Nonlocal Adhesion Models for Microorganisms on Bounded Domains

Abstract

In 2006 Armstrong, Painter and Sherratt formulated a non-local differential equation model for cell-cell adhesion. For the one dimensional case we derive various types of adhesive, repulsive, and no-flux boundary conditions. We prove local and global existence and uniqueness for the resulting integro-differential equations. In numerical simulations we consider adhesive, repulsive and neutral boundary conditions and we show that the solutions mimic known behavior of fluid adhesion to boundaries. In addition, we observe interior pattern formation due to cell-cell adhesion.

A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis

“…The model relies on the presence of an advective flux of the cell density at position x, calculated nonlocally by scouting for availability of adhesion sites in a sensing region centered in x. We refer the interested reader to [46,49,100] for results on the existence and uniqueness of solutions to the model equation proposed by Armstrong et al [7]. This has become a popular continuum description of cell adhesion, with a number of variations proposed to study tumour invasion and cell movement across the ECM [15,22,25,37,43,79,100], development and cell-sorting dynamics [8,23,43,80].…”

Stability of a non-local kinetic model for cell migration with density dependent orientation bias

“…In Figure 2(b), again one then has instability in the region of the V b − kR plane to the left of the red dashed curve with the most unstable wave identified by the lowest blue curve. 4. Numerical tests.…”

“…Recently, this model was derived in [6] from a space jump process, while the steady states and bifurcations of the macroscopic adhesion model and their stability were studied in [3,5]. Further studies concerning these models on bounded domains are proposed in [4]. Other macroscopic models describing cell migration with non-local measures of the environment were proposed in [20,21,22].…”

Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

“…Still, this offers the easiest way to properly define the output of the nonlocal operator in the boundary layer where the sensing region is not fully contained in Ω. Very recently various other boundary conditions have been derived and compared in the context of a single equation modeling cell-cell adhesion in 1D (Hillen and Buttenschön 2020). Few previous works focus on solvability for models with nonlocality in a taxis term.…”

“…Few previous works focus on solvability for models with nonlocality in a taxis term. Some of them deal with single equations that only involve cell-cell adhesion (Dyson et al 2010(Dyson et al , 2013Hillen and Buttenschön 2020), others study nonlocal systems of the sort considered here for two (Hillen et al 2007) or more components (Engwer et al 2017). The global solvability and boundedness study in Hillen et al (2018) is obtained for the case of a nonlocal operator with integration over a set of sampling directions being an open, not necessarily strict subset of R n .…”