Maximum principle

Dans le document The DART-Europe E-theses Portal (Page 48-52)

1.5 Appendix

1.5.3 Maximum principle

'

'

&

'

'

%

˜ τεpxq B

Bxφ˜pxq pβ˜εpxq λεqφ˜pxq2 ˜βεpxq ˆ R

0

κεpy, Rxqφ˜εpyqdyδφ˜εpRq, 0 x R, φ˜εp0q0,

(1.40)

where ˜τεpxqτεpRxqand ˜βεpxqβεpRxq.Then the same method than for the direct equation give the result, namely the existence ofλand ˜φsolution to (1.40).

Finally we have proved existence ofpλU,Uεq andpλφ, φεqsolution to the direct and adjoint equations of (1.38). It remains to prove thatλU λφ but it is nothing but integrating the direct equation against the ajoint eigenvector, what gives

λU

ˆ

Uεφελφ ˆ

Uεφε.

To have existence of solution for (1.19), it remains to doεÑ0.For this we can prove uniform bounds inL8forUεandφεbecause we are on the fixed compactr0, Rs.Then we can extract subsequences which converge L8weak towardU and φ, solutions to (1.19) because τε and βε converge in L1 toward τ andβ.Concerningκε, we have that for allϕPCc8, ´

ϕpxqκεpx, yqdxÝÑ´

ϕpxqκpx, yqdx y, and it is sufficient to pass to the limit in the equations.

1.5.3 Maximum principle

Lemma 1.5.3. If there exists A0 ¡ 0 such that φ ¥ φ on r0, A0s and φ a supersolution of (1.3) on

rA0, Rswith φpRq¥φpRq, thenφ¥φon r0, Rs.

Proof. The proof is based on the same tools than to prove uniqueness (see above) or to establish GRE principles (see [98, 99] for instance).

We know thatφ¥φonr0, A0sand thatφis a supersolution to the equation satisfied byφonrA0, Rs, i.e.

there exists a functionf ¥δφp0qsuch that

τpxq B

Bpxq pλ βpxqqφpxqpxq ˆ x

0

κpy, xqφpyqdy fpxq, xPrA0, Rs. So we have for allxPrA0, Rs

τpxq B

Bxpφpxqφpxqq pλ βpxqqpφpxqφpxqqpxq ˆ x

0

κpy, xqpφpyqφpyqqdyfpxq. Then, multiplying by 1lφ¥φ,we obtain (see [110] for a justification)

τpxq B

Bxpφφq pxq pλ βpxqqpφφq pxq¤pxq ˆ x

0

κpy, xqpφφq pyqdyfpxq1lφ

¥φpxq, and this inequality is satisfied onr0, Rssincepφφq 0 on r0, A0s.

If we test againstU we have, using the fact that φpRq0 φpRq, ˆ R

0

pφφq pxq B

BxpτpxqUpxqqdx ˆ R

0

pλ βpxqqpφφq pxqUpxqdx

¤ 2 ˆ R

0

pφφq pyq ˆ R

y

βpxqκpy, xqUpxqdxdy ˆ R

0

fpxq1lφ¥φpxqUpxqdx.

But if we test the equation (1.3) satisfied byU againstpφφq ,we find ˆ R

0

pφφq pxq B

BxpτpxqUpxqqdx ˆ R

0

pλ βpxqqpφφq pxqUpxqdx

2 ˆ R

0

pφφq pyq ˆ R

y

βpxqκpy, xqUpxqdxdy, and finally, substracting,

0¤ ˆ R

0

fpxq1lφ

¥φpxqUpxqdx, so

δφp0q ˆ R

0

1lφ

¥φpxqUpxqdx¤0 and this can hold only if 1lφ

¥φ0 orφp0q0.But we deal with the truncated problem withτpxq¥η¡0, so 1τ PL10 andφp0q¡0 thanks to the lemma 1.3.1. Thus 1lφ

¥φ0 and the lemma 1.5.3 is proved.

Chapitre 2

D´ ependance par rapport aux param` etres : premi` ere application

Dans cet article nous nous int´eressons au ph´enom`ene de souche des maladies `a pri-ons. Plus pr´ecis´ement, nous nous demandons quel est le coefficient dans le syst`eme prion qui d´epend le plus de la souche. Il est sugg´er´e dans [136] qu’il s’agit du terme de fragmentation, mais une d´ependance du terme de transport leur est malgr´e tout n´ecessaire pour coller correctement aux donn´ees exp´erimentales. Nous montrons ici, en utilisant une classe plus large de coefficients, qu’il est possible d’expliquer les valeurs exp´erimentales en ne supposant qu’une d´ependance de la fragmenation par rapport `a la souche de prion. Pour ce faire, nous consid´erons la classe de coeffi-cients en puissance de x alors que [136] consid´erait un transport constant et une fragmentation lin´eaire. L’´etude nous permet en outre d’obtenir la forme du taux de polym´erisationτpxq dans la classe des fonctions en puissance dex.L’article est paru dans Mathematical and Computer Modelling sous le titre The shape of the polymerization rate in the prion equation [59].

2.1 Introduction

Polymerization of prion proteins is a central event in the mechanism of prion diseases. The cells produce naturally the normal form of this protein (PrPc) but there exists a pathogenic form (PrPsc) present in infected cells as polymers. These polymers have the ability to transconform and attach the normal form by a polymerization process not yet very well understood. Mathematical modelling is a promising tool to investigate this mechanism provided we can find accurate parameters.

