Macroscopic dynamics of gene regulatory networks revealed by individual entropy

Complex systems are usually high-dimensional with intricate interactions among internal components, and may display complicated dynamics under different conditions. While it is {difficult} to measure detail dynamics of each component, proper macroscopic description of a complex system is crucial for quantitative studies. In biological systems, each cell is a complex system containing a huge number of molecular components that are interconnected with each other through intricate molecular interaction networks. Here, we consider gene regulatory networks in a cell, and introduce individual entropy as a macroscopic variable to quantify the transcriptional dynamics in response to changes in random perturbations and/or network structures. The proposed individual entropy measures the information entropy of a system at each instant with respect to a basal reference state, and may provide temporal dynamics to characterize switches of system states. Individual entropy provides a method to quantify the stationary macroscopic dynamics of a gene set that is dependent on the gene regulation connections, and can be served as an indicator for the evolution of network structure variation. Moreover, the individual entropy with reference to a preceding state enable us to characterize different dynamic patterns generated from varying network structures. Our results show that the proposed individual entropy can be a valuable macroscopic variable of complex systems in characterizing the transition processes from order to disorder dynamics, and to identify the critical events during the transition process.