Conclusion and Perspectives

2.5 3 3.5 4 4.5 5

Rate of growth (r) [GdnHCl]1/2values (G)

data G(r)=A.r^1/ν−1+b G(r)=A.(r+µ

0)^1/ν−1+b

Figure 2.1: Experimental values ofGandr for differ-ent strains are plotted (circles). Then these data are fitted using the reduced model (2.22) withµ0 0 and µ0 free.

Ar^ν11 b A^pr µ0^q

1 ν1 b

ν -0.482 0.316

µ0 0 0.023

A 0.083 0.01

b 1.54 1.69

R² 0.7 0.72

Table 2.2: Best fitting parameters obtained with the two different models. The correlation coefficientR² is also reported in the last row.

First we assume as in [136] that the death rate µ0 is equal to zero. Then we obtain a model already considered by [136] (Table 3, page 5). In our framework, we can also letµ0 be free and fit the data with the model (2.22). As we can see in Table 2.2, the accuracy of the fitting is a little bit better with this generalisation than in the caseµ00.But the most significant difference between the two fittings is the shape ofτ : it is a decreasing function ofxin the first case^pν 0^qand it is an increasing function in the second case^pν^¡0^q.

2.4 Conclusion and Perspectives

Thanks to a larger class of parameters than [136], we are able to fit the experimental data by considering only the fragmentation rate to be dependent on strains. We even obtain a more general model and so a more accurate fitting. But, because the parameters profiles are nota prioriknown, we cannot find ¯xfrom experimental data and have to postulate a particular dependency (2.11) whereas this was a consequence of the model in [136]. To validate it, new data are necessary, for instance values of ¯xobtained by an experimental size distribution.

A consequence of the method is the prescription of the shape of the polymerization rate from exper-imental data, and also the death rate µ0. Nevertheless, we notice a significant difference between the two recovered exponents ν obtained by assuming that µ0 0 or not. Particularly one is negative and the other one positive (see Table 2.2). The idea to compare parameters from different strains has been deeply exploited here. But the fact that there are only few different strains in comparison to the number of fitting parameters is a weakness for precise fitting and precise determination of τ.Some parameters such asµ0orbshould be previously determined by another method and other data. Furthermore the pa-rameters of model (2.1) could have a size-dependency more general than power laws. A complete inverse problem is to recover all the size-dependent parameters from experimental data without the restriction to a class of shape. This problem is very complicated even though first elementary models are treated in [40, 114, 109]. Moreover it requires more information such as the size distributionU and not only the mean size ¯x.

Aknowledgment

This work has been partly supported by ANR grant project TOPPAZ.

Chapitre 3

D´ ependance par rapport aux param` etres : ´ etude approfondie

Dans ce travail en collaboration avecVincent Calvez et Marie Doumic, nous analysons la d´ependance de la valeur propre principale de l’´equation de croissance-fragmentation en fonction de param`etres. Cette question apparait dans la recherche des ´etats d’´equilibre pour le syst`eme du prion avec la d´ependance par rapport au terme de transport, et dans le probl`eme d’optimisation de la PMCA avec la d´epen-dance par rapport `a un param`etre de fragmentation. Grˆace `a une dilatation ap-propri´ee des vecteurs propres, nous donnons le comportement asymptotique de la valeur propre quand les param`etres tendent vers z´ero ou l’infini. Pour des coefficients d´eg´en´er´es nous obtenons des d´ependances non monotones, contrairement `a ce qui avait pu ˆetre conjectur´e auparavant en se basant sur des cas simples.

3.1 Introduction

Growth and division of a population of individuals structured by a quantity conserved in the division process may be described by the following growth and fragmentation equation:

$

'

'

'

'

'

'

&

'

'

'

'

'

'

% B

Btu^px, t^{q B}

Bx τ^px^qu^px, t^q

β^px^qu^px, t^q2 ˆ ⁸

x

β^py^qκ^px, y^qu^py, t^qdy, x^¥0, u^px,0^qu0^px^q,

u^p0, t^q0.

(3.1)

This equation is used in many different areas to model a wide range of phenomena. The quantityu^pt, x^q may represent a density of dusts ([45]), polymers [19, 20], bacteria or cells [10, 11]. The structuring variablexmay be size ([110] and references), label ([6, 5]), protein content ([37, 88]), proliferating parasite content ([7]); etc. In the literature, it is refered to as the ”size-structured equation”, ”growth-fragmentation equation’, ”cell division equation”, ”fragmentation-drift equation” or Sinko-Streifer model.

The growth speed τ ^dx_dt represents the natural growth of the variable x, for instance by nutrient uptake or by polymerization, and the rateβ is called the fragmentation or division rate. Notice that ifτ is such that _τ¹ is non integrable at x0, then the boundary conditionu^p0, t^q 0 is useless. The so-called fragmentation kernelκ^px, y^qrepresents the proportion of individuals of sizex^¤y born from a given dividing individual of sizey; more rigorously we should write κ^pdx, y^q,with κ^pdx, y^qa probability

45

measure with respect tox. The factor “2“ in front of the integral term highlights the fact that we consider here binary fragmentation, namely that the fragmentation process breaks an individual into two smaller ones. The method we use in this paper can be extended to more general cases where the mean number of fragments isn0^¡1 (see [41]).

