BCM Protein crystallography - I. Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS

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1 BCM Protein crystallography - I Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS

2 What SAXS can do Small-angle X-ray scattering (SAXS) yields information on biological structure that is difficult or impossible to obtain by other methods. Ab initio determination of the macromolecular envelope (shape); Compositional analysis of macromolecular assemblies in equilibrium with constituents; Solution modeling of a newly discovered protein from its structurally related homologs; Detection of conformational differences in protein-protein interactions or ligand binding; Validate macromolecular complexes reconstructed from high resolution structures of their isolated components; Characterization of lipid and membrane protein systems; Corroborate macromolecular conformations and assemblies predicted from computational modeling. SAXS is a low-resolution technique - it does not provide direct information on atomic coordinates BCM

3 SAXS Embodiment SAXS requires no special sample preparation prior to analysis and is a label-free low sample volume technique that is fairly rapid to perform (seconds to minutes). Macromolecule can be studied in their solution state in a physiologically relevant environment with reasonable throughput in seconds to minutes. High-quality data from small volumes (10-20 l) and protein concentrations 1 mg ml 1 Radially averaged data BCM

4 SAXS : Theory Calculation of the phase difference δ between the waves from scatterers at points O and P in a particle. In the drawing, s and s 0 are unit vectors in the directions of scattered and incident beams, respectively, and P is displaced from O by the vector r, θ denotes the scattering angle and q the scattering vector. Diagram illustrates how the phase difference δ between the waves from scatterers at point O and P in a particle can be calculated, where the angle between the incident and scattered beams is the scattering angle θ and the vector r goes from point O to point P. The phase difference between the waves scattered by the two scatterers will be q r. δ = AO + PB = r s 0 + r s = r q q is the momentum transfer where λ is the wave length; s and s 0 are 2 length unit vectors in the direction of the scattered and incident beams, respectively BCM

5 SAXS : Theory The angle-dependent scattering amplitude is related to the electron density distribution ρ(r) of the scatterer by a Fourier transformation. ρ(r) is the number of electrons per unit volume at the position r. A volume element dv at r contains ρ(r)dv electrons. The scattering amplitude A of the whole irradiated volume V is then given by : A q = A e ρ r exp ir q dr V F S = ρ r exp 2πir S dv cell where A e denotes the scattering amplitude of one electron. The scattering intensity of one single particle I 0 (q) is the absolute square given by the product of the amplitude and its complex conjugate A(q) * I 0 q = A q A q = I e ρ r ρ r (exp i(r r ) q drdr The electron scattering intensity I e is given from the classical Thomson formula V V I 0 (q) has units of electrons e.u. where I P denotes the primary intensity and d the distance between sample and detector. The numerical factor is the classical electron radius squared. I e depends only slightly on the scattering angle θ by the polarization factor, it is practically 1 for small angles. BCM

6 SAXS Assumptions The particles are statistically isotropic and there is no correlation between particles at great spatial distances. 1. Ideality : no intermolecular interactions o Non-ideality: interactions between particles 2. Monodispersity: identical particles o Polydispersity: Size and shape variation N I 0 q = n j i j (q) j i j q = i(q) Ideality & Monodispersity I 0 q = N i(q) Interactions and mixtures can be dealt with in a limited fashion BCM

7 Simplifications The particles are randomly oriented in a matrix (solution) and thermal motion is present during measurement process. The matrix is considered to be a homogeneous medium with the electron density ρ 0 and makes a scattering contribution The electron density of interest in the equations is then the difference in electron density Δρ(r) = ρ(r) ρ 0 which can take positive or negative values. ρ(r) ρ 0 Particle The average over all orientations due to thermal motion then leads to the fundamental formula of Debye : e iqr Ω = sin(qr) qr Debye factor with dv = r 2 sin θdrdθdφ BCM

