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

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.


Introduction.
1.1. Alzheimer's disease and interaction with prions. According to the World Alzheimer Report, in 2015 more than 46 million people were living with dementia worldwide [?]. With 60% to 80% dementia cases, Alzheimer's disease (AD) is considered as the most common dementia subtype [?]. AD is a fatal incurable disease leading to progressive neuron destruction, with memory impairment, issues to perform daily tasks and behavior changes as main consequences.
1.3. Objectives. Our aim herein is to introduce and study a new model describing the evolution of Aβ polymers and their interactions with PrP c . We study these mechanisms at the protein level and in a in vitro context. We describe Aβ polymerization process and the role of Aβ in the misconformation of PrP sc . We also distinguish Aβ-40 from Aβ-42, as they oligomerize in different ways [?]. Indeed, Aβ-42 monomers tend to aggregate faster than Aβ-40 monomers and to form larger polymers [?]. Moreover, presence of Aβ-42 polymers prevails in amyloid plaques, although Aβ-42 monomer concentration is around 10% of Aβ-40 concentration [?], and that impacts the emergence of AD. These are the reasons why we are interested in modeling distinctly the two dynamics, with different parameter values. This paper is organized as follows. We first present the mathematical model proposed to describe in vitro dynamics of Aβ and prions. We then investigate its well-posedness. Finally, we present some numerical simulations and discuss our results.
2. Mathematical modeling. We choose to build our model in an in vitro context, as, to the best of our knowledge, only in vitro data seem to be available. And so, to obtain a consistent qualitative behavior in a first step, then to quantitatively estimate parameters, we decide to study only in vitro mechanisms. We therefore consider no source term of monomers or prions and no degradation of any proteins involved either. We study evolution and impact of Aβ seeded at time t = 0 in an environment containing PrP c . Aβ-40 (respectively Aβ-42) monomers are able to aggregate to form small polymers [?] that can polymerize and depolymerize into bigger structures, by attaching or loosing one monomer. These structures as referred to as Aβ-40 (respectively Aβ-42) proto-oligomers. Once these proto-oligomers reach a maximal size x 0 , they are supposed to become stable structures called oligomers. We also consider that Aβ-40 (respectively Aβ-42) monomers can form Aβ-40 (respectively Aβ-42) fibrils in addition to proto-oligomers. These fibrils can polymerize and depolymerize, and can be carried out to β-amyloid plaques ( in vivo by astrocytes). In our model, we assume the existence of one big amyloid plaque in which fibrils can still depolymerize. Therefore monomers can be released from there. For both proto-oligomers and fibrils, we assume that they cannot be composed by a mix of Aβ-40 and Aβ-42 monomers. Once they have reached the maximal size x 0 , Aβ oligomers are able to interact with prions PrP c , and misfold them into PrP ol . It requires a fixed duration τ during which Aβ oligomer and PrP c form a complex. Once the process ends, the oligomer is released and can bind to an other prion. Aβ oligomers can also be carried out to β-amyloid plaque (in vivo by astrocytes). We assume that they are gathered into the same plaque as fibrils, with the difference that oligomers cannot depolymerize.
2.1. Notations. To study the evolution of different concentrations, defined at time t 0, let us denote by: -m i (t): concentration of Aβ monomers, -u i (t, x): size density of Aβ proto-oligomers, with 0 x < x 0 , -f i (t, x): size density of Aβ fibrils, with x 0, -f a,i (t, x): size density of Aβ fibrils inside Aβ plaque, with x 0, -u 0 i (t): concentration of Aβ oligomers, -u a,i (t): concentration of Aβ oligomers inside Aβ plaque, -p c (t): concentration of PrP c , -p sc (t): concentration of PrP sc , -C i (t): concentration of complex Aβ/PrP c , where i = 1 (respectively i = 2) stands for Aβ-40 (respectively Aβ-42). Definitions of model parameters (rates and growth velocities) are reported in Table 1.
