Scaling limit of a discrete prion dynamics model
aa r X i v : . [ m a t h . A P ] J u l Scaling limit of a discrete prion dynamics model
Marie Doumic ∗ Thierry Goudon † Thomas Lepoutre ‡§∗
October 24, 2017
Abstract
This paper investigates the connection between discrete and continuous modelsdescribing prion proliferation. The scaling parameters are interpreted on biological groundsand we establish rigorous convergence statements. We also discuss, based on the asymptoticanalysis, relevant boundary conditions that can be used to complete the continuous model.
Keywords
Aggregation fragmentation equations, asymptotic analysis, polymerization pro-cess.
AMS Class. No.
Introduction
The modelling of intracellular prion infection has been dramatically improved in the pastfew years according to recent progress in molecular biology of this pathology. Relevantmodels have been designed to investigate the conversion of the normal monomeric form ofthe protein (denoted PrPc) into the infectious polymeric form (denoted PrPsc) according tothe auto-catalytic process: PrPc + PrPsc −→ , in fibrillar aggregation of the protein. These models are based on linear growth of PrPscpolymers via an autocatalytic process [Eig96].The seminal paper by Masel et al. [MJN99] proposed a discrete model where the prionpopulation is described by its distribution with respect to the size of polymer aggregates.The model is an infinite-dimensional system of Ordinary Differential Equations, taking intoaccount nucleated transconformation and polymerization, fragmentation and degradationof the polymers, as well as production of PrPc by the cells. This model consists in anaggregation fragmentation discrete model. In full generality, it writes as follows: d v d t = λ − γv − v ∞ X i = n τ i u i + 2 X j ≥ n X i
1) for a constant β and k i,j was a uniform repartition over { , . . . j − } , i.e., k i,j = j − . These laws express that all polymers behave in the same way, and that any joint pointof any polymer has the same probability to break. It allowed the authors to close thesystem into an ODE system of three equations, which is quite simple to analyze. However,following recent experimental results such as in [SRH + +
09, CLD + admissible solutions, i.e., solutions obtained by taking the limit oftruncated systems (see Appendix A.3).Recent work by Greer et al. analyzed this process in a continuous setting [GPMW06].They proposed a Partial Differential Equation to render out the above-mentioned polymer-ization/fragmentation process. It writesd V d t = λ − γV − V Z ∞ x τ ( x ) U ( t, x ) d x + 2 Z ∞ x = x Z x y =0 yk ( y, x ) β ( x ) U ( t, x ) d x d y, (2) ∂U∂t = − µ ( x ) U ( t, x ) − β ( x ) U ( t, x ) − V ∂∂x ( τ U ) + 2 Z ∞ x β ( y ) k ( x, y ) U ( t, y ) d y. (3)The coefficients of the continuous model (2)(3) have the same meaning than those of thediscrete one (1); however, some questions about their scaling remain, and in particular aboutthe exact biological interpretation of the variable x. The aim of this article is to investigate the link between System (1) and System (2)(3).We discuss in details the convenient mathematical assumptions under which we can ensurethat the continuous system is the limit of the discrete one and we establish rigorously theconvergence statement. We also want to discuss possible biological interpretations of ourasymptotic analysis, and see how our work can help to define a proper boundary conditionat x = x for System (2)(3). Indeed, Eq. (3) holds in the domain x > x and, due to theconvection term, at least when V ( t ) τ ( x ) > ε appear, and state the main result: the asymptotic convergence of the rescaled discrete systemtowards the continuous equations. Section 3 is devoted to its proof, based on moments apriori estimates. Sections 4 and 5 discuss how these results can be interpreted on physicalgrounds. We also comment the issue of the boundary condition for the continuous model.2 Basic properties of the equations
All the considered coefficients are nonnegative. We need some structural hypothesis on k and k j,i to make sense. Obviously, the hypothesis take into account that a polymer can onlybreak into smaller pieces. We also impose symmetry since a given polymer of size y breaksequally into two polymers of size respectively x and y − x . Summarizing, we have k i,j ≥ , k ( x, y ) ≥ ,k j,i = 0 for j ≥ i k ( x, y ) = 0 for x > y, (4) k j,i = k i − j,i , k ( x, y ) = k ( y − x, y ) , (5) j − X i =1 k i,j = 1 , Z y k ( x, y ) d x = 1 . (6)(Note that (4) and (6) imply that 0 ≤ k i,j ≤ j − X i =1 ik i,j = j, Z y xk ( x, y ) d x = y. (7)The discrete equation belongs to the family of coagulation-fragmentation models (see[BC90],[BCP86]). Adapting the work of [BC90, BCP86] to this system, we obtain thefollowing result. It is not optimal but sufficient for our study. Theorem 1
