Size distribution dependence of prion aggregates infectivity

We consider a model for the polymerization (fragmentation) process involved in infectious prion self-replication and study both its dynamics and non-zero steady state. We address several issues. Firstly, we extend a previous study of the nucleated polymerization model [M.L. Greer, L. Pujo-Menjouet, G.F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol. 242 (2006) 598; H. Engler, J. Pruss, G.F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl. 324 (2006) 98] to take into account size dependent replicative properties of prion aggregates. This is achieved by a choice of coefficients in the model that are not constant. Secondly, we show stability results for this steady state for general coefficients where reduction to a system of differential equations is not possible. We use a duality method based on recent ideas developed for population models. These results confirm the potential influence of the amyloid precursor production rate in promoting amyloidogenic diseases. Finally, we investigate how the converting factor may depend upon the aggregate size. Besides the confirmation that size-independent parameters are unlikely to occur, the present study suggests that the PrPsc aggregate size repartition is amongst the most relevant experimental data in order to investigate this dependence. In terms of prion strain, our results indicate that the PrPsc aggregate repartition could be a constraint during the adaptation mechanism of the species barrier overcoming, that opens experimental perspectives for prion amyloid polymerization and prion strain investigation.