A qualitative analysis of an Aβ-monomer model with inflammation processes for Alzheimer’s disease

A qualitative analysis of an Aβ-monomer model with inflammation processes for Alzheimer’sdisease Ionel Ciuperca1, Laurent Pujo-Menjouet1, Leon MatarTine1, Nicolas Torres2 and Vitaly Volpert1,3 1CNRS, Ecole Centrale de Lyon, Université Jean Monnet, Universite Claude Bernard Lyon 1, ICJ UMR5208, INSA Lyon, Lyon, Villeurbanne 69622, France 2Departamento de Matemática Aplicada, Universidad de Granada, Granada, Andalusia, Spain 3Peoples’ Friendship University of Russia, Moscow 117198, Russia NT,0000-0001-6059-9754 We introduce and study a new model for the progression of Alzheimer’s disease (AD) incorporating the interactions of Aβ-monomers, oligomers, microglial cells and interleukins with neurons through different mechanisms such as protein polymerization, inflammation processes and neural stress reactions. To understand the complete interactions between these elements, we study a spatially homogeneous simplified model that allows us to determine the effect of key parameters such as degradation rates in the asymptotic behaviour of the system and the stability of equilibrium. We observe that inflammation appears to be a crucial factor in the initiation and progression of AD through a phenomenon of hysteresis with respect to the oligomer degradation rate d. This means that depending on the advanced state of the disease (given by the value of the Aβ-monomer degradation rate d: large value for an early stage and low value for an advanced stage) there exists a critical threshold of initial concentration of interleukins that determines if the disease persists or not in the long term. These results give perspectives on possible anti-inflammatory treatments that could be applied to mitigate the progression of AD. We also present numerical simulations that allow us to observe the effect of initial inflammation and monomer concentration in our model. 1. Introduction Understanding the origin and development of Alzheimer’s disease (AD) has been a challenging problem for biologists © 2024 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. Research Cite this article: Ciuperca I, Pujo-Menjouet L, Matar-Tine L, Torres N, Volpert V. 2024 A qualitative analysis of an Aβ-monomer model with inflammation processes for Alzheimer’s disease. R. Soc. Open Sci. 11: 231536. https://doi.org/10.1098/rsos.231536 Received: 13 October 2023 Accepted: 13 February 2024 Subject Category: Mathematics Subject Areas: applied mathematics, computational biology, biomathematics Keywords: Alzheimer’s disease, persistence, bifurcation analysis, hysteresis, inflammation Author for correspondence: Nicolas Torres e-mail: [email protected] Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 during the past few decades. As in many neurodegenerative diseases, AD is known to be associated with the misconformation, aggregation and propagation of different proteins in the nervous system [1–5]. They form stable oligomers that eventually accumulate in the so-called amyloid plaques and this phenomenon is believed to lead to a progressive irreversible neuronal damage. One of these proteins that appears to be relevant in the early stages of the development of AD are the Aβ-monomers, whose precise mechanisms of aggregation and diffusion are yet to be discovered. In this context, mathematical models arise as a useful approach to understand the different processes underlying AD. Several types of models have been considered, including from simple systems of ordinary differential equations to more complex partial differential equations, such as transport equations [6], reaction–diffusion models [7–10] and stochastic control models [11]. The goal of this article is to understand the complete interactions between Aβ-monomers, oligomers, microglial cells and interleukins through a new system of partial differential equations, involving the development of AD in the brain. Neurons produce Aβ-monomers that almost instantaneously start to polymerize into proto-oligomers. In this aggregation process, proto-oligomers are able to polymerize or depolymerize and once they reach a critical size they become stable under the form of Aβ-oligomers. These latter are assumed to be totally stable in the sense that neither polymerization nor depolymerization is possible for Aβ-oligomer equilibrium [12,13]. This mechanism on Aβ-oligomers is known as the amyloid cascade hypothesis and there is a general consensus that it is a key factor in the progression of AD [1,5]. Besides the mechanism of polymerization, oligomers interact