Let k i.j satisfy Assumptions (4)–(6). We assume the following growth assump-tions on the coefficients (cid:26) There exists
K > , α ≥ , m ≥ and ≤ θ ≤ such that ≤ β i ≤ Ki α ≤ µ i ≤ Ki m , ≤ τ i ≤ Ki θ . (8) The initial data v ≥ , u i ≥ satisfies, for σ = max(1 + m, θ, α ) ∞ X i = n i σ u i < + ∞ . Then there exists a unique global solution to (1) which satisfies for all t ≥ v ( t ) + ∞ X i = n iu i ( t ) = v + ∞ X i = n iu i + λt − Z t γv ( s ) d s − Z t ∞ X i = n iµ i u i ( s ) d s. (9)A sketch of the proof is given in Appendix A.3. Let us introduce the quantity ρ ( t ) = v ( t ) + ∞ X n iu i ( t ) , (10)which is the total number of monomers in the population. Equation (9) is a mass balanceequation, which can be written asdd t ρ = λ − γv ( t ) − ∞ X i = n iµ i u i ( t ) . (11)Similarly for the continuous model we define ̺ ( t ) = V ( t ) + Z ∞ x xU ( t, x ) d x. ̺ ( t ) − ̺ (0) = λt − Z t γV ( s ) d s − Z t Z ∞ x xµ ( x ) U ( t, x ) d x. (12)In fact, the argument to deduce (12) from the system (2)(3) is two–fold: it relies bothon the boundary condition on { x = x } for (3) and on the integrability properties of thefragmentation term x × (cid:16) Z ∞ x β ( y ) k ( x, y ) U ( t, y ) d y − β ( x ) U ( t, x ) (cid:17) , the integral of which has to be combined to (2) by virtue of (7). The question is actuallyquite deep, as it is already revealed by the case where µ = 0, τ = 0 and x = 0. In thissituation it can be shown that (3) admits solutions that do not satisfy the conservation law: R ∞ xU ( t, x ) d x = R ∞ xU (0 , x ) d x , see [DS96]. Hence, (12) has to be incorporated in themodel as a constraint to select the physically relevant solution, as suggested in [DS96] and[LW07]. Nevertheless, the integrability of the fragmentation term is not a big deal since itcan be obtained by imposing boundedness of a large enough moment of the initial data as itwill be clear in the discussion below and as it appeared in [DS96, LW07]. More interesting ishow to interpret this in terms of boundary conditions; we shall discuss the point in Section4. (Note that in [LW07] the problem is completed with the boundary condition U ( t, x ) = 0while x > τ ( x ) > Definition 1
We say that the pair ( U, V ) is a “monomer preserving weak solution of theprion proliferation equations” with initial data ( U , V ) if it satisfies (2) and if for any ϕ ∈ C ∞ c (( x , ∞ )) , we have Z ∞ U ( t, x ) ϕ ( x ) d x − Z ∞ U ( x ) ϕ ( x ) d x = − Z t Z ∞ µ ( x ) U ( s, x ) ϕ ( x ) d x d s − Z t Z ∞ β ( x ) U ( s, x ) ϕ ( x ) d x d s + Z t V ( s ) Z ∞ τ ( x ) U ( s, x ) ∂ x ϕ ( x ) d x d s + 2 Z t Z ∞ x β ( y ) U ( s, y ) Z yx k ( x, y ) ϕ ( x ) d x d y d s, (13) and V ( t ) + Z ∞ x xU ( t, x ) d x = V + Z ∞ x xU ( x ) d x + λt − Z t γV ( s ) d s − Z t Z ∞ x xµ ( x ) U ( s, x ) d x d s. (14)A break is necessary to discuss the functional framework to be used in Definition 1. Westart with a set up of a few notation. We denote by M ( X ) the set of bounded Radonmeasures on a borelian set X ⊂ R ; M ( X ) stands for the positive cone in M ( X ). Thespace M ( X ) identifies as the dual of the space C ( X ) of continuous functions vanishing atinfinity in X , endowed with the supremum norm, see [Mal82]. Given an interval I ⊂ R ,we consider measure valued functions W : y ∈ I W ( y ) ∈ M ( X ). Denoting W ( y, x ) = W ( y )( x ) , we say that W ∈ C ( I ; M ( X ) − weak − ⋆ ), if, for any ϕ ∈ C ( X ), the function y R X ϕ ( x ) W ( y, x ) d x is continuous on I . We are thus led to assume U ∈ C ([0 , T ]; M ([0 , ∞ )) − weak − ⋆ ) , V ∈ C ([0 , T ]) , φ ∈ C ( X ) means hat φ is continuous and for any η >
