Background
The complex biological processes, such as pericyte-to-neuron transition, pluripotency-to-hepatocyte transition, and epithelial-to-mesenchymal transition, involve pre-transition or critical states where significant and qualitative shifts occur. From a dynamics viewpoint, complex biological processes are typically seen as a time-dependent nonlinear dynamical system, characterized by three main phases: a before-transition state with stability and resilience, a pre-transition state marked by high sensitivity and instability where cell state transition occurs, and a subsequent stable after-transition state. Identifying the critical state and associated key molecules is pivotal for many biological phenomena, such as cell differentiation, where it is crucial for cellular reprogramming and holds great significance for progress in regenerative medicine. However, the high dimensionality, non-linearity, and dynamic complexity of biological systems make it difficult to detect precursory signs of the pre-transition state, especially when working with inherently sparse, noisy, and heterogeneous single-cell datasets. Although various methods have achieved effective outcomes in single-cell data analysis, particularly in cell clustering, trajectory inference, and cellular heterogeneity [6-8], the task of analyzing molecular causal regulatory relationships and predicting pre-transition states continues to present a significant challenge.

Research Progress
To address this challenge, Professor Liu Rui's team from South China University of Technology introduces a novel quantitative method, cell-specific causal network entropy (CCNE), which analyzes gene-level causal relationships to infer distinct causal networks for individual cells and identify pre-transition states in complex biological systems. Specifically, the cell-specific causal network is constructed based on the continuity scaling law, a rigorous mathematical framework consistent with the natural interpretation of functional dependency, allowing for the quantification of 'continuous causality' and its causal strength between two molecules through varying neighbor size (Fig. 1A). Furthermore, the local CCNE is calculated to evaluate the dynamic alterations of causal relationships among molecules within localized causal networks, and the CCNE at each time point serves to quantify the criticality of complex biological process, with a significant increase indicating an impending pre-transition or critical state (Fig. 1B). To demonstrate the reliability and effectiveness of CCNE, Liu Rui's team conducted a verification using a numerical simulation and five distinct real-world single-cell datasets. The results show that CCNE is more effective and accurate in pinpointing pre-transition states compared to existing methods, with the predicted pre-transition states aligning well with experimental observations. Furthermore, by converting the sparse single-cell expression matrix into a non-sparse entropy matrix, CCNE can effectively distinguish cellular heterogeneity over time and facilitate analysis of temporal clusters.

Future Prospects
Although the current CCNE method demonstrates significant advantages in capturing critical signals in biological processes, classifying cellular heterogeneity, and identifying "dark genes," there are still areas for improvement. For example, the CCNE method currently relies on the PPI network as a background network. Improving the generalizability and accuracy of the method without relying on existing networks will be a key focus of future research. Additionally, although CCNE can effectively identify critical points, interpreting the biological significance of each identified critical point in multi-stage biological processes remains a challenge.

The complete study is accessible via DOI:10.34133/research.0852