Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions

Journal of Mathematical Biology

Volumn 69, Issue 5, 2014, Pages 1207-1235

  Helal, Mohamed ^a   Hingant, Erwan ^b   Pujo Menjouet, Laurent ^c   Webb, Glenn F ^d  

^a UNIVERSITY OF SIDI BEL ABBES   (Algeria)

^b UNIVERSITY OF CONCEPCIÓN   (Chile)

^c UNIVERSITÉ LYON 1   (France)

^d VANDERBILT UNIVERSITY   (United States)

Author keywords

Alzheimer; Mathematical model; Prion; Stability; Well posedness

Indexed keywords

AMYLOID BETA PROTEIN; PRION;

ALZHEIMER DISEASE; BIOLOGICAL MODEL; HUMAN; METABOLISM; PRION;

ALZHEIMER DISEASE; AMYLOID BETA-PEPTIDES; HUMANS; MODELS, BIOLOGICAL; PRIONS;

EID: 84885495431     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-013-0732-0     Document Type: Article

References (42)

1. Achdou Y, Franchi B, Marcello N, Tesi M (2012) A qualitative model for aggregation and diffusion of $$\beta $$β-amyloid in alzheimers disease. J Math Biol 1–24
2. Allen LJS (2007) An introduction to mathematical biology. Pearson/Prentice Hall, Upper Saddle River
3. Calvez V, Lenuzza N, Oelz D, Deslys JP, Laurent P, Mouthon F, Perthame B (2009) Size distribution dependence of prion aggregates infectivity. Math Biosci 217(1):88–99
4. Calvez V, Lenuzza N, Doumic M, Deslys JP, Mouthon F, Perthame B (2010) Prion dynamics with size dependency-strain phenomena. J Biol Dyn 4(1):28–42
5. Chen S, Yadav SP, Surewicz WK (2010) Interaction between human prion protein and amyloid-$$\beta $$β (a$$\beta $$β) oligomers role of N-terminal residues. J Biol Chem 285(34):26,377–26,383
6. Chung E, Ji Y, Sun Y, Kascsak R, Kascsak R, Mehta P, Strittmatter S, Wisniewski T (2010) Anti-prpc monoclonal antibody infusion as a novel treatment for cognitive deficits in an alzheimer’s disease model mouse. BMC Neurosci 11(1):130
7. Cissé M, Mucke L (2009) Alzheimer’s disease: a prion protein connection. Nature 457(7233):1090–1091
8. Cissé M, Halabisky B, Harris J, Devidze N, Dubal DB, Sun B, Orr A, Lotz G, Kim DH, Hamto P et al (2011) Reversing ephb2 depletion rescues cognitive functions in alzheimer model. Nature 469(7328):47–52
9. Collet JF, Goudon T (2000) On solutions of the lifshitz-slyozov model. Nonlinearity 13(4):1239
10. Craft DL, Wein LM, Selkoe DJ (2002) A mathematical model of the impact of novel treatments on the a$$\beta $$β burden in the alzheimers brain, csf and plasma. Bull Math Biol 64(5):1011–1031
11. Craft DL, Wein LM, Selkoe DJ (2005) The impact of novel treatments on a$$\beta $$β burden in alzheimers disease: insights from a mathematical model. In: Brandeau ML, Sainfort F, Pierskalla WP (eds) Operations research and health care, international series in operations research and management science, vol 70. Springer, USA, pp 839–865
12. Duyckaerts C, Delatour B, Potier MC (2009) Classification and basic pathology of alzheimer disease. Acta Neuropathol 118(1):5–36
13. Fawzi NL, Okabe Y, Yap EH, Head-Gordon T (2007) Determining the critical nucleus and mechanism of fibril elongation of the alzheimers a$$\beta _(1-40)$$β1-40 peptide. J Mol Biol 365(2):535–550
14. Freir DB, Nicoll AJ, Klyubin I, Panico S, Risse E, Asante EA, Farrow MA, Sessions RB, Saibil HR, Clarke AR, Rowan MJ, Walsh DM, Collinge J (2011) Interaction between prion protein and toxic amyloid $$\beta $$β assemblies can be therapeutically targeted at multiple sites. Nat Commun 2:336
15. Gabriel P (2011) The shape of the polymerization rate in the prion equation. Math Comput Model 53(7):1451–1456
16. Gallion SL (2012) Modeling amyloid-beta as homogeneous dodecamers and in complex with cellular prion protein. PLoS One 7(11):e49,375
17. Gimbel DA, Nygaard HB, Coffey EE, Gunther EC, Laurén J, Gimbel ZA, Strittmatter SM (2010) Memory impairment in transgenic Alzheimer mice requires cellular prion protein. J Neurosci 30(18): 6367–6374
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
19. Greer ML, Van den Driessche P, Wang L, Webb GF (2007) Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation. SIAM J Appl Math 68(1):154–170
