Scaling limit of a discrete prion dynamics model

Doumic, Marie; Goudon, Thierry; Lepoutre, Thomas

doi:10.4310/cms.2009.v7.n4.a3

Cited by 30 publications

(52 citation statements)

References 17 publications

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“…As such, rather than an infinite system of ordinary differential equations, the system consisted of a single ODE for protein in the normal configuration and a PDE specifying the distribution of aggregate sizes. While this formulation departs from the physically discrete nature of aggregates, in the limit of large aggregate sizes these formalisms are provably equivalent [47] and the use of PDEs permits a wider array of mathematical techniques. Most notably, the continuous relaxation on aggregate sizes has permitted determination of the explicit asymptotic density [44,46] ] prion system [53][54][55], but linking experimental outcomes uniquely to specific kinetic parameters remains challenging.…”

Section: Establishing a Mathematical Framework Of Prion Aggregate Dynmentioning

confidence: 99%

Mathematical Modeling of Prion Disease

Sindi¹

2017

Prion - An Overview

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The prion hypothesis, once a heretical violation of the central dogma of molecular biology, has become an accepted mechanism used to explain a host of progressive neurodegenerative diseases in mammals and heritable phenotypes in yeast. From the beginning, mathematical models have been an essential tool in studying prion and other protein misfolding/aggregation processes. In this work, we review some of the major mathematical studies that have contributed to our understanding of prion disease and discuss trends in current and future studies.

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Section: Establishing a Mathematical Framework Of Prion Aggregate Dynmentioning

confidence: 99%

Mathematical Modeling of Prion Disease

Sindi¹

2017

Prion - An Overview

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“…Compared to earlier completely discrete models, it has the advantage of taking into account two scales, as depicted by Figure 1; (i) a small scale (of the order of several PrPsc molecules) for continuous aggregation represented by the x-derivative and (ii) a large scale for the total length of the fibrils represented by the integral term. It can be derived through an asymptotic analysis departing from the single PrPsc scale with a discrete model [11,13,18]. Well-posedness, in the class of weak solutions, has been studied in great generality by Laurençot and Mischler [19] and Simonett and Walker [34].…”

Section: Introductionmentioning

confidence: 99%

Prion dynamics with size dependency–strain phenomena

Cálvez

Lenuzza

Doumic

et al. 2010

Journal of Biological Dynamics

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Models for the polymerization process involved in prion self-replication are well-established and studied [H. Engler, J. Pruss, and G.F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl. 324 (2006), pp. 98-117; M.L. Greer, L. Pujo-Menjouet, and G.F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol. 242 (2006), pp. 598-606; J. Pruss, L. Pujo-Menjouet, G.F. Webb, and R. Zacher, Analysis of a model for the dynamics of prions, Discrete Cont. Dyn. Sys. Ser. B 6(1) (2006), pp. 215-225] in the case where the dynamics coefficients do not depend on the size of polymers. However, several experimental studies indicate that the structure and size of the prion aggregates are determinant for their pathological effect. This motivated the analysis in Calvez et al. [Size distribution dependence of prion aggregates infectivity, Math Biosci. 217 (2009), pp. 88-99] where the authors take into account size-dependent replicative properties of prion aggregates. We first improve a result concerning the dynamics of prion aggregates when a pathological state exists (high production of the normal protein). Then we study the strain phenomena and more specifically we wonder what specific replicative properties are determinant in strain propagation. We propose to interpret it also as a dynamical property of size repartitions.

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“…Let n ∈ N, p ≥ 1, α ∈ (0, 1), the function ρ ∈ C ∞ c (R) and the coefficient m satisfying Assumptions (9). We define the function ρ α (x) = 1 α ρ(…”

Section: Appendices 6 Mathematical Proofsmentioning

confidence: 99%

“…Oligomer dynamics can be modelled by System (1), [9]. We remark that in this model there are two unknowns: the depolymerisation rate b and the initial condition u 0 .…”

Section: Initial State Estimation Without a Priorimentioning

confidence: 99%

See 1 more Smart Citation

Estimation from moments measurements for amyloid depolymerisation

Armiento

Doumic

Moireau

et al. 2016

Journal of Theoretical Biology

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Estimating reaction rates and size distributions of protein polymers is an important step for understanding the mechanisms of protein misfolding and aggregation, a key feature for amyloid diseases. This study aims at setting this framework problem when the experimental measurements consist in the time-dynamics of a moment of the population (i.e. for instance the total polymerised mass, as in Thioflavine T measurements, or the second moment measured by Static Light Scattering). We propose a general methodology, and we solve the problem theoretically and numerically in the case of a depolymerising system. We then apply our method to experimental data of degrading oligomers, and conclude that smaller aggregates of ovPrP protein should be more stable than larger ones. This has an important biological implication, since it is commonly admitted that small oligomers constitute the most cytotoxic species during prion misfolding process.

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Scaling limit of a discrete prion dynamics model

Cited by 30 publications

References 17 publications

Mathematical Modeling of Prion Disease

Mathematical Modeling of Prion Disease

Prion dynamics with size dependency–strain phenomena

Estimation from moments measurements for amyloid depolymerisation

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