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^{ν}^{}^{1}^{1} b A^{p}r µ0^{q}

1
ν^{}1 b

ν -0.482 0.316

µ0 0 0.023

A 0.083 0.01

b 1.54 1.69

R^{2} 0.7 0.72

Table 2.2: Best fitting parameters obtained with the
two different models. The correlation coefficientR^{2} 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 caseµ0^{}0.But the most significant difference between the two fittings is the
shape ofτ : it is a decreasing function ofxin the first case^{p}ν^{ }0^{q}and it is an increasing function in the
second case^{p}ν^{¡}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^{p}x, t^{q} ^{B}

Bx τ^{p}x^{q}u^{p}x, t^{q}

β^{p}x^{q}u^{p}x, t^{q}^{}2
ˆ ^{8}

x

β^{p}y^{q}κ^{p}x, y^{q}u^{p}y, t^{q}dy, x^{¥}0,
u^{p}x,0^{q}^{}u0^{p}x^{q},

u^{p}0, t^{q}^{}0.

(3.1)

This equation is used in many different areas to model a wide range of phenomena. The quantityu^{p}t, 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 _{τ}^{1} is non integrable at x^{}0, then the boundary conditionu^{p}0, t^{q}^{} 0 is useless. The
so-called fragmentation kernelκ^{p}x, y^{q}represents the proportion of individuals of sizex^{¤}y born from a
given dividing individual of sizey; more rigorously we should write κ^{p}dx, y^{q},with κ^{p}dx, y^{q}a 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^{p}τ^{p}x^{q}U^{p}x^{qq} ^{p}β^{p}x^{q} λ^{q}U^{p}x^{q}^{}2
ˆ ^{8}

x

β^{p}y^{q}κ^{p}x, y^{q}U^{p}y^{q}dy, x^{¥}0,
τU^{p}x^{}0^{q}^{}0, U^{p}x^{q}^{¡}0 forx^{¡}0, ´^{8}

0 U^{p}x^{q}dx^{}1.

(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 PMCA^{1} (see [83] and references therein). Moreover, when
modeling polymerization processes, Equation (3.1) is coupled with the density of monomersV^{p}t^{q}, which
appears as a multiplier for the polymerization rate (i.e.,τ^{p}x^{q}is replaced byV^{p}t^{q}τ^{p}x^{q}, andV^{p}t^{q}is 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^{p}τ^{p}x^{q}U_{α}px^{qq} ^{p}β^{p}x^{q} λ_{α}^{q}U_{α}px^{q}^{}2
ˆ ^{8}

x

β^{p}y^{q}κ^{p}x, y^{q}U_{α}py^{q}dy, x^{¥}0,
τU_{α}^{p}x^{}0^{q}^{}0, U_{α}^{p}x^{q}^{¡}0 forx^{¡}0, ´^{8}

0 U_{α}^{p}x^{q}dx^{}1,

(3.3)

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

$

'

'

&

'

'

% B

Bx^{p}τ^{p}x^{q}Va^{p}x^{qq} ^{p}aβ^{p}x^{q} Λa^{q}Va^{p}x^{q}^{}2a
ˆ ^{8}

x

β^{p}y^{q}κ^{p}x, y^{q}Va^{p}y^{q}dy, x^{¥}0,
τVa^{p}x^{}0^{q}^{}0, Va^{p}x^{q}^{¡}0 forx^{¡}0, ´^{8}

0 Va^{p}x^{q}dx^{}1,

(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_{α}andVa 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_{α}px^{q}for different values ofαwhenβ^{p}x^{q}^{}1, κ^{p}x, y^{q}^{}^{1}_{y}1l0^{¤}x^{¤}yandτ^{p}x^{q}^{}x.

In this case we have an explicit expressionU_{α}px^{q}^{}2^{?}α

?

αx ^{αx}_{2}^{2} exp

?

