Random walks on binary strings applied to the somatic hypermutation of B-cells
Résumé
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 foreignpathogen is confronted to our immune system, the mutation mechanism thatallows B-cells to adapt to it is called somatic hypermutation: a programmed process of mutation affecting B-cell receptors at extremely high rate. By con-sidering random walks on graphs, we introduce and analyze a simplified math-ematical model in order to understand this extremely efficient learning process.The structure of the graph reflects the choice of the mutation rule. We focuson 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 agiven 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 discussits numerical outputs within our mathematical framework. We provide as welllimitations and possible extensions of our approach.