  The century-long Michaelis-Menten rate law and its modifications in the
modeling of biochemical rate processes stand on the assumption that the
concentration of the complex of interacting molecules, at each moment, rapidly
approaches an equilibrium (quasi-steady state) compared to the pace of
molecular concentration changes. Yet, in the case of actively time-varying
molecular concentrations with transient or oscillatory dynamics, the deviation
of the complex profile from the quasi-steady state becomes relevant. A recent
theoretical approach, known as the effective time-delay scheme (ETS), suggests
that the delay by the relaxation time of molecular complex formation
contributes to the substantial breakdown of the quasi-steady state assumption.
Here, we systematically expand this ETS and inquire into the comprehensive
roles of relaxation dynamics in complex formation. Through the modeling of
rhythmic protein-protein and protein-DNA interactions and the mammalian
circadian clock, our analysis reveals the effect of the relaxation dynamics
beyond the time delay, which extends to the dampening of changes in the complex
concentration with a reduction in the oscillation amplitude against the
quasi-steady state. Interestingly, the combined effect of the time delay and
amplitude reduction shapes both qualitative and quantitative oscillatory
patterns such as the emergence and variability of the mammalian circadian
rhythms. These findings highlight the drawback of the routine assumption of
quasi-steady states and enhance the mechanistic understanding of rich
time-varying biomolecular activities.