0, there exists a compact set K ⊂ X such thatsup X \ K | φ ( x ) | ≤ η . We denote C c ( X ) the space of continuous functions with compact support in X . (cid:0) U ( t, . ) (cid:1) ⊂ [ x , ∞ ) , Z ∞ x xU ( t, x ) d x < ∞ , which corresponds to the physical meaning of the unknowns. Hence, formula (13) makessense for continuous coefficients µ, β, τ ∈ C ([ x , ∞ )) . Concerning the fragmentation kernel, it suffices to suppose y k ( · , y ) ∈ C ([ x , ∞ ); M ([0 , ∞ )) − weak − ⋆ ) . We first rewrite system (1) in a dimensionless form, as done for instance in [CGPV02] (seealso [LM07]). We summarize here all the absolute constants that we will need in the sequel: • T characteristic time, • U characteristic value for the concentration of polymers u i , • V characteristic value for the concentration of monomers v , • T characteristic value for the polymerisation rate τ i , • B characteristic value for the fragmentation frequency β i , • d characteristic value for the degradation frequency of polymers µ i , • Γ characteristic value for the degradation frequency of monomers γ , • L characteristic value for the source term λ, The dimensionless quantities are defined by¯ t = tT , ¯ v (¯ t ) = v (¯ tT ) V , ¯ u i (¯ t ) = u i (¯ tT ) U , ¯ β i = β i B , ¯ τ i = τ i T , ¯ µ i = µ i d , ¯ λ = λL , ¯ γ = γ Γ . We remind that k i,j is already dimensionless. The following dimensionless parameters appear a = LT V , b = BT, c = Γ
T, d = d Ts = UV , ν = T T V . (15)Omitting the overlines, the equation becomes d v d t = aλ − cγv − νsv P τ i u i + 2 bs X j ≥ n X i
There exists
K > (cid:12)(cid:12) β i +1 − β i (cid:12)(cid:12) ≤ Ki α − (cid:12)(cid:12) µ i +1 − µ i (cid:12)(cid:12) ≤ Ki m − , (cid:12)(cid:12) τ i +1 − τ i (cid:12)(cid:12) ≤ Ki θ − , (22)where the exponents α, θ, m are defined in (8). For the fragmentation kernel we assumefurthermore There exists
K > i, j (cid:12)(cid:12)(cid:12) i − X p =0 p − X r =0 k r,j +1 − i − X p =0 p − X r =0 k r,j (cid:12)(cid:12)(cid:12) ≤ K. (23)These assumptions will be helpful for investigating the behavior of (19) as ε goes to 0since they provide compactness properties. We summarize these properties in the followinglemmata. Lemma 1
Let (cid:0) z i (cid:1) i ∈ N be a sequence of nonnegative real numbers verifying ≤ z i ≤ Ki κ , (cid:12)(cid:12) z i +1 − z i (cid:12)(cid:12) ≤ Ki κ − for some K > and κ ≥ . For x ≥ , we set z ε ( x ) = P i ε κ z i χ [ εi,ε ( i +1)) ( x ) . Then thereexist a subsequence ε n → , and a continuous function z : x ∈ [0 , ∞ ) z ( x ) such that z ε n converges to z uniformly on [ r, R ] for any < r < R < ∞ . If κ > , the convergence holdson [0 , R ] for any < R < ∞ and we have z (0) = 0 . We shall apply this statement to the sequences β ε , µ ε , τ ε . A similar compactness propertycan be obtained for the fragmentation coefficients. Lemma 2
Let the coefficients k i,j satisfy Assumptions (5),(6) and (23). Then there ex-ist a subsequence (cid:0) ε n (cid:1) n ∈ N and k : y ∈ [0 , ∞ ) k ( · , y ) ∈ M ([0 , ∞ )) which belongs to C ([0 , ∞ ); M ([0 , ∞ )) − weak − ⋆ ) satisfying also (5) and (6) (in their continuous version)and such that k ε n converges to k in the following sense: for every compactly supportedsmooth function ϕ ∈ C ∞ c ([ x , ∞ )) , denoting φ ε n ( y ) = Z yn ( ε n ) ε n k ε n ( x, y ) ϕ ( x ) d x, φ ( y ) = Z yx k ( x, y ) ϕ ( x ) d x, (24) we have φ ε n → φ uniformly locally in [ x , + ∞ ) . The detailed proofs of Lemma 1 and Lemma 2 are postponed to Appendix A.2.7 .3 Main results
We are now ready to state the main results of this article.
Theorem 2
Assume (8) and (22) . Suppose the fragmentation coefficient fulfill (4) – (6) and (23) .Then, there exist a subsequence, denoted (cid:0) ε n (cid:1) n ∈ N , continuous functions µ, τ, β , and anonnegative measure-valued function k verifying (5) and (6), such that µ ε n , τ ε n , β ε n , k ε n → µ, τ, β, k in the sense of Lemma 1 and Lemma 2.Let the initial data satisfy (21) . Then we can choose the subsequence (cid:0) ε n (cid:1) n ∈ N such thatthere exists ( U, V ) for which (cid:26) u ε n ⇀ U, in C ([0 , T ]; M ([0 , ∞ )) − weak − ⋆ )) ,v ε n ⇀ V uniformly on [0 , T ] . We have xU ( t, x ) ∈ M ([0 , ∞ )) , the measure U ( t, . ) has its support included in [ x , + ∞ ) for all time t ≥ , and ( U, V ) satisfies (13) – (14) . Theorem 3
The limit ( U, V ) exhibited in Theorem 2 is a monomer preserving weak solution( i.e. satisfies also Equation (2)) in the following situations:i) x = 0 and either θ > (so that the limit τ satisfies τ (0) = 0 ), or the rates τ i = τ areconstant.ii) x > and the discrete fragmentation coefficients fulfill the following strengthenedassumption: for any i, j we have (cid:12)(cid:12)(cid:12)(cid:12) X i ′ ≤ i (cid:16) k i ′ j +1 − k i ′ ,j (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Kj , k i,j ≤ Kj . (25)
