1 X-Ray Solution Scattering (SAXS)

Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 b,d,† , John A. Tainer A. , John c,† *, Greg L. Hura b, (J.A. Tainer) or [email protected] (G. L. Hura). Tainer) (J.A. u *Contributed equally La Jolla, CA 92037, USA La Jolla, CA *, Michal Hammel a, [email protected] Research Institute, 10550 North Torrey Pines Road, Torrey Research Institute, 10550 North macromolecular structures, conformations and assemblies in solution and assemblies conformations structures, macromolecular Correspondence: Christopher D. Putnam † Ludwig Institute for Cancer Research, 9500 Gilman Drive, La Jolla, 92093-0669, USA Drive, La Jolla, 92093-0669, for Cancer Research, 9500 Gilman Ludwig Institute a Life Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 94720, CA Berkeley, Laboratory, Lawrence Berkeley National Life Science Division, b

Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA CA Berkeley, National Laboratory, Division, Lawrence Berkeley Physical Biosciences Department of Molecular Biology MB4 and The Skaggs Institute for Chemical Biology, The Scripps for Chemical Biology, The Skaggs Institute Biology MB4 and Department of Molecular c d

Crystallography supplies unparalleled detail on structural information critical for mechanistic analyses; detail on structural information critical for mechanistic analyses; Crystallography supplies unparalleled X-ray solution scattering (SAXS) combined with crystallography and computation: defining accurate defining accurate computation: and with crystallography (SAXS) combined scattering X-ray solution mechanisms in structural cell biology and nanotechnology. membrane protein, RNA, DNA, and protein-nucleic acid complexes. The methods discussed provide the basis membrane protein, RNA, DNA, and protein-nucleic acid complexes. flexibility and conformational to examine molecular interactions in solution and to study macromolecular simulation, and prediction of changes that have become increasing relevant for accurate understanding, these techniques in conjunction with computational methods to productively combine solution scattering data these techniques in conjunction with computational methods to productively results are considered with a focus with high-resolution structures. Detailed aspects of SAXS experimental acid macromolecular structures, including on data interpretation tools suitable to model protein and nucleic eukaryotic DNA replication, recombination and repair to microbial membrane secretion and assembly systems. assembly and secretion membrane microbial to repair and recombination replication, DNA eukaryotic concerns and considerations for using This six-part review addresses both theoretical and practical concepts, limitations inherent in NMR and electron microscopy studies. Together these techniques can allow multi-scale these techniques Together electron microscopy studies. limitations inherent in NMR and accurate images of macromolecules for modeling allosteric mechanisms, modeling to create complete and from molecular machines acting in diverse processes ranging supramolecular complexes, and dynamic Small angle X-ray scattering (SAXS) offers complementary information about macromolecular folding, offers Small angle X-ray scattering (SAXS) and assembly conformations, flexibly linked domains, shape, conformation, unfolding, aggregation, extended size resolution range of about 50 to 10 Å resolution, but without the state in solution, albeit at the lower however, it is restricted to describing low energy conformations of macromolecules within crystal lattices. conformations it is restricted to describing low energy however, Abstract Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 SAXS structures ab initio SAXS envelopes ab initio Flexibility, oligomerization, or aggregation? Flexibility, Rigid-body docking into low resolution envelopes Flexible docking into low resolution envelopes Multiple well-defined conformations Conformational disorder explored by SAXS Computational docking combined with SAXS data in crystallographic and solution structures Differences Determining biological assemblies in crystal structures Calculation of Calculation of scattering curves from atomic models Calculation of scattering curves Rigid-body modeling with SAXS data Contrast variation and assessing crystallographic targets sample quality Validating for phasing Using low resolution SAXS envelopes Structural database of SAXS scattering Measuring SAXS data Measuring SAXS macromolecules Scattering from function Pair-distribution curves Information content in scattering state in solution Molecular weight and multimerization The Patterson function Phase determination Structure determination Structure refinement in crystals Flexibility and disorder Crystal lattices – unit cells – symmetry – unit cells Crystal lattices conditions and Laue from crystals Diffraction arrangements Intensities and atomic Data collection Data evaluation Data analysis Structure modeling and interpretation Modeling atomic assemblies using Flexibility and conformational disorder measured by SAXS Time scales of macromolecular motion Time structures Motion from experimentally determined Normal mode analysis Molecular dynamics simulations Monte Carlo simulations assessing agreement Comparing SAXS profiles and structures with SAXS data Direct comparison of crystallographic SAXS characterizations that aid crystallography SAXS characterizations that aid Small angle X-ray scattering Small angle X-ray X-ray crystallography Interactions of X-rays with matter of X-rays with Interactions 4.4.3. 4.3.2. 4.3.3. 4.4.1. 4.4.2. 4.2.3. 4.2.4. 4.2.5. 4.3.1. 4.2.1. 4.2.2. 2.3.6. 2.4.1. 2.4.2. 2.4.3. 2.3.1. 2.3.2. 2.3.3. 2.3.4. 2.3.5. 2.2.4. 2.2.5. 2.2.6. 2.2.7. 2.2.8. 2.2.1. 2.2.2. 2.2.3. 5.3. 5.4. 5.1. 5.2. Strategy and tactics for SAXS experiments Strategy and tactics for 4.4. 4.3. 4.1. 4.2. 3.4. 3.5. to model structures Using crystallography and SAXS 3.1. 3.2. 3.3. Computational techniques for modeling macromolecular flexibility macromolecular modeling Computational techniques for 2.4. 2.3. 2.2. Introduction techniques SAXS and of crystallography Comparison 2.1. 5. 4. 3. 1. 2. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 ��  Baumeister, 2005; Scheres et & al., Baumeister, 2007). , meaning to “form”, “shape”, or “organize”. informare �� (Craig et al., 2006; Dubochet et al., 1988; Lucic, Forster General biological and biophysical implications biological and biophysical General prospects Needs and and predictions Expectations Criteria for evaluation of SAXS results of SAXS for evaluation Criteria and models measurements, experiments, of SAXS resolution and Accuracy Macromolecules are inherently near-sighted. Stable macromolecular interfaces involve forces that We We believe that the major challenge for structural biology in the next decade will be in providing a Cellular coordination, in which macromolecules serve as bit players, functions as a gestalt, where Genomic information plays only an indirect role in organizing the spatial and temporal order in cells 6.3. 6.1. 6.2. 5.5. 5.6. and conclusions Prospects We We predict, however, that the combination of X-ray crystallography and SAXS is well poised Both Å. 10 to 50 roughly from range resolution a with solution to in structures generating for technique important become an tomography of very large complexes has led to important advances in understanding important macromolecular macromolecular important understanding in advances important to led has complexes large very of tomography complexes resolution information about large complexes and conformational states that are too flexible, too large, or too difficult to stabilize as homogeneous samples for these techniques. Recentspectrometers and synchrotron substantial facilities combined with impressive advances in investmentselectron microscopy and cryo- in NMR of the most important problems will involve macromolecular complexes whose structures and complexes will not be easily solved by any single biophysical technique. Thus advances will require the use and development of methods to bridge atomic resolution structures determined by X-ray crystallography and NMR with lower biotechnological, and pharmaceutical fields. Addressing these challenges is fundamentally important fromscientific a standpoint, yet tremendously difficult from a practical one. Our experience has suggested that many mechanistic understanding of the macromolecular and supramolecular complexes changes that and underlie cell their biology and conformational in using these structures to provide new opportunities in the medical, of shape is control of information. Thus it seems highly appropriate derived from the Latin that the English word “information” is or inactivation of the macromolecule of interest, or indirectly through pathways that the affect macromolecule. At its core, each of these levels of control is expressed through the shapes of specific macromolecules. Control destruction and through reversible modifications, such phosphorylation,as formation the requiring mechanisms are degradation and of synthesis control even and modification Functional methylation,activation through directly, and ubiquitination. either macromolecule the control that states conformational and interfaces dynamic of the whole is greater than the sum of its parts. Macromolecules are controlled through their creation and their an appropriate milieu in which the macromolecules can act and be acted on. Macromolecules can be thought of thought be can Macromolecules on. acted be and act can macromolecules the which in milieu appropriate an pathways and networks are the result of the availability of substrates. as “cogs in the machine” in which Conformational changes, driven by small molecule binding, allostery, or complex formation can be propagated Conformational changes, driven by small molecule binding, allostery, through long distances. But even requires these pathways changes into complexes are macromolecular only and macromolecules the these sum of integration of the short-range even interactions Ultimately, between atoms. typically are only effective in short ranges that can be measured in Ångstroms, and these interfaces typically fit typically interfaces these and Ångstroms, in measured be can that ranges short in effective only are typically together like pieces of a jigsaw puzzle that exclude bulk solvent and leave very few gaps at the shared surface. modern, mechanistic understanding of cells, therefore, requires detailed knowledge of the three-dimensional the atoms involved in these processes. configuration of stay static – ultimately lie with macromolecules encoded by DNA. Both directly proteins control and the RNA cell through reactions they perform, conformations they adopt, and interactions that they make in solution. A and organisms. Cellular functions – the decisions to grow and divide, to die by programmed cell death, or to 1. Introduction 6. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 SAXS models, and appropriately consider flexibility and disorder SAXS in ab initio SAXS and X-ray crystallography are fundamentally similar techniques and can share most of the A fundamental difference between solution scattering and X-ray crystallography lies in the relative This review aims to provide a general framework for making informed decisions about experimental As a solution technique, SAXS offers the potential for obtaining some information with every sample, beam of X-rays irradiates a sample, and the intensities of the scattered (SAXS) or diffracted (crystallography) X-rays are measured by an X-ray detector (Fig. 1A). hardware required to generate, prepare, and detect X-rays. In most experiments, a collimated, monochromatic 2. Comparison of crystallography and SAXS techniques relevance to the understanding, simulation, and prediction of macromolecular interactions and conformations in solution. benefit from a directed research effort. An overall goal of this review is to provide the framework for improved benefit from a directed research effort. collaborative efforts involving SAXS with other techniques with the belief that problem-driven developments will help push substantial improvements in SAXS technologies and software with obvious and important prospects for further developments and applications of SAXS to define experimentally validated macromolecular validated experimentally define to SAXS of applications and developments further for prospects by obtained typically be can than information complete more provide and solution in conformations and shapes At the same time either we technique note alone. concerns and areas where we believe SAXS in particular will experiments. Section 5 details specific experimental strategies and tactics and provides avalue basisof to different assessinterpretations the of SAXS data for a given experiment. Section 6 outlines our views on the of the methods used for fitting and deforming atomic structures in the context of low resolution information. Section 4 focuses on the principal means to directly compare SAXS data and crystal structures, employ SAXS experiments to derive crystallography through computational methods. For considers the 3 Section techniques. scattering and diffraction X-ray of assessment comparative a provides purpose 2 section of bringing everyone to most the understanding for important are which flexibility, macromolecular modeling same for techniques computational level, design, data processing, and data interpretation to combine SAXS with atomic-resolution structures from structures provides more detailed three-dimensional information. We therefore expect that SAXS will return any return will SAXS that expect therefore We information. three-dimensional detailed more provides structures of experimental resources or additional computational techniques. investments made into development conformations, interactions, and assemblies in solution. The accessible experimental resolution can thus be made be thus can resolution experimental accessible The solution. in assemblies and interactions, conformations, and shape global the define directly can measurements SAXS asked. being question biological the to appropriate component high-resolution plus computation with SAXS of combination the whereas solution, in conformation SAXS data can be used to determine the low resolution structures of macromolecules without any additional experimental information. SAXS Moreover, is not only likely to be more powerful in conjunction with atomic resolution structures to provide more accurate and complete models of protein, RNA, and DNA structures, curves is substantially less than that in crystallography, which is an inherent limitation of this technique. However, However, technique. this scattering in content information The solved. are structures resolution high after and to prior both useful be can of limitation inherent an is which crystallography, in that than less substantially is curves microscopy methods. The combination of much performed be can Experiments SAXS. of utility the increased substantially has techniques computational current third-generation synchrotron sources and data SAXS from derived information addition, In experiments. crystallographic or EM either than rapidly more sophisticated requires modest sample preparation and material relative to understanding crystallography, and systems is possessing a substantial natural flexibility. SAXS technique can for characterize shapesolution for quite small macromolecular to systems, very spanning large the ranges limiting NMR and electron and conformation in techniques are becoming increasingly accessible to a broad range of investigators. Sample preparation for SAXS SAXS for preparation Sample investigators. of range broad a to accessible increasingly becoming are techniques analysis is particularly accessible to a variety of laboratories which otherwise may have techniques. structural thus far never used Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & Johnson, 1976; Drenth, . B. Scattering from a solution of yeast PCNA with θ ) for SAXS data collection. Both images collected at beamline -1 Crystallography provides substantially more information content

=0.6 Å q & Svergun, 2003) and excellent books (Blundell X-ray interactions with sample for SAXS and crystallography A. Both SAXS and X-ray crystallography The theoretical underpinnings for both of these techniques are well understood and have been the subject the been have and understood well are techniques these of both for underpinnings theoretical The Department of Chemistry, Texas A&M University. Texas Department of Chemistry, Fig. 1. involve placing a sample (orange) into a highly collimated X-ray beam (red) and measuring the scattered X-rays. The angle of any scattered position with the direct beam is 2 a maximum resolution of 23.9 Å. C. Diffraction from a nickel superoxide The equivalent dismutase position of crystal the highest at resolution of 2.0 the SAXS experiment Å is indicated resolution. (red circle). The blue circle indicates the highest resolution achievable ( 12.3.1 (SIBYLS) at the Lawrence Berkeley National Laboratories. Diffraction image courtesy David Barondeau, will be important for using these paired X-ray techniques for the growing number of multi-resolution structure determination problems. introduction to SAXS and that SAXS specialists will benefit most from highlight areas We of theoverlap crystallography. with the expectation introductionthat some appreciation of toboth techniques macromolecular of recent reviews (Koch, Vachette 1994; Giacovazzo et al., 1992). Our goal here is therefore to not to exhaustively address each technique, to but introduce expect and that draw crystallographers parallels will between benefitWe them. primarily from the than SAXS scattering, allowing atomic resolution structures to be determined; packing however, the in requirement the of crystal lattice can lead to molecules whose interactions (section 4.2.4). biologically relevant conformations are inappropriately fixed by non- due to the atomic positions and provides enormously greater signal. Moreover, in the crystallography retains lack information about of specific orientationsin radial the molecule symmetry and requires that crystals be rotated during 1997). data collection (Dauter, molecules molecules are highly organized within a crystal lattice. Diffraction from a crystal lattice gives rise to discrete diffraction maxima that are caused by the convolution of the crystal lattice onto the continuous transform organization organization of target molecules during data collection. In solution scattering, the signal from all orientations Solution together. averaged are apparatus, experimental the and another one to relative molecules, target the of the crystallography X-ray in contrast, In 1B,C). (Fig. (isotropic) symmetric radially and continuous is scattering Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 Macromolecular . & McSweeney, 2000). angles. Atomic scattering Atomic angles. θ ), reaching a maximum when maximum a reaching ), θ 2 ) at which two objects can be 2 min & Steiner, 2001). Neutrons however, in differ, /2 λ = 0°) and falling off at large 2 large at off falling and 0°) = = θ min d relative to the direct beam is proportional to (1+cos to proportional is beam direct the to relative θ is the X-ray wavelength. In practice, the wavelengths typically chosen at synchrotron radiation sources radiation synchrotron at chosen typically wavelengths the practice, In wavelength. X-ray the is Crystals are ordered arrays of atoms related by pure translation (a transformation with only a change in X-ray crystallography requires the generation of crystals, and macromolecular crystals are typically Neutrons can also be used in both crystallographic (neutron diffraction) and solution (small-angle The theoretical limit for the resolution, the minimum distance (d Both SAXS and X-ray crystallography exploit coherent (Thomson) X-ray scattering. In coherent λ ’s of these ’s wavelengths are typically much smaller than can be measured (typically 3-1 Å) from crystals of min position but not orientation or rotation) in one-dimension (fibers), two-dimensions (sheets), and three-dimensions and (sheets), two-dimensions (fibers), one-dimension in rotation) or orientation not but position (lattices). Although fiber diffraction of samples such as DNA can be studied with X-rays, the mostcommon grown under conditions where molecules are 1997) reversibly driven out of solution (Weber, samples require that these conditions are gentle and do not cause crystal lattice forces are much weaker than macromolecular folding energies. unfolding Typically, or disassociation of complexes. 2.2.1. Crystal lattices – unit cells – symmetry 2.2. X-ray crystallography (Gutberlet et al., 2001). So we herein consider X-ray based experiments, which have broader general utility and utility general broader have which experiments, based X-ray consider herein we So 2001). al., et (Gutberlet applicability to macromolecular systems. spallation sources that currently generate thermal et neutrons al., (Taylor 2007). Signal to noise problems with neutron diffraction and scattering experiments are significant challenges to obtaining highSuch neutron qualityexperiments datainvolve more sets. specialized efforts and have been discussed well in detail elsewhere and increase sensitivity of biological samples to X-rays the Unfortunately, recent increases in (Burmeister, X-ray source 2000; intensities has not been Ravelli mirrored by the fission reactors or neutron scattering; SANS) experiments (Gutberlet, Heinemann the fact that they interact with atomic nuclei and thus generate substantially fewer radicals than X-rays during the experiment. Radical formation is known to reduce redox-active sites, such as disulfides and metal centers, meaningfully recorded in SAXS scattering experiments (typically 50-10 Å, depending on the sample). meaningfully recorded in SAXS d be can that those that smaller significantly are and crystal, the of size or disorder internal to due macromolecules (0.8-1.5 Å) are selected to limit damage to the crystal or to take advantage of anomalous scattering. The theoretical The scattering. anomalous of where advantage take to or crystal the to damage limit to selected are Å) (0.8-1.5 distinguished, is based on the wave properties of the X-rays: distinguished, is based on the wave method for solving protein crystal structures (section 2.2.5). protein crystal structures (section method for solving atomic orbital binding energies unique to each atom. The atomic scattering factors at these energies are described are energies these at factors scattering atomic The atom. each to unique energies binding orbital atomic by additional terms accounting for this Use behavior. of this “anomalous scattering” has become an important factors have been accurately calculated for all of the elements and are influenced by the number of electrons for electrons of number the by influenced are and elements the of all for calculated accurately been have factors decreases X-rays scattered coherently of intensity the general, In occupy. electrons the orbitals the and atom the with increasing X-ray energies (decreasing X-ray wavelength). This decrease is discontinuous at energies near sources, sources, emitting X-rays with the same wavelength as the incident beam, but 180° out of The phase. scattering 2 of angle an at measured (2 beam X-ray incident the to parallel is scattering the scattering, scattering, electrons oscillating under the influence of the electric field of the X-ray beam act as secondary 2.1. Interactions of X-rays with matter X-rays of Interactions 2.1. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 α, reconstruction ab initio is the vector corresponding vector the is S is the experimentally measured experimentally the is θ = l = k = h S S S * * * c a b with incident beam, and thus 2 thus and beam, incident with θ are vectors corresponding to the orientation of the unit cell edges, edges, cell unit the of orientation the to corresponding vectors are to the incident X-rays, and hence the diffraction maxima measured during the crystallographic during the measured maxima diffraction the X-rays, and hence incident to the c θ , and , b ). , As electromagnetic waves, X-rays possess both a wavelength and phase. In the crystal, X-rays are X-rays diffracted from crystals can be mathematically treated as if they were being reflected from a In many cases the biologically relevant complex possesses symmetry. These biological symmetries may symmetries biological These symmetry. possesses complex relevant biologically the cases many In Frequently, internal symmetry exists within the unit cell when the unit cell contains multiple molecules. molecules. multiple contains cell unit the when cell unit the within exists symmetry internal Frequently, γ a and index, and process diffraction data prior to knowing the atomic structure. index, and process diffraction and orientation, but not the positions of atoms in the unit cell, control which reflections are in the diffraction condition and where these diffracted reflections occur This on allows the crystallographers detector. to collect, that scatter X-rays are determined by the wavelength of the incident X-rays (or wavelengths in the case of Laue of case the in wavelengths (or X-rays incident the of wavelength the by determined are X-rays scatter that Laue The crystal. the of orientation the and parameters, cell unit the experiments), diffraction multi-wavelength conditions give rise to a regularized the lattice unit of spots cell diffraction (Fig. size, 1C). shape, Importantly, to the path difference between incident and scattered X-rays, and h, k, and l must be integers to ensure constructive ensure to integers be must where l and k, h, and X-rays, scattered and incident between difference path the to planes the Thus, reflection. each identify to used are and indices Miller the are values l and k, h, The interference. ray, then the diffracted X-rays undergo constructive (in-phase) X-rays then interference undergo the and diffracted can be detected experimentally ray, as diffraction maxima. If not, then the X-rays interfere destructively (out-of-phase) and are not observed. This requirement can be expressed mathematically using the Laue conditions: diffracted from multiple parallel planes simultaneously, and from this diffracted multiple leads parallel to planes a simultaneously, path through difference which these X-rays travel. When the path difference corresponds to an integral number of wavelengths of the incident X- angle between the direct beam position and the diffraction maximum on the X-ray detector (Fig. 1A). and the diffraction angle between the direct beam position plane of angle plane experiment are frequently termed “reflections”. By this definition and due2 of to angle an themake X-rays geometry diffracted the ofevent, the diffraction the accuracy of the final models. conditions crystals and the Laue 2.2.2. Diffraction from Analogously, Analogously, particle symmetry provides important constraints on model-based and of three-dimensional shapes from one-dimensional increases substantially SAXS crystallography, data in symmetries (sections non-crystallographic the 4.2 of use like and constraints, symmetry 4.3). Application of these the appropriate assembly is present in the crystal structure. For crystallographic analysis, non-crystallographic constraints of generation as well as 2.2.6), (section improvement map for useful tremendously be can symmetries 1998). al., et (Brunger CNS program the in implemented been have as such models, atomic of refinement the for be observed by some combination of both the crystallographic and non-crystallographic symmetry operators, if operators, symmetry non-crystallographic and crystallographic the both of combination some by observed be crystallographic symmetries and lattice translations is termed the asymmetric unit. In contrast, symmetries that lattice are non-crystallographic. do not apply to the If these symmetries apply to the entire lattice, they are crystallographic symmetries and constrain the parameters parameters the constrain and symmetries crystallographic are they lattice, entire the to apply symmetries these If of the unit cell. The smallest portion of structural information required to reconstruct the entire lattice through β, crystals crystals studied by macromolecular crystallography are three-dimensional. The smallest repeating unit of the crystal that is related only by translations is called the unit cell. For three-dimensional crystals, the shape and size of the unit cell is defined by the length of three axes (a, b, and c)and three angles between these axes ( Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 is the resolution- j atom in the unit cell. th λ = n θ are the fractional positions of the j j 2 d sin , and z j , y j , but not the relative phase information necessary to calculate the electronic distribution in is the structure factor, h, k, l are the Miller indices of the structure factor, f F(h,k,l) The crystal lattice has two effects. First, the orientation of the molecules allows the diffraction data to In general, measured intensities are on a relative scale, not an absolute one. From a theoretical standpoint, standpoint, theoretical a From one. absolute an not scale, relative a on are intensities measured general, In In contrast to the positions of the reflections, the intensities of the diffracted X-rays are dictated by the Bragg’s Bragg’s law provides a measure of the distance between the theoretical planes giving rise to X-ray F(h,k,l) defined by unit cell, which also increases signal-to-noise. The measured X-ray intensities in crystallography are crystallography in intensities X-ray measured The signal-to-noise. increases also which cell, unit by defined the square of the summed amplitudes from the atoms in the unit cell. In contrast, SAXS intensities are the sum molecules in solution that are orientationally averaged. Second, the scattering from the atoms in the unit cell is cell unit the in atoms the from scattering the Second, averaged. orientationally are that solution in molecules positions discrete at only sampled is diffraction crystal the that so lattice the from scattering the with convoluted retain information about atomic positions in three-dimensional space, which is lost in SAXS data collected on a scale factor between calculated and observed intensities. This subtraction is the basis of contrast techniques variation where matching the average electron density of bulk solvent to specific components of a scattering SAXS data (section 2.3.6). complex causes their contributions to be eliminated from the processed against low 1995).resolution Inreflections contrast, (Urzhumtsev the & contribution Podjarny, of bulk solvent is explicitly subtracted out during SAXS data processing and typically is involved in the modeling process as plot (Wilson, 1942) using data between 3.0 – 1.5 Å resolution. Placing quite difficult. Inhowever, addition to the ordered atoms, thethe non-ordered bulk solvent must be included in the data on a true absolute scale is, structures X-ray refining in solvent bulk modeling of importance the by demonstrated be can which calculation, I(0,0,0) is the square of the sum of the number of electrons in the unit cell and important is SAXS directly comparable result to where the I(0) Wilson is the using scale proportional quasi-absolute a to only on put the generally are crystallography square in Data of 2.3.2). (section particle the number of electrons in the scattering the unit cell. Measurement of the relative phase of X-rays striking the detector at any two diffraction spots has not been This possible. problem is the “phase problem” of protein crystallography that must be solved in order for structures to be determined. dependent atomic scattering factor, and x data Unfortunately, collection only allows measurement of the intensities, I(h,k,l), which are the square of the amplitude of where amplitude and phase. Mathematically, atomic arrangements in the unit cell. The positions and types of atoms within the unit cell control both the 2.2.3. Intensities and atomic arrangements if a crystal is merohedrally or pseudo-merohedrally twined (Yeats, 1997), rather X-rays. diffract than whether or not it can for data collection and structure determination. In contrast, evaluating structural reconstruction samples is in more determining difficultof SAXS reminiscent becausemore is every for 5) (section macromolecularanalysis suitability further solutionfor suitable will for is curve scatter SAXS a X-rays.if In determining sense, this By analogy to light scattering through slits, in the case of a theoretical perfect crystal of infinite size the first order spectrum will occur when n=1, a second order spectrum will occur at n=2, and so forth. The maximum resolution (smallest spacing between planes) measured from the crystal can provide insights into its suitability scattering: Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 3 1-3 3-1 1-4 4-1 2 3-4 4-3 2-3 3-2 2-4 4-2 Solution 1 1-2 2-1 r (distance)

0 Pair-distribution Function P(r) 4 2 1 Distances 3 2 3-4 1 2-4 1-4 3-1 3-2 0 2-1 1-2 origin 1 4-1 4-2 2-3 1-3 Crystal u (distances along a) -2 4-3 Patterson Function a Comparison of the Patterson autocorrelation function in X-ray crystallography and Comparison of the Patterson autocorrelation function in X-ray crystallography

2 1 0

u -1 -2 b b) along (distances v v are not shown in the expanded view due to the size of the unit cell. The pair-distribution The pair-distribution are not shown in the expanded view due to the size of the unit cell. function, on the other hand, resolves distances but not directions within each scattering molecule add together. all equivalent distances in the four-atom Thus, unit. Fig. 2. theoretical two-dimensional A autocorrelation function in SAXS. the pair-distribution molecule of four atoms is placed in an arbitrary two-dimensional crystal in solution. the crystal The Patterson function contains cross peaks for every interatomic distance in and these crosspeaks in the u, v plane, are indicated by circles and retain directional The crosspeaks between symmetry mates information about their positions in the crystal. 2.2.4. The Patterson function 2.2.4. techniques techniques that rely upon over sampling to determine the phases of measured reflections (Koch et al., 2003). (section 2.2.5). information use additional the phase problem to solve Hence, techniques of the squared amplitudes from each scattering event. SAXS data are also continuous and vastly over sampled in comparison to the independent data content as derived from theorem. Shannon’s Even with higher signal- “super-resolution” of advantage take cannot and sampled under substantially are data crystallographic to-noise, Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 , are used are , 2+ 12 Br 6 , the largest peaks in F(h,k,l) peaks in the Patterson function. It has been been has It function. Patterson the in peaks 2 10 . atoms with a large number of electrons such as mercury or uranium, e.g peaks in the u, w v, space of the Patterson function and the symmetry of the crystal, the N 2 is the observed amplitude and V is the volume of the unit cell. The importance of the second Experimental phasing techniques systematically perturb the intensities in ways that can be used to extract to used be can that ways in intensities the perturb systematically techniques phasing Experimental To build a map of the electron density in the unit cell by adding together the diffracted X-ray waves, it is it waves, X-ray diffracted the together adding by cell unit the in density electron the of map a build To When the Patterson function is calculated using the measured amplitudes The autocorrelation function for the electron density, which is essentially a three-dimensional map of all of map three-dimensional a essentially is which density, electron the for function autocorrelation The F(h,k,l) instead of single heavy atoms (Knablein et al., 1997). quickly become challenging to solve. Moreover, as the size of the unit cell increases, the height of each Patterson Patterson each of height the increases, cell unit the of size the as Moreover, solve. to challenging become quickly Ta as such clusters, atom heavy cells, unit large particularly for weaker; relatively becomes peak used. From the N increase, binds atom heavy the where sites of number the as Importantly, calculated. be can space z y, x, in peaks so do the number of Patterson peaks. Thus Patterson-based methods for solving heavy atom positions can information, the positions of these atoms must heavy determining firstFor below). described (as methods direct through or beintensity in differences from calculated identified either through analysis of Patterson maps atom positions, the differences in the amplitudes between the heavy atom derivative and the native crystal are the introduction of “heavy atoms”, phasing for atoms heavy the of advantage take To 1997). (Ke, lattice overall the perturbing without crystals into information about the relative phases of the measured reflections. Isomorphous replacement (IR) depends upon depends (IR) replacement Isomorphous reflections. measured the of phases relative the about information are not constrained mathematically unless they possess special symmetry relationships. reflection needs to be determined andThe one of three techniques are used: experimental phase methods, direct methods, for each and molecular replacement. necessary to determine phases for each of the reflections whose intensity is measured. The phases of reflections of phases The measured. is intensity whose reflections the of each for phases determine to necessary 2.2.5. Phase determination (Fig. 2). Additionally, the Patterson function includes all vectors between atoms for all molecules within the crystal. The P(r) function, on the other hand, is a histogram of distances that are orientationally averaged and particle. correspond only to the scattering (section 2.3.3). Unlike the SAXS P(r) function, the crystallographic Patterson function is calculated from molecules from calculated is function Patterson crystallographic the function, P(r) SAXS the Unlike 2.3.3). (section vectors interatomic the about information three-dimensional retains thus and rotations their in restricted are that the autocorrelation are the positions of direct translations between molecules in the crystal lattice; this Patterson this lattice; crystal the in molecules between translations direct of positions the are autocorrelation the function is typically called the “self-Patterson” intensities (Fig. SAXS the 2). from calculated The P(r), function, self-Patterson autocorrelation the to function related is in and function autocorrelation crystallography is an important for several macromolecular phasing techniques (section 2.2.5). macromolecular phasing techniques important for several where is function this and information, phasing any of absence the in calculated be can function this that is expression where u, w v, corresponds to some difference vector between the position x, and y, z and x+u, N and to y+v, rise z+w. give will (atoms) map density electron the in peaks N Thus the expression: formulation is equivalent to shown that the above of the atom-atom vectors in the crystal, can be written as: written can be the crystal, in vectors atom-atom of the Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 1.0 map calculated using calculated map

c -F o 2F 0 u 5 v = 0.00 F. Met10 Anomalous difference Patterson Anomalous difference experimental electron density after MAD phasing and phasing MAD after density electron experimental Met11 c -F o 2F 0 origin 0.00 0.0 Final refined structure 0.0 w – f´ + if˝ (yellow). E. Final refined structure and structure refined Final E. (yellow). . 0 E. σ 11 1.0 B f = f is the wavelength-independent component of the scattering, 0 (red). F. Overall structure of YgbM, TIM-barrel, YgbM, is a shown zinc-containing as Overall structure of (red). F. σ 12,670 (lime), and 5 and (lime), σ ) are not the same. Phasing experiments that use multiple wavelengths, and wavelengths, multiple use that experiments Phasing same. the not are ) olts) Experimental (green), 3 (green), (orchid), and 5 Experimental density σ 12,660 σ D. YgbM determined by selenomethionine MAD. A. Comparison of theoretical and measured X-ray F(-h,-k,-l) and (blue), 3 E. coli σ Theoretical 12,650 Phe12 F(h,k,l) density X-ray energy (electron-V The other major experimental phasing technique relies upon introducing atoms with anomalous scattering scattering anomalous with atoms introducing upon relies technique phasing experimental major other The

