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About this book

This monograph considers the mathematical modeling of cellular adhesion, a key interaction force in cell biology. While deeply grounded in the biological application of cell adhesion and tissue formation, this monograph focuses on the mathematical analysis of non-local adhesion models. The novel aspect is the non-local term (an integral operator), which accounts for forces generated by long ranged cell interactions. The analysis of non-local models has started only recently, and it has become a vibrant area of applied mathematics. This monograph contributes a systematic analysis of steady states and their bifurcation structure, combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the symmetries of the non-local term. These methods allow readers to analyze and understand cell adhesion on a deep level.

Table of Contents

Frontmatter

Introduction

Frontmatter

Chapter 1. Introduction

Abstract
Cellular adhesion is one of the most important interaction forces in tissues. Cells adhere to each other, to other cells, and to the extracellular matrix (ECM). Cell adhesion is responsible for the formation of tissues, membranes, vasculature, muscle tissue, as well as cell movement and cancer spread.
Andreas Buttenschön, Thomas Hillen

Chapter 2. Preliminaries

Abstract
In this section, we present some basic results that are needed later. We give a summary of the derivation of the non-local adhesion model from biological principles as presented.
Andreas Buttenschön, Thomas Hillen

The Periodic Problem

Frontmatter

Chapter 3. Basic Properties

Abstract
In this chapter, we define the non-local operator \({\mathcal {K}}[u]\), and we collect some basic properties of \({\mathcal {K}}[u]\) in one spatial dimension. We prove results on integrability, continuity, regularity, positivity, and a priori estimates, and we show that \({\mathcal {K}}[u]\) is a compact operator. We analyze the corresponding spectrum of \({\mathcal {K}}\), and we use these properties to derive properties of steady-state solutions such as symmetries, regularities, and a priori estimates. We find that the non-local term \({\mathcal {K}}[u]\) acts like a non-local derivative and the term \({\mathcal {K}}[u]'\) acts like a non-local curvature, in a sense made precise later.
Andreas Buttenschön, Thomas Hillen

Chapter 4. Local Bifurcation

Abstract
The success of the Armstrong–Painter–Sherratt adhesion model (2.​14) is that it can replicate the complicated patterns observed in cell sorting experiments.
Andreas Buttenschön, Thomas Hillen

Chapter 5. Global Bifurcation global bifurcation

Abstract
For each \(\bar {u}>0\), we found local bifurcations at \((\alpha _n, \bar {u})\) with non-trivial eigenfunctions e n of \({\mathcal {D}}_u {\mathcal {F}}(\alpha _n, \bar {u})\) in \(H^2_P\) be given by
$$\displaystyle \alpha _n = \frac {n \pi }{\bar {u} h'(\bar {u}) L M_n(\omega )} , \qquad e_n(x) = \cos \left (\frac {2 \pi n x}{L}\right ), $$
where M n(ω) are the Fourier sine coefficients of ω (see 3.​4).
Andreas Buttenschön, Thomas Hillen

Non-local Equations with Boundary Conditions

Frontmatter

Chapter 6. No-Flux Boundary Conditions no-flux boundary conditions for Non-local Operators

Abstract
It is a challenge to define boundary conditions for non-local models on bounded domains. The periodic case, which we studied in the previous chapters, is an exception, since we can work with periodic extensions outside of the domain. However, no-flux conditions or Dirichlet or Robin boundary conditions need special attention.
Andreas Buttenschön, Thomas Hillen

Chapter 7. Discussion and Future Directions

Abstract
The central building block to include adhesive interactions between cells in reaction-advection-diffusion models of tissues is to use a non-local term.
Andreas Buttenschön, Thomas Hillen

Backmatter

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