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

by Kathrin Hellmuth, Christian Klingenberg, Qin Li and Min Tang

Computation 2021, 9(11), 119; https://doi.org/10.3390/computation9110119 - 11 Nov 2021

Cited by 5 | Viewed by 3612

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. Full article

(This article belongs to the Special Issue Inverse Problems with Partial Data)

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22 pages, 5419 KB

Open AccessArticle

Anisotropic Network Patterns in Kinetic and Diffusive Chemotaxis Models

by Ryan Thiessen and Thomas Hillen

Mathematics 2021, 9(13), 1561; https://doi.org/10.3390/math9131561 - 2 Jul 2021

Cited by 4 | Viewed by 2834

Abstract

For this paper, we are interested in network formation of endothelial cells. Randomly distributed endothelial cells converge together to create a vascular system. To develop a mathematical model, we make assumptions on individual cell movement, leading to a velocity jump model with chemotaxis. We use scaling arguments to derive an anisotropic chemotaxis model on the population level. For this macroscopic model, we develop a new numerical solver and investigate network-type pattern formation. Our model is able to reproduce experiments on network formation by Serini et al. Moreover, to our surprise, we found new spatial criss-cross patterns due to competing cues, one direction given by tissue anisotropy versus a different direction due to chemotaxis. A full analysis of these new patterns is left for future work. Full article

(This article belongs to the Special Issue Mathematical Models for Cell Migration and Spread)

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