DNFE: Directed-network flow entropy for detecting the tipping points during biological processes

There generally exists a critical state or tipping point from a stable state to another in dynamic biological processes, beyond which a significant qualitative transition occurs. Identifying this tipping point and its driving network is essential to prevent or delay catastrophic consequences. However, most traditional approaches based on undirected networks still suffer from the problem of the robustness and effectiveness when applied to high-dimensional small sample data, especially for single-cell data. To address this challenge, we developed a directed-network flow entropy (DNFE) method which can transform measured omics data into a directed network. This method is applicable to both single-cell RNA-sequencing (scRNA-seq) and bulk data. By applying this method to five real datasets, including three single-cell datasets and two bulk tumor datasets, the method can not only successfully detect the critical states as well as their dynamic network biomarkers, but also help explore regulatory relationships between genes. Numerical simulation indicates that the DNFE method is robust and superior to existing methods. Furthermore, DNFE has predicted active transcription factors (TFs), and further identified 'dark genes', which are usually overlooked by traditional methods.