Corso di dottorato "FIRST-PASSAGE PHENOMENA IN BIOPHYSICAL SYSTEMS " GLEB OSHANIN - SORBONNE UNIVERSITY / CNRS - Didattica di eccellenza VP
Stochastic analysis of various processes encountered in diverse biophysical and biochemical systems, as well as in molecular biology, is a rapidly growing field of research which attracts nowadays the scientists with a background in applied mathematics and probability theory. Indeed, it is well understood by now that the standard rate equations approach based on the evolution of the mean densities of particles, which is appropriate for the analysis of standard chemical kinetic schemes and is usually conveniently used by chemists or chemical physicists, fails badly to provide a comprehensive understanding of processes taking place in extra- or in intracellular environments, in neural networks, and so on. The point is that here not only the dynamics is usually very different, as compared to the one that takes place in chemical systems, - one encounters the phenomenon of the so-called active Brownian motion or cooperative random motion hindered by a crowded molecular environment (the so-called “molecular soup”), but also in the important aspect that the numbers of particles involved in a given process are typically relatively small. As a consequence, there are large realization-to-realization fluctuations which render the descriptions in terms of mean values inappropriate, and reactions usually occur when some extreme events happen: first arrival to a given site, first penetration through an energy or an entropy barrier, or when a random signal reaches for the first time some prescribed level. In this short advanced course I am going to present a succinct but instructive and insightful overview of the available mathematical techniques, including the celebrated fluctuation theorems, and of our current understanding of the role of first-passage phenomena in biophysical systems. In my lectures I will talk about a variety of different biophysical processes with underlying stochastic dynamics, discuss the peculiar features of the dynamical behavior, different geometrical settings, different scenarios for dynamics in geometrically complex environments as well as about some spectacular reaction mechanisms, culminating at the analysis of multi-stage processes of intracellular signal transduction and cell-to-cell communications. For each situation I will derive the exact probability density functions of the times of corresponding extreme events, whenever possible, describe some approximate alternative approaches, as well as introduce the concept of an effective broadness of distributions and appropriate characteristic time scales.
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