Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.

When invading the tissue, malignant tumour cells (i.e. cancer cells) need to detach fromneighbouring cells, degrade the basement membrane, and migrate through the extracellularmatrix. These processes require loss of cell-cell adhesion and enhancement of cell-matrixadhesion. In this paper we present a mathematical model of an intracellular pathway forthe interactions between a cancer cell and the extracellular matrix. Cancer cells usesimilar mechanisms as with normal cells for their interactions with the extracellularmatrix. We develop a model of cell-matrix adhesion that accounts for reactions between thecell surface receptor integrins, the matrix glycoprotein fibronectin, and the actinfilaments in the cytoskeleton. Each represents components for an intermediate compartment,the extracellular compartment, and the intracellular compartment, respectively. Binding offibronectin with integrins triggers a clustering of protein complexes, which thenactivates and phosphorylates regulatory proteins that are involved in actin reorganisationcausing actin polymerization and stress fibre assembly. Rearrangement of actin filamentswith integrin/fibronectin complexes near adhesion sites and interaction with fibrillarfibronectin produces the force necessary for cell migration, accounting for cell-matrixadhesion.