Size of fibrils and proto-oligomers x 0 Maximal size of Aβ proto-oligomers µ(x) Spontaneous creation of proto-oligomers or fibrils v i (t, x) Polymerization/depolymerization rate of Aβ proto-oligomers v f,i (t, x) Polymerization/depolymerization rate of Aβ fibrils g i (x) Rate at which Aβ monomers are added to proto-oligomers g f,i (x) Rate at which Aβ monomers are added to fibrils b i Rate at which Aβ monomers are lost from proto-oligomers b f,i Rate at which Aβ monomers are lost from fibrils b a,i (t) Rate of Aβ monomers escaping amyloid plaque γ i Displacement rate of Aβ oligomers into the plaque γ f,i Displacement rate of Aβ fibrils into the plaque δ i Reaction rate between Aβ oligomers and PrP c τ Duration of PrP ol catalysis, with Aβ oligomers  2.2. Model for Aβ-40 and -42 polymerization. The first submodel describing the process of Aβ-40 and Aβ-42 polymerization formally consists of six partial differential equations and two ordinary differential equations (system (I)). As the equations are similar for both Aβ-40 and Aβ-42, we give the model for i = 1, 2, where i = 1 refers to the model for Aβ-40. Our model is based on Lifshitz-Slyozov equations [?], describing the growth process of grains, with a continuous size x.
Equations (1)-(2) describe Aβ polymerization in proto-oligomers or fibrils, through standard size structured advection-reaction equations. As proposed in [?], the polymerization rates are given by: for i = 1, 2. We further assume that g i and g f,i are increasing functions of x. Thus, these rates express a constant depolymerization of all polymers, while the polymerization process is accelerated by a high concentration of monomers and facilitated for longer polymers. Further assumptions on polymerization rates are given in Hypothesis 1.

Hypothesis 1. Polymerization rates
Rates v i and v f,i , for i = 1, 2, are required to satisfy the following conditions: It is important to note that for each time t, there exists a critical size x(t) > 0, for which polymerization rate is null, this critical size depending of the monomer concentration at time t. Therefore, polymers of size smaller than x(t) depolymerize whereas polymers of size greater than x(t) tend to attach more monomers. This phenomenon is referred to as Ostwald ripening [?]. Let us remark that x(t) can be greater than x 0 , and all proto-oligomers depolymerize in this case.
Finally, the term µ(x) in equations (1)-(2) represents the ability of monomers to spontaneously aggregate in polymers smaller than x 0 , to start the polymerization process. In our model, this function allows the creation of small proto-oligomers that could otherwise not exist due to the depolymerization of small polymers.

Hypothesis 2. Function µ
We assume that µ is a positive function with compact support, defined for all x in [0, +∞). Moreover, the function µ is in Finally, equation (4), describing the evolution of Aβ monomers, is given by the gain and loss in monomer from every fibrils and proto-oligomers.
To complete the system, initial conditions are given by: We further assume that: We also need boundary conditions in x = x 0 , for proto-oligomers: This condition represents the fact that no oligomers of size x 0 depolymerize, even if the rate of polymerization is negative. From Hypothesis 1 let mention that no boundary condition is required at x = 0, for the simple reason that the polymerization/depolymerization rates for proto-oligomers and fibrils are negative when x tends to 0.

2.3.
Model for Aβ-Prion interaction. We now introduce the second submodel, describing the interactions between Aβ oligomers and PrP c . Misconformation process of PrP c into PrP ol takes an incompressible duration, denoted τ , during which Aβ oligomer and PrP c form a complex. The oligomer is then released and can bind to another PrP c . This reaction leads to a system of 6 CIUPERCA, DUMONT, LAKMECHE ET AL.
