Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local well-posedness for a novel nonlocal model for cell-cell adhesion via receptor binding

Mabel Lizzy Rajendran, Anna Zhigun

TL;DR

The paper develops a rigorous local well-posedness theory for a novel nonlocal cell-adhesion model coupling a degenerate porous-medium diffusion–advection equation with a nonlinear nonlocal receptor-binding integral equation. The authors decouple the system and solve each part via Banach fixed point arguments, introducing the Kantorovich-Rubinstein norm to handle measure-valued inputs in the integral equation. They establish local well-posedness for the integral equation with measure parameters, prove well-posedness for the diffusion-adhesion PDE with degenerate diffusion and fixed advection, and then treat the coupled system by iterative fixed point arguments, ensuring stability and non-explosion of the support for small times. The results provide a mathematically robust framework for macroscopic models that encode microscopic receptor binding dynamics within cell-cell adhesion, with potential implications for upscaling and biological interpretation of early invasion dynamics.

Abstract

Local well-posedness is established for a highly nonlocal nonlinear diffusion-adhesion system for bounded initial values with small support. Macroscopic systems of this kind were previously obtained by the authors through upscaling in [32] and can account for the effect of microscopic receptor binding dynamics in cell-cell adhesion. The system analysed here couples an integro-PDE featuring degenerate diffusion of the porous media type and nonlocal adhesion with a novel nonlinear integral equation. The approach is based on decoupling the system and using Banach's fixed point theorem to solve each of the two equations individually and subsequently the entire system. The main challenge of the implementation lies in selecting a suitable framework. One of the key results is the local well-posedness for the integral equation with a Radon measure as a parameter. The analysis of this equation utilizes the Kantorovich-Rubinstein norm, marking the first application of this norm in handling a nonlinear integral equation.

Local well-posedness for a novel nonlocal model for cell-cell adhesion via receptor binding

TL;DR

The paper develops a rigorous local well-posedness theory for a novel nonlocal cell-adhesion model coupling a degenerate porous-medium diffusion–advection equation with a nonlinear nonlocal receptor-binding integral equation. The authors decouple the system and solve each part via Banach fixed point arguments, introducing the Kantorovich-Rubinstein norm to handle measure-valued inputs in the integral equation. They establish local well-posedness for the integral equation with measure parameters, prove well-posedness for the diffusion-adhesion PDE with degenerate diffusion and fixed advection, and then treat the coupled system by iterative fixed point arguments, ensuring stability and non-explosion of the support for small times. The results provide a mathematically robust framework for macroscopic models that encode microscopic receptor binding dynamics within cell-cell adhesion, with potential implications for upscaling and biological interpretation of early invasion dynamics.

Abstract

Local well-posedness is established for a highly nonlocal nonlinear diffusion-adhesion system for bounded initial values with small support. Macroscopic systems of this kind were previously obtained by the authors through upscaling in [32] and can account for the effect of microscopic receptor binding dynamics in cell-cell adhesion. The system analysed here couples an integro-PDE featuring degenerate diffusion of the porous media type and nonlocal adhesion with a novel nonlinear integral equation. The approach is based on decoupling the system and using Banach's fixed point theorem to solve each of the two equations individually and subsequently the entire system. The main challenge of the implementation lies in selecting a suitable framework. One of the key results is the local well-posedness for the integral equation with a Radon measure as a parameter. The analysis of this equation utilizes the Kantorovich-Rubinstein norm, marking the first application of this norm in handling a nonlinear integral equation.
Paper Structure (18 sections, 30 theorems, 276 equations)

This paper contains 18 sections, 30 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Analytical challenges of and their handling
  3. Preliminaries
  4. Miscellaneous notation
  5. Radon measures and the Kantorovich-Rubinstein norm
  6. Problem setting and main results
  7. Well-posedness of for fixed u
  8. Operator Y
  9. Properties of Y for a small ball
  10. Local well-posedness of for a small ball (proof of )
  11. Solvability of for a point mass
  12. Well-posedness of a PDE with PM diffusion and fixed advection direction
  13. Well-posedness of
  14. Non-explosion of support for small times
  15. Well-posedness of for fixed w
  16. ...and 3 more sections

Key Result

Theorem 4.5

Let and $(\mu_0,w_0)$ satisfy where the partial derivative $\partial_w$ is taken in the Fréchet sense. Define sets Then, there exist constants $\Cr{R2},\Cr{R1},\Cl{F-contin-mu}>0$ that depend only on the parameters from DefG2AssPhiKpmaabb as well as such that for all $\mu\in \Cr{SU2}$ there exists a unique $w\in \Cr{W3}$ for which and the solution map is well-defined and Lipschitz continuous

Theorems & Definitions (66)

  • Definition 4.1: Solutions to \ref{['M_PLnewIBVP', 'PLnewIBVP']}
  • Definition 4.2: Time-independent solutions to \ref{['M_LambdaEq', 'LambdaEq']}
  • Remark 4.3
  • Definition 4.4: Solutions to \ref{['M_PLnewIBVP']}-\ref{['M_LambdaEq']} and \ref{['PLnewIBVP']}-\ref{['LambdaEq']}
  • Theorem 4.5: Local well-posedness of \ref{['LambdaEq']}
  • Remark 4.6
  • Remark 4.7
  • Theorem 4.8: Local well-posedness of \ref{['PLnewIBVP']}-\ref{['LambdaEq']}
  • Theorem 4.9: Time-independent solutions to \ref{['M_LambdaEq']} vs. \ref{['LambdaEq']}
  • Theorem 4.10: Solutions to \ref{['M_PLnewIBVP']}-\ref{['M_LambdaEq']} vs. \ref{['PLnewIBVP']}-\ref{['LambdaEq']}
  • ...and 56 more