with microglial cells, considered as auxiliary cells in the nervous systems regulating brain development. They induce an inflammation reaction through a chemical cascade in microglial cells, releasing interleukins [14,15]. These interleukins then activate an increase of Aβ-monomer production from the neurons. However, if the concentration of Aβ-oligomers is high enough, then a reaction of stress called unfolded protein response (UPR) [4] is triggered which leads to a decrease of Aβ-monomer production, while the rest of oligomers diffuse in the neuronal environment. In this context, two opposed mechanisms of stimulation and inhibition will determine the persistence of AD or not. Inflammatory reaction 4 Oligomers Proto-oligomers Interleukins IL-1 Aβ-monomers Diffusion Activation of microglial cells Microglial cells displace oligomers toward amyloid plaques Oligomers Amyloid plaques APP Neuron Increase of Aβ-monomers production Microglial cell 3 2 1 6 7 5 Figure 1. Schematic representation of Aβ-monomers and inflammation cycle. Neurons produce Aβ-monomers (1) that polymerize into proto-oligomers (2). These proto-oligomers eventually reach a critical size to become stable oligomers (3). They activate microglial cells triggering an inflammatory reaction (4) by producing interleukins. The interleukins stimulate neurons (5) to increase Aβ-monomer production, closing the positive feedback cycle. Moreover, when oligomer concentration is high, neurons are stressed (6) and decrease the Aβ-monomer production, while oligomers are displaced by microglial cells towards the amyloid plaques (7). 2 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 Moreover, oligomers are brought and displaced by microglia to the amyloid plaques, that is, an aggregate of Aβ-oligomers that becomes an inert element (no diffusion, no polymerization, no depolymerization). Each element of the system (monomers, proto-oligomers, and oligomers except those in the amyloid plaques) diffuses, with a size-dependent rate. Microglial cells can also have random motility, but they displace free Aβ-oligomers to the amyloid plaques through a chemotactic process and amyloid plaques will more likely develop where the concentration of microglial cells is high. These cells are known indeed to be very reactive to neuronal insults [16–19]. Inflammation processes seem to be crucial to control the disease progression [15] and to find possible therapeutic strategies to mitigate the negative effects of AD. For example, it is suggested by Rivers-Auty et al. [20] that diclofenac-based drugs might be associated with slower cognitive decline with possible perspectives on AD progression. However, despite epidemiological evidence, robust clinical trials have not been successful in providing efficacy evidence of such anti-inflammatory treatments [21–23]. On the other hand, in Ali et al. and Imbimbo et al. [24,25], it is suggested that anti-inflammatory treatments might be effective if they are applied years before the development of clinical symptoms. Furthermore, in Imbimbo et al. [25], it is mentioned that some anti-inflammatory treatments decrease the levels of Aβ by allosterically inhibiting the γ-secretase complex, which could give interesting perspectives in finding efficient cures. Other treatment suggestions include actions on multiple targets besides neuroinflammatory and neuroprotective effects such as anti-amyloid and anti-tau effects [26,27]. Bertsch et al. and Andrade-Restrepo et al. [9,10], using reaction–diffusion type equations, describe the initiation and progression of AD under the hypothesis of amyloid cascade where the Aβ in its oligomeric form is toxic for neurons. In our paper, in addition to the amyloid cascade hypothesis, we take into account the effect of inflammation on the progression of the disease. This inflammation appears through the process of recruitment of microglial cells and then the activation of interleukins (IL-1). As a general goal, we aim to understand the progression of AD through an analysis-compatible simplified version of this base model. The article is organized as follows. In §2, we introduce the main system of partial differential equations and we describe the reactions involving monomers, (proto-)oligomers, microglial cells and interleukins, which are summarized in figure 1. Then, in §3, we deal with a spatially homogeneous version of the main model, where polymerization and depolymerization processes are simplified. For this simplified model, we analyse the existence of steady states depending on the parameters. Finally, in §4, we present numerical simulations of the simplified model in order to observe the different possible dynamics of solutions and the stability of the steady states. 