Determination of the critical exponents for FUS of the scaling invariance.

(A,C)Determination of the exponent φ for FUS (A) and SNAP-tagged FUS (C). The ratios of the average moments of the droplet sizes (<sk+1>/<sk>, at k=0.5, 1, 1.5, 2, Eq. (4)) are represented at various distances from the critical concentration . The exponent φ for each value of k was determined by error weighted linear regressions. The exponent and its error were determined as mean and standard deviation of the three independent measurements (Eqs. (15) and (16)). Error bars are shown in inset for graphical clarity. (B,D) Determination of the exponent α for FUS (B) and SNAP-tagged FUS (D). The mean of the droplet size distributions is plotted at various distances from the critical concentration . The value of the exponent α was determined by error weighted linear regression (Eq. (6)), using 𝜑 =1, where the errors were standard deviations of the three independent measurements (Eq. 16). Error bars, which were obtained as the standard deviation of the three independent measurements are shown in inset for graphical clarity. Error weighted linear regressions are performed in both cases excluding the data point at the lowest concentration ρ = 0.125 μM. The fit corresponding to the scaling ansatz, compatible with φ=1 and α=0, is represented by a dashed gray line with a slope of 1.

Log-normal behavior of FUS and SNAP-tagged FUS size distributions below the critical concentration.

(A, B)Variation of the size distribution with protein concentration: lns0 (A) and σ (B). lns0 and σ (inset) were computed for FUS (blue) and SNAP-tagged FUS (red) using Eq. (11). Error bars (in inset for graphical clarity) are estimated from three independent measurements. While droplet sizes increase with concentration, the width of distribution does not change considerably. (C) The collapse of the droplet size distribution functions is consistent with a log-normal behaviour. The droplet size distribution functions for both untagged FUS (blue) and SNAP-tagged FUS (red) are plotted after rescaling the sizes by the lns0 and σ values, the first and second moment of the logarithm of the droplet size distribution, which are a function of the concentration. The rescaled curves for both the untagged and the tagged protein collapse to the normal distribution (gray dashed), as expected when the non-rescaled droplet sizes follow a log-normal distribution.

Estimation of the critical concentration of FUS using the scale invariance.

(A,B) Critical concentration of FUS (A) and SNAP-tagged FUS (B). The scaling model predicts that the function of the moments plotted versus the concentration ρ becomes a straight line near the critical concentration ρc and intersects the ρ-axis at ρc, independently of the value of k. Error weighted linear regressions are performed in both cases excluding the data point at the lowest concentration ρ = 0.125 μM. The resulting estimate of the critical concentration is shown in green along with the corresponding standard deviation, estimated from three independent measurements.

Collapse of the droplet size distributions of FUS as predicted by the scale invariance.

If the scaling ansatz of Eq. (2) holds, the standard deviation 0 of the log-normal distribution should not depend on the distance from the critical concentration, and a collapse should be achieved by rescaling the size with the distance from the critical concentration. (A,C) Droplet size distributions derived from the experimental data of untagged FUS at 0.125, 0.25, 0.5, 1.0 and 2.0 μM concentrations (A) and SNAP-tagged FUS at 0.125, 0.25, 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0 μM concentrations (C), and their standard error of the mean from three independent measurements (26). (B,D) Collapse of the droplet size distributions rescaled by the estimated critical concentration. The error bars show the standard error of the mean from the three independent measurements (26).

Estimation of the critical concentration of α-synuclein using the scale invariance.

(A) Determination of the critical exponent 𝒳 ρ. The ratios of the average moments of the droplet sizes (<sk+1>/<sk>, at k=0.25, 0.75, 1.25, 1.75, Eq. (4)) are represented at various distances from the critical concentration . The exponent φ and its error for each value of k were determined as a mean and standard deviation of the three independent measurements (Eqs. (15) and (16)). Error bars are shown in inset for clarity. (B) Determination of the critical exponent α. The mean of the droplet size distributions is plotted at various distances from the critical concentration . The value of the exponent α was determined by error weighted linear regression (Eq. 6), using 𝜑 =1, where the errors were standard deviations of the five independent measurements (Eq. (16)). Error bars, which were obtained as the standard deviation of the five independent measurements are shown in inset for clarity. Error-weighted linear regressions were performed. The fit corresponding to the scaling ansatz, compatible with φ=1 and α=0, is represented by a scattered gray line with a slope of 1. (C,D) Determination of the critical concentration for α-synuclein using the scaling ansatz in two different ways (see Methods). The scaling model predicts that the function of the moments plotted versus the concentration ρ becomes a straight line near the critical concentration ρc and intersects the ρ-axis at ρc, independently of the value of k. Error-weighted linear regressions were performed. The resulting estimates of the critical concentration are shown in green along with the corresponding standard deviation, estimated from five independent measurements. In panel D, 𝒳 ρ was constrained to 1.0 using Eq. (19).