Quantifying the Dynamics of Prion Infection: a Bifurcation Analysis of Laurentʹs Model

Laurent (1996a, Médecine/sciences12, 774–785; 1996b, Biochem. J.318, 35–39; 1998, Bio-phys. Chem.72, 211–222) proposed a model for the dynamics of diseases of the central nervous system caused by prions. It is based on the protein-only hypothesis (Prusiner et al., 1981, Proc. Natl. Acad. Sci. U.S.A.78, 6675–6679), which assumes that infection can be spread by particular proteins (prions) that can exist in two forms that share the same sequence, but have a different structure. The normal form is harmless, while the infectious isoform of the prion protein catalyses a transconformation from the native isoform to itself within a specialized compartment of the brain cells. This paper systematically explores the model behavior with the aim of quantifying the fundamental parameters characterizing the dynamics of prion infection. To this end we use data from the literature to fix orders of magnitude for the rates of synthesis and degradation of the native form of prion protein and for the shape of the autocatalytic function. The dynamical behavior is classified with respect to two unknown parameters (bifurcation analysis): the rate of spontaneous transconformation and the rate of output of the infectious isoform from the specialized compartment. We thus find that the bistability properties evidenced by Laurent are confined to a certain range of parameters and that permanent oscillations of the two isoforms concentrations are possible. The bifurcation analysis allows us to estimate approximate ranges for the values of the two unknown parameters and consequently to derive incubation times and compare them with actual data for hamster. Also, our study predicts that the output rate of the infectious isoform is relatively insensitive to variations of model parameters.