# Modelling Population-Level Hes1 Dynamics: Insights from a Multi-framework Approach

## TL;DR

This paper models Hes1 dynamics in neural development using mathematical approaches to understand population-level behavior and fate decisions.

## Contribution

The novel contribution is a multi-framework approach linking deterministic and stochastic models of Hes1 dynamics in cell populations.

## Key findings

- A detailed spatial ODE model captures Hes1 oscillations and fate decisions in neural development.
- The study links deterministic and stochastic grid-based models to enhance interpretability and tractability.
- The approach shows promise for modeling various biological processes in cell populations.

## Abstract

Mathematical models of living cells have been successively refined with advancements in experimental techniques. A main concern is striking a balance between modelling power and the tractability of the associated mathematical analysis. In this work we model the dynamics for the transcription factor Hairy and enhancer of split-1 (Hes1), whose expression oscillates during neural development, and which critically enables stable fate decision in the embryonic brain. We design, parametrise, and analyse a detailed spatial model using ordinary differential equations (ODEs) over a grid capturing both transient oscillatory behaviour and fate decision on a population-level. We also investigate the relationship between this ODE model and a more realistic grid-based model involving intrinsic noise using mostly directly biologically motivated parameters. While we focus specifically on Hes1 in neural development, the approach of linking deterministic and stochastic grid-based models shows promise in modelling various biological processes taking place in a cell population. In this context, our work stresses the importance of the interpretability of complex computational models into a framework which is amenable to mathematical analysis.