Journal of Mathematical Biology

, Volume 78, Issue 1–2, pp 57–81 | Cite as

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-MenjouetEmail author
  • Human Rezaei
  • Léon M. Tine


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.


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

Mathematics Subject Classification

34D23 92B05 



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


  1. Achdou Y, Franchi B, Marcello N, Tesi MC (2013) A qualitative model for aggregation and diffusion of \(\beta \)-amyloid in Alzheimer’s disease. J Math Biol 67(6–7):1369–92 ISSN 1432-1416MathSciNetCrossRefzbMATHGoogle Scholar
  2. Barz B, Liao Q, Strodel B (2017) Pathways of amyloid-\(\beta \) aggregation depend on oligomer shape. J Am Chem Soc 140(1):319–327CrossRefGoogle Scholar
  3. Becker R, Döring W (1935) Kinetische Behandlung der Keimbildung in übersättigten Dämpfen. Ann Phys 24:719–752CrossRefzbMATHGoogle Scholar
  4. Bertsch M, Franchi B, Marcello N, Tesi MC, Tosin A (2016) Alzheimer’s disease: a mathematical model for onset and progression. Math Med Biol 34(2):193–214MathSciNetzbMATHGoogle Scholar
  5. Bitan G, Kirkitadze MD, Lomakin A, Vollers SS, Benedek GB, Teplow DB (2003) Amyloid \(\beta \)-protein (A\(\beta \)) assembly: A\(\beta \)40 and A\(\beta \)42 oligomerize through distinct pathways. Proc Natl Acad Sci USA 100(1):330–335 ISSN 0027-8424CrossRefGoogle Scholar
  6. Calvez V, Lenuzza N, Oelz D, Deslys J-P, Laurent P, Mouthon F, Perthame B (2009) Size distribution dependence of prion aggregates infectivity. Math Biosci 217(1):88–99 ISSN 025-5564MathSciNetCrossRefzbMATHGoogle Scholar
  7. Canter RG, Penney J, Tsai L-H (2016) The road to restoring neural circuits for the treatment of Alzheimer’s disease. Nature 539(7628):187CrossRefGoogle Scholar
  8. Carulla N, Caddy GL, Hall DR, Zurdo J, Gairí M, Feliz M, Giralt E, Robinson CV, Dobson CM (2005) Molecular recycling within amyloid fibrils. Nature 436(7050):554CrossRefGoogle Scholar
  9. Cissé M, Mucke L (2009) A prion protein connection. Nature 457:1090–1091 ISSN 08866236CrossRefGoogle Scholar
  10. Ciuperca IS, Dumont M, Lakmeche A, Mazzocco P, Pujo-Menjouet L, Rezaei H, Tine LM (2018) Alzheimer’s disease and prion: analysis of an in vitro mathematical model. AIMS’ Journals. (submitted)
  11. Craft D (2002) A mathematical model of the impact of novel treatments on the A\(\beta \) burden in the Alzheimer’s brain, CSF and plasma. Bull Math Biol 64(5):1011–1031 ISSN 00928240CrossRefzbMATHGoogle Scholar
  12. Engler H, Prüss J, Webb GF (2006) Analysis of a model for the dynamics of prions II. J Math Anal Appl 324:98–117MathSciNetCrossRefzbMATHGoogle Scholar
  13. Francis PT, Palmer AM, Snape M, Wilcock GK (1999) The cholinergic hypothesis of Alzheimers disease: a review of progress. J Neurol Neurosurg Psychiatry 66(2):137–147CrossRefGoogle Scholar
  14. Freir DB, Nicoll AJ, Klyubin I, Panico S, Mc Donald JM, Risse E, Asante EA, Farrow MA, Sessions RB, Saibil HR et al (2011) Interaction between prion protein and toxic amyloid \(\beta \) assemblies can be therapeutically targeted at multiple sites. Nat Commun 2:336CrossRefGoogle Scholar
  15. Gabriel P (2011) The shape of the polymerization rate in the prion equation. Math Comput Model 53(7–8):1451–1456 ISSN 08957177MathSciNetCrossRefzbMATHGoogle Scholar
  16. Gallion SL (2012) Modeling amyloid-beta as homogeneous dodecamers and in complex with cellular prion protein. PLoS ONE 7(11):e49375CrossRefGoogle Scholar
  17. Gimbel DA, Nygaard HB, Coffey EE, Gunther EC, Lauren J, Gimbel ZA, Strittmatter SM (2010) Memory Impairment in transgenic Alzheimer mice requires cellular prion protein. J Neurosci 30(18):6367–6374 ISSN 0270-6474CrossRefGoogle Scholar
  18. Greer ML, Pujo-Menjouet L, Webb GF (2006) A mathematical analysis of the dynamics of prion proliferation. J Theor Biol 242(3):598–606 ISSN 0022-5193MathSciNetCrossRefGoogle Scholar