Equation (7) describes the evolution of Aβ-40 and Aβ-42 oligomers, with time. The first term S i (t) stands for the source term of oligomers. It represents the creation rate of Aβ oligomers from proto-oligomers that reached the maximal size x 0 , and is the coupling with the previous system. The last terms describe the interaction between Aβ oligomers and PrP c , leading to the formation of PrP ol after a duration of τ units of time. Equation (8) describe the displacement of Aβ oligomers into amyloid plaques. Finally, equations (9)-(11) describe the evolution of prions and complexes.
We assume that PrP c are the only prion proteins initially in the experiment, ie all initial conditions at t = 0 are null, except for p c (0), which is equal to p 0 c and is positive. Then, on [0, τ ), the model is described using equations (7)-(11), without any delayed part, as no PrP ol and oligomer are released from a complex during the first τ units of time.
We now want to determine the expressions of S 1 (t) and S 2 (t), representing the source terms of Aβ-40 and Aβ-42 oligomers, that is the coupling between the first submodel and this second one. To do so, we use the property of mass conservation of the system. Indeed, as we are in an in vitro context, the total mass Q(t) remains the same during the study (no source term and no loss). We first compute the value of Q, denoting m p the size of a prion PrP c or PrP ol : + u a,i (t)) + m p (p c (t) + p sc (t)) + (x 0 + m p )(C 1 (t) + C 2 (t)).
We then computeQ(t), using equations (1)-(4) and (7)-(11). We finally obtain: which must be equal to zero. This equation gives sufficient conditions on S i : This condition gives an expression for the source term of oligomers, which is exactly the flow of proto-oligomers reaching the size x 0 . We can note that these source terms are non-negative, thanks to condition (6), and continuous.

Existence of solutions for the system (I).
To show the existence of solutions for the system (I), we based our analysis on the notion of "mild" solutions by introducing the characteristic curves associated to the kinetic rates at which monomers are added to or removal from fibrils or proto-oligomers. In the following definition we specify how "mild" solutions to equations (1)-(3) should be understood: We assume that a is a continuous function and satifies We also assume that b is a continuous function with respect to t and x. Let consider the linear transport problem that consists to find a solution U : where in the case L < ∞ we add the following boundary condition: Considering So, we call U to be a "mild" solution of (14)-(15) if for all (t, x) ∈ [0, T ] × (0, L) we have that the function s ∈ V t,x → U (s, X(s, t, x)) satisfies the following system With the previous definition, one can remark that X(s, t, x) is defined as the continuous extension of X(s, t, x) ats ∈ V t,x . Such extension always exists.
Theorem 1. Existence of solutions for system (I) Let Hypotheses 1, 2 and 3 hold. Then, for non-negative initial conditions, there exists T in (0, +∞) such that the system (I) has a unique non-negative "mild" solution Proof of existence of solutions follows an iterative process which is based on the fact that for a given functionm i , we can compute the mild solutionsũ i ,f i andf a,i , i = 1, 2. Using these mild solutions, we can now compute m i as the solution of equation (4). We build an application h that links each functionm i to the function m i , and show that it admits a fixed point, using Schauder fixed point theorem. This implies the existence of at least one solution of our model, corresponding to this fixed point. The whole proof is presented in section 4.
3.2. Existence of solutions for the system (II). We now focus on the system of delayed differential equations. We state our main results for this submodel.
Theorem 2. System (II) admits a unique solution on [0, +∞). Besides, these solutions are non-negative for non-negative initial conditions.
We first prove existence and uniqueness of solutions on [0, τ ) with Cauchy-Lipschitz theorem, and extend this result to well-chosen time intervals, likewise for the non-negativity. The whole proof is given in section 5.
Proof. Equation (1): For i = 1, 2, we rewrite as follow the equation (1) which models the dynamics of the two family of proto-oligomers (Aβ-40 and Aβ-42) Using the method of characteristics as depicted in Definition 1 of "mild" solution, we obtain is the Jacobian and X u,i is the characteristic curve associated v i .
