Insights into the dynamic trajectories of protein filament division revealed by numerical investigation into the mathematical model of pure fragmentation

• Published in: PLoS computational biology Vol. 17; no. 9; p. e1008964
• Main Authors: Tournus, Magali, Escobedo, Miguel, Xue, Wei-Feng, Doumic, Marie
• Format: Journal Article
• Language: English
• Published: United States Public Library of Science 03.09.2021; PLOS; Public Library of Science (PLoS)
• Subjects: Alzheimer's disease; Amyloid; Amyloid - chemistry; Approximation; Biology and Life Sciences; Biopolymers - chemistry; Cell division; Cytoskeleton - chemistry; Depletion; Division; Engineering and Technology; Experiments; Filaments; Fragmentation; Intermediate filament proteins; Life Sciences; Mathematical analysis; Mathematical models; Mathematicians; Models, Theoretical; Numerical analysis; Physical Sciences; Physics; Physiological aspects; Population; Power; Prion protein; Prions; Proteins; Proteins - chemistry; Research and Analysis Methods; Robustness (mathematics); Seeds; Size distribution; France; Amyloid Fibrils; Long-time dynamics; Inverse problem approach; Fragmentation Models; Amyloid assembly; Self-similar behaviour; Numerical experiments
• ISSN: 1553-7358; 1553-734X; 1553-7358
• DOI: 10.1371/journal.pcbi.1008964

The dynamics by which polymeric protein filaments divide in the presence of negligible growth, for example due to the depletion of free monomeric precursors, can be described by the universal mathematical equations of 'pure fragmentation'. The rates of fragmentation reactions reflect the stability of the protein filaments towards breakage, which is of importance in biology and biomedicine for instance in governing the creation of amyloid seeds and the propagation of prions. Here, we devised from mathematical theory inversion formulae to recover the division rates and division kernel information from time-dependent experimental measurements of filament size distribution. The numerical approach to systematically analyze the behaviour of pure fragmentation trajectories was also developed. We illustrate how these formulae can be used, provide some insights on their robustness, and show how they inform the design of experiments to measure fibril fragmentation dynamics. These advances are made possible by our central theoretical result on how the length distribution profile of the solution to the pure fragmentation equation aligns with a steady distribution profile for large times.