The structure of Anomalous difference X-ray Fluorescence X-ray D. structure. refined final the with superimposed mean the above Met11 σ C. 12,640 . with phasing information from other sources such as MAD or SAD experiments or even partial structures (Blow structures partial even or experiments SAD or MAD as such sources other from information phasing with & Crick, 1959; Sim, 1959). Anomalous dispersion techniques have become a method of choice for solving of anomalous scatters can also be determined by Patterson-based techniques (Fig. 3B; (Hendrickson & Ogata, is not it combined, readily be can information phases (Fig. 3C,D). Because phase used to determine 1997)) and unusual for the anomalous signal from heavy atom derivatives (MIRAS) to be used and potentially combined Both SAD and MAD experiments use wavelengths positions the MIR, for As pairs. Friedel measured carefully have at must differences f˝ using experiments Further, the atomic transitions to maximize the information. Friedel pairs ( pairs Friedel can take advantage of both f´ and f˝ differences (multiwavelength anomalous dispersion, MAD) or at a single wavelength that can only use (single f˝ wavelength differences anomalous dispersion, SAD) can be performed. and f´ and f˝ are the real and imaginary parts of the atomic scattering. At any that wavelength, f´ differences f´ can only is be measured constant, by comparing so data at different the wavelengths. of amplitude The the imaginary part, that so the law Friedel’s in breakdown a to leads however, 3A), (Fig. f˝ absorption photoelectric where f is the scattering factor of the atom, f or nuclear excitations and are typically described as: or nuclear excitations and are typically a cartoon. Anomalous scattering occurs or when are dispersion X-ray near into electronic energies the (typically) crystal. at 5 at 1 at contoured modification density final refined phases at 1 fluorescence at the selenium edge for the crystal. Anomalous B. difference Patterson map identifying the selenomethionine Met11- Met11 and Met105-Met105 crosspeaks generated by crystallographic symmetry operators contoured density difference Anomalous (at C. the A). (panel point Harker inflection fluorescence selenium the at pairs section) Friedel between differences calculated using Fig. 3. A Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

12 Schneider, 1997; Hope, 1990), and & the Schneider, techniques to introduce Molecular replacement solutions introduce the possibility of model-biased phases, which generates Unlike the other techniques described above, molecular replacement attempts to computationally position computationally to attempts replacement molecular above, described techniques other the Unlike Direct methods, on the other hand, have been primarily used in macromolecular crystallography as an will increasingly become a reasonable strategy to solve new structures. To this end, the ability to identify overall identify to ability the end, this To structures. new solve to strategy reasonable a become increasingly will structural similarities through comparison of experimental and calculated SAXS scattering (see section 2.4.3) be observed in these omitted regions. As the number of solved protein structures increases, the ability to use molecular replacement to rapidly screen through all reasonable or all possible molecular replacement targets model decreases, but can be detected through the use of omit maps, which are calculated with portions of the model omitted in the phase calculation. For true solutions, the electron still be observed. density For phase-biased results that are for entirely dependent upon the the model, no electron density will omitted regions will maps that do not show the differences between the atomic model used to solve the structure and the electron density that gives rise model to bias the tends scattering. to Importantly, increase as homology with the atomic functions, faster algorithms that do the equivalent searches are used in practice (for example (Navaza, 2001)). functions, faster algorithms that do the equivalent searches are used translation function can also be calculated by comparing the Patterson function from the experimental data with data experimental the from function Patterson the comparing by calculated be also can function translation the Patterson functions calculated from rotated molecules to Patterson calculated explicitly which overlaying and calculating different by performed be translations can replacement molecular have Although been applied. however, however, the close intermolecular vectors between atoms in the atomic packing and non-crystallographically related molecules with other orientations result in “noise” in this search. Increasing the number of molecules in the asymmetric unit typically makes the molecular replacement problem more difficult. Similarly, the into a three-dimensional rotational search, followed by a three-dimensional translational search. In the rotation search, typically only vectors within a radius similar to the longest intramolecular distance are considered; and any potential conformational changes that can occur. Molecular replacement is a vectors six-dimensional interatomic all search; containing function Patterson calculated a allow will model a of rotation proper however, to correlate well with one calculated from the experimental data. This allows the problem to be broken down an atomic model using experimental intensities. From the positioned molecule or molecules, phases can then be calculated for the calculation of electron density The maps. atomic model must be “similar” to the structure of the crystallized molecule, where the degree of similarity depends on the number of molecules to be found the use between intensities of rather the than differences intensities introduces some noise into the substructure this for roles major the of one be to continue likely will substructures these of determination The determination. technique in macromolecular crystallography. direct methods can be readily applied to substructure determination, as atoms in these substructures typically are much more than 1.2 Å apart and fit the “atomaticity” requirementsmethods even have been able atto determine moderatelarge anomalous substructures et resolutions.(Weeks al., 2002), despite Direct the fact that et al., 2002). Direct methods take advantages of relationships (Giacovazzo et between al., 1992) and phases require that data between are complete, multiple accurate, and are reflections high enough resolution so that individual scatterers can be resolved (resolutions of better than 1.2 Å). In macromolecular crystallography, alternative to Patterson-based methods for solving heavy atom or anomalous scattering substructures (Weeks (Andresen et al., 2004), and anomalous scattering is not yet as important in SAXS as it is in macromolecular crystallography. component, component, ASAXS so has that anomalous can differences only be measured between wavelengths. In theory, a number of potential applications such as monitoring the distance between two anomalous scatters; however, DNA of solvation ion as such systems biological simple involved have applications most and small is signal the 1997). Anomalous scattering has also been used in conjunction with SAXS (ASAXS) (Miake-Lye, & Doniach Hodgson, 1983; Stuhrmann, 1981); however, the orientational averaging of SAXS data eliminates the f˝ crystal crystal structures due to the tunability of synchrotron radiation (Helliwell, 1997), the ability to cryo-cool and collect datasets from single crystals (Garman anomalous scatters such as selenomethionine in proteins and bromouridine in DNA molecules (Doublie, Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 , where c -F o F set can be difficult, free parameter, which calculates which parameter, free | ) l , k , h ) ( l c , F k , − h is a global parameter and while it can help determine help can it while and parameter global a is ) ( l o free , F k are amplitudes calculated from the model. This number , h 13 ∑ ( o F (h,k,l) | c are calculated amplitudes. Placement of residues into maps can F ∑ = (h,k,l) R c F , and the difference electron density map, calculated with the coefficients c -F o are observed amplitudes and 2F (h,k,l) o are observed amplitudes and One important advance in helping detect the problem of over fitting is the R the is fitting over of problem the detect helping in advance important One The overall agreement of the model of the asymmetric unit with the experimental data is measured by From the initial interpretable maps, an atomic structure is usually fit through rounds of atom placement Given an initial set of phases, either from experimental or computational sources, electron density maps maps density electron sources, computational or experimental from either phases, of set initial an Given F (h,k,l) o Moreover, Moreover, choosing the number of reflections and whichreflections to include in an R an R factor for the current model using a set of reflections, typically several thousand, that are withheld from R However, 1992). (Brunger, calculation refinement the over fitting, it is not sensitive enough to evaluate the validity of small changes to the crystallographic model. providing a numerical target for minimization by automated refinement packages. providing a numerical target where allows the crystallographer to monitor the effects of making manual modifications to the structure as well as the “R-factor”: refinement into local minima. automated density modeling and model refinement programs, given initial phases withthese cases, sufficientthe crystallographer quality. supervises In the process and corrects regions that are wrong or are trapped by now be automated, such as by the program suite ARP/wARP (Perrakis, Morris & Lamzin, 1999) and RESOLVE RESOLVE and 1999) Lamzin, & Morris (Perrakis, ARP/wARP suite program the by as such automated, be now 2003), (Terwilliger, which can allow for rapid building of macromolecular structures by cycling between an the phases calculated from the model itself can The crystallographers also normally be evaluate combined the experimental with electron experimental density phasing map, calculated information. coefficients with the Fourier errors in the phases. Thus it is quite common, although not required, to use maps calculated from phases from the updated model itself (Fig. 3E). In the case of atomic partial models or excellent experimental phases, followed by automated substantial retain that maps refinement into atoms place to possible is it quality, (Kleywegt excellent of be can techniques modification Jones, & 1997). Although experimental maps after phase techniques not only allows for the improvement of initial phases, but also allows for phase extension in cases is at lower resolution than the native data set (Fig. 3D). where experimental phasing information regions in crystals should have uniform density and that both positive and negative deviations should either be other the on averaging, symmetry Non-crystallographic magnitude. in flipped or density average this to flattened modification phase these of Use molecules. related non-crystallographically between density the averages hand, These density modification techniques mainly operate by directly modifying the electron density maps and back and maps density electron the modifying directly by operate mainly techniques modification density These calculating new phases. Solvent flattening and solvent flippingoperate on the assumption that the bulk solvent to generate a map of sufficiently good quality so that an atomic model can be built. Multiple density modification modification density Multiple built. be can model atomic an that so quality good sufficiently of map a generate to techniques can be used to improve phase information. The two most important 1997). Main, & ones Cowtan Zhang, 1997; Read, are & solvent flattening (Vellieux averaging symmetry or non-crystallographic and flipping of the unit cell can be calculated. Frequently initial phase information has substantial errors; however, the goal is goal the however, errors; substantial has information phase initial Frequently calculated. be can cell unit the of 2.2.6. Structure determination 2.2.6. Structure could greatly reduce the number of atomic models to be screened and thereby improve the efficiency and success and efficiency the improve thereby and screened be to models atomic of number the reduce greatly could crystallography. for methods replacement of molecular F Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & Pryor, 1975). And is imperfect in some ways, it is it ways, some in imperfect is free & Podjarny, 2004)). In each of these cases, the 14 (section 6). (section Huber, These 1991). constraints & are Huber, required to maintain a ratio of free Importantly, Importantly, the decision on how to model the ADFs is not dictated by whether or not anisotropic At lower resolutions (below 3 Å), individual ADFs isotropic for all non-hydrogen atoms can introduce Traditionally, a crystallographic model is constructed of primarily one conformation. Additional alternate Additional conformation. one primarily of constructed is model crystallographic a Traditionally, Crystallographers are aware that molecules in crystals are not completely rigid and are composed of atoms of composed are and rigid completely not are crystals in molecules that aware are Crystallographers many parameters. For example, more economical parameterizations of non-isotropic motion have been recently been have motion non-isotropic of parameterizations economical more example, For parameters. many applied, recognizing that much of the anisotropic motion of atoms can be correlated to domain motions within motions (or valence electrons) are present in the crystal, but whether or not any particular model introduces too too introduces model particular any not or whether but crystal, the in present are electrons) valence (or motions non-spherical treatment of electrons corresponding to the atoms introduces additional parameters that require these extraordinarily high resolutions to be fit. modeling can be used to fit the experimental data using “multipolar models” for fitting the non-spherical valence non-spherical the fitting for models” “multipolar using data experimental the fit to used be can modeling and scatterers, Gaussian additionally with electrons bond valence for account to atoms” “dummy electrons, shell quantum mechanics modeling methods (reviewed in (Petrova inappropriate as valence electrons in the protein backbone atoms and unpaired electrons on oxygen atoms become atoms oxygen on electrons unpaired and atoms backbone protein the in electrons valence as inappropriate At these resolutions, additional electron density maps visible in (Jelsch difference et al., 2000; Ko et al., 2003). side chains, residues, or even whole domains). In contrast, very high resolution structures (1.3 use Å or anisotropic better), ADFs that describe probability ellipsoids with six parameters at (Willis subatomic resolutions (0.7 Å or better), the treatment of atoms as spheres of electrons begins to become too many fittable parameters, and typically a ADF single isotropic is then refined for groups of atoms (such as such as implemented in CNS (Brunger et al., 1998), as well as generalized least-squares-based refinement, such refinement, least-squares-based generalized as well as 1998), al., et (Brunger CNS in implemented as such 1997). & Schneider, as implemented in SHELX (Sheldrick torsion angles, chiral volumes, and planar restraints, plays an important role in constraining atomic positions to chemically reasonable positions (Engh refinement, dynamics-based molecular both to applied is and over-refinement prevents that constraints and data model fails to capture atomic displacement directed along a single direction (anisotropic displacements) or if the if or displacements) (anisotropic direction single a along directed displacement atomic capture to fails model disorder is due to the superposition of multiple static conformations. Thus, in this resolution range, each atom is typically described by four parameters. Introducing geometric restraints, such as bond lengths, bond angles, called the B-factor, the temperature factor, or the factor, Debye-Waller assumes that all atoms can be treated as an isotropic Gaussian distribution of atomic positions centered at an equilibrium position. However, this atom is normally only refined for structures with alternate conformations or partially bound ligands. For most motion Gaussian the describe to used is parameter single a Å), (3.0-1.3 resolutions high to moderate at structures of each atom about their equilibrium positions. This isotropic atomic displacement factor (ADF), alternately observed in difference electron density maps. Atomic positions in the using model three positional parameters, are x, and y, z most and some number frequently of parameters to described describe the displacement of the atom from an equilibrium position. An additional parameter describing the “occupancy” of a particular conformations are typically added at high resolutions (<2.0 Å) when clear evidence for their existence can be density ranges (Matthews, 1968), “rules of thumb” of how the molecules can be modeled based on the data-to- from the crystal. highest resolution data measured be given as a function of the parameter ratio can held together by electrons in specific orbitals. crystallographers However, are constrained in their ability to fit these features by the amount Thus, of crystallographers unique are data rarely observed able in the experiment. to fit all of the features of the molecules that are present. Since macromolecular crystals have fairly consistent 2.2.7. Structure refinement 2.2.7. Structure currently is no suitable SAXS analog to R analog to is no suitable SAXS currently particularly when substantial non-crystallographic symmetry exists. Though R Though exists. symmetry non-crystallographic substantial when particularly a universally accepted measure of quality which is useful for both crystallographers and external reviewers. In contrast an to appropriate X-ray analog crystallography, to the R factor is under debate (section 4.1), and there Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

. & Go, 1990) and have been Smith, 2007), they hold potentially hold they 2007), Smith, & Go, 1994). This diffuse scatter arises from arises scatter diffuse This 1994). Go, & 15 & Papiz, 2003). The decisions on how to properly model Gruner, 1997) (Meinhold 1997) Gruner, & & Trueblood, 1968) or normal mode models (Kidera Phillips, 1997; Wall, Ealick Ealick Wall, 1997; Phillips, & & Murshudov, 2001; Winn, Murshudov These methods for understanding flexibility of molecules in the contextof a crystal lattice can directly In addition to alternative modeling techniques to fit potential information about disorder in the X-ray The crystallographic Debye-Waller or B-factor has been used as a surrogate for flexibility and local to monitor over fitting (Rejto & Freer, 1996). However, in at least one case, TLS refinement performed free 2.3. Small angle X-ray scattering complement more direct measurements of flexibilitySAXS, and fluorescence studies. care must However, be taken as the crystal lattice derived can directly influence what from solution experiments includingconformations can be observed and what range of motions are possible. NMR, valuable information regarding biologically relevant motions. to streaks due to lattice Although distortions. these experiments have been largely restricted to model systems Clarage (Wall, scatter is not radially averaged as it is in SAXS scattering. Fitting of this diffuse scatter by techniques like normal like techniques by scatter diffuse this of Fitting scattering. SAXS in is it as averaged radially not is scatter mode analysis (section 3.3) has suggested that they involve scatter does distances not this between include diffuse unit distortions cells affecting that give rise correlated Importantly, motions of domains of the proteins. experiments, have emerged (Faure et al., 1994; Mizuguchi, Kidera Mizuguchi, 1994; al., et (Faure emerged have experiments, transient and static imperfections in the crystal lattice and causes scattered X-ray intensities to be observed at positions other than the Bragg peaks. Since these motions occur in crystals trapped in the lattice, the diffuse diffraction data, theories for the interpretation of diffuse scatter, which is normally data, ignored theories in diffraction for X-ray the diffraction scatter, interpretation of diffuse R 2000). & Brunger, (Wilson better than multiple model refinement the resolution of diffraction. Unfortunately, these ensembles also introduce the real possibility of introducing far more parameters than can be justified by the raw data. Attempts to ensure that individual refinements are restrained have been performed by ADFsrefining at only a single time while models with monitoring isotropic second sort of crystallographic ensemble, multiple structures can be simultaneously calculated against the raw This data. refinementwould be most appropriate for structures that have the types of disorder that tend to limit NMR refinement. From a number of test cases, it has been suggested that multiplecan fit the distinct experimental data equally well isotropic and thereby suggests models that classic measures of model accuracy fail to the In 2004). Blundell, & Bakker de (dePristo, refinements model single in ambiguities and inaccuracies capture analyze disorder within crystal structures (Furnham et al., 2006). Two types of “crystallographic ensembles” can be envisioned. In the first, the different structures represent independent refinements againstduring thegenerated models calculated rawindependently the data to equivalent most are These other. each “see” not do and disorder in crystals. A number of alternative ways to model these features in crystal structures have understand better to as well emerged as R-factors improve to disorder the fit better to seek schemes These recently. more disorder in the crystallized molecules. For example, the use of multiple models provides a possible means to crystallographic refinement derived from chemistry or non-crystallographic symmetry refinement derived from chemistry crystallographic in crystals and disorder 2.2.8. Flexibility it is in X-ray crystallography. For SAXS, the application of use of external constraints, such as symmetry or atomic structures of individual domains, can be very important to ensure solution the structures reproducible (section 4), reconstruction and of these constraints are analogous to the use of geometric constraints during Isupov the structure given the information content of the data is as important in the generation of SAXS models as crystals. crystals. These schemes simultaneously model the motion of groups of atoms (TLS) by models translation-libration-screw (Schomaker used to explain motion and help refinecrystal structuresat moderate resolutions (Howlin et al., 1993;Winn, Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 , and bulk solvent solvent bulk and , ) models r and I(0) and PF1282 ( G as well as ρ max ab initio 11 Å; 11 magenta) than G Pyrococcus Pyrococcus furiosis , of solute molecules of interest, of molecules solute of , ) r ( 16 ∆ρ . D. Each protein has both a substantially D different G 82 Å; green). C. The linear region of the Guinier plot, from which R which from plot, Guinier the of region linear The C. green). Å; 82 G SAXS curves and parameters measured for the

Experimental of the scattering particle is a simple function of molecular weight for perfect spheres (spheres), spheres perfect for weight molecular of function simple a is particle scattering the of max for pure water): 3 protein show that the intensity of scattering falls more slowly for rubredoxin (R (R minicellulosome the can be derived, is function of the R atomic arrangements. function, reflecting the different pair-distribution Fig. 4. “designed” the 2005), al., et (Hammel (red) S4 protein scaffoldin “designed” the (magenta), rubredoxin (blue). kinase protein DNA-dependent the and (green), subunits catalytic 3 containing minicellulosome D A. but not for proteins that adopt different shapes (diamonds). Envelopes correspond to calculated from experimental curves using GASBOR. B. The experimental scattering curves for each /Å - Unlike X-ray crystallography, SAXS is inherently a contrast method where the scattering signal is derived is signal scattering the where method contrast a inherently is SAXS crystallography, X-ray Unlike (~0.33 e

S from the difference in the average electron density, density, electron average the in difference the from ρ 2.3.1. Measuring SAXS data Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 ), is S with is the r λ . Confusingly Confusingly . q /d, where 1/d is the π ) is a function of the q ), is radially symmetric radially is ), as the scattering angle. scattering the as q are the inverse of units = 2 θ, q q values give rise to larger ) r can be used for used be can ∆ρ( h should be identical for the same the for identical be should q ), is the square root of the average G and s rather than 2 than rather ) vs. ) q . Larger θ, 3 /Å - , like the hydrodynamic or Stokes’ radius , (R like the hydrodynamic or Stokes’ G . R r 1/2 , and others define others and , 17 S as the scattering angle. The units of = (3/5) θ G is the scattering angle, as in X-ray and crystallography, = 2π = , and the value is a measure of the directional momentum change q θ -1 or nm , so that so , -1 , where 2 λ λ )/ )/ θ θ = (2sin = S as defined above with 2 = (4πsin q q is the maximum distance present in the scattering particle. max From a practical standpoint, the lowest resolution portion of the SAXS scattering curve is dictated by a The scattering curve of a homogeneous sample can be derived from the electron distribution of the The theoretical basis for solution scattering has been the subject of an excellent review (Koch et al., Unlike X-ray crystallography, where diffraction provides a clear measure of quality, it can be more The scattering curve resulting from the subtraction of the buffer from the sample, I( sample, the from buffer the of subtraction the from resulting curve scattering The uniform electron density, for example, has uniform a electron R density, the radius of gyration (R single This size size parameter parameter, (Fig. 4). squared distance of each scatterer from the particle center (Table 1). For example, a sphere of radius where D function, P(r), section 2.3.3): particle (the pair-distribution commonly used relationships relevant to this case are tabulated in 1. Table More complicated or recalcitrant (section 5). samples require additional experimental and theoretical treatment 2003). Here we briefly consider the most common situation for structure reconstruction (section 4), in which samples are homogeneous, monodisperse, and lacking long-range interactions in solution. Many of the most 2.3.2. Scattering from macromolecules 2.3.2. Scattering from will benefit from a growinggroup of researchersWe adopting encourage SAXS researchers methodologies. to in the literature. describe problems as well as solutions unsolved problem; however, some empirical guidelines do exist for assessing data quality (sections 5.2-5.5). Many Many 5.2-5.5). (sections quality data assessing for exist do guidelines empirical some however, problem; unsolved issues are primarily and understood a anecdotally, directed effort on the best methods to assess sample quality difficult to confirm that a measured scattering curve is appropriate for further analysis. In general this is an that the photons undergo. By comparison with Bragg’s law in X-ray crystallography, law in X-ray crystallography, By that comparison the with photons Bragg’s undergo. I( of plot a wavelength, incident the of Regardless resolution. reciprocal anomalous scattering of atoms within the sample occurs. sample, except at wavelengths where will consistently use used in the wavelength, typically Å wavelength of the incident X-ray beam. In various treatments, the symbols the treatments, various In beam. X-ray incident the of wavelength define treatments other Each of these definitions is equivalent; however, so the convention being followed must be defined. Here we (isotropic) due to the randomly-oriented distribution of particles in solution (Fig. 4). I( momentum transfer Subtracting scattering of the blank from the sample must be done as precisely as possible to accurately measure accurately to possible as precisely as done be must sample the from blank the of scattering Subtracting of magnitude (section 5.1). of over three orders differences structures, structures, which have higher contrasts than proteins. In practice, data is collected on a buffer a blank sample. and Subtraction on of observed scattering yields the signal from the scattering due to the macromolecule. Proteins, Proteins, for example, have variation an contrast average for electron density as of well ~0.44 as e solutions dilute from scattering maximize to important is which 1), (Table signals techniques (section This 2.3.6). result also makes SAXS particularly attractive and for DNA determining RNA Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 ). is is max data. data. V ; D P(r). I(0) q may be X C X 2 max . and and R G D M is is goes to 0 as as to 0 goes R P(r) . Measured. . 2 2 G XC q 2 R R , and Comments ] vs.q ] ] vs. ] max D must be on an absolute an absolute be on must qI(q) I(q) , G goes to 0 in regimes in regimes 0 to goes however, portion of a plot of of a plot of portion ln[ The slope of the linear linear the of slope The Calculation of of Calculation of version this using to susceptible less inter- and aggregation than correlations particle low of extrapolation I(0) by normalized and scale molar not and mass/volume concentration. The co-axial length rather rather length co-axial The ( hypotenuse the than complicated by flexibility or multimerization. however and absolute scale concentrationinformation are unnecessary. Particularly useful elon- for Particularlyuseful gated proteinsthe where approximation Guinier is narrower valid over range. Good consistency check for R transform Indirect Fourier methods have been developed calculatingfor Assignment of Most common method of estimating The integral this portion of integral The equation as is the known AccuracyPorod invariant. and size; varies shape for q is scattering where by dominated via thevia slope theof plot ln[ : < S =0 G max is ȡ q . : Inte- /D qR (0) XC max ʌ R exp values. values. I D q q I(0): range. range. range. =0 and =0 does and and q q q : Avogadro’s : Avogadro’s G A R range. range. N Definitions q q q and Variable < elongatedfor1.4 Entire Entire Entire See See <1.3 globular,<1.3 G Particle electron electron Particle G Range of Data Used Used of Data Range : Intermediate Intermediate : Average mass per per mass : Average p Requires data Requires data qR Solvent electron density density electron Solvent ȡ Maximum dimension of dimension Maximum particle density density ȝ electrons. of number Entire theexperimental intensity at qR Entire number. number. 0.8 elongated. pr- the scattering nsity of ofile extrapolated to not require an not absolute an require scale. ibly linked ibly linear macromolecules ) · ¸ ¸ ¹ 22 G Rq 18 drrPdrrPrR dqqrqqI 22 XC dqqqI )()( )sin()( 22 2 G 2 Rq 2 Rq · ¸ ¸ ¹ 3 eRq 2122 )()0(2 ))(1( drrP pSA 0 max ³³ )) )(4)0( 1( D UU 2 )]0(ln[)](ln[ XCG 0 ³ f )]0(ln[)](ln[ P 22 G RRL ¨ © § ¨ )0( 0 max ³ D IqI qIqqI I § ¨ ¨ © 0 ³ f Common formulas applied in SAXS analysis SAXS in applied formulas Common S (12( )0(2 44 2 G Formula for large r

