Fractional-Order Dynamical Modeling and Stability Analysis of the Core Circadian Clock in Arabidopsis thaliana

Existing models of the circadian clock in Arabidopsis thaliana are conventionally formulated by integer-order ordinary differential equations (ODEs), which inherently lack the capacity to adequately capture non-local history-dependent regulatory dynamics that arise from sequential biochemical processes such as transcription, translation and protein degradation. Here we construct a Caputo fractional-order model of the Arabidopsis core circadian system and establish local existence and uniqueness, non-negativity, and boundedness of solutions under non-negative initial conditions. Model parameters are fitted to wild-type mRNA expression profiles collected under a standard 12 h light/12 h dark (12L12D) photoperiod. Without any subsequent refitting of parameters, the predictive performance of the fractional-order model is validated on two independent test datasets: wild-type expression time series under three additional photoperiod regimes, and publicly available expression data for major circadian clock loss-of-function mutants. Compared with the original ODE counterpart, the fractional-order formulation exhibits substantially improved performance in reproducing the post-peak decay kinetics and extended tough phase of the PRR5/TOC1 regulatory module. Quantitative error evaluation confirms that the fractional-order model achieves consistently lower mean squared errors across all four photoperiod conditions, and outperforms the integer-order counterpart in two of the four tested mutant backgrounds, indicating that performance gains are not uniformly distributed across all genetic perturbations. Through Matignon-type stability analysis and extensive numerical simulations, we identify a well-defined critical fractional-order threshold. When the fractional order exceeds this critical value, the system’s unique positive equilibrium loses its stability, giving rise to sustained oscillations. From a systems biology perspective, the introduction of fractional-order operators provides a compact phenomenological representation of aggregated historical memory effects and may influence the amplitude, phase, and long-term robustness of the core circadian-clock oscillations. This work offers a new mathematical framework for refining the dynamical characterization of eukaryotic circadian pacemakers.