8 Solution scattering amplitude and intensity Redefine the scattering amplitude and intensity in terms electron density difference, Δρ(r), where I 0 q = A(q) A (q) from A q = ρ r exp ir q dv V P Particle in solution undergoes thermal motion during the measurement, and therefore adopts all orientations with respect to X-ray beam. Therefore, only the spherical average of the scattered intensity is experimentally accessible. Applying the conditions of ideality and monodispersity and using time spatial i q = i q = A(q) A (q) then I 0 q = N i(q) BCM

9 Autocorrelation function The scattering equation then reduces to where γ(r) is called an autocorrelation function. The autocorrelation function can be obtained by the inverse Fourier transform of the scattering data as: γ(r) can also be defined from the electron density (r) as follows: γ r = ρ r ρ r = ρ r + u ρ r dv u V P which has a crystallographic equivalent known as the Patterson function. γ 0 r = γ(r)/γ(0) γ 0 r : probability of finding a density within the particle at a distance r from a given point Then taking the spherical average as before: γ r = 1 2π 2 γ r V P = γ(r) ρ r + u ρ r u2 du BCM

10 SAXS constants As the Debye factor equals to unity for q = 0 and r = 0, we have then and For q = 0, all secondary waves are in phase, so that I 0 (0) is equal to the square of the total number of electrons in the irradiated volume V (N x volume of one single particle). I 0 0 = ( ρ) 2 V However, this quantity is not directly available experimentally but can be estimated through the Guinier approximation shown below The equation for (0) shows that the integral of the intensity over all the reciprocal space is directly related to the mean square fluctuation of excess electron density, irrespective of special features of the structure. Note: if is expressed in terms of electron density, then I 0 0 = κ N V 2 P ρ 2 where is an instrumental constant, N is the number of solute molecules per unit volume; V P is the volume of the solute molecule while ρ is the contrast in mean electron density BCM

11 Hydrated particle volume The hydrated particle volume (V p ) can be obtained from the data on a relative scale thereby avoiding inaccuracies in parameter estimation caused by errors in concentration measurement. Assuming a uniform electron density inside the particle, V p is estimated following Porod s equation V p = 2π 2 I 0 0 Q where Q is the so-called Porod invariant. Q = q 2 I q dq = 2π 2 k N V P ( ρ) 2 0 I 0 0 and Q can be determined from the SAXS data BCM

12 Guinier s Law The Taylor series of the Debye factor can be written as At low q region, the intensity equation then reduces to sin (qr) qr = 1 qr 2 3! + O 4. where R g is the radius of gyration given by R g 2 = 1 2 γ(r)r4 dr γ r r 2 dr o R g is also related to the electron density ρ(r) of the particle and can be represented as R g 2 = Δρ(r)r2 dr Δρ r dr where r is defined as the vector taken from the center of gravity of Δρ(r) Given e -x 1-x, for qr << 1, I 0 q can be also expressed as I q I 0 exp R g 2 q 2 ln I q ln I 0 R g 2 q 2 or 3 3 This relation is called Guinier s law, which is a most useful relation in SAXS analysis. Allows R2 g and I 0 (0) to be obtained from scattering data in the region of smallest angles. No prior assumption is made regarding shape and internal structure of the particles Frequently valid up to qr 1.3 BCM

13 SAXS Experiment with radially averaged data Note: s q (A) Schematic representation of a typical SAXS experiment. (B) X-ray scattering patterns from a solution of BSA measured at beamline X33 (DORIS, Hamburg) in 50 mm HEPES, ph 7.5 (1), buffer only scattering (2) and the difference curve (containing the contribution from the protein alone, scaled for the solute concentration, 5 mg/ml) (3). H.D.T. Mertens, D.I. Svergun. (2010) Structural characterization of proteins and complexes using small-angle X- ray solution scattering Journal of Structural Biology 172, BCM

14 Sample Preparation Ideality : reached by working at infinite dilution In practice : perform measurements at decreasing concentrations and check whether scattering pattern is independent of concentration. Monodispersity: from purification protocol eg SEC Mass Spec, DLS, AUC, MALS + RI, etc. Accurate protein concentration SDS: Purity SEC: Monodisperse High MW species must be absent Dialyze sample: matched solvent for accurate solvent subtraction BCM

