Alzheimer's disease and prion: An &lt;i&gt;in vitro&lt;/i&gt; mathematical model

Ciuperca, Ionel Sorin; Dumont, Matthieu; Lakmeche, Abdelkader; Mazzocco, Pauline; Praly, Laurent; Rézaei, Human; Tine, Léon Matar

doi:10.3934/dcdsb.2019057

Cited by 16 publications

(14 citation statements)

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“…A more recent discovery is that Aβ oligomers interact with the prion protein (PrP) to induce neurotoxicity, and different models of joint PrP-Aβ dynamics have been introduced [6,15,16]. These models propose an elaborate description of the degenerative effect oligomers have on the neurons, but they do not analyze the effects of their spatial spreading.…”

Section: Previous Mathematical Workmentioning

confidence: 99%

“…These models propose an elaborate description of the degenerative effect oligomers have on the neurons, but they do not analyze the effects of their spatial spreading. In [6,15], a size-continuous description of the aggregates is used, and while the latter can be a good approximation for very large aggregates it is less relevant for small oligomers. The relation between continuous and discrete protein sizes has been studied rigorously in [8,26,28].…”

Section: Previous Mathematical Workmentioning

confidence: 99%

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Modeling the spatial propagation of Aβ oligomers in Alzheimer’s Disease

et al. 2020

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Recent advances in the study of Alzheimer’s Disease and the role of Aβ amyloid formation have caused the focus of biologists to progressively shift towards the smaller protein assemblies, the oligomers. These appear very early on in the disease progression and they seem to be the most infectious species for the neurons. We suggest a model of spatial propagation of Aβ oligomers in the vicinity of a few neurons, without considering the formation of large ﬁbrils or plaques. We also include a simple representation of the oligomers neurotoxic effect. A numerical study reveals that the oligomers spatial dynamics are very sensitive to the balance between their diffusion and their replication, and that the outcome in terms of the progression of AD strongly depends on it.

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Section: Previous Mathematical Workmentioning

confidence: 99%

Section: Previous Mathematical Workmentioning

confidence: 99%

Modeling the spatial propagation of Aβ oligomers in Alzheimer’s Disease

et al. 2020

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“…Following Refs. 32,37 , we suppose that (1) the monomers undergo conformational changes into misfolded monomers which then are polymerized into small polymers involving diameter (two monomers), trimer (three monomers), etc. ; (2) the small polymers polymerize (depolymerize) into bigger (smaller) structures by attaching (losing) one misfolded monomer; (3) the maximum oligomer size is limited to six since this is relevant for transthyretin oligomers 31,32 .…”

Section: Chemical Master Equation Modelmentioning

confidence: 99%

Stochastic master equation for early protein aggregation in the transthyretin amyloid disease

Ruonan

Kang

2020

Sci Rep

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It is significant to understand the earliest molecular events occurring in the nucleation of the amyloid aggregation cascade for the prevention of amyloid related diseases such as transthyretin amyloid disease. We develop chemical master equation for the aggregation of monomers into oligomers using reaction rate law in chemical kinetics. For this stochastic model, lognormal moment closure method is applied to track the evolution of relevant statistical moments and its high accuracy is confirmed by the results obtained from Gillespie’s stochastic simulation algorithm. Our results show that the formation of oligomers is highly dependent on the number of monomers. Furthermore, the misfolding rate also has an important impact on the process of oligomers formation. The quantitative investigation should be helpful for shedding more light on the mechanism of amyloid fibril nucleation.

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“…Therefore an interdisciplinary collaboration between mathematicians and biologists lead to a new mathematical model where these considerations have been discussed and supported by biological assumptions and observations. While a number of studies focus either on aggregation of Aβ monomers (see for instance Lomakin et al (1997); Urbanc et al (1999); Craft (2002); Achdou et al (2013); Bertsch et al (2016)) or on PrP C proliferation (Greer et al (2006); ; Engler et al (2006); Calvez et al (2009); Gabriel (2011), to cite a few), to the best of our knowledge, only two mathematical models have been proposed to study Aβ/ PrP C interactions (Helal et al, 2014;Ciuperca et al). In the first one, authors describe plausible in vivo interactions between Aβ oligomers and PrP C , but did not focus on the whole process of polymerization (Helal et al, 2014).…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Stability analysis of a steady state of a model describing Alzheimer’s disease and interactions with prion proteins

Helal¹,

Igel-Egalon

Lakmeche³

et al. 2018

J. Math. Biol.

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Alzheimer's disease (AD) is a neuro-degenerative disease affecting more than 46 million people worldwide in 2015. AD is in part caused by the accumulation of A[Formula: see text] peptides inside the brain. These can aggregate to form insoluble oligomers or fibrils. Oligomers have the capacity to interact with neurons via membrane receptors such as prion proteins ([Formula: see text]). This interaction leads [Formula: see text] to be misfolded in oligomeric prion proteins ([Formula: see text]), transmitting a death signal to neurons. In the present work, we aim to describe the dynamics of A[Formula: see text] assemblies and the accumulation of toxic oligomeric species in the brain, by bringing together the fibrillation pathway of A[Formula: see text] peptides in one hand, and in the other hand A[Formula: see text] oligomerization process and their interaction with cellular prions, which has been reported to be involved in a cell-death signal transduction. The model is based on Becker-Döring equations for the polymerization process, with delayed differential equations accounting for structural rearrangement of the different reactants. We analyse the well-posedness of the model and show existence, uniqueness and non-negativity of solutions. Moreover, we demonstrate that this model admits a non-trivial steady state, which is found to be globally stable thanks to a Lyapunov function. We finally present numerical simulations and discuss the impact of model parameters on the whole dynamics, which could constitute the main targets for pharmaceutical industry.

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Alzheimer's disease and prion: An <i>in vitro</i> mathematical model

Cited by 16 publications

References 0 publications

Modeling the spatial propagation of Aβ oligomers in Alzheimer’s Disease

Modeling the spatial propagation of Aβ oligomers in Alzheimer’s Disease

Stochastic master equation for early protein aggregation in the transthyretin amyloid disease

Stability analysis of a steady state of a model describing Alzheimer’s disease and interactions with prion proteins

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