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Research Article

A continuous-time mathematical model and discrete approximations for the aggregation of β-Amyloid

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Pages 109-136 | Received 04 May 2020, Accepted 11 Dec 2020, Published online: 11 Jan 2021

References

  • A.S. Ackleh, H.T. Banks, K. Deng, and S. Hu, Parameter estimation in a coupled system of nonlinear size-structured populations, Math. Biosci. Engin. 2 (2005), pp. 289–315.
  • A.S. Ackleh, J. Carter, K. Deng, Q. Huang, N. Pal, and X. Yang, Fitting a structured juvenile-Adult model for Green tree frogs to population estimates from capture-mark-recapture field data, Bull. Math. Biol. 74 (2012), pp. 641–665.
  • A.S. Ackleh, K. Deng, and X. Yang, Sensitivity analysis for a structured juvenile-adult model, Comput. Math. Appl. 64 (2012), pp. 190–200.
  • R.W. Atherton, R.B. Schainker, and E.R. Ducot, On the statistical sensitivity analysis of models for chemical kinetics, AZChE J. 21(3) (1975), pp. 441–448.
  • M. Bertsch, B. Franchi, N. Marcello, M.C. Tesi, and A. Tosin, Alzheimer's disease: a mathematical model for onset and progression, Math. Med. Biol. J. IMA 34(2) (2016), pp. 193–214.
  • L.M. Besser, D.P. Gill, S.E. Monsell, W. Brenowitz, D. Meranus, W. Kukull, and D.R. Gustafson, Body mass index, weight change, and clinical progression in mild cognitive impairment and Alzheimer's disease, Alzheimer Dis. Assoc. Disorders 28(1) (2014), pp. 36.
  • R.J.H. Beverton and S.J. Holt, On the Dynamics of Exploited Fish Populations, Fishery Investigations, Series II Volume XIX, Ministry of Agriculture, Fisheries and Food, 1957.
  • K. Beyreuther and C.L. Masters, Amyloid precursor protein (APP) and beta a4 amyloid in the etiology of alzheimer's disease: precursor-Product relationships in the derangement of neuronal function, Brain Pathol. 1 (1991), pp. 241–251. Brain Pathol. 1, pp. 241–251.
  • M.F. Bishop and F.A. Ferrone, Kinetics of nucleation-controlled polymerization. A perturbation treatment for use with a secondary pathway, Biophys. J. 46(5) (1984), pp. 631–644.
  • J. Busciglio, D.H. Gabuzda, P. Matsudaira, and P.A. Yankner, Generation of beta-amyloid in the secretory pathway in neuronal and nonneuronal cells, Proc. Natl. Acad. Sci. USA 90 (1993), pp. 2092–2096.
  • V. Calvez, N. Lenuzza, D. Oelz, J.P. Deslys, P. Laurent, F. Mouthon, and B. Perthame, Size distribution dependence of prion aggregates infectivity, Math. Biosci. 217(1) (2009), pp. 88–99.
  • H. Caswell, Matrix Population Models, Sinauer, Sunderland, MA, Vol. 1, 2000.
  • B. Caughey and P.T. Lansbury, Protofibrils, pores, fibrils, and neurodegeneration: separating the responsible protein aggregates from the innocent bystanders, Annu. Rev. Neurosci. 26 (2003), pp. 267–298.
  • F. Chiti and C.M. Dobson, Protein misfolding, functional amyloid, and human disease, Annu. Rev. Biochem. 75 (2006), pp. 333–366.
  • I.S. Ciuperca, M. Dumont, A. Lakmeche, P. Mazzocco, L. Pujo-Menjouet, H. Rezaei, and L.M. Tine, Alzheimer's disease and prion: an in vitro mathematical model, (2018).
  • S.I. Cohen, M. Vendruscolo, M.E. Welland, C.M. Dobson, E.M. Terentjev, and T.P. Knowles, Nucleated polymerization with secondary pathways. I. Time evolution of the principal moments, J. Chem. Phys. 135(6) (2011), pp. 08B615.
  • S.I.A. Cohen, M. Vendruscolo, C.M. Dobson, and T.P.J. Knowles, From macroscopic measurements to microscopic mechanisms of protein aggregation, J. Molecular Biol. 421(2–3) (2012), pp. 160–171.
  • C.M. Cowan, S. Quraishe, and A. Mudher, What is the pathological significance of tau oligomers?, Biochem. Soc. Trans. 40(4) (2012), pp. 693–697.
  • D.L. Craft, L.M. Wein, and D.J. Selkoe, A mathematical model of the impact of novel treatments on the Aβ burden in the Alzheimer's brain, CSF and plasma, Bull. Math. Biol. 64(5) (2002), pp. 1011–1031.
