## 5 Solution Scattering
### Intrinsically Disordered Proteins  
One of the common mechanisms that drives cellular organization is the formation of biomolecular condensates via liquid-liquid phase separation (LLPS). By using SAXS to determine the structural features of proteins that encode LLPS, one can obtain critical insights into many biological processes. The driving force behind the biomolecular phase separation is thought to be IDPs. IDPs lack a fixed three-dimensional structure under physiological conditions, existing instead as dynamic ensembles of conformations. IDPs can interact with multiple other proteins and adapt to changing conditions crucial for signaling, recognition, and regulation. They fold when binding to specific sites, bridging their dynamic nature with essential biological functions [120]. Figure 10a shows an example of raw SAXS data. These featureless data are typical of solution scattering. The size of the scattering particle, IDP, is most easily obtained in the form of the radius of gyration \(R_{g}\) (\(\AA\)) using the Guinier approximation. \[I(q)\approx I(0)\exp\ (-\frac{q^{2}R_{g}^{2}}{3}) \tag{18}\]  
where \(I(q)\) is the scattering intensity at the scattering vector \(q\) and \(I(0)\) is the scattering intensity at the zero angle (\(q=0\)). \(R_{g}\) can be obtained from the slope, \(-(R_{g})^{2}/3\) of the linear region of the curve in the \(\ln(I(q))\) vs. \(q^{2}\) plot, Guinier plot (Figure 10b). The intercept yields \(I(0)\), which is determined by the concentration and is proportional to the molecular weight and electron density contrast of the particle. Thus, particles with larger molecular mass and larger contrast scatter more X-rays, resulting in a higher \(I(0)\). The equation is valid at low \(q\), \(qR_{g}<1.3\) for globular particles or \(qR_{g}<0.8\) for elongated particles [27]. \(R_{g}\) measures the size of the scattering particle, representing the root-mean-square distance of all scattering points from their center of mass. The molecular mass equation is shown in Equation (19), where MM is the molecular mass (g/mol) and \(c\) is the sample concentration (mol/L). The subscript \(st\) corresponds to a standard. \[MM_{p}=\frac{I(0)}{c}*\frac{MM_{st}}{I(0)_{st}/c_{st}} \tag{19}\]  
The volume of the particle can be obtained from the Porod's Invariant \(Q\) (Equation (20)). Porod's invariant quantifies the total scattering power of the system; it is independent of the particles' size, shape, or arrangement, and depends on the electron density contrast and the volume of the particle. Information about the surface area and the volume of the particle is obtained by calculating Equation (21). \[Q=\int_{0}^{\infty}q^{2}I(q)dq \tag{20}\]  
\[Q=2\pi^{2}(\Delta\rho)^{2}V \tag{21}\]  
For a system with sharp boundaries, \(\Delta\rho\) is the electron density difference between the sample and solvent. Information about the internal order, flexibility, and the shape of the particle are obtained from a Kratky plot, a plot of \(I(q)\)\(q^{2}\) vs. \(q\) (Figure 10c) [121]. Well-folded, globular, monodomain proteins have bell-shaped Kratky plots, whereas disordered proteins show continuous increase and unfolded proteins appear hyperbolic [122]. Another parameter useful in characterizing particles in solution is the fractal dimension, \(D_{s}\), which is associated with the interfaces or boundaries between particles and the surrounding medium. \(D_{s}\) can be determined from the scattered intensity at the tail end, i.e., the high \(q\)-region, of the SAXS curve using the Equation (22)  
\[I(q)\ \alpha\ q^{-(6-D_{s})} \tag{22}\]  
In the limit of when the surface of the particle has sharp and smooth interface, then \(D_{s}=2\), and the Equation (2) reduces to [PERSON]'s law states that \(I(q)\propto q^{-4}\). The generalized form of [PERSON]'s law is used to charcereize surface roughness in terms of fractals of dimension \(D_{s}\), with (\(2\leq D_{s}<\)3) for surface fractals, and \(I(q)\propto q^{-(6-D_{m})}\), with (\(1\leq D_{m}<\) 3) for mass fractals.