Typically, in dynamic biological processes, there is a critical state or tipping point that marks the transition from one stable state to another, surpassing which a considerable qualitative shift takes place. Identifying this tipping point and its driving network is essential to avert or delay disastrous outcomes. However, most traditional approaches built upon undirected networks still suffer from a lack of robustness and effectiveness when implemented based on high-dimensional small-sample data, especially for single-cell data. To address this challenge, we develop 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. Applying this algorithm to six real datasets, including three single-cell datasets, two bulk tumor datasets, and a blood dataset, the method is proved to be effective not only in identifying critical states, as well as their dynamic network biomarkers, but also in helping explore regulatory relationships between genes. Numerical simulation results demonstrate that the DNFE algorithm is robust across various noise levels and outperforms existing methods in detecting tipping points. Furthermore, the numerical simulations for 100-node and 1000-node gene regulatory networks illustrate the method's application for large-scale data. The DNFE method predicts active transcription factors, and further identified "dark genes", which are usually overlooked with traditional methods.

There generally exists a critical state or tipping point from a stable state to another in the development of colorectal cancer (CRC) beyond which a significant qualitative transition occurs. Gut microbiome sequencing data can be collected non-invasively from fecal samples, making it more convenient to obtain. Furthermore, intestinal microbiome sequencing data contain phylogenetic information at various levels, which can be used to reliably identify critical states, thereby providing early warning signals more accurately and effectively. Yet, pinpointing the critical states using gut microbiome data presents a formidable challenge due to the high dimension and strong noise of gut microbiome data. To address this challenge, we introduce a novel approach termed the specific network information gain (SNIG) method to detect CRC's critical states at various taxonomic levels via gut microbiome data. The numerical simulation indicates that the SNIG method is robust under different noise levels and that it is also superior to the existing methods on detecting the critical states. Moreover, utilizing SNIG on two real CRC datasets enabled us to discern the critical states preceding deterioration and to successfully identify their associated dynamic network biomarkers at different taxonomic levels. Notably, we discovered certain 'dark species' and pathways intimately linked to CRC progression. In addition, we accurately detected the tipping points on an individual dataset of type I diabetes.

BACKGROUND: There exists a critical transition or tipping point during the complex biological process. Such critical transition is usually accompanied by the catastrophic consequences. Therefore, hunting for the tipping point or critical state is of significant importance to prevent or delay the occurrence of catastrophic consequences. However, predicting critical state based on the high-dimensional small sample data is a difficult problem, especially for single-cell expression data. RESULTS: In this study, we propose the comprehensive neighbourhood-based perturbed mutual information (CPMI) method to detect the critical states of complex biological processes. The CPMI method takes into account the relationship between genes and neighbours, so as to reduce the noise and enhance the robustness. This method is applied to a simulated dataset and six real datasets, including an influenza dataset, two single-cell expression datasets and three bulk datasets. The method can not only successfully detect the tipping points, but also identify their dynamic network biomarkers (DNBs). In addition, the discovery of transcription factors (TFs) which can regulate DNB genes and nondifferential 'dark genes' validates the effectiveness of our method. The numerical simulation verifies that the CPMI method is robust under different noise strengths and is superior to the existing methods on identifying the critical states. CONCLUSIONS: In conclusion, we propose a robust computational method, i.e., CPMI, which is applicable in both the bulk and single cell datasets. The CPMI method holds great potential in providing the early warning signals for complex biological processes and enabling early disease diagnosis.

Complex biological systems often undergo sudden qualitative changes during their dynamic evolution. These critical transitions are typically characterized by a catastrophic progression of the system. Identifying the critical point is critical to uncovering the underlying mechanisms of complex biological systems. However, the system may exhibit minimal changes in its state until the critical point is reached, and in the face of high throughput and strong noise data, traditional biomarkers may not be effective in distinguishing the critical state. In this study, we propose a novel approach, mutual information weighted entropy (MIWE), which uses mutual information between genes to build networks and identifies critical states by quantifying molecular dynamic differences at each stage through weighted differential entropy. The method is applied to one numerical simulation dataset and four real datasets, including bulk and single-cell expression datasets. The critical states of the system can be recognized and the robustness of MIWE method is verified by numerical