Dynamic patterns of gene regulation I: simple two-gene systems

Area of Science:

• Systems Biology
• Computational Biology
• Biophysics

Background:

• Gene regulation is a fundamental biological process.
• Mathematical models, particularly ordinary differential equations (ODEs), are crucial for understanding complex biological dynamics.
• Analyzing bifurcation patterns in ODEs reveals critical transitions in system behavior.

Purpose of the Study:

• To investigate the cross-regulation of two genes using mathematical analysis.
• To classify different gene regulatory states based on activator and repressor binding.
• To derive analytical expressions for bifurcation points and understand their dependence on regulatory interactions.

Main Methods:

• Computer-assisted mathematical analysis of ordinary differential equations (ODEs).
• Modeling gene-regulator complexes with Hill coefficients up to n=4.
• Analysis of bifurcation patterns (one-dimensional and two-dimensional Hopf bifurcations).
• Introduction of a 'regulatory determinant' (D) for classifying regulatory states.

Main Results:

• Identified two main classes of regulatory states based on the sign of the regulatory determinant D.
• Systems with D<0 (e.g., dual activators or repressors) exhibit one-dimensional bifurcations, leading to bistability for n≥2.
• Systems with D>0 (combinations of activation and repression) sustain Hopf bifurcations and undamped oscillations for n>2.
• Described the influence of basal transcription activity on bifurcation patterns.

Conclusions:

• The mathematical framework provides a classification of gene regulatory states.
• The sign of the regulatory determinant D predicts system dynamics, distinguishing between bistability and oscillations.
• Complex binding dynamics, including intermediates, can lead to richer regulatory behaviors and a mixed sign for D.