2. Mathematical model Let us detail each equation of the system. In this model, we consider that dynamics occur in a part of the brain considered as an open bounded domain Ω⊂ ℝd (with d∈{2, 3}) and the main variables of the system are given in table 1. Table 1. Variables of the mathematical model. variable definition ui(t,x)concentration of Aβ-proto-oligomers of size i u(t,x)concentration of Aβ-oligomers up(t,x)concentration of oligomers in the amyloid plaques m(t,x)concentration of Aβ-monomers M(t,x)concentration of microglial cells I(t,x)concentration of interleukins 3 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 2.1. Proto-oligomers Aβ-proto-oligomers have a size ranging from i= 2 to i=i0−1 and become oligomers when they reach the size i=i0 (see point (2) in figure 1). Equations for proto-oligomers with size i= [2, ⋯,i0−1] are given by ∂ui ∂t(t,x) = ri−1ui−1(t,x)m(t,x) + biui+ 1(t,x)−riui(t,x)m(t,x)−bui(t,x) + νiΔui(t,x), where r1 is the bi-monomeric nucleation rate (with the notation u1=m(t,x) the monomer concentration), bi is the rate of monomer loss from proto-oligomers and ri is the rate of monomer gain. The couple (ri,bi), i∈[2, ⋯,i0−1] is called kinetic coefficients with the notation bi=b if i≤i0−2 and bi0−1= 0. The first term of the right-hand side stands on the one hand for the bi-monomeric nucleation when i= 2 and on the other hand for the polymerization with rate ri−1 (i≥3) of a proto-oligomer of size i−1 with the contact of a monomer, giving then a proto-oligomer of size i. The second term describes the depolymerization with rate bi of a proto-oligomer of size i+ 1 to a proto-oligomer of size i. The third and fourth terms are related to the symmetric process respectively of polymerization and depolymerization of a proto-oligomer of size i. Finally, each proto-oligomer can diffuse with a size-dependent coefficient (the smaller the size, the faster the diffusion). 2.2. Free oligomers The variation of the Aβ-oligomer population is described as follows (see point (3) in figure 1): ∂u ∂t(t,x)=ri0−1ui0−1(t,x)m(t,x)−γ(M(t,x))u(t,x)−τ0u(t,x)+νi0Δu(t,x), where the first term of the right-hand side stands for the polymerization with rate ri0−1 of a protooligomer of size i0−1 with the contact of a monomer giving then an oligomer of size i0. The second term describes the recruitment of oligomers to the amyloid plaques by microglial cells M with a rate γ given by γ(M)=γ0+γ1M 1+γ2M, depending on M through a Michaelis–Menten function with parameters γi (i∈{0, 1, 2}) and the third term corresponds to the degradation of oligomers with rate τ0. Finally, each oligomer diffuses with rate νi0. It is important to remember here that oligomers neither polymerize nor depolymerize, unlike proto-oligomers. 2.3. Oligomers in the amyloid plaques The variation of the Aβ-oligomer population stuck in the amyloid plaques is described as follows (see point (7) in figure 1): ∂up ∂t(t,x) = γ(M(t,x))u(t,x)−τpup(x,t), where the first term of the right-hand side stands for the recruitment of free oligomers to the amyloid plaques by microglial cells M with a rate γ and the second term represents the corresponding loss with rate τp. We remind here that oligomers in the amyloid plaques neither polymerize, depolymerize nor diffuse. 2.4. Monomers The variation of the Aβ-monomer population is described as follows (see point (1) in figure 1): 4 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 ∂m ∂t(t,x)=−r1m2−∑ i= 2 i0−1 riui(t,x)m(t,x) + b∑ i= 2 i0−1 ui(t,x) +S(u(t,x), I(t,x)) −d m(t,x) + ν1Δm(t,x), where I(t,x) is the concentration of interleukins and the function S is given by (2.1)S(u,I) = τS 1+CunI,n≥1. The term S(u,I) is called the stress function. According to the form of this function, under a high concentration of oligomers u surrounding the neuron, this latter will be stressed and stop the production of Aβ-monomers, which means that S(u,I) is close to 0 (see point (6) in figure 1). We remark that the neuron can be torn between the decision of producing Aβ-monomers due to the inflammation (caused by the interleukins) and the stress caused by the amount of oligomers surrounding the neurons causing the UPR process that stops this Aβ production. Note that this object is one of the major key properties in our model. For simplicity, we do not take into account the fact that microglia produce Aβ-monomers and this will be considered in a future work with a more complex model. The first and second terms of the right-hand side stand, respectively, for the bi-monomeric nucleation and the polymerization of proto-oligomers of all sizes, while the third term describes the corresponding processes of depolymerization of proto-oligomers. The fourth term is the source term depending on the inflammation reaction caused by the interaction of Aβ-oligomers with microglial cells. The fifth term describes the degradation of the monomers with a rate d. This rate d may depend on oligomer concentration and behave as a Hill function, but for simplicity, we consider in the sequel that d is a given positive constant. Finally, the last term stands for the monomer diffusion ability with rate ν1. 