## Theoretical Considerations

Scattering of X-rays at small angles can be used to obtain parameters such as the hydrated volume, the external surface area, the electron density, and the degree of hydration of proteins and other macromolecules in solution. The basic theory has been presented by Luzzati, Witz, and Nicolaieff  and Van Holde ,

X-ray scattering measurements can be made at angles (0) small enough to allow a good extrapolation to 6 = 0. The molecular weight of a solute can be estimated from this extrapolated scattering , The intensity of X-ray scattering at a distance r from a single electron is where e is the charge on an electron and m is its mass, /0 is the X-ray source intensity, and c is the speed of light. For N molecules, each with n electrons, in a cubic centimeter of solution, at 6 = 0,

For experiments in solution, n is replaced by (n - n0), representing the excess electrons in a solute molecule over the volume of solvent it displaces. Recasting eq. (9.2) in terms of molecular weight (M), the weight concentration of solute (C) and Avogadro's number (NA), we have 

The quantity (n — n(JM) is essentially a ratio of atomic number to atomic weight and depends on the nature of the macromolecule. Thus, the scattering intensity depends on of the types of molecules doing the scattering and may be analyzed to obtain a variety of useful molecular parameters.

To obtain the radius of gyration (Rc) of the macromolecule, the angular dependence of the X-ray scattering at small angles is used. The equations and the notation that follow  apply to globular particles and to so-called infinite slit collimation. These experimental conditions must be satisfied for the systems under examination.

The excess scattered intensity j„(s) is the scattered intensity of the sample normalized with respect to the intensity of the incident beam and corrected for the scattering of the blank. It is related to the scattering angle (6) by 

Us) = ]„(()) exp[-(4/3)7r2R2as2] + <b(s) (9.4)

where the symbols are as follows:

6 = scattering angle,

A = wavelength of the source radiation (e.g., 1.542A for the Cu-Ka doublet),

/„(0) = normalized intensity extrapolated to 8 = 0, Figure 9.1 Theoretical scattering intensity curve according to eq. (9.4) for (4/3)ir2i?„ = 1/0.000012, /„(0) = 0.4, and <£(\$) = 0.

Figure 9.1 Theoretical scattering intensity curve according to eq. (9.4) for (4/3)ir2i?„ = 1/0.000012, /„(0) = 0.4, and <£(\$) = 0.

### Table 9.1 General Model for Analysis of SAXS Data

Assumptions: Solvent or buffer background has been subtracted. This can be done by fitting background curves to the model that follows with p = 2, then subtracting the fitting equation from the raw data  Regression equation:

Regression parameters: Data:

b0 and ;„,,(0); Ra i for i = 1 to p /«.¿(O) vs. s

Special instructions: Fit data to p = 1, p = 2, p = 3. . . . , models in succession. Test for goodness of fit by comparing residual plots and by using the extra sum of squares F test (see Section 2.C.1); use proper weighting, such as for a counting detector Wj = l/y; in eq. (2.9) (cf. Section 2.B.5)

 Symbol Parameter Equation mapp Apparent molecular mass, in e-/molecule, at concentration ce m (tpp = /„(0)(1 - Pllh)-2/Ce m Molecular mass extrapolated to zero concentration m = "tapp + 2 Bm2ce Weight-average molecular weight M = mNA/q V Hydrated volume of particle V=' n(0) I j J lTtsjn(s)ds A p Electron density difference; = Pi P2 ,, ,, + cepi(i Pi>h) C.(l - Pi(s) = residual error between the model and the observed scattering. The exponential term on the right-hand side of eq. (9.4) represents the traditional Guinier approximation , The radius of gyration Ra and /„(0) can be obtained by nonlinear regression of jn(s) vs. 5 data. By comparing eq. (9.4) with the first entry in Table 3.8, we see that the model for j„(s) vs. s is simply a Gaussian peak with x0 = s0 = 0, so that (.x - x0)2 = (s - s0)2 = s2- The maximum of the Gaussian occurs at s = 0 (Figure 9.1), so that we see only the right-hand half of the peak. In practice, proteins are often encountered in solutions in equilibrium with their aggregates or as mixtures of proteins. Furthermore, asymmetry associated with rod-shaped particles, for example, or spherically symmetric particles with inhomogeneous regions of differing electron density may be encountered. Any of these situations can lead to models with multiple Gaussian components , A model for such systems can be formulated by summing terms similar to the exponential in eq. (9.4). Such a general model is summarized in Table 9.1. The important parameter, Ra, the apparent radius of gyration of the macromolecule, can be obtained from nonlinear regression analysis of the scattering data onto the model in Table 9.1. If the theoretical point-source scattering function at zero angle /„(0) is obtained from these data, additional parameters relating to the system can be found. Traditionally, /„(0) is obtained by deconvolution of the smeared infinite-slit data j„(s) and extrapolating to zero angle. A computer program for this deconvolution has been published , Later in this chapter, we will present a method to obtain ¿„(0) directly. The additional parameters that can be estimated from /„(0) are summarized in Table 9.2.