We start by establishing a priori estimates uniformly with respect to ε . These estimateswill induce compactness properties on the sequence of solutions. As described in [Lau02]for general coagulation fragmentation models, the model has the property of propagatingmoments. Lemma 3
Let the assumptions of Theorem 2 be fulfilled. Then for any
T > , there existsa constant C < ∞ which only depends on M , M σ , K and T , such that for any ε > : sup t ∈ [0 ,T ] Z ∞ (1 + x + x σ ) u ε ( t, x ) d x ≤ C, ≤ v ε ( t ) ≤ C Proof . For r ≥
0, we denote M εr ( t ) = ε ∞ X i = n ( iε ) r u εi ( t ) . As in [CGPV02], we can notice that Z ∞ (cid:16) x (cid:17) r u ε ( t, x ) d x ≤ M εr ( t ) ≤ Z ∞ x r u ε ( t, x ) d x. M ε ( t ) , M ε ( t ) and M ε σ ( t ) . We notice the obvious butuseful inequality, for 0 ≤ r ≤ σ, ( iε ) r ≤ iε ) σ , and therefore, | M εr | ≤ | M ε | + | M ε σ | . In the sequel, we use alternatively two equivalent discrete weak formulations of Equation(19) in the spirit of [LW07]. We multiply the second equation of (19) by ϕ i and summingover i , we first obtaindd t ∞ X i = n u εi ϕ i = − ε m ∞ X i = n µ i u εi ϕ i − ε α ∞ X i = n β i u εi ϕ i − ε θ − ∞ X i = n v ε ( τ i u εi − τ i − u εi − ) ϕ i + 2 ε α ∞ X i = n ϕ i X j>i β j k i,j u εj , = − ε m ∞ X i = n µ i u εi ϕ i − ε α ∞ X i = n β i u εi ϕ i + ε θ − ∞ X i = n τ i u εi ( ϕ i +1 − ϕ i )+2 ε α ∞ X i = n ϕ i X j>i β j k i,j u εj . (26)Using the properties of k i,j , we rewrite the fragmentation terms as follows ∞ X i = n β i u εi ϕ i = 2 ∞ X j = n +1 β j j − X i =1 ik i,j u εj ϕ j j + β n u εn ϕ n = 2 ∞ X j = n +1 j − X i = n ik i,j β j u εj ϕ j j + 2 ∞ X j = n +1 n − X i =1 ik i,j β j u εj ϕ j j + β n u εn ϕ n , ∞ X i = n ϕ i X j>i β j k i,j u εj = 2 ∞ X j = n +1 j − X i = n ik i,j β j u εj ϕ i i . By using (7), we obtain2 ∞ X i = n ϕ i X j>i β j k i,j u εj − ∞ X i = n β i u εi ϕ i = − ∞ X j = n n − X i =1 ik i,j β j u εj ϕ j j +2 ∞ X j = n +1 j − X i = n ik i,j β j u εj (cid:18) ϕ i i − ϕ j j (cid:19) . Replacing in the weak formulation we getdd t ∞ X i = n u εi ϕ i = − ε m ∞ X i = n µ i u εi ϕ i + ε θ − v ε ∞ X i = n τ i u εi ( ϕ i +1 − ϕ i )+2 ε α ∞ X j = n +1 j − X i = n ik i,j β j u εj (cid:18) ϕ i i − ϕ j j (cid:19) − ε α ∞ X j = n n − X i =1 ik i,j β j u εj ϕ j j . (27)This last formulation makes the estimates straightforward (the computations are formal butcan be understood as uniform bounds on solutions of truncated systems and therefore on9ny admissible solution). Taking φ i = iε, we obtain the first moment, that is, the previouslyseen mass balance:dd t (cid:18) v ε + ε ∞ X i = n iu εi (cid:19) = − γv ε − ε m ∞ X i = n µ i iu εi + λ ≤ λ. (28)Therefore, we get ( u εi and v ε are nonnegative)0 ≤ v ε ( t ) + M ε ( t ) ≤ ρ + λT for 0 ≤ t ≤ T < ∞ and Z t ε m ∞ X i = n µ i iu εi ( s, x ) d s ≤ ρ + λT for 0 ≤ t ≤ T < ∞ . To obtain an estimate on the 0th order moment, we take ϕ i = ε . The term with τ i vanishes.Considering only the nonnegative part of the derivative, we derive from (27)dd t M ε ( t ) ≤ ε α ∞ X j = n +1 j − X i = n ik i,j β j u εj i , ≤ ε α ∞ X j = n +1 β j u εj ≤ KM εα ( t ) . To give the bound on the (1+ σ )th moment, we choose ϕ i = ε ( εi ) σ in the weak formulation.Thanks to the mean value inequality, we have(( ε ( i + 1)) σ − ( εi ) σ ) ≤ (1 + σ ) ε ( ε ( i + 1)) σ ≤ (1 + σ )2 σ ε ( εi ) σ , therefore (27) yieldsdd t M ε σ ( t ) + ε m ∞ X i = n µ i ( εi ) σ u εi ≤ v ε (1 + σ )2 σ ∞ X i = n ε θ τ i u εi ε ( εi ) σ , ≤ K ( ρ + λT )(1 + σ )2 σ M εθ + σ ( t ) . Since 0 ≤ θ ≤
1, denoting C = max( K ( ρ + λT )(1 + σ )2 σ , K ), it leads todd t (cid:18) M ε ( t ) + M ε σ ( t ) (cid:19) ≤ C (cid:18) M εα ( t ) + M εθ + s ( t ) (cid:19) ≤ C (cid:18) M ε ( t ) + M ε σ ( t ) (cid:19) , and we conclude by the Gronwall lemma. It ends the proof of Lemma 3.Hereafter, we denote by C a constant depending only on T, M , ρ , M σ , K and λ suchthat M ε , v ε , M ε , M ε σ ≤ C. Lemma 4
Under the assumptions of Lemma 3, the sequence of monomers concentration ( v ε ) ε> is equicontinuous on [0 , T ] . Proof . We use the estimates of Lemma 3 to evaluate the derivative of v ε . We recall theequation satisfied by v ε d v ε d t = λ − γv ε + ε θ v ε X τ i u εi + 2 ε α X i ≥ n X j
