No abstract

Alzheimer's disease (AD) is a neuro-degenerative disease affecting more than 46 million people worldwide in 2015. AD is in part caused by the accumulation of A[Formula: see text] peptides inside the brain. These can aggregate to form insoluble oligomers or fibrils. Oligomers have the capacity to interact with neurons via membrane receptors such as prion proteins ([Formula: see text]). This interaction leads [Formula: see text] to be misfolded in oligomeric prion proteins ([Formula: see text]), transmitting a death signal to neurons. In the present work, we aim to describe the dynamics of A[Formula: see text] assemblies and the accumulation of toxic oligomeric species in the brain, by bringing together the fibrillation pathway of A[Formula: see text] peptides in one hand, and in the other hand A[Formula: see text] oligomerization process and their interaction with cellular prions, which has been reported to be involved in a cell-death signal transduction. The model is based on Becker-Döring equations for the polymerization process, with delayed differential equations accounting for structural rearrangement of the different reactants. We analyse the well-posedness of the model and show existence, uniqueness and non-negativity of solutions. Moreover, we demonstrate that this model admits a non-trivial steady state, which is found to be globally stable thanks to a Lyapunov function. We finally present numerical simulations and discuss the impact of model parameters on the whole dynamics, which could constitute the main targets for pharmaceutical industry.

Carboxyfluorescein diacetate succinimidyl ester (CFSE) labelling has been widely used to track and study cell proliferation. Here we use mathematical modelling to describe the kinetics of immune cell proliferation after an in vitro polyclonal stimulation tracked with CFSE. This approach allows us to estimate a set of key parameters, including ones related to cell death and proliferation. We develop a three-phase model that distinguishes a latency phase, accounting for non-divided cell behaviour, a resting phase and the active phase of the division process. Parameter estimates are derived from model results, and numerical simulations are then compared to the dynamics of in vitro experiments, with different biological assumptions tested. Our model allows us to compare the dynamics of CD4+ and CD8+ cells, and to highlight their kinetic differences. Finally we perform a sensitivity analysis to quantify the impact of each parameter on proliferation kinetics. Interestingly, we find that parameter sensitivity varies with time and with cell generation. Our approach can help biologists to understand cell proliferation mechanisms and to identify potential pathological division processes.