The role of mutants in the spatio-temporal progression of inflammatory bowel disease: three classes of permanent form travelling waves

TL;DR

This paper models the spread of mutant epithelial cells in inflammatory bowel disease using reaction-diffusion equations, identifying three classes of traveling waves that influence disease progression.

Contribution

It introduces a novel mathematical framework combining geometric singular perturbation theory and matched asymptotic expansions to analyze mutant cell propagation in IBD.

Findings

Identifies three classes of traveling wave solutions in the model.

Shows mutant epithelial cells are critical but have a subdominant effect on progression speed.

Provides a bifurcation parameter controlling wave dynamics.

Abstract

Despite its high prevalence and impact on the lives of those affected, a complete understanding of the cause of inflammatory bowel disease (IBD) is lacking. In this paper, we investigate a novel mechanism which proposes that mutant epithelial cells are significant to the progression of IBD since they promote inflammation and are resistant to death. We develop a simple model encapsulating the propagation of mutant epithelial cells and immune cells which results from interactions with the intestinal barrier and bacteria. Motivated by the slow growth of mutant epithelial cells, and relatively slow response rate of the adaptive immune system, we combine geometric singular perturbation theory with matched asymptotic expansions to determine the one-dimensional slow invariant manifold that characterises the leading order dynamics at all times beyond a passive initial adjustment phase. The dynamics on this manifold are controlled by a bifurcation parameter, $\sigma$, which depends on the ratio of growth to decay rates of all components except mutants and determines three distinct classes of permanent-form travelling waves that describe the propagation of mutant epithelial and immune cells. These are obtained from scalar reaction-diffusion equations with the reaction being (i) a bistable nonlinearity with a cut-off, (ii) a cubic Fisher nonlinearity and (iii) a KPP or Fisher nonlinearity. Our results suggest that mutant epithelial cells are critical to the progression of IBD. However, their effect on the speed of progression is subdominant.