Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation

Abstract

We consider a stochastic individual-based model in continuous time to describe a sizestructured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We address here the problem of nonparametric estimation of the kernel ruling the divisions based on the eigenvalue problem related to the asymptotic behavior in large population. This inverse problem involves a multiplicative deconvolution operator. Using Fourier techniques we derive a nonparametric estimator whose consistency is studied. The main difficulty comes from the non-standard equations connecting the Fourier transforms of the kernel and the parameters of the model. A numerical study is carried out and we pay special attention to the derivation of bandwidths by using resampling.

Insights into the dynamic trajectories of protein filament division revealed by numerical investigation into the mathematical model of pure fragmentation

“…More recently, a theory was developed [20] to estimate both the division rate and the division kernel from the measurement of the particle distribution profile at the end of the experiment, under assumptions on the division rate being given by the simple power law αx γ . Another approach emerging to estimate the division kernel is the use of stochastic individual based models by studying the underlying stochastic branching processes [21].…”

Individual and population approaches for calibrating division rates in population dynamics: Application to the bacterial cell cycle

“…We follow the same notations as above. We do not treat here the interesting question of estimating the fragmentation kernel b(y, x), and refer for instance to [102,9,103,10].…”

Estimating the division rate from indirect measurements of single cells

“…At this stage, the reconstruction formula is formal: to give it a rigorous meaning and ensure its validity, we would have to prove that all the quantities are well-defined, in particular that the Fourier transform G * B never vanishes. This requires a full study per se, and is beyond the scope of this work: in another case study, this has been done for instance for the estimation of the fragmentation kernel of the growth-fragmentation equation in the article [13], using the Cauchy integral to prove that a Mellin transform never vanishes, proof adapted to another case in [23]. For these two cases however, the proofs used strongly an explicit formulation of the solution with the use of Mellin or Fourier transforms, thanks to the fact that B was a power law in [13], and constant in [23].…”

“…First, we have to prove that the denominator of our ratios in our inverse Fourier transforms, namely G * B , does not vanish. This has been done for related problems (estimation of the fragmentation kernel) in two recent papers, by using complex analysis methods (Lemma 1.iii in [23], Theorem 2.i in [13]). In [13], it was the central and most technical point of the study.…”

“…In [13], it was the central and most technical point of the study. Here however, the proof carried out in [23] based on the argument principal could probably be adapted.…”