[1]
|
Y. Achdou, B. Franchi, N. Marcello and M. C. Tesi, A qualitative model for aggregation and diffusion of $\beta$-amyloid in Alzheimer's disease, J. Math. Biol., 67 (2013), 1369-1392.
doi: 10.1007/s00285-012-0591-0.
|
[2]
|
M. Ahmed, J. Davis, D. Aucoin, T. Sato, S. Ahuja, S. Aimoto, J. I. Elliott, W. E. Van Nostrand and S. O. Smith, Structural conversion of neurotoxic amyloid-$\beta$1-42 oligomers to fibrils, Nat. Struct. Mol. Biol., 17 (2010), 561-567.
|
[3]
|
B. Barz, Q. Liao and B. Strodel, Pathways of amyloid-$\beta$ aggregation depend on oligomer shape, Journal of the American Chemical Society, 140 (2018), 319-327.
|
[4]
|
V. R. Becker and W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen, Ann. Phys., 24 (1935), 719-752.
doi: 10.1002/andp.19354160806.
|
[5]
|
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., 34 (2017), 193-214.
doi: 10.1093/imammb/dqw003.
|
[6]
|
G. Bitan, M. Kirkitadze, A. Lomakin, S. S. Vollers, G. B. Benedek and D. B. Teplow, Amyloid $\beta$-protein (A$\beta$) assembly: A$\beta$40 and A$\beta$42 oligomerize through distinct pathways, Proc. Natl. Acad. Sci. U. S. A., 100 (2013), 330-335.
|
[7]
|
G. Bitan, M. D. Kirkitadze, A. Lomakin, S. S. Vollers, G. B. Benedek and D. B. Teplow, Amyloid $\beta$-protein (A$\beta$) assembly: A$\beta$40 and A$\beta$42 oligomerize through distinct pathways, Proc. Natl. Acad. Sci. U. S. A., 100 (2003), 330-335.
|
[8]
|
V. Calvez, N. Lenuzza, D. Oelz, J.-P. Deslys, P. Laurent, F. Mouthon and B. Perthame, Size distribution dependence of prion aggregates infectivity, Math. Biosci., 217 (2009), 88-99.
doi: 10.1016/j.mbs.2008.10.007.
|
[9]
|
N. Carulla, G. Caddy, D. R. Hall, J. Zurdo, M. Gairí, M. Feliz, E. Giralt, C. V. Robinson and C. Dobson, Molecular recycling within amyloid fibrils, Nature, 436 (2005), 554-558.
doi: 10.1038/nature03986.
|
[10]
|
M. Cisse and L. Mucke, A prion protein connection, Nature, 457 (2009), 1090-1091.
doi: 10.1038/4571090a.
|
[11]
|
D. Craft, A mathematical model of the impact of novel treatments on the a$\beta$burden in the Alzheimer's brain, CSF and plasma, Bull. Math. Biol., 64 (2002), 1011-1031.
|
[12]
|
H. Engler, J. Prüss and G. F. Webb, Analysis of a model for the dynamics of prions Ⅱ, J. Math. Anal. Appl., 324 (2006), 98-117.
doi: 10.1016/j.jmaa.2005.11.021.
|
[13]
|
D. Freir, A. Nicoll, S. Klyubin, I. qnd Panico, J. Mc Donald, E. Risse, E. Asante, M. Farrow, R. Sessions and H. e. a. Saibil, Interaction between prion protein and toxic amyloid $\beta$ assemblies can be therapeutically targeted at multiple sites, Nature Communications, 2 (2011), 336.
|
[14]
|
P. Gabriel, The shape of the polymerization rate in the prion equation, Math. Comput. Model., 53 (2011), 1451-1456.
doi: 10.1016/j.mcm.2010.03.032.
|
[15]
|
S. Gallion, Modeling amyloid-beta as homogeneous dodecamers and in complex with cellular prion protein, PloS One, 7 (2012), e49375.
doi: 10.1371/journal.pone.0049375.
|
[16]
|
J. B. Gilbert, The role of amyloid beta in the pathogenesis of Alzheimer's disease., J. Clin. Pathol., 66 (2013), 362-366.
|
[17]
|
D. A. Gimbel, H. B. Nygaard, E. E. Coffey, E. C. Gunther, J. Lauren, Z. A. Gimbel and S. M. Strittmatter, Memory impairment in transgenic alzheimer mice requires cellular prion protein, J. Neurosci., 30 (2010), 6367-6374.
doi: 10.1523/JNEUROSCI.0395-10.2010.
|
[18]
|
T. Goudon, F. Lagoutière and L. M. Tine, The Lifschitz-Slyozov equation with space-diffusion of monomers, Kinet. Relat. Model., 5 (2012), 325-355.
doi: 10.3934/krm.2012.5.325.
|
[19]
|
M. L. Greer, L. Pujo-Menjouet and G. F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theor. Biol., 242 (2006), 598-606.
doi: 10.1016/j.jtbi.2006.04.010.
|
[20]
|
M. Helal, A. Igel-Egalon, A. Lakmeche, P. Mazzocco, A. Perrillat-Mercerot, L. Pujo-Menjouet, H. Rezaei and L. Tine, Single molecule imaging reveals A$\beta$42: A$\beta$40 ratio-dependent oligomer growth on neuronal processes, Journal of Mathematical Biology, 104 (2018).
