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The spatial propagation of complex populations can depend on some structuring variables. In particular, recent developments in microscopy have revealed the impact of bacteria heterogeneity on the population motility. Biofilms of Myxococcus xanthus bacteria have been shown to be structured in clusters of various sizes, which remarkably, tend to move faster when they consist of a larger number of bacteria. We propose a minimal reaction-diffusion discrete-size structure model of a population of Myxococcus with two possible cluster sizes: isolated and paired bacteria. Numerical experiments show that this model exhibits travelling waves whose propagation speed depends on the increased motility of clusters, and the exchange rates between isolated bacteria and clusters. Notably, we present evidence of the existence of a characteristic threshold level θ*. on the ratio between cluster motility and isolated bacteria motility, which separates two distinct regimes of propagation speed. When the ratio is less or equal than θ*, the propagation speed of the population is constant with respect to the ratio. However, when the ratio is above θ*, the propagation speed increases. We also consider a generalised model with continuous-size structure, which also shows the same behaviour. We extend the model to include interactions with a resource population, which show qualitative behaviours in agreement to the biological experiments.

The goal of the present paper is to explore the long-time behavior of the growth-fragmentation equation formulated in the case of equal mitosis and variability in growth rate, under fairly general assumptions on the coefficients. The first results concern the monotonicity of the Malthus parameter with respect to the coefficients. Existence of a solution to the associated eigenproblem is then stated in the case of a finite set of growth rates thanks to the Kreĭn-Rutman theorem and a series of estimates on the moments. Providing additionally enough mixing in the population expressed in terms of irreducibility of the transition kernel, uniqueness of the eigenelements and asynchronous exponential growth are stated through entropy methods. Notably, convergence in shape to the steady state holds in the case of individual exponential growth where, in the absence of variability, the solution is known to exhibit oscillations at large times. We eventually perform a few numerical approximations to illustrate our results and discuss our mixing condition. [ABSTRACT FROM AUTHOR]

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]

In this paper, we design and analyze a numerical scheme that approximates a Jin–Xin linear system with implicit equilibrium on a bounded domain. This scheme relaxes toward the asymptotic limit of the linear system. The main properties of the limiting scheme are that it does not require to invert the implicit function defining the manifold and that it provides an accurate discretization of the boundary conditions. [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]

We study a nonlinear stationary system of transport equations with specific boundary conditions describing the transport of solutes dissolved in a fluid circulating in a countercurrent tubular architecture, which constitutes a simplified model of a kidney nephron. We prove that for every Lipschitz and monotonic nonlinearity (which stems from active transport across the ascending limb), the dynamic system, a PDE which we study through contraction properties, relaxes toward the unique stationary state. A study of the linearized stationary operator enables us, using eigenelements, to further show that under certain conditions regarding the nonlinearity, the relaxation is exponential. We also describe a finite volume scheme which allows us to efficiently approach the numerical solution to the stationary system. Finally, we apply this numerical method to illustrate how the countercurrent arrangement of tubes enhances the axial concentration gradient, thereby favoring the production of highly concentrated urine. [ABSTRACT FROM AUTHOR]