  19. Helal M, Hingant E, Pujo-Menjouet L, Webb GF (2014) Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions. J Math Biol 69(5):1207–35 ISSN 1432-1416MathSciNetCrossRefzbMATHGoogle Scholar
  20. Hilser VJ, Wrabl JO, Motlagh HN (2012) Structural and energetic basis of allostery. Ann Rev Biophys 41:585–609CrossRefGoogle Scholar
  21. Iooss B, Lemaître P (2015) A review on global sensitivity analysis methods. In: Meloni C, Dellino G (eds) Uncertainty management in simulation-optimization of complex systems. Springer, Berlin, pp 101–122CrossRefGoogle Scholar
  22. Jack CR Jr, Knopman DS, Jagust WJ, Petersen RC, Weiner MW, Aisen PS, Shaw LM, Vemuri P, Wiste HJ, Weigand SD et al (2013) Update on hypothetical model of Alzheimers disease biomarkers. Lancet Neurol 12(2):207CrossRefGoogle Scholar
  23. James GB (2013) The role of amyloid beta in the pathogenesis of Alzheimer’s disease. J Clin Pathol 66:362–366 ISSN 1472-4146CrossRefGoogle Scholar
  24. Johnson RD, Schauerte JA, Chang C, Wisser KC, Althaus JC, Carruthers CJL, Sutton MA, Steel DG, Gafni A (2013) Single-molecule imaging reveals A\(\beta \)42:A\(\beta \)40 ratio-dependent oligomer growth on neuronal processes. Biophys J 104(4):894–903 ISSN 00063495CrossRefGoogle Scholar
  25. Kandel N, Zheng T, Huo Q, Tatulian SA (2017) Membrane binding and pore formation by a cytotoxic fragment of amyloid \(\beta \) peptide. J Phys Chem B 121(45):10293–10305CrossRefGoogle Scholar
  26. Karran E, Mercken M, De Strooper B (2011) The amyloid cascade hypothesis for Alzheimer’s disease: an appraisal for the development of therapeutics. Nat Rev Drug Discov 10(9):698–712 ISSN 1474-1776CrossRefGoogle Scholar
  27. Kessels HW, Nguyen LN, Nabavi S, Malinow R (2010) The prion protein as a receptor for amyloid-\(\beta \). Nature 466(7308):E3CrossRefGoogle Scholar
  28. Laurén J, Gimbel DA, Nygaard HB, Gilbert JW, Strittmatter SM (2009) Cellular prion protein mediates impairment of synaptic plasticity by amyloid-\(\beta \) oligomers. Nature 457(7233):1128–1132 ISSN 0028-0836CrossRefGoogle Scholar
  29. Lomakin A, Chung DS, Benedek GB, Kirschner DA, Teplow DB (1996) On the nucleation and growth of amyloid beta-protein fibrils: detection of nuclei and quantitation of rate constants. Proc Natl Acad Sci 93(3):1125–1129CrossRefGoogle Scholar
  30. Lomakin A, Teplow DB, Kirschner DA, Benedek GB (1997) Kinetic theory of fibrillogenesis of amyloid beta-protein. Proc Natl Acad Sci USA 94:7942–7947CrossRefGoogle Scholar
  31. Maccioni RB, Farías G, Morales I, Navarrete L (2010) The revitalized tau hypothesis on Alzheimer’s disease. Arch Med Res 41(3):226–231CrossRefGoogle Scholar
  32. Nick M, Wu Y, Schmidt NW, Prusiner SB, Stöhr J, DeGrado WF (2018) A long-lived A\(\beta \) oligomer resistant to fibrillization. Biopolymers.
  33. Nunan J, Small DH (2000) Regulation of APP cleavage by alpha-, beta- and gamma-secretases. FEBS Lett 483(1):6–10 ISSN 00145793CrossRefGoogle Scholar
  34. Prince M, Wimo A, Guerchet M, Ali G, Wu Y, Prina M (2015) World Alzheimer Report 2015 The Global Impact of Dementia. Alzheimer’s Disease InternationalGoogle Scholar
  35. Prüss J, Pujo-Menjouet L, Webb GF, Zacher R (2006) Analysis of a model for the dynamics of prions. Discrete Contin Dyn Syst Ser B 6(1):225–235MathSciNetzbMATHGoogle Scholar
  36. Small SA, Duff K (2008) Linking A\(\beta \) and tau in late-onset Alzheimer’s disease: a dual pathway hypothesis. Neuron 60(4):534–542CrossRefGoogle Scholar
  37. Sosa-Ortiz AL, Acosta-Castillo I, Prince MJ (2012) Epidemiology of dementias and Alzheimer’s disease. Arch Med Res 43(8):600–608 ISSN 01884409CrossRefGoogle Scholar
  38. Urbanc B, Cruz L, Buldyrev SV, Havlin S, Irizarry MC, Stanley HE, Hyman BT (1999) Dynamics of plaque formatio in Alzheimer’s disease. Biophys J 76:1330–1334 ISSN 0006-3495CrossRefGoogle Scholar

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
    Email author
  • 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

Personalised recommendations