A novel trait-structured Keller–Segel model that explores the dynamics of a migrating cell population guided by chemotaxis in response to an external ligand concentration is derived and analysed. Unlike traditional Keller–Segel models, this framework introduces an explicit representation of ligand-receptor bindings on the cell membrane, where the percentage of occupied receptors constitutes the trait that influences cellular phenotype. The model posits that the cell’s phenotypic state directly modulates its capacity for chemotaxis and proliferation, governed by a trade-off due to a finite energy budget: cells highly proficient in chemotaxis exhibit lower proliferation rates, while more proliferative cells show diminished chemotactic abilities. The model is derived from the principles of a biased random walk, resulting in a system of two non-local partial differential equations, describing the densities of both cells and ligands. Using a Hopf–Cole transformation, we derive an equation that characterises the distribution of cellular traits within travelling wave solutions for the total cell density, allowing us to uncover the monotonicity properties of these waves. Numerical investigations are conducted to examine the model’s behaviour across various biological scenarios, providing insights into the complex interplay between chemotaxis, proliferation, and phenotypic diversity in migrating cell populations.

Collective cell migration is a multicellular phenomenon that arises in various biological contexts, including cancer and embryo development. ‘Collectiveness’ can be promoted by cell-cell interactions such as co-attraction and contact inhibition of locomotion. These mechanisms act on cell polarity, pivotal for directed cell motility, through influencing the intracellular dynamics of small GTPases such as Rac1. To model these dynamics we introduce a biased random walk model, where the bias depends on the internal state of Rac1, and the Rac1 state is influenced by cell-cell interactions and chemoattractive cues. In an extensive simulation study we demonstrate and explain the scope and applicability of the introduced model in various scenarios. The use of a biased random walk model allows for the derivation of a corresponding partial differential equation for the cell density while still maintaining a certain level of intracellular detail from the individual based setting.

We extend a previously presented mesoscopic (i.e. colony scale) mathematical model of the reaction of bacterial biofilms to antibiotics. In that earlier model, exposure to antibiotics evokes two responses: inactivation as the antibiotics kill the bacteria, and inducing a quorum sensing based stress response mechanism upon exposure to small sublethal dosages. To this model we add now quorum quenching as an adjuvant to antibiotic therapy. Quorum quenchers are modeled like enzymes that degrade the quorum sensing signal concentration. The resulting model is a quasilinear system of seven reaction-diffusion equations for the dependent variables volume fractions of upregulated (protected), downregulated (unprotected) and inert (inactive) biomass [particulate substances], and for concentrations of a growth promoting nutrient, antibiotics, quorum sensing signal, and quorum quenchers [dissolved substances]. The biomass fractions are subject to two nonlinear diffusion effects: (i) degeneracy, as in the porous medium equation, where biomass vanishes, and (ii) a super-diffusion singularity where as it attains its theoretically possible maximum. We study this model in numerical simulations. Our simulations suggest that for maximum efficacy quorum quenchers should be applied early on before quorum sensing induction in the biofilm can take place, and that an antibiotic strategy that by itself might not be successful can be notably improved upon if paired with quorum quenchers as an adjuvant.