interactions, modelling Petr V. Konarev 1 A.V. Shubnikov Institute - PowerPoint PPT Presentation

Practical course “Solution scattering from biological macromolecules” 19-26 November 2018 Hamburg Form and structure factor, interactions, modelling Petr V. Konarev 1 A.V. Shubnikov Institute of Crystallography, Federal Scientific Research Center “Crystallography and photonics” Russian Academy of Sciences, Moscow, Russia 2 National Research Center “Kurchatov Institute”, Moscow, Russia

SAXS experimental facilities in Russia Laboratory setup AMUR-K (ICRAS) Synchrotron beamline “BioMUR” (Kurchatov Institute, Moscow) (built from former X33) in operation from Dec 2017

Outlines Parametric modelling using least-squares methods and information content of SAS data Form factors of simple geometrical bodies (spheres, cylinders, spherical core-shells, ellipsoids etc.) Concentration effects, interactions and structure factors Polydisperse & interactive systems in ATSAS Equilibrium oligomeric mixtures (OLIGOMER) Assembly/disassembly processes (SVDPLOT, MIXTURE) Restoring intermediates in evolving systems (DAMMIX) Graphical package for interactive processing (POLYSAS) Dissociation processes (GASBORMX, SASREFMX) Ensemble characterization of flexible systems (EOM)

Least-squares methods and parametric modelling Experimental data: Data calculated from the model described by parameters {a j }, j=1, M Azimuthally averaged intensities (s i , I exp (s i ),  (s i ) ), i=1, N ( s i , I mod (s i ) ) , i =1,N 𝒕 𝒋 = 𝟓𝝆𝒕𝒋𝒐(𝜾 𝒋 ) 𝝁 𝑶 𝟑 𝟐 𝑱 𝒇𝒚𝒒 𝒕 𝒋 − 𝑱 𝒏𝒑𝒆 (𝒕 𝒋 ) 𝝍 𝟑 = 𝑶 − 𝟐 Fit quality 𝝉(𝒕 𝒋 ) 𝒋=𝟐  2 = 1 for N>>M corresponds to | I exp (s i ) - I mod (s i ) | =  (s i ) means statistical agreement between model and data Information content of SAS data

Sampling formalism Shannon sampling theorem: the scattering intensity from a particle with the maximum size D max is defined by its values on a grid s k = k π /D max (Shannon channels):      sin D ( s s ) sin D ( s s )    max max n n   sI ( s ) s a   n n   D ( s s ) D ( s s )  max max n 1 n n Shannon sampling was utilized by many authors (e.g. Moore, 1980). An estimate of the number of channels in the experimental data range (N s =s max D max / π ) is often used to assess the information content in the measured data.

Determination of a useful data range Given a (noisy, especially at high angles) experimental data set, which part of this set provides useful information for the data interpretation? A usual practice is to cut the data beyond a certain signal-to-noise ratio but • there is no objective estimation of the threshold • this cut-off does not take into account the degree of oversampling

Application of sampling theorem to small-angle scattering data from monodipserse systems Due to a finite experimental angular range, the data can be approximated by a truncated Shannon expression     M sin D ( s s ) sin D ( s s )     max max n n   sI ( s ) U ( s ) s a   M n n   D ( s s ) D ( s s )  max max 1 n n n M     ( ) ( ) 8 sin( ) p r p r r s a s r M n n n  n 1 The best approximation should minimize the discrepancy N   1     2 2 ( M ) s I ( s ) U ( s )  i i M i 2 2 2 s  i 1 i i

Interpolation with different number of Shannon channels Ellipsoid with half-axes 1, 15, 15 nm contains M=38 channels within angular range up to s=4 nm -1

Interpolation with different number of Shannon channels Ellipsoid with half-axes 1, 15, 15 nm contains M=38 channels within angular range up to s=4 nm -1

Interpolation with different number of Shannon channels 2 D   max ( ) dp r    M ( p ) dr     dr 0 f(M)=  2 (M) + α  (p M )  =  2 (M max ) /  (p(M min ))

Algorithm for determination of effective number of Shannon channels (program Shanum ) 1. Automatically estimate D max (using AutoRg and AutoGnom (Petoukhov et.al., 2007)) 2. Calculation of the nominal number of Shannon channels N S = s max π/ D max and set up the search range [Mmin;Mmax], where M min = max(3, 0.2*N S ), M max = 1.25*N S 3. For M min <M<M max , calculate the coefficients of Shannon approximation a n (n=1,…M) by solving system of equations using a non-negative linear least-squares procedure (Lawson & Hanson, 1974). 4. For each Shannon fit, calculate the discrepancy  2 (M) and the integral derivative  (p M ) . 5. Evaluate the scaling coefficient α as the ratio between  2 (M max ) and  (p(M min )) 2 D   max dp ( r )  =  2 (M max ) /  (p(M min ))  f(M)=  2 (M) + α  (p M )   M ( p ) dr     dr 0 6. Determine the optimum value M S corresponding to the minimum of the target function f(M) P.V. Konarev & D.I. Svergun A posteriori determination of the useful data range for small-angle scattering experiments on dilute monodisperse systems . IUCr Journal (2015) V. 2 , p. 352-360

