Figure 4.

Illustration of The Key Insights Emerging from the Mathematical Analysis of the Division Model

The behavior of the division equation Equation 1 is entirely and uniquely dictated by a set of three properties: α, γ, and κ₀. Several key predictive insights emerged from the analytical solution of Equation 1 with regard to these three properties.

(A) The three example length distributions in the left panel can be rescaled to show the same distribution shape in the right panel, illustrating the concept of self-similar length distributions.

(B) After a period of time, the self-similar length distribution shape is reached. From this point, the reduction in the average length of the fibril length distribution can be described as a power law versus time. The decay of mean length of a sample is predicted to tend toward a straight line, the asymptotic line, when plotted on a log-log plot with the slope of the line representing −1/γ (black line in B, D, and E). The asymptotic line denoting mean fibril lengths decay also does not depend on the initial length distribution (colored lines in B).

(C) The self-similar length distribution shape contains information about κ₀, which describes how likely a fibril will divide in the middle versus shedding a small fragment from the edge. A κ₀ indicative of fibril types that are more likely to divide in the middle will result in fibril length distributions with a distinct peak and low relative population of small fragments (dark blue and light blue curves). In contrast, κ₀ indicative of fibril types and conditions that promote equal likelihood of division along the fibril or even favor shedding of small fragments from fibril edges will result in self-similar fibril length distributions that have a larger relative population of small fibril fragments (orange and red curves) compared with κ₀ values favoring division in the center of the fibrils.

(D and E) (D) and (E) illustrate how the black asymptotic line describing the decay of fibril lengths in (A) is dictated by the parameters α and γ, respectively. For each panel, the color bar to the right illustrates the different properties associated with the colors in the panel (e.g., division in the center versus at the edge of a fibril for (C), and division of a long versus a short fibril in (E)).