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 detailed 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 enables 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.

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.

Using single-cell entropy to describe the dynamics of reprogramming and differentiation of induced pluripotent stem cells

Induced pluripotent stem cells (iPSCs) provide a great model to study the process of stem cell reprogramming and differentiation. Single-cell RNA sequencing (scRNA-seq) enables us to investigate the reprogramming process at single-cell level. Here, we introduce single-cell entropy (scEntropy) as a macroscopic variable to quantify the cellular transcriptome from scRNA-seq data during reprogramming and differentiation of iPSCs. scEntropy measures the relative order parameter of genomic transcriptions at single cell level during the process of cell fate changes, which show increase tendency during differentiation, and decrease upon reprogramming. Hence, scEntropy provides an intrinsic measurement of the cell state, and can be served as a pseudo-time of the stem cell differentiation process. Moreover, based on the evolutionary dynamics of scEntropy, we construct a phenomenological Fokker-Planck equation model and the corresponding stochastic differential equation for the process of cell state transitions during pluripotent stem cell differentiation. These equations provide further insights to infer the processes of cell fates changes and stem cell differentiation. This study is the first to introduce the novel concept of scEntropy to quantify the biological process of iPSC, and suggests that the scEntropy can provide a suitable macroscopic variable for single cells to describe cell fate transition during differentiation and reprogramming of stem cells.

Using single-cell entropy to describe the dynamics of reprogramming and differentiation of induced pluripotent stem cells

Induced pluripotent stem cells (iPSCs) provide a great model to study the process of stem cell reprogramming and differentiation. Single-cell RNA sequencing (scRNA-seq) enables us to investigate the reprogramming process at single-cell level. Here, we introduce single-cell entropy (scEntropy) as a macroscopic variable to quantify the cellular transcriptome from scRNA-seq data during reprogramming and differentiation of iPSCs. scEntropy measures the relative order parameter of genomic transcriptions at single cell level during the process of cell fate changes, which show increase tendency during differentiation, and decrease upon reprogramming. Hence, scEntropy provides an intrinsic measurement of the cell state, and can be served as a pseudo-time of the stem cell differentiation process. Moreover, based on the evolutionary dynamics of scEntropy, we construct a phenomenological Fokker-Planck equation model and the corresponding stochastic differential equation for the process of cell state transitions during pluripotent stem cell differentiation. These equations provide further insights to infer the processes of cell fates changes and stem cell differentiation. This study is the first to introduce the novel concept of scEntropy to quantify the biological process of iPSC, and suggests that the scEntropy can provide a suitable macroscopic variable for single cells to describe cell fate transition during differentiation and reprogramming of stem cells.

Thermodynamics of porous media

Equilibrium thermodynamics for porous media is considered with special emphasis on its basis in pore-scale thermodynamics. It is shown that porosity, the new purely macroscopic variable, enters the relations on the same footing as mass densities and the strain tensors. Biot’s use of elastic energy potential, which lies at the foundation of his theory of poroelasticity, is examined in light of the results obtained here.