Well-posedness of this problem as well as the existence of eigenelements has been proved in [47, 41].

Here we focus on the first eigenvalueλassociated to the eigenvectorU defined by

$

'

'

&

'

'

% B

Bx^pτ^px^qU^px^{qq p}β^px^q λ^qU^px^q2 ˆ ⁸

x

β^py^qκ^px, y^qU^py^qdy, x^¥0, τU^px0^q0, U^px^q¡0 forx^¡0, ´⁸

0 U^px^qdx1.

(3.2)

The first eigenvalue λis the asymptotic exponential growth rate of a solution to Problem (3.1) (see [98, 99]). It is often called the malthusian parameter or the fitness of the population. Hence it is of deep interest to know how it depends on the coefficients: for given parameters, is it favorable or unfavorable to increase fragmentation ? Is it more efficient to modify the transport rateτor to modify the fragmentation rateβ? Such concerns may have deep impact on therapeutic strategy (see [10, 11, 37, 23]) or on experimental design of devices such as PMCA¹ (see [83] and references therein). Moreover, when modeling polymerization processes, Equation (3.1) is coupled with the density of monomersV^pt^q, which appears as a multiplier for the polymerization rate (i.e.,τ^px^qis replaced byV^pt^qτ^px^q, andV^pt^qis governed by one or more ODE - see for instance [63, 64, 20]). Asymptotic study of such polymerization processes thus closely depends on such a dependency (see [20, 19], where asymptotic results are obtained under the assumption of a monotonic dependency ofλwith respect to the polymerization rateτ).

Based on simple cases already studied (see [63, 64, 117, 59]), the first intuition would be that the fitness always increases when polymerization or fragmentation increases. Nevertheless, a closer look reveals that it is not true.

To study the dependency of the eigenproblem on its parameters, we depart from given coefficientsτ and β, and study how the problem is modified under the action of a multiplier of either the growth or the fragmentation rate. We thus consider the two following problems: first,

$

'

'

&

'

'

%

α^B

Bx^pτ^px^qU_αpx^{qq p}β^px^q λ_α^qU_αpx^q2 ˆ ⁸

x

β^py^qκ^px, y^qU_αpy^qdy, x^¥0, τU_α^px0^q0, U_α^px^q¡0 forx^¡0, ´⁸

0 U_α^px^qdx1,

(3.3)

whereα^¡0 measures the strength of the polymerization (transport) term, as in the prion problem (see [63]), and second

$

'

'

&

'

'

% B

Bx^pτ^px^qVa^px^{qq p}aβ^px^q Λa^qVa^px^q2a ˆ ⁸

x

β^py^qκ^px, y^qVa^py^qdy, x^¥0, τVa^px0^q0, Va^px^q¡0 forx^¡0, ´⁸

0 Va^px^qdx1,

(3.4)

where a ¡ 0 modulates the fragmentation intensity, as for PMCA or therapeutics applied to the cell division cycle (see discussion in Section 3.4).

To make things clearer, w give some intuition on the dependency of Λ^a and λ_α on their respective multipliersaandα.First of all, one suspects that ifavanishes or ifαexplodes, since transport dominates, the respective eigenvectorsU_αandVa tend to dilute, and on the contrary ifa explodes or ifαvanishes, since fragmentation dominates, they tend to a Dirac mass at zero (see Figure 3.1 for an illustration). But

1PMCA, Protein Misfolded Cyclic Amplification, is a protocol designed to amplify the quantity of prion protein aggregates due to periodic sonication pulses. In this application,urepresents protein aggregates density andxtheir size; the division rateβis modulated by sound waves. See Section 3.4.3 for more details.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

0.5 1 1.5 2 2.5 3 3.5

α=0.05 α=0.1 α=0.2 α=0.5 α=1 α=2 α=4

Figure 3.1: EigenvectorsU_αpx^qfor different values ofαwhenβ^px^q1, κ^px, y^q1_y1l0^¤x^¤yandτ^px^qx.

In this case we have an explicit expressionU_αpx^q2^?α

?

αx ^αx₂² exp

?

αx ^αx₂² . One sees that ifαvanishes,U_αtends to a Dirac mass, whereas it dilutes whenα^{Ñ 8}.

what happens to the eigenvaluesλα and Λ^a ? Integrating Equation (3.4), we obtain the relation Λaa

ˆ ⁸

0

β^px^qVa^px^qdx

which could give the intuition that Λ^a is an increasing function ofa,what is indeed true ifβ^px^qβ is a constant since we obtain in this case Λ^aβa.However, whenβis not a constant, the dependency of the distributionVa^px^qonacomes into account and we cannot conclude so easily. A better intuition is given by integration of Equation (3.4) against the weightx,which gives

Λa

ˆ

xVa^px^qdx ˆ

τ^px^qVa^px^qdx and as a consequence we have that

xinf^¡0

τ^px^q

x ^¤Λ^a¤sup

x^¡0

τ^px^q x .

