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

Magali Tournus,Miguel Escobedo,Wei-Feng Xue,Marie Doumic,Philip K Maini

doi:10.1371/journal.pcbi.1008964

Abstract

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.

Highlights

How can we extract information on the stability and dynamics of proteins nano-filaments from population distribution data? This general question is of topical interest due to the everincreasing evidence to suggest that the fragmentation of amyloid and prion protein fibrils [1] are associated with their biological response ranging from being inert, functional to toxic, infectious and pathological [2]
We presented the mathematical analysis of the pure fragmentation equation
Based on the theoretical analysis, inversion formulae to directly recover information regarding division rates α and γ parameters, and division kernel κ from time dependent experimental measurements of filament size distribution are derived

Summary

Introduction

How can we extract information on the stability and dynamics of proteins nano-filaments from population distribution data? This general question is of topical interest due to the everincreasing evidence to suggest that the fragmentation of amyloid and prion protein fibrils [1] are associated with their biological response ranging from being inert, functional to toxic, infectious and pathological [2]. How can we extract information on the stability and dynamics of proteins nano-filaments from population distribution data? To analyze the division of protein filaments when the experimental information we have is at the level of the population distribution, for instance when the type of data we currently can acquire are time-point samples of fibril length distributions and individual dividing particles cannot yet be isolated and tracked, the pure fragmentation equation reveals to be a powerful mathematical tool. The pure fragmentation equation describes the time evolution of a population of fibril particles structured by their size x that divide into smaller particles. Though the fragmentation equation describes the dynamics at the level of the whole population, the properties B and κ have a natural interpretation in terms of the microscopic stability of the polymers.

Methods

Results

Conclusion