Mathematical modeling and analysis of the Notch-Delta pathway

TL;DR

This paper develops and analyzes mathematical models of the conserved Notch-Delta pathway to understand cell differentiation, focusing on symmetry bifurcations and pathway features without specific reaction kinetics.

Contribution

It introduces a structure-based modeling approach that avoids defining reaction kinetics, enabling analysis of pathway features and bifurcations in a parameter-rich context.

Findings

Identification of symmetry-induced bifurcations in the pathway

Analysis of conditions for singular Jacobian formation

Highlighting pathway features with greater biological relevance

Abstract

In this paper mathematical models for the evolutionary conserved Notch-Delta pathway are developed and analyzed in order to better understand how two neighboring biological cells can become different. We pursue a structure-based stoichiometric type of approach, such that no specific reaction kinetics have to be defined. Only their dependencies on the relevant species participating in the model network are taken into account. Reaction networks and their related systems of ODEs are presented and analyzed with respect to their capacity for symmetry-induced bifurcations. The possibility to obtain a singular Jacobian is analyzed symbolically. This approach is valid for parameter-rich kinetics, where the parametrization of the steady-state fluxes and of the first derivatives of the reaction rates evaluated at the steady state are independent. In this context, also with the help of abstract minimal models, we could mathematically identify some of the Notch pathway's features being more relevant than others.