The landscape and flux of a minimum network motif, Wu Xing* *Project supported by the National Natural Science Foundation of China (Grant No. 21721003) and the Fund from the Ministry of Science and Technology of China (Grant No. 2016YFA0203200).

Abstract

Wu-Xing theory is an ancient philosophy that serves as a guiding principle in the traditional Chinese medicine (TCM). It has been used to explain the unbalance among the TCM organ systems in disease states and provide treatment philosophy qualitatively. Until now, it is still a challenge to explore the Wu-Xing theory beyond its philosophical nature. In this study, we established a quantitative framework using the landscape and flux theory to characterize the nature of the Wu-Xing theory from a perspective of a minimal network motif and leave certain specific functional aspects of Wu-Xing theory for future exploration. We uncovered the irregular ring shape of projection landscape for the Wu-Xing network with several local basins and barriers. We found that the dynamics of the self-organized Wu-Xing system was determined by the underlying negative landscape gradient force and the nonequilibrium rotational flux. While the shape of the Wu-Xing landscape determines the stabilities of the states, the rotational flux guarantees the persistent periodic oscillation and the stability of the flow. This provides a physical and quantitative basis for Yin–Yang duality of the driving forces for determining the dynamics and behaviors of the living systems. Applying landscape and flux analysis, we can identify the key parameter for the dynamics/function of Wu-Xing network. These findings allow us to have a deeper understanding of the scientific merits of the ancient Wu-Xing theory from the network motif perspective.