Analytic stability analysis of three-component self-regulatory genetic circuit

TL;DR

This paper performs a detailed stability analysis of a three-component self-regulatory genetic circuit, revealing oscillatory behaviors and fixed point stability using eigenvalue analysis of the full system.

Contribution

It introduces a comprehensive three-dimensional stability analysis of the genetic circuit, including eigenvalues of the Hessian matrix, extending beyond previous two-component models.

Findings

Identification of stable and saddle points in the three-component system

Existence of complex eigenvalues leading to oscillations

Oscillatory convergence due to slow DNA binding/unbinding

Abstract

A self-regulatory genetic circuit, where a protein acts as a positive regulator of its own production, is known to be a simplest form of biological network with a positive feedback loop. Although at least three components, DNA, RNA, and the protein, are required to form such a circuit, the stability analysis of fixed points of the self-regulatory circuit has been performed only after reducing the system into to a two-component system consisting of RNA and protein only, assuming a fast equilibration of the DNA component. Here, the stability of fixed points of the three-component positive feedback loop is analyzed by obtaining eigenvalues of full three dimensional Hessian matrix. In addition to rigorously identifying the stable fixed points and the saddle points, detailed information can be obtained, such as the number of positive eigenvalues near a saddle point. In particular, complex eigenvalues is shown to exist for sufficiently slow binding and unbinding of the auto-regulatory transcription factor to DNA, leading to oscillatory convergence to the steady states, a novel feature unseen in the two-dimensional analysis.