1. Introduction 1482. Basics of X-ray and neutron scattering 1492.1 Elastic scattering of electromagnetic radiation by a single electron 1492.2 Scattering by assemblies of electrons 1512.3 Anomalous scattering and long wavelengths 1532.4 Neutron scattering 1532.5 Transmission and attenuation 1553. Small-angle scattering from solutions 1563.1 Instrumentation 1563.2 The experimental scattering pattern 1573.3 Basic scattering functions 1593.4 Global structural parameters 1613.4.1 Monodisperse systems 1613.4.2 Polydisperse systems and mixtures 1633.5 Characteristic functions 1644. Modelling 1664.1 Spherical harmonics 1664.2 Shannon sampling 1694.3 Shape determination 1704.3.1 Modelling with few parameters: molecular envelopes 1714.3.2 Modelling with many parameters: bead models 1734.4 Modelling domain structure and missing parts of high-resolution models 1784.5 Computing scattering patterns from atomic models 1844.6 Rigid-body refinement 1875. Applications 1905.1 Contrast variation studies of ribosomes 1905.2 Structural changes and catalytic activity of the allosteric enzyme ATCase 1916. Interactions between molecules in solution 2036.1 Linearizing the problem for moderate interactions: the second virial coefficient 2046.2 Determination of the structure factor 2057. Time-resolved measurements 2118. Conclusions 2159. Acknowledgements 21610. References 216A self-contained presentation of the main concepts and methods for interpretation of X-ray and neutron-scattering patterns of biological macromolecules in solution, including a reminder of the basics of X-ray and neutron scattering and a brief overview of relevant aspects of modern instrumentation, is given. For monodisperse solutions the experimental data yield the scattering intensity of the macromolecules, which depends on the contrast between the solvent and the particles as well as on their shape and internal scattering density fluctuations, and the structure factor, which is related to the interactions between macromolecules. After a brief analysis of the information content of the scattering intensity, the two main approaches for modelling the shape and/or structure of macromolecules and the global minimization schemes used in the calculations are presented. The first approach is based, in its more advanced version, on the spherical harmonics approximation and relies on few parameters, whereas the second one uses bead models with thousands of parameters. Extensions of bead modelling can be used to model domain structure and missing parts in high-resolution structures. Methods for computing the scattering patterns from atomic models including the contribution of the hydration shell are discussed and examples are given, which also illustrate that significant differences sometimes exist between crystal and solution structures. These differences are in some cases explainable in terms of rigid-body motions of parts of the structures. Results of two extensive studies – on ribosomes and on the allosteric protein aspartate transcarbamoylase – illustrate the application of the various methods. The unique bridge between equilibrium structures and thermodynamic or kinetic aspects provided by scattering techniques is illustrated by modelling of intermolecular interactions, including crystallization, based on an analysis of the structure factor and recent time-resolved work on assembly and protein folding.
Topological Signatures in Nodal Semimetals through Neutron Scattering