Many bacteria use rotating helical flagella in swimming motility. In the search for food or migration towards a new habitat, bacteria occasionally unbundle their flagellar filaments and tumble, leading to an abrupt change in direction. Flexible flagella can also be easily deformed by external shear flow, leading to complex bacterial trajectories. Here, we examine the effects of flagella flexibility on the navigation of bacteria in two fundamental shear flows: planar shear and Poiseuille flow realized in long channels. On the basis of slender body elastodynamics and numerical analysis, we discovered a variety of non-trivial effects stemming from the interplay of self-propulsion, elasticity and shear-induced flagellar bending. We show that in planar shear flow the bacteria execute periodic motion, whereas in Poiseuille flow, they migrate towards the centre of the channel or converge towards a limit cycle. We also find that even a small amount of random reorientation can induce a strong response of bacteria, leading to overall non-periodic trajectories. Our findings exemplify the sensitive role of flagellar flexibility and shed new light on the navigation of bacteria in complex shear flows.

We consider the fragmentation equationand address the question of estimating the fragmentation parameters -i.e. the division rate B(x) and the fragmentation kernel k(y, x) -from measurements of the size distribution f (t, ·) at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate B(x) = αx γ and a self-similar fragmentation kernel k(y, x) = 1 y k 0 ( x y ), we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet (α, γ, k 0 ) and a representation formula for k 0 . To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

International audienceWe study a non linear stationary system describing the transport of solutes dissolved in a fluid circulating in a counter-current 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 counter-current arrangement of tubes enhances the axial concentration gradient, thereby favoring the production of highly concentrated urine

(Ca 2ϩ ) 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 Ca 2ϩ 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 Ca 2ϩ from the vasa recta and loops of Henle generates a significant axial Ca 2ϩ concentration gradient in the medullary interstitium. In the base case, the urinary Ca 2ϩ concentration and fractional excretion are predicted as 2.7 mM and 0.32%, respectively. Urinary Ca 2ϩ excretion is found to be strongly modulated by water and NaCl reabsorption along the nephron. Our simulations also suggest that Ca 2ϩ molar flow and concentration profiles differ significantly between superficial and deep nephrons, such that the latter deliver less Ca 2ϩ to the collecting duct. Finally, our results suggest that the DCT and CNT can act to counteract upstream variations in Ca 2ϩ transport but not always sufficiently to prevent hypercalciuria. calcium; finite volume; kidney; mathematical model; transport CALCIUM (CA 2ϩ ) IS THE MOST abundant cation in the body. It plays an essential role in cardiac, skeletal, and smooth muscle function, and extra-and intracellular Ca 2ϩ concentrations ([Ca 2ϩ ]) must therefore be kept within a narrow range. Calcium homeostasis is maintained by the concerted action of the intestine, the parathyroid glands, and the kidneys. Preserving low Ca 2ϩ concentrations in the urinary filtrate is especially important, since hypercalciuria is one of the major risk factors for the formation of kidney stones. Recent studies on Ca 2ϩ handling by the kidney have focused on the molecular transporters and sensors of Ca 2ϩ (2, 9), but our understanding of Ca 2ϩ transport at the organ level remains limited. Whether there is a corticomedullary interstitial [Ca 2ϩ ] gradient has yet to be determined, and the contribution of each nephron segment to perturbations in the renal Ca 2ϩ balance is not fully understood. To address these questions, we developed a mathematical model of renal Ca 2ϩ transport across all nephron segments below the proximal tubules from the descending limbs of Henle to the inner medullary collecting ducts (IMCD). Our model is based on that of Hervy and Thomas (17), referred to below as the HT model, which describes the transport of water, NaCl, glucose, and lactate in the renal medulla. We modified the HT model to 1) make it dynamic, 2) explicitly represent the distal cortical segments, and 3) incorporate Ca 2ϩ transport. For this purpose, we developed a new finite-volume scheme that is robust and fast.About t...

To survive in harsh conditions, motile bacteria swim in complex environments and respond to the surrounding flow. Here, we develop a mathematical model describing how flagella bending affects macroscopic properties of bacterial suspensions. First, we show how the flagella bending contributes to the decrease in the effective viscosity observed in dilute suspension. Our results do not impose tumbling (random reorientation) as was previously done to explain the viscosity reduction. Second, we demonstrate how a bacterium escapes from wall entrapment due to the self-induced buckling of flagella. Our results shed light on the role of flexible bacterial flagella in interactions of bacteria with shear flow and walls or obstacles.

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-Perthame in the BV framework, under the assumptions that guarantee uniqueness of the solution, plus one additional technical assumption. We prove that existence in the BV framework fails withouth this additional technical assumption, by providing a counter-example.

Abstract. We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail in this context and in particular BV estimates. They are the most standard and simplest way to prove compactness and convergence.We provide a novel and simple method to obtain partial BV regularity and strong compactness in this framework. The standard notion of entropy is not convenient either and we also indicate another, but closely related, notion. We give two examples motivated by renal flows which consist of 2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more general.Key words. Hyperbolic relaxation; spatial heterogeneity; entropy condition; boundary conditions; strong compactness.Subject classifications. 35L03, 35L60, 35B40, 35Q92 IntroductionThe usual framework of hyperbolic relaxation [4,13,3] concerns the convergence of a general hyperbolic system with stiff source terms toward a conservation law, when the relaxation parameter ǫ goes to zero. More specifically, Jin and Xin [10] introduced a 2 × 2 linear hyperbolic system with stiff source term that approximates any given conservation law. The problem of interest is to prove the convergence of the microscopic quantities depending on ǫ toward the macroscopic quantities. A problem entering the framework of hyperbolic relaxation is motivated by very simplified models of kidney physiology [18,19,17] and fits the Jin and Xin framework with two major differences, boundary conditions, and spatial dependence of the source term. The type of boundary condition and the spatial dependence constitute the main novelty of the present study.The system represents two solute concentrations u ǫ (x,t) and v ǫ (x,t) and is written, for t ≥ 0 and x ∈ [0,L],

Summary The division of amyloid protein fibrils is required for the propagation of the amyloid state and is an important contributor to their stability, pathogenicity, and normal function. Here, we combine kinetic nanoscale imaging experiments with analysis of a mathematical model to resolve and compare the division stability of amyloid fibrils. Our theoretical results show that the division of any type of filament results in self-similar length distributions distinct to each fibril type and the conditions applied. By applying these theoretical results to profile the dynamical stability toward breakage for four different amyloid types, we reveal particular differences in the division properties of disease-related amyloid formed from α-synuclein when compared with non-disease associated model amyloid, the former showing lowered intrinsic stability toward breakage and increased likelihood of shedding smaller particles. Our results enable the comparison of protein filaments' intrinsic dynamic stabilities, which are key to unraveling their toxic and infectious potentials.