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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. Author summary: Amyloid fibrils are fibrillar protein structures involved in many neurodegenerative illnesses, such as Parkinson's disease or Alzheimer's disease. To propagate in disease, these misfolded protein aggregates must grow and divide to proliferate. Therefore, the intrinsic characteristics of their division, including the division rate and the pattern of division in terms of whether the fibrils are likely to break in the middle or at the edges, impact the disease aetiology. Here, we discovered mathematical formulae that can be used to directly extract the fibril division characteristics from recent experiments data obtained from time-dependent fibril length distribution measurements. We explain how these formulae can be used, and prove the robustness of the division rate formula where small errors in the measurement leads to small errors in the division rate. We also demonstrate that the mathematical formula is not robust enough to precisely decipher the pattern of division in the data, and suggest instead new future experimental design with short time measurements in experiments starting with fibril suspensions where all fibrils have similar size, which would be suitable to provide an improved estimates. [ABSTRACT FROM AUTHOR]

We consider a scalar conservation law with a flux containing spatial heterogeneities of bounded variation, where the number of discontinuities may be infinite. We prove existence of an adapted entropy solution in the sense of Audusse and Perthame in the BV framework, under assumptions that guarantee uniqueness of the solution, plus one additional technical assumption. In a counterexample, we prove that existence in the BV framework fails without this additional technical assumption. [ABSTRACT FROM AUTHOR]

We developed a mathematical model of calcium (Ca2+) transport along the rat nephron to investigate the factors that promote hypercalciuria. The model is an extension of the flat medullary model of Hervy and Thomas (Am J Physiol Renal Physiol 284: F65-F81, 2003). It explicitly represents all the nephron segments beyond the proximal tubules and distinguishes between superficial and deep nephrons. It solves dynamic conservation equations to determine NaCl, urea, and Ca2+ concentration profiles in tubules, vasa recta, and the interstitium. Calcium is known to be reabsorbed passively in the thick ascending limbs and actively in the distal convoluted (DCT) and connecting (CNT) tubules. Our model predicts that the passive diffusion of Ca2+ from the vasa recta and loops of Henle generates a significant axial Ca2+ concentration gradient in the medullary interstitium. In the base case, the urinary Ca2+ concentration and fractional excretion are predicted as 2.7 mM and 0.32%, respectively. Urinary Ca2+ excretion is found to be strongly modulated by water and NaCl reabsorption along the nephron. Our simulations also suggest that Ca2+ molar flow and concentration profiles differ significantly between superficial and deep nephrons, such that the latter deliver less Ca2+ to the collecting duct. Finally, our results suggest that the DCT and CNT can act to counteract upstream variations in Ca2+ transport but not always sufficiently to prevent hypercalciuria. [ABSTRACT FROM AUTHOR]