On September 29, 2023, the research team led by Professor Lin Wei and Associate Researcher Qin Bowei published an academic paper titled Modulating Biological Rhythms: A Noncomputational Strategy Harnessing Nonlinearity and Decoupling Frequency and Amplitude in Physical Review Letters[1]. In this study, the researchers unveil a universal theory for decoupling the frequency and amplitude in a noncomputational manner. Based on this theory, they then propose an easy-to-use design methodology for frequency and amplitude modulation controllers.
Oscillations are prevalent in living systems, such as circadian rhythms, neuronal firing, and heartbeat. Frequency and amplitude play crucial roles in representing biological oscillations. The former characterizes the upstream signal and determines the activation of downstream functions, while the latter regulates their strength [2]. Maintaining oscillations with appropriate frequency and amplitude is essential for the functioning of living systems. Abnormal frequencies or amplitudes can lead to rhythm disturbances and trigger various physical or psychological disorders, including sleep disorders, type 2 diabetes and depression [3,4]. Therefore, it becomes necessary to adjust the frequency and amplitude to normal levels using either the system's inherent modulation mechanism or manual intervention. Researchers from life sciences and related interdisciplinary fields have long been devoted to understanding how to independently control frequency or amplitude while keeping other factors constant. Over decades of research in this direction, significant progress has been made. Some scholars have revealed correlations between frequency adjustability and feedback loop modes in biological oscillatory systems [5]. Others have employed mathematical models to propose feasible FM controllers for specific systems [6], which have been validated by biological experiments conducted by other researchers [7]. Additionally, some scientists have put forth methods and theories on linear feedback regulation of biological oscillations [8,9]. Although these efforts have achieved success at times, they often involve computationally intensive procedures that are complex and resource-demanding or are only applicable to specific systems. These limitations pose challenges for practical applications. Henceforth it holds great theoretical value as well as practical significance to develop a general principle of frequency and amplitude modulation along with an easily implementable controller design.
图 1 非线性控制器实现独立调幅与独立调频，避免了繁琐、复杂的计算。
In order to address this issue, the Fudan University team in their latest study constructed precise quantitative characterizations of frequency and amplitude using rigorous mathematical theories based on a universal dynamic model of biological oscillations. Through a series of theoretical and computational analyses, they unveiled an easily applicable principle for decoupling frequency and amplitude. By leveraging this principle, researchers can intelligently harness the nonlinear dynamics prevalent in biological systems and devise feedback controllers capable of independently adjusting frequencies or amplitudes without necessitating intricate calculations (Figure 1).
图 2非线性控制器以更小的代价实现更广范围的振幅调控。
To validate the efficacy of this controller, the proposed nonlinear feedback controller is implemented on various typical biological oscillating systems. Initially, the authors compare the novel approach with an existing linear feedback controller in a classical genetic oscillator. The new method not only circumvents intricate calculations but also facilitates a wider range of amplitude modulation at a reduced expense (Figure 2). Subsequently, the effectiveness of the nonlinear feedback controller is further substantiated in protein-protein interaction systems. By manipulating the kinetic behavior of one such protein, simultaneous regulation of all protein concentrations can be achieved (Figure 3). This phenomenon is also supported by corresponding mathematical theory. Finally, the author demonstrates the distinctive impact of nonlinear controllers in complex networks. The amplitude control for all nodes can be accomplished solely by adjusting the input signal of a single node in Watts-Strogatz small-world neural network.
This research not only establishes a comprehensive mathematical foundation for the frequency and amplitude modulation of biological oscillatory systems, but also provides a fundamental guideline for biologists to achieve the frequency and amplitude modulation of synthetic living systems in the near future. This study is part of a series of accomplishments by the Fudan team in the field of Basic theory and key algorithm of frequency and amplitude modulation of biological oscillations over the past decade [8,9]. Zhong Zhaoyue, a doctoral student at Fudan University's School of Mathematical Sciences and Research Institute of Intelligent Complex Systems, serves as the first author while Professor Lin Wei and Associate Researcher Qin Bowei are co-corresponding authors. This research received support from various sources including Science and Technology Innovation 2030- Brain Science and Brain-like Research major project, National Natural Science Foundation of China, Shanghai Municipal Science and Technology Commission, Shanghai Municipal Education Commission, and Shanghai Artificial Intelligence Laboratory.
[Reference]
[1]Z. Zhong, W. Lin, and B.-W. Qin, Modulating biological rhythms: A noncomputational strategy harnessing nonlinearity and decoupling frequency and amplitude, Phys. Rev. Lett.131, 138401 (2023).
[2]A.S. Hansen, and E.K. O’Shea, Limits on information transduction through amplitude and frequency regulation of transcription factor activity, eLife4, e06559 (2015).
[3]J. Bass, and J.S. Takahashi, Circadian integration of metabolism and energetics, Science330, 1349-1354 (2010).
[4]F. Rijo-Ferreira, and J.S. Takahashi, Genomics of circadian rhythms in health and disease, Genome Med.11, 82 (2019).
[5]T.Y.-C. Tsai, Y.S. Choi, W. Ma, J.R. Pomerening, C. Tang, and J.E. Ferrell, Robust, tunable biological oscillations from interlinked positive and negative feedback loops, Science321, 126-129 (2008).
[6]M. Tomazou, M. Barahona, K.M. Polizzi, and G.-B. Stan,Computational re-design of synthetic genetic oscillators for independent amplitude and frequency modulation. Cell Syst.6, 508-520.e5 (2018).
[7]F. Zhang, Y. Sun, Y. Zhang et al., Independent control of amplitude and period in a synthetic oscillator circuit with modified repressilator, Commun. Biol.5, 23 (2022).
[8]T. Ge, X. Tian, J. Kurths, J. Feng, and W. Lin, Achieving modulated oscillations by feedback control, Phys. Rev. E90, 022909 (2014).
[9]B.-W. Qin, L. Zhao, and W. Lin, A frequency-amplitude coordinator and its optimal energy consumption for biological oscillators, Nat. Commun.12, 5894 (2021).
