Multiscale Convergence of the Inverse Problem for Chemotaxis in the Bayesian Setting

Kathrin Hellmuth,Christian Klingenberg,Qin Li,Min Tang

doi:10.3390/computation9110119

Kathrin Hellmuth, Christian Klingenberg + Show 2 more

Open Access

https://doi.org/10.3390/computation9110119

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Abstract

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe the organism’s movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller–Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms’ population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions.

Highlights

Chemotaxis is the phenomenon of organisms directing their movements upon certain chemical stimulation
Our result suggests that the solution to the Keller–Segel inverse problem is close to the kinetic result and qualifies as a ‘good’ initial guess, for the full reconstruction on the kinetic level
Given a prior distribution μ0 on A and an additive centered Gaussian noise in the data, the posterior distributions for the tumbling kernel derived from the kinetic chemotaxis equation and the macroscopic Keller–Segel equation as underlying models are asymptotically equivalent in the Kullback–Leibler divergence y ε →0 y dKL −−→ 0

Summary

Introduction

Chemotaxis is the phenomenon of organisms directing their movements upon certain chemical stimulation. There are various ways to conduct inverse problems, and, in this paper, we take the viewpoint of Bayesian inference This is to assume that the coefficients are not uniquely configured in reality but, rather, follow a certain probability distribution. The rest of the paper is organized as follows: in Section 2, we present the asymptotic relation between the two forward models. This can be seen as an adaption of the results. 2. Asymptotic Analysis for Kinetic Chemotaxis Equations and the Keller–Segel Model. Since only the boundedness of (K0 , K1 ) is seen in the proof, the convergence is uniform in A

Bayesian Inverse Problem Setup

Convergence of Posterior Distributions

Summary and Discussion