[Paper Review] Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

Maria Krasnianski, Kevin J. Painter+2 more|arXiv (Cornell University)|Aug 27, 2019

Mathematical Biology Tumor Growth50 references4 citations

TL;DR

This paper establishes a rigorous mathematical limit procedure linking nonlocal models for cell migration—governed by integral operators acting on signal gradients—to their local counterparts (classical chemotaxis and haptotaxis). By reformulating nonlocalities via gradient-based integral operators, it proves convergence as the sensing radius $ r \to 0 $, demonstrating that the new formulation avoids spurious singularities and enables global existence of weak solutions, offering a more biologically plausible and analytically robust framework.

ABSTRACT

A rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

Motivation & Objective

• To establish a mathematically rigorous limit procedure connecting nonlocal and local models for cell migration.
• To address the issue of spurious singularities in existing nonlocal models near domain boundaries when $ r \to 0 $.
• To unify the treatment of nonlocal adhesion and nonlocal chemotaxis through a common reformulation of nonlocal operators.
• To extend the analytical framework to allow for solution-dependent coefficient functions in the model.
• To provide a biologically more plausible and mathematically stable alternative to previous nonlocal formulations.

Proposed method

• Reformulate nonlocal operators by applying integral operators directly to the gradients of signal-dependent quantities, rather than to the signals themselves.
• Introduce two new nonlocal operators: $ \mathcal{T}_r $, which averages signal gradients along line segments within the sensing radius, and $ \mathcal{S}_r $, which averages over directions.
• Prove convergence of solutions of the nonlocal models to those of the local models as the sensing radius $ r \to 0 $, under appropriate assumptions.
• Use a unified framework to treat both adhesion and chemotaxis models by embedding the nonlocal drift terms in a common structure involving $ \nabla \cdot (c \chi \mathcal{A}_r(g)) $.
• Establish global existence of weak solutions for the nonlocal models using the new operator formulation.
• Conduct 1D numerical simulations to illustrate theoretical findings and compare solution behavior between different nonlocal formulations.

Experimental results

Research questions

• RQ1 Can a rigorous limit procedure be established that connects nonlocal models for cell migration to their local counterparts as the sensing radius $ r \to 0 $?
• RQ2 How do different formulations of nonlocal operators affect the regularity and boundary behavior of solutions, especially near domain boundaries?
• RQ3 Does reformulating nonlocality via integral operators applied to gradients improve the mathematical and biological plausibility of the model?
• RQ4 Can the new formulation support general solution-dependent coefficient functions while preserving convergence to the local limit?
• RQ5 What is the impact of the integration path (via $ s \in [0,1] $) on the resulting drift direction and solution smoothness?

Key findings

• The new nonlocal operator $ \mathcal{T}_r $, which averages signal gradients along line segments, leads to smoother solutions and avoids spurious singularities at the domain boundary as $ r \to 0 $, unlike the previous $ \mathcal{A}_r $ formulation.
• Numerical simulations in 1D confirm that the $ \mathcal{T}_r $-based model produces qualitatively more stable and biologically plausible solutions compared to the $ \mathcal{A}_r $-based model.
• The reformulated nonlocal operators enable a rigorous passage to the limit as $ r \to 0 $, yielding the classical local chemotaxis and haptotaxis models as the limit system.
• The new formulation ensures global existence of weak solutions for the nonlocal models, even in the presence of solution-dependent coefficients.
• The integration over the path $ s \in [0,1] $ in $ \mathcal{T}_r $ captures the continuous probing of the environment by cellular protrusions, making it more biologically realistic than pointwise evaluation.
• The $ \mathcal{S}_r $ formulation, which averages over directions only, leads to jump discontinuities in the drift field at unit distance from signal concentration points, while $ \mathcal{T}_r $ preserves continuity and weakens singularities.