αx ^{αx}_{2}^{2} . One sees that
ifαvanishes,U_{α}tends 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
Λa^{}a

ˆ ^{8}

0

β^{p}x^{q}Va^{p}x^{q}dx

which could give the intuition that Λ^{a} is an increasing function ofa,what is indeed true ifβ^{p}x^{q}^{}β is a
constant since we obtain in this case Λ^{a}^{}βa.However, whenβis not a constant, the dependency of the
distributionVa^{p}x^{q}onacomes 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^{p}x^{q}dx^{}
ˆ

τ^{p}x^{q}Va^{p}x^{q}dx
and as a consequence we have that

xinf^{¡}0

τ^{p}x^{q}

x ^{¤}Λ^{a}^{¤}sup

x^{¡}0

τ^{p}x^{q}
x .

This last relation highlights the link between the first eigenvalue Λa and the growth rate τ^{p}x^{q}, or more
precisely ^{τ}^{p}_{x}^{x}^{q}.For instance, if ^{τ}^{p}_{x}^{x}^{q} is bounded, then Λ^{a} is bounded too, independently ofa.Notice that
in the constant case β^{p}x^{q} ^{} β, there cannot exist a solution to the eigenvalue problem (3.2) for ^{τ}^{p}_{x}^{x}^{q}
bounded since we have Λ^{a} ^{}βa which contradicts the boundedness of ^{τ}^{p}_{x}^{x}^{q}. In fact we check that the
existence condition (3.27) in Section 3.2.2 imposes, forβ constant, that _{τ}^{1} is integrable at x^{}0 and so

τ^{p}x^{q}

x cannot be bounded.

Similarly, concerning Equation (3.3), an integration against the weightxgives
λ_{α}^{} 1

α

´ τU_{α}dx

´ xU_{α}dx,

that could lead to the (false) intuition thatλ_{α}decreases withαwhich is indeed true in the limiting case
τ^{p}x^{q}^{}x.A simple integration gives more insight: it leads to

λα^{}

ˆ

β^{p}x^{q}U_{α}^{p}x^{q}dx, inf

x^{¡}0β^{p}x^{q}^{¤}λα^{¤}sup

x^{¡}0

β^{p}x^{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_{α}px^{q}is expected to concentrate into a Dirac mass inx^{}0,so the identityλ_{α}^{}´

β^{p}x^{q}U_{α}px^{q}dx
indicates thatλ_{α} should tend toβ^{p}0^{q}.Similarly, whenαtends to infinity,λ_{α}should behave asβ^{p} ^{8q}.

These intuitions on the link between ^{τ}^{p}_{x}^{x}^{q} 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^{p}x^{q} and β^{p}x^{q} have
power-like behaviours in the neighbourhood ofL^{}0 orL^{} ^{8},namely that

Dν, γ^{P}R such that τ^{p}x^{q} ^{}

x^{Ñ}Lτ x^{ν}, β^{p}x^{q} ^{}

x^{Ñ}Lβx^{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, forL^{}0 or L^{} ^{8},

αlim^{Ñ}Lλα^{} lim

x^{Ñ}Lβ^{p}x^{q} and lim

aÑLΛa^{} lim

x^{Ñ}_{L}^{1}

τ^{p}x^{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^{Ñ}^{8}). 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^{Ñ}0β^{p}x^{q} ^{}limx^{Ñ8}β^{p}x^{q} ^{}0, then the function α^{ÞÑ} λα satisfies limα^{Ñ}0λα ^{}

limα^{Ñ8}λ_{α} ^{} 0 and is positive on ^{p}0, ^{8q}, because λ_{α} ^{} ´

βU_{α} ¡ 0 for α ^{¡} 0. If limx^{Ñ}0τ^{p}x^{q}

x ^{}

limx^{Ñ8}τ^{p}x^{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^{Ñ}Lβ^{p}x^{q}or limx^{Ñ}Lτ^{p}x^{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.