15 Estimation - I(0) and R g I(0) Estimates of I(0) and R g are taken from SAXS data close to the incident beam scattering direction satisfying qr 1.3 Note: s q Standard plots for characterization by SAXS. (A and B) SAXS curves and Guinier plots for BSA samples measured at X33 (DORIS, Hamburg) in different buffers showing (1) aggregation, (2) good data and (3) inter-particle repulsion. The Guinier fits for estimation of Rg and I(0) are displayed, with the linear regions defining s min and s max used for parameter estimation indicated by the thick lines. H.D.T. Mertens, D.I. Svergun. (2010) Structural characterization of proteins and complexes using small-angle X-ray solution scattering Journal of Structural Biology 172, BCM

16 Detection of conformational changes Escherichia coli maltose binding protein (MBP) Sucrose R g = 23.8 Å Guinier plot R g = 22.2 Å Guntas, G., Mansell, T.J., Kim, J.R. & Ostermeier, M. Directed evolution of protein switches and their application to the creation of ligand binding proteins. Proc Natl Acad Sci U S A 102, (2005) BCM

17 Kratky plot Scattering curves for some important shapes are known compact object rodlike object random coil Plotting q 2 I(q) vs q (a Kratky plot) distinguishes between these cases The appearance of the SAXS curve in the Kratky plot allows to infer protein chain folding characteristics BCM

18 Protein folding Note: s q Standard plots for characterization by SAXS. (C and D) SAXS curves and Kratky plots for lysozyme samples measured at X33 (DORIS, Hamburg) showing (1) folded lysozyme, (2) partially unfolded lysozyme (in 8 M urea), (3) partially unfolded lysozyme at 90 C and (4) unfolded lysozyme (in 8 M urea at 90 C). Plots are arbitrarily displaced on the vertical axis for clarity with the exception of (D), where all curves have been scaled to the same forward scattering intensity, I(0). H.D.T. Mertens, D.I. Svergun. (2010) Structural characterization of proteins and complexes using small-angle X- ray solution scattering Journal of Structural Biology 172, BCM

19 Scattering is Shape Dependent Pair Distance Distribution Function p(r) is the Fourier Transform of the SAXS curve with p(r) = r 2 (r) p(r) is a real-space function and intuitively better suited for modeling p(r) provides information on the maximum particle dimension, D max D max is the value of r at p(r)=0 for large r; requires data q π/d max Scattering intensities and distance distribution functions, p(r) calculated for typical geometric shapes: Solid sphere (black), prolate ellipsoid (red), oblate ellipsoid (blue), twodomain (green) and long rod (cyan). The bead models used for the calculation of scattering intensities are shown above the plots. Bump in two-domain p(r) is characteristic of modular structure Note: s q H.D.T. Mertens, D.I. Svergun. (2010) Structural characterization of proteins and complexes using small-angle X- ray solution scattering Journal of Structural Biology 172, BCM

20 The Achilles Heel: Protein Aggregates Distance distribution function p r = r2 2π 2 0 In theory, the calculation of p(r) from I(q) is simple. Problem : I(q) - is only known over [q min, q max ] : range is sensitive to experimental errors Solution : Indirect Fourier Transform. First proposed by O. Glatter in 1977 p(r) is parameterized on [0, D Max ] by a linear combination of orthogonal basis functions. q2 I(q) M sin qr qr p r = c n φ n (r) n=1 The coefficients c n are found by least-squares methods. Ill-posed problem solved using stabilization methods dq Guinier Plots Note: s q Distance Distribution BCM

21 Reciprocity: Size Scattering Angle compact object BCM

22 SAXS Size Range Experimental SAXS curves and parameters measured for P. furiosis rubredoxin (magenta), a scaffoldin protein S4 (red), a minicellulosome containing three catalytic subunits (green), and a DNA-dependent protein kinase (blue). Envelopes correspond to ab-initio models calculated from experimental curves using GASBOR. From d = λ/(2 sinθ) = 2π/q, q min sets D max, the largest dimensions observable in an experiment For q min = Å -1, we have D max = 2π/q min = 1000 Å Putnam CD, Hammel M, Hura GL, Tainer JA. (2007). Q Rev Biophys. Aug;40(3): Review. The limiting experimental factor at q min is how close reliable data can be determined near the primary beam, while at q max it is the noise in the experimental data. BCM