By the Arzela-Ascoli theorem and Lemma 4, there exists afunction V ∈ C ([0 , T ]) and a subsequence that we still denote v ε such that v ε ( t ) −→ V ( t ) in C ([0 , T ]) . In the same way, the moment estimates of Lemma 3 give uniform boundedness for (1 + x + x σ ) u ε in M ([0 , ∞ )) . Pick a function ϕ ∈ C ∞ c ([0 , ∞ )). We define ϕ εi = Z ( i +1) εiε ϕ ( x ) d x, so that ∞ X n ε u εi ϕ εi = Z ∞ u ε ( t, x ) ϕ ( x ) d x, and also for y ∈ [ jε, ( j + 1) ε [, Z y k ε ( x, y ) ϕ ( x ) dx = Z jε k ε ( x, jε ) ϕ ( x ) dx = j X i =0 k i,j ϕ εi ε Thanks to the moment estimates of Lemma 3, and using (26), we have (cid:12)(cid:12)(cid:12)(cid:12) dd t Z u ε ( t, x ) ϕ ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( k ϕ k ∞ + k ϕ ′ k ∞ ) and (cid:12)(cid:12)(cid:12)(cid:12) Z u ε ( t, x ) ϕ ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ϕ k ∞ for some constant C depending only on K, M , M σ , λ, T . Therefore, for any function ϕ ∈ C ∞ c ([0 , ∞ )) , the integral R u ε ( · , x ) ϕ ( x ) d x is equibounded and equicontinuous. Using adensity argument, we can extend this property to ϕ ∈ C ([0 , ∞ )), the space of continuousfunctions on [0 , ∞ ) that tend to 0 at infinity. This means that (cid:0)R ∞ u ε ( ., x ) ϕ ( x ) d x (cid:1) ε belongsto a compact set of C (0 , T ). As in [CGPV02], by using the separability of C ([0 , ∞ )) and theCantor diagonal process, we can extract a subsequence u ε n and U ∈ C ([0 , T ]; M ([0 , ∞ )) − weak − ⋆ ), such that the following convergence Z ∞ u ε n ( t, x ) ϕ ( x ) d x → Z ∞ U ( t, x ) ϕ ( x ) d x, as ε n →
0, holds uniformly on [0 , T ], for any ϕ ∈ C ([0 , ∞ )). As u ε ( t, x ) = 0 for x ≤ εn ( ε ),we check that U ( t, . ) has its support in [ x , ∞ [. It remains to prove that ( U, V ) satisfies (13)(14).Let ϕ be a smooth function supported in [ δ, M ] with x < δ < M < + ∞ , choosing εn ( ε ) + 2 ε < δ (what is possible due to (20)). By using Lemma 1 and Lemma 5, we checkthat, for a suitable subsequence, one has Z ∞ µ ε n ( x ) u ε n ( t, x ) ϕ ( x ) d x −−−−→ ε n → Z ∞ µ ( x ) U ( t, x ) ϕ ( x ) d x, Z ∞ β ε n ( x ) u ε n ( t, x ) ϕ ( x ) d x −−−−→ ε n → Z ∞ β ( x ) U ( t, x ) ϕ ( x ) d x, Z ∞ τ ε n ( x ) u ε n ( t, x ) ϕ ( x ) d x −−−−→ ε n → Z ∞ τ ( x ) U ( t, x ) ϕ ( x ) d x, (29)11niformly on [0 , T ]. Equation (26) can be recast in the following integral formdd t Z ∞ u ε ( t, x ) ϕ ( x ) d x = − Z ∞ x µ ε u ε ( t, x ) ϕ ( x ) d x − Z ∞ τ ε u ε ∆ ε ϕ ( x ) d x − Z ∞ β ε u ε ( t, x ) ϕ ( x ) d x + 2 Z ∞ Z ∞ x ϕ ( x ) β ε ( y ) u ε ( t, y ) k ε ( x, y ) d x d y (30)where we have defined∆ ε ϕ ( x ) = Z ( i +1) εiε ϕ ( s + ε ) − ϕ ( s ) ε d s, for x ∈ [ iε, ( i + 1) ε [ . The first and third terms are treated in (29). Using (29) again and remarking that | ∆ ε ( x ) − ϕ ′ ( x ) | ≤ ε k ϕ ′′ k ∞ , we have Z ∞ τ ε n ( x ) u ε n ( t, x )∆ ε n ϕ ( x ) d x −−−−→ ε n → Z ∞ τ ( x ) U ( t, x ) ϕ ′ ( x ) d x, (31)uniformly on [0 , T ]. Let us now study the convergence of the last term in (30). To this end,we use the notation φ and φ ε as defined in (24) of Lemma 2 and we rewrite2 Z ∞ x Z yx ϕ ( x ) k ε ( x, y ) u ε ( t, y ) β ε ( y ) d x d y = 2 Z ∞ x u ε ( t, y ) β ε ( y ) φ ε ( y ) d y. Owing to (23) we use Lemma 2 which leads to φ ε n −−−−→ ε n → φ uniformly on [ x , R ] for any R > , and thus also β ε n φ ε n −−−−→ ε n → βφ uniformly on [ x , R ] for any R > , for a suitable subsequence. Finally, we observe that φ ε n and therefore φ are bounded by k ϕ k ∞ . Thus, by using the boundedness of the higher order moments of u ε in Lemma 3 with1 + σ > α , we show that the fragmentation term passes to the limit (see Lemma 5 in theAppendix). We finally arrive at Z ∞ x U ( t, x ) ϕ ( x ) d x − Z ∞ x U (0 , x ) ϕ ( x ) d x = − Z t Z ∞ x µU ( t, x ) ϕ ( x ) d x − Z t V ( s ) Z ∞ x τ ( x ) U ( s, x ) ϕ ′ ( x ) d x − Z t Z ∞ x β ( x ) U ( s, x )( t, x ) ϕ ( x ) d x + 2 Z t Z ∞ x β ( y ) U ( s, y ) Z y ϕ ( x ) k ( x, y ) d x d y, (32)which is the weak formulation (13). Moreover, (28) recasts as v ε ( t ) + Z ∞ e ε ( x ) u ε ( t, x ) d x = v ,ε ( t ) + Z ∞ e ε ( x ) u ε (0 , x ) d x + λt − γ Z t v ε ( s ) d s − Z t Z ∞ e ε ( x ) µ ε ( x ) u ε ( s, x ) d x d s e ε ( x ) = ∞ X i =0 εi χ [ iε, ( i +1) ε ) ( x ) . Clearly e ε ( x ) converges to x uniformly. Using the moment estimate in Lemma 3, with σ > v ε n ( t ) + Z ∞ e ε n ( x ) u ε n ( t, x ) d x −−−−→ ε n → V ( t ) + Z ∞ xU ( t, x ) d x uniformly on [0 , T ] as well as Z t Z ∞ e ε n ( x ) µ ε n ( x ) u ε n ( s, x ) d x d s −−−−→ ε n → Z t Z ∞ xµ ( x ) U ( s, x ) d x d s. (We refer again to Lemma 5, or a slight adaptation of it.) As ε n → Proof of Theorem 3.
