## 5 Solution Scattering
### Intrinsically Disordered Proteins
If the particle is a sphere with well-defined boundaries, \(d_{f}=4\), where intensity decays \(q^{-4}\), characteristic of smooth, solid surfaces. For fractal surfaces, \(d_{f}<4\) represents rough, irregular surfaces. For 2D (sheet) or 1D (rod-like) particles, \(d_{f}=2\) or 1, respectively. Fractal dimensions also provide insights into the structural organization and solvation of intrinsically disordered proteins (IDPs). The fractal dimension (\(D_{m}\)) reflects how mass scales within the protein, with values around 1.7 indicating a well-solvated, expanded random coil. For the bacteriophage \(\lambda\)N protein, a fractal dimension of 1.76 confirmed its extensively unfolded and well-solvated nature, even under varying urea concentrations or ionic strength conditions. This highlights the protein's flexibility and suitability for dynamic interactions, making \(D_{m}\) a valuable metric for studying protein behavior and solvation in diverse environments [125]. The parameters, such as the \(R_{g}\) related to size, the Porod exponent, and the shape of the Kratky plot related to internal order, are implicit in the pair distribution function, \(P(r)\) (PDF) (Equation (23)), which can be obtained from the scattered intensity, \(I(q)\), by carrying  
Figure 10: (**a**) Raw data for solution scattering. (**b**) Guinier transformation. (**c**) Kratky plot; reprinted from [PERSON] et al. (2021) [123] with permission from Elsevier. (**d**) Image of an antibody. (**e**) Scattering intensity of various immunoglobins. (**f**) Pair distribution functions of the transformed scattering intensity; reprinted from [PERSON] et al. (2015) [124] with permission from International Union of Crystallography. out indirect Fourier transform (IFT, Equation (16)), where \(D_{max}\) is the maximum dimension of the particle. \[P(r)=\frac{1}{2\pi^{2}}\int_{0}^{\infty}q^{2}I(q)\frac{\sin\left(qr\right)}{qr}dq \tag{23}\]  
\[I(q)=4\pi\int_{0}^{D_{max}}P(r)\frac{\sin\left(qr\right)}{qr}\ dr \tag{24}\]  
\(P(r)\) describes the frequency of occurrence of the distance, \(r\), between pairs of scattering centers within the particle. Peaks in \(P(r)\) correspond to interatomic or intermolecular distances. At \(r>D_{max}\), \(P(r)\) goes to 0. When \(P(r)\) is graphed against \(r\), the shape of the plot gives insight into the particle's geometry. A symmetrical \(P(r)\) function indicates a spherical particle. An asymmetric \(P(r)\) indicates an elongated or irregular shape. The \(P(r)\) function is valuable for unraveling the details of multidomain proteins. This information is used for shape determination for well-folded systems [126]. However, since the use of IFT requires the \(P(r)\) function to converge to zero at a fixed maximum dimension (\(D_{max}\)), its use is difficult with the data from highly flexible, disordered systems, such as IDPs. It is often recommended that IFT be used to obtain a more reliable value of radii of gyration compared to Guinier analysis because it uses scattering data from the full \(q\) range. Although the problems resulting from the use of \(P(r)\) calculated from IFT outweigh the benefits for IDPs, these functions are useful with multidomain proteins as will be discussed in the next section.