Junghun Chae1,7, Roktaek Lim1,2,7, Thomas L. P. Martin2, Cheol-Min Ghim1,3,4,,and Pan-Jun Kim2,4,5,6,1- Department of Physics, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea2- Department of Biology, Hong Kong Baptist University, Kowloon, Hong Kong 3- Department of Biomedical Engineering, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea4- Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Republic of Korea 5- Center for Quantitative Systems Biology & Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon, Hong Kong6- Abdus Salam International Centre for Theoretical Physics, 34151 Trieste, Italy7- J.C. and R.L. contributed equally to this work.*Author to whom correspondence should be addressed:C.-M.G. (email: cmghim@unist.ac.kr) or P.-J.K. (email: extutor@gmail.com).AbstractThe 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, thecombined 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 quasisteady states and enhance the mechanistic understanding of rich time-varying biomolecular activities.