Fors, on can easily check, by using the argument that characteristics not cross each other, that: i) For all fixed t ∈ (0, T ], the function x ∈ (0, x 0 ) →s(t, x) is increasing. Therefore, for all t ∈ (0, T ] the following limit exists: lim x→x0,x<x0s (t, x) and we denote it bys 0 (t).
ii) For all fixed t ∈ (0, T ], ∀x 1 , Lemma 2. With the additional assumption: u in i continuous on [0, x 0 ], one obtains for all t ∈ (0, T ] the existence of the following limit lim x→x0,x<x0 The proof of the lemma 2 stands on two cases: case 1: Let assumes 0 (t) = 0. So, we haves(t, x) = 0 ∀ x < x 0 and the "mild" solution take the form Using the dominated convergence theorem of Lebesgue, one has the existence of the limit lim x→x0,x<x0 We apply again the dominated convergence theorem of Lebesgue and deduce from the previous form of the "mild solution" the existence of the limit lim x→x0,x<x0 case 2: Let assumes 0 (t) > 0. For this case there exists x t ∈ (0, x 0 ) such thats(t, x) > 0 ∀ x ∈ (x t , x 0 ). So we get the following expression for the "mild" solution Let consider the sequence (x k ) k∈N → x 0 , with x k < x 0 and let prove the following convergence To prove the relation (20) we know thats(t, x k ) <s 0 (t), so one can compute The first term converge to 0 thanks to the dominated convergence theorem of Lebesgue and the second term goes to 0 thanks to the fact thats(t, x k ) →s 0 (t) and that the term under the integral is bounded. That achieves the proof of the convergence result.
Proof. In this proof we drop the index i for sake of simplicity. Let first prove the measurability ofū(t) thanks to the fact that the function (t, x) is measurable (see Annexe 1 for the proof).
Step 1: Let's prove that the "mild" solution given by (19) is a measurable function at (t, x). Let introduce the sets We split the solution as follows From the measurability ofs we deduce that A + and A 0 are measurable. Knowing that (t, is a continuous function on A 0 , so it is also measurable on A 0 . That achieves the proof of the measurability for u 1 . For the measurability of u 2 , we put The relation of equation (23) is based on the dominated convergence theorem of Lebesgue. The fact that the functions under the integral are bounded, it suffices to prove that Case 1. Let assume s / ∈ (y, t). In this case the result is straightforward because all terms vanish when k is high. Case 2. Let assume s ∈ (y, t). So, one need just to show Knowing that y k → y and t k → t then for k large enougth we have s ∈ (y k , t k ) that implies s belongs either to V t,x and to V t k ,x k . So X u (s, t k , x k ) −→ k→+∞ X u (s, t, x) thanks to the continuity of the characteristic equation. It remains to prove the convergence of the sequence of Jacobian functions and for that we need to prove the following result Here also we base our reasoning on the Lebesgue's dominated convergence theorem and achieve the proof by showing the pointwize convergence of the function under the previous integral for almost every σ ∈ (s, t). Subcase 2.1. If σ / ∈ [s, t) then the result is straightforward because one obtains 0 → 0. Subcase 2.2. If σ ∈ [s, t) then the fact that σ ∈ [s, t k ] for large k implies that one needs just t, x)). The proof stands on the fact that σ is Then from the continuity of ∂v ∂x with respect to X u we achieve the proof of the continuity of φ on D.
For the measurability of u 2 , one can write x,s(t, x)). We remark that g([0, T ] × (0, x 0 )) ⊂ D and is also measurable becauses is measurable. Then from Rudin'book [Theorem I. 7, page 10] we obtain that u 2 is a measurable function. What achieves the Step 1 of the proof.
Step 2 Knowing that u(t, x) is measurable, we apply the Fubini theorem and deduce the existence . Thenū(t) is measurable as limit of measurable sequence. Now we easily see that u is bounded since we integrate bounded function on bounded intervals. Then we haveū ∈ L ∞ (0, T ).
