Multi-type Galton-Watson Processes with Affinity-Dependent Selection Applied to Antibody Affinity Maturation.

We analyze the interactions between division, mutation and selection in a simplified evolutionary model, assuming that the population observed can be classified into fitness levels. The construction of our mathematical framework is motivated by the modeling of antibody affinity maturation of B-cells in germinal centers during an immune response. This is a key process in adaptive immunity leading to the production of high-affinity antibodies against a presented antigen. Our aim is to understand how the different biological parameters affect the system's functionality. We identify the existence of an optimal value of the selection rate, able to maximize the number of selected B-cells for a given generation.

Mathematical modeling of lymphocytes selection in the germinal center.

Lymphocyte selection is a fundamental process of adaptive immunity. In order to produce B-lymphocytes with a target antigenic profile, mutation selection and division occur in the germinal center, a specific part of lymph nodes. We introduce in this article a simplified mathematical model of this phenomenon, taking into account the main mechanisms. This model is written as a non-linear, non-local, inhomogeneous second order partial differential equation, for which we develop a mathematical analysis. We assess, mathematically and numerically, in the case of piecewise-constant coefficients, the performance of the biological function by evaluating the duration of this production process as a function of several parameters such as the mutation rate or the selection profile, in various asymptotic regimes.

On the role of the epithelium in a model of sodium exchange in renal tubules.

In this study we present a mathematical model describing the transport of sodium in a fluid circulating in a counter-current tubular architecture, which constitutes a simplified model of Henle's loop in a kidney nephron. The model explicitly takes into account the epithelial layer at the interface between the tubular lumen and the surrounding interstitium. In a specific range of parameters, we show that explicitly accounting for transport across the apical and basolateral membranes of epithelial cells, instead of assuming a single barrier, affects the axial concentration gradient, an essential determinant of the urinary concentrating capacity. We present the solution related to the stationary system, and we perform numerical simulations to understand the physiological behaviour of the system. We prove that when time grows large, our dynamic model converges towards the stationary system at an exponential rate. In order to prove rigorously this global asymptotic stability result, we study eigen-problems of an auxiliary linear operator and its dual.

Random walks on binary strings applied to the somatic hypermutation of B-cells.

Within the germinal center in follicles, B-cells proliferate, mutate and differentiate, while being submitted to a powerful selection: a micro-evolutionary mechanism at the heart of adaptive immunity. A new foreign pathogen is confronted to our immune system, the mutation mechanism that allows B-cells to adapt to it is called somatic hypermutation: a programmed process of mutation affecting B-cell receptors at extremely high rate. By considering random walks on graphs, we introduce and analyze a simplified mathematical model in order to understand this extremely efficient learning process. The structure of the graph reflects the choice of the mutation rule. We focus on the impact of this choice on typical time-scales of the graphs' exploration. We derive explicit formulas to evaluate the expected hitting time to cover a given Hamming distance on the graphs under consideration. This characterizes the efficiency of these processes in driving antibody affinity maturation. In a further step we present a biologically more involved model and discuss its numerical outputs within our mathematical framework. We provide as well limitations and possible extensions of our approach.

Analysis of the fiber architecture of the heart by quantitative polarized light microscopy. Accuracy, limitations and contribution to the study of the fiber architecture of the ventricles during fetal and neonatal life.

OBJECTIVE: To address the advantages and drawbacks of quantitative polarized light microscopy for the study of myocardial cell orientation and to identify its contribution in the field. METHODS: Quantitative polarized light microscopy allows to measure the orientation of myocardial fibers into the ventricular mass. For each pixel of a horizontal section, this orientation is the mean value of the directions of all myosin filaments contained in the thickness of the section for each pixel of the section and is accounted for by two angles, the azimuth angle, which is the angle of the fiber in the plane of the section, and the elevation angle, which measures the way the fiber escapes from the section. The azimuth is accurately measured, and its range of definition is complete from 0 degrees to 180 degrees . The elevation angle can be defined only in the range 0 degrees to 90 degrees . It is accurately measured between 20 degrees and 70 degrees . From 0 degrees to 20 degrees , there is a systematic bias raising the measured values, and from 70 degrees to 90 degrees , the angle is not accurately measured. RESULTS: With this method, we validated Streeter's conjecture concerning the architecture of the left ventricle. We formulated a pretzel conjecture about the fiber architecture of the whole ventricular mass during fetal period. In our model, elaborated by visual analysis of registered maps of orientation, the fibers run like geodesics on a nested set of 'pretzels'. Next, the validity of the helical ventricular myocardial band model of Torrent-Guasp has been examined. It appears that the band model does not account for the patterns observed in our data during the fetal period. However, after the major events of postnatal cardiovascular adaptation, our data can neither discard nor confirm Torrent-Guasp's model. CONCLUSIONS: Present limitations of quantitative polarized light analysis can neither confirm nor discard the existing models of fiber orientation in the whole ventricular mass after the neonatal period. However, the problems of mathematical and experimental validation of these two models have been posed in a rigorous manner. Non-ambiguous fiber tracking and demonstration of these models will require significant improvement of the definition range of the elevation angle that should be extended to 180 degrees .