecular mass 66 kDa, no symmetry imposed A test ab initio ab initio shape determination run shape determination run A test Program Slow mode DAMMIN Bovine serum albumin: comparison of the ab initio model with the crystal structure of human serum albumin

DAMMIF, a fast DAMMIN DAMMIF, a fast DAMMIN DAMMIF is a completely reimplemented DAMMIN written in object-oriented code • About 25-40 times faster than DAMMIN (in fast mode, takes about 1-2 min on a PC) • Employs adaptive search volume Makes use of multiple • CPUs Franke, D. & Svergun, D. I. (2009) J. Appl. Cryst. 42 , 342–346 Limitations of shape determination Limitations of shape determination � Very low resolution Very low resolution � � Ambiguity of the models Ambiguity of the models � Accounts for a restricted portion of the data lg I(s) 2.00 1.00 0.67 0.50 0.33 Resolution, nm 8 Atomic 7 structure Shape How to construct ab initio 6 models accounting for F old higher resolution data? 5 0 5 10 s, nm -1 15

Ab initio dummy residues model dummy residues model Ab initio � Proteins Proteins typically consist of folded polypeptide typically consist of folded polypeptide � chains composed of amino acid residues chains composed of amino acid residues At a resolution of 0.5 nm a protein can be represented by an ensemble of K dummy residues centered at the C α positions with coordinates { r i } Scattering from such a model is computed using the Debye (1915) formula. Starting from a random model, simulated annealing is employed similar to DAMMIN GASBOR run on C subunit of V- GASBOR run on C subunit of V -ATPase ATPase Starting from a random “gas” of 401 dummy residues, fits the data by a locally chain- compatible model