Estimating the division rate of the growth-fragmentation equation with a self-similar kernel

Bourgeron, Thibault; Doumic, Marie; Escobedo, Miguel

doi:10.1088/0266-5611/30/2/025007

Cited by 18 publications

(36 citation statements)

References 32 publications

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“…The cells are typically rod-shaped and are about 2μm long, with 0.5μm diameter. Then, the average volume falls between 0.6 − 0.7μm 3 . These unicellular organisms are extensively studied in vitro and in vivo.…”

Section: Structured Population Modelsmentioning

confidence: 99%

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Structured Models for Cell Populations: Direct and Inverse Problems

Albani

Zubelli

2015

ITM Web of Conferences

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Abstract. Structured population models in biology lead to integro-differential equations that describe the evolution in time of the population density taking into account a given feature such as the age, the size, or the volume. These models possess interesting analytic properties and have been used extensively in a number of areas. In this article, we consider a size-structured model for cell division and revisit the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such question as an inverse problem for an integro-differential equation posed on the half line. The focus is to compare from a computational view point the performance of different algorithms. Finally, we will discuss real data reconstructions with E. coli data.

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Section: Structured Population Modelsmentioning

confidence: 99%

“…In one spatial dimension, the amount of data needed to estimate the division kernel k(·, ··) would be unrealistic in practical applications without further hypothesis. See [1][2][3] …”

Section: Structured Population Modelsmentioning

confidence: 99%

Structured Models for Cell Populations: Direct and Inverse Problems

Albani

Zubelli

2015

ITM Web of Conferences

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“…Proposition 1. Let B and g be such that there exists a unique positive eigentriplet (λ, U B , φ B ) solution of the eigenproblem (3)- (6). Let us furthermore assume λU B,x + d dx (gU B,x ) ∈ L 2 (x p dx) and define L B as the unique solution of Equation (9) given by Lemma 1.…”

Section: Reconstruction Formula In a Deterministic Settingmentioning

confidence: 99%

“…We obtain L n (y) an estimator of L B (y). For this step, we follow [6], and concatenate the inverse given in Lemma 1 in L 2 (dx) for x ≤x, that we denote L n,l (y), with the inverse in L 2 (x 4 dx), denoted L n,r (y), for x ≥x, for a given (to be determined numerically)x > 0: we set L n (y) = L n,l (y)1l x≤x + L n,r (y)1l x>x .…”

Section: Estimation Of G Bmentioning

confidence: 99%

Estimating the division rate from indirect measurements of single cells

Doumic¹,

Olivier

2020

Discrete &Amp; Continuous Dynamical Systems - B

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Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? This article adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.

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“…This leading idea -how to use measurements on the asymptotic distribution to estimate functional parameters of the equation -has been first initiated in [21] and continued e.g. in [6,10] for the growth-fragmentation equation. However, up to now, the studies were focused on the question of estimating the division rate, whereas the division kernel was assumed to be known.…”

Section: Introductionmentioning

confidence: 99%

Estimating the division rate and kernel in the fragmentation equation

Doumic

Escobedo

Tournus

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider the fragmentation equationand address the question of estimating the fragmentation parameters -i.e. the division rate B(x) and the fragmentation kernel k(y, x) -from measurements of the size distribution f (t, ·) at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate B(x) = αx γ and a self-similar fragmentation kernel k(y, x) = 1 y k 0 ( x y ), we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet (α, γ, k 0 ) and a representation formula for k 0 . To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

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Estimating the division rate of the growth-fragmentation equation with a self-similar kernel

Cited by 18 publications

References 32 publications

Structured Models for Cell Populations: Direct and Inverse Problems

Structured Models for Cell Populations: Direct and Inverse Problems

Estimating the division rate from indirect measurements of single cells

Estimating the division rate and kernel in the fragmentation equation

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