Small-angle scattering: a view on the properties, structures and structural changes of biological macromolecules in solution

1. Introduction 148 2. Basics of X-ray and neutron scattering 149 2.1 Elastic scattering of electromagnetic radiation by a single electron 149 2.2 Scattering by assemblies of electrons 151 2.3 Anomalous scattering and long wavelengths 153 2.4 Neutron scattering 153 2.5 Transmission and attenuation 155 3. Small-angle scattering from solutions 156 3.1 Instrumentation 156 3.2 The experimental scattering pattern 157 3.3 Basic scattering functions 159 3.4 Global structural parameters 161 3.4.1 Monodisperse systems 161 3.4.2 Polydisperse systems and mixtures 163 3.5 Characteristic functions 164 4. Modelling 166 4.1 Spherical harmonics 166 4.2 Shannon sampling 169 4.3 Shape determination 170 4.3.1 Modelling with few parameters: molecular envelopes 171 4.3.2 Modelling with many parameters: bead models 173 4.4 Modelling domain structure and missing parts of high-resolution models 178 4.5 Computing scattering patterns from atomic models 184 4.6 Rigid-body refinement 187 5. Applications 190 5.1 Contrast variation studies of ribosomes 190 5.2 Structural changes and catalytic activity of the allosteric enzyme ATCase 191 6. Interactions between molecules in solution 203 6.1 Linearizing the problem for moderate interactions: the second virial coefficient 204 6.2 Determination of the structure factor 205 7. Time-resolved measurements 211 8. Conclusions 215 9. Acknowledgements 216 10. References 216 A 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 be