4 research outputs found
Effects of General Incidence and Polymer Joining on Nucleated Polymerization in a Model of Prion Proliferation
Two processes are incorporated into a new model for transmissible prion diseases. These are general incidence for the lengthening process of infectious polymers attaching to and converting noninfectious monomers, and the joining of two polymers to form one longer polymer. The model gives rise to a system of three ordinary differential equations, which is shown to exhibit threshold behavior dependent on the value of the parameter combination giving the basic reproduction number R0. For R00 \u3e1, the system is locally asymptotic to a positive disease equilibrium. The effect of both general incidence and joining is to decrease the equilibrium value of infectious polymers and to increase the equilibrium value of normal monomers. Since the onset of disease symptoms appears to be related to the number of infectious polymers, both processes may significantly inhibit the course of the disease. With general incidence, the equilibrium distribution of polymer lengths is obtained and shows a sharp decrease in comparison to the distribution resulting from mass action incidence. Qualitative global results on the disease free and disease equilibria are proved analytically. Numerical simulations using parameter values from experiments on mice (reported in the literature) provide quantitative demonstration of the effects of these two processes
Global stability for the prion equation with general incidence
We consider the so-called prion equation with the general incidence term
introduced in [Greer et al., 2007], and we investigate the stability of the
steady states. The method is based on the reduction technique introduced in
[Gabriel, 2012]. The argument combines a recent spectral gap result for the
growth-fragmentation equation in weighted spaces and the analysis of a
nonlinear system of three ordinary differential equations
High-order WENO scheme for Polymerization-type equations
Polymerization of proteins is a biochimical process involved in different
diseases. Mathematically, it is generally modeled by
aggregation-fragmentation-type equations. In this paper we consider a general
polymerization model and propose a high-order numerical scheme to investigate
the behavior of the solution. An important property of the equation is the mass
conservation. The fifth-order WENO scheme is built to preserve the total mass
of proteins along time
Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation
The Greer, Pujo-Menjouet andWebb model [Greer et al., J. Theoret. Biol., 242
(2006), 598-606] for prion dynamics was found to be in good agreement with
experimental observations under no-flow conditions. The objective of this work
is to generalize the problem to the framework of general
polymerization-fragmentation under flow motion, motivated by the fact that
laboratory work often involves prion dynamics under flow conditions in order to
observe faster processes. Moreover, understanding and modelling the
microstructure influence of macroscopically monitored non-Newtonian behaviour
is crucial for sensor design, with the goal to provide practical information
about ongoing molecular evolution. This paper's results can then be considered
as one step in the mathematical understanding of such models, namely the proof
of positivity and existence of solutions in suitable functional spaces. To that
purpose, we introduce a new model based on the rigid-rod polymer theory to
account for the polymer dynamics under flow conditions. As expected, when
applied to the prion problem, in the absence of motion it reduces to that in
Greer et al. (2006). At the heart of any polymer kinetical theory there is a
configurational probability diffusion partial differential equation (PDE) of
Fokker-Planck-Smoluchowski type. The main mathematical result of this paper is
the proof of existence of positive solutions to the aforementioned PDE for a
class of flows of practical interest, taking into account the flow induced
splitting/lengthening of polymers in general, and prions in particular.Comment: Discrete and Continuous Dynamical Systems - Series B (2012) XX-X