2.5. Microglial cells The evolution of the microglial cell population is described as follows (see point (4) in figure 1): ∂M ∂t(t,x) = D1ΔM(t,x)−α∇ ⋅ (M(t,x)∇u(t,x)) +λM+α1u(t,x) 1+α2u(t,x)M ^−M(t,x)M(t,x)−σM(t,x), where the first term of the right-hand side stands for the diffusion of microglial cells with rate D1. The second term represents the chemotaxis of microglial cells in response to the increase of oligomer population. This chemotactic effect results in an activation of microglial cells due to the presence of oligomers which causes an inflammatory reaction with the production of interleukins (IL-1). The third term describes the proliferation of microglial cells at a constant rate λM. In the fourth term, M ^ is the maximum capacity of microglial cells in the neuron environment and the last term characterizes the loss of microglial cells with rate σ. 2.6. Interleukins The equation for the evolution of interleukins is (see point (5) in figure 1) ∂I ∂t(t,x) = DIΔI(t,x) + τ1u(t,x) 1+τ2u(t,x)M(t,x)−τ3I(t,x), where the first term of the right-hand side is the diffusion of the interleukins, and the second term represents the proliferation which depends on the concentration of oligomers through a Michaelis–Menten function with parameters τ1,τ2 and the microglial cells. The third term represents the loss of interleukins with rate τ3. We note that all equations are complemented with Neumann boundary conditions with zero flux through ∂Ω and the parameters of the system are non-negative real numbers. The main interactions of this system are summarized in figure 1. 5 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 3. A bi-monomeric simplified model In order to proceed to a full mathematical analysis, and understand the qualitative dynamics of the actors of this problem, we consider a simplified model version of the full system of partial differential equations. We assume a bi-monomeric nucleation, that is, two monomers can merge to form a free oligomer (m+m→u) and the intermediate proto-oligomer phase is absent. For this case, we assume that when a monomer attaches to a free oligomer, the latter does not change and the monomer is consumed (u+m→u). The equations of the simplified PDE system are the following: (3.1) ∂u ∂t=ν2Δu+r1m2−γ(M)u−τ0u, ∂up ∂t=γ(M)u−τpup, ∂m ∂t=ν1Δm+τS 1 + CunI−dm −r2um −r1m2, ∂M ∂t=D1ΔM−α∇ ⋅ (M∇u) + α1u 1+α2u(M ^−M)M−σM +λM, ∂I ∂t=DIΔI+τ1u 1+τ2uM−τ3I. We also assume that when a monomer binds to an oligomer, then the monomer is consumed with rate r2 and the number of oligomer molecules does not change. Under these assumptions, we notice that there is no term involving the rate r2 in the equation of oligomers. We could recall here that it is essential to consider all intermediate stages of oligomer formation, as they are important and could potentially play a significant role in the disease dynamics. However, we chose to study a simplified problem initially to conduct a thorough analysis of stability and highlight the phenomenon of hysteresis (see §4.1). This decision could, however, be biologically explained by the fact that the polymerization and depolymerization process is much faster compared with the degradation process, as was considered in previous works [10,11] with the choice of parameter values. Therefore, we consider the intermediate stages at equilibrium. This is obviously a significant simplification, and in our future work, we will incorporate all stages into a more comprehensive study. 