Biochemical Oscillations in Delayed Negative Cyclic Feedback: Harmonic Balance Analysis with Applications

TL;DR

This paper introduces a harmonic balance analysis framework to analytically determine the frequency, phase, and amplitude of oscillations in gene regulatory networks with negative cyclic feedback, applicable to various biochemical systems.

Contribution

It develops a novel analytical method for predicting oscillation profiles in biochemical networks, bridging nonlinear dynamics with biological insights.

Findings

Analytical expressions for oscillation frequency, phase, and amplitude derived.

Validation with Pentilator and Hes7 networks shows accurate predictions.

Waveform of oscillations obtained analytically for broad biochemical systems.

Abstract

Oscillatory chemical reactions often serve as a timing clock of cellular processes in living cells. The temporal dynamics of protein concentration levels is thus of great interest in biology. Here we propose a theoretical framework to analyze the frequency, phase and amplitude of oscillatory protein concentrations in gene regulatory networks with negative cyclic feedback. We first formulate the analysis framework of oscillation profiles based on multivariable harmonic balance. With this framework, the frequency, phase and amplitude are obtained analytically in terms of kinetic constants of the reactions despite the nonlinearity of the dynamics. These results are demonstrated with the Pentilator and Hes7 self-repression network, and it is shown that the developed analysis method indeed predicts the profiles of the oscillations. A distinctive feature of the presented result is that the waveform of oscillations is analytically obtained for a broad class of biochemical systems. Thus, it is easy to see how the waveform is determined from the system's parameters and structures. We present general biological insights that are applicable for any gene regulatory networks with negative cyclic feedback.