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

Ionel S. Ciuperca; Matthieu Dumont; Abdelkader Lakmeche; Pauline Mazzocco; Laurent Pujo-Menjouet; Human Rezaei; Léon M. Tine

doi:10.3934/dcdsb.2019057

Article Contents

2019, Volume 24, Issue 10: 5225-5260. Doi: 10.3934/dcdsb.2019057

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

Received: September 30, 2017

Revised: July 31, 2018

Early access: April 2019

Published: August 2019

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

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Abstract

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 PrP^c proteins. This interaction leads to the misconformation of PrP^c into pathogenic oligomeric prions, PrP^ol.

We develop here a model describing in vitro Aβ polymerization process. We include interactions between oligomers and PrP^c, causing the misconformation of PrP^c into PrP^ol. 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.

Keywords:

Mathematics Subject Classification: Primary: 35Q92, 82D60; Secondary: 35A02.

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Figure 1. Schematic representation of A$ \beta $ polymerization processes and interactions with PrP^c prions. All parameters, quantities and interactions are described in the main text

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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

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Figure 3. Evolution with time of A$ \beta $ monomers, A$ \beta $ oligomers, oligomers in plaque, prions PrP^c, prions PrP^ol and complexes, with only monomers and PrP^c initially

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Figure 4. Time evolution of the concentrations of A$ \beta $ monomers, oligomers, oligomers in plaque, PrP^c and PrP^sc, A$ \beta $ / PrP^c complexes and total mass

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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 $)

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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 $)

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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} $)

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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 PrP^c
$ \tau $	Duration of PrP^ol catalysis, with A$ \beta $ oligomers

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