3.1. Spatially homogeneous model In addition to the previous subsection, and to simplify the analysis in this work, we focus on spatially homogeneous solutions of the bi-monomeric model equation (3.1). We recall that this present work has two main objectives: (i) to introduce the most comprehensive possible model, which, in our view, is the most biologically realistic; and (ii) to propose an initial simplification to provide clear insights into a novel dynamics of this process. Of course, this simplification comes at the expense of tissue realism, especially spatial heterogeneity. The biological interpretation of the choice of homogeneity here is to localize the disease in a specific tissue (a region of the brain), where anti-inflammatory signals would have a notable effect. Naturally, in our future works, we will explore much more heterogeneous regions with potentially richer dynamics, but much more challenging to obtain and interpret. For simplicity, we assume that the rate of recruitment of oligomers to the amyloid plaques γ(M) is constant, which corresponds essentially to considering an average rate of oligomers being recruited and we consider that oligomers have a highly stable structure and their degradation is negligible, which means τ0= 0. However, the results of the qualitative analysis of the system do not change if we consider the degradation of oligomers. Under this setting, the model is reduced to the following system of ordinary differential equations: 6 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 (3.2) du dt =r1m2−γ0u, dup dt =γ0u−τpup, dm dt =τS 1+CunI−dm −r2um −r1m2, dM dt =α1u 1+α2u(M ^−M)M−σM +λM, dI dt =τ1u 1 + τ2uM−τ3I. Because of this simplification, we obtain the following result. Proposition 3.1. For any non-negative initial condition (u0,up0,m0,M0,I0), the system has a unique global solution which is bounded. Proof. Existence and uniqueness of a local solution are straightforward from the Cauchy–Lipschitz theorem for ordinary differential equations. For the positivity of solutions, consider the vector field F= (f1, …, f5) for y= (y1, …, y5)∈ ℝ5 given by f1(y)=r1y3 2−γ0y1, f2(y) = γ0y1−τpy2, f3(y) = τS 1+Cy1 nI−dy3−r2y1y3−r1y3 2, f4(y) = α1y1 1+α2y2 (M ^−y4)y4−σy4+λM, f5(y) = τ1y1 1 + τ2y2 y3−τ3y5, and observe that F satisfies the quasi-positivity property, that is, for all indices i∈{1, …, 5} we have ∀(yj)j≠i∈(ℝ+)4,fi(y1, …, yi−1, 0, yi+ 1, …, y5)≥0. Thus, from proposition 2.1 in Haraux [28], we conclude that the solution remains non-negative because of this property. We now assert that the solution remains bounded. Indeed, from the fourth equation of system (3.2), we conclude that if M is large enough then dM/dt< 0 and, therefore, M(t) remains bounded. By reapplying the same argument, we subsequently conclude the same result for the rest of the variables of the system. Since the solutions of system (3.2) are bounded, they are defined for all t> 0.∎ 3.2. Steady states The stationary points of system (3.2) correspond to solutions of the following system: (3.3) r1m2−γ0u= 0, γ0u−τpup= 0, τS 1 + CunI−dm −r2um −r1m2= 0, α1u 1+α2u(M ^−M)M−σM +λM= 0, τ1u 1 + τ2uM−τ3I= 0. One of the solutions of this system is the disease-free equilibrium, given by 0, 0, 0, λM σ, 0 . Besides this equilibrium, there may be other steady states depending on the parameter values of our system, whose existence will be studied in this section. Concerning the disease-free equilibrium, we get the following result. Proposition 3.2. For the system (3.2), the disease-free equilibrium 0, 0, 0, λM σ, 0 is locally asymptotically stable for every choice of positive parameters. 7 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 Proof. The Jacobian matrix around the vector 0, 0, 0, λM σ, 0 is given by J= −γ00 0 0 0 γ0−τp0 0 0 0 0 −d0τS α1M ^−λM σ λM σ0 0 −σ0 τ1 λM σ0 0 0 −τ3 , whose set of eigenvalues is given by {−γ0,−τp,−d,−σ,−τ3}. Since they are all negative, then the disease-free equilibrium is locally asymptotically stable. ∎ An interesting question is to determine under which parameter values the existence of non-trivial steady states (i.e. AD persists) holds. In this regard, we have the following result. Theorem 3.3. Assume that the parameters satisfy the condition (3.4)σγ0τ3<τ1τSλM. Then for d> 0 small enough, there exist at least two positive steady states of system (3.3). If d> 0 is large enough, then there are no positive solutions of system (3.3), regardless of condition (3.4). Proof. From system (3.3), we solve for u and up in terms of m and we get the following relation: u=ρm2, up=r1 τpm2with ρ=r1 γ0. From the equation of microglial cells, we solve the quadratic equation of M in terms of u and by taking the positive root we get the following equality: (3.5)M=Δ(u)−σ−(σα2−M ^α1)u 2α1u, with Δ(u)=(σ+ (σα2−M ^α1)u)2+ 4λMα1u(1+α2u). For the interleukins we get the relation I=τ1 τ3 ρm2 1 + τ2ρm2M. Substituting these expressions into the equation of m in equation (3.3), we obtain the equation with respect to m: (3.6) m(P(m) + d) = mF(m), where the functions P and F are given by (3.7) P(m)=r2ρm2+r1m, F(m) = 2τ1τSλM τ3 ρm(1+α2ρm2) [Δ(ρm2)+σ+ (σα2−M ^α1)ρm2](1 + τ2ρm2)(1 + Cρnm2n) . The disease-free equilibrium corresponds to the case when m= 0 in equation (3.6) . In order to get a positive steady state of system (3.2), we must determine the values where P(m)+d=F(m). From the definition of Δ(u), we remark that the denominator is strictly positive in the function F. We observe that F(0) = 0, F(m)>0 for m> 0 and F(m)→0 as m→∞, since the numerator is of order O(m3) and the denominator is of order O(m2n+ 4). Moreover F′(0) is given by F′(0) = r1τ1τSλM σγ0τ3> 0. From condition (3.4), we observe that P′(0) < F′(0), hence there exists m ~> 0 such that (3.8) P(m) < F(m) for all m∈(0, m ~) . 