|
[21]
|
M. Helal, E. Hingant, L. Pujo-Menjouet and G. F. Webb, Alzheimer's disease: Analysis of a mathematical model incorporating the role of prions, J. Math. Biol., 69 (2014), 1207-1235.
doi: 10.1007/s00285-013-0732-0.
|
[22]
|
V. Hilser, J. Wrabl and H. Motlagh, Structural and energetic basis of allostery, Annual review of biophysics, 4 (2012), 585-609.
doi: 10.1146/annurev-biophys-050511-102319.
|
[23]
|
R. D. Johnson, J. A. Schauerte, C.-C. Chang, K. C. Wisser, J. C. Althaus, C. J. Carruthers, M. A. Sutton, D. G. Steel and A. Gafni, Single-molecule imaging reveals A$\beta$42: A$\beta$40 ratio-dependent oligomer growth on neuronal processes, Biophys. J., 104 (2013), 894-903.
|
[24]
|
N. Kandel, T. Zheng, Q. Huo and S. Tatulian, Membrane binding and pore formation by a cytotoxic fragment of amyloid $\beta$ peptide, The Journal of Physical Chemistry B, 121 (2017), 10293-10305.
|
[25]
|
E. Karran, M. Mercken and B. D. Strooper, The amyloid cascade hypothesis for Alzheimer's disease: An appraisal for the development of therapeutics, Nat. Rev. Drug Discov., 10 (2011), 698-712.
doi: 10.1038/nrd3505.
|
[26]
|
H. Kessels, L. Nguyen, S. Nabavi and R. Malinow, The prion protein as a receptor for amyloid-$\beta$, Nature, 446 (2010), E3-E5.
|
[27]
|
J. Laurén, D. A. Gimbel, H. B. Nygaard, J. W. Gilbert and S. M. Strittmatter, Cellular prion protein mediates impairment of synaptic plasticity by amyloid-$\beta$ oligomers, Nature, 457 (2009), 1128-1132.
|
[28]
|
I. Lifshitz and V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids, 19 (1961), 35-50.
doi: 10.1016/0022-3697(61)90054-3.
|
[29]
|
A. Lomakin, D. S. Chung, G. Benedek, D. A. Kirschner and D. B. Teplow, On the nucleation and growth of amyloid beta-protein fibrils: Detection of nuclei and quantitation of rate constants, Proc. Natl. Acad. Sci. U. S. A., 93 (1996), 1125-1129.
doi: 10.1073/pnas.93.3.1125.
|
[30]
|
A. Lomakin, D. B. Teplow, D. A. Kirschner and G. B. Benedek, Kinetic theory of fibrillogenesis of amyloid beta -protein, Proc. Natl. Acad. Sci. U. S. A., 94 (1997), 7942-7947.
|
[31]
|
S. Nath, L. Agholme, F. R. Kurudenkandy, B. Granseth, J. Marcusson and M. Hallbeck, Spreading of neurodegenerative pathology via Neuron-to-Neuron Transmission of -Amyloid, J. Neurosci., 32 (2012), 8767-8777.
doi: 10.1523/JNEUROSCI.0615-12.2012.
|
[32]
|
M. Nick, Y. Wu, N. Schmidt, S. Prusiner, J. Stöhr and W. DeGrado, A long-lived a$\beta$ oligomer resistant to fibrillization, Biopolymers.
|
[33]
|
J. Nunan and D. H. Small, Regulation of APP cleavage by alpha-, beta- and gamma-secretases, FEBS Lett., 483 (2000), 6-10.
|
[34]
|
W. Ostwald, Studien über die bildung und umwandlung fester körper, Z. phys. Chem., 22 (1897), 289-330.
doi: 10.1515/zpch-1897-2233.
|
[35]
|
M. Prince, A. Wimo, M. Guerchet, G.-C. Ali, Y.-T. Wu and M. Prina, World alzheimer report 2015 the global impact of dementia, Alzheimer's Dis. Int., (2015).
|
[36]
|
J. Prüss, L. Pujo-Menjouet, G. F. Webb and R. Zacher, Analysis of a model for the dynamics of prions, Discret. Contin. Dyn. Syst. Ser. B, 6 (2006), 225-235.
doi: 10.3934/dcdsb.2006.6.225.
|
[37]
|
M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Zeitschriftf. Phys. Chemie, (1917), 129.
doi: 10.1515/zpch-1918-9209.
|
[38]
|
A. L. Sosa-Ortiz, I. Acosta-Castillo and M. J. Prince, Epidemiology of dementias and alzheimer's disease, Arch. Med. Res., 43 (2012), 600-608.
doi: 10.1016/j.arcmed.2012.11.003.
|
[39]
|
B. Urbanc, L. Cruz, S. V. Buldyrev, S. Havlin, M. C. Irizarry, H. E. Stanley and B. T. Hyman, Dynamics of plaque formation in Alzheimer's disease, Biophys. J., 76 (1999), 1330-1334.
doi: 10.1016/S0006-3495(99)77295-4.
|