We rewrite the rescaled ODE as dv ε dt = λ − γv ε − Z ∞ n ε τ ε ( x ) u ε ( t, x ) dx + 2 Z ∞ n ε β ε ( y ) u ε ( t, y ) Z n ε e ε ( x ) k ε ( x, y ) dx, Depending on the value of x , we have to care about the last term ( x >
0) or the next twolast term ( x = 0). As already remarked in the proof of Lemma 4, in case where x = 0,the fragmentation term can be dominated by2 ε α X i ≥ n X j
0, the above calculation gives solid intuitive ground to choose Equation (34)as a boundary condition, defining the incoming flux by means of a weighted average of thesolution over the size variable. In particular if the Dirac part vanishes we obtain
V τ ( x ) U ( t, x ) = 0 , the boundary condition proposed in [GPMW06], for constant coefficient τ . It is also theboundary condition used in [LW07].If x = 0 , the problem is still harder, since Equation (34) is empty. Dividing it by x > V τ (0) U (0) = 2 Z ∞ ψ + ( x ) β ( x ) U ( t, x ) d x. (35)Here again, it generalizes what has been proposed in [GPMW06] for τ constant and k ( x, y ) = y χ x ≤ y , though without any rigorous justification, and if ψ + = 0 it imposes a vanishingincoming flux. ε A biological discussion upon the parameter values can be found in [Len09] and is based on[MJN99, MGA05] and references therein.To carry out the previous scaling limit theorem, we made the following assumptions: s = UV = ε , ν = 1 ε , lim ε → εn ( ε ) = x , η = a = c = d = 1 . i the average size of polymers. Even if there still exists much uncertaintyupon its value, we can estimate that the typical size of polymers ranges between 15 and1000 , so we can write ε = 1 i ≪ . It is also known that the “conversion rate” of PrPc is around 5 to 10% at most (depending onthe experiment, on the stage of the disease, etc) ; it means that the mass of proteins presentin the monomeric form is much larger than the mass of proteins involved in polymers. Interms of characteristic values, it writes ε = i UV ≪ . Finally, we get: ε = r UV = √ ε ε ≪ . Hence, it legitimates the assumption on the parameters s and ε. Concerning the parameter a, we have a = L V ≈ , which is in the order of 1. We have only d ≤ . − : this shouldlead to neglect the degradation rate of polymers and simplify the equation.For the fragmentation frequency, it is in the order of the exponential growth rate, foundexperimentally to be in the order of 0 .
1; in the case of Masel’s articles [MJN99, MGA05], it issupposed that α = 1 , so it seems relevant (it leads to a fragmentation frequency in the orderof ε ). However, it has to be precisely compared to the other small parameters which aregiven by the typical size i and the conversion rate to justify the approximation. Moreover,the assumption of a linear fragmentation kernel β has to be confrounted to experiments.Concerning the aggregation rate T , and its related parameter ν = τ V, as shown in [Len09],in most cases we have ν in the range of [0 . , . , so it seems justified to suppose it small ;what has to be explored is its link with the other previously seen small parameters.To conclude (or open the debate), it seems that each specific experiment, like PMCAprotocole, in vitro or in vivo measures, or yet for the case of recombinant P rP (see [Rez08]),the orders of magnitude of each parameter should be carefully estimated, in order to adaptthe previous model and stick to the biological reality - which proves to be very different in in vivo , ex vivo or in vitro situations, or yet at the beginning (when there are still very fewpolymers) and at the end of experiences. The following discussion illustrates this idea, andgives some possible extensions to the previously seen models. k i,j To illustrate the central importance of a good estimate of the orders of magnitude, weexhibit here a case where the limit is not the continuous System (2)(3), but another one.Our calculation is formal, but a complete proof can be deduced from what preceeds andfrom [CGPV02].Let us take, instead of b = ε α : b = ε α − , and suppose also that the fragmentation kernel verifies: k ,i = k i − ,i = 12 (1 − εr i ) , k i,j = εk i,j r j , ≤ i ≤ j − . It means that the polymers are much more likely to break at their ends than in the middle oftheir chain. In that case, under Assumption (23) on k i,j and (22) on r j and β j , if α − ≤ σ, d v d t = λ − γv + v Z ∞ x τ ( x ) U ( t, x ) d x − Z ∞ x β ( x ) U ( t, x ) d x + 2 Z ∞ x = x Z x y =0 yk ( y, x ) r ( x ) U ( t, x ) d y d x,∂u∂t = − µ ( x ) U ( t, x ) − r ( x ) U ( t, x ) − v ∂∂x ( τ U ) + ∂∂x ( βu ) + 2 Z ∞ x r ( y ) k ( x, y ) U ( t, y ) d y. (36)Notice also that System (2)(3) includes the case of “renewal” type equations (refer to [Per07]for instance), meaning that the ends of the polymers are more likely to break. For instance,if we have, in the above setting: k i − ,i = k ,i = m i , k i,j = 0( 1 j ) , ≤ i ≤ j − , then Equation (36) remains valid, but we have to write the boundary condition as: τ ( x = 0) U ( x = 0) = Z m ( y ) U ( t, y ) d y. (37)This indeed can also be written as the measure of 0 by dµ y = k ( x, y ) d x : m ( y ) = µ y ( { } ) . Both of these cases mean that the ends of polymers are more likely to break. Whatchanges is the order of magnitude of what we mean by “more likely to break”: is it inthe order of ε , in which case System (2)(3) is valid but with a boundary condition of type(37) ? Or is the difference of the order of ε , in which case Equation (36) is more likely ?Refer to [Len09] for a more complete investigation of what model should be used in whatexperimental context. n We have seen above that to have x = 0 , it suffices to make Assumption (20). Having alsoseen that the typical size i is large, and that ε = 1 i Mm V , Mm V ≪ , it is in any case valid to suppose that1 i = ε c , < c < . Hence, Assumption (20) can be reformulated as: n ≪ i c . (38)For c = 1 , it means n ≪ i , which is true. On the contrary, if we suppose that x > , it means that n ≈ i c : in most cases, where for instance i = 100 or i = 1000 , it seemsirrelevant. A Appendix
A.1 Compactness of the coefficients
Proof of Lemma 1.
