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doi: 10.3934/dcdsb.2019057

Alzheimer's disease and prion: An in vitro mathematical model

1. 

Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex, France

2. 

Inria Team Dracula, Inria Grenoble Rhône-Alpes Center, 69100 Villeurbanne, France

3. 

University Sidi Bel Abbes, Laboratory of Biomathematics, Sidi Bel Abbes, Algeria

4. 

INRA, UR892 Virologie Immunologie Moléculaires, 78352 Jouy-en-Josas, France

* Corresponding author: pujo@math.univ-lyon1.fr

Received  October 2017 Revised  August 2018 Published  April 2019

Fund Project: L. P-M. has been supported by Association France-Alzheimer, SM 2014

Alzheimer's disease (AD) is a fatal incurable disease leading to progressive neuron destruction. AD is caused in part by the accumulation in the brain of Aβ monomers aggregating into oligomers and fibrils. Oligomers are amongst the most toxic structures as they can interact with neurons via membrane receptors, including PrPc proteins. This interaction leads to the misconformation of PrPc into pathogenic oligomeric prions, PrPol.

We develop here a model describing in vitro Aβ polymerization process. We include interactions between oligomers and PrPc, causing the misconformation of PrPc into PrPol. The model consists of nine equations, including size structured transport equations, ordinary differential equations and delayed differential equations. We analyse the well-posedness of the model and prove the existence and uniqueness of the solution of our model using Schauder fixed point and Cauchy-Lipschitz theorems. Numerical simulations are also provided to some specific profiles.

Citation: Ionel S. Ciuperca, Matthieu Dumont, Abdelkader Lakmeche, Pauline Mazzocco, Laurent Pujo-Menjouet, Human Rezaei, Léon M. Tine. Alzheimer's disease and prion: An in vitro mathematical model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019057
References:
[1]

Y. AchdouB. FranchiN. 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. AhmedJ. DavisD. AucoinT. SatoS. AhujaS. AimotoJ. I. ElliottW. 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. BarzQ. 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. BertschB. FranchiN. MarcelloM. 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. BitanM. KirkitadzeA. LomakinS. S. VollersG. 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. BitanM. D. KirkitadzeA. LomakinS. S. VollersG. 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. CalvezN. LenuzzaD. OelzJ.-P. DeslysP. LaurentF. 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. CarullaG. CaddyD. R. HallJ. ZurdoM. GairíM. FelizE. GiraltC. 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. EnglerJ. 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. GimbelH. B. NygaardE. E. CoffeyE. C. GuntherJ. LaurenZ. 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. GoudonF. 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. GreerL. 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. HelalE. HingantL. 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. HilserJ. 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. JohnsonJ. A. SchauerteC.-C. ChangK. C. WisserJ. C. AlthausC. J. CarruthersM. A. SuttonD. 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. KandelT. ZhengQ. 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. KarranM. 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. LaurénD. A. GimbelH. B. NygaardJ. 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. LomakinD. S. ChungG. BenedekD. 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. LomakinD. B. TeplowD. 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. NathL. AgholmeF. R. KurudenkandyB. GransethJ. 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. PrüssL. Pujo-MenjouetG. 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-OrtizI. 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. UrbancL. CruzS. V. BuldyrevS. HavlinM. C. IrizarryH. 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.

show all references

References:
[1]

Y. AchdouB. FranchiN. 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. AhmedJ. DavisD. AucoinT. SatoS. AhujaS. AimotoJ. I. ElliottW. 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. BarzQ. 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. BertschB. FranchiN. MarcelloM. 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. BitanM. KirkitadzeA. LomakinS. S. VollersG. 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. BitanM. D. KirkitadzeA. LomakinS. S. VollersG. 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. CalvezN. LenuzzaD. OelzJ.-P. DeslysP. LaurentF. 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. CarullaG. CaddyD. R. HallJ. ZurdoM. GairíM. FelizE. GiraltC. 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. EnglerJ. 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. GimbelH. B. NygaardE. E. CoffeyE. C. GuntherJ. LaurenZ. 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. GoudonF. 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. GreerL. 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. HelalE. HingantL. 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. HilserJ. 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. JohnsonJ. A. SchauerteC.-C. ChangK. C. WisserJ. C. AlthausC. J. CarruthersM. A. SuttonD. 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. KandelT. ZhengQ. 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. KarranM. 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. LaurénD. A. GimbelH. B. NygaardJ. 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. LomakinD. S. ChungG. BenedekD. 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. LomakinD. B. TeplowD. 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. NathL. AgholmeF. R. KurudenkandyB. GransethJ. 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. PrüssL. Pujo-MenjouetG. 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-OrtizI. 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. UrbancL. CruzS. V. BuldyrevS. HavlinM. C. IrizarryH. 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.

