Fig. 2: Scaling law for the most common cluster size.

The most common cluster size plotted as a function of total protein concentration, \(C\). Theoretical fits are represented by dashed lines, and experimental data by circles. A FUS cluster radius obtained from the NTA measurements, which are most sensitive to the smaller clusters. Theoretical fit to \({R}^{* }=a\cdot {C}^{\frac{1}{3}}\) (corresponding to Eq. 12), the fitting parameter \(a\) is found to be 5.25 ∙ 10³ nm². Data for \(C\) and \({R}^{* }\) are as listed in Table S3. The inset is the same data however the y-axis was rescaled to show the linear relation between \({N}^{* }\) (estimated as \(\frac{4\pi }{3}\frac{{{R}^{* }}^{3}}{v}\), with \(v\) the estimated volume of FUS monomer, taken as 180 nm³) and \(C\) as indicated in Eq. 13. The solid black lines are the \({R}^{* }\) (or \({N}^{* }\) in the insert) values taken from three different measurements, these are not error bars. The mean value of these is shown by the blue circles. B CPEB4: Data for \(C\) and \({R}^{* }\) are as listed in Table S2. The theoretical curve is derived using a linear fit to \({{R}^{* }}^{-3}=a+b\cdot C\), as described in Eq. 14. \(a\) and \(b\) were found to be 2 ∙ 10⁻⁴ nm⁻³ and −0.018 nm⁻³ μM⁻¹. The curve is generated by inverting the relationship, \({R}^{* }={\left(a+b\cdot C\right)}^{-\frac{1}{3}}\). C, D Ddx4n1 and α-synuclein: cluster mass data taken from ref. ¹³. This measurement is also most accurate for the smaller clusters (in the range of 15 kDa to 5 MDa, corresponding to \(N\) in the range of 10⁰–10²). \({N}^{* }\) is the number of monomers in a cluster calculated as \({N}^{* }={M}_{{cluster}}/{M}_{{mono}}\), where \({M}_{cluster}\) is the mass of the cluster obtained from mass photometry¹³ and \({M}_{mono}\) is the mass of a single protein based on its amino acid sequence. The fitting follows a linear relationship, \({N}^{* }=a\cdot v\cdot C\) (Eq. 13), where \(a\) is the fitting parameter and \(v\) is the monomer volume (48 nm³ for Ddx4n1 and 22 nm³ for α-synuclein). The values of \(a\) were found to be 4.63 ∙ 10⁴ for Ddx4n1, 1.72 ∙ 10⁴ for α-synuclein at 20% PEG, and 8.59 ∙ 10⁴ at 5% PEG. D Two sets of measurements were done for α-synuclein, represented by the solid lines (that are not error bars). The average is shown by the open circles.