We refer to [CGPV02] for the case κ = 0. We prove here the case κ >
0. First, we show that z ε is close to a subsequence satisfying the requirements of the17rzela–Ascoli theorem on [ r, R ]. We define ˜ z ε by˜ z ε ( x ) = ε κ z i + ε κ z i +1 − z i ε ( x − iε ) for iε ≤ x ≤ ( i + 1) ε. We have | ˜ z ε ( x ) − z ε ( x ) | = | ε κ z i +1 − z i ε ( x − iε ) | , ≤ ε κ | z i +1 − z i | , ≤ εK ( εi ) κ − ≤ ε ( Kr κ − + KR κ − ) . Furthermore ˜ z ε has a bounded derivative since (cid:12)(cid:12)(cid:12)(cid:12) d˜ z ε d x (cid:12)(cid:12)(cid:12)(cid:12) = ε κ z i +1 − z i ε , ≤ K ( εi ) κ − , ≤ Kr κ − + KR κ − . Therefore, the family ˜ z ε satisfies the requirements of Arzela Ascoli theorem for any interval[ r, R ] with 0 < r < R < + ∞ . We can extract a subsequence converging uniformly to z . Thelimit is continuous and satisfies z ( x ) ≤ Kx κ . When κ > , R ]owing to the remark sup x ∈ [0 ,r ] (cid:12)(cid:12) ( z ε − z )( x ) (cid:12)(cid:12) ≤ Kr.
This concludes the proof.During the proof of Theorem 2 and Theorem 3 we made repeated use of the followingclaim.
Lemma 5
Let z n converge to a continuous function z uniformly on [0 , M ] for any A.2 Compactness of the fragmentation kernel We look for conditions on the coefficients guaranteeing some compactness of k ε . We use afew classical results of convergence of probability measures (see [Bil99] for instance). Let usintroduce a few notations. Given a probability-measure-valued function y ∈ R k ( ., y ) ∈M ( R ), we denote F ( ., y ) its repartition function: F ( x, y ) = R x −∞ k ( s, y ) d s and G ( x, y ) thefunction R x −∞ F ( z, y ) d z . We shall deduce the compactness of k ε from the compactness ofthe associated G ε . To this end, we need several elementary statements. Lemma 6 Let { P n , n ∈ N } be a family of probability measures on R , having their supportincluded in some interval [ a, b ] . We denote F n the repartition function of P n , and G n thefunctions defined by R x −∞ F n ( s ) d s . The following assertions are equivalent:1. P n → P weakly ( i.e. , ∀ f ∈ C b ( R ) , P n f → P f )2. F n ( x ) → F ( x ) for all x at which F is continuous3. G n → G uniformly locally Lemma 7 (Conditions for F ) Let F be a nondecreasing function on R . There exists aunique probability measure P on R , such that F ( x ) = P (] − ∞ , x ]) , iff • F is rightcontinuous everywhere, • lim x →−∞ F ( x ) = 0 , lim x → + ∞ F ( x ) = 1 .Furthermore P has its support included in [ a, b ] iff F ≡ on ] − ∞ , a [ and F ( b ) = 1 . Lemma 8 (Conditions for G ) Let G be a convex function on R . There exists a probabilitymeasure P on R , having its support included in [ a, b ] , such that G ( x ) = R x −∞ F ( s ) d s , where F ( x ) = P (] − ∞ , x ]) , iff • G is increasing, • for x > b , G ( x ) = G ( b ) + x , • G ≡ on ] − ∞ , a ] . Corollary 1 Let (cid:0) G n (cid:1) n ∈ N a sequence satisfying the assumptions of lemma 8. Suppose G n → G uniformly locally on R , then G also satisfy these assumptions and we have P n → P weakly. Proof . We define the function F as F ( x ) = lim δ → + G ( x + δ ) − G ( x ) δ , it is then easy to checkthat F satisfies assumptions of lemma 7, and G ( x ) = R x −∞ F ( s ) d s . Proof of Lemma 2. We prove the following result, which contains Lemma 2.19 emma 9 Suppose that the discrete coefficients satisfy (23). Then there exist a subsequence ε n and k ∈ C ([0 , ∞ ) , M ([0 , ∞ )) − weak − ⋆ ) such that • k satisfies (6),(5) (and therefore (7)), • for every y > , k ε n ( ., y ) → k ( ., y ) in law, • for every ϕ ∈ C ∞ c ([0 , ∞ )) , φ ε n → φ uniformly on [0 , R ] for any < R < ∞ . For any y ≥ k ε ( x, y ) d x defines a probability measure on [0 , ∞ ), supported in [0 , y ]. Weset F ε ( x, y ) = R x k ε ( z, y ) d z and G ε ( x, y ) = R x F ε ( z, y ) d z . Let ϕ ∈ C ∞ c ( R ∗ + ). We start byrewriting, owing to integration by parts, φ ε ( y ) = ϕ ( y ) − Z y F ε n ( x, y ) ϕ ′ ( x ) d x = ϕ ( y ) − G ε ( y, y ) ϕ ′ ( y ) + Z y G ε ( x, y ) ϕ ′′ ( x ) d x, where we used the fact that F ε ( y, y ) = R y k ε ( z, y ) d z = 