Figure 1.  Schematic representation of A$ \beta $ polymerization processes and interactions with PrPc prions. All parameters, quantities and interactions are described in the main text
Figure 2.  Evolution of size density repartition of fibrils $ f(t,x) $, proto-oligomers $ u(t,x) $ and fibrils in plaque $ f_a(t,x) $ for different times ($ t = 10, 20, 30, 40 $). The last figure displays the evolution of the total mass
Figure 3.  Evolution with time of A$ \beta $ monomers, A$ \beta $ oligomers, oligomers in plaque, prions PrPc, prions PrPol and complexes, with only monomers and PrPc initially
Figure 4.  Time evolution of the concentrations of A$ \beta $ monomers, oligomers, oligomers in plaque, PrPc and PrPsc, A$ \beta $ / PrPc complexes and total mass
Figure 5.  Evolution of size density repartition of fibrils $ f(t,x) $, proto-oligomers $ u(t,x) $ and fibrils in plaque $ f_a(t,x) $ for different times ($ t = 0,10,20 $)
Figure 6.  Evolution of size density repartition of fibrils $ f(t,x) $, proto-oligomers $ u(t,x) $ and fibrils in plaque $ f_a(t,x) $ for different times ($ t = 20,30,40 $)
Figure 7.  Evolution of A$ \beta $-42 type of species fibrils (after a time period of 10 units) (top left) and oligomers (top right) taking into account the same assumptions as in section 6 and oligomers with a faster kinetics (bottom). Here we consider the ratio of $ 10 \% $ of A$ \beta $-40 type species proposed in [23] with the same kinetics and then a faster kinetics for oligomers ($ g(x) = x^{1/2} $)
Table 1.  Description of model parameters. Parameters are given for $ i = 1,2 $, $ i = 1 $ corresponding to parameters related to A$ \beta $ -40
Parameter/Variable Definition
$ t $ Time
$ x $ Size of fibrils and proto-oligomers
$ x_0 $ Maximal size of A$ \beta $ proto-oligomers
$ \mu(x) $ Spontaneous creation of proto-oligomers or fibrils
$ v_i(t, x) $ Polymerization/depolymerization rate of A$ \beta $ proto-oligomers
$ v_{f, i}(t, x) $ Polymerization/depolymerization rate of A$ \beta $ fibrils
$ g_i(x) $ Rate at which A$ \beta $ monomers are added to proto-oligomers
$ g_{f,i}(x) $ Rate at which A$ \beta $ monomers are added to fibrils
$ b_i $ Rate at which A$ \beta $ monomers are lost from proto-oligomers
$ b_{f,i} $ Rate at which A$ \beta $ monomers are lost from fibrils
$ b_{a,i}(t) $ Rate of A$ \beta $ monomers escaping amyloid plaque
$ \gamma_i $ Displacement rate of A$ \beta $ oligomers into the plaque
$ \gamma_{f, i} $ Displacement rate of A$ \beta $ fibrils into the plaque
$ \delta_i $ Reaction rate between A$ \beta $ oligomers and PrPc
$ \tau $ Duration of PrPol catalysis, with A$ \beta $ oligomers
Parameter/Variable Definition
$ t $ Time
$ x $ Size of fibrils and proto-oligomers
$ x_0 $ Maximal size of A$ \beta $ proto-oligomers
$ \mu(x) $ Spontaneous creation of proto-oligomers or fibrils
$ v_i(t, x) $ Polymerization/depolymerization rate of A$ \beta $ proto-oligomers
$ v_{f, i}(t, x) $ Polymerization/depolymerization rate of A$ \beta $ fibrils
$ g_i(x) $ Rate at which A$ \beta $ monomers are added to proto-oligomers
$ g_{f,i}(x) $ Rate at which A$ \beta $ monomers are added to fibrils
$ b_i $ Rate at which A$ \beta $ monomers are lost from proto-oligomers
$ b_{f,i} $ Rate at which A$ \beta $ monomers are lost from fibrils
$ b_{a,i}(t) $ Rate of A$ \beta $ monomers escaping amyloid plaque
$ \gamma_i $ Displacement rate of A$ \beta $ oligomers into the plaque
$ \gamma_{f, i} $ Displacement rate of A$ \beta $ fibrils into the plaque
$ \delta_i $ Reaction rate between A$ \beta $ oligomers and PrPc
$ \tau $ Duration of PrPol catalysis, with A$ \beta $ oligomers
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