8 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 Let us denote m0= sup {m ~> 0: property holds}> 0 in equation (3.8). Since P(m)→∞ as m→∞, we conclude that m0<∞ and from continuity, we get (3.9)P(m) < F(m) for all m∈(0, m0), P(m0) = F(m0) . Let us now denote d ~= max y∈[0, m0](F(y)−P(y)), which is strictly positive by condition (3.9). Let y0∈(0, m0) such that F(y0)−P(y0)=d ~. We now take an arbitrary d such that 0<d<d ~. And the following inequalities hold: P(0) + d>F(0), P(y0)+d<F(y0), P(m0) + d>F(m0) . Therefore, there exists a positive solution of equation (3.6) in (0, y0) and another positive solution in (y0,m0). This proves the existence result. For the non-existence result, observe that F reaches a maximum, since F(0) = 0 and F(m)→0 as m→∞, and this maximum is independent of d. Hence, for d large enough, we have that P(m)+d> max y> 0 F(y)≥F(m) for all m≥0, and we conclude that there is no solution in that case. ∎ From the previous result, we assert that when the rest of the parameters are fixed, there exists a critical value of degradation rate of monomers d=dc, such that for d>dc the system (3.2) has only the disease-free equilibrium and for d<dc there are at least two positive solutions. From a biological point of view, this means that a high degradation of monomers can avoid the persistence of AD, while a lower degradation of monomers is not sufficient to stop the pathogenic cycle of monomers, oligomers and interleukins. Table 2. Parameter values for the numerical simulations of equation (3.2). parameter value units description r110−1l (mol)−1(months)−1bi-monomeric polymerization rate r210−1l (mol)−1(months)−1 polymerization rate of monomers attaching to oligomers dvariable (months)−1degradation rate of monomers γ05 × 10−2(months)−1recruitment rate of oligomers to the amyloid plaques τ11l (mol)−1(months)−1growth coefficient of interleukins τ21l (mol)−1growth coefficient of interleukins τ31(months)−1degradation rate of interleukins τp3 × 10−2(months)−1degradation rate of oligomers in the amyloid plaques τS1(months)−1coefficient of neural stress C1ln(mol)−ncoefficient of stress function n2 — power coefficient of stress function α11l2(mol)−2(months)−1growth coefficient of microglial cells α21l (mol)−1growth coefficient of microglial cells λM10−3mol l−1(months)−1rate of proliferation of microglial cells M ^1 capacity of microglial cells σ10−3(months)−1degradation rate of microglial cells 9 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 destruction due to the accumulation of oligomers in the amyloid plaques. In particular, the stress function equation (2.1) will also depend on the neural population. One example of a possible anti-inflammatory treatment is docosahexaenoic acid (DHA). It has been demonstrated that the onset of brain diseases is linked to a deficiency in DHA, the primary omega-3 fatty acid in the brain. DHA is an essential polyunsaturated fatty acid crucial for the proper functioning of our metabolism; since it is synthesized in insufficient quantities de novo, it needs to be included in our diet (found in fatty fish or nuts). DHA is a bioactive nutrient crucial for brain development and reduces the progression of cognitive decline [31]. It also enhances synaptosomal membrane fluidity, and reduces the accumulation of Aβ peptides, fibril formation and the pro-apoptotic effects of oligomers. Note that even if diet high in omega-3 does not necessarily reflect the level of omega-3 crossing the blood–brain barrier, some studies have highlighted a more significant passage of esterified DHA in phospholipids through a specific transporter, especially in the form of structured phospholipids [32]. This form has demonstrated pro-neurogenic and anti-oxidant effects [33]. Furthermore, DHA possesses anti-inflammatory properties, which could appear as a good therapeutical hope for future research. Concerning the dynamics of the full model incorporating the spatial