This last relation highlights the link between the first eigenvalue Λa and the growth rate τ^px^q, or more precisely ^τp_x^xq.For instance, if ^τp_x^xq is bounded, then Λ^a is bounded too, independently ofa.Notice that in the constant case β^px^q β, there cannot exist a solution to the eigenvalue problem (3.2) for ^τp_x^xq bounded since we have Λ^a βa which contradicts the boundedness of ^τp_x^xq. In fact we check that the existence condition (3.27) in Section 3.2.2 imposes, forβ constant, that _τ¹ is integrable at x0 and so

τ^px^q

x cannot be bounded.

Similarly, concerning Equation (3.3), an integration against the weightxgives λ_α 1

α

´ τU_αdx

´ xU_αdx,

that could lead to the (false) intuition thatλ_αdecreases withαwhich is indeed true in the limiting case τ^px^qx.A simple integration gives more insight: it leads to

λα

ˆ

β^px^qU_α^px^qdx, inf

x^¡0β^px^q¤λα^¤sup

x^¡0

β^px^q.

This relation connectsλ_α to the fragmentation rateβ when the parameterαis in front of the transport term. Moreover, we have seen that when the growth parameterαtends to zero, for instance, the distri-butionU_αpx^qis expected to concentrate into a Dirac mass inx0,so the identityλ_α´

β^px^qU_αpx^qdx indicates thatλ_α should tend toβ^p0^q.Similarly, whenαtends to infinity,λ_αshould behave asβ^{p 8q}.

These intuitions on the link between ^τp_x^xq and Λa on the one hand, β andλα on the other hand, are expressed in a rigorous way below. The main assumption is that the coefficients τ^px^q and β^px^q have power-like behaviours in the neighbourhood ofL0 orL ⁸,namely that

Dν, γ^PR such that τ^px^q

x^ÑLτ x^ν, β^px^q

x^ÑLβx^γ. (3.5)

Theorem 3.1.1. Under Assumption (3.5), Assumptions (3.12)-(3.13) on κ, and Assumptions (3.22)-(3.28) given in [41] to ensure the existence and uniqueness of solutions to the eigenproblems (3.3) and (3.4), we have, forL0 or L ⁸,

αlim^ÑLλα lim

x^ÑLβ^px^q and lim

aÑLΛa lim

x^Ñ_L¹

τ^px^q x .

This is an immediate consequence of our main result, stated in Theorem 3.2.2 of Section 3.2.1. As expected by the previous relations, for Problem (3.3) the eigenvalue behavior is given by a comparison between β and 1 in the neighbourhood of zero if polymerization vanishes (α ^Ñ 0), and in the neigh-bourhood of infinity if polymerization explodes (α^Ñ8). For Problem (3.4), it is given by a comparison betweenτ and x(in the neighbourhood of zero when aÑ 8or in the neighbourhood of infinity when aÑ0).

One notices that these behaviors are somewhat symmetrical. Indeed, the first step of our proof is to use a properly-chosen rescaling, so that both problems (3.3) and (3.4) can be reduced to a single one, given by Equation (3.18). Theorem 3.2.2 studies the asymptotic behaviour of this new problem, which allows us to quantify precisely the rates of convergence of the eigenvectors toward self-similar profiles.

A consequence of these results is the possible non-monotonicity of the first eigenvalue as a function of αor a. Indeed, if limx^Ñ0β^px^q limx^Ñ8β^px^q 0, then the function α^ÞÑ λα satisfies limα^Ñ0λα

limα^Ñ8λ_α 0 and is positive on ^p0, ^8q, because λ_α ´

βU_α ¡ 0 for α ^¡ 0. If limx^Ñ0τ^px^q

x

limx^Ñ8τ^px^q

x 0,we have the same conclusion foraÞÑΛ^a (see Figure 3.2 for examples).

This article is organised as follows. In Section 3.2 is devoted to state and prove the main result, given in Theorem 3.2.2. We first detail the self-similar change of variables that leads to the reformulation of Problems (3.3) and (3.4) in Problem (3.18), as stated in Lemma 3.2.1. We then recall the assumptions for the existence and uniqueness result of [41]. We need these assumptions here not only to have well-posed problems, but also because the main tool to prove Theorem 3.2.2 is given by similar estimates to the ones used in [41] to prove well-posedness. In Section 3.3, we give more precise results in the limiting cases, i.e. when limx^ÑLβ^px^qor limx^ÑLτ^px^q

x is finite and positive, and conversely more general results under assumptions weaker than Assumption (3.5). Finally, Section 3.4 proposes possible use and interpretation of the results in various fields of application.

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