Examples of practical applications: SAXS data (Importin  /  ) Importins  и  mediate the import of nucleoplasmins through the nuclear pore, the latter ones interact with histones regulating the formation and shape of nucleosome estimates the effective Shanum number of Shannon channels M=8 and thus determines the useful angular range up to s=1.3 nm -1 Complex Importin  /  Taneva, S.G., Bañuelos, S., Falces, J., Arregi, I., Muga, A., Konarev, P.V., Svergun, D.I., Velázquez-Campoy, A., Urbaneja, M.A. (2009) J Mol Biol. 393, 448-463

Intensity from a system of monodisperse particles 𝒆𝝉(𝒕) = 𝑱 𝒕 = 𝒐𝚬𝝇 𝟑 𝑾 𝟑 𝑸 𝒕 𝑻 𝒕 = 𝒅𝑵𝚬𝝇 𝒏 𝟑 𝑸 𝒕 𝑻(𝒕) 𝒆𝛁 Number of scattered neutrons or photons per unit time, relative to the incident flux of neutron or photons per unit solid angle at s per unit volume of the sample where n - the number density of particles  - the excess scattering length density given by electron density differences V - volume of the particles P(s) - the particle form factor , P(s=0)=1 the particle structure factor , S(s=  )=1 S(s) - V  M n = c/M  can be calculated from partial specific density, composition

Form factor of a solid sphere 𝛁 = 𝒕𝒋𝒐𝒕𝒔 𝑱 𝒕𝒒𝒊𝒇𝒔𝒇 𝒕 = < 𝑩 𝒕 𝟑 > 𝛁 𝒇 𝒋𝒕𝒔 𝒕𝒔 ∞ 𝑺 𝝇 𝒔 𝒕𝒋𝒐 𝒕𝒔 𝝇(𝒔) 𝒕𝒋𝒐(𝒕𝒔) < 𝑩 𝒕 > = 𝟓𝝆 𝒔 𝟑 𝒆𝒔 = 𝟓𝝆 𝒔 𝟑 𝒆𝒔 = 𝒕𝒔 𝒕𝒔 𝟏 𝟏 𝑺 = 𝟓𝝆 𝒕 𝒕𝒋𝒐 𝒕𝒔 𝒔𝒆𝒔 = 𝒗𝒕𝒇 𝒒𝒃𝒔𝒖𝒋𝒃𝒎 𝒋𝒐𝒖𝒇𝒉𝒔𝒃𝒖𝒋𝒑𝒐 = 𝟏 𝑺 = 𝟓𝝆 − 𝑺𝒅𝒑𝒕(𝒕𝑺) + 𝒕𝒋𝒐(𝒕𝒔) = 𝟓𝝆 − 𝑺𝒅𝒑𝒕(𝒕𝑺) + 𝒕𝒋𝒐(𝒕𝑺) = 𝒕 𝟑 𝒕 𝟑 𝒕 𝒕 𝒕 𝒕 𝟏 = 𝟓𝝆 𝟒 𝑺 𝟒 𝟒 𝒕𝒋𝒐 𝒕𝑺 − 𝒕𝑺𝒅𝒑𝒕(𝒕𝑺) = 𝑾𝚾(𝒕𝑺) (𝒕𝑺) 𝟒

Ellipsoid of revolution 𝟐 𝟐 𝚾 𝟑 [𝒕𝑺 𝟐 + 𝒚 𝟑 𝜻 𝟑 − 𝟐 𝟑 ]𝒆𝒚 𝑸 𝒕 = 𝟏

Measured data from spherical particles (SANS) Bacteriophage T7 is a large bacterial virus with MM of 56 MDa consisting of an icosahedral protein capsid (diameter of about 600A ) that contains a double- stranded DNA molecule. Instrumental smearing is routinely included in SANS data analysis

Core-shell particles 𝑩 𝒕 = 𝚬𝝇 𝒕𝒊𝒇𝒎𝒎 𝑾 𝒑𝒗𝒖 𝚾 𝒕𝑺 𝒑𝒗𝒖 − 𝚬𝝇 𝒕𝒊𝒇𝒎𝒎 − 𝚬𝝇 𝒅𝒑𝒔𝒇 𝑾 𝒋𝒐 𝚾 𝒕𝑺 𝒋𝒐 Where 𝚾(𝒚) = 𝟒 𝒕𝒋𝒐 𝒚 − 𝒚𝒅𝒑𝒕(𝒚) V out = 4  R out 3 /3 and V in = 4  R in 3 /3 𝒚 𝟒  core – the excess scattering length density of the core  shell – the excess scattering length density of the shell

Cylinder R H H P(s)= J 1 (x) is the Bessel function of the first order and the first kind S(x)=sin(x)/x

Form factors of spheres and cylinders

Fitting data using geometrical bodies Primus-qt interface Primus interface

Library of form-factors from geometrical bodies and polymer systems Literature SASFIT software (J.Kohlbrecher, I.Bressler, PSI)

Scattering from mixtures Scattering from monodisperse (shape polydispersity) systems    D I ( s ) v I ( s ) sin sr   I ( s ) 4 p ( r ) dr k k sr 0 k The scattering is proportional to that of a single particle averaged over all For equilibrium and non-equilibrium orientations, which allows one to mixtures, solution scattering permits determine size, shape and internal to determine the number of structure of the particle at low ( 1-10 nm ) components and, given their resolution. scattering intensities I k (s), also the volume fractions