Chapter 13
Use of Ensemble Methods to
Describe Biomolecular Dynamics
by Small Angle X-Ray Scattering
Giancarlo Tria, Dmitri I. Svergun, and Pau Bernadó
Introduction 366
SAXS Basis 367
Ensemble Approaches for SAXS Data Analysis 370
Ensemble Optimization Method 371
Minimal Ensemble Search 373
Basis-Set Supported SAXS 374
Ensemble Renement of SAXS 374
Application of Ensemble Models to Study Biomolecular Dynamics by SAXS 376
IDPs 376
Multidomain Proteins 381
Concerted Motions in Multidomain Proteins 383
Highly Flexible Multidomain Proteins 385
Ensemble Methods in Nucleic Acids 387
Biomolecular Complexes 390
Dynamics in Biomolecular Complexes 390
Transient Biomolecular Complexes 393
Systems with Complex Dynamics 395
Final Remarks 396
Acknowledgments 397
References 397
Use of Ensemble Methods to Describe Biomolecular Dynamics by SAXS
In this chapter, we present modern computational approaches to use small-
angle x-ray scattering (SAXS) in the analysis of exible proteins. SAXS is a
powerful method for analyzing the structure and structural changes of bio-
logical macromolecules in solution (Svergun et al. 2013). e method is most
often used for puried solutions of globular proteins and complexes where
all particles can be considered identical and the experimental scattering
can be related to the structure of a single particle (although the scattering
is typically isotropic due to the average over particle orientations). In this
case, SAXS provides information about the overall shape and can construct
three-dimensional (3-D) low-resolution models, either ab initio or using rigid
body analysis in terms of known high resolution structures of domains or
For the solutions of nonidentical particles, the measured scattering reects
the average not only over the orientations but also over the dierent structures
of the particles present in solution. In this case, shapes of individual compo-
nents cannot be reconstructed given only the experimental scattering from
the mixture. However, if the scattering intensities from the components are
known a priori, their volume fractions can be determined. A typical applica-
tion is the characterization of oligomeric mixtures, where only a few types of
particles coexist in solution (e.g., monomer-dimer equilibrium) but much more
complicated systems can also be analyzed. In particular, SAXS can be readily
employed for the analysis of exible systems including multidomain proteins
with exible linkers or intrinsically disordered proteins (IDPs).
e major problem of the SAXS data analysis for exible systems is that the
solution may contain an astronomic number of components (individual macro-
molecules having dierent conformations). Until recently, it was only possible
to qualitatively distinguish between globular and unfolded macromolecules
from the SAXS data. e situation has changed with the advent of methods
utilizing ensemble analysis, which allows the extraction of quantitative infor-
mation about the exibility from the scattering data. In the ensemble approach
one from the very beginning admits that the scattering data cannot be repre-
sented by a single model. Instead, one generates a pool of possible structures
covering the entire conformational space explored by the macromolecule and
attempts to t the experimental data by a subensemble of the general pool.
e overall properties of the subensemble compatible with the experiment can
be subsequently analyzed and provide quantitative information about exibil-
ity and dynamics of the macromolecule. e results of this analysis are often
complemented by other methods providing structural information on such
systems (methods such as nuclear magnetic resonance [NMR], circular dichro-
ism [CD], calorimetry, and structure prediction).

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