Trait-structured chemotaxis: Exploring ligand-receptor dynamics and travelling wave properties in a Keller-Segel model

TL;DR

This paper introduces a new trait-structured Keller-Segel model that incorporates ligand-receptor binding dynamics, revealing how phenotypic trade-offs influence chemotaxis and proliferation in migrating cell populations through analytical and numerical analysis.

Contribution

It develops a novel trait-structured Keller-Segel model with explicit ligand-receptor interactions and analyzes travelling wave solutions, advancing understanding of phenotypic diversity in chemotactic migration.

Findings

Travelling wave solutions exhibit monotonicity properties.

Phenotypic trade-offs significantly affect migration dynamics.

Numerical simulations reveal complex interactions between chemotaxis and proliferation.

Abstract

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.