Small-Angle X-ray Scattering (SAXS)

The small-angle scattering technique can be used to study the structure and interactions of colloidal solutions. Particularly the properties of biological macromolecules such as DNA and proteins, which may be regarded as (charged) colloids, can be determined by SAXS measurements.

Basic Principles: From Bragg's law to small-angle scattering

According to Bragg's law and for wavelengths λ around 1 Å the scattering angle 2θ corresponding to atomic distances d of a few Å is relatively large, i.e. typically 20°. Since colloids, biological macromolecules such as polymers, amphiphilic systems, membranes and liquid crystals exhibit typical distances d or domain sizes of 100 Å, the corresponding scattering angle 2θ is much smaller, i.e. typically 0.5°. All structural information is contained in a small angular range - which is why the technique is called "small-angle scattering".

Figure 1: Geometry of the path length difference and definition of the scattering vector q.

The total scattering intensity I(q) for a monodisperse spherical system at a scattering angle 2θ as a function of the scattering vector q can be expressed by:

where

• N_P is the number of protein molecules per unit volume in the solution,
• V_P is the volume of a single protein,
• Δρ = (ρ_p - ρ_s), is the difference between the electron density of protein molecules and that of the solvent (usually called the scattering contrast),
• P(q) is the form factor of a given protein, i. e. the scattering from a single protein molecule after orientational averaging,
• S(q) is the structure factor, which contains information on the protein interactions

For a polydisperse or non-spherical system S(q) may be replaced by an effective structure factor, which is calculated using a monodisperse structure factor at an effective sphere diameter. Further details on small-angle scattering and the data analysis can be found in this tutorial.

Experimental Setup

We have a lab-based small-angle X-ray scattering setup (Details).

Figure 2: The Xeuss 2.0 system.

Moreover, we have access to small-angle beamlines at different synchrotron radiation sources, such as the ID02 at the ESRF.

Example

We recently studied interactions of a model protein, bovine serum albumin (BSA) in aqueous solutions as a function of protein concentration and ionic strength. First, the form factor of single protein molecule can be determined from the scattering of dilute protein solution with moderate ionic strength as shown in Figure 3. An oblate ellipsoidal form factor gives the best fit of the scattering int ensity. BSA molecules at pH 7 are negatively charged, therefore, at low ionic st rength, the interactions between proteins are dominated by electrostatic repulsion. Combined with screened Coulomb structure factor, the scattering profiles from moderate to high protein concentration solution can be well-fitted and the corresponding structure factors are also evaluated from data fitting.

Figure 3: Scattered intensity and simulation by three form factors, for a protein solution of 10mg/ml with 0.3 M NaCl. It was shown that an oblate ellipsoid sha pe provided the best fit. The inset is the front view of space-filling model of serum albumin with basic residues colored in blue, acidic residues in red and ne utral ones in yellow. Figure 4: (a) Scattered intensity and theoretical fit by an ellipsoidal form fac tor and screened Coulomb potential model (E+SC), for a wide range of protein con centration at lower ionic strength; (b) Structure factor SSC(q) (screened Coulomb potential) calculated from (a) as a function of protein concentration.

References

[1] R. J. Roe, "Methods of X-ray and Neutron Scattering in Polymer Science", Oxford, Oxford University Press, 2000.
[2] P. Lindner, Th. Zemb, Ed. "Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter", North-Holland, Elsevier, 2002.
[3] F. Zhang, M. Skoda, R. Jacobs, R. A. Martin, C. M. Martin, F. Schreiber, J. Phys. Chem. B 2007, 111, 251-259.

For our previous work on SAXS, see our list of publications.