  • M.A. Dayeh, G. Livadiotis, and S. Elaydi, A discrete mathematical model for the aggregation of β-Amyloid, PLoS ONE 13(5) (2018), p. e0196402. Available at https://doi.org/https://doi.org/10.1371/journal.pone.0196402.
  • E.P. Dougherty and H. Rabitz, Computational kinetics and sensitivity analysis of hydrogen-oxygen combustion, J. Chem. Phys. 72(12) (1980), pp. 6571–6586.
  • S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer, New York, NY, 2005.
  • H. Engler, J. Prüss, and G.F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl. 324(1) (2006), pp. 98–117.
  • S.T. Ferreira, M.N. Vieira, and F.G. De Felice, Soluble protein oligomers as emerging toxins in alzheimer's and other amyloid diseases, IUBMB Life 59 (2007), pp. 332–345.
  • B. Franchi and M.C. Tesi, A qualitative model for aggregation-fragmentation and diffusion of β-amyloid in Alzheimer's disease, Rend. Semin. Mat. Univ. Politec. Torino 70 (2012), pp. 75–84.
  • P.E. Fraser, L.K. Duffy, M.B. O'Malley, J. Nguyen, H. Inouye, and D.A. Kirschner, Morphology and antibody recognition of synthetic beta-amyloid peptides, J. Neurosci. Res. 28 (1991), pp. 474–485.
  • D. Games, D. Adams, R. Alessandrini, R. Barbour, and P. Borthelette, Alzheimer-type neuropathology in transgenic mice overexpressing V717F β-amyloid precursor protein, Nature 373(6514) (1995), pp. 523.
  • C. Haass and D.J. Selkoe, Soluble protein oligomers in neurodegeneration: lessons from the Alzheimer's amyloid β-peptide, Nat. Rev. Mol. Cell. Biol. 8 (2007), pp. 101–112.
  • W. Hao and A. Friedman, Mathematical model on Alzheimer's disease, BMC Syst. Biol. 10 (2016), p. 108.
  • J.A. Hardy and D. Allisop, Amyloid deposition as the central event in the aetiology of Alzheimer's disease, Trends Pharmac 12 (1991), pp. 383–388. Trends in Pharmaca 12, 383–388.
  • J.A. Hardy and G.A. Higgins, Alzheimer's disease: the amyloid cascade hypothesis, Science 256 (1992), pp. 184–185.
  • D.M. Hartley, D.M. Walsh, C.P. Ye, T. Diehl, S. Vasquez, P.M. Vassilev, D.B. Teplow, and D.S.Selkoe, Protofibrillar intermediates of amyloid β-protein induce acute electrophysiological changes and neurotoxicity in cortical neurons, J. Neurosci. 19 (1999), pp. 8876–8884.
  • M. Helal, E. Hingant, L. Pujo-Menjouet, and G.F. Webb, Alzheimer's disease: analysis of a mathematical model incorporating the role of prions, J. Math. Biol. 69(5) (2014), pp. 1207–1235.
  • S.E. Hill, J. Robinson, G. Matthews, and M. Muschol, Amyloid protofibrils of lysozyme nucleate and grow via oligomer fusion, Biophys. J. 96(9) (2009), pp. 3781–3790.
  • L. Holcomb, M.N. Gordon, E. McGowan, X. Yu, S. Benkovic, P. Jantzen, K. Wright, I. Saad, R.Mueller, D. Morgan, S. Sanders, C. Zehr, K.O. Campo, J. Hardy, C.M. Prada, C. Eckman, S. Younkin, K. Hsiao, and K. Duff, Accelerated Alzheimer-type phenotype in transgenic mice carrying both mutant amyloid precursor protein and presenilin-1 transgenes, Nat. Med. 4 (1998), pp. 97–100.
  • K. Hsiao, P. Chapman, S. Nilsen, C. Eckman, Y. Harigaya, S. Younkin, F. Yang, and G. Cole, Correlative memory deficits, Aβ elevation, and amyloid plaques in transgenic mice, Science 274 (1996), pp. 99–102.
  • C.R. Jack Jr, D.S. Knopman, W.J. Jagust, R.C. Petersen, M.W. Weiner, P.S. Aisen, L.M. Shaw, P. Vemuri, H.J. Wiste, S.D. Weigand, and T.G. Lesnick, Tracking pathophysiological processes in Alzheimer's disease: an updated hypothetical model of dynamic biomarkers, Lancet Neur. 12(2) (2013), pp. 207–216.
  • H.G. Lee, G. Casadesus, X. Zhu, J.A. Joseph, G. Perry, and M.A. Smith, Perspectives on the amyloid-β cascade hypothesis, J. Alzheimer's Dis. 6 (2004), pp. 137–145.