20. Hardy J, Selkoe DJ (2002) The amyloid hypothesis of Alzheimer’s disease: progress and problems on the road to therapeutics. Science 297(5580):353–356
21. Holmes C, Boche D, Wilkinson D, Yadegarfar G, Hopkins V, Bayer A, Jones RW, Bullock R, Love S, Neal JW, Zotova E, Nicoll JA (2008) Long-term effects of a$$\beta $$β42 immunisation in alzheimer’s disease: follow-up of a randomised, placebo-controlled phase i trial. The Lancet 372(9):216–223. doi:10.1016/S0140-6736(08)61075-2. http://www.sciencedirect.com/science/article/pii/S0140673608610752
22. Khalil HK (1996) Nonlinear systems. Prentice Hall, Upper Saddle River
23. LaSalle JP (1976) The stability of dynamical systems. In: CBMS-NSF regional conference series in applied mathematics SIAM
24. Laurén J, Gimbel DA, Nygaard HB, Gilbert JW, Strittmatter SM (2009) Cellular prion protein mediates impairment of synaptic plasticity by amyloid-$$\beta $$β oligomers. Nature 457(7233):1128–1132
25. Laurençot P, Walker C (2007) Well-posedness for a model of prion proliferation dynamics. J Evol Equ 7(2):241–264
26. 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 USA 93(3):1125–1129
27. Lomakin A, Teplow DB, Kirschner DA, B BG, (1997) Kinetic theory of fibrillogenesis of amyloid$$\beta $$β-protein. Proc Natl Acad Sci USA 94(15):7942–7947
28. Nath S, Agholme L, Kurudenkandy FR, Granseth B, Marcusson J, Hallbeck M (2012) Spreading of neurodegenerative pathology via neuron-to-neuron transmission of $$\beta $$β-amyloid. J Neurosci 32(26):8767–8777
29. Nygaard HB, Strittmatter SM (2009) Cellular prion protein mediates the toxicity of beta-amyloid oligomers: implications for Alzheimer disease. Arch Neurol 66(11):1325
30. Portet S, Arino J (2009) An in vivo intermediate filament assembly model. Math Biosci Eng 6(1):117–134
31. Prigent S, Ballesta A, Charles F, Lenuzza N, Gabriel P, Tine LM, Rezaei H, Doumic M (2012) An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation. PLoS One 7(11):e43,273
32. Prüss J, Pujo-Menjouet L, Webb G, Zacher R (2006) Analysis of a model for the dynamics of prions. Discret Contin Dyn Syst Ser B 6(1):225–235
33. Resenberger UK, Harmeier A, Woerner AC, Goodman JL, Müller V, Krishnan R, Vabulas RM, Kretzschmar HA, Lindquist S, Hartl FU, Gerd M, Winklhofer KF, Tatzelt J (2011) The cellular prion protein mediates neurotoxic signalling of $$\beta $$β-sheet-rich conformers independent of prion replication. EMBO J 30(10):2057–2070
34. Rubenstein R, Merz PA, Kascsak RJ, Scalici CL, Papini MC, Carp RI, Kimberlin RH (1991) Scrapie-infected spleens: analysis of infectivity, scrapie-associated fibrils, and protease-resistant proteins. J Infect Dis 164(1):29–35
35. Selkoe DJ (2008) Soluble oligomers of the amyloid $$\beta $$β-protein impair synaptic plasticity and behavior. Behav Brain Res 192(1):106–113
36. Serpell LC (2000) Alzheimers amyloid fibrils: structure and assembly. Biochim Biophys Acta Mol Basis Dis 1502(1):16–30
37. Simonett G, Walker C (2006) On the solvability of a mathematical model for prion proliferation. J Math Anal Appl 324(1):580–603
38. Urbanc B, Cruz L, Buldyrev S, Havlin S, Irizarry M, Stanley H, Hyman B (1999) Dynamics of plaque formation in Alzheimer’s disease. Biophys J 76(3):1330–1334
39. Vincent B, Cisse MA, Sunyach C, Guillot-Sestier MV, Checler F (2008) Regulation of $$\beta $$β app and prpc cleavage by $$\alpha $$α-secretase: mechanistic and therapeutic perspectives. Curr Alzheimer Res 5(2):202–211
40. Walsh DM, Lomakin A, Benedek GB, Condron MM, Teplow DB (1997) Amyloid $$\beta $$β-protein fibrillogenesis detection of a protofibrillar intermediate. J Biol Chem 272(35):22,364–22,372
41. Wimo A, Prince M (2010) World Alzheimer report 2010: the global economic impact of dementia. Technical report, Alzheimer’s Disease International
42. Zou WQ, Zhou X, Yuan J, Xiao X (2011) Insoluble cellular prion protein and its association with prion and Alzheimer diseases. Prion 5(3):172–178