Physical Constraints on the Rhythmicity of the Biological Clock

TL;DR

This study models the KaiABC system to understand how biochemical and physical factors like energy cost and noise influence the emergence and stability of circadian rhythms in biological clocks.

Contribution

It provides a minimal model linking biochemical interactions, thermodynamic costs, and noise to the conditions necessary for circadian oscillations.

Findings

Oscillations occur within a narrowly bounded phase space of protein concentrations.

Higher energy costs are required for increased rhythmic precision due to noise.

An optimal noise level can induce oscillations beyond traditional bifurcation points.

Abstract

Circadian rhythms in living organisms are temporal orders emerging from biochemical circuits driven out of equilibrium. Here, considering the KaiABC system, a minimal model in the synthetic biology, we study how the oscillation emerges from the circuit made of three Kai proteins and ATP alone. The phase diagram constructed in terms of KaiC and KaiA concentrations reveals a narrowly bounded oscillatory phase, which naturally explains arrhythmia upon protein over-expression. As dictated by the cost-precision trade-offs of the thermodynamic uncertainty relations, the presence of intrinsic noise, amplified in small systems, demands higher free energy cost to achieve greater rhythmic precision. The cost-minimizing condition within the oscillatory phase is found to generate $\sim$21-hr rhythm, which is entrained to 24-hr environmental signals as long as the forcing amplitude is greater than $\sim 10$ \% of the metabolic rate. An optimal level of intrinsic noise can also induce oscillations even beyond the Hopf bifurcation, effectively expanding the oscillatory phase. Our study clarifies how the physical factors, such as regulatory mechanism, energy cost, and stochastic noise contribute to the operation of biological clocks.