We consider the following model of prion proliferation (see [19, 20, 63, 80])

$

'

'

'

'

'

'

'

'

'

&

'

'

'

'

'

'

'

'

'

%

dVptq

dt λVptq

δ ˆ 8

0

τpxqupx, tqdx

,

B

Btupx, tq VptqB

Bx τpxqupx, tq

rµ0 βpxqsupx, tq 2 ˆ 8

x

βpyqκpx, yqupy, tqdy, up0, tq 0,

upx,0q u0pxq.

(2.1)

This equation is the continuous version of the discrete model of [92] (see [38] for the derivation). Sys-tem (2.1) is a quite general model able to reproduce biological experimental results (see [83]). In this equation,Vptqis the quantity of PrPc which is produced by the cell with a rateλ,and degraded with the

39

rateδ.The quantity of aggregates of PrPsc with sizexat timetis denoted byupx, tq.Polymers of PrPsc can attach monomers (PrPc) with rateτ and can split themselves in two smaller polymers with rateβ.

To fit with experimental observations [124], the polymerization and fragmentation rates should depend on the polymer sizex(see [19, 20]). When fragmentation occurs, a polymer of sizeyproduces two pieces of size xandyxwith the probability κpx, yq. The natural assumptions (see [98, 99, 110, 80]) on the kernelκare then

Supp κtpx, yq, 0¤x¤yu since fragmentation produces smaller aggregates than the initial one,

y¡0, ˆ

κpx, yqdx1 pκp, yqis a probability measureq (2.2) and

x¤y, κpx, yqκpyx, yq (2.3)

because of indistinguishability of the two pieces of sizesxandyxafter fragmentation. A consequence of the two assumptions (2.2) and (2.3) is that the mean size of the fragments obtained from a polymer of sizey isy{2

y, ˆ

px, yqdxy

2. (2.4)

In the discrete model of [91, 92, 93], a critical size n of polymers is considered. Under this size, the polymers are unstable so the proteins that compose them go back to their normal form PrPc. This effect is taken into account in the continuous model of [63, 64] with the critical sizex0.As a consequence, there is one more integral term of apparition of monomers in the equation on V.Nevertheless, even ifn ¡0 in the discrete model, the value of x0 can be taken equal to 0 in the continuous one as justified in [38].

Then the fact that small polymers are unstable, and so do not exist, is taken into account by the bounday conditionup0, tq0.The parameterµ0 is a death term which will be discussed later.

In the article [136], model (2.1) with the specific parameters of [43, 63, 64, 91, 92, 93] is used. It assumes that the polymerization rate does not depend on polymer size (τpxqτ0) and the fragmentation is described by βpxq β0x and κpx, yq 1l0¤x¤y1

y. In this case, equation (2.1) can be reduced to a non-linear system of ODEs (see [43, 63, 64] or even [91, 92, 93]). This system is used in [136] to investigate the parameters involved in the so-called “prion strain phenomenon”. This phenomenon is the ability of the same encoded protein to encipher a multitude of phenotypic states. Prion strains are characterized by their incubation period (time elapsed between experimental inoculation of the PrPsc infectious agent and clinical onset of the disease) and their lesion profiles (patterns produced by the death of neurones) in brain [57]. A reason for the strain diversity can be that there exist various conformation states of PrPsc, namely various misfoldings of the prion protein, which lead to various biological and biochemical properties [102, 118]. These different conformations could be characterizable by their different stability against denaturation. Indeed, recent studies on prion disease have confirmed that the incubation time is related not only to the quantity of pathogenic protein inoculated (´

u0pxqdx) and the level of expression of PrPc (λ), but also to the resistance of prion strains against denaturation [82] in terms of the concentration of guanide hydrochloride (Gdn-HCl) required to denaturate 50% of the disease-causing protein. Other studies have highlighted a strong relationship between the stability of the prion protein against denaturation and neuropathological lesion profiles [81, 123]. A natural question is then to know how the conformation states of the PrPsc protein affect the parameters of model (2.1) and more precisely which of these parameters is the most strain-dependent. One of the conclusions of [136]

is that the key parameter describing strain-dependent replication kinetics is the fibril breakage rateβ0. Using the model, some coefficients already defined in [92] are calculated from experimental data and the dependency between them indicates that the fragmentation rateβ0is the most adapted parameter to be strain-dependent. But, in this framework, the fragmentation dependency is not sufficient to fit precisely the data and the polymerization rateτ0 is also assumed to change with the strain. In particular, to keep a one-parameter family of models, the polymerization rate is assumed to depend on the fragmentation one through a power relation

Dφ s.t. τ0β0φ

. (2.5)

However, polymerization and fragmentation are two different biochemical processes which have no reason to be linked together.

In this paper we prove that we can get rid of such a relation by considering a larger class of parameters for model (2.1). We take advantage of a general framework developped in [19, 20, 80], where coefficients can be more general than in [43, 63, 64, 91, 92, 93, 136], and we consider power laws as in [45, 47, 80, 95].

Thanks to this extension, we find a shape for the polymerization rate allowing us to fit experimental data with only the fragmentation rate dependent on strain. The experimental data we use are the stability against denaturationGand the growth rate of the quantity of PrPsc (r) which is linked to the incubation time. These data are strain-dependent and we can find in [136] estimated empirical values for different strains. In Section 2.2 we present the class of coefficients we consider and give the necessary assumptions for mathematical investigations. Most of these assumptions are already made in [92, 136] and, thanks to them, we are able to link r and Gto the mathematical model. In Section 2.3 we give the dependency of these parameters on polymerization and fragmentation rates. Then we give a relation between them using the only fragmentation dependency on strain and are able to fit data with an accurate shape of the polymerization rate.

Dans le document The DART-Europe E-theses Portal (Page 48-52)