dependence, the chemotaxis of microglial cells and the whole polymerization process of proto-oligomers are far from being fully understood. For the whole and complete model, we expect a similar phenomenon of hysteresis to the one observed in the spatial-homogeneous simplified model, though the analysis to prove the existence of steady states becomes way more challenging. Degradation d (1/months) Persistence Disease-free Monomer concentration m (mol/l) 2 Concentration (mol/l) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 00 50 100 150 Time (months) 200 250 1.8 1.6 1.4 u m M I up 1.2 1 0.4 0.2 0.8 0.6 0 0 0.35 0.3 0.2 0.1 0.4 dc 0.6 0.7 0.8 Critical mc Figure 10. Example 6. (Left) Numerical solution of system (3.2) with d= 0.35 (months)−1 and m0= 0.7 mol l−1. The parameters correspond to those in table 2 and the initial data in table 4. (Right) Asymptotic behaviour in terms of degradation rate of monomers d and the initial concentration of monomers m0. The value of m0 is indicated with an arrow and d by a vertical line. Degradation d (1/months) Persistence Disease-free Monomer concentration m (mol/l) 2 Concentration (mol/l) 1.2 1 0.8 0.6 0.4 0.2 00 50 100 150 Time (months) 200 300250 1.8 1.6 1.4 u m M I up 1.2 1 0.4 0.2 0.8 0.6 0 0 0.35 0.3 0.2 0.1 0.4 dc 0.6 0.7 0.8 Critical mc Figure 11. Example 7. (Left) Numerical solution of system (3.2) with d= 0.35 (months)−1 and m0= 1 mol l−1. The parameters correspond to those in table 2 and the initial data in table 4. (Right) Asymptotic behaviour in terms of degradation rate of monomers d and the initial concentration of monomers m0. The value of m0 is indicated with an arrow and d by a vertical line. 16 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 Ethics. This work did not require ethical approval from a human subject or animal welfare committee. Data accessibility. Codes are available in the Dryad Data Repository [34]. Declaration of AI use. We have not used AI-assisted technologies in creating this article. Authors’ contributions. I.C.: formal analysis, supervision; L.P.-M.: conceptualization, project administration, supervision, validation; L.M.-T.: conceptualization, supervision, validation; N.T.: formal analysis, writing—original draft; V.V.: conceptualization, supervision, validation. All authors gave final approval for publication and agreed to be held accountable for the work performed therein. Conflict of interest declaration. We declare we have no competing interests. Funding. This project has received support from Agence National de la Recherche PrionDiff Project-ANR-21CE15-0011. N.T. was supported by the grant Juan de la Cierva FJC2021-046894-I funded by MCIN/AEI and the European Union Next Generation EU/PRTR. V.V. has been supported by the RUDN University Strategic Academic Leadership Program. References 1. Haass C, Selkoe DJ. 2007 Soluble protein oligomers in neurodegeneration: lessons from the Alzheimer’s amyloid β-peptide. Nat. Rev. Mol. Cell. Biol. 8, 101–112. (doi:10.1038/nrm2101) 2. Sakono M, Zako T. 2010 Amyloid oligomers: formation and toxicity of Aβ oligomers. FEBS J. 277, 1348–1358. (doi:10.1111/j.1742-4658.2010. 07568.x) 3. Sengupta U, Nilson AN, Kayed R. 2016 The role of amyloid-β oligomers in toxicity, propagation, and immunotherapy. EBioMedicine 6, 42–49. (doi:10.1016/j.ebiom.2016.03.035) 4. Soto C. 2003 Unfolding the role of protein misfolding in neurodegenerative diseases. Nat. Rev. Neurosci. 4, 49–60. (doi:10.1038/nrn1007) 5. Cohen S, Linse S, Luheshi LM, Knowles TPJ. 2013 Proliferation of amyloid-β42 aggregates occurs through a secondary nucleation mechanism. Proc. Natl Acad. Sci. USA 110, 9758–9763. (doi:10.1073/pnas.1218402110) 6. Ciuperca IS, Dumont M, Lakmeche A, Mazzocco P, Pujo-Menjouet L, Rezaei H, Tine L. 2019 Alzheimer’s disease and prion: an in vitro mathematical model. Discrete Continuous Dyn. Syst. B 24, 5225. (doi:10.3934/dcdsb.2019057) 7. Matthäus F. 2006 Diffusion versus network models as descriptions for the spread of prion diseases in the brain. J. Theor. Biol. 240, 104–113. (doi:10.1016/j.jtbi.2005.08.030) 8. Matthäus F. 2009 The spread of prion diseases in the brain models of reaction and transport on networks. J. Biol. Syst. 17, 623–641. (doi:10. 1142/S0218339009003010) 9. Bertsch M, Franchi B, Marcello N, Tesi MC, Tosin A. 2016 Alzheimer’s disease: a mathematical model for onset and progression. Math. Med. Biol. 34, 193–214. (doi:10.1093/imammb/dqw003) 10. Andrade-Restrepo M, Lemarre P, Pujo-Menjouet L, Tine LM, Ciuperca SI. 2020 Modeling the spatial propagation of Aβ oligomers in Alzheimer’s disease. ESAIM: ProcS. 