  • F. Kametani and M. Hasegawa, Reconsideration of Amyloid hypothesis and tau hypothesis in Alzheimer's Disease, Front. Neurosci. 12 (2018), p. 25. 1–21.
  • K. Kepp, Ten challenges of the amyloid hypothesis of alzheimer's disease, J. Alzheimer's Dis.55 (2017), pp. 447–457.
  • W.L. Klein, G.A. Krafft, and C.E. Finch, Targeting small β-Amyloid oligomers: the solution to an Alzheimer's disease conundrum?, Trends Neurosci. 24 (2001), pp. 219–224.
  • T.P. Knowles, C.A. Waudby, G.L. Devlin, S.I. Cohen, and A. Aguzzi, An analytical solution to the kinetics of breakable filament assembly, Science 326(5959) (2009), pp. 1533–1537.
  • S. Kumar and J. Walter, Phosphorylation of amyloid beta (Aβ) peptides : A trigger for formation of toxic aggregates in Alzheimer's disease, AGING 3(8) (2011).
  • E. Kwessi, S. Elaydi, G.B. Dennis, and G. Livadiotis, Nearly exact discretization of single species, Nat. Res. Model. 31(4) (2018), p. e12167.
  • K.J. Laidler, Chemical Kinetics, 3rd ed., Harper & Row, New York, NY, 1987. p. 42.
  • F.M. Laferla, K.N. Green, and S. Oddo, Intracellular amyloid-beta in Alzheimer's disease, Nat. Rev. Neurosci. 8 (2007), pp. 499–509.
  • M.P. Lambert, A.K. Barlow, B.A. Chromy, C. Edwards, R. Freed, M. Liosatos, T.E. Morgan, I.Rozovsky, and B. Trommer, Diffusible, nonfibrillar ligands derived from ABeta-Amyloid1-42 are potent central nervous system neurotoxins, Proc. Natl. Acad. Sci. USA 95 (1998), pp. 6448–6453.
  • H.G. Lee, P.I. Moreira, X. Zhu, M.A. Smith, and G. Perry, Staying connected: synapses in Alzheimer disease, Am. J. Pathol 165 (2004), pp. 1461–1464.
  • S. Lesne, M.T. Koh, L. Kotilinek, R. Kayed, C.G. Glabe, A. Yang, M. Gallagher, and K.H. Ashe, A specific amyloid-beta protein assembly in the brain impairs memory, Nature 440 (2006), pp. 352–357.
  • C. Letellier, S. Elaydi, L.A. Aguirre, and A. Alaoui, Difference equation versus differential equations, a possible equivalence for the Rössler system, Physica D 195 (2004), pp. 29–49.
  • S. Linse, Monomer-dependent secondary nucleation in amyloid formation, Biophys. Rev. 9(4) (2017), pp. 329–338.
  • A. Lomakin, D.B. Teplow, D.A. Kirschner, and G.B. Benedek, Kinetic theory of fibrillogenesis of amyloid β-protein, Proc. Natl. Acad. Sci. USA. 94 (1997), pp. 7942–7947.
  • J. Masel and V.A. Jansen, Designing drugs to stop the formation of prion aggregates and other amyloids, Biophys. Chem. 88(1–3) (2000), pp. 47–59.
  • M.P. Mattson, B. Cheng, D. David, K. Bryant, I. Lieberburg, and R.E. Rydel, β-Amyloid peptides destabilize calcium homeostasis and render human cortical neurons vulnerable to excitotoxicity, J. Neurosci. 12 (1992), pp. 376–389.
  • R.E. Mickens, Applications of Non-standard Finite Difference Schemes, World Scientific, River Edge, NJ, 2000.
  • R.E. Mickens, Nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl. 8(9) (2002), pp. 823–947.
  • G. Morris, I. Clark, and B. Vissel, Inconsistencies and controversies surrounding the amyloid hypothesis of Alzheimer's disease, Acta Neuropath Communi. 2 (2014), p. 135. 1–40.
  • L. Mucke and D.J. Selkoe, Neurotoxicity of amyloid beta-protein: synaptic and network dysfunction, Cold Spring Harb. Perspect. Med. 2 (2012), p. a006338.
  • H. Naiki and K. Nakakuki, First-order kinetic model of Alzheimer's β-amyloid fibril extension in vitro, Lab. Invest. 74 (1996), pp. 374–383.
  • K.M. Rodrigue, K.M. Kennedy, and D.C. Park, Beta-amyloid deposition and the aging brain, Neuropsychol Rev. 19 (2009), pp. 436–450.
  • C.L. Ni, H.P. Shi, H.M. Yu, Y.C. Chang, and Y.R. Chen, Folding stability of amyloid-beta 40 monomer is an important determinant of the nucleation kinetics in fibrillization, FASEB J. 25 (2011), pp. 1390–1401.