1. The proof is based on thefollowing argument: G ε is close to a ˜ G ε which satisfies the assumptions of the Arzela-Ascolitheorem. Given x, y ≥ ε > i, j denote the integers satisfying x ∈ [ iε, ( i + 1) ε [, y ∈ [ jε, ( j + 1) ε [ and a short computation leads to F ε ( x, jε ) = S i,j + x − iεε k i,j ,G ε ( x, jε ) = ε i − X p =0 S p,j + ( x − iε ) S i,j + ε S i,j + ( x − iε ) ε k i,j , where S i,j = i − X r =0 k r,j . We define ˜ k ε ( x, y ) = ( j + 1) ε − yε k ε ( x, jε ) + y − jεε k ε ( x, ( j + 1) ε )and we have ˜ G ε ( x, y ) = ( j + 1) ε − yε G ε ( x, jε ) + y − jεε G ε ( x, ( j + 1) ε ) . Observe that | ˜ G ε ( x, y ) − G ε ( x, y ) | = y − jεε | G ε ( x, ( j + 1) ε ) − G ε ( x, jε ) |≤ (cid:12)(cid:12)(cid:12) ε i − X p =0 ( S p,j +1 − S p,j ) + ( x − iε )( S i,j +1 − S i,j ) + ε S i,j +1 − S i,j )+ ( x − iε ) ε ( k i,j +1 − k i,j ) (cid:12)(cid:12)(cid:12) . Due to (6), we have 0 ≤ k i,j ≤ | k i,j +1 − k i,j | ≤ 1. Similarly 0 ≤ S i,j ≤ | S i,j +1 − S i,j | ≤ 1. Hence, since (23) can also be written (cid:12)(cid:12)(cid:12) i − X p =0 S p,j +1 − S p,j (cid:12)(cid:12)(cid:12) ≤ K, it allows us to obtain | ˜ G ε ( x, y ) − G ε ( x, y ) | ≤ ε ( K + 1 + 1 / / 2) = ε ( K + 2) . 20e also deduce that (cid:12)(cid:12) ∂ y ˜ G ε ( x, y ) (cid:12)(cid:12) = (cid:12)(cid:12) G ε ( x, jε ) − G ε ( x, ( j + 1) ε ) (cid:12)(cid:12) ε ≤ K + 2while | ∂ x ˜ G ε ( x, y ) | ≤ . Moreover, we have | ˜ G ε ( x, y ) | ≤ ε ( i + 2)which is bounded uniformly with respect to ε and 0 ≤ x, y ≤ R < ∞ . As a consequence ofthe Arzela-Ascoli theorem we deduce that, for a subsequence, G ε n converges uniformly to acontinuous function G ( x, y ) on [0 , R ] × [0 , R ] for any 0 < R < ∞ . It follows that φ ε n ( y ) −−−−→ ε n →∞ φ ( y ) = ϕ ( y ) − G ( y, y ) ϕ ′ ( y ) + Z y G ( x, y ) ϕ ′′ ( x ) d x uniformly on [0 , R ]. We conclude by applying Lemma 8 to the function x G ( x, y ), with y ≥ Lemma 10 Suppose that the discrete coefficients satisfy (25). Then F ε and G ε are uni-formly bounded and converge (up to a subsequence) uniformly on compact sets. Proof . Assumption (25) rewrites (cid:12)(cid:12)(cid:12) S i,j +1 − S i,j (cid:12)(cid:12)(cid:12) ≤ Kj , k i,j ≤ Kj , so, with the same notation for ˜ F ε as for ˜ G ε and ˜ k ε , we get | F ε ( x, y ) − ˜ F ε ( x, y ) | = y − jεε | F ε ( x, ( j + 1) ε ) − F ε ( x, jε ) | ≤ Kj ≤ Ky ε, together (the computations are very similar to those above) with | ∂ x ˜ F ε | ≤ k i,j + k i,j +1 ε ≤ Kjε ≤ Kn ε ≤ Kx , and | ∂ y ˜ F ε | ≤ ε | F ε ( x, ( j + 1) ε ) − F ε ( x, jε ) | ≤ Ky , which leads to Ascoli assumptions and therefore the suitable compactness.With such assumptions, we can take into account any k of the form k ( x, y ) dx = y k ( x/y ) dx ,including Dirac mass. If we consider such a distribution on [0 , 1] (taken symmetric), thenwe can define k i,j as k i,j = k (cid:16)i i − j − , ij − h(cid:17) + 12 k (cid:16)n i − j − o(cid:17) + 12 k (cid:16)n ij − o(cid:17) + 12 k (cid:16)n o(cid:17) δ i + 12 k (cid:16)n o(cid:17) δ j − i with δ ji the Kronecker symbol. With these notations, we have for p ≥ j − S p,j = p X i =0 k i,j = k (cid:16)h , pj − h(cid:17) + 12 k (cid:16)n pj − o(cid:17) and S j − ,j = S j,j = 1, which leads to S p,j +1 − S p,j = k (cid:16)h pj , pj − h(cid:17) + 12 k (cid:16)n pj o(cid:17) − k (cid:16)n pj − o(cid:17) , if p < j − , j − ,j +1 − S j − ,j = − k (cid:16)i j − j , i(cid:17) − k (cid:16)n j − j o(cid:17) , S j,j +1 − S j,j = 0 , as 0 ≤ i ≤ j , we have for any p ≤ i , p − j − ≤ pj , the intervals (cid:2) pj , pj − (cid:2) and (cid:2) p − j , p − j − (cid:2) are disjoint. This leads to (cid:12)(cid:12)(cid:12) i X p =0 S p,j +1 − S p,j (cid:12)(cid:12)(cid:12) ≤ k (cid:16) i [ p =0 h pj , pj − h(cid:17) + 12 k (cid:16) i [ p =0 n pj o(cid:17) + 12 k (cid:16) i [ p =0 n pj − o(cid:17) ≤ , which gives the criterion (23). The limit is then obviously given by k ( x, y ) dx = y k ( x/y ) dx . A.3 Discrete system We discuss here briefly the existence theorem for the discrete system. 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