Operon dynamics with state dependent transcription and/or translation delays

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 statedependent 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.

Complex dynamics in an unexplored simple model of the peroxidase–oxidase reaction

“…The fixed points of the model were computed for a range of parameter values and were found to be saddle-like. 37 Furthermore, the saddle quantity σ 0 (sum of leading stable and unstable eigenvalues) was found to be positive throughout the domains of the 2D isospike stability diagrams shown below, which suggests the possibility of Shilnikov chaos. 38 Fig.…”

The dynamic-process characterization and prediction of synthetic gene circuits by dynamic delay model

“…Delay differential equations (DDEs) are proposed to simplify the multistep reactions into a single time delay, which may reduce the number of parameters that are needed to predict dynamic behaviors. 46 , 47 , 48 , 49 , 50 , 51 , 52 DDEs for modeling gene expression have been proposed in many works to study the influences of delays in gene expression networks, 53 , 54 , 55 , 56 or predict experimental behaviors of lac operon dynamics. 55 The methods of introducing delay time in these works can be summarized as follows: (1) Directly assume a total delay time τ without the precise formula link to the detailed model 53 , 54 ; (2) Consider a delay time at each step of transcription, translation and protein folding/maturation.…”

“… 55 The methods of introducing delay time in these works can be summarized as follows: (1) Directly assume a total delay time τ without the precise formula link to the detailed model 53 , 54 ; (2) Consider a delay time at each step of transcription, translation and protein folding/maturation. 55 , 56 The first method was simple and convenient for theoretical analysis of influences of delay times, but how specific parameters in detailed model affects the delay time was still not clear. The second method showed how to calculate the delay times in each step of transcription, translation and protein folding/maturation, but with many parameters need to be determined if it is used to predict synthetic gene expression dynamics.…”