23 Resolution ranges to identify structural features in SAXS data The resolution ranges are highlighted with blue box and the upper axis of the inset graphs indicate the spatial resolution (Δ=2π/q) of this range. Putnam CD, Hammel M, Hura GL, Tainer JA. (2007). Q Rev Biophys. Aug;40(3): Review. BCM

24 Ab initio modeling procedure using DAMMIN (A) Starting from a spherical search volume, a fitting procedure is conducted until a final model is generated satisfying not only a fit to the experimental data, but also forming a compact and connected model of dummy atoms/beads. (B) The spherical search volume with beads assigned to the particle (yellow) and solvent (blue). The new implementation is the program DAMMIF where F refers to fast. H.D.T. Mertens, D.I. Svergun. (2010) Structural characterization of proteins and complexes using small-angle X-ray solution scattering Journal of Structural Biology 172, BCM

25 SAXS Analysis BSA modeled in 15 min using a home source BCM

26 Structure validation Validate known crystal structure data for protein in solution CRYSOL is a program for evaluating solution scattering from macromolecules with known atomic structure and fitting it to experimental scattering curves from SAXS. For input, one can use a PDB file of an X-ray or NMR structure of a protein or a protein- DNA(RNA) complex BCM

27 Rigid body modeling Mertens HDT, Svergun DI. (2010) J. Struct. Biol. 172, Computation of scattering from highresolution models is often used to identify the biologically active conformations of crystal structures and help distinguish between alternative crystallographic dimers and/or higher oligomers For example, a new crystallographic form of the protein complex of Cdt1 and Geminin was validated in solution by SAXS (De Marco et al., 2009) A heterohexameric structure was observed in the crystal whereas a previous crystallographic study had shown only the existence of a heterotrimer. From the fit of the theoretical scattering curves computed with the program CRYSOL to the SAXS data, the heterohexamer was identified as the correct model in solution. Mertens et al, 2010 Identification of heterohexameric solution state of the human Cdt1 Geminin complex by SAXS. Experimental scattering pattern for the complex in solution and the fits calculated from the crystal structures of the heterotrimer (red broken line) and heterohexamer (blue line). It is clear that the heterohexamer fits the experimental data while the heterotrimer does not. BCM

28 MW determination from I(0)/c The forward scattered intensity where - s is the contrast, defined as the difference in the mean electron densities of the solute and solvent. Using the weight concentration c (mg/ml) instead of number of molecules per unit volume (N), an alternate expression of the forward scatter can be derived by substituting N = N A c M r, the volume of the solute is V = v M r N A, then ρ = n e N A M r v where N A is Avogadro s number, n e is the number of electrons, M r is the molecular weight, and v is the partial specific volume of the solute. The electron density of the solvent ρ s is assumed to be the same as that of pure water, namely x e/cm 3. The forward scattered intensity becomes I 0 = κ M r cn A I 0 c n e M r v ρ s N A However as n e M r then 2 n e M r vρ s N A 2 I 0 = κ N V P 2 ρ 2 = κ N V P 2 (ρ ρ s ) 2 and vs M r should yield a straight line I 0 n e M r v ρ s N A c M r const MW Graziano V et al. SARS (2006) CoV main proteinase: The monomer-dimer equilibrium dissociation constant. Biochemistry. 45(49): BCM

29 SAXS determines solution assembly state The experimental scattering curve for protein PF1787 is shown with calculated scattering curves for monomeric and dimeric atomic resolution structures of homologs. The best fit (best fit multimer) to the experimental SAXS data are calculated from a threefold symmetric trimer derived from a monomeric homolog (PDB 1WR2). The trimeric form of PF1787 was confirmed using the extrapolated intensity at 0 scattering angle (I(0)), normalized for concentration (inset). The indicated protein standards were used to place the data on a relative scale. Relevant structures from analysis of PF1787 are shown on the right. The crystallographic dimer is a flexibly linked two-domain protein. Models with threefold symmetry enforced match the SAXS results. Hura GL & 14 others. (2009) Robust, high-throughput solution structural analyses by small angle X-ray scattering (SAXS) Nature Methods 6, BCM

