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Stability analysis of a steady state of a model describing Alzheimer’s disease and interactions with prion proteins

  • Mohammed Helal
  • Angélique Igel-Egalon
  • Abdelkader Lakmeche
  • Pauline Mazzocco
  • Angélique Perrillat-Mercerot
  • Laurent Pujo-Menjouet
  • Human Rezaei
  • Léon M. Tine
Article

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\(\beta \) 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 (\(\hbox {PrP}^\mathrm{{C}}\)). This interaction leads \(\hbox {PrP}^\mathrm{{C}}\) to be misfolded in oligomeric prion proteins (\(\hbox {PrP}^\mathrm{{ol}}\)), transmitting a death signal to neurons. In the present work, we aim to describe the dynamics of A\(\beta \) assemblies and the accumulation of toxic oligomeric species in the brain, by bringing together the fibrillation pathway of A\(\beta \) peptides in one hand, and in the other hand A\(\beta \) 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.

Keywords

Mathematical model analysis Steady state Alzheimer’s disease Prions Numerical simulations 

Mathematics Subject Classification

34D23 92B05 

Notes

Acknowledgements

The authors thank Prof Glenn F. Webb for his valuable reading and corrections.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Mohammed Helal
    • 1
  • Angélique Igel-Egalon
    • 2
  • Abdelkader Lakmeche
    • 1
  • Pauline Mazzocco
    • 3
  • Angélique Perrillat-Mercerot
    • 4
  • Laurent Pujo-Menjouet
    • 5
    • 6
  • Human Rezaei
    • 2
  • Léon M. Tine
    • 5
    • 6
  1. 1.Laboratory of BiomathematicsUniversity Sidi Bel AbbesSidi Bel AbbèsAlgeria
  2. 2.UR892 Virologie Immunologie MoléculairesINRAJouy-en-JosasFrance
  3. 3.CNRS UMR 5558, Laboratoire de Biométrie et Biologie Evolutive, Université Claude Bernard Lyon 1Université de LyonVilleurbanneFrance
  4. 4.Laboratoire de Mathématiques et Applications, UMR CNRS 7348, SP2MI Equipe DACTIM-MISUniversité de PoitiersChasseneuil Futuroscope CedexFrance
  5. 5.CNRS UMR 5208 Institut Camille Jordan, Université Claude Bernard Lyon 1Université de LyonVilleurbanne CedexFrance
  6. 6.Inria Team DraculaInria Grenoble Rhône-Alpes CenterVilleurbanneFrance

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