  • J.M. Ortega, Matrix Theory: A Second Course, University Series in Mathematics, Springer, New York, NY, 1987.
  • M.M. Pallitto and R.M. Murphy, A mathematical model of the kinetics of β-Amyloid fibril growth from the denatured state, Biophys. J. 81 (2001), pp. 1805–1822.
  • C.J. Pike, D. Burdick, A.J. Walencewicz, C.G. Glabe, and C.W. Cotman, Neurodegeneration induced by β-amyloid peptides in vitro: the role of peptide assembly state, J. Neurosci. 13 (1993), pp. 1676–1687.
  • C.J. Proctor, D. Boche, D.A. Gray, and J.A. Nicoll, Investigating interventions in Alzheimer's disease with computer simulation models, PloS One 8(9) (2013), pp. e73631.
  • J. Pruss, L. Pujo-Menjouet, G. Webb, and R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. Ser. B 6(1) (2006), pp. 225–235.
  • I.K. Puri and L. Li, Mathematical modeling for the pathogensis of Alzheimer's disease, Plosone 5(12) (2010), pp. c15176.
  • D. Purves, G. Augustine, D. Fitzpatrick, W.C. William, L. Anthony-Samuel, L.E. White, R.D. Mooney, and M.L. Platt, Neuroscience, 5th ed., Sinauer Associates, Sunderland, MA, 2012. p. 713–
  • M. Sakono and T. Zako, Amyloid oligomers: formation and toxicity of Ab oligomers, FEBS 277 (2010), pp. 1348–1358.
  • B. Seilheimer, B. Bohrmann, L. Bondolfi, F. Muller, D. Stuber, and H. Dobeli, The toxicity of the Alzheimer's β-amyloid peptide correlates with a distinct fiber morphology, J. Struct. Biol. 119 (1997), pp. 59–71.
  • D.J. Selkoe, The molecular pathology of Alzheimer's disease, Neuron 6 (1991), pp. 487–498.
  • D.J. Selkoe and J. Hardy, The Amyloid hypothesis of Alzheimer's disease at 25 years, EMBO Molecular Med. 8 (2016), pp. 595–608.
  • S.L. Shammas, G.A. Gonzalo, S. Kumar, M. Kjaergaard, M.H. Horrocks, N. Shivji, E. Mandelkow, E. Knowles, and D. Klenerman, A mechanistic model of tau amyloid aggregation based on direct observational oligomers, Nat. Commun. 6 (2015), pp. 7025.
  • L.K. Simmons, P.C. May, K.J. Tomaselli, R.E. Rydel, K.S. Fuson, E.F. Brigham, S. Wright, I.Lieberburg, G.W. Becker, D.N. Brems, and W.Y. Li, Secondary structure of amyloid β peptide correlates with neurotoxic activity in vitro, Mol. Pharmacol. 45 (1994), pp. 373–379.
  • P.F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Corresp. Math. Phys. 10 (1838), pp. 113–121. Retrieved 3 December 2014.
  • D.M. Walsh, I. Klyubin, J.V. Fadeeva, W.K. Cullen, R. Anwyl, M.S. Wolfe, M.J. Rowan, and D.J.Selkoe, Naturally secreted oligomers of amyloid beta protein potently inhibit hippocampal long-term potentiation in vivo, Nature 416 (2002), pp. 535–539.
  • H.W. Wang, J.F. Pasternak, H. Kuo, H. Ristic, M.P. Lambert, B. Chromy, K.L. Viola, W.L. Klein, W.B. Stine, G.A. Krafft, and B.L. Trommer, Soluble oligomers of beta amyloid (1-42) inhibit long-term potentiation but not long-term depression in rat dentate gyrus, Brain Res. 924 (2002), pp. 133–140.
  • R.V. Ward, K.H. Jennings, R. Jepras, W. Neville, D.E. Owen, J. Hawkins, G. Christie, J.B. Davis, A. George, E.H. Karran, and D.R. Howlett, Fractionation and characterization of oligomeric, protofibrillar and fibrillar forms of β-amyloid peptide, Biochem. J. 348 (2000), pp. 137–144.
  • B.A. Yankner, L.K. Duffy, and D.A. Kirschner, Neurotrophic and neurotoxic effects of amyloid b-protein: reversal by tachykinin neuropeptides, Science J. 250 (1990), pp. 279–282.
  • Y. Yoshiike, R. Minai, Y. Matsuo, Y.R. Chen, T. Kimura, and A. Takashima, Amyloid oligomer conformation in a group of natively folded proteins, PLoS One 3 (2008), pp. e3235.