30 Solution Structure Modeling Ab initio modeling Rigid body modeling Putnam CD et al. Q Rev Biophys (3): Envelope representation using spherical harmonics Envelope from densely packed dummy beads Envelope from dummy residues forming a chain compatible model Missing domain represented by ensemble of dummy residues for a chain compatible model Rigid body mod + missing loop represented by ensemble of dummy residues Atomic model derived from rigid body modeling applying conformational sampling SAXS models of a cellulase complex reconstructed using different methods. (A) Ab-initio model reconstructed by spherical harmonics using SASHA. (B) Densely packed beads model using DAMMIN. (C) Model calculated with GASBOR. (D) Reconstruction of one missing module using dummy residues with CREDO (green). The secondary structure elements of known atomic structure are in gray. (E) Rigid-body modeling from knoiwledge of the atomic structures (gray) in combination with ab-initio modeling of the linker region (cyan) using BUNCH. (F) Rigid-body modeling using conformational sampling. Thousands of possible atomic models, produced from performing molecular dynamics on the linker region (blue) were used in search of the best-fit conformation. One hundred conformations (yellow) are shown superimposed on the catalytic module (gray). BCM

31 Proteomic Scale, Shape and Assembly Hura et al. (2009) Robust, high-throughput solution structural analyses by small angle X-ray scattering (SAXS). Nat. Meth. 6, SAXS provides accurate shape and assembly in solution for most samples. (a) For ten proteins with structural homologs or existing structures, the experimental scattering data (colors) were compared with the scattering curve calculated for the matching structure (black). (b) For monodisperse samples, the envelope determinations (colored as in a) were overlaid with the existing structures (ribbons). All monomeric units had a 9-amino-acid His tag attached. (c) For the 9 proteins with no available structural information, envelope predictions from two independent programs were compared and generally agreed. The DAMMIN results (black mesh) were generated without symmetry. The GASBOR results used twofold symmetry for the PF0014- PF0015 protein complex, PF0965-PF0966- PF0967- pf0971 protein complex, PF1911 (dimer), PF00716 (dimer), PF0699 (dimer) and PF1950 (dimer). Fourfold symmetry was imposed on tetrameric PF1291 and PF1372. (d) Plotting the SAXS data as I (q) x q 2 versus q (Kratky plot) highlights proteins with large unfolded regions. The Kratky plot of PF0715 is shown for comparison of a folded protein and shows characteristic parabolic behavior at wide angles. In contrast PF0706.1, PF and PF1282-PF1205 fusion protein have SAXS data consistent with unfolded regions as reflected in the nonparabolic wide angle properties. BCM

32 High quality SAXS data SAXS models of full-length p97 (AAA family ATPase) in different nucleotide states: (A) AMP-PNP, (B) ADP-AlF x, (C) ADP, and (D) No Nucleotide. Each model is represented as an isosurface and viewed from the (row i) side, (row ii) bottom, (row iii) oblique angle, and (row iv) top. Domains are assigned to density in the ADP-AlF x state, as indicated by the tags N, D 1, and D 2. The program GASBOR with simulated annealing was used to model the SAXS data. A number of independent GASBOR runs were performed for each nucleotide state. All runs used data to 8.6 Å, and the 6-fold symmetry observed in crystal structures and cryo-em analyses of p97 was imposed in the modeling. The models were aligned, averaged, and filtered based on occupancy by using DAMAVER to obtain a most probable model for each state. Davies et al. Conformational Changes of p97 during Nucleotide Hydrolysis Determined by Small-Angle X-Ray Scattering. Structure 13: (2005) BCM

