Constrained Maximum Likelihood Estimation of Relative Abundances of Protein Conformation in a Heterogeneous Mixture from Small Angle X-Ray Scattering Intensity Measurements

A direct approach to estimate the anisotropy of protein structures from small-angle X-ray scattering

In the field of small-angle X-ray scattering (SAXS), the task of estimating the size of particles in solution is usually synonymous with the Guinier plot. The approximation behind this plot, developed by Guinier in 1939, provides a simple yet accurate characterization of the scattering behavior of particles at low scattering angle or momentum transfer q, together with a computationally efficient way of inferring their radii of gyration R G. Moreover, this approximation is valid beyond spherical scatterers, making its use ubiquitous in the SAXS world. However, when it is important to estimate further particle characteristics, such as the anisotropy of the scatterer's shape, no similar or extended approximations are available. Existing tools to characterize the shape of scatterers rely either on prior knowledge of the scatterers' geometry or on iterative procedures to infer the particle shape ab initio. In this work, a low-angle approximation of the scattering intensity I(q) for ellipsoids of revolution is developed and it is shown how the size and anisotropy information can be extracted from the parameters of that approximation. The goal of the approximation is not to estimate a particle's full structure in detail, and thus this approach will be less accurate than well known iterative and ab initio reconstruction tools available in the literature. However, it can be considered as an extension of the Guinier approximation and used to generate initial estimates for the aforementioned iterative techniques, which usually rely on R G and D max for initialization. This formulation also demonstrates that nonlinearity in the Guinier plot can arise from anisotropy in the scattering particles. Beyond ideal ellipsoids of revolution, it is shown that this approximation can be used to estimate the size and shape of molecules in solution, in both computational and experimental scenarios. The limits of the approach are discussed and the impact of a particle's anisotropy in the Guinier estimate of R G is assessed.

Direct shape determination of intermediates in evolving macromolecular solutions from small-angle scattering data

Many important biological processes like amyloid formation, viral assembly etc. can be monitored in vitro. Small-angle X-ray scattering (SAXS) is one of the most effective techniques to structurally characterize these processes in solution. For monodisperse systems and some oligomeric mixtures, low-resolution shapes can be determined ab initio from the SAXS data, but for evolving systems, such analysis is hampered by the presence of multiple species and no direct reconstruction procedures are available. The authors consider a frequently occurring case where the scattering from the initial and final states of the process are known but there exists a major (unknown) intermediate component. A method is presented to directly reconstruct the low-resolution shape of this transient component together with its volume fractions from multiple scattering patterns recorded from an evolving system. The method is implemented in the computer program DAMMIX freely available to academic users and its effectiveness is illustrated in several synthetic and experimental examples.

What Can We Learn from Wide-Angle Solution Scattering?

Extending collection of x-ray solution scattering data into the wide-angle regime (WAXS) can provide information not readily extracted from small angle (SAXS) data. It is possible to accurately predict WAXS scattering on the basis of atomic coordinate sets and thus use it as a means of testing molecular models constructed on the basis of crystallography, molecular dynamics (MD), cryo-electron microscopy or ab initio modeling. WAXS data may provide insights into the secondary, tertiary and quaternary structural organization of macromolecules. It can provide information on protein folding and unfolding beyond that attainable from SAXS data. It is particularly sensitive to structural fluctuations in macromolecules and can be used to generate information about the conformational make up of ensembles of structures co-existing in solution. Novel approaches to modeling of structural fluctuations can provide information on the spatial extent of large-scale structural fluctuations that are difficult to obtain by other means. Direct comparison with the results of MD simulations are becoming possible. Because it is particularly sensitive to small changes in structure and flexibility it provides unique capabilities for the screening of ligand libraries for detection of functional interactions. WAXS thereby provides an important extension of SAXS that can generate structural and dynamic information complementary to that obtainable by other biophysical techniques.

Consensus Bayesian assessment of protein molecular mass from solution X-ray scattering data

Molecular mass (MM) is one of the key structural parameters obtained by small-angle X-ray scattering (SAXS) of proteins in solution and is used to assess the sample quality, oligomeric composition and to guide subsequent structural modelling. Concentration-dependent assessment of MM relies on a number of extra quantities (partial specific volume, calibrated intensity, accurate solute concentration) and often yields limited accuracy. Concentration-independent methods forgo these requirements being based on the relationship between structural parameters, scattering invariants and particle volume obtained directly from the data. Using a comparative analysis on 165,982 unique scattering profiles calculated from high-resolution protein structures, the performance of multiple concentration-independent MM determination methods was assessed. A Bayesian inference approach was developed affording an accuracy above that of the individual methods, and reports MM estimates together with a credibility interval. This Bayesian approach can be used in combination with concentration-dependent MM methods to further validate the MM of proteins in solution, or as a reliable stand-alone tool in instances where an accurate concentration estimate is not available.

Effects of Catalytic Action and Ligand Binding on Conformational Ensembles of Adenylate Kinase

Crystal structures of adenylate kinase (AdK) from Escherichia coli capture two states: an "open" conformation (apo) obtained in the absence of ligands and a "closed" conformation in which ligands are bound. Other AdK crystal structures suggest intermediate conformations that may lie on the transition pathway between these two states. To characterize the transition from open to closed states in solution, X-ray solution scattering data were collected from AdK in the apo form and with progressively increasing concentrations of five different ligands. Scattering data from apo AdK are consistent with scattering predicted from the crystal structure of AdK in the open conformation. In contrast, data from AdK samples saturated with Ap5A do not agree with that calculated from AdK in the closed conformation. Using cluster analysis of available structures, we selected representative structures in five conformational states: open, partially open, intermediate, partially closed, and closed. We used these structures to estimate the relative abundances of these states for each experimental condition. X-ray solution scattering data obtained from AdK with AMP are dominated by scattering from AdK in the open conformation. For AdK in the presence of high concentrations of ATP and ADP, the conformational ensemble shifts to a mixture of partially open and closed states. Even when AdK is saturated with Ap5A, a significant proportion of AdK remains in a partially open conformation. These results are consistent with an induced-fit model in which the transition of AdK from an open state to a closed state is initiated by ATP binding.