Estimation of the critical concentration of α-synuclein using the scale invariance.
(A) Determination of the critical exponent φ. 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 m. 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, either using Eq. (18) (C), resulting in ρc = 137 ± 10 μM, or Eq. 19 (D), resulting in ρc = 125 ± 7 μM, which are consistent within errors. 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 four independent measurements. In panel D, φ was constrained to 1.0 using Eq. (19).