The depolymerization (i.e. progressive shortening) of large molecules can be modeled by discrete Becker-Döring-type equations, or by continuous equations. In many applications, the dynamic nature of the experiments, as well as their nanometric scale, makes it difficult to estimate quantitatively, or even simply to decipher the mechanisms involved.
In this talk, I will discuss two problems inspired by experiments carried out by Human Rezaei's team at INRAE on the depolymerization of PrP protein fibers (responsible for prion diseases). Starting from a discrete depolymerization model, we first evaluate the impact of using continuous approximations to solve the initial state estimation problem. At second order, the asymptotic model becomes an advection-diffusion equation, where diffusion is a corrective term. This approximation is much more accurate, but we are faced with a trade-off between accuracy and stability: the inverse reconstruction turns out to be "severely ill-posed". Using Carleman-type inequalities and log-convexity, we prove an observability result and an error estimate. This is joint work with Philippe Moireau.
A second project, in collaboration with Klemens Fellner, Mathieu Mezache and Juan Velazquez, involved the design and analysis of an oscillating depolymerization model - the standard models being unable to account for the sustained oscillations observed experimentally.
Jérémy Heleine, David Lafontaine