Using the change of variables y = X u,i (0, t, x) in the expression (19), we deduce: Equation (2) For i = 1, 2, characteristic curves associated to the growth velocity of fibrils v f,i are defined by: As done previously (here there is no maximal size for the fibrils, L = ∞), we obtain the unique mild solution: x))dσ) is the Jacobian. We then have: The estimations are directly derived from this relation.

Equation (3)
For the last equations, characteristic curves are defined as follow, for i = 1, 2 (no maximal size: So the unique mild solution reads which gives us the last estimation.

4.2.
Proof of theorem 1. Let denote by Σ T the subset of C([0, T ]) such as: where T is in (0, +∞) and M T is given by the subset above. We build the following mapping h: with m i (t) the solution of the following equation: whereÃ and functions (ũ i ,f i ,f a,i ) are solutions of the following system of PDE: To prove the existence of solutions, we follow a Schauder fixed point theorem.
Lemma 4. If with b a,i = sup Proof. Let (ũ i ,f i ,f a,i ) be mild solutions of system (31). Then, for all t in [0, T ],Ã i andB i are well-defined thanks to lemma 1. Their non-negativity is obvious as soon as initial data verify condition (5). Equation (28) is an ordinary differential equation and admits a continuous solution on [0, T ]. This implies that m i (t), i = 1, 2 is bounded by a constant M T , that can be computed.

ALZHEIMER'S DISEASE AND PRION 13
As functionÃ i is non-negative, we obtain We have to determine an upper bound forB i (t), using equation (30): Estimations (18) provide the needed upper bounds. Moreover,m i is upper-bounded by M T for all t lower than T , as it is in Σ T . We obtain: Because T verifies relation (32), we have: and the upper bound M T is well defined. To lighten notations, we drop out subscript i for now. We havė whereÃ m ,B m ,Ã n andB n are obtained from system (31). We are interested in the following quantity:ṁ We can transform this equality: We thus have: According to Grönwall's inequality, we obtain: where C T > 0.
Lemma 6. There exists α and β real positive constants, such that Lemma 6 and relation (36) are sufficient to prove the uniform equicontinuity of h. Then Ascoli theorem gives that h(Σ T ) is a relatively compact subspace of C 0 b ([0, T ]). Proof of lemma 6 is given in Appendix B.
Lemma 7. The application h defined in system (27) is a continuous application.
Proof. Let (m i,n ) n∈N a sequence of elements from Σ T which tends tom i in Σ T . Is the limit of h(m i,n ) equal to h(m i ) when n tends to infinity? We define sequences (ũ i,n ) n∈N , (f i,n ) n∈N and (f a,i,n ) n∈N , solutions of the following system of equations: These sequences are used to computeÃ i,n andB i,n such as: We proceed in the same way as in the proof of lemma 5 to obtain the following relation: We then apply lemma 6 and show that ifm i,n tends tom i when n tends to infinity, then it implies that h(m i,n ) tends to h(m i ), which obviously is in Σ T .
Then, according to Schauder fixed point theorem, the application h admits a fixed point m * i = h(m * i ). This implies that system (I) admits at least one solution. Uniqueness: to prove uniqueness of the solution let us assume that (u 1 , f 1 , f a,1 , m 1 ) and (u 2 , f 2 , f a,2 , m 2 ) are two solutions of the system (I) with the same initial data (u in , f in , f in a , m 0 ) as in equation (5). Using the same arguments as in the proof of lemma 5 (see equation (35)) one deduces Now, using the result of lemma 6 to estimate the right hand side of the previous inequality, we have So the Grönwall lemma gives then using the fact that we have the same initial data, means m 1 (0) = m 2 (0) = m 0 , we deduce m 1 (t) = m 2 (t) so f 1 ≡ f 2 , u 1 ≡ u 2 and f a1 ≡ f a2 . That concludes the uniqueness of the solution of (1)-(4).
