Figure 3

Schematic representation of the cell loss kinetics predicted by stretched-exponential decay models. (A) The stretched exponential is a generalization of the exponential function with one additional parameter: the stretching exponent, β (1 > β > 0). With β = 1, the simple exponential function (linearly on semi-log plots) is recovered (Eq. 16). The parameter β depends on the distribution width. (B-D) PolyQ-length dependence of neuronal cell loss by the stretched-exponential decay model. (B) Schematic representations of the cell-loss kinetics per polyQ-length in stretched-exponential functions with different β values (β = 1.0, 0.9, 0.8). Assuming that the extent of neuronal cell loss at the onset of disease (red line: 30% cell loss is defined as symptom onset) is identical irrespective of polyQ-length (B), the relationships between polyQ-length and nucleation lag time at disease onset, calculated by using Eq. 13 with different β values (β = 1.0, 0.9, 0.8), show a first-order exponential function regardless of the β values (C). (D) Semi-log plots of polyQ-length versus nucleation lag time at an identical ratio of cell loss for each polyQ-length (with β = 0.8) show that the slope of linear line (the risk of cell death) slightly changes during disease progression [t(0.7) → t(0.3)]. Red dotted line: nucleation lag time at disease onset for each polyQ-length [30% neuronal loss, t(0.7)].