67, 30–45. (doi:10.1051/proc/202067003) 11. Hu J, Zhang Q, Meyer-Baese A, Ye M. 2022 Finite-time stability and optimal control of a stochastic reaction-diffusion model for Alzheimer’s disease with impulse and time-varying delay. Appl. Math. Model. 102, 511–539. (doi:10.1016/j.apm.2021.10.004) 12. Murphy RM, Pallitto MM. 2000 Probing the kinetics of β-amyloid self-association. J. Struct. Biol. 130, 109–122. (doi:10.1006/jsbi.2000.4253) 13. Nag S, Sarkar B, Bandyopadhyay A, Sahoo B, Sreenivasan VKA, Kombrabail M, Muralidharan C, Maiti S. 2011 Nature of the amyloid-β monomer and the monomer-oligomer equilibrium. J. Biol. Chem. 286, 13827–13833. (doi:10.1074/jbc.M110.199885) 14. Forloni G, Balducci C. 2018 Alzheimer’s disease, oligomers, and inflammation. J. Alzheimers Dis. 62, 1261–1276. (doi:10.3233/JAD-170819) 15. Kinney JW, Bemiller SM, Murtishaw AS, Leisgang AM, Salazar AM, Lamb BT. 2018 Inflammation as a central mechanism in Alzheimer’s disease. Transl. Res. Clin. Interventions 4, 575–590. (doi:10.1016/j.trci.2018.06.014) 16. Hansen DV, Hanson JE, Sheng M. 2018 Microglia in Alzheimer’s disease. J. Cell Biol. 217, 459–472. (doi:10.1083/jcb.201709069) 17. Mazaheri F et al. 2017 TREM2 deficiency impairs chemotaxis and microglial responses to neuronal injury. EMBO Rep. 18, 1186–1198. (doi:10. 15252/embr.201743922) 18. Ransohoff RM. 2016 A polarizing question: do M1 and M2 microglia exist? Nat. Neurosci. 19, 987–991. (doi:10.1038/nn.4338) 19. Sarlus H, Heneka MT. 2017 Microglia in Alzheimer’s disease. J. Clin. Invest. 127, 3240–3249. (doi:10.1172/JCI90606) 20. Rivers-Auty J, Mather AE, Peters R, Lawrence CB, Brough D. 2020 Anti-inflammatories in Alzheimer’s disease: potential therapy or spurious correlate? Brain Commun. 2, fcaa109. (doi:10.1093/braincomms/fcaa109) 21. Group AR et al. 2008 Cognitive function over time in the Alzheimer’s Disease Anti-inflammatory Prevention Trial (ADAPT): results of a randomized, controlled trial of naproxen and celecoxib. Arch. Neurol. 65, 896–905. (doi:10.1001/archneur.2008.65.7.nct70006) 22. ADAPT Research Group. 2007 Naproxen and celecoxib do not prevent AD in early results from a randomized controlled trial. Neurology 68, 1800– 1808. (doi:10.1212/01.wnl.0000260269.93245.d2) 23. Ozben T, Ozben S. 2019 Neuro-inflammation and anti-inflammatory treatment options for Alzheimer’s disease. Clin. Biochem. 72, 87–89. (doi: 10.1016/j.clinbiochem.2019.04.001) 17 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024 24. Ali MM, Ghouri RG, Ans AH, Akbar A, Toheed A. 2019 Recommendations for anti-inflammatory treatments in Alzheimer’s disease: a comprehensive review of the literature. Cureus 11, e4620. (doi:10.7759/cureus.4620) 25. Imbimbo BP, Solfrizzi V, Panza F. Are NSAIDs useful to treat Alzheimer’s disease or mild cognitive impairment? Front. Aging Neurosci. 2010, 19. (doi:10.3389/fnagi.2010.00019) 26. Huang LK, Chao SP, Hu CJ. 2020 Clinical trials of new drugs for Alzheimer disease. J. Biomed. Sci. 27, 1–13. (doi:10.1186/s12929-019-0609-7) 27. Zhu M, Wang X, Sun L, Schultzberg M, Hjorth E. 2018 Can inflammation be resolved in Alzheimer’s disease? Ther. Adv. Neurol. Disord. 11, 1756286418791107. (doi:10.1177/1756286418791107) 28. Haraux A. 2016 A simple characterization of positivity preserving semi-linear parabolic systems. (doi:https://arxiv.org/abs/1610.09909) 29. Lakshmikantham V, Bainov DD, Simeonov PS. 1989 Theory of impulsive differential equations. Singapore: World Scientific. (doi:10.1142/0906) 30. Samoilenko AM, Perestyuk NA. 1995 Impulsive differential equations. Singapore: World Scientific. (doi:10.1142/2892) 31. Belkouch M, Hachem M, Elgot A, Van AL, Picq M, Guichardant M, Lagarde M, Bernoud-Hubac N. 2016 The pleiotropic effects of omega-3 docosahexaenoic acid on the hallmarks of Alzheimer’s disease. J. Nutr. Biochem. 38, 1–11. (doi:10.1016/j.jnutbio.2016.03.002) 32. Hachem M etal. 2016 Efficient docosahexaenoic acid uptake by the brain from a structured phospholipid. Mol. Neurobiol. 53, 3205–3215. (doi: 10.1007/s12035-015-9228-9) 33. Lo Van A, Sakayori N, Hachem M, Belkouch M, Picq M, Fourmaux B, Lagarde M, Osumi N, Bernoud-Hubac N. 2019 Targeting the brain with a neuroprotective omega-3 fatty acid to enhance neurogenesis in hypoxic condition in culture. Mol. Neurobiol. 56, 986–999. (doi:10.1007/ s12035-018-1139-0) 34. Torres Escorza, Nicholas. 2024 Data from: A qualitative analysis of an Aβ-monomer model with inflammation processes for Alzheimer’s disease. Dryad. (doi:10.5061/dryad.mcvdnck6d) 18 royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 11: 231536 Downloaded from https://royalsocietypublishing.org/ on 21 June 2024