The non-negativity of the unique solution of (1)-(4) is obvious as soon as initial data fulfill relation (5). The reader can easily check this point from explicit relations of mild solutions.

System (II)-Proof of the main results.
5.1. Existence and uniqueness of solutions. We first prove the existence of initial conditions on [0, τ ), defined by the following system, with i = 1, 2: with S i given by (13). Due to the non continuity of S i we can't directly apply the Cauchy-Lipschitz theorem. So, in order to prove the existence result we use the following change of unknown ψ i (t) = ϕ i (t) − t 0 S i (σ)dσ which is relevant because S i ∈ L ∞ (0, T ) thanks to Lemma 3. We 16 CIUPERCA, DUMONT, LAKMECHE ET AL.
rewrite the system (39) as follow For the existence let us note the vector . We have to solve the following Cauchy problem: where F (t, X) is defined by Function F is continuous for t and Lipschitz with respect to the second variable X. Indeed components F i , i = 1 to 7, are continuously differentiable with respect to the second variable. Cauchy-Lipschitz theorem gives the local existence and uniqueness of solution for problem (41). Thereby we have the local existence of solution for the system (39). The global existence of the solution of (39) on [0, τ ) requires the solution X(t) = t (ϕ 1 (t), ϕ 2 (t), ϕ a,1 (t), ϕ a,2 (t), ϕ pc (t), ϕ C1 (t), ϕ C2 (t)) to be bounded and non-negative on [0, τ ). To prove that, let us start with the initial conditions defined in system (39). We know that: This equation can easily be written as: ϕ pc (t) = p 0 c e (− t 0 (δ1ϕ1(s)+δ2ϕ2(s))ds) , which is positive for all t in [0, τ ], as p 0 c is greater than 0. In addition it is straightforward that ϕ pc (t) ≤ p 0 c .
Then, for i = 1, 2, we havė As ϕ i (0) = 0, we have the non-negativity of ϕ i (t) for t in [0, τ ], and i = 1, 2. In addition, one can deduce that ϕ(t) ≤ τ 0 S i (σ)dσ. As ϕ 1 (t), ϕ 2 (t) and ϕ pc (t) are greater or equal to 0, for all t in [0, τ ], functions ϕ a,i and ϕ Ci , i = 1, 2 are increasing. This implies the non-negativity of these functions for all t in [0, τ ], as ϕ a,i (0) and ϕ Ci , i = 1, 2 are null. One can easily verifies that ϕ a,i and ϕ Ci , i = 1, 2 are bounded. We further define X(τ ) as X(τ ) = lim t→τ − X(t). We prove existence and uniqueness of solutions of system (II) on [τ, +∞) with a method of steps. We first study the system (II) on [τ, 2τ ). We have to solve the following Cauchy problem: where Y (t) = t u 0 1 (t), u 0 2 (t), u a,1 (t), u a,2 (t), p c (t), p sc (t), C 1 (t), C 2 (t) , X(t) = t (ϕ 1 (t), ϕ 2 (t), ϕ a,1 (t), ϕ a,2 (t), ϕ pc (t), 0, ϕ C1 (t), ϕ C2 (t)), and G is defined by: Here, we perform again a change of variable as done previously in order to overcome the non continuity of S i , i = 1, 2. With the same strategy, we can actually re-write system (43) as follow:   Ẏ (t) =G(t,Ỹ (t), X (t − τ )) = G (t,Ỹ (t)), τ t < 2τ, Y (t) = X (t), 0 t < τ whereỸ (t) and X are respectively the same vectors as Y (t) andX(t) when replacing u 0 . As we did previously, we easily show that G is a continuous function, and continuously differentiable with respect to the second variable. So, Cauchy-Lipschitz theorem gives the local existence and uniqueness of solutions on [τ, 2τ ) for the above problem. That implies the local existence of solution to system (43). To prove that the solution is global we investigate again the positivity and the finite bounds of Y (t) = t u 0 1 (t), u 0 2 (t), u a,1 (t), u a,2 (t), p c (t), p sc (t), C 1 (t), C 2 (t) . For that, we begin with the relationṗ which is positive or null for all t in [τ, 2τ ), as p c (τ ) = ϕ pc (τ ) is greater or equal to 0. For the non-negativity of u 0 i (t), i = 1, 2 for all t in [τ, +2τ ), we havė For the upper bound of u 0 i one can remark thaṫ is known and correspond to the initial function ϕ i which is already bounded. Then a simple integration on [τ, t] with t < 2τ achieves the proof that u 0 i is bounded. Given the non-negativity of p c , u 0 1 and u 0 2 , we have, for all t in [τ, 2τ ): In addition, it is straightforward to verify that u a,i and p sc are bounded. We finally consider functions C i : Thanks to non-negativity of u 0 i and p c , this proves that C i , i = 1, 2 is non-negative on [τ, 2τ ) and obviously bounded. That achieves the global existence solution on [τ, 2τ ). We then iterate this process on intervals [nτ, (n + 1)τ ), n ≥ 2 and n ∈ N * , and obtain existence and uniqueness of solutions of system (II) on [0, T ].
