Investigation of a Structured Fisher's Equation with Applications in Biochemistry

TL;DR

This paper extends Fisher's Equation by incorporating biochemical pathway activity, deriving a structured PDE model, proving self-similar solutions, and numerically analyzing how biochemical patterns influence cell migration.

Contribution

It introduces a structured Fisher's Equation with biochemical activity, proves the existence of traveling wave solutions, and explores the impact of biochemical patterns on migration rates.

Findings

Existence of self-similar traveling wave solutions.

Biochemical activity patterns can modulate migration speed.

Numerical results show complex effects of MAPK activation on cell migration.

Abstract

Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's Equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's Equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.