Operon Dynamics with State Dependent Transcription and/or Translation Delays

TL;DR

This paper models operon dynamics incorporating state-dependent delays in transcription and translation, revealing complex behaviors like stable oscillations and multistability, with implications for understanding gene regulation.

Contribution

It introduces a novel model with state-dependent delays for operon regulation, providing detailed analysis and numerical simulations that highlight differences from constant delay models.

Findings

Stable periodic orbits in inducible systems

Multistability in repressible systems

Potential Shilnikov-type bifurcations

Abstract

Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.