6. Numerical simulations. In this section, we give illustrations of the dynamics of our model, through numerical simulations, using only one type of Aβ. The numerical scheme is based on a finite volumes method for the size discretization of the advection-reaction equations combined with a second order Runge-Kutta time discretization. We use the Van Leer flux limiters for the advection part which is known to be of order two. So, the numerical solutions of our model are TVD (Total Variation Diminishing) and of order two. We neglect any difficulties due to truncation of the computational domain and introduce the regular mesh with constant size step ∆x > 0: the cells are the intervals [x k−1 , x k ], k ∈ N with x k = (k + 1/2)∆x and x −1 = 0. We denote by F n k on of the numerical unknown (it can be the fibrils or the proto-oligomers or the fibrils inside the plaque). In the particular case where F = f , f n k is intended to be an approximation of 1 ∆x x k x k−1 f (t (n) , z)dz, where t (0) = 0 < t (1) < · · · < size-density (see equation (2)) is defined by the relation The interface fluxes, f lux n k = (vf ) n k and f lux * k = (v * f * ) n k are computed by using Van Leer approximation respectively with vf evaluated at time t (n) and at intermediate time t * thanks to the second order Runge-Kutta method. Here m (n) and m * are the numerical approximations of the monomers concentration respectively at first and second stage of the Runge-Kutta method based on the equation (4). For the flux with Van Leer limiter method, we compute: where the limiter function φ given by φ(θ) = 1 2 |θ| + θ 1 + |θ| .
We apply this scheme for the part of the model dealing with partial differential equations. For the other part of the model dealing with ordinary differential equation, the approximation is done thanks to the second order Runge-Kutta method. Boundaries conditions are taken into account thanks to fictious mesh added at the domain. For the parameters of the simulations we consider the followings: For all the simulations we take initial conditions for the quantities involved in the prion catalysis process as follow u 0 (t = 0) = 0, p c (t = 0) = 1, p sc (t = 0) = 0, C(t = 0) = 0, u a (t = 0) = 0, if 0 < x < 1.9, 0 elsewhere.
6.1. Results with free initial size-density repartition for fibrils, proto-oligomers and plaque. We first consider the case where there are only Aβ monomers and prions PrP c initially, which corresponds to what can be done experimentally.