33 Liposomes and Lipoplexes Lipid self-assembly SAXS measures bilayer repeat distance: the distance from the center of one bilayer to the center of its neighbor, which includes the thickness of associated water layers. Representative x-ray diffraction patterns obtained from DOPE/PC and DOPE/SPM bilayer samples. Sample diffraction orders can be observed to shift to the left going from back to front, consistent with an increase in d- space associated with increasing acyl chain length. Yuan C, O'Connell RJ, Jacob RF, Mason RP, Treistman SN. (2007 ) Regulation of the gating of BKCa channel by lipid bilayer thickness. J Biol Chem. 282(10): BCM

34 Structural modifications of phosphocholine bilayers SAXS WAXS X-ray diffraction patterns (SAXS and WAXS) at 20 C for DPPC at ph 5.0 and subsequent mixtures with with tolmetin (NSAID) at different concentrations (mol %): 0 (A), 20 (B), 40 (C), and 60 (D). Interaction of tolmetin in the lipid phase (L β )is drug/lipid ratio-dependent. q(nm -1 ) q(nm -1 ) Model To mimic the cellular membrane, liposomes of 1,2- dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) were used. SAXS measures the bilayer repeat distance WAXS (Wide-Angle X-ray Scattering) measures lattice packing of phospholipids and the sharp reflection at 40% indicates a lattice change Drug penetration into the headgroup region decreases bilayer thickness and tilt angle Schematic representation of the interaction of 40 mol % of tolmetin with the phospholipid bilayer Nunes C, Brezesinski G, Lima JL, Reis S, Lúcio M. (2011) Synchrotron SAXS and WAXS study of the interactions of NSAIDs with lipid membranes. J Phys Chem B. Jun 23;115(24): BCM

35 Detergent organization in Aquaporin DDM: n-dodecyl-beta-d-maltoside SAXS signal of the AQP0 surrounded by its detergent corona (open dots). The contribution from free detergent micelles (top light gray curve) has been subtracted from the raw data (top dark curve). The red line is the best fitting curve from the complex with a cylindrical detergent torus. The green line is the best fitting curve from the complex with an elliptical detergent torus. 3D representations of the AQP0 tetramer and its complex with DDM. Top view (above) and side view (beneath) of the elliptical model of the lipid bilayer plane, represented with a section inside the detergent corona. Berthaud A, Manzi J, Perez J, Mangenot S. (2012) Modeling detergent organization around aquaporin-0 using Small Angle X-ray Scattering. JACS, 134(24): BCM

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38 Standard characterization Guinier plot Rg, L(0) (AUTORG) Sample characteristic Purity (eg. Monodispersity) Advanced characterization Component analysis (OLIGOMER/SVD) Porod invariant (Particle volume) Oligomeric composition Rigid body modeling (SASREF/BUNCH) p(r) Rg:Dmax ratio (GNOM/AUTOGNOM) Shape/anisometry Ab initio modeling (DAMMIF) Kratky plot Flexibility Folded/Non-folded Multiple confomations (EOM) Concentration series Interactions/ideality Interaction potentials Overall parameters (Rg, Dmax, MM) Structure validation/ biologically active assembly Predicted scattering of atomic models (CRYSOL) BCM

39 Road map for interpretable data Putnam CD et al. Q Rev Biophys (3): BCM

40 Road map controls!!! Putnam CD, Hammel M, Hura GL, Tainer JA. (2007). Q Rev Biophys. Aug;40(3): Review. BCM

41 and more controls!!!!!!!!!! Putnam CD et al. Q Rev Biophys (3): BCM

42 and controls again!!!! Putnam CD, Hammel M, Hura GL, Tainer JA. (2007). Q Rev Biophys. Aug;40(3): Review. BCM

43 Remember The method is simple but deceptively so: analysis and modeling require a monodispersed and ideal solution. it is critical to check the validity of these assumptions. Otherwise SAXS IN OUT BCM

44 Recipe for Good Data Gel filtration of sample prior to data collection Ensure buffer is matched Reduce aggregates 1) Add glycerol (up to 10%) and reducing agent Only small reduction in contrast 2) Increase salt concentration Watch out for contrast loss Attenuate incident beam to reduce radiation damage BCM