Modeling Polydisperse Oligomeric Protein Mixtures

The experimental SAXS profiles of SPOP report on the average of a polydisperse mixture of oligomeric species in solution. The concentration of each oligomer should follow the isodesmic model where the concentration of the smallest subunit, the BTB-BTB dimer, is given by: ${\displaystyle c_1=\frac{2c_{tot}{K}_A+1-\sqrt{4c_{tot}{K}_A+1}}{2c_{tot}{K}_A^2}}$
The concentration c_i of any larger oligomer with $i$ subunits can be calculated given c₁ and the concentration of oligomer $i$ –1, c_i-1: $c_i=K_A{c}_{i-1}{c}_1$
$K_A$ is the isodesmic association constant and $c_{tot}$ is the total concentration of protomers. Here we assume that the SPOP BTB-BTB dimer is always fully formed (Marzahn et al., 2016 (link)) and $c_{tot}$ in Equation 1 is thus half of the total protein concentration reported for the SAXS experiments, which refers to the SPOP monomer concentration. Given the concentration c_i of each oligomer $i$ from the isodesmic model, we can calculate the volume fraction ${\varphi }_i$ of the oligomer: ${\displaystyle {\varphi }_i=\frac{i{c}_i}{\sum _i^{N}i{c}_i}}$
The average SAXS intensities from the mixture of oligomers ${⟨I⟩}_{\text{mix}}$ are then given by: ${\displaystyle ⟨I{⟩}_{\text{mix}}=\sum _i^N⟨I{⟩}_{i,\text{ensemble}}{\varphi }_i}$
where ${⟨I⟩}_{i,\text{ensemble}}$ is the conformationally averaged SAXS intensity of oligomer $i$ . Note that the magnitude of the SAXS intensities calculated with Pepsi-SAXS were set to be proportional to the number of subunits in the oligomer, so given Equations 3 and 4 the total contribution of each oligomer to the averaged SAXS intensity is proportional to $i^2{c}_i$ .