In terms of initial conditions, we therefore have: f in (x) = 0, f in a (x) = 0 and u in (x) = 0. Figure 2 displays the evolution in time of the size density repartition of fibrils, proto-oligomers and fibrils in plaque, as well as the evolution of the total mass, which remains constant as expected. One can observe the creation of fibrils and proto-oligomers is only due to function µ, which allows to create small polymers. In this case, there are very few polymers with a large size. Evolutions of concentration of Aβ monomers, oligomers, oligomers in plaque, PrP c , PrP ol and complexes are presented in Figure 3. One can note that, because there is no polymer initially, few oligomers are created and thus, the emergence of PrP ol prions remains quite slow. 6.2. Results with gaussian initial distribution for fibrils, proto-oligomers and plaque. We now assume that proto-oligomers and fibrils are present initially with monomers and PrP c . Initial conditions are given by: In a first time we observe an increase in monomers, meaning that proto-oligomers and fibrils initially depolymerize. Then monomer concentration decreases, which corresponds to the formation of larger polymers. Oligomers appear after a certain time, and their concentration decreases  after a while, meaning that proto-oligomers do not reach the size x 0 . With the increase of Aβ oligomers, we notice the emergence of Aβ/ PrP c complexes and of PrP sc .
Figures 5 and 6 display the evolution of size density repartition of fibrils f (t, x), proto-oligomers u(t, x) and fibrils in plaque f a (t, x) for given times. One observes that fibrils become larger with time, but after a certain time there are more small fibrils due to the spontaneous term µ than large ones. Likewise, because fibrils in plaque only depolymerize, we notice a larger concentration of small ones. For proto-oligomers, we observe the impact of µ function as small proto-oligomers rapidly appear. Some proto-oligomers finally reach the maximal size x 0 and become oligomers.
7. Discussion. The role of Aβ oligomers and PrP c prions in Alzheimer's disease remains to be fully understood. Recent evidence suggests that Aβ oligomers can interact with PrP c to induce cytotoxic damages to neurons, increasing their apoptosis. Moreover, this interaction could misfold PrP c into pathogenic prions PrP sc , potentially leading to the emergence of prion diseases such as Creutzfeldt-Jakob disease. Mathematical modeling can help to qualitatively explain polymerization kinetics and evolution of polymer length that are involved in the emergence of AD.
In this work, we propose a mathematical model to describe the polymerization of Aβ monomers, and the interactions between Aβ oligomers and PrP sc . Polymerization process is modeled with partial differential equations, based on Lifshitz-Slyozov equations [?]. One can note that in our model, we study the evolution of three different species (proto-oligomers, fibrils and fibrils in plaque) through advection-reaction equations, making the analysis more complex. PrP ol catalysis, through interactions with Aβ oligomers, is described using ordinary and delayed differential equations. These two submodels are linked through the source term of oligomers coming from proto-oligomers, and can be studied one at a time. For the first one, we use Schauder fixed point theorem to prove existence and uniqueness of mild solutions, even in the case of singular polymerization rates. Existence and uniqueness of solutions for the second submodel are obtained with Cauchy-Lipschitz theorem. Numerical simulations with different initial conditions are given to illustrate the different profiles that can be obtained with this model. Because we have no experimental data available, we only provide simulations with one type of Aβ.
To the best of our knowledge, this is the first model describing both Aβ polymerization process and interactions with PrP c . However, because it is developed in an in vitro context, some in vivo processes are not included in the model. For instance, one could add the production of Aβ monomers on diseased neuronal membranes, as proposed in [?, ?]. Neurons could also be damaged due to the binding of Aβ oligomers to PrP c , as done in [?]. Nevertheless, we believe that our model gives insights on Aβ polymerization and on the interactions between Aβ and PrP c . It remains to compare our numerical simulations to experimental data and to find optimal parameter estimates. This can help to highlight differences between Aβ-40 and Aβ-42 and to identify new possible therapeutic targets to slow down or even avoid the emergence of Alzheimer's disease or prion diseases.
We now have to study I B1 , the first term in equation (56): In I 11 , we make the following substitution: y = X(s; t, x) which can be written as x = X(t; s, y). We then have: According to (57), there exists β 2 such as: I 11 e T GM T β 2 sup [0,T ] |m(t) −ñ(t)|.
We now go back to I B1 , and find that: Therefore, there exists a constant β 1 such as: Combining relations (57)