2 2 exp Rq r

N S IV I 2 2 0 max

D S is the value of at r P(r)=0

I M )(

)( Table 1. Table rP

max qI 2 G D Formulas for elongated or flex ) ) max ) ) D L P(r) M ) XC : Debye R P(r) : Guinier ) ( ): Porod ) G : defined by G : Intensity at V Invariant ) Mass ( Mass Maximum Maximum ( Length ( Length Parameter G (R (R proportional to Approximation Approximation Function ( of cross-section cross-section of I(0) Particle Volume (R Pair Distribution q=0 which is also also is q=0 which Dimension ( Mass and Volume Mass and Volume Radius of Gyration Gyration of Radius Radius of Gyration Radius of Gyration Radius of Gyration Radius of Gyration Radius of Gyration Radius of Gyration Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 and and G = 1 in 1 = f values q can alter the alter can G = 4, flat (oblate) (oblate) flat 4, = f <1.3 for globular G qR values is I(0), the intensity q and I(0) can be extracted (Fig. 4C; (Fig. extracted be can I(0) and G shows some dependence on the contrast G should not vary with concentration for well- for concentration with vary not should and thus cannot directly provide information provide directly cannot thus and G )/3] G 2 G R f 2 d 4 q − − =5/3. q q f ∝ ∝ ) ) q q 19 ( ( than larger particles (Fig. 4C). In practice, this estimation of estimation this practice, In 4C). (Fig. particles larger than I I G ) = I(0) exp[-( q I( will give a straight line from which R which from line straight a give will 2 ), which can be calculated directly from the scattering curve, provides an q q is the radius of an equivalent sphere that diffuses identically to the molecule molecule the to identically diffuses that sphere equivalent an of radius the is -range over which the Guinier approximation is valid ( S -range, whereas scattering from needle-like (prolate) ellipsoids has a d a has ellipsoids (prolate) needle-like from scattering whereas -range, q q )) against against )) = 0), which must be determined by extrapolation, as it is coincident with the direct q q for a perfect sphere. for a perfect r = S ) as a function of = 2 in the high the in 2 = may be more appropriate (Table 1). Similarly, R Similarly, 1). (Table appropriate more be may f q values contain details regarding molecular shape. For folded macromolecules, the intensity of G q I( 2 q -range. Random coils in “good solvent” have d & Fournet, 1955)). The q is the fractal degrees of freedom. For example, scattering comprised of spheres has a d a has spheres of comprised scattering example, For freedom. of degrees fractal the is f Higher The second important parameter that can be evaluated from the lowest At low resolution, the scattering can be described by the Guinier approximation: by the can be described the scattering At low resolution, Thus, SAXS is an ideal method for identifying and characterizing polymers without folded domains. The domains. folded without polymers characterizing and identifying for method ideal an is SAXS Thus, must be performed iteratively or interactively (Konarev et al., 2003), since new estimates of R of estimates new since 2003), al., et (Konarev interactively or iteratively performed be must are different, however, in that R that in however, different, are G S -range for which the estimate can be made. Lack of linearity in the Guinier plot is a sign that more care needs where d where d a has ellipsoids approximation, holds only in a portion of the scattering curve, and we have observed some samples that possess that samples some observed have we and curve, scattering the of portion a in only holds approximation, termed typically is scattering of region this polymers, arbitrary For law. Porod’s following scattering no or little scattering can be expressed as: the “power law regime”, where the resolution-dependence of the This relationship, however, assumes a uniform density for the scatterer, which breaks down at high when atomic resolution information begins to contribute significantly. Hence, Porod’s law, like the Guinier law (Porod, 1951): by Porod’s the scattering falls off a straight line. Additionally, since I(0) depends on the square of the number of electrons (molecular weight), the assembly state of the scatterers. SAXS is particularly sensitive to particle shape and is useful for molecular weight determination (section 2.3.4). I(0) is equivalent to the value of value the to equivalent is I(0) 2.3.4). (section determination weight molecular for useful is and shape particle gives concentration vs. I(0) of plot a samples, well-behaved For 2.2.3). (section crystallography X-ray in I(0,0,0) measured at zero angle ( beam. On an absolute scale, I(0) is the square of the number of electrons in the scatterer and is unaffected by difference difference between bulk solvent and the sample comparisons of different samples should be performed in the same buffer. to be taken to evaluate the sample (section 5), or that samples are elongated. For these samples, other methods R estimating for behaved samples with no interparticle interference or aggregation. R R q The Guinier plot of log(I( of plot Guinier The (Guinier R small with particles for larger much is proteins) shape-dependent and a poor measure of the actual molecular weight (volume) of the molecule of interest. R interest. of molecule the of (volume) weight molecular actual the of measure poor a and shape-dependent R hence R of interest, shaped like a parabola. The position of the peak provides some information about its overall size; however, our however, size; about its overall The position of the peak provides some information a parabola. shaped like R like shape-dependent is position the that shown has experience Kratky plot ( excellent tool for evaluating the folding of samples. For folded domains, the Kratky plot yields a peak roughly the high Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 for G corresponds to the maximum max -region. At low resolutions the scattering the resolutions low At -region.    q , where D 2 x 1 max 8 + D ≥ x 1 4 − in the large large the in 4 1 q − x 6 <3/b. For peptides, b varies between 19-25 Å, yielding an 20    q 2 b & Porod, 1949). Random coil or unstructured peptides lack the = value for unfolded or chemically denatured samples is so large that large so is samples denatured chemically or unfolded for value 2 ) G G R ( . is the radius of gyration of the cross section, L is the total length of the polymer, and b and polymer, the of length total the is L section, cross the of gyration of radius the is c Lb/6 and R and Lb/6 2 q Theoretically, the Theoretically, P(r) function is zero at r=0 and at r The important differences between P(r) and the Patterson function are that the P(r) is radially averaged The pair-distribution function The P(r), pair-distribution also called the distribution function pair-density (PDDF; Fig. 4D) is samples and can be an indicator of good quality data (section 5.3). On the other hand unfolded proteins are linear dimension in the scattering particle. For the processing of real data, the P(r) function is typically constrained typically is function P(r) the data, real of processing the For particle. scattering the in dimension linear in the calculation to be zero at these values. This constraint is often not necessary for well-behaved (globular) Additionally, Additionally, GIFT also simultaneously fits contributions from the scattering due to interparticle interactions 1997) & Glatter, (Brunner-Popela 2000), the inverse transformation is solved using Boltzmann simplex simulated annealing to solve the non-linear the solve to annealing simulated simplex Boltzmann using solved is transformation inverse the 2000), parameters. the fits iteratively and parameters factor structure P(r) the with curve scattering the of dependencies trial P(r) functions that are Fourier transformed and evaluated in comparison with the experimental scattering. In scattering. experimental the with comparison in evaluated and transformed Fourier are that functions P(r) trial the GNOM 1991), program a (Semenyuk regularizing & multiplier Svergun, is used to balance the smoothness of the trial P(r) functions with the goodness of fit to the data. Fritz In program & the (Bergmann, Glatter, GIFT at (0,0,0) and other positions corresponding discrete to due problems avoid to transformation Fourier to indirect an by calculated is function P(r) the pure constructs Typically, crystallographic essentially transform translations Fourier indirect in The 1977). the (Glatter, range Patterson finite a function. over curve I(q) the of sampling and lacks vectors corresponding to vectors between scattering particles, which gives rise to large “origin peaks” “origin large to rise gives which particles, scattering between vectors to corresponding vectors lacks and information about the distances between electrons in the scattering particles in the sample, in a manner similar can also be calculated directly from the electron density: The P(r) function to the Patterson function. the SAXS function corresponding to the Patterson function (section 2.2.4). This autocorrelation function can be can function autocorrelation This 2.2.4). (section function Patterson the to corresponding function SAXS the directly calculated through a Fourier transform of the scattering 1), curve and (Table the result provides direct 2.3.3. Pair-distribution function 2.3.3. Pair-distribution the scattering region following Guinier’s approximation is typically not recorded in normal beamline geometries beamline normal in recorded not typically is approximation Guinier’s following region scattering the (Calmettes et al., 1994). where x=L/b. This equation is useful as the R the as useful is equation This x=L/b. where average persistence length of 9.5-12.5 Å or roughly 3-4 amino acids (Perez et al., 2001). The expected R the equation: can be calculated with the unfolded polypeptide where y = y where is twice the persistence length, the maximum length that the polymer chain persists in any one direction (Table 1). This relationship holds in the resolution range Cotton, 1996): Boue & Cotton, by (Brulet, can be described regarding molecular weight. In contrast, extended semi-stiff polymers, such as random coil peptides, follow the the follow peptides, coil random as such polymers, semi-stiff extended contrast, In weight. molecular regarding Porod-Kratky worm-like chain model (Kratky to respect with linear are and peak folded characteristic Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

max by max Ab initio Moore, & ab initio reconstructions is 260 Å, whereas the D , giving the value more . More problematically, . More problematically, max max max B-DNA may indicate aggregation or aggregation indicate may /D C π max 2 ≥

q and I(0) can be calculated from the G 200 values and evaluation of the resulting P(r) max π / r (Å) max D ) 100 Experimental min q Theoretical - , and hence contribute little to the Thus, overall scattering. max 21 q max

= ( s P(r) N 0.3 0.2 0.1 0.0 B is useful for characterizing the sample; however, accurately determining determining accurately however, sample; the characterizing for useful is max reconstructions generated by GASBOR (red; (Svergun et al., 2001a)). The experimental 0.2 ab initio ) -1 can be difficult to identify, including extended structures and globular structures with structures globular and structures extended including identify, to difficult be can Theoretical Å max ( q 0.1 ab initio reconstructions of 69 Å as compared to the 67 Å for the structure. theoretical B-DNA B. Comparison of the P(r) functions calculated Experimental G The P(r) function has many important usages. First, a value for R One of the central strengths of SAXS is that measurements are done in solution with little preparation 0 1 Theoretical and experimental SAXS scattering from a 76mer double-stranded DNA fragment. A. Theoretical scattering A. from fragment. Theoretical and experimental SAXS scattering from a 76mer double-stranded DNA

100 for samples can be difficult. The scatteringfor data samples shouldcan be be measureddifficult. at 1000 Intensity max A 1949): relative to other techniques. The downside is that the measured data are rotationally averaged and cannot, for example, be used to distinguish between enantiomorphs. Furthermore, (Shannon channels the Shannon of number scattering the by estimated typically curve points, data independent only of number has a small 2.3.4 Information content in scattering curves shapes (Fig. 5) and are particularly useful in conjunction with Kratky plots (section 2.3.2). shapes (Fig. 5) and are particularly useful in conjunction with Kratky This has important methods implications of for using many atomic different models in conjunction with SAXS data (section 4.2). The P(r) function can also give some initial indication of the overall shape from its overall shape, for example spherical objects with bell-shaped P(r) functions can be readily distinguished from rod-like better estimate for samples complicated by small amounts of aggregation that most strongly affect the lowest resolution information. Second, P(r) functions can be readily calculated from atomic models (section 4.2.1). a linear or extended molecule and can be seen with protein samples as well (Fig. 10). The observed D calculated for the fragment theoretical is DNA 255 Å, and the linear length expected for 76 bases is of 250 B-DNA Å. C. GASBOR are quite similar in thickness and length to the size of B-DNA. generated by reconstructions of the DNA experimental scattering (black) or data has a R of characteristic are P(r) the of off fall linear roughly and peak early The GNOM. by scattering experimental and theoretical the from Fig. 5. a 76mer fragment B-DNA without counter ions was calculated using (green; CRYSOL (Svergun et al., 1995)) and compared to the direct beam that is used in the Guinier approximation. Thus, this real space approximation is likely to be a P(r) function that takes into account all of the collected data and is not limited to the small region about the the inverse Fourier transformation involves choosing multiple D fit to the experimental scattering. functions for their P(r) curves are typically small in the vicinity of D D of estimates in errors disordered extensions, such as unstructured N- and C-termini in proteins. In practice, estimation of D the Fourier indirect methods transformation to P(r) calculate rely upon the value of D importance than if the P(r) function could be calculated from a direct Fourier transformation. Moreover, the often not zero at r=0 in unconstrained P(r) functions, and non-zero values at r=D at values non-zero and functions, P(r) unconstrained in r=0 at zero not often D subtraction. background improper D Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 is the the is

in the in ψ max q between two q ∆ of the molecular M/m , is reported as mass per . This fact has been used c max /D π is the ratio µ . An alternative measure for the number max ), I(0) can be directly related to the molecular /D -1 π

min q is the average electron density of the solvent, and solvent, the of density electron average the is 0 22 ρ is Avogadro’s number, and Avogadro’s is A N algorithms include constraints that attempt to force final solutions to have to solutions final force to attempt that constraints include algorithms , where A m/N ought to be measured at µ ab initio ab = 1.87). Therefore, = 1.87). min q c=N & Hansen, 2006). SAXS data are dramatically over sampled, in that M/m Koch, 2001b). The number of restraints added by this type of external information is not is information external of type this by added restraints of number The 2001b). Koch, & are known, and the intensity of the incident beam is known on an absolute scale, then the intensity the then scale, absolute an on known is beam incident the of intensity the and known, are c is the number of electrons of the particle, the of electrons of number the is An experimentally more tractable measurement of mass can be obtained using relative I(0) values In a monodisperse, ideal solution of identical particles, the observed scattering is linearly related to the Given this estimate of information content in a solution scattering curve it is remarkable that accurate , and , m 0 ρ , ), the first data point G usually does not exceed 10-15. As the lowest-resolution value describes the overall size of the scatterer ψ s If 2000). & Svergun, Vachette et al., 2003; at the origin provides a determination of the molecular weight (Koch

volume (mg/mL) and is weight to the number of electrons, which depends on the chemical composition of the particle (for proteins a good approximation is in units of the molarity of the particle, then I(0) is proportional to the mass squared. However typically only the only typically However squared. mass the to proportional is I(0) then particle, the of molarity the of units in molarity of the monomeric unit is known and the concentration of the particle, target where concentration by scaled is curve scattering the If electron. the of number its to particle the of volume the of ratio weight of the particle: the scattering of the single particle and it is proportional to the square of the total excess scattering length in the in length scattering excess total the of square the to proportional is it and particle single the of scattering the particle. If the measurements are made on an absolute scale (cm number of particles, N, in the sample. The measured I(0) obtained after scaling for concentration corresponds to corresponds concentration for scaling after obtained I(0) measured The sample. the in N, particles, of number theoretical and practical methods to do this in sections 4 and 5. theoretical and practical methods state in solution weight and multimerization 2.3.5. Molecular using known crystal structures as a basis for fitting low resolutionchemical bonding parameters SAXSin the case of moderate scatteringor “low” resolution X-ray diffraction data, dataand we detail mirrors the use of easily estimated. Regardless of the precise number of fitted parameters that can be justified, we would suggest that there is a clear analogy with the difficulties of(section refining2.2.7) in X-raywhich crystal the structuresbest atuse of differentthe resolutions available experimental data includes external information. Thus, the reconstructions. Modern reconstructions. the protein-like properties. For example GASBOR enforces penalties on its shape reconstructions for compactness Petoukhov (Svergun, shapes. This intuitively daunting feature may explain why the development of at SAXS the has same not rate been of pursued crystallography. Studies have demonstrated situations structural in information can which be extracted more utilizing SAXS and data when more additional constraints detailed are been imposed on adjacent points measured in the scattering curve are, however, much less than to argue that the effective information content is higher than predicted from the number of Shannon channels (Koch et al., 2003). Hansen & Puerschel, 1996). This method accounts for the problem of determining a uniquely defined uniquely a determining of problem the for accounts method This 1996). Puerschel, & Hansen this measure does not substantially change the number of parameters presence of experimental noise; however, available (Vestergaard (R of experimentally-determined parameters has been suggested that uses a maximum entropy method (Mueller, The number of independent values that can be extracted from the scattering (reciprocal space) has been shown to be equivalent to the number of independent data points in real space (Moore, 1980). For most SAXS curves, N Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

-4 q values q ab initio are typically 3 using the fall off of intensity with q & Svergun, 2007). Samples composed of shape determination algorithms (section 4.3) (Hammel 4.3) (section algorithms determination shape 23 ab initio ab range (due to extrapolation of high q , and this contributes error to the calculation. Thus, the large angle portions of the curves should be should curves the of portions angle large the Thus, calculation. the to error contributes this and , -1 Importantly, Importantly, the I(0) method for molecular weight determination and volumes derived from The volume of the macromolecule undergoing scattering can be calculated from I(0) and the Porod undergoing The volume of the macromolecule Since most macromolecules have fairly uniform densities, the molecular weight is also directly related Each of these techniques requires accurate determinations of I(0) values from Guinier or Debye provides a powerful approach to determining the molecular weight and assembly state in solution that can be extremely useful for crystallization efforts (section 2.4), for modeling solution assemblies (section 4), and for 2006; Nemeth-Pongracz et al., 2007; Petoukhov et al., 2006; Petoukhov et al., 2003; Qazi thus each et can be al., used to independently 2007), confirm theand results from a single sample 1). (Table Inpractice, SAXS shape determination (Hammel et al., 2002; Krebs et al., 2004) should yield consistent results (Gherardi et al., discarded in the computation (Glatter, 1982). Additionally, this technique for extracting mass is very inaccurate inaccurate very is mass extracting for technique this Additionally, 1982). (Glatter, computation the in discarded (Konarev PRIMUS as such programs using calculated easily be can volumes Excluded particles. asymmetric for et al., 2003). ). The contribution of internal particle structure to scattering at larger angles becomes significant at Å 0.2 above twice the molecular masses in kDa and is a valuable conformation of mass estimates using I(0) al., (Gherardi 2006; Petoukhov et et al., 2003). These volume determinations are, however, subject to error as they rely on the accurate data over the entire This calculation does not require data normalization. For globular proteins, Porod volumes in nm where the invariant is calculated by: where the invariant is calculated invariant Q (Porod, 1982): et al., 2002; Krebs, Durchschlag & Zipper, 2004) and/or volumes derived from Porod law (Porod, 1982). the scattering according the nor reference samples have been measured. In this approach the theoretical excluded volume calculated from sequence or from an atomic model, such as reported by the (Svergun, program Baraberato CRYSOL & Koch, by generated volumes to compared be can 1995), to volume. Volume information to can volume. Volume be derived from SAXS curves in experiments where neither absolute I(0) the I(0) measured from Guinier plots, particularly for data where only a few points have been measured in the The P(r)-based I(0) value is by Guinier interparticle region interactions. or where the Guinier region is affected 1992). et al., 2000; Svergun, P(r) calculations (Bergmann typically reported by programs performing The P(r) is calculated from the scattering curve. Thus, extracting I(0) from the P(r) has several advantages over advantages several has P(r) the from I(0) extracting Thus, curve. scattering the from calculated is P(r) The approximations (Table 1), or from estimates using the P(r) function: 1), or from estimates using the (Table approximations above the critical micelle concentration of the solubalizing detergent (Columbus et al., 2006). (Columbus et al., detergent the solubalizing concentration of critical micelle above the problematic. problematic. Nevertheless even with mixed electron density systems, bounding the mass between values may collected systems protein membrane for demonstrated been has as state multimeric the establish to sufficient be after after proper calibration with reference samples, such as lysozyme (14.3 kDa), bovine 66.2 serum kDa), and glucose albumin isomerase (172 kDa)(Kozak, 2005; (BSA, Mylonas multiple components with different average electron densities, such as protein-DNA complexes, can be more Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 = q H), makes neutron scattering (SANS) with (SANS) scattering neutron makes H), 2 model of the detergent solubilized apoB-100 protein. apoB-100 solubilized detergent the of model 24 H) and deuterons ( deuterons and H) 1 & Svergun, 2003), and the secondary structure information ab initio ab Reconstructed three-dimensional Reconstructed was mapped onto the low resolution model. C. Hypothetical model of apo-100B in LDL. D. the Model of the particle LDL spatial with apoB-100 and a arrangement superimposed 250 Å of sphere, representing the lipid components (after (Johs et al., 2006)). Fig. 6. Fig. (Svergun, DAMMIN by restored were fit of goodness similar with models single independent Ten A. 1999). B. The average envelope of apoB-100 was calculated for 10 independent DAMMIN models using (Volkov the program DAMAVER Measurable Measurable scattering from a solute is contingent on the contrast in scattering density between the . SAXS can be used to extract internal structural features of systems comprised of two or more components more or two of comprised systems of features structural internal extract to used be can SAXS . -1 the interaction of neutrons with hydrogen atoms ( atoms hydrogen with neutrons of interaction the of choice for contrast variation studies. specifically deuterated components, and not SAXS, the technique or salt can mask out the scattering of one of the components (Pilz, 1982), and contrast variation studies have been successfully performed with SAXS (Muller et al., 1978). In practice, the however, dramatic difference in with distinct average electron densities. 1973; Since (Stuhrmann, systems multicomponent SAXS of structure inner is the about information a extract can density contrast electron method, variation of the average solvent glycerol, sugars, of concentrations high with densities electron solvent appropriate Choosing 1982). Stuhrmann, solute and solvent. For length scales larger then 15 Å the scattering density for biomolecules is approximately than lower resolutions at ignored successfully be can features structural internal most Thus, homogeneous. Å 0.2 interpreting biochemical and mutational results. results. mutational and biochemical interpreting variation 2.3.6. Contrast Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 , 2007). Schizosaccharomyces Schizosaccharomyces structure had not been

subunit of polymerase DNA III α , features of the SAXS scattering were �� (Chamberlain et al., 2001). One recent SANS study used used study SANS recent One 2001). al., et (Chamberlain Pyrococcus Pyrococcus furiosis 25 structure reconstruction as discussed below (section 4.2) and direct ab initio �� (Stuhrmann (Stuhrmann et al., 1976; et Svergun al., 1997a; et Svergun al., 1997b), human plasma lipoprotein Finally, SAXS is also sensitive to the overall shape of the macromolecule, and samples that are unfolded are that samples and macromolecule, the of shape overall the to sensitive also is SAXS Finally, Similarly, Similarly, for some molecules that form complexes, control of the molecular assembly state has been Experimental evidence has shown that homogenous solutions of macromolecules with little or no A A frequently underappreciated feature of a SAXS experiment is that even in the absence of an atomic Taken Taken together, these uses of SAXS data can be tremendously helpful for characterizing proteins SANS SANS contrast variation studies on two component systems have been performed for samples such as cell cycle-regulatory protein suc1 for crystallization (Bourne et al., 1995; Parge et al., 1993), although aggregation or unfolding (section 5), SAXS can be used to provide initial information regarding the shape and the assembly of complexes from being expressed and purified for biochemical, biophysical and crystallographic studies. In general, the typical concentrations and total amounts of sample required for SAXS are readily accessible when crystallography or other biophysical experiments are being pursued. In addition to evaluating constructs and conditions buffer for are clearly visible in the This Kratky evaluation plot is (section particular important 2.3.2). for engineering and characterizing truncation mutations for crystallographic studies, such as the proteins (Shell, Putnam & Kolodner (Lamers et al., 2006), as well as identification of natively unfolded particular assemblies for crystallization. pombe As many has others been examples described also in exist section based 2.3.4, on SAXS anecdotal is evidence. exquisitely sensitive to the assembly state and thus can be used to identify buffer conditions that help stabilize important for crystallization, such as for the requirement of multimerization by the between samples where aggregation can be ameliorated by varying conditions, centrifugation, or size-exclusion size-exclusion or centrifugation, conditions, varying by ameliorated be can aggregation where samples between are hopelessly aggregated. chromatography from those that & D’Arcy, 1992) and screen for buffer conditions for crystallization (Jancarik et al., 2004) as it can evaluate the overall aggregation of the sample. As SAXS uses wavelengths that probe atomic resolution the information, presence of structure in the SAXS curves as compared to featureless decay has been used to distinguish aggregation have a tendency to readily crystallize both when 1994; multiple Ferre-D’Amare proteins (D’Arcy, 1997) or multiple & preparations Burley, of the same protein are compared (Habel, Ohren & 2001). Borgstahl, (Zulauf macromolecules characterize to technique important an as emerged has (DLS) scattering light Dynamic project involving protein complexes isolated from excellent indicators of sample folding, and assembly, aggregation. In turn each of these features was a useful Adams unpublished). Tainer, that were readily crystallized (Hura, predictor for identifying samples resolution structure, SAXS experiments are useful in directly assessing sample quality. In a structural genomics genomics structural a In quality. sample assessing directly in useful are experiments SAXS structure, resolution 2.4. SAXS characterizations that aid crystallography 2.4. SAXS characterizations quality and assessing crystallographic targets sample Validating 2.4.1. forming the lipoprotein particle (Fig. 6BCD). forming the lipoprotein well characterized. The authors have calculated low resolution models (Fig. 6AB) that cavity in the center of reveal the molecule with alternating a wide and narrow sections, which pronounced have been interpreted as when rearrange to apoB-100 allowing flexibility, confer may linkers These linkers. by connected domains folded membrane-binding membrane-binding protein apolipoprotein B-100 (apoB-100) (Johs component et of human low-density lipoproteins (LDL), al., which triggers the receptor-mediated cellular 2006). uptake of ApoB-100 is the LDL. protein Due to its size (550 kDa and 4,536 residues) and hydrophobicity, the apoB-100 (Stuhrmann et al., 1975), and DNA-protein complexes complexes DNA-protein and 1975), al., et (Stuhrmann hydrophobic the solubilize to required molecules detergent the of contribution the eliminate to variation contrast the ribosome Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 Main, & & Svergun, 2003b). & Quispe, 2005). Many Main, 1996; Cowtan 1996; Main, & & Svergun, 2003a; Sokolova, Volkov 26 phasing from fairly simple envelope models has been used for phasing for used been has models envelope simple fairly from phasing Ab initio Ab Podjarny, 1995). 1995). Podjarny, & These difficulties may be responsible for the mixed success in studies using the current version of DARA DARA of version current the using studies in success mixed the for responsible be may difficulties These There are a number of difficulties in performing these comparisons, even in the absence of experimental of absence the in even comparisons, these performing in difficulties of number a are There With With the increase in speed of modern computers, one potential crystallographic phasing strategy is a Recently, a database of calculated scattering curves, DARA, was created for a large portion of the From a one-dimensional experimental SAXS scattering curve, it is possible to reconstruct a three- For average macromolecular crystals lacking the extensive non-crystallographic symmetry of viruses, (Hamada et al., 2007). Nevertheless we have found it useful in several cases (Fig. 7), and it is currently unclear if unclear currently is it and 7), (Fig. cases several in useful it found have we Nevertheless 2007). al., et (Hamada possibility the However, investigated. being macromolecules the of features specific upon dependent is success

Second, many PDB entries lack appropriate or correct information for the biologically relevant assemblies in the in assemblies relevant biologically the for information correct or appropriate lack entries PDB many Second, file meta-data, which are problematic for DARA searches. Moreover, multiple conformations of residues and the theoretical scattering. heavy atoms in the crystal structures may cause problems for calculating 4.1). (section difficult be can curves scattering compare to which by criteria appropriate determining First, noise. features. For a brute force molecular replacement search, these phasing. hits are likely to be much more useful for structures deposited in the PDB (Sokolova, Volkov DARA ranks experimental scattering curves relative to their fit structural secondary to and the shape overall precalculated both DARA scatteringfit to curves. tended similar as scored were that curves scattering Theoretical solution scattering curve might be able to provide sufficient structural information to focus such a search. able to provide sufficient structural information to focus such solution scattering curve might be (PDB). These searches still take substantial amounts of computer time and focus the molecular search replacement on models that are most like the protein of question. Measurement of a one useful strategy would be to molecular replacement search using all solved domains from the Protein Data Bank (Bernstein et al., 1977) identified from difference Fourier maps rather than the more difficult Patterson maps. Fourier maps rather identified from difference scattering 2.4.3. Structural database of SAXS resolution phasing by envelopes has the potential to help locate heavy atoms, particularly heavy atom clusters used for phasing crystals with large asymmetric units, and thereby allow the firstheavy atom positions to be for general crystallographic problems might be to leverage the power of multicrystal averaging, if (Cowtan crystal molecular non-isomorphous each for found be can solutions replacement 1998). However, even in the absence of appropriate density modification techniques for phase extension, low the major challenge for the use of low resolution phasing techniques is to extend the resolution high enough that atomic models can be built. One intriguing phase possibility that may be more accessible information to a viral particles. For these systems, success is due to the ability to perform rounds of phase extension with non- in (Rossmann, 1995)). symmetry averaging (reviewed crystallographic using structure factors calculated from a low resolution envelope (Urzhumtsev in an appropriate cell with P1 symmetry replacement (Hao, 2001; Hao, 2006; Hao et al., 1999; Ockwell et al., 2000). This has been successful for low resolution phasing of phytase (Liu et al., 2003) and lobster clottable protein (Kollman replacements molecular do also can 2001), (Navaza, AMoRe as such programs, replacement molecular standard dimensional dimensional envelope (section 4.3.1). Even in the may be useful for absence low resolution phasing of of crystallographic data. For example, a specialized programs such as high-resolution structure, molecular by crystal the into SAXS or these (EM) microscopy electron from derived envelopes place can FSEARCH envelopes information for establishing the biologically relevant solution assemblies as discussed below (section 4.1.5) (section below as discussed assemblies solution relevant the biologically for establishing information phasing envelopes for SAXS low resolution 2.4.2. Using Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

ab initio seconds -4 to 10 -8 -helices in proteins occur in the sub- α A fundamental problem in combining solution SAXS with A 27 3. Computational techniques for modeling macromolecular modeling macromolecular techniques for 3. Computational flexibility atomic resolution structures determined by crystallography is to find ways to model motions that are accessibleto molecules outside of a crystal lattice (section 3.1). Motions can be derived of identifying similarly shaped molecules using SAXS curves high-throughput for powerful particularly be to potential the has applications. Fine tuning parameters data for may also be analysis aided by applying of the analysis on SAXS structures identified by DARA. For example,help database the identify hollow protein may search shells like ferritin (Trikha, & Allewell, Theil 1995), which tend to be biased against in (section 4.3). shape restoration -sheets and the flexing of β GASBOR envelope was calculated ab initio One method of understanding macromolecular flexibility is to extract information from experimentally Slower conformational changes typically involve concerted flexing or movement of domains and are Macromolecular motions have been best studied in proteins, and the time scale at which the motion An example of a successful (Sokolova DARA that superimposes well with the crystal structure of the of structure crystal the with well superimposes that hit. best DARA et al., 2003a; Sokolova et al., 2003b) hit from a protein a from hit 2003b) al., et Sokolova 2003a; al., et of unknown structure and function. data, an Using SAXS Fig. 7. determined crystal structures in different conformations. For systems where many structures are available, it 3.2. Motion from experimentally determined structures 3.2. Motion from Grishaev, 2005). (Bax & Grishaev, dynamic information for these kinds of motions in the biologically important time scale of 10 Stabilization can occur through binding ligands, mutagenizing individual residues, or, in crystallography, finding crystallography, in or, residues, individual mutagenizing ligands, binding through occur can Stabilization residual interpreting and measuring in advances Recent environments. packing different with forms crystal new internuclear dipolar couplings from weak alignment NMR, however, has the potential to provide a wealth of for both fluorescence and classical NMR experiments. At present, the best understood large-scale conformational large-scale understood best the present, At experiments. NMR classical and fluorescence both for changes are results of solving independent structures in which each structure is stabilized in a particular state. during the probing steps of both NMR and fluorescence experimentsan collect to takes it milliseconds areof tens the than faster be to tend also motions these lost Moreover, occurs. change faster than the conformational problematic are motions large-scale slower Thus, resonances. separate as resolved not are thus and signal, NMR more difficult to study, as these motions are typically(nanosecond time slowerscale). thanThus, the the information rotationregarding orientation of theof specificmolecule in solution reporters that are imparted nanosecond time scale. to picosecond time scales are primarily vibrational. Switching between side-chain rotamers tends to occur in the picosecond time range, and is best understood in tryptophan Quasi-harmonic motions, such as twisting of residues using their intrinsic fluorescence. occurs is strongly dependent upon the nature of the conformational change. Fast motions on the femtosecond 3.1. Time scales of macromolecular motion scales of macromolecular Time 3.1. computational tools for modeling the results in any particular experiment. We expect that leveraging SAXS data using computation will not only substantially improve SAXS interpretation tools, but the accuracy of modeling techniques. experimental feedback to improve will also provide when multiple conformations are experimentally observed at high resolution (section 3.2) or can be through inferred computational techniques such appropriate as the use normal and modes identify analysis to is (section challenge 3.3), current The molecular 3.5). dynamics (section (section techniques Carlo-based Monte and 3.4), Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & & Karplus, & Erman, 2001; Hinsen, & Berendsen, 1993) to analyze the & Gerstein, 2003) attempt to extrapolate 28 -only representations of proteins (Bahar, Atilgan α Molecular dynamics (MD) has played an important role in many areas of structural biology (Karplus Normal mode analysis (NMA) is an effective computational tool for exploring the slow, large-scale In general, techniques that attempt to derive the details of conformational changes from known crystal Most applications focus on the lowest frequency scale motions motions, larger that describe the slower, generate a wide range of macromolecular conformations from which experimental signals are calculated and compared to measured results (section 4.2.2, 4.4.2). & McCammon, 2002). For bridging atomic resolution structures with SAXS data, it is most commonly used to 3.4. Molecular dynamics simulations 3.4. Molecular molecules for agreement with experimental and 2004), data Dumas, & (Delarue structures and crystal of refinement for 2004), have Sanejouand, & (Suhre crystallography been used for molecular replacement in protein SAXS data (see section 4.3.3). have real advantages for deforming atomic models to fit EM and observed in high resolution structures that can be modeled using combinations of the lowest frequency normal modes (Krebs et al., 2002). Normal modes can also provide the ability to generate systematically deformed (Lindahl & Delarue, 2005). Additionally, experimental (Lindahl verification& hasDelarue, Additionally, been2005). challenging, as the time scales of physically be to likely is NMA 3.1). (section times correlation rotational than slower are motions NMA large the meaningful based on NMR relaxation data (Korzhnev et al., 2001) and analysis of conformational changes although the absolute magnitudes of these frequencies lack physical meaning as the force constants are arbitrary are constants force the as meaning physical lack frequencies these of magnitudes absolute the although backbone atoms in reduced C 1998). in which the molecular dynamics trajectories can be inaccurate (Smith et al., 1995). The geometry of the atoms the of geometry The 1995). al., et (Smith inaccurate be can trajectories dynamics molecular the which in is more important for controlling the calculated normal modes than the details of the force field used, allowing between or 1996) (Tirion, atoms adjacent between measured only are interactions which in simplification further the static structure. These calculations can be performed in Cartesian or torsion angle space (Levitt, Sander Stern, 1985). This description reduces the potentials to quadratic functions, making normal mode calculations much less computationally expensive than molecular dynamics (section 3.4), particularly for long time scales 1985). In essence, each atom in the structure is treated as if it were separated by springs from all other atoms in atoms other all from springs by separated were it if as treated is structure the in atom each essence, In 1985). the structure, with the equilibrium position for each of these springs being defined as the distance observed in motions by which macromolecules move. NMA derives “fundamental” or “essential” motions of a macromolecule a of motions “essential” or “fundamental” derives NMA move. macromolecules which by motions via a simplifying assumption: distortions can be described by harmonic energy potentials (Brooks 3.3. Normal mode analysis changes will be relevant and are not, for example, artifacts stabilized by crystal-packing interactions. Despite these potential problems, any modeling of experimental SAXS data in which flexibility is suspected ought to conformations (section 4). include investigation of known experimental to the solution conditions have been stabilized and conformations are unlikely to The have analysis been also observed. assumes that the observed conformational characterized. For very flexible targets, all appropriate structures have a number of limitations. The analysis assumes that the appropriate conformations relevant intermediates intermediates between analysis two direct the structurally than determined satisfying endpoints less is by intermediates generating these theoretical of intermediates introduction the that sense, are some In minimized. energy structures. analysis only requires two endpoint this structures; however, of experimental (over 15) individual structures are known (van Aalten et al., 1997). In contrast to ED, other as techniques such those implemented in the MolMovDB server (Echols, Milburn is possible to apply the essential dynamics (ED) method (Amadei, Linssen major motions that underlie conformational changes. ED was originally derived to extract information about motions from molecular dynamic simulations of proteins, but has been applied to situations for which multiple Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & Abagyan, 1997). Monte Carlo 29 methods have been developed to generate low resolution envelopes, similar to those in EM in those to similar envelopes, resolution low generate to developed been have methods quantum chemical calculations. The force fields can also be user-defined, which allows additional allows which user-defined, be also can fields force The calculations. chemical quantum ab initio ab Monte Carlo simulations have also been used to sample configuration space proteinsin (Binder & Molecular dynamics calculates the energies and of dynamics with molecules atoms being influenced by dynamics Molecular the calculates energies A variety of tools and techniques are employed to model SAXS data in conjunction with atomic models. atomic with conjunction in data SAXS model to employed are techniques and tools of variety A ab initio ab 4.1 Comparing SAXS profiles and assessing agreement 4.1 Comparing SAXS profiles strong constraints on possible solutions, making this type of modeling particularly powerful for understanding biological systems. curve is generated from a population the of assessing different conformers modeling, (section SAXS 4.4). This of final casetype iseach perhaps theIn most developed. actively being are methods and SAXS, of use powerful provides information resolution atomic of inclusion the however, crucial; is solution particular any of uniqueness 4.2). Several 4.2). reconstructions, that can be used for docking atomic structures (section 4.3). More complicated analysis must be performed in cases where the macromolecules of interest have substantial flexibility and where the SAXS A critical A and non-trivial task is comparing scattering curves to each another (section 4.1). This comparison is most commonly used to compare observed profiles to those calculated from atomic resolution models (section 4. Using crystallography and SAXS to model structures simulations are proving quite useful for modeling of SAXS data, and we expect their use to allow continued advances in interpretation tools (section 4.4.2, and section 5). Carlo algorithms have been extensively used in the study of protein folding (Hansmann & Okamoto, 1999), as well as modeling large rearrangements between protein domains (Maiorov rejection to steps that increase the energy of the system. As with molecular dynamics simulations, a user-defined user-defined a simulations, dynamics molecular with As system. the of energy the increase that steps to rejection energetic force field description is required to evaluate the conformation at each step; however, Monte Carlo simulations do not necessarily have to follow physically relevant trajectories to reach their final states. Monte the basis of an calculated the energy for Metropolis the criterion new is conformation. used, Classically, which probabilistic simulation the in a system the of “temperature” applying current the and change but energy their of system basis the on steps the evaluates of energy the decreases that steps allowing always 1953), al., et (Metropolis Heerman, 1992). The name comes from the application of random numbers to perturb features of the structure on generally accepted are structure the of Perturbations minima. energy local in structures of trapping prevent to 3.5. Monte Carlo simulations subset of important motions dominating the simulation (Amadei et al., 1993). The large conformational changes conformational large The 1993). al., et (Amadei simulation the dominating motions important of subset conjunction with SAXS data (sections 4.2.2, 4.4.2). have been useful for modeling in to those force field terms necessary to satisfy chemical parameters. Deconvolution of the important motions from calculated trajectories (time-dependent changes in conformation) is also a non-trivial problem. Essential dynamics (ED), as described above (section 3.2), has been used to reduce the population of structures into a method of refining structures in X-ray crystallography and NMR (Brunger, Kuriyan & Karplus, 1987; Nilsson et et Nilsson 1987; Karplus, & Kuriyan (Brunger, NMR and crystallography X-ray in structures refining of method addition in constraints experimentally-determined satisfy to system the to “pseudo-energy” a applies 1986) al., bonded distances (MacKerell Jr et al., 1998), as these features do not emerge from the calculation as they do from terms to be added or even changed during the course of the calculation. For example, the simulated annealing force fields used in molecular dynamics are, by necessity, approximations as most systems studied by molecular by studied systems most as approximations necessity, by are, dynamics molecular in used fields force created are fields force Thus, methods. mechanical quantum by treated be can that sizes the exceed far dynamics non- and restraints, planarity angles, torsion angles, bond lengths, bond including parameters, chemical define to most cases, the forces are not amenable to analytical solution, so they are solved numerically by evaluating the displacement of each atom for some tiny increment in time after which the forces are then re-evaluated. The the classical equations of motion and forces due to bonding, electrostatics, and van der interactions. Waals In Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & : 2 χ ) and q ( 1 discrepancy function

is used to apply external n P n α proposed to analyze separately analyze to proposed et al. et . the normalized

formulate the requirements of the absence the of requirements the formulate calc n ) P q . n ( α I exp ) − q . ( I exp ) i q shape restoration programs (section 4.3), the penalty ∑ ( I i 30 ∑ = ab initio R , corresponding to the weighting in the Shannon sampling theorem i is the discrepancy and the penalty term =q 2 i χ ), corresponding to the overall shape of the protein, and medium angle range angle medium and protein, the of shape overall the to corresponding ), -1 ) is the experimental error. This measure clearly weights the lowest resolution lowest the weights clearly measure This error. experimental the is ) i (q σ , where Svergun, 2006), the penalty terms penalty the 2006), Svergun, & n P n = 0 - 0.15 Å 0.15 - 0 = ∑α q +

2 χ ), which corresponds to information corresponding to internal structure. -1 Moore, 1949), being the most sensitive. Furthermore, Sokolova Sokolova Furthermore, sensitive. most the being 1949), Moore, & The suite of SAXS modeling programs developed by Svergun and coworkers use target fitness functions fitness target use coworkers and Svergun by developed programs modeling SAXS of suite The The programs developed by Svergun and coworkers minimize and coworkers The programs developed by Svergun A SAXS of version the R-factor has in been employed solution structures, which combining crystal and A Fitting theoretical models to SAXS scattering curves requires that a measure be established for determining determining for established be measure a that requires curves scattering SAXS to models theoretical Fitting ): q = 0.4 - 0.9 Å ( q 2 Svergun, 2005; Petoukhov 2005; Svergun, of the overlaps between the subdomains. For of the form E = constraints on the solutions. For rigid body modeling programs (section 4.2), such as SASREF (Petoukhov ( with the weighting function Weight (Shannon ( range angle low the where the scaling multiplier ScaFac yielding the best least-squares fit is where the scaling multiplier ScaFac yielding the best least-squares I 1995). A different measurement has been used in the database of theoretical protein scattering, DARA ((Sokolova data most DARA strongly and scattering, is protein used in theoretical comparing of experimental database and theoretical the curves in in (Svergun et CRYSOL used al., been has measurement different A 1995). et al., 2003b), section 2.4.3), implemented a “weighted R factor” to compare the scattering profiles I where c is a scaling factor and factor scaling a is c where

was defined by analogy with X-ray crystallography (Smith, Harrison & Perkins, 1990): (Smith, Harrison & Perkins, analogy with X-ray crystallography was defined by been employed for SAXS. Although all of these measures are minimized on exact fits between two curves, these curves, two between fits exact on minimized are measures these of all Although SAXS. for employed been weightings can strongly impact Different portions of the scattering curve differently. measures weight different protocols. the results of modeling factor (section 2.2.7) is well established. Lower R-factors correspond to the better fits between the calculated and experimental diffraction data. In contrast to a X-ray multitude crystallography, of different measures have the agreement between two scattering curves. In X-ray crystallography, this goodness of fit measure, the R- Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

q ab initio ab ab initio & Svergun, programs have made it difficult to , the F-factor gives a larger weight to the 2 χ ab initio 31 . tested several different fitness functions during the development of an of development the during functions fitness different several tested . models to SAXS scattering curves. models to SAXS scattering curves. et al et ab initio ) being prone to amounts large of The both. existence of a standard R-factor measure in X-ray -1 can be used to enforce model The connectivity. program BUNCH, which combines both n is the number of points of the profile. In comparison to p P n > 0.4 Å In principle, the P(r) function and hence the scattering from the solute can be calculated by evaluating Theoretical SAXS scattering curves can be directly calculated from atomic models. The observed If the crystal structure of a macromolecular sample is known, a theoretical solution scattering pattern can be can pattern scattering solution theoretical a known, is sample macromolecular a of structure crystal the If We We are not currently convinced that a “best” measure has been adequately developed; however, the α q the scattering from it as if it were filled by an electron density equivalent to bulk solvent (Fraser, MacRae et al., 1995). & Suzuki, 1978; Lattman, 1989; Svergun scattering profile is largely the difference between the scattering of the target molecule with its ordered solvation ordered its with molecule target the of scattering the between difference the largely is profile scattering existence the to due solvent bulk of scattering missing the account into takes that volume excluded the and layer calculating and molecule the of shape the defining by determined be can term volume excluded The solute. the of 4.2.1. Calculation of scattering curves from atomic model 4.2.1. Calculation of scattering curves from evaluate how solution structures differ from crystal structures (section 4.2.4), and to establish which contacts in contacts which establish to and 4.2.4), (section structures crystal from differ structures solution how evaluate (section 4.2.5). a crystal structure are the biologically relevant solution assemblies models. These structures can be used to perform rigid body modeling in situations where individual components, individual where situations in modeling body rigid perform to used be can structures These models. but not the assembly are known (section 4.2.2). Computed scattering curves are can also extremely be used valuable to screen as through they potential solutions generated by computational docking (section 4.2.3), to significant deviations from the experimental scattering. Though this is one of evaluated, the most be straightforward uses also of must data SAXS fit that structures resolution atomic of arrangements of uniqueness the SAXS, SAXS possible constrain greatly structures Atomic linkers. flexible have that molecules of case the in especially calculated from the atomic coordinates (section 4.2.1). This provides the opportunity to evaluate computationally computationally evaluate to opportunity the provides This 4.2.1). (section coordinates atomic the from calculated generated models under situations where the curve computed from the crystallographic coordinates displays 4.2. Direct comparison of crystallographic structures with SAXS data structures comparison of crystallographic 4.2. Direct function used for refinement. Thus adoption of a standard need not constrain the development of new protocols new of development the constrain not need standard a of adoption Thus refinement. for used function for fitting atomic or that establishing a standard measure for SAXS will be important for the community as a whole. We note that as that note We whole. a as community the for important be will SAXS for measure standard a establishing that fitness particular the of independent be can value fit” of “goodness standard a reporting crystallography X-ray in crystallography has been tremendously important in evaluating different crystallographic methods as well as providing insight for crystallographers as to “acceptable” values for final refined structures. Hence, we believe the possibility of avoiding problematic fitting of the entire scattering profile. This parameter can deal with the problem that different parts of the scattering curve have different systematic and statistical errors with high data ( proposed partitioning of the scattering profile in the “weighted R-factor” scheme (Sokolova et al., 2003b) has compare the scores of the final solution. In one study, a separate reciprocal space R-factor was used to Nishikawa & Fujisawa, 2003). make (Takahashi, direct comparisons where N higher resolution data. The differences in the fitness functions in reconstruction reconstruction program DALAI_GA (Chacon et al., 2000; Chacon et al., 1998). Of them, the comparison results: gave the most promising by the F-factor a linear scale scale rather than on a logarithmic intensities of terms terms and rigid body modeling, has penalty terms that combine both Chacon contrast, In 2005). types of constraints (Petoukhov Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 , 3 , average 3 Smith, 2002a) Smith, & is typically in a range range a in typically is max ) is more problematic for spherical –1 > 0.4 Å q , where the default values of L of values default the where , 2 ) or adjusted parameters (blue line; solvation shell contrast 0.005e/ 3 ). Theoretical ). scattering calculated with program solX using all-atom 3 Å +1) 5 32 Å 5 ), or solX using all atom-methods (green line). ), or solX using all atom-methods (green max 3 Å 4 & Doniach, 1988) or by surrounding the particle by a continuous ), CRYSOL with adjusted parameters (blue line; solvation shell contrast 0.01e/Å contrast shell solvation line; (blue parameters adjusted with CRYSOL ), 3 Å 4 Fitting experimental scattering at higher resolutions ( Although solute atoms dominate the scattering signal at small angles, the scattering from ordered A. Comparison of experimental scattering curves of glucose isomerase (black) with the scattering curve calculated from the , average atomic radius 1.61 Å, excluded volume 2.13x10 , average atomic radius 1.4 Å, exclude volume 2.29x10 3 3 harmonic reconstructions, as they do not account for the internal structure of the scattering particles. Even scattering from the solute and the excluded volume. envelope representing the solvation shell of 3 Å with a density that can differ both from bulk density and the solute (Svergun et al., 1995). Including the hydration shell improves the accuracy of the calculated scattering the than lower magnitude of orders several is layer solvent ordered the from contribution the however, profiles; channels stabilizing their structures and smoothing the excluded volume (Kuhn et al., 1992). Computationally, channels stabilizing their structures and smoothing the excluded volume (Kuhn et al., 1992). Computationally, 1994; al., et (Fujisawa surface the on molecules water placing explicitly by modeled been has shell hydration the Grossmann et al., 1993; Hubbard, Hodgson due to both geometric effects and changes in the water structure such as shorter oxygen-oxygen distances and increased water coordination numbers. Strongly bound water molecules are known to fill surface grooves and corresponds to 0.3 grams of water per gram of protein and is 15% denser than bulk water (Merzel water layer solvation ordered this For proteins, 2001)). (Perkins, in (reviewed considered be also must atoms solvent bulk than denser 15% is and protein of gram per water of grams 0.3 to corresponds methods is shown in green line (Tiede & Zuo, 2007). B. Comparison of the experimental scattering curves of truncated Cel5A (black) Cel5A truncated of curves scattering experimental the of Comparison B. 2007). Zuo, & (Tiede line green in shown is methods with the scattering curves calculated using CRYSOL with default parameters (red line; solvation shell 5.35x10 volume excluded Å, 1.61 radius contrast atomic 0.03 e/Å volume 5.48x10 average atomic radius 1.8 Å, excluded atomic model using the program CRYSOL (Svergun et al., 1995) with the default parameters (red line; solvation shell contrast 0.03e/ (Svergun atomic model using the program CRYSOL Å Å Fig. 8. The spherical harmonics procedures scale as N(L as scale procedures harmonics spherical The programs. different from 15 to 17 with (Zhao et al., 1998) and is 1995). al., implemented et (Svergun in CRYSOL ORNL_SAS in implemented as model (Tjioe entire the cover & that envelopes expansion) (multipole Heller, 2007) or from spherical harmonics all of the interatomic distances in the structure: a procedure that scales by the square of the number of thousands atoms. calculated be must scattering calculated which for purposes fitting for inappropriate is algorithm This of times. Faster alternatives include the calculation of the P(r) function using Monte Carlo integration routines Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

q & Trewhella, 2000)). In rigid- When atomic resolution structures Evaluation of each trial rotation and rotation trial each of Evaluation 4.2.2. Rigid-body modeling with SAXS 4.2.2. Rigid-body modeling with data of individual domains are known, SAXS can be used domains these of placement and orientation to determine the agreement the maximizing by complex a in relative experimentally and theoretical the between observed scattering (reviewed in Gallagher (Wall, considered are domains the modeling, body static objects and orientations only are their building an changed. assembly with M subunits has relative In 6(M-1) fittable parameters, general, as one subunit can be fixed and M-1 subunits are mobile with three translations and three rotations each. The number of parameters can be with homo-oligomers. Depending on the problem, rigid-body modeling with atomic the of dimensionality the reduce can models fitting problem and be a more economical and powerful way to use the information content in the SAXS curve. reduced if symmetry is present, such as ab ) is influenced by atomic displacement factors (section –1 33 > 0.25 Å & Svergun, 2005)) and q ) well, but fits involving multiple -1 values is more problematic, as exemplified by a truncated form of Cel5A (Fig. Cel5A of form truncated a by exemplified as problematic, more is values q < 0.03 Å q region ( q In many of the modeling schemes described in the following sections, the methods for manipulating Experimental SAXS curves (black circles) and scattering profiles model using the program GASBOR (green line (Svergun et al., 2001a)). model using the program GASBOR (green line (Svergun et al., 2001a)). modeled as the globular region at the position correlated with its position in the the in position its with correlated position the at region globular the as modeled beads gold as shown is model GASBOR One arrow). the with (marked sequence blue beads representation and varied from run to run. did Single not fit the best-fit smallest models been has domain small the of structure unknown the of position The did. models The secondary structural elements of the known atomic structures of the single modules are shown by a multicolor ball red). Restored representation linker conformations (yellow, between gold, the gray, modules are displayed by a computed from the models of a multidomain protein reconstructed by the rigid body modeling using BUNCH (red line (Petoukhov initio Fig. 9. integration so that other programs can be readily integrated into SAXS modeling even if these programs were in mind. never written with X-ray scattering et al., 1998) were designed to be highly modifiable, which we believe isimportant crystallographic partrefinement of package. Thus, its we success are excited as about the becoming stated intent an of the authorsof for scriptable easily is engine calculation curve scattering the that ensure to 2007) Heller, & (Tjioe ORNL_SAS the structures of interest are directly tied to the pratical introduces have calculation abilities scripting limited and coupling tight this sense, some In models. the for curves engines for generating the theoretical scattering X-PLOR and its successor CNS (Brunger limitations for altering modeling schemes. In X-ray crystallography, 2.2.8) in the atomic model (ADFs; (Zhang, Thiyagarajan & scattering resolution Tiede, higher 2000a)). of calculation proper The hence, solution; crystallographically in molecules refined for appropriate be not may ADFs unsolved problem. curves remains an proteins with flexibility at high at flexibility with proteins 8b), for which multiple conformations must be explicitly included calculation (section of higher resolution scattering 4.4). ( Moreover, the theoretical & Smith, 2002b). For example the program solX (Tiede & Zuo, 2007) shows & Smith, good 2002b). agreement throughout For the example the program solX (Tiede Fitting range with measured scattering profiles using default settings and values even without a solvation layer. globular globular proteins, such as glucose isomerase, typically profiles scattering of Calculation 8a). (Fig. curve experimental the of require portion resolution high the fit to extreme CRYSOL values for adjustable (Merzel computation intensive parameters more require but in accurate more are atoms all include explicitly that algorithms by Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 , where where , 6(M-1) & Svergun, n Trewhella, 2002). Trewhella, & Reference Reference Svergun, 2005) (Petoukhov & Svergun, (Konarev et al., 2001; Kozin et al., 1997) 2005) (Petoukhov & Svergun, 2005) (Petoukhov & Svergun, 2005; (Petoukhov & Svergun, 2006) Petoukhov & Svergun, (Petoukhov et al., 2002) (Petoukhov et al., 2002) (Petoukhov et al., 2002) (Petoukhov et al., 2002) & 2005; Svergun, Svergun, 2007). 34 Method Method & Svergun, 2001); however, in many cases, being able to generate Table 2. Rigid-body modeling SAXS programs 2. Rigid-body Table Table 3. Reconstruction of missing components 3. Reconstruction Table Individual residues are modeled with spring forces between neighbors Dummy atoms with spring forces between neighbors Restore structures as random-walks with secondary-structural features added when known Dummy atom construction of missing loops Brute force rigid body modeling asymmetric oligomers with reconstruction of missing linkers and domains. Fast algorithm for modeling globular homo- or heterodimeric Fast algorithm for modeling globular homo- structures. oligomers Brute-force rigid body modeling of symmetric can simultaneously fit Rigid body modeling of a complex that scattering from subcomplexes Manual domain docking CREDO BUNCH CHADD Program Program SASREF With With symmetry constraints, rigid body modeling has been successful at constructing the dimeric Several programs are freely available for rigid-body docking using these spherical harmonic envelopes GLOOPY CHARGE DIMFOM GLOBSYMM MASSA, ASSA MASSA, is the number of steps searched for Different simplifications, can each however, beparameter. performed for 2006) or glyceraldehyde-3-phosphate dehydrogenase (Ferreira da Silva et al., 2006) and Even with the small number of parameters of rigid- glutamate (Petoukhov et al., 2003) with p222 point-group symmetry. synthase complexes of hepatocyte growth factor and tyrosine kinase with p2 point-group symmetry (Gherardi et al., ray and neutron scattering curves that included contrast variation data from selectively deuterated complexes (Gherardi et al., 2006). heuristic search algorithm has been implemented in the program An SASREF. advantage of SASREF ability is to use the additional constraints to incorporate other information about the system, such as known subunit interfaces (Grishaev et al., 2005; Mattinen et al., 2002). Recently SASREF was used to simultaneously fit X- For two component complexes, DIMFOM exhaustively searches rotation space by rolling one monomer over the other. For larger symmetric complexes that reduce the dimensionality of the search space, GLOBSYMM is appropriate. For complexes containing several subunits that may or may not be symmetrically related, a 1997) or MASSHA (Konarev, Petoukhov automated rigid-body modeling is preferable (Koch et al., 2003; Petoukhov of atomic domains (Table 2). Interactive docking can be performed using ASSA (Kozin, Volkov used as a basis for construction of an atomic model by energy minimization (Tung, Walsh used as a basis for construction of Walsh an atomic model by minimization energy (Tung, 1995) al., et (Svergun applied been also have models atomic from calculated representations harmonic Spherical & Stuhrmann, 1991). and Stuhrmann (Svergun of Svergun using the algorithm profiles are easily computed. Forexample, individual components have beenmodeled usingtriaxial ellipsoids (Gallagher et al., 1999), such as in the case of a heterodimeric cAMP-dependent protein kinase which was translation in the rigid body search can be computationally expensive. A brute-force search scales as scales search brute-force A expensive. computationally be can search body rigid the in translation n scattering which from forms in bodies rigid of representation the is simplification important An screening. rapid Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 �� & Ritchie,  (Dominguez et (Dominguez �� & Perkins, 2001; crystal structures & Eisenstein, 2003;  (Mustard �� Histone SOS and cat DH-PH- Additionally, docking Additionally, has also employed �� Weng, 2003; Heifetz & Weng,  modeling (Table 3). The unknown portions of portions unknown The 3). (Table modeling closest to the experimental values (Hammel et al., G (Chen, Li �� ab initio ab 35 Perkins, 2004) and the cellulosome (Hammel et al., 2005) 2005) al., et (Hammel cellulosome the and 2004) Perkins, & methods (section 4.3.1) where each amino acid is represented is acid amino each where 4.3.1) (section methods Bates, 2005) from NMR structure determination structure NMR from 2005) Bates, & ab initio ab Weng, & 2003; Weng, Palma et al., 2000; Schnecke et al., 1998). �� Atomic structures have long held the promise of allowing the structures of unknown complexes to be SAXS can provide an experimental target function to score hits from the docking search and can Rigid-body modeling can also be combined with combined be also can modeling Rigid-body was performed separately from comparisons to SAXS results. Of the top dimeric 40 docking solutions, 100 the top-scoring generate to used also was docking Computation scattering. SAXS the to fit best the also was solution of the Ras activator son of sevenless folds onto a helical linker between two other son of sevenless (SOS) (Sondermann domains et al., 2005). In this work, the docking of the SOS substantially aid in the problem of development of scoring good and functions bad that solutions. successfully For example, distinguish this between combination has been used to show that the histone-like domain the protocol involved, an energy minimization is frequently run for the top protocol solutions involved, to an resolve energy steric clashes at the interface (Li et al., 2003). 2005), or residual dipolar coupling 2004). (Tai, A number of web servers generate conformational ensembles specifically for docking (Barrett & Noble, 2005; Lei et al., 2004; Suhre & Sanejouand, 2004). Regardless of Li, Chen Sternberg (Smith, conformation of ensembles al., 2004), molecular dynamics simulations (Rajamani et al., 2004), normal mode analysis distinguishes correct solutions. A number of different strategies have been developed to introduce flexibility. Protocols allowing some degree of interpenetration of the molecules (“soft docking”), and truncating side-chain surface side chains have been flexibility, developed to be simple. Computational docking has two major challenges: (1) dealing with flexibility in macromolecules in a computationally practical manner and (2) determining an appropriate scoring function that successfully proven not determined has computationally. Substantial problem effort this has however, been 2001); directed al., towards et using Mandell only 2006; chemical (Bonvin, and sites geometric docking determine to parameters 4.2.3. Computational docking combined with SAXS data 4.2.3. Computational docking combined 2005). This modeling, however, is likely to be inappropriate for macromolecules with substantial flexibility that flexibility substantial with macromolecules for inappropriate be to likely is however, modeling, This 2005). ensembles lacking a “best-fit” structure (section 4.4). must be treated as conformational were produced at regular intervals along discrimination enabled data experimental the the with configurations MD-generated trajectory the of comparison The of profiles. subsequent calculations of the theoretical SAXS of a finite number of structures with the best fit and with R Gilbert et al., 2006; Gilbert et al., 2005; Sun, Reid Sun, 2005; al., et Gilbert 2006; al., et Gilbert the simulation is performed on the linker regions Frequently, at very high temperature (~1500K) to prevent the molecule from becoming trapped in a local minimum (Leach, 2001). Different conformations of the protein and an “automated constrained fit” procedure generates thousands of possible models by applying molecular dynamics (MD) on the linker region in exhaustive search of the applied best-fitto a conformationnumber (Boehmof ethuman complement al., regulating 1999) proteins (Aslam et al., 2003; Aslam determines a best-fit conformation through a simulated annealing protocol (for example, Fig. 9). Flexible regions Flexible 9). Fig. example, (for protocol annealing simulated a through conformation best-fit a determines have also been modeled using a Monte Carlo dihedral angle sampling of hinge regions (Akiyama et al., 2004) the macromolecule may be modeled as in as modeled be may macromolecule the by “dummy residues” (also called “beads” or “big atoms”). The program BUNCH models full-length proteins as rigid-body domains linked by their termini to flexible chains or domains (Petoukhov & Svergun, 2005) and generated generated multiple solutions with similar fits et to al., the 2005). These experimental models data (Vestergaard cryo-EM studies. constructs and by deletion analysis of additional by SAXS were evaluated body body modeling, the small number of independent data values can lead to practical overfitting standpoint, many models should be (section generated independently and examined 2.3.4).for uniqueness and tested Froma (RF1) 1 factor release bacterial of modeling body rigid automated example, For techniques. biophysical other by Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 Tainer, 2003). Tainer, (McMurray &

36 Crystallization typically requires the formation of symmetric contacts that exist solely in the crystal The differences between solution and crystal structures can be important for deciphering biological A A naïve assumption is that crystallization will mostly force macromolecules to adopt a more compact The ability to directly compare atomic models to SAXS scattering curves is a remarkably useful tool for tool useful remarkably a is curves scattering SAXS to models atomic compare directly to ability The & Kolodner, 2005). Crystal structures analogs of have or ATP MutS-DNA complexes yet with ATP to Kolodner, 2002) and studies in related systems for which both states are known (Hopfner et al., 2000). al., et (Hopfner known are states both which for systems related in studies and 2002) Kolodner, & and may not be important in biology. However biologically relevant multimers can also be symmetric, and 4.2.5. Determining biological assemblies in crystal structures 4.2.5. Determining biological assemblies in crystal structures efficient DNA repair machinery is important as toxins that interfere with the repair machinery are expected to be to expected are machinery repair the with interfere that toxins as important is machinery repair DNA efficient DNA more dangerous to public health than toxins that directly damage Regardless of whether or not these conformational changes are incompatible with the crystal lattice or due to the to due or lattice crystal the with incompatible are changes conformational these not or whether of Regardless been yet not have changes conformational these decipher to attempts DNA, trapped the of effects concentration successful through crystallography alone. Properly assessing these conformational changes in the case of the bound nucleotide have been anticipated by the effects of dominant mutations in the eukaryotic homologs (Hess, homologs eukaryotic the in mutations dominant of effects the by anticipated been have nucleotide bound Gupta Mazur reveal the conformational changes anticipated from biochemical studies (Lamers et al., These 2004). predicted domain motions that would allow the two halves of the pockets composite ATP-binding to engage with their results. MutS recognizes mispairs in DNA when in a nucleotide-free or ADP-bound state. by Binding of ATP rigid a like acts protein the which in state a induces complex DNA-MutS the of versions eukaryotic or bacterial ring that can freely diffuse along the length of the DNA and is no longer restricted to the mispair (Mendillo, mechanisms. For example, the single conformation of the dimeric bacterial DNA mispair recognition protein MutS likely constrained by lattice forces, despite the fact that multiple states are expected from biochemical interactions was clearly more compact in solution (Svergun et al., 2000). in solution (Svergun interactions was clearly more compact (reviewed in (Koch et al., 2003)). Despite these sorts of results, compaction by the lattice is not the only possible only the not is lattice the by compaction results, of sorts these Despite 2003)). al., et (Koch in (reviewed identical be to found was one enzymes, diphosphate-dependent thiamine four of study a in example, For effect. subunit weaker with complex one and solution, in compact less be to appeared two solution, in and crystal the in with the ligand-bound “relaxed” state (but not the unliganded “tense” state) could be modeled by small rigid-body subunit rotations on the order of 10 degrees however, of These differences, aspartate transcarbamoylase. structure. SAXS has revealed that crystallographically induced compaction is likely to be frequent, such as 2003)) that has two molecules in the asymmetric unit with substantially domain different arrangements tend to be the exceptions, so we favor continued research to optimize the combination of computational searches and conformations. SAXS data to characterize solution crystal lattices and conformational states is the cracking of reduced deoxyhemoglobin crystals upon the addition the upon crystals deoxyhemoglobin reduced of cracking the is states conformational and lattices crystal of oxygen (Perutz et al., 1964). By their very nature, crystal structures al., tend et not (Nagar 1opl; to id reveal (PDB kinase the protein flexibilityc-Abl ofthe for the as such structures, Crystal macromolecules. crystallized Moreover, Moreover, proteins that undergo allosteric rearrangements and have multiple stable conformations separated by small energy differences can be affected by lattice forces. The classic example for incompatibility between lowest energy state within the lattice and under crystallization conditions; however, it not necessarily the lowest the necessarily not it however, conditions; crystallization under and lattice the within state energy lowest folding the alter not do lattice crystal the of effects the that suggested have studies Many solution. in state energy of domains, but rather influence the conformations adopted by flexible termini or linkers between domains. deciphering the influences that the crystal lattice has on the observed structure. The crystal structure is likely the likely is structure crystal The structure. observed the on has lattice crystal the that influences the deciphering 4.2.4. Differences in crystallographic and solution structures and solution in crystallographic 4.2.4. Differences structures of purine nucleoside phosphorylase and to verify that the crystallographic trimer was the assembly state state assembly the was trimer crystallographic the that verify to and phosphorylase nucleoside purine of structures supramolecular the of docking computational that anticipate We 2003). al., et Azevedo de (Filgueira solution in powerful combination. a particularly will become of models by SAXS and validation complexes Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 , is , 2 Å ) ) Å 2 2 Å Å = 90 120 = 100 max max Dimer 2 100 Dimer 4 Dimer 3 = 28.3 Å D = 30.4 Å D G 80 Dimer 1 G Dimer 4, (B/2 = 255 Dimer 3, (B/2 = 406 R R 60 r (Å) 37 . Dimer 1, with a buried surface area of the monomer of monomer the of area surface buried a with 1, Dimer . 40 Å max Å ) ) 2 2 Å Å and D and G = 80 = 120 20 max max

1 5 4 3 2 P(r), arbitrary units arbitrary P(r), 0 6 = 26.08 Å D = 34.04 Å D G Dimer 1, (B/2 = 755 Dimer 2, (B/2 = 923 G R R SAXS could have readily distinguished between alternative dimer structures of the C-terminus the of structures dimer alternative between distinguished readily have could SAXS , is the asymmetric unit of the crystal, whereas dimer 2, with a buried surface area of 923 Å 923 of area surface buried a with 2, dimer whereas crystal, the of unit asymmetric the is , 2 scattering and parameters such as R as such parameters and scattering Å 755 dimer for calculated functions P(r) Theoretical B. 2005). al., et (Kosinski assembly dimer solution the is that P(r) globular characteristic a has 1 Dimer distinguished. readily are (red) 2 dimer and (black) 1 peak and a long tail. bell-shaped, whereas dimer 2 has a characteristic extended P(r) with an early Fig. 10. Fig. of MutL observed in the crystal structure (PDB id 1x9z; (Guarne SAXS et in al., 2004)). differences A. measurable Each to of rise the giving four shapes, overall different remarkably has dimers different B A In many cases identifying the biological multimer can be a difficult problem. Systematic investigation of atomic resolution structures has shown that authentic interfaces tend to be large and involve hydrophobic

contacts. biologically relevant symmetry operators can also be part of the underlying symmetry of the crystal structure so structure crystal the of symmetry underlying the of part be also can operators symmetry relevant biologically crystallographers Thus, assembly. biological the for guide a as used be cannot unit asymmetric the of contents the must decipher which macromolecular contacts mediate multimerization in solution and which are only crystal Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & & 28.1 G & Thornton, 120 Å vs. R vs. Å 120 max Identifying biological 22 with four different dimer 3 has the potential help to information experimental to provide test and identify the biologically relevant interfaces in the crystal. dimers is particularly difficult, as crystallographic symmetries tend to generate many dimeric interactions. For example, protein repair mismatch bacterial the functional form to known is MutL is dimers. comprised MutL of an N-terminal ATPase domain, a flexible internal linker, and a C- interactions interactions on the surfaces of the protein (Jones packed better be to tend and 1996) than crystal contacts (Bahadur et al., 2004; Getzoff, Tainer Olson, 1986; Keskin, 2005; Nussinov, Li et al., 1985). Ma & Additionally, these are less interfaces hydrated (Rodier et al., 2005) and tend to have greater sequence conservation (Valdar & Thornton, 2001a; Valdar Thornton, 2001b) with frequent and phenylalanines, tryptophans, methionines (Ma et in useful al., although analyses, 2003). These information, other of absence the are theoretical in nature. SAXS biologically the on collected data relevant assemblies in solution et al., 2004)). Two interfaces have interfaces Two 2004)). al., et remarkably small buried surface areas, but two others were more sizable. The dimer interface terminal domain that mediates constitutive (Drotschmann et The dimeric al., C-terminal domain dimerization 1998). crystallized in the space group P4 (Guarne 1x9z; id (PDB interfaces 34.1 Å and D and Å 34.1 G Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å 10 95 95 1 125 80 70 90 120 30.0 70 29.5 32.0 35.5 25.3 24.8 Å 31.0 G max G max G max max G G max 24.6 G max G max R D R D R D G max D R 0 R D R D R D 8 R D Crystal dimer Crystal dimer Crystal dimer Crystal dimer Model dimer Model dimer Model dimer Model dimer 40 r (Angstroms)

0 P(r), arbitrary units arbitrary P(r), MutS 38 Rad50-Modeled Dimer later proved to be incorrect, whereas the other large dimer interface was consistent et al. Rad50 Crystallographic Dimer SAXS has the potential to indicate mismatches between the dimers in the ABC HisP MalK B A The biologically relevant dimer is more extended than the dimers generated by crystal contacts, and theoretical (R size by it distinguished easily have could SAXS that suggest calculations proposed by Guarne with crosslinking experiments and identifiable by computations analysis of packing (Kosinski et al., 2005). MJ1267 MJ0796 of the signature motifs. Theoretical P(r) scattering curves clearly distinguish between the crystallographically observed dimers and those modeled using the Rad50 assembly. the ATP-binding the ATP-binding surface. B. Different crystallographic dimers were MalK, observed MJ0796, for and HisP, MJ1267 (PDB ids 1b0u, 1g29, 1f3o, 1g6h; (Diederichs et Hung et al., al., 1998; 2000; Karpowich et et al., al., 2001; Yuan 2001)) that lack special positioning (Obmolova et al., 2000)) that were normally stable as displayed as dimers yellow and in red ribbons; solution. bound nucleotides are orange Subunits ball-and-stick are models, and blue ribbons indicate the position of the signature motif that forms the second half of ATPase assemblies observed in crystal and with their solution conformation, if dimeric states dimeric if conformation, solution their with and crystal in observed assemblies ATPase could have been stabilized for solution A. scattering. The biological dimeric assemblies of the were ABC ATPases first observedwith the DNA repair proteins Rad50 (PDB id 1f2u, (Hopfner et al., 2000)) and MutS (PDB id 1e3m, (Lamers et al., 2000); PDB id 1ewq, Fig. 11. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 Kolodner, Kolodner, & histidine permease histidine maltose transporter MalK ABC ATPases MJ0796 ABC ATPases (PDB id BtuCD heterotetramer (PDB id 1l7v, 1l7v, id (PDB heterotetramer BtuCD Salmonella typhimurium Salmonella E. coli E. mismatch repair protein MutS protein forms a Thermococcus litoralis 80 Å) (Fig. 10). (Fig. 80 Å) max E. coli 39 Methanococcus jannaschii 2 with two symmetry-related dimers in the crystal packing (PDB packing crystal the in dimers symmetry-related two with 2 1 2 3 28.1 Å and D Å and 28.1 G George, 1999). Dimerization generates an “ATP & sandwich” George, 1999). with Dimerization the generates active an site “ATP being 80 Å) as well as by the P(r) function that is characteristic of an elongated shape, having a peak at A more insidious challenge is when the authentic multimer does not exist in the crystal at all. Generating Generating all. at crystal the in exist not does multimer authentic the when is challenge insidious more A max Finally, SAXS Finally, can establish the validity of weak assemblies observed in crystal packing that might be SAXS can be used to distinguish between alternative assemblies in different crystal structures. The One particularly difficult problem wasATP-binding identifying cassette the (ABC) biological dimer of 2007)). A 2007)). A potential tetramer contact was observed in the crystal form; however, the interface weakly was packed. Despite small this, and SAXS data collected on the MBP fusion protein and subsequent mutagenesis of maltose binding protein (MBP) mediated its tetramerization, and the crystal structure revealed a helix-loop-helix a revealed structure crystal the and tetramerization, its mediated (MBP) protein binding maltose Putnam (Mendillo, 2ok2; id (PDB dimer tight a form to interaction symmetric a made that domain ignored as artifacts of the crystallization process. The upon depend that tetramers forms MutS concentrations, protein high At DNA. in mispairs recognizes that dimer the presence of the C-terminal 53 amino acids (Bjornson et al., 2003). Fusions of this C-terminal domain onto against HslV and that the extended HslU domain extends into solvent (Sousa et al., 2000). and that the extended HslU domain extends into solvent (Sousa against HslV (Bochtler et al., The 2000)). P(r) functions for these assemblies different and hence their X-ray scattering were pack HslU of domains C-terminal and N- globular the solution in that revealed SAXS and distinguishable easily (Fig. 12; (Bochtler et al., 2000; Sousa et al., 2000)). In the center of both assemblies was an HslV hexamer, but hexamer, was an HslV of both assemblies center 2000)). In the al., 2000; Sousa et al., et (Fig. 12; (Bochtler in the two different crystal forms the globular N- and C-terminal domains of 1doo, HslU id were (PDB either domain packed internal against extended an by HslV from away held or 2000)) al., et (Sousa 1g3i, id (PDB HslV bacterial HslUV chaperone/protease complex had been observed in two distinct crystallographic assemblies dimer assembly was later established by the crystal structure of the of structure crystal the by established later was assembly dimer Lee & Rees, 2002)). (Locher, from the DNA repair proteins Rad50 (PDB id 1f2u, (Hopfner et al., 2000)) and MutS (PDB id 1e3m, (Lamers et (Lamers 1e3m, id (PDB MutS and 2000)) al., et (Hopfner 1f2u, id (PDB Rad50 proteins repair DNA the from used subsequently was particular in structure Rad50 the and 2000)), al., et (Obmolova 1ewq, id PDB 2000); al., for modeling of the ABC transporters. The inference that the membrane transporters shared the Rad50/MutS HisP monomer HisP (Jones motif “signature” ATPase ABC an and molecule one on site hydrolysis and binding ATP A Walker a of up made from the other subunit. These interfaces were first observed crystallographicallyATPase in domains the ABC of these ABC had ATPases dimers generated by crystallographic symmetry, yet none shared common dimer interfaces. The biologically relevant dimer was first proposed using sequence analysis and the structure of the transporter ABC ATPases were solved later, including the (PDB id 1g29; (Diederichs et al., 2000)), and the 1f3o; et (Yuan al., 2001)) and MJ1267 (PDB id 1g6h and 1gaj, (Karpowich et al., 2001)). Remarkably, each transporters were known to dimerize. The first of these structures, the structures, these of first The dimerize. to known were transporters P4 group space the in crystallized was HisP Other one. relevant biologically the be to proposed was interface dimer largest the 1998)); al., et (Hung 1b0u; id hydrolysis with the transport of ligands across membranes ATP ABC transporters couple transporters (Fig. 11). and include the cystic fibrosis transmembrane The regulator. nucleotide binding domainsATPase of the ABC biologically relevant screw assembly (Putnam et al., 2001). screw assembly (Putnam et biologically relevant on multimers in solution and theoretical scattering curves RuvB motor calculated from ATPase assembles observed junction Holliday in the for the assembly hexameric crystals the example, For clue. important an be could could be characterized by SAXS results in combination with the crystal structure of the subunit from a non- and modeling assemblies with SAXS data will be discussed But below. before modeling can be performed, it is critical to identify that the biologically relevant assembly is not present in the crystal and to recognize that modeling must be done in the first place. In these cases, the lackof agreement between SAXS data collected a solution assembly model given only the high resolution structure of a subunit is possible with SAXS data, Å and D (R tail long extended and a distances short Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

2 /Rad50 ab initio 2 shapes from ab initio SAXS structures Substantial theoretical ab initio envelope reconstruction protocols establishing that reconstructions three-dimensional can from the be scattering one-dimensional derived curve SAXS Successful reconstructions (section generate low-resolution 4.3.1). envelopes reconstructions averaged to analogous that are generated by electron (EM). Similarly, microscopy many of the tools and practical work has gone into salt bridges that stabilized it revealed that the tetramer in solution resembled closely the crystallographically. assembly observed a ultracentrifugation, Similarly, combination microscopy and was used to validate the biologically of electron relevant Mre11 SAXS, heterotetrameric heterotetrameric DNA head used processing in repair, whose double-strand structure was break inferred from the atomic resolution structures of Mre11, Rad50, and et al., 2001). results (Hopfner biochemical 4.3. Modeling atomic assemblies using ab initio 40 of 220 Å (PDB id 1g3i; (Sousa et al., 2000)). SAXS envelopes solely refers to lack of a pre-defined input structure. These programs implicitly max of 265 Å (PDB id 1doo; (Bochtler et al., 2000)). In the max ab initio ab initio of 65 Å and a D G The two different HslUV protease/chaperone assemblies observed models are best treated as hypotheses to be tested by additional experimentation (Fig. 13). models are best treated as hypotheses to be tested by additional experimentation of 90 Å and a D Several programs have been created for the purpose of calculating so called G independent experimental values extracted from the SAXS curves are used for generating an overall envelope, which frequently has a uniqueness problem. Depending on the system, we suggest that rigid-body docking of a small number of subunits is a more powerful use of the information content in SAXS curves than the three-dimensional reconstruction involves a state of the that was used macromolecule for determining the that atomic resolution structure. The is different from the state substantially from differ those using atomic models described above (section 4.2). In these protocols, all of the structures in a rigid body fashion (section 4.3.2) or using methods that allow flexibility to be introduced into the atomic models (section 4.3.3). Introducing flexibility in the atomic model is valuable in situations where used in EM for fitting known atomic assemblies into these envelopes can also be used for docking atomic The experimental P(r) function (not shown; (Sousa et al., 2000)) strongly argues for a monomodal distribution and the more compact 1g3i-like assembly (after (Sousa et al., 2000)). globular domains of HslU are separated from giving HslV, a bimodal P(r) function with a R second, the globular domains of HslU pack against giving HslV, a monomodal P(r) function with a R Fig. 12. crystallographically have different orientations of the HslU hexamers interact (green) on both that sides of the double-ringed HslV dodecamer (gold). In the first, the attempt to provide detailed descriptions of these programs here and the citations listed referenced. The in general approach Table taken 4 by these should programs is be to propose shapes, calculate scattering curves or profiles and the term or explicitly assume the shape is a continuous object, which substantially reduces the do search not space. We scattering profiles(Table 4). In practice many of these programs use information external to the scattering ab initio 4.3.1. Calculation of envelope reconstruction; however, these modeling methods are independent and can be performed in parallel. Similar answers deduced by each technique can help give confidence in the solutions; although we believe that Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & model reconstructed shape determination Ab initio modeling of the linkers region ab initio ab initio model with a single parameter: a spherical envelope 41 ab initio Svergun, 2005). F. Rigid body modeling using conformational Thousands sampling. & of 2005). Svergun, possible F. . Whole-body methods approach the problem by attempting to fit X-rayscattering G R 1/2 can be thought of as describing an G & Harding, 1985). A modern version of this algorithm that can also combine several simple shapes Shape restoration programs using bead models do not uniquely define the position of each bead within a within bead each of position the define uniquely not do models bead using programs restoration Shape Recently, Recently, a number of over-parameterized methods have been used for Many of the early programs restricted searches to shapes defined by a small number of parameters. For SAXS models of the complexed cellulase can be reconstructed using different methods. A.

computationally intensive and cheap methods of calculating scattering curves from proposed shapes have been developed (section 4.1). arrangements increases combinatorially, so all programs use computational tricks to search through the solution the through search to tricks computational use programs all so combinatorially, increases arrangements space including Monte Carlo becomes This approaches, proposed. shape each for transform Fourier a involving data genetic scattering the to directly fit programs algorithms, and simulated annealing (Table 4). Several (Volkov & (Volkov Svergun, 2003). These methods attempt to generate a bead model (or dummy atom model) to fill a volume consistent with the experimental scattering. External constraints, such as smoothness, connectivity, and particle symmetry have been used to reduce the search space (Koch et al., 2003). The number of possible atomic models, by applying molecular dynamics on the linker region (blue) have been used in an on the catalytic module (gray) (Hammel et al., 2004a). conformation. One hundred conformations (yellow) are shown superimposed exhaustive search of the best-fit CREDO (green) (Petoukhov et al., 2002). The secondary structural elements of fixed known atomic structure are in gray. E. Rigid body modeling applied on the knowing atomic structures (gray) in the combination (cyan) using BUNCH (Petoukhov with Fig. 13. by spherical harmonics using (Svergun SASHA et al., 1996). B. Densely packed beads model using DAMMIN (Svergun, 1999). C. Model calculated with GASBOR (Petoukhov et al., 2002) D. Reconstruction of one missing module using dummy residues with

resolution structures (Zheng, Doerschuk & Johnson, 1995). (Zheng, Doerschuk & Johnson, resolution structures Stuhrmann, 1991) as implemented (Fig in 13a; SASHA (Svergun et al., 1996)). spherical However, harmonics descriptions cannot properly represent all shapes, such as low structures describe with to employed cavities been also or have holes. harmonics icosohedral In particles, virus the symmetric specialized icosohedrally of case using spheres, oblate, prolate, and triaxial ellipsoids and are also used for modeling hydrodynamic properties (Rallison 2006). (Heller, Spherical harmonics is also available have with been the used program (Svergun ELLSTAT example, example, R of radius r = (5/3) P(r) functions and optimize the agreement to the experimental data (Fig. 13). data (Fig. the experimental to agreement the and optimize functions P(r) Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

, i 1 & to 1 (p P ρ , in ab initio 1i

, and and , p i P is the average the is d , i P is defined as 2 P and 1 reconstructions and the average P ab initio envelopes has not been conducted. Fig. & Chacon, 2001; Wriggers, Milligan is the number of points in points of number the is

i One parameter used to characterize the N is the distance from arbitrary points, ab initio ) The final resolution attainable from 2 6 shows 10 shape from the highly flexible detergent solubilized variation larger A DAMMIN. by generated apoB-100 exists between the models in comparison to OGG1 (Fig. 14); although, the overall dimensions agreement are and in suggest flexible relative structures typically have fewer features Very domain motions. in their scattering profiles, which comparison of the output A cangreater variety of shapes. be fit by a of multiple independent modeling runs (at least 6- 10) provides some measure of the uniqueness of the models. agreement among models is the normalized Briefly, 2001). spatial Svergun, & (Kozin (NSD) discrepancy if models 1 and 2 are expressed in two sets of points, then, NSD between where P distance between neighboring points in points neighboring between distance 2003); however, a systematic study on the accuracy of volume, nor do the beads represent positions of specific specific of positions represent beads the do nor volume, and non-unique are positions bead the Rather, residues. illustrated as particle, scattering the for volume a define by six independent GASBOR reconstructions of the Each 14). (Fig. glycosylase/lyase DNA OGG1 human of the reconstructions fits the experimental scattering equally well and describes similar shapes. GASBOR modeling (though proteins for specifically designed is of other macromolecules is possible); dummy atoms have radii approximating amino acids with penalties imposing chain connectivity. In the case of OGG1, the envelope generated truncated the shape by and volume averaging the fits runs the GASBOR different (Fig. 14). crystal structure envelopes varies with data 5.6). (section flexibility and shape size, particle used, quality, the program Several studies have compared available programs (Takahashi et al., 2003; Zipper & Durschschlag, 42 are very different. In these cases, using the program SITUS, which d and

i N SAXS envelope defined by the average the by defined envelope SAXS . NSD is 0 for identical models. NSD enables quantitative comparison of similarities of 2 P ab initio ab GASBOR reconstructions of OGG1, a 39kD protein critical protein 39kD a OGG1, of reconstructions GASBOR was originally developed for EM, is more McCammon, 1999). appropriate (Wriggers the nearest point in models if the modes have the same resolutions, but it is not straightforward to compare similarities of models with different resolutions because computers typically found in most labs. exposure. B. The crystal structure of a truncated version of OGG1 the within well fits of the 6 GASBOR runs. Each run takes around two hours on in agreement with the experimental data. Each model is composed of dummy atoms whose radii approximate second 30 those a with mg/mL of 5 of concentration a amino at collected were acids. Data reconstructions for showing GASBOR of runs functional independent Six A. McMurray). Cynthia analyses in collaboration the variation among models all of which produce scattering curves with to the recognition and repair of oxidized guanine in double-stranded double-stranded in guanine oxidized of repair and recognition the to DNA by the DNA base-excision repair (Protein pathway. courtesy of Tapas Hazra and Sankar Mitra University of Texas, Galveston, Fig. 14. Fig. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 Reference (Heller, (Heller, Krueger & Trewhella, 2003) et al., (Svergun 1996) (Zhao et al., 1998) (Walther, Cohen & 2000) Doniach, (Chacon et al., (Chacon et 2000; Chacon et al., 1998) (Svergun, 1999) 2006) (Heller, et al., (Svergun 2001b) a priori Advantage region Fast calculation, not an over- parameterized reconstruction of the observed scattering constraints Volume Straightforward modeling without any information, fast calculation Missing connectivity penalty penalty Missing connectivity fit for lead to excellent unfolded protein Optional use of symmetry constraints, loose and are disconnected shapes mode penalized, in the expert adjusting of penalties Not an over-parameterized observed reconstruction of the scattering Beads are on a chain with centers 3.8 Å apart to simulate protein and provide additional constraints, realistic models for multidomain particles with extended linker or hinge agreement to the experimental data. Imposing 2 χ agreement and NSD values (Fig. 16F). We note that the agreement and NSD values (Fig. We 16F). 2 χ 43 values and marginally poorer fits to the data. On the other 2 χ SAXS envelope reconstruction programs reconstruction envelope SAXS Characterization Ab initio atoms in the protein structure. The number atoms in the protein structure. α Table 4. Table simply shaped homogeneous particle applying the simply shaped homogeneous particle applying is required. spherical harmonics. No extra information ellipsoids that Uses Monte Carlo approaches, chain of grid. rotate about attachment point and no fixed any Based on Monte Carlo approach without take” limitation of search space using “give and approach. Similar to DAMMIN. The protein structure is Similar to DAMMIN. beads but represented not by densely packed dummy forming a rather by an ensemble of dummy residues The spatial positions of these chain-compatible model. to those residues aim at approximately corresponding of the C protein. of residues should be equal to that in the approaches, Uses a genetic algorithm and Monte Carlo fixed by no fixed grid and dummy beads radii are resolution of data. shape of a Allows determining the low-resolution beads models of the molecule, using the Debye of the molecule, beads models with a genetic models are optimized formula. Bead the huge space of possible algorithm that searches models. evolves convergent mass distributions and dummy beads configuration Searches for a compact function applying the minimizing the discrepancy using the simulating non-compactness penalty a random configuration. annealing starting from combination of simple shapes Builds models using and cylinders including ellipsoids Fits the scattering profile computed from dummy from dummy profile computed Fits the scattering SASHA The combination of multiple runs and external constraints has been remarkably successful for the low Program GASBOR ELLSTAT DAMMIN SAXS_3D DALAI_GA SASMODEL GA_STRUCT proper symmetry. mass was a critical parameter and required the as ability input to for determine overall this symmetry study, by SAXS will likely depend greatly on the overall shape of the target. In this case, the dramatic X-shape of the potassium channel, which is apparent even at extremely low resolutions, was likely critical for determining the hand symmetries of four-fold and higher had low NSD values, implying a much smaller variation in shapes. An symmetry imposed optimized four-fold both the shape could be reconstructed even when four-fold symmetry was not explicitly enforced. Imposed symmetries below and including four-fold derived shapes with similar higher symmetries resulted in noticeably higher fold symmetry could be derived from the scattering using GASBOR, as shown by the result of imposing various imposing of result the by shown as GASBOR, using scattering the from derived be could symmetry fold symmetries (Fig. 16). Each of the five symmetries shownwas run 8 times resulting in a total of 40 GASBOR runs. Each run required 12 hours of computer time for the 1288 amino-acid complex. In this case, the overall resolution of final results. In general, symmetry information will need to be introduced from external information. external from introduced be to need will information symmetry general, In results. final of resolution four- the 2006), al., et (Pioletti channel potassium voltage-gated a of portion cytosolic the of case the in However resolution envelope reconstruction of many systems (Fig. 15). Imposing correct symmetry greatly enhances the enhances greatly symmetry correct Imposing 15). (Fig. systems many of reconstruction envelope resolution Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & Hanein, 1999; Wriggers Wriggers 1999; Hanein, & derived models can support models be modeled before based vector fitting can be applied & (Volkmann quantization accurate more Additionally, 2003). Hanein, correlation-based approaches have been proposed. In these methods the solution sets leads to the possibility confidence intervals and error margins of for defining particularly is which parameters, fitting the Vector quantization based fitting is limited Vector to cases in which all density in the SAXS envelope is accounted for by the atomic model. In practice missing or disordered regions of the atomic model need to structures with dummy however, this atom problem is essentially models; same the as that resolution faced when structures fittingSubstantial progress has been into made high in the EM EM community maps. in this area Navaza 2006; al., et Craig 2002; Wriggers, (Chacon et al., 2002; Roseman, 2000; Rossmann, Volkmann 2000; & Chacon, 2001), and the EM SITUS program (Wriggers et al., 1999) been has used also for SAXS (Rosenberg 2005). In SITUS et the distribution of atoms al., well as structure high-resolution the within as the low-resolution reconstructions are vectors of number small a by approximated that are calculated by vector quantization. be a surprisingly challenging process. The program SUPCOMB (Kozin & Svergun, 2001) automatically superimposes atomic Ab initio structures can be used directly to build atomic ab initio 44 programs is that they require much less information and are easier ab initio SAXS structure is essentially a low resolution envelope to which an atomic structure atomic an which to envelope resolution low a essentially is structure SAXS ab initio ab with lower symmetry models but also has low NSD values. NSD low has also but models symmetry lower with 2 χ apart. Models were calculated with the point group symmetries group point the with calculated were Models apart. ° ; section 4.1). The model generated with 4-fold symmetry has 2 χ SAXS envelopes calculated with different point-group symmetries One One of the temptations of A candidate A All models suggest a 4-fold symmetric structure, excepting the model forced to have p8 point-group symmetry. section 4.3.1) is plotted against the average agreement with the experimental scattering ( equivalent nearly of A. p1 (no symmetry), B. p2 (one 2-fold), C. p222 (three perpendicular 2-folds), perpendicular (three p222 C. 2-fold), (one p2 B. symmetry), (no p1 A. of indicated symmetry applied each For 8-fold). (one p8 E. and 4-fold), (one p4 D. (NSD; models individual the between discrepancy the circle, the in value the by potassium channel (a tetramer generated from two heterodimers) and overlaid is the result et al., 2006). Each of the envelopes on the crystal structure (Pioletti of averaging eight GASBOR (Svergun et al., 2001a) runs and is shown in two 90 orientations, Fig. 16. voltage-gated a of portion cytosolic the from data experimental from determined et al., 1997) or with standard molecular graphics programs using envelopes expressed as atomic positions can may be docked. Finding a manual best fit using specifically developed SAXS software (Konarev et al., 2001; Kozin 2001; al., et (Konarev software SAXS developed specifically using fit best manual a Finding docked. be may 4.3.2. Rigid-body docking into low resolution envelopes docking into low resolution 4.3.2. Rigid-body models through rigid-body (section 4.3.2) or flexible (section 4.3.3) docking into the envelopes. (section 4.3.3) docking into rigid-body (section 4.3.2) or flexible models through built with or proposed by other methods. Moreover, and solutions possible on restraints more substantially provides information resolution atomic However, run. to is in our opinion the future of SAXS computational development. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & & & Sanejouand, SAXS envelopes, bound state, different Brooks, 2002). Thus 2002). Brooks, & x reconstruction of SAXS of reconstruction ab initio Ab initio Ab , ADP, and no nucleotide were readily were nucleotide no and ADP, , -RNA polymerase complex to activate -RNA x 54 σ Wriggers, 2003; Tama, Wriggers Wriggers Tama, 2003; Wriggers, & 45 enhancer binding protein NtrC1 in various nucleotide-bound various in NtrC1 protein binding enhancer -RNA polymerase complex, while subsequent hydrolysis -RNA and phosphate 54 σ Aquifex aeolicus Aquifex In addition to rigid-body fitting of domains into low-resolution envelopes, The elastic normal modes computed based on the low resolution envelopes compare well with the normal the with well compare envelopes resolution low the on based computed modes normal elastic The Modeling atomic resolution structures into SAXS envelopes has been used to study the conformational The ability to align crystal structures into SAXS envelopes is particularly informative when there are conformations can be calculated. Sanejouand, 2004). Computational methods other than NMA could be used to deform molecules to fit into these into fit to molecules deform to used be could NMA than other methods Computational 2004). Sanejouand, low resolution has envelopes; proven NMA however, to be quite useful due to the efficiency for which altered from SAXS or EM, as has been shown for the DNA-dependent protein (Boskovic et al., 2003; has Kong also et been al., kinase used 2003). to NMA study the motion in free and complexed and pyruvate dehydrogenase cellulase built from SAXS models (Hammel et al., 2004a) when submitted to the ElNémo server (Suhre modes obtained at atomic resolution (Chacon, Tama Tama (Chacon, resolution atomic at obtained modes derived envelopes resolution low from extracted directly be can assemblies macromolecular large of motions the with SITUS first (Wriggers et al., 1999). with SITUS first (Wriggers fitting EM density maps (Hinsen et al., 2005; Mitra et al., 2005; Tama, Miyashita & Brooks, 2004). For example, For 2004). Brooks, & Miyashita Tama, 2005; al., et Mitra 2005; al., et (Hinsen maps density EM fitting the NORMA software package (Suhre et al., 2006) allows flexible fitting to EM maps andSAXS isenvelopes; however, the wellSAXS envelopes suitedgenerally need for to be converted into synthetic EM envelopes 2006). Large conformational changes have been shown to frequently correspond to highly collective movements movements collective highly to correspond frequently to shown been have changes conformational Large 2006). that can be described by a small number of low-frequency normal Sanejouand, 2001)), and structures modes modified by normalof mode analysis (NMA; sectionprotein 3.3) have been(section used for 3.3; (Tama potential flexibility during the docking protocols become quite useful. This problem has been faced frequently in frequently faced been has problem This useful. quite become protocols docking the during flexibility potential EM studies and a variety of computational methods are currently being applied (Suhre, Navaza like experimental EM maps, may correspond to states macromolecule. Hence, that computational techniques are that can different take from single conformational the states and crystallized account form for of the 4.3.3. Exploring macromolecular motions using normal mode analysis (NMA) motions using normal 4.3.3. Exploring macromolecular release drove the conformational changes necessary to generate an open polymerase/promoter complex (Chen et al., 2007). low-resolution envelopes derived from SAXS with enforced seven-fold symmetry could be used to model ATP- the an demonstrated authors the EM, with conjunction In states. conformational different of control nucleotide bound state that stabilized the EBP- states of the ATPase domain of the of domain ATPase the of states acts upon a quiescent states. AAA+-ATPase, NtrC1, which is also an with domain ATPase heptameric the of structure crystal ADP-bound the of Superposition initiation. transcription hydrolysis through two parallel rings of domains. two parallel rings of domains. hydrolysis through Rouiller et al., 2002; Zhang et al., 2000b). Using the crystal structure of the ADP-AlF conformations were proposed, revealing a coordinated mechanism for transferring the energy of nucleotide conformational changes induced upon nucleotide binding (Davies ADP-AlF et mimic like transition-state the al., AMP-PNP, analog hydrolysable 2005). States induced by the non- 2003; al., et (Beuron studies EM previous with consistent and envelopes GASBOR-generated in distinguishable structures structures combined with changes. SAXS analysis conformational have provided ATP-coupled substantial information undergoes regarding p97 the mechanism by AAA+-ATPase the which with p3 or envelopes p6 for point ATPase this group hexameric symmetry provided substantial information on functionally-related conformational changes. For example, both symmetry and the existence of atomic resolution resolution atomic of existence the and symmetry both example, For changes. conformational functionally-related important important in the context of docking atomic 2003). Hanein, & structures look (Volkmann should fit a correct on how is available information into low-resolution density maps as no independent Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & Wright, While samples Several forms biological systems. containing multiple acids in length (Vucetic et acids al., in length (Vucetic 2003), and growing evidence suggests that macromolecular important an be will flexibility part of mechanism in many different the regulatory This response to changes in ionic strength was linked to primase function (Corn et al., 2005). A more dramatic form of flexibility isby exemplified a new class of important regulatory proteins, do not which have a dimensional single three- and conformation are either unstructured intrinsically unfolded or (Dyson 2005). natively It has been estimated that over 50% of eukaryotic proteins contain unstructured amino 40 over are that regions 4.4. Flexibility and Flexibility 4.4. disorder conformational by SAXS measured of flexibilityfor the are function macromolecular crucial of complexes many and enzymes (Boehr et 2006). al., A common example are flexible linkers, suchthose as controlling the overall domain conformations activation and of protein kinases another In 2006). al., et (Nagar example, analysis of the P(r) function demonstrated the that flexibly linkedsubunits primase dissociated ionic strength was when increased. 46 Determination of the solution conformation of the hexameric archaeal secretion ATPase ATPase secretion archaeal hexameric the of conformation solution the of Determination crystallography and NMR and recent advances in computational approaches, SAXS has the potential to provide to potential the has SAXS approaches, computational in advances recent and NMR and crystallography realistic information regarding large-scale structural rearrangements. The techniques described in sections 4.2 conformations can be challenging to crystallize and tend to reach a single conformation low-energy when they X-ray from structures domain with combination In SAXS. by collection data for difficulties no present they do, model while a is solution ADP best with described excess as a mixture of all the models (after 2007)). Tainer, & (Yamagata determined via crystallography. B. Top views of the crystal structure and two models proposed Top B. determined via crystallography. by modifying subunits to adopt either all open (brown) or all closed (green) conformations. C. closed all the to similar most conformation a adopts AMPPNP excess in ATPase the solution In Fig.17. Fig.17. structure hexameric the in found GspE AMPPNP from a to bound combination monomer of GspE ATPase SAXS and the crystallography. A. of configurations Side view of the 2 alternating Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 corresponds were readily 2 G max Tainer, & 2007). Tainer, binding (Rosenberg ’s ’s that are the square 2+ G reconstructions of mixed Ab initio of 0.13 Å, were lost when noise was G and 20 Å differences in D G and AMP-PNP bound enzyme fit the experimental values will yield observed R 2+ G 47 values of the different components. Thus, mixed populations continue to 2 G /calmodulin-dependent protein kinase II (CaMKII) converts from a compact 2+ Identifying that a solution contains a mixed population of macromolecules can be challenging and often and challenging be can macromolecules of population mixed a contains solution a that Identifying Alternatively, direct comparisons of different conformational states with theoretical scattering calculated scattering theoretical with states conformational different of comparisons direct Alternatively, In general, the characterization of these mixed states by SAXS is most straightforward if the different Assuming that different conformational states do not interact, the resulting scattering from a mixed to large to changes large in the SAXS scattering will be the most straightforward. For example, scattering power in the small angle region goes by the square of the mass; hence, mixed multimeric populations states of will different conformational changes involving protein A, kinase with a change in R added to the theoretical scattering 2005). Therefore, (Heller, SAXS studies of mixed populations that give rise the collapse of calmodulin, for example, the 2.6 Å difference in R apparent, even when simulated noise was added to the scattering the contrast, In states. curves. collapsed and extended both of components conformational substantial show populations root of the weighted sum of the R have linear Guinier regions (Heller, 2005). Moreover, the signal indicating heterogeneity is highly dependent upon how much the conformational changes alter the SAXS scattering (Heller, 2005). In the calculation of to the average square distance of each scatterer from the center of the particle contributing mixed to samples the containing scattering, components with different R R Since electrophoresis. gel native as such techniques other from information additional requires alone. Moreover, the alone. ADP-bound scattering Moreover, of state the was well fit by a mixture of scattering from the crystal only system the in flexibility considerable demonstrating model, closed all the and model, open all the structure, using experimentally determined states of the monomer for modeling. Fitting of the solution X-ray scattering curves of the Mg crystal structure less well than a computational hexamer generated from the closed state of the monomers from atomic-resolution structures has been quite successful in identifying and deconvoluting the relative fractions fractions relative the deconvoluting and identifying in successful quite been has structures atomic-resolution from GspE (Fig. The 17). in structure the was sample, solved such as as a ATPase with mixed the archaeal secretion hexamer of open and closed conformational states of the component monomers (Yamagata et al., 2005). example, the dodecameric Ca autoinhibited state to a loosely tethered state with independent kinase domains upon Ca states can be isolated independently and treated as homogeneous samples. Driving the population to a single conformation might be induced by buffer conditions or through binding specific ligands or substrates. For SAXS DNA bound and free structures of the important DNA damage response protein p53, which is mutated in mutated is which p53, protein response damage DNA important the of structures free and bound DNA SAXS et al., 2007 in press). 50% of all human cancer (Tidow challenge for all shape reconstruction techniques. Nevertheless, data determined from the consensus population consensus the from determined data Nevertheless, techniques. reconstruction shape all for challenge in solution, under a variety of conditions can be advantageous. A concern with cryo-EM is minor that an population observed overly biases the determined shape. This was dramatically demonstrated by cryoEM and sample is a population-weighted average of scattering from individual states. This scattering poses a significant significant a poses scattering This states. individual from scattering of average population-weighted a is sample 4.4.1. Multiple well-defined conformations 4.4.1. Multiple well-defined such as are observed during allosteric rearrangements. In contrast, assemblies for free, continuous motions containing of individual domains are more linkers challenging to model (section that 4.4.2) and must be allow (section 4.4.3). from aggregation SAXS from samples suffering distinguished in parameters parameters and the difficulties in incorporating multiple models during The model simplest refinement. case of conformational heterogeneity in SAXS is the presence of multiple well-definedconformations (section 4.4.1), and 4.3 focus on determining a single “best” conformation. For particularly flexible samples, this conformation conformation this samples, flexible particularly For conformation. “best” single a determining on focus 4.3 and is unlikely to exist, and any single conformation of the macromolecule would be predicted to fit the scattering poorly. The real challenges for modeling this flexibility are the dramatic increasein the number of fittable Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 is also an MxN an also is U where virus, which give rise to N) ω ×

& Doniach, 1996; Perez et al., (N B N) × experimentally determined scattering determined experimentally th (N W N) , the n the , Johnson, 2001; Lee et al., 2004). Moreover, Moreover, 2004). al., et Lee 2001; Johnson, & × (q) n (M Nudaurelia Nudaurelia capensis I U are reduced to a common minimal basis set as described as set basis minimal common a to reduced are N) = N) × contains N vectors, only important basis vectors will have (q) will fit noise in the data rather than the significant parts of n 48 I U (M U A basis vector to vector basis j u is a diagonal NxN matrix composed of the so called singular values . Though Though . j n W values. Many vectors in basis vectors, j j �� (Akiyama et al., 2004; Goettig et al., 2005; Graille et al., 2005; Nowak et al., 2006a; u where the N columns are the intensity values of the scattering profiles determined at M N) × (M A is the weighting contribution of contribution weighting the is j . Such a matrix may be represented as represented be may matrix a Such . n q b is an NxN matrix containing b j SVD will identify the minimum number of curves required to describe all the scattering profiles and can and profiles scattering the all describe to required curves of number minimum the identify will SVD In SVD, all the collected scattering profiles scattering collected the all SVD, In In addition to systems of linear equations, single value decomposition (SVD; (Press et al., 1992)) can Using experimentally determined SAXS scattering and theoretical scattering from individual components components individual from scattering theoretical and scattering SAXS determined experimentally Using The best way to identify heterogeneity is by following it through titration of the states from one form B and j measured in the presence of various substrate analogs, activators, and inhibitors (Fetler et al., 1995). measured in the presence of various substrate analogs, activators, readily distinguish a system with a single transition from one with multiple transitions. Only two basis vectors were vectors basis two Only transitions. multiple with one from transition single a with system a distinguish readily necessary to describe the scattering data sets collected from the allosteric enzyme aspartate transcarbamoylase scattering profiles. Several commercially available mathematical packages have built in SVD routines. scattering profiles. Several commercially available mathematical matrix containing the w associated significant w MxN matrix of values where w where profile. Determining the values of basis vectors and the relative weights is accomplished through creating an in the following equation: curves from individual components, but in order to extract real scattering curves external information may be models for transitions. required, such as thermodynamic 2001) as well as deconvoluting mixtures complexes of (Bilgin protein–RNA et al., 1998) and transient protein conformations (Fetler et al., 1995). SVD requires multiple scattering curves collected from samples containing different populations of states. Unlike solving systems of linear equations, SVD does not require scattering deconvolute SAXS data from mixtures. SVD was introduced into SAXS 1983) in and has the been applied to early the problem 1980s of protein (Fowler folding (Chen, et Hodgson al., being open and 39% being compact and active (Graille et al., 2005). being open and 39% being compact a compact structure in the crystal, but unlike the crystal, the solution scattering is consistent with a population containing 92.5% in an open conformation and only 7.5% Similarly, the in transcriptional the antiterminator compact LicT form exhibits (Vestergaard et a heterogeneous al., population 2005). in solution with 61% (form factors), volume fractions in each conformation can be determined by solving a systems of linear equations linear of systems a solving by determined be can conformation each in fractions volume factors), (form with the program OLIGOMER (Konarev et al., 2003). The bacterial class I release factors, for example, adopt Nowak et al., 2006b; Vestergaard et al., 2005). Vestergaard Nowak et al., 2006b; substantial changes in the SAXS scattering curve (Canady, Tsuruta Tsuruta (Canady, curve scattering SAXS the in changes substantial several schemes have been used to prove the existence of transient, substrate-induced conformational changes and characterize them to another or through situations in which atomic models are available so that they can be directly compared to the observed scattering. For example, SAXS has been used to follow large-scale changes due the pH-induced maturation of viral capsids of the HK97 bacteriophage and be more readily identified. unfoldingSimilarly, of macromoleculesinvolves large-scalechanges in theoverall structure and will dramatically change the observed scattering; thus, SAXS has been a method 1983). of Glockner, Provencher & Perez et al., 2001; (Doniach, 2001; protein folding studying choice for Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 (Hammel et al., 2004a)).

49 restoration are highlighted with the orange arrows. C. Experimental ab initio models of free and complexed cellulase Cel48F. A. Five typical CREDO ab initio Partial Doniach, 2007). The experimental matrix for SVD used SAXS data collected from different different from collected data SAXS used SVD for matrix experimental The 2007). Doniach, & subdomains are schematically represented below each construct. The observed displacements between individual runs of partial of factors form averaged the by fitted curve) (upper complexed and curve) (bottom free of profiles SAXS SAXS and 2003)), al., et (Konarev line; (red OLIGOMER program the by obtained models CREDO the profiles calculated from the single CREDO models (green line) (after Fig. 18. (Petoukhov et al., 2002) models of linker – dockerin region of free Cel48F models restored Two are B. domain. catalytic Cel48F of structure crystal the on superposed together colors displayed in different (green and blue) of the dockerin/cohesin complex of the complexed Cel48F using the program CREDO corresponding The 2000)). al., et (Parsiegla 1fbw; id (PDB Cel48F of structure crystal the on superposed Another promising application of SVD is toward characterizing detergent solubilized membrane In an illustrative use of SVD, the temperature denaturation of neocarzinostatin was characterized by their experimental application using membrane protein TM0026 did not provide details on structural parameters structural on details provide not did TM0026 protein membrane using application experimental their complex, micelles, a Though complex. micelle-micelle protein/detergent membrane the is interaction interest of component the course component Of component. interaction and a membrane protein/detergent to micelle proteins (Lipfert (Lipfert proteins concentration ratios of membrane All protein concentrations to of detergent. detergents were above the critical micelle concentration. They reprojected the SVD vectors as scattering due to membrane protein/detergent the basis set. However, the the derived basis scattering set. curve However, was degenerate, and the authors concluded that the folding states. of related flexible intermediate involves an ensemble pathway of neocarzinostatin SAXS (Perez et al., 2001). At protein least starting the three from scattering SVD the basis With state. vectors intermediate one were least required at of for presence the a suggesting unfolding, complete description of the and the final unfolded state, they attempted to derive the scattering from the intermediate state by reprojecting Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 50 models of the different cellulosomes constructs calculated with ab initio tube representation. The secondary structural elements of superimposed atomic structures α tubes. C. Best-fit models were calculated by MD of different constructs and superimposed on cohesin α , the potential to isolate the scattering profile of amembrane protein/detergent complex is very The solution structure of the cellulosome. A. The average envelope shape of different cellulosome constructs G SVD is potentially very powerful; however, some care must be taken in employing the technique. Robust technique. the employing in taken be must care some however, powerful; very potentially is SVD from S4 or complexed Cel48F (highlighted with the Three black models circle). colored by blue, brown, green for each model, except for FtS4Fc complex where only two models (green and blue) for better visualization, are presented (after (Hammel et al., 2005)). the top of each known fixed The shape. (yellow). representation surface B. in displayed are Restored models CREDO The partial 2002). al., et (Petoukhov CREDO atomic structures are in C are shown as C Fig. 19. were calculated with GASBOR (Svergun et al., 2001a). The corresponding modules are schematically represented on induced conformational changes. Unfortunately, time resolved experiments with short exposures also from weaker SVD signals. identifiessuffer Finally, the minimum number of scattering curves andnot necessarily all in the SVD analysis, great care must be taken in titration experiments to guarantee the buffer is properly matched matched properly is buffer the guarantee to experiments titration in taken be must care great analysis, SVD the in to the solution for background subtraction. SVD is also particularly powerful for time-resolved experiments of analysis requires many scattering curves and relatively error-free subtraction among data. the data sets Systematic may cause particularly errors insidious problems for SVD, from as some of background used the is basis data subtracted sets background If solutions. of representative truly be than rather error fit to required be may other other than R effort. research further well worth and exciting Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 model calculated model ab initio ab GPI (left) is shown with attached sugars (red). B. (red). sugars attached with shown is (left) GPI GPI with the indicated fractional occupancies in 2 2 β β models has been correlated with flexibility. A number of number A flexibility. with correlated been has models shape restoration. 51 ab initio ab ab initio GPI). A. The crystal structure of structure crystal The A. GPI). 2 β GPI conformations were constructed by simple rotations between domains CCP2 and CCP3 of the best the of CCP3 and CCP2 domains between rotations simple by constructed were conformations GPI -glycoprotein I ( I -glycoprotein 2 2 β β , respectively). Three different atomic conformations of -3 and 8.5x10 -3 In some cases, the lack of convergence of convergence of lack the cases, some In In contrast to samples that can be described as containing multiple well-defined conformations,many Solution structure of structure Solution models calculated by CREDO from scattering data collected from the cellulase Cel48F did not generate a single a generate not did Cel48F cellulase the from collected data scattering from CREDO by calculated models conformation (Fig. 18; (Hammel et al., 2004a)). Combining and weighting the scattering of individual models (Nagar et al., 2003), fit the averaged SAXS envelope well, despite the fact that the molecule was proposed to be be to proposed was molecule the that fact the despite well, envelope SAXS averaged the fit 2003), al., et (Nagar extent the about interpretation Further states. heterogeneous of range wide a with ensemble conformational a in of flexibility would require analysis beyond a binding pocket on the protein generates a compact autoinhibitated state. SAXS data predicted to disrupt the autoinhibited revealed state were much more Abl elongated that than closely wild-type matched that mutants the more compacted crystal structure. Fully extended models, created from the crystal structure (PDB id 1opl; is comprised of three domains connected by flexible linkers. Binding of the myristoylated N-terminus within of the c-Abl tyrosine kinase that is dysregulated by gene fusions in chronic myelogenous leukemia (Nagar et al., 2006) have provided additional insight into the autoinhibition of the kinase. The N-terminal half of Abl for large numbers of eukaryotic proteins. For these types of samples, attempting shape reconstructions to derive to reconstructions shape attempting samples, of types these For proteins. eukaryotic of numbers large for a single model with a “best-fit” conformation can be misleading and at best provides amodel representing an average of the conformations. At times an averaged model can be informative. For example, SAXS studies multidomain proteins and protein complexes contain flexible linkers that allow them to numbers adopt of large it is likely to be quite common This situation is substantially more complex to model; however, conformations. 4.4.2. Conformational disorder explored by SAXS explored disorder 4.4.2. Conformational states states that may exist (Koch et al., 2003), which might be particularly problematic when a continuum of states peptides. unfolded with partially such as exist, are 3.0x10 Different shown. are solution fit single structure (after (Hammel et al., 2002)). The best single-conformation model derived from experimental SAXS data is superposed with the averaged the with superposed is data SAXS experimental from derived model single-conformation best The by program DAMMIN (Svergun, 1999) displayed as grey cages. C. The theoretical scattering profile calculated for different theconformations improved mixturethe fit of to the experimental data (root mean squared deviation for single and multiconformation fit Fig. 20. Fig. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 - 2 β shape ab initio ab model (section 4.3.1), was unable to to unable was 4.3.1), (section model ab initio ab & Perkins, 2001; Boehm et al., 1999; Gilbert et 52 GPI likely to be important to its function, but it also illustrates illustrates also it but function, its to important be to likely GPI 2 β GPI), the SAXS scattering fits poorly with the crystallized conformation or individual models individual or conformation crystallized the with poorly fits scattering SAXS the GPI), 2 β We are enthusiastic about a new approach to analyze the presence of multiple conformations of proteins proteins of conformations multiple of presence the analyze to approach new a about enthusiastic are We Once generated from atomic structures, these models can then be evaluated for their relative contribution relative their for evaluated be then can models these structures, atomic from generated Once Conventional MD methods are computationally intensive; however, several advances have increased the increased have advances several however, intensive; computationally are methods MD Conventional The biggest challenge in trying to model conformationally very flexible systems using SAXS data is generated to cover the possible conformational space. A genetic algorithm is then applied to select subsets (N=50) subsets ensemble select to applied optimization then is method algorithm genetic (EOM) A space. in conformational possible the which cover to generated a pool of possible conformations (N>1000) is randomly contributing to the experimental scattering profile (Bernado et al., 2007). Bernado and coworkers define an the common case in which external information must be introduced in order to generate the models to fit the observed scattering. correct types of glycoprotein I ( I glycoprotein approach model a multiple using well fit does but 2002) al., et (Hammel conformations modified systematic with in flexibility inherent the is only Not 20). (Fig. to the scattering using SVD or the algorithm underlying OLIGOMER (section 4.4.1). In the case of additional experimental evidence, the combination of SAXS and atomic models generated by MD trajectories cannot prove the existence of any particular conformation. the average overall shape of the minicellulosomes and for proposing plausible atomic conformations they may these best the experiments, SAXS from values data independent of amount restricted the Given 19). (Fig. adopt techniques can provide is a set of conformations that are consistent with experimental data. In the absence of reconstruction, and reconstruction of missing domains using CREDO have given similar results to MD sampling MD to results similar given have CREDO using domains missing of reconstruction and reconstruction, (Marino et al., 2006). SAXS profiles calculated using MD trajectories are particularly useful for determining at very high temperatures (~1000 K) to prevent molecules from becoming 2001). trapped Similarly, many in of local the minima simulations (Leach, modeling, are body rigid of sped Comparison 1999). al., up et (Losonczi terms by electrostatic not but including restraints, only van der Waals terms and distance size of tractable conformational changes (Yuzawa et al., 2001). These new techniques include multiple timestep timestep multiple include techniques new These 2001). al., et (Yuzawa changes conformational tractable of size MD and high temperature MD (Aslam et al., 2003; Aslam al., 2006; Hammel et al., 2005; von Ossowski et al., et 2005; Yuzawa al., 2001) in which simulations are run Monte Carlo techniques (Buey et al., 2007; Shell et al., 2007), molecular and 2001) (Schlick, sets, conformation generating of method exploring non-dynamical a CONCOORD, 2004), the dihedral space of linkers (Tai, 2002) have all been employed. dynamics (Levy & Becker, data. A data. A number of creative techniques have begun to address the problem of quantitatively modeling flexible macromolecules observed in SAXS experiments (Akiyama et al., modeling 2004). approaches Various can be used to generate atomic models that sample conformational space for use in fitting experimental SAXS curves. leverage existing atomic structures to reduce the parameter space of the model by describing the ensemble as a set of most probable structures. minimize To potential problems with overfitting, individual conformations to be tested as probably components of the population ought to be generated independently of the SAXS to avoid overfitting the raw data. One strategy to avoid overfitting the raw data with multiple models is to and those model features, such as partial occupancy, necessary to completely describe this population-weighted population-weighted this describe completely to necessary occupancy, partial as such features, model those and electron density. the measured scattering is derived from the population-weighted thermodynamic ensemble and describes some population-weighted distribution of electrons. The inability of CREDO to generate and converge fit”to model likely derives from incompatibilitiesa between the constraints that“best CREDO has in generating models fact that CREDO, a program that generates an over-parameterized over-parameterized an generates that program a CREDO, that fact population, heterogeneous a from scattering of case the In data. raw the to fit model single better a with up come with with OLIGOMER improved the overall fit (Fig. 18). Inthis case,more parameters wereadded whichmakes an improvement in the the is fit however, unsurprising;remarkable, truly however,is itWhat issteps. merging noteworthy and that weighting the thefrom and individualother each modelsof independently were generated Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 values. Guinier plots of plots Guinier values. max SAXS model, whereas the <1.3 which is more typical for G and D and R G q ab initio We We also are enthusiastic about the interpretation. In the study of the first two potential offered by the recently which NMR, developed (RDC) coupling dipolar residual has been used to identify relative orientations external an to relative proteins multidomain in coordinate system (the ‘‘alignment tensor’’). For multidomain proteins, the orientation of each domain can be determined within this separately coordinate frame, and the relative interdomain orientation can (Bernado et be al., deduced for 2005). unbiased RDC determination data of orientations interdomain allow in solution, albeit with a 4-fold but degeneracy, require structural models for Ig domains of titin, Z1Z2, SAXS used data to resolve the RDC degeneracy was (Marino et al., 2006). the Calculations with models of 50 the that the showed conformers 200 RDC well-defined a were values lowest-discrepancy cluster of conformations that superimposed with the <0.8 instead of G R q 53 The experimental P(r) of the rabbit 15-lipoxyganase-1 (black). Studies on the soybean lipoxygenase-1 and rabbit 15-lipoxygenase-1 illustrate the need for careful In general multidomain proteins with long linkers or that adopt extended conformations have smooth terminal domain (Hammel et al., 2004b). Rigid body modeling was applied, and the improvement of the fit in the in fit the of improvement the and applied, was modeling body Rigid 2004b). al., et (Hammel domain terminal interpretation of SAXS data in cases where flexibility is proposed (Fig. 21). The major discrepancies between the experimental scattering curve of rabbit lipoxygenase in solution and the curve calculated from the atomic coordinates were interpreted in terms of a large movement of the N-terminal domain with respect to the C- amounts of aggregation, which primarily affects the same low resolution region of the scattering curve. the same low resolution region amounts of aggregation, which primarily affects these macromolecules are linear over a smaller region: globular samples (Table 1). Unfortunately, many of these features are also observed in the presence of small scattering profiles with few prominent features at high resolution and extended tails inR P(r)of variety a with conformations heterogeneous have typically also proteins functions. These 4.4.3. Flexibility, oligomerization, or aggregation? oligomerization, or 4.4.3. Flexibility, Thus, the synergy of SAXS with molecular modeling, and crystallography, NMR promises Mattinen to 2005; provide unique Grishaev, & (Bax flexibility intrinsic with proteins of characterization structural the into insights et al., 2002). agreement of experimental and simulated RDC to SAXS data validated the conformational sampling results and results sampling conformational the validated data SAXS to RDC simulated and experimental of agreement protein. unfolded of ensemble the of dimensions average and dynamics structure, local of description a provided more compact crystal structure failed to fit the solution structures. Furthermore, conformational sampling using sampling conformational Furthermore, structures. solution the fit to failed structure crystal compact more close The protein. unfolded partially of profiles SAXS of analysis the in applied was RDCs averaged simulated (PDB id 1lox; (Gillmor et al., 1997)) using in the P(r) calculation are shown are calculation P(r) the in using 1997)) al., et (Gillmor 1lox; id (PDB (Hammel et al., 2004b). as surfaces using the same coloring (after adopting different extensions in the N-terminal regions with the indicated oligomeric 20% and monomeric 80% of mixture a for and (red) occupancies calculation P(r) the in used structures crystal The shown. are (blue) assemblies Fig. 21. Theoretical P(r) calculated for a mixture of two protein conformations unstructured unstructured and multidomain proteins, EOM was able to distinguish between rigid and flexible proteins and contacts. assess interdomain of configurationsthat fit theexperimental scattering.The advantage of this method is the use of criteria for quantitative analyzing the EOM-selected models and for determining the optimal number of conformers in the for data experimental and theoretical both Using evolution. further for selected then are subsets best The subset. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 54 Schematic for SAXS data collection, evaluation, analysis, modeling, and interpretation. We favor the Fig. 22. combination of atomic structures with SAXS envelope modeling (lower right) where possible. Using and conformation in solution. experimental observations to identify possible flexibility along with shape, assembly the SAXS evaluation evaluation of rabbit lipoxygenase indicated that partial (20%) aggregation could also explain the scattering of (Fig. 21). rabbit lipoxygenase workers found that the modeled structure was quite similar to the crystal structure and that slightly aggregated samples would give rise to elongated signals, as reported for rabbit lipoxygenase (Dainese et al., 2005). Re- entire entire scattering profile wasobserved using swing a outmixture conformations of(Figure different 21 red, two conformations are shown, Hammel et The al., modeling 2004b). of rabbit lipoxygenase was entirely dependent co- and Dainese lipoxygenase, soybean of study a In monodisperse. was studied solution the that establishing on Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 c )/ 94 40 (0 I 0.10 7.5 mg/ml 5.0 mg/ml 2.5 mg/ml 1.25 mg/ml buffer 0.15 limit = 1.3 Å Å G .004 ) G R ) -1 R be ameliorated 22 30 2 Å q ) ( Å 0.10 -1 indicates the q ( 2 Å 2 - q ( 2 0 q q aggregation that may 0.05 00 5 l l I(0) 0. .0 infinite dilution )) vs. 5 mg/ml 10 mg/ml 20 mg/ml 40 mg/ml q buffer subtracted subtracted buffer sample

1 mg/m 6 mg/m

0 0

Intensity (log scale) (log Intensity 0.0 scale) (log Intensity 0.000

0.1

0.01 Intensity 0.001 004 0. 0.15 .004 l ) 2 Å

) ( 10 2 ) with the 2 - ) 0. q q 6 mg/m Å -1 Aggregation in the sample can be reduced or eliminated ( 2 0 Å 2 - ( 0.002 q aggregation 00 q

0. raw data raw l A non-linear dependence of log(I( 55 0.05 aggregation

Guinier fit Superposition of scaled 1 mg/m

). Intensity (log scale) (log Intensity

0.000 q

0.00 Intensity (log scale) (log Intensity 0.000

0.1

0.01 Intensity

Scattering from aggregated samples strongly influences the entire data set 0.004 ) 2 Å ( 2 - q no aggregation and I(0) from the Guinier plot. X-ray sensitivity with .002 G limit = 1.3 G 0 G I(0) R q . -1 Å 00

and I(0). X-ray damage Intensity (log scale) (log Intensity

G 0.0

<0.1

Calculate R presence of aggregation. and no further processing should be performed! by varying buffer conditions, centrifugation, and filtration. typically lead to increases in R can be minimized by adding 5-10% glycerol. Measure four different concentrations in the range 1-10 mg/ml. can be detected by changes in scattering observed in repeated exposures and changes in multimerization state or increasing amounts of aggregation in the sample. Watch for increasing I(0)/c and R increasing concentration. These can be due to

concentration-dependent scattering. Decreases in intensity at concentration-dependent scattering. indicates very small q with increasing protein concentration be dealt with by dilution, interparticle repulsion. Repulsion can the data to infinite increasing ionic strenght, and/or extrapolating dilution. Scaling the data to extrapolated scattering is only is visible, typically required in the range where the interference q Evaluate the effects of interparticle interference F( Evaluate the effects of interparticle scattering of individual particles S( can reveal scattering curves from multiple concentrations

COLLECTION EVALUATION

Data collection and evaluation.

DATA DATA DATA DATA Fig. 23. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 protein =0 (I(0)/c) q kDa max 200 20 D SA 66 66 SA 0.4 folded unfolded partially unfolded B 172 kDa) 172 unfolded protein multidomain globular protein proceed. 0 1 3 interparticle interparticle 100 0. ) -1 ) ) can can Å Lysozyme 14.3 kDa Å Glucose Isomerase (43x4 = -1 0 9 ( Å q r ( ( .2 0 q 0 without without 0.2

50 Normalized Intensity at

150 100 0 6 Molecular Weight (kDa) Weight Molecular infinite dilution highest protein concentration 1 0 structure structure 0.

0 3 0 5 0

samples samples

1. 0. 0. P(r) Intensity (log scale) (log Intensity 0.0 0.0 0 0 5

0. 1. 0.

) q I(

solution solution q a 2 of . a e the the The the with

of of can of q values

have calculated

onodisperse onodisperse oligomerization and position 56 plot the peaks. primarily m I(0) and ) and large the in law function . This can be validated -1 at nd Å that is accuracy samples. , a from molecules, peak P(r) Guinier G Molecular weight from I(0) requires a calibration curve and can be used to determine state using calculation of the excluded volum multiple <0.1 the zero a R max q Porod's this D the to of increasing macromolecules and have reconstruction reconstruction have unfolded lack from Extended aggregation follow can

data collected collected returns of particle, values then then slightly and and the are are for elongated r are peptides, curves. identifies in or function amounts resolution while function data data large plot , P(r) at length low macromolecules macromolecules P(r) unfolded range. small plot.

plateau peak tail the q the interference, interference, bell-shaped a as SAXS SAXS the Kratky If If Disagreements , a measure of the

Data extrapolation, merging, and analysis. Proper data analysis as outlined is essential to any further modeling and exposure the sample-to- times, and/or decreasing detector distance. To obtain an ideal scattering curve for the entire curve for the entire an ideal scattering To obtain to must be extrapolated scattering profile range, the infinite dilutions resolution ( at low merged angles. profile for larger with the scattering can be obtained by large angle data Accurate using longer higher concentrations, measuring maximum where r. from indicate affect Guinier Globular single longer Globular have such have larger The can be determined from the Guinier plot. R overall size of the molecule,

DATA ANALYSIS DATA IDEAL SCATTERING IDEAL Fig. 24. interpretation. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

), 1 - 2 Å a .3 in 0 number body crystal > domain ab initio 3 structure . the max rigid q shapes. ( performed q When with linker 0.2

2 0 of ab initio be 0. ) ) 1 1 - tart - Å Å can S ( ( q anisotropic 1 q values 0. .1 1 for alculations. large experiment multiconformational fit experiment CRYSOL fit c modeling

crystal

assembly.

0 0

a scale) (log Intensity Intensity (log scale) (log Intensity structure

0.0 0. initio b

advantages crystal structure a structure reasonably overall runs. ensure the to to the

to with some the deletion is individual with loops that ultiple but and domain not multiple domains of with m atomic from multi- essentially extends Quaternary solution. 57 identify but a

provides missing Investigate scattering envelope scattering fitting multiple assembly multidomain missing it are from in fiting indicate data from b) symmetry known, data appropriate not

If as individual is between SASREF. combination over and possibility

combining the known SAXS in with complex. to onstruct averaged SAXS does at different building the calculate C using are modeled matches using nvelopes resolution structures of the states. due linkers Investigate domain e calculated be experimental domains when protein way a) known low use way. CREDO to

potentially available. GLOBSYMM. one Perfect solution, crystal initio BUNCH recommended and the modeling

are missing. the nvestigate try and/or subunits assemblies are and mixtures Investigate of and experimental is I curves of ab need problems assembly betwen using and c) with are fragment. scattering

body with automated disagree. may long. avoid residues

known candidate scattering are models structures scatering constructs Structure others domains flexibility that conformational Individual modeling fully Rigid structures. solution identical. Theoretical can alternate suspected, fitting states. Average To reproducible envelopes Compare SAXS Determine of GASBOR ab initio ab reconstruction reconstruction reconstruction

rigid body modelling + modelling body rigid ab initio ab shape evaluate theoretical scattering theoretical evaluate

Partial atomic structure: atomic Partial No atomic structure: atomic No Available atomic structure: atomic Available

Solution structure modeling and interpretation. SOLUTION STRUCTURE MODELING STRUCTURE SOLUTION Fig. 25. Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 58 at a wavelength of 1 Å in a 30 second exposure. As described As exposure. second 30 a in Å 1 of wavelength a at -1 = 0.3 Å 0.3 = q shape restorations only to later become disillusioned because of perceived of because disillusioned become later to only restorations shape ab initio ab Based upon our experience with many types of samples, we recommend the strategy and tactics shown With With increased general use of SAXS as a powerful tool to examine biological samples, it becomes We designed and built the SIBYLS synchrotron beamline (http://www.bl1231.als.lbl.gov) at the Advanced Advanced the at (http://www.bl1231.als.lbl.gov) beamline synchrotron SIBYLS the built and designed We A A number of parameters affect the amount of sample required for data collection. The difference in At the SIBYLS beamline, of 15 a µL well behaved 100kD protein in typical can buffers provide SAXS Therefore, Therefore, samples that are suspected of possessing intrinsic flexibilitymust becarefully characterized average electron density between the sample and bulk solvent plays a key role. For and example, DNA, RNA times are 1-100 seconds although millisecond time resolved experiments are possible. times are 1-100 seconds although millisecond time resolved experiments below collecting data as a function of concentration is an important quality minimum amount of control sample A at the concentrations same between volume 10 are and recommended. 1 in mg/mL collection data SAXS Typical analysis mg/mL. and 1 at 4 µL 15 is conditions same the under data interpretable for protein same the for data with noise levels below 1% out to out 1% below levels noise with data 5.1. Data collection. schematically in Figs. 22-25. Conceptually the SAXS experiment can be divided into four major collection, data evaluation (Fig. 23), data analyses steps: (Fig. 24), and solution structure modeling (Fig. data 25), each of which is detailed below. other possible data problems. role (Fig. 22). It is difficult toand errors systematic possible overto regard in diligent remaining stressand tests, several by thequality of level the validating paramount importance of collecting high quality SAXS data, necessary to have both strategic concepts and tactical plans for the assessment, processing, and interpretation of interpretation and processing, assessment, the for plans tactical and concepts strategic both have to necessary proceed to basis strategic a provide we Here quality. of regardless sample every from collected results SAXS the essential an play assessments quality data where data, of types different for processing and evaluation data with the steps along the way there is a real danger of generating incorrect results from problematic data. the steps along the way there is a boxes” and are frequently designed using ideal data and results giving judging for few metrics the indicators of and about resolve cannot and problematic can SAXS data. information Without what of a understanding clear poor images in the diffraction experiment and thereby indicate data collection and processing will not be fruitful. be not will processing and collection data indicate thereby and experiment diffraction the in images poor In contrast, every solution sample gives scattering that could be processed results are and valid. Both SAXS and modeled crystallographic analysis packages are whether becoming more and or more like “black not the uncertainty. uncertainty. In particular, some crystallographers feel that they have biased their results in a certain As way. mentioned throughout this article, in contrast SAXS to suffers from crystallography, few standardized ways of validating results. In crystallography, visibly attractive but poorly ordered crystals will immediately result in our goals in the stewardship of this beamline has been to encourage the larger community of crystallographers to crystallographers of community larger the encourage to been has beamline this of stewardship the in goals our excited are crystallographers Typically process. this in pitfalls several demonstrated has Experience SAXS. use from results initial the about of One hour). an (under quickly endstation crystallography a and SAXS a between interconvert to Source Light 5. Strategy and tactics for SAXS experiments tactics for 5. Strategy and experiments. experiments. We have used DLS heterogeneous flexibility, as intrinsic with dealing to approaches based empirically Our an monodispersity. maximum important pre-screening tool to optimize samples below. are presented in section 5 and multidomain proteins conformations, aggregation, and buffers for homogeneity of samples. We homogeneity have of also samples. We found dynamic light scattering (DLS) useful for providing overall size and sample polydispersity measurements under the same concentrations and buffer conditions as for SAXS to ensure monodispersity prior to any SAXS modeling. Native gels can be very useful as an indicator of the Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 model of the ab initio ranges than smaller than ranges q model of the SAM-riboswitch (red) superposed on the ab initio Batey, 2006)) (green and cyan). Average 2006)) & (green Batey, Average and cyan). 59 range over which data may be collected in one single exposure. single one in collected be may data which over range q , and aggregation, lower sample concentrations will be required. The amount of sample required also Many SAXS instruments can adjust the adjust can instruments SAXS Many SAXS scattering from RNA samples. A. Experimental scattering curves of SAM-riboswitch at 1 mg/mL (red) and a 95-bp max , D G This may be done by adjusting the sample to detector distance, the incident wavelength, or the offset of the (Svergun, 1999) were transformed into wireframe model using SITUS (Wriggers et al., 1999). 1999) were transformed into wireframe model using SITUS (Wriggers (Svergun, A are A shown with the maximum value normalized to 1. Average C. theoretical dimer of the crystal structure (PDB id 2gis; (Montange DAMMIN with reconstructed models beads The beads). (blue of model DAMMIN single on superposed wireframe) (blue RNA 95-bp Fig. 26. molecule RNA at 0.7 mg/mL (courtesy of Robert Rambo) in comparison to lysozyme 2.5 mg/mL (gray). For better comparison the curves have been normalized by concentration and molecular weight. B. The calculated P(R) functions for the samples shown in varies with incident beam size, wavelength and whether static or flow cells are used. static or flow cells are used. beam size, wavelength and whether varies with incident the same mg/mL concentration, larger macromolecules have stronger scattering in smaller in scattering stronger have macromolecules larger concentration, mg/mL same the macromolecules. If small angle information is desired from larger macromolecules for determination of mass, R below the critical micelle concentration is required macromolecular for size most standard analysis. Similarly, is also very important. The total scattering from the same mg/mL concentrations of a small macromolecule is At the angular distribution of scattering is not the same. macromolecule; equivalent however, to that of a larger equivalent signals. Detergents, which are used commonly for and stabilizing purifying hydrophobic molecules, signals. equivalent Detergents, can also be problematic, as detergent micelles scatter very well. Collecting data at detergent concentrations which which have greater scattering contrasts can decrease the osmolites scattering in contrast buffers with macromolecules. Salt signals (Fig. 26). Similarly, than protein, require lower for buffers salt low than concentrations sample higher sample require will NaCl, M 1 above particularly concentrations, concentrations for equivalent Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

q max data, q and D G from Guinier from = 22.1 Å and Å 22.1 = G G <1.3 for globular samples and G R min q = 250.0 Å and requires a minimum minimum a requires and Å 250.0 = data for extremely noisy high q max ; however, for determining R determining for however, ; is required. For complexes, large such as the max max data can be very valuable, especially for particle q /D ≤π/D π min q ≥2 = 80.8 Å and D and Å 80.8 = max G is range can require a prohibitive amount of time and one q G q , which is readily achievable in most default SAXS station 60 -1 of over the time frames in which samples are measured and the is typically not a problem for normal geometries; however, for is typically not a problem for normal geometries; however, q -4 -1 or 10 -range necessary to accurately determine values for both R region. While high region. -3 q q ≥0.025 Å max , a maximum q max fold higher intensity than the scattered X-rays. Stray rays of the primary beam MutS tetramer has a R a has tetramer MutS of less than 0.059 Å 10

min q data by increasing the sample-to-detector distance or decreasing the incident X-ray E. coli E. necessary for determining R determining for necessary q q (Mendillo et al., 2007), which may require more careful experimental setup. In contrast, in contrast, In setup. experimental careful more require may which 2007), al., et (Mendillo -1 values inherently provide more information and give structural reconstruction algorithms more algorithms reconstruction structural give and information more provide inherently values q and I(0) as a function of total also X-ray routinely exposure. check We samples before and after G One of the first quality checks that must be performed during data collection is to determine the sensitivity the determine to is collection data during performed be must that checks quality first the of One Much like UV spectroscopy, the scattering from the buffer must be subtracted from the sample. There are There sample. the from subtracted be must buffer the from scattering the spectroscopy, UV like Much Larger Larger Properly Properly choosing beamline geometries to collect the appropriate scattering information is important. <0.8 for elongated samples) can be more constrictive than the theoretical constraints. For large complexes, large For constraints. theoretical the than constrictive more be can samples) elongated for <0.8 G = 52.0 Å, requires a R max min deal with samples that have inherent Cooling radiation the sensitivity. sample, diluting the sample, and adding free radical scavenging compounds or protectants like glycerol have helped prevent aggregation (Kuwamoto, increasing R longer exposures even for samples that pass initial screening of radiation sensitivity. Several options exist to of the sample to X-ray irradiation damage. Changes between SAXS profiles collected by multiple short exposures short multiple by collected profiles SAXS between Changes damage. irradiation X-ray to sample the of can indicate radiation sensitivity. Additionally, samples with radiation damage tend to aggregate, showing Fifth, correction for the difference in X-ray absorption by the sample and the buffer can be applied, although the although applied, be can buffer the and sample the by absorption X-ray in difference the for correction Fifth, correction is small and is commonly ignored. X-ray beam can easily be larger than 10 cuvette is washed. Thus, the resulting scattering must be normalized by the intensity of the incident X-rays. the same cuvette, which has windows made of a material that scatters X-rays such poorly, as mica, kapton, or beryllium. Fourth, the fact that the identical cuvette is used for the sample and blank prevents them from being measured Even simultaneously. at relatively stable synchrotron X-ray sources, changes in the intensity of the can cause large differences relative to signal at high resolutions. Thus, the sample is typically exchanged into to Third, chromatography. size-exclusion or dialysis by experiment the to prior blank a as used is that buffer the avoid systematic errors due to small in differences X-ray path lengths, the samples and blanks are measured in can dominate the scattering. Often some portion of these stray rays are present during data collection but can be can but collection data during present are rays stray these of portion some Often scattering. the dominate can concentration, salt of difference micromolar a even differences, buffer small Second, buffer. the with subtracted typically on the order of a meter, and the incident X-ray beam must be tightly collimated and focused and carefully carefully and focused and collimated tightly within be a must few beam degrees) X-ray of incident the the and incident beam. meter, a In of order order to the measure on the typically data, the sample to detector distance is blocked, as it typically has 10 several technical challenges in the subtraction for SAXS. First, the signal in SAXS data lies very close (typically (typically close very lies data SAXS in signal the First, SAXS. for subtraction the in challenges technical several reconstruction, care should be taken so as not to sacrifice accurate low samples. especially in the case of X-ray sensitive information to fit which helps constrain the resulting solutions (Fig. 23). Unfortunately, since scattering intensity intensity scattering since Unfortunately, 23). (Fig. solutions resulting the constrain helps which fit to information falls off higher rapidly, concentrations (which may not be possible) and/or longer exposures may be required for adequate signal to noise in the high small proteins this limit should be checked. small proteins this limit should be of less than 0.013 Å 0.013 than less of order to sufficiently constrain D this required limit MutS tetramer, D the whereas geometries, q getting sufficiently low R with protein maltose-binding well-folded small the example, For required. be may wavelength while obtaining sufficient information for beingable touse thedata forstructural modeling purposes (Fig. 23). minimum the theory In plots (section 2.3.2), multiple points are required and the linear region ( The first consideration is to capture the detector detector relative to the direct beam. Changing the required. is often configuration and well-calibrated stable Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 ) 2 q )) vs. q with increasing protein q values where aggregation is q (Table 1), which can be determined be can which 1), (Table G indicates the presence of aggregation. G Often the first parameter extracted from data from extracted parameter first the Often One of the attractive features about SAXS <1.3/R evaluation is R is evaluation from the slope in a Guinier plot (log(I( (Fig. 23). A Guinier plot can be easily generated from the raw data using the Nonlinear behavior in the Guinier plot in the range program PRIMUS. q The scattering from the the entire aggregation dataset and any influences further data processing should proceed with caution. Some samples may show non-linearity over a small region Data within the linear. is data remaining the and region Guinier clipped at the lowest an intermediate concentration, higher concentration higher concentration, intermediate an A 23). (Fig. fruitful be to unlikely are measurements dilute to be might material available the of use better the concentrated samples into different buffers to scattering. screen for conditions that give better using buffer blanks measured both before and after from collected Data identical. be should sample the “de-smearing” require setups SAXS of types several required not is This properties. beam incident to due point focusing at the detector. for sources with 5.2. Data evaluation. data collection particularly is This is collection. data during performed that data evaluation useful can be with limited example if time samples shows signs of and/or aggregation at material. For 61 ). For example, decreasing intensity at very small -1 <0.1 Å q Characteristic scattering of aggregates in SAXS. A fully With With ideal samples, scattering profiles from a concentration gradient should be superimposable when For samples without radiation damage, individual frames can be averaged. Background/buffer scattering scattering Background/buffer averaged. be can frames individual damage, radiation without samples For calculated from the Guinier plot and from the P(r) function by function P(r) the from and plot Guinier the from calculated G concentration indicates the presence of repulsion forces. In these cases, the observed scattering is interactions and treated particle) as the of a factor” “form (the solution ideal in particle the of curve scattering the of product additional and unwanted correlations in solution. These correlations affect the scattering profile, usually in the lowest resolution region ( scaled by concentration. In this case individual scattering particles are not interacting with one another 23). (Fig. In some cases target macromolecules interact with one another either repulsively or attractively adding filtration should be attempted to remove the aggregation for further analysis as the aggregation may have subtle have may aggregation the as analysis further for aggregation the remove to attempted be should filtration throughout the scattering profile (Fig. 27). effects most apparent may be processed further. However, when possible, varying buffer conditions, centrifugation and centrifugation conditions, buffer varying possible, when However, further. processed be may apparent most indirect Fourier transformation. Passage of the sample through a filter a through sample the of Passage transformation. Fourier indirect removes the aggregated material and with allows a an 100 kDa cutoff aggregation-free scattering curve to be collected (green line). affected, affected, and this type of scatter can be observed low both resolution shape of the scattering through and disagreement between I(0) the R and of features in the scattering curve can also be observed with unfolded samples. Unfolded and aggregated samples using can a be Kratky distinguished plot. A red. partially In this scattering curve, aggregated only the lowest resolution sample scattering is is shown in Fig. 27. aggregated sample is shown in black. This aggregate has no features in the scattering curve and indicates a poorly behaved sample. Lack should be measured both before and after the measurement of the sample. Scattering due to macromolecules is then calculated by subtracting the buffer blanks from the sample. The resulting scattering curve calculated Akiyama & Fujisawa, 2004). Flow cells can also significantly improve data collection for very sensitive samples, samples, sensitive very for collection data improve significantly also can cells Flow 2004). Fujisawa, & Akiyama although flow cells can require much more dataAlternatively, material. frommultiple short exposurescan be on a fresh sample. is performed where each exposure summed Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 to reconstruct

and I(0) can be G 1/3 , excluded volume, and ) ] max is substantially than larger π G ) * (3/4 , I(0), D 3 G can be calculated by: G shape determination can be applied ) lacking bell shaped peaks and having a plateau or a q Ab initio vs. 62 2 q ) q [ MW(Daltons)/(0.44 Daltons/Å 1/2 = (3/5) values (section 2.3.2, Fig. 24). For proteins, if R q G, globular protein R shape-determining programs are known to provide accurate solutions given the correct models. shape restoration programs allow further constraint on the solution by enforcing known Ab initio ab initio Ab initio Determining the oligomeric state requires a determination of molecular weight, such as methods relying methods as such weight, molecular of determination a requires state oligomeric the Determining If a solution of folded macromolecules is monodisperse, then the measured scattering profiles can be Extended and globular macromolecules can be distinguished by P(r), as globular particles have bell- For SAXS data that is free of aggregation, radiation damage, and long-range interactions, data analysis for the indirect Fourier transformation or other problems such as heterogeneity or unfolding. such as heterogeneity or transformation or other problems for the indirect Fourier max symmetry. symmetry. models. final of resolution the improves symmetry higher enforced correctly fact, In 25). Fig. 4; (Table symmetry

the solution. Multiple repetition of the modeling process significantly decreases the risk of inferring erroneous of map probability a computes and uniformity, the checks averages, aligns, DAMMAVER program The shapes. the given protein is known, the program GASBOR shapes anisotropic other and cavities of has generation the prohibiting not of and models the the constrain that penalties advantages of providing significantly more sophisticated of stability the verify to important extremely is reconstructions multiple of Comparison 2001b). al., et (Svergun the low resolution envelopes of the protein if atomic structures are not available (section 4.3). For this purpose programs DAMMIN, DALAI_GA, SAXS3D, or GA_STRUCT can be used. If the number of residues of the used for solution structure determination (Fig. 25). shapes and is particularly problematic for small proteins. shapes and is particularly problematic modeling and interpretation. 5.4. Structure dividing the excluded volume derived this way by 2. This accounts for the hydration layer as well as the protein the as well as layer hydration the for accounts This 2. by way this derived volume excluded the dividing volume. This estimate holds up well for large globular proteins (>70kD) but fails for proteins with unusual upon I(0) (section 2.3.5). Most often the monomeric molecular weight is known and non-integral stoichiometries non-integral and known is weight molecular monomeric the often Most 2.3.5). (section I(0) upon from I(0) imply heterogeneous multimerization. The excluded volume calculated through the Porod invariant can be converted to a mass (Table 1; section 2.3.5). For proteins, a rule of thumb is to estimate the mass by conformation or potentially an oligomer (Fig. 24). The theoretical R (Fig. 24). conformation or potentially an oligomer they also possess characteristic Kratky plots (I( slowly increasing curve at large extended an in likely is protein the then weight molecular same the of protein globular a for value theoretical the extended tail (Figs. 5 and 10). Unfolded samples not only have P(r) functions consistent with extended molecules, shaped extended functions, with consistent whereas functions extended P(r) have particles only have not samples functions Unfolded with 10). and a 5 maximum (Figs. at tail extended short distances and a long P(r) (section 2.3.3; Table 1, Fig. 24). Disagreement between the values may be a sign of improper assignment of assignment improper of sign a be may values the between Disagreement 24). Fig. 1, Table 2.3.3; (section P(r) D molecular weight is the first stepfunction, pair-distribution the of calculation through or plot Guinier the in reconstructionusing curve SAXS the from determined of the solution structure (Table 1). R may proceed (Fig. 24). Determination of the experimental parameters R increasing salt concentration. increasing 5.3. Data analysis. these curves can be used to extrapolate the scattering to infinite dilution. Long-rangeby interactions electrostatic interactionsscreening by canas such also be buffer, the changing by experimentally diminished or removed between particles (the “structure factor” of the solution). An interference-free scattering due to the “form factor” “form the to due scattering interference-free An solution). the of factor” “structure (the particles between increases concentration the Lowering concentrations. different three least at measuring by extracted be then can Thus, interactions. interparticle of effects and strength the decreases and particles between distance average the Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 . max values derived from derived values 2 χ agreement to evaluate the 2 χ for well behaved P(r) max but also assigned D G with multiple methods (Table 1) with multiple methods (Table Assessment G or atomic models agreement with experiment or atomic models agreement with programs will often find shapes that also fitthe scattering 63 ab initio ab initio Atomic model validation Atomic model validation Validates Validates of modeling stability and convergence Verifies Atomic model validation Non-linear behavior indicates aggregation or inappropriate q-range. Non-linear behavior indicates aggregation Consistency of extracted R R increases confidence in not only weight and concentration Should correlate with molecular D Proper description of the range of functions in P(r) may indicate High frequency oscillations or discontinuities problematic Fourier transform process. Range must be suitable through the entire spatial resolution through the entire spatial Range must be suitable models required for determined shapes and atomic resolution models may be superimposed by using the using by superimposed be may models resolution atomic and shapes ) 2 χ Ab initio Ab G model Table 5. SAXS parameters for data validation and interpretation data validation for SAXS parameters 5. Table max model

G -range max model R I(0) P(r) D NSD R P(r) q D Parameters Guinier plot Experimental Structure modeling One possible reason for disagreement between atomic resolution models and SAXS results is the presence the is results SAXS and models resolution atomic between disagreement for reason possible One If the atomic structure of the sample is known or an atomic model has been proposed, comparison of Goodness of fit (R or <3.0 are not uncommon. not are <3.0 2 χ of heterogeneous multimerization. This may be apparent if the molecular weight is determined. Nevertheless the Nevertheless determined. is weight molecular the if apparent be may This multimerization. heterogeneous of of programs SUPCOMB or SITUS. theoretical SAXS profile can be calculated with the using program the CRYSOL, data SAXS experimental and structures crystal from curves SAXS theoretical between Agreement models. best theoretical SAXS profile with the experimental data is the first step in the structure evaluation (Fig.25). The the SAXS data alone (Fig. 16) may not be generally applicable, especially for less anisotropic assemblies. assemblies. for less anisotropic especially generally applicable, 16) may not be data alone (Fig. the SAXS Unfortunately Unfortunately given incorrect symmetries Assumingprofiles. higher symmetrywith knowledge of only the molecular weight must be carefully justified. and NSD comparing by channel potassium the of symmetry four-fold the of Identification Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 value should G modeling of the missed linker may also be calculated from the P(r) the from calculated be also may G shape restorations are also very useful very also are restorations shape can be validated by calculating the P(r) the calculating by validated be can and/or by calculating the P(r) function P(r) the calculating by and/or ab initio max max ab initio ab , but also to compare multiple modeling runs (Fig. 25). 2 χ 64 )=0 (section 2.3.3). I(0) and R and I(0) 2.3.3). (section )=0 max is greater than the expected D expected the than greater is may not be possible and a possible symptom of heterogeneity (Fig. 24). max max value calculated if necessary (section 5.2). The P(r) function and derived G In solution structure modeling from SAXS measurements, it is important not only to validate the fit of the of fit the validate to only not important is it measurements, SAXS from modeling structure solution In The first crucial parameters for SAXSdata validation comes from the linearity of the Guinier plot and In preparing or evaluating publications with SAXS results, measurements and analyses it is critical to If the sample is a multimeric assembly and atomic models of all subunits are available but arrangement Molecular weight determination using I(0) is directly related to the excluded volume, and if SAXS is value provide a second set of important parameters. If the P(r) function gives an extremely elongated value calculated from this plot (Fig. 23-24). Ideally the concentration dependence of the R max G calculated to the experimental SAXS data using R or underdetermined of interpretation over of risk the decrease significantly process modeling the of trials Multiple

24). These calibrations are particularly useful in identifying sub-stoichiometric amounts of aggregation or mixed or aggregation of amounts sub-stoichiometric identifying in useful particularly are calibrations These 24). assembly states that need to be analyzed and modeled as mixed populations and not homogenous samples that can be easily used for low resolution structure reconstruction. used to determine the oligomerization state, then accurate calibration of I(0) is necessary (section 2.3.5; Fig. without the constraints that enforce P(D enforce that constraints the without plot. function and should correspond to those derived from the Guinier tail, accurate characterization of D D suspect. be may data SAXS these of interpretation structural Further r where range r larger the in function R be presented and a corrected R D

we suggest here some empirical criteria with concerned studies in presented be should to information this propose consider We 5). (Table for results SAXS interpreting SAXS data quality control and to help avoid of biological macromolecules. SAXS solution structure modeling over SAXS data analyses will be important for judging accurate SAXS measurements and larger modeling, community and in for the the interpretation and publication of SAXS data. Whereas universally control parameters accepted for quality SAXS results comparable to those currently used for crystallography do not yet exist, have metrics identifying the reliability of extracted information. In the long term, establishing standards for 5.5 Criteria for evaluation of SAXS results evaluation of SAXS results 5.5 Criteria for are evaluated in terms of the goodness of the fit of their calculated scattering curves to the experimental data. and modeling body rigid from generated models of Comparison results (Fig. 25). in establishing the validity of the three-dimensional three-dimensional structures of multidomain assemblies based on multiple scattering data sets from deletion mutants when the structure(s) of individual domains are available. Other can conformational be applied sampling 3; (Table Fig. methods 25) to model the quaternary structure of multidomain assemblies. Final models regions, a combination of rigid body modeling of the subdomains and using flexible chains of interconnected residues canbe The applied. program BUNCH allows determination of is unknown, then rigid body modeling can be applied (Table 2; Fig. automatically fully or 25). MASSHA Quaternary program the using structure manually performed be modeling can data SAXS the of against complex a using program If SASREF. the full-length protein represents an assembly of the domains connected by linker OLIGOMER may improve the final fit. may improve OLIGOMER models and SAXS data are flexibility and truncatedloops regions (common are not built in crystal into structures models). Generating disordered where a variety of conformations of the flexible regionsand using program program OLIGOMER can take the calculated scattering profilesof several proposed multimers and determine and OLIGOMER with fit excellent An data. the fit best to required multimer each of concentration fractional the a poor impliedfit heterogeneous with multimerization. CRYSOL Other possibilities for disagreement between Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

<< sym sets the sets the min max q q is more difficult . Thus, . q per se )) = 2π/ θ /(2sin( λ , and the P(r) functions of the models to the to models the of functions P(r) the and , A SAXS resolution A A A critical issue for both SAXS and or d = 10Å is often possible). The limiting max -1 or d = 1000 Å is often possible). The limiting , D , -1 G curve can described in a manner related to that of 50%). to define,nominal asthe 1) addressed: be to thereneed that resolution are different two resolution of types of the experimental scattering curve, and 2) the effective resolution of the model. The nominal resolution of the experimental scattering resulting structure, even if the exact values of some of values exact the if even structure, resulting of set common a example, For debated. are criteria criteria to define a 2 Å resolution crystal structure would expect over 50% of the data spacings with of 2 Bragg Å to be measured with a signal-to- noise ratio of more than 2 and between good agreement multiply observed reflections (R of accuracy of level the defining is crystallography resolution the is what Essentially, models. resulting of resolution the crystallography, For model? the of data the of resolution the as defined is structure the quantitative Robust structure. the determine to used measures are used to define the resolution of the models. We suggest that at models. We least 10 repetitions are required to obtain an accurate average model for evaluation. Probably the parameter easiest NSD a way use to is to models the validate in differences to validate the stability and convergence multiple of modeling rounds (section 4.3). the For cases of comparisons used, is refinement body rigid where R provide also can values determined experimentally substantial insight into the quality of the as model well as the differences between any atomic pseudo- model and the experimentally scattering observed (section cannot be resolved using SAXS structures 4.2.4-4.2.5). alone. be should important is chirality where Conclusions Handedness carefully examined. of SAXS Accuracy and resolution 5.6 and models experiments, measurements, = 0.6 Å max q = 0.006Å 65 min q and Bragg spacing (d) is d = q /q) of this range. A. The A. /q) of this range. π = 2 ∆ & Hura, 2002), as the between the solution difference buffer and protein solution becomes very The resolution range required to identify structural features The resolution range required to identify due to organization of bulk solvent begins to overpower the signal from the relatively small population of solutes of population small relatively the from signal the overpower to begins solvent bulk of organization to due (Head-Gordon smallest Bragg spacing observable in an experiment ( signal the Å, 10 than smaller spacings Bragg At data. experimental the in noise the is consideration experimental crystallography. The relationship between crystallography. largest dimensions observable in an experiment ( experimental factor is how close reliable data can be determined near the primary beam. Similarly, (16–7 Å resolution) provides information about small structural scattering (40–16 Å resolution) provides information about domain motions, as illustrated by the scattering profiles of human glucokinase without (black; PDB id 1v4s) and with (red) a glucose analog (PDB id The high angle scattering information 1v4t; (Kamata et al., 2004)). C. particle size and overall shape, as illustrated by a monomeric (black) and dimeric (red) assemblies of the extracellular fibrinogen-binding The medium angle protein (PDB id 2gom; (Hammel et al., 2007)). B. ranges are highlighted with blue box and the upper axis of the inset ranges are highlighted with blue box and graphs indicate the spatial resolution ( low angle scattering information (500–30 Å resolution) is determined X-ray profile calculated by CRYSOL (Svergun et al., 1995) for (Svergun X-ray profile calculated by CRYSOL domain different oligomerization states, B. A. different proteins with The resolution fluctuations. structural conformations and C. different Fig. 28. showing the theoretical in SAXS data is qualitatively illustrated Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 model (orange beads) matches beads) (orange model SAXS-based models should be used be should models SAXS-based Tainer, 2007), see for example Fig. Tainer, & ab initio ab Ab initio Ab ); however, the model’s structural details the ); do model’s however, not envelopes. Averaging is needed to construct a -1 ab initio =0.47 Å 66 max q range and data quality, as well as on the molecular size, shape, and shape, size, molecular the on as well as quality, data and range q . q We note that the resolution required in any experiment depends upon the biological question of interest. interest. of question biological the upon depends experiment any in required resolution the that note We The resolution of models derived from SAXS reconstructions is frequently not described in the literature, literature, the in described not frequently is reconstructions SAXS from derived models of resolution The Although the nominal experimental resolution sets a limit on the maximum possible resolution for the & Gosling, 1953b). In cases involving conformational flexibility and transitions, lower resolution results that can be used to describe the flexibility can potentially answer biological questions of interest better than high The lower resolution diffraction from B-form DNA turned out to be more A-form useful DNA diffraction than for definingthe thehigher structureresolution of the double helix (Franklin & Gosling, 1953a; Franklin over determined by the SAXS experiment alone. recommend that publications of SAXS models make explicit the sources of error, intended level of accuracy, and accuracy, of level intended error, of sources the explicit make models SAXS of publications that recommend stated. estimates the for bases the with resolution model estimated sufficiently not generally are they as information, of types other with combination in or design experimental for whether to trust or ignore the finer features of SAXS shapesthem to identify to what andextent additional runs of shape We modeling programs will conformationalenhance their final model. changes as well as enable approaches used for EM, where the solution sets are used to define confidence intervals and error margins for the fitting parameters & (Volkmann Hanein, 2003). The definition of valid and robust measures of resolution will be important advance for SAXS experiments. A true measure of resolution will allow researchers know variance of the single model volume from the averaged volume envelope. As we favor the use of high-resolution of use the favor we As envelope. volume averaged the from volume model single the of variance structures as restraints in the SAXS experiments, another approach would be to define the variability of a fit of known domain structures into the SAXS envelope. This would be analogous to the accurate correlation-based structures are not definedas having any particular resolution, but are rather given a confidence level equivalent by the root mean squared SAXS derivation equivalent from would A the be averaged to structure. examine the processed sets This of assumes data. coherence in Fourier space and random Gaussian error in electron density maps. In principle, this method could be used for SAXS, but validation of this approach is currently a research problem, as the degree of systematic error is not as well defined as itis for EM or crystallography. In NMR, the effective resolution is more difficult and has strong similarities for the determination of the resolutionmodels of derived from electron microscopy (EM). In EM, resolution is estimated by the Fourier shell method, which indicates the highest resolution shells that show agreement in reciprocal space for two independently SAXS model, the resolution of the data does not define the effective resolution of the model. Determining 17. In contrast, flexible multi-domain systems will show broad variations from the averaged structures but can models (Fig. 19). be improved by introducing atomic changes in protein conformation can be quantitatively monitored (Tiede, Zhang & Seifert, 2002). Thus in a well- a in Thus 2002). Seifert, & Zhang (Tiede, monitored quantitatively be can conformation protein in changes questions address to sufficient certainly and impressive be can resolution effective the experiment, SAXS defined Yamagata of conformational state (Chen et al., 2007; Davies et al., 2005; with SAXS data (Fig. 9), and can allow details of the model to more closely match the experimental resolution, experimental the match closely more to model the of details allow can and 9), (Fig. data SAXS with synchrotron- the if general, In 28). (Fig. details structural of types different determining for critical be can which based high-angle scattering is combined with crystal structure even the dynamic aspects of reaction-linked reach this spatial resolution, and the averaged conjunction model in models (blue atomic-resolution of transparent incorporation beads) the through is visualized be at can details an more even Alternately, lower resolution. flexibility (section 4.3.1). For example, averaging of different modelscan eliminate higher resolution details, single The 9. Fig. in shown as such flexible are molecules if particularly the experimental data to highest resolution 13 Å ( does not provide a unique solution, particularly for reasonable model for the observed scattering. The resolution of measured by defined resolution reciprocal the final averaged model depends both on and models are simply termed “low resolution” structures. Part of the difficulty in determining the resolution of SAXS envelopes or SAXS-based models arises from the fact that fitting of the SAXS scattering curve small at larger values of values at larger small Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 Teichmann, & Teichmann, 2001). & Hodge, 1998)). Understanding these 67 Lundblad, 1999). The presence of these scaffolding domains domains scaffolding these of presence The 1999). Lundblad, & & Lundblad, 2001). a Similarly, Cdc13-Est2 fusion protein eliminates the requirement for Combining data from solution scattering with atomic resolution structures holds tremendous promise Decision-making by biological pathways involves dynamic molecular interactions. These interactions The use of both X-ray crystallography and SAXS is poised to be tremendously relevant for addressing Biophysical studies have indicated that macromolecular complexes are formed through local contacts 6.2 Needs and Prospects 6.2 Needs and Prospects exchange, switching of states in chemo-mechanical assemblies, and systems flexingthese by unstructuredof regions.many Studying Thus, challenging. remarkably be can technique biophysical single any by processes these for the combination of SAXS with high-resolution structures and computational techniques. are ideal targets are pathway specific and can include the formation of cooperative ensembles, use of interface mimicry and conformational states these proteins adopt, regardless of whether or not a particular molecule or conformational or molecule particular a not or whether of regardless adopt, proteins these states conformational state is suitable for high resolution structural studies. Diffley, 2007). However, this fusion comes at the cost of the regulation that the cyclin-dependent kinase normally kinase cyclin-dependent the that regulation the of cost the at comes fusion this However, 2007). Diffley, performs in coordinating the yeast cell cycle (reviewed in (Mendenhall types of mechanisms in the context of the native systems will require structural characterization of the various and proteins in normal cells, however, provides the opportunity for additional cellular levels example, fusion of of control. the Sld3 For and Dpb11 proteins bypasses the requirement for the phosphorylation of Sld3 by the budding yeast cyclin-dependent kinase and the phospho-binding BRCT domains of Dpb11 (Zegerman & (Pennock, Buckley (Evans function telomerase for protein Est1 the within complexes and may possess substantial domains are flexibility.dispensable with the Inintroduction of appropriate fusions. For some example, telomeres can be properly cases these scaffoldingmaintained moleculeswhen Cdc13 and and Stn1 are replaced with a fusion of the Cdc13 DNA binding domain onto Stn1 features that control the mechanisms of assembly. Thus, the control of interface shape provides simple structure- simple provides shape interface of control the Thus, assembly. of mechanisms the control that features molecules of existence the predict also studies these Moreover, processes. cellular control to mechanisms based that are primarily required as molecular scaffolds and jigs that bring together appropriate active components dominated by short-range interactions at the binding surface. Shape and chemical complementarity are critical and pathways, 3) to create highly effective local concentrations of components at target sites, allow a degree of self-regulation. pathway coordination, and 5) to 4) to promote between multiple partners are exchanged to coordinate important cellular processes. Dynamic interactions within interactions Dynamic processes. cellular important coordinate to exchanged are partners multiple between the enlarging significantly via probability, recognition 1) to increase respects: in several be important may cells area and three-dimensional size of the 2) target, to reduce disruptive interference among macromolecular steps (Bahattacharyya (Bahattacharyya et al., 2006), but so are the proteins that make them up be to (Apic, activities independent allow proteins Gough individual within and complexes within architectures modular These interactions where roles dynamic playing proteins important are there extreme, the At coordinated. functionally from bio-energy to medicine. The current initiatives in genome sequencing along genome bioinformatics studies have with revealed that not only the are the components of resulting signaling pathways modular whole- the different biological pathways that control cells and offers enormous potential impacts in areas ranging 6. Prospects and conclusions and 6. Prospects and biophysical implications 6.1 General biological of SAXS with a high-resolution method such as crystallography becomes so powerful. becomes such as crystallography method with a high-resolution of SAXS resolution resolution experiments that enforce a single conformation. In general, the great strength of SAXS experiments is that they characterize the overall molecular shape and assembly in solution, and delineate the architectural arrangements needed to place high-resolution structures of This components. is precisely why the combination Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 & Engelman, 68 . However, a perfect analog is as unlikely, the limited number of independent parameters free SAXS appears to offer significant potential advantages for examining the structure of membrane proteins membrane of structure the examining for advantages potential significant offer to appears SAXS We also We anticipate that the use of SAXS in the study of nucleic acids, particularly RNA enzymes and Additionally, we believe that efforts to automate SAXS data collection and processing will be profoundly be will processing and collection data SAXS automate to efforts that believe we Additionally, One issue that would benefit from sustained theoretical and practical study is the development of a mixture all have different electron mixture densities, all complicating have the different analysis. Matching the scattering of the lipid and detergent in SAXS or SANS provides one way to address these multicomponent systems (Bu and NMR experiments, but provide potential advantages for SAXS experiments by providing enhanced contrast enhanced providing by experiments SAXS for advantages potential provide but experiments, NMR and lipid/detergent and proteins, buffer, the that is SAXS for challenge central The complexes. aqueous to compared and size The technology. emerging an still is SAXS of application this although detergents, and bilayers lipid in complexity of membrane protein complexes with detergent and lipids are major challenges to crystallization aid in the discovery of proteins that regulate processes by mimicry of DNA and RNA structures aid (Putnam in et the al., discovery of proteins and RNA that regulate processes by mimicry of DNA 2005). Tainer, 1999; Putnam & conformation, whereas the low energy states determined in macromolecular crystallography have frequently been of inactive conformations. As SAXS provides direct measures of shape in solution, it may furthermore folding algorithms such as MFOLD (Zuker, Mathews & Turner, 1999) and frequently use sequence conservation sequence use frequently and 1999) Turner, & Mathews (Zuker, MFOLD as such algorithms folding and nuclease sensitivity to be validated. Third, RNA activity is often well involves defined by conformational SAXS.switching Fourth, SAXSthat allows the examination of multiple RNA states to define the active (Fang et al., 2000), allowing for useful characterizations of less concentrated samples. Further, SAXS can provide and the can SAM riboswitch SAXS (Fig. 26). scatters RNA X-rays Further, approximately fivesamples. times moreconcentrated stronglyless than proteins of characterizations useful for allowing 2000), al., et (Fang helpful constraints onto the large number of potential secondary and tertiary structures that are predicted by 2007b) al., et (Lipfert riboswitch glycine by illustrated as such important, increasingly become will riboswitches types of technological advances, as our experience has suggested that the ease for which experiments can be and scale of experiments that will be attempted. automated directly controls the type control the formation of active states, or control the folding and separation for allow may system chromatography size-exclusion inline an with conjunction in SAXS using of specific targets. Similarly, the possibility of characterization of samples with dynamic heterogeneity in their assembly states. We are excited about these buffer buffer conditions to control aggregation, titration conditions to understand specificor conformationalsmall changes, molecules from libraries to search for lead compounds that inhibit or promote complex formation, washing and filling take up a substantial fraction of the synchrotron time allocate for experiments. Thus,use the of flow cells, robotics, and temperature-controlled sample holders offer the possibility of streamlining the different include can These conditions. many of screening rapid for allow can and process collection data SAXS important. The requirement of using the same cuvette for buffers and samples typically means that cuvette crystallographic R reference unrefined an as use for number sufficient a extract to unwise it make experiment SAXS a in measured set. solved problem, and we expect that multiple measures might be required depending on the types of modeling that are performed. An equally important statistical measure for evaluating this fitting is an equivalent to the universally accepted SAXS scattering assessment to successfully measure the in the modeling is to be able requirements of the fundamental One factor crystallography. or factors that are equivalent fit of proposed models to to solution scattering As curves. detailed in section the 4.1, we do not consider this to R-factor be a in positioned to become important developments in the near future. in the near developments to become important positioned techniques so that they can be productively applied in the context of this problem. We note that advances continue advances that note We problem. this of context the in applied productively be can they that so techniques are processing and collection data SAXS in areas several that predict we and fields, these of each in made be to for addressing biophysical details of how specific complexes and flexibility challenge has drive been that this combination requires structures, understanding solution biologicalof scattering, biology, and processes. The computational methods. Thus, the goal of this review has been to introduce sufficient detail for each of these Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 (Kwok et (Kwok from the

G Tainer, 2007; Tainer, & & Sligar, 2007). This & Sligar, 2006; Denisov et al., 2004) to encircle 69 One major class of such disease-causing mutations is thought to cause defects in folding and structure, Using SAXS data as a source of experimental restraints for modeling macromolecular flexibility and A A fundamental strength of SAXS is that it provides the overall structure including both architectural as occurs for eukaryotic and microbial Cu, Zn superoxide prion dismutases isoform misfolded (Bourne a into protein et prion cellular al., the of conversion 1996). structural the Similarly examine to basis SAXS a provides et al., 1996). The ability to do rapid SAXS analyses such organisms, different among activities similar maintain will that assemblies different distinguish to experiments also allow effective use of comparative genomics neurodegenerative disease (Deng et al., 1993; DiDonato et al., 2003; Foster et al., 1999; Perry, Fan Perry, 1999; al., et Foster 2003; al., et DiDonato 1993; al., et (Deng disease neurodegenerative to means efficient an provide experiments SAXS 2006). Hickson, & Wu 2003; al., et Williams 2006; al., et Perry (Borgstahl dismutase superoxide mitochondrial the for seen as defects structural cause may polymorphisms if ask predisposition to breast cancer, the DNA damage response protein p53, where mutations are associated with predispositions to numerous cancer types, the with Werners associated are mutations where dismutase, superoxide syndrome enzyme control oxygen reactive the and aging rapid helicase-nuclease where mutations cause which can be readily observed in SAXS. Mutations in this class have already been identified to affect a number a affect to identified been already have class this in Mutations SAXS. in observed readily be can which of medically relevant proteins such as the DNA repair protein BRCA1, where mutations are associated with identified in humans cause structural defects in macromolecules. the information needed to place high-resolution Thus, structures of components in macromolecular complexes. SAXS has the potential for addressing many important believe We that issues SAXS can play an in important role in fundamental identifying which of biology the over 10 and million polymorphisms human disease. arrangements and conformations in the 50 to 10 Å resolution range in near physiological conditions to yield 6.3. Expectations and Predictions. 6.3. Expectations and Predictions. new weak-field aligned NMR and fluorescence experiments that can probe samples in the biologically interesting biologically the in samples probe can that experiments fluorescence and NMR aligned weak-field new millisecond time frame. With appropriate resources for directed efforts, SAXS in macromolecular interactions with widespread impacts. experimental data on flexibility can provide complementary spectrometry, which can approach single residue resolution as shown for the photocycle changes of photoactive photoactive of changes photocycle the for shown as resolution residue single approach can which spectrometry, with along player important an become to positioned well is SAXS Thus, 2006). al., et (Brudler proteins yellow highest intensity region of the scattering curve during the folding of RNAs and proteins (for example, (for proteins and RNAs of folding the during curve scattering the of region intensity highest al., 2006) and (Uzawa et al., natural A 2006)). complement to the global shape and conformation from SAXS will be residue level information from advancing techniques of enhanced hydrogen/deuterium exchange mass unlike traditional NMR and fluorescence experiments, is not affected by the molecular rotation times, so time- resolved SAXS can be performed in an equivalent manner to the traditional static experiments. Millisecond- resolved SAXS experiments performed to date have primarily focused on following changes in R samples can provide important experimental feedback, and very require measurements Time-resolved could experiments. time-resolved by characterized changes be conformational usefully extended to include dynamic high X-ray flux and fast detectors designed for rapid electronic shuttering. Both are now available, and SAXS, computational docking is an exciting and relatively underdeveloped possibility. SAXS data from appropriate While all of these current approaches will benefit from and detergents. membrane proteins in lipid bilayers to aid in the characterization of potential of SAXS further use and development, they demonstrate the the hydrophobic lipid/membrane protein complexes to form “nano-disks” Atkins (Nath, varying by controlled be can size whose particles homogeneous and monodisperse yield potentially can system lipo-protein size; however, sample preparation and control of particle homogeneity are non-trivial challenges. of more closely matched blank samples (Caffrey, 2000; Lunde et al., 2006). Another approach use has of been the mutant lipid-carrying proteins (Bayburt, Grinkova 1999). SVD has also been used to extract the scattering profile of protein-detergent complexes at concentrations concentrations at complexes protein-detergent of profile scattering the extract to used been also has SVD 1999). above the critical micelle concentrations (section 4.4.1; (Lipfert et al., 2007a)). Embedding membrane proteins in cubic lipid phases can also control the nature of the lipid/detergent systems allowing for the generation Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 70 SAXS is an effective and important complement to crystallography as it can provide information on every on information provide can it as crystallography to complement important and effective an is SAXS Results from the structural genomics initiatives provide compelling evidence for the utility of advanced A second major class of disease-causing mutations is likely to localize to macromolecular interfaces. Since interfaces. macromolecular to localize to likely is mutations disease-causing of class major second A throughput structural analyses, it is tragic that SAXS, throughput technique, which has been certainly neglected has by comparison. the The investment capacity in to SAXS is be surprisingly small, a as true high- results provide an efficient and powerful way to identify experimentally testable models for macromolecular interactions and conformations in solution. With the millions invested in crystallography to develop high- sample, faster data collection than EM or NMR, and native structural analysis in solution. Furthermore, SAXS basis to identify structural similarity even cross-genomic of in relevance the the validate also absence and data structure of crystal phasing sequence for efforts homology replacement molecular and thereby direct efficient structural modeling. of conformational landscapes, docking, denatured proteins, and the folding of proteins and RNA, which should which RNA, and proteins of folding the and proteins, denatured docking, landscapes, conformational of experimental an provides furthermore SAXS predictions. and simulations computational of accuracy the improve structural characterization of membrane proteins in their hydrophobic environments with the added advantage that lipids and detergents scatter less than protein. Their contribution to the scattering may be appropriate minimized choice by of buffer. SAXS provides appropriate experimental feedback for computational modeling 2.5 to 3.5 Å diffraction data. Similarly, for RNA and multi-domain proteins that undergo functionally important functionally undergo that proteins multi-domain and RNA for Similarly, data. diffraction Å 3.5 to 2.5 conformational changes, SAXS can test and validate active conformational states in membrane bound proteins solution. that remain difficult to crystallize, SAXS providesFor a possible general solution to the the many solved, the best use of available SAXS experimental from information models atomic accurate will obtain to constraints incorporate stereochemical residue of these use the to domains analogously information, as external details of biological pathways will require the integration of overall shape, conformation, and architecture, including unstructured regions and those structures that may techniques, not like SAXS. SAXS can define the adopt single states suitable for high-resolution structural studies. As more folded domain structures are being SAXS technologies. These efforts indicate that about half of all eukaryotic proteins have unstructured regions of over 40 residues in length and many are at least partially unfolded without their Given specific protein the partners. ubiquity of flexibility in macromolecules, it is almost certain that decipheringmechanistic the region (Tsutakawa et al., 2007). Thus, the characterization of static and dynamic be equally complexes or more by important SAXS than high-resolution may structures of component active sites for understanding the in cell biology and human health. implications of the genome project difficult to identify by other techniques. For example, SAXS shape predictions for PCNA accurately predicted a predicted accurately PCNA for predictions shape SAXS example, For techniques. other by identify to difficult trimeric ring assembly while also repair allowing accurate prediction of the folded region of a full-length DNA unstructured large a of truncation and contacts crystal from resulting alterations structural including glycosylase a third component such as DNA such as by the Rad51 nature altered the resolve clearly can that information additional provide can polymerization SAXS from derived arrangements interface (Shin et al., 2003). Atomic of these assemblies, such as distinguishing between open trimeric versus dimeric PCNA rings that are more mimicry and exemplified by the PCNA peptide binding surface (Chapados et al., 2004; Dore et al., 2006; Pascal 2006; al., et Dore 2004; al., et (Chapados surface binding peptide PCNA the by exemplified and mimicry et al., 2006; Sakurai et al., 2005) or by exchange of similar interactions by multiple proteins as controlled by these interfaces involve more residues than active site regions, more polymorphisms will likely alter interface residues. Polymorphisms that cause these surfaces to become altered or disrupted will be readily identifiable with SAXS. An important interface type involves acceptor sites that can bind multiple partners by interface the right problems, SAXS can be an important tool for identifying lead compounds. lead compounds. tool for identifying can be an important problems, SAXS the right stabilize stabilize the native shape and assembly. SAXS offers obvious powerful advantages for identifying even low affinity small molecule binders without specific requiring assays.any labeling For or the development of target shape shape that causes human disease (Redecke As et SAXS al., has 2007). intrinsic advantages for examining both also but defects, such of characterization for technology critical only not offers it solution, in shape and folding an efficient and powerfulmeans of drug discovery by SAXS-based screening for compounds that bind to and Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07 71 We thank Betsy Goldsmith, Lorena Beese, Wei Yang, Anna-Maria Hays Putnam, Meindert Lamers, Marc Marc Lamers, Meindert Putnam, Hays Anna-Maria Yang, Wei Beese, Lorena Goldsmith, Betsy thank We grant CA92584. J.A.T. research on flexible protein interactions and assemblies is supportedgrants AI22160, inCA81967, partCA104660, byand NIH GM46312. C.D.P. was funded as a Robert Foundation. Damon Runyon Cancer Research Black fellow by the DE-AC02-05CH11231 DE-AC02-05CH11231 with the U.S. Department of Energy. Efforts to apply characterize SAXS eukaryotic pathways relevant to human cancers are and supported in part by National Cancer crystallography Institute to shapes and conformations in solution at the SIBYLS beamline at Lawrence Berkeley National Laboratory are supported in part by the DOE Number Contract under program (MAGGIE) Efficiently Integrated Genomics and Integrated Genes Assemblies Molecular Diffraction program Analysis Technologies (IDAT) and the DOE co-workers and in particular Jane Tanamachi, to contributions for Fierobe Henri-Pierre and Rambo Robert Richard Tsuruta, Hiro Ellenberger, Kolodner, Tom Brechot, Receveur Marc Mendillo, Scarlet Shell, Veronique the projects described. X-ray scattering and diffraction technologies and their applications to macromolecular Mendillo, Ken Frankel, Susan Tsutakawa, Elizabeth Getzoff, Niels Volkmann, Michael Wall, Eva Scott Nogales and Brown, Brian Chapados for comments on the manuscript. We are grateful to our collaborators and 7. Acknowledgements Acknowledgements 7. these SAXS tools and technologies evolve and become widely adopted, we expect that they will be applied in pushing in role active an play to also but biology structural in problems existing solve to only not to ways novel of research in structural biology. the cutting edge new experimenters to use SAXS and interpret SAXS data in their biophysical experiments. In the next decade, any funding in SAXS will more than pay for themselves in real results and substantial technological advances that address important unsolved problems in medicine, biology, and nanotechnology, Thus, biotechnology. as and the modeling of structures using a combination of SAXS and atomic resolution technological and computational infrastructure structures. is currently in place, which substantially lowers Much the barriers for of the SAXS can evaluate samples and to aid in the efficient optimization of constructs for crystallography. However, However, SAXS can evaluate samples and to aid in the efficient optimizationof constructs for crystallography. given the limited funding for SAXS in the US, it is no surprise that much of the development and software for SAXS has been accomplished in Europe this and is Japan. Yet, an exciting time in the development of SAXS Putnam, Hammel, Hura, Tainer: Submitted to Quarterly Reviews in Biophysics 9/17/07

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