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

Blaine van Rensburg, David J. Needham, Fabian Spill, Alexandra Tzella

TL;DR

This work develops a spatio-temporal reaction–diffusion framework for inflammatory bowel disease that includes mutant epithelial cells interacting with bacteria, the barrier, and immune cells. By exploiting a separation of time scales with geometric singular perturbation theory and matched asymptotics, the authors reduce a four-variable system to a two-dimensional slow manifold and derive a planar M–I dynamics whose long-time behavior is governed by three travelling-wave regimes determined by the bifurcation parameter $\sigma$. Depending on $\sigma$, the model yields a full-transition PTW for $0<\sigma<1$, a cubic-FKPP-type PTW at $\sigma=1$, and two transition waves (LPTW and UPTW) for $\sigma>1$, with explicit asymptotics for wave speeds and thresholds such as the initial mutant density $M_0 > \max(K_M(1-\sigma),0)$. The results show that mutant epithelial cells can influence IBD progression, but their impact on propagation speed is subdominant, and reducing $\sigma$—through targeting immune recruitment, barrier damage, or bacterial influx—offers a potential therapeutic route to halt disease spread.

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, $σ$, 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.

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

TL;DR

This work develops a spatio-temporal reaction–diffusion framework for inflammatory bowel disease that includes mutant epithelial cells interacting with bacteria, the barrier, and immune cells. By exploiting a separation of time scales with geometric singular perturbation theory and matched asymptotics, the authors reduce a four-variable system to a two-dimensional slow manifold and derive a planar M–I dynamics whose long-time behavior is governed by three travelling-wave regimes determined by the bifurcation parameter . Depending on , the model yields a full-transition PTW for , a cubic-FKPP-type PTW at , and two transition waves (LPTW and UPTW) for , with explicit asymptotics for wave speeds and thresholds such as the initial mutant density . The results show that mutant epithelial cells can influence IBD progression, but their impact on propagation speed is subdominant, and reducing —through targeting immune recruitment, barrier damage, or bacterial influx—offers a potential therapeutic route to halt disease spread.

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, , 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.
Paper Structure (26 sections, 18 theorems, 214 equations, 6 figures)

This paper contains 26 sections, 18 theorems, 214 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Summary of results and outline
  4. Model Reduction
  5. The Initial Value Problem (LIVP)
  6. (LIVP) with and
  7. (LIVP) with , or, and
  8. Permanent Form Travelling Waves
  9. Travelling waves when
  10. Travelling waves when
  11. Travelling waves when
  12. Existence of [LPTW]'s
  13. Existence of [UPTW]'s
  14. Nonexistence of [FPTW]'s
  15. Numerics
  16. ...and 11 more sections

Key Result

Theorem 3.1

Let $\delta$ be small and $\sigma\ge \delta$. Then for each positive $(M_0,I_0)\in R(\delta)$, (LIVP) has a unique, global, classical solution, $(M,I):\overline{D}_{\infty}\to R(\delta)$. Moreover, for each $T>0$, uniformly for $t\in [0,T]$, where $(M_{-\infty},I_{-\infty})\equiv (m,i):\overline{\mathbb{R}}^+\to R(\delta)$ is the unique global solution to the temporal evolution problem, subject

Figures (6)

  • Figure 1: Qualitative sketches of the phase portraits for the planar temporal dynamical system (DS), depending on $\sigma$, showing the temporal dynamics confined to the clipped region $R(\delta)$. (a) $\sigma\in[\delta,1]$; the stable slow manifold $\mathcal{S}_{\delta}$ connects $\textbf{e}_F$ to $\textbf{e}(0)$ via the continuum of equilibrium points $\{(m_e,0): m_e\in[0,1-\sigma)]\}$. (b) $\sigma\in(1,\infty)$; the stable slow manifold connects $\textbf{e}_F$ to $\textbf{e}_T$.
  • Figure 2: Propagation speed $v^{*}(\sigma)$ obtained by solving \ref{['eqn:EVP1']} numerically via a shooting method (solid line) and the explicit asymptotic expressions provided in \ref{['eqn:vstar_sigma_asym']} for $v^{*}(\sigma)$ (dashed lines). Here, $\alpha_2=\beta_2=1$ so that $b(\sigma)=(1+\sigma)^{-1}$.
  • Figure 3: Numerically computed PTW =(minimum) propagation speed of the full system obtained as a function of $\sigma$ for $\delta=0.5$ (thick solid) and $\delta=0.05$ (thin solid) by solving the set of boundary value problems (\ref{['eqn:EVP1']}), (\ref{['eqn:EVP2']}), and (\ref{['eqn:EVP3']}). This is compared against the asymptotic (dashed) speed of propagation obtained from the one-dimensional boundary value problems (\ref{['eqn:MVP1']}), (\ref{['eqn:MVP2']}) and (\ref{['eqn:MVP4']}).
  • Figure 4: Numerically determined PTW profiles for $M_T(z)$ (black solid line) and $I_T(z)$ (red solid line) obtained by solving the set of boundary value problems (\ref{['eqn:EVP1']}), (\ref{['eqn:EVP2']}), and (\ref{['eqn:EVP3']}) and plotted as a function of $z$ for selected values of $\sigma$, and $\delta=0.05$ (thin solid line), $\delta=0.5$ (thick solid line). These are compared against the corresponding leading order asymptotic approximations for $M_T(z)$ (black dashed line) and $I_T(z)$ (red dashed line) using (\ref{['eqn:MVP1']}) for (a), (\ref{['eqn:MVP2']}) for (b) and (\ref{['eqn:MVP4']}) for (c) and (d). The inset in (a) focuses on the transition region near $M_T(z)=1-\sigma$ and additionally shows the first order correction to the asymptotic approximation for $I_T(z)$ (blue dashed line) in the transition region near $M_T(z)=1-\sigma$ for the two values of $\delta$.
  • Figure 5: LPTW profiles $I_T(z)$ computed for $\sigma=1.25$ (black lines) and $\sigma=4$ (red lines) and $\delta=0.05$ (thin lines), $\delta=0.5$ (thick lines). As $\sigma$ increases, the width of front decreases, and its maximum height increases.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Corollary 3.1
  • Theorem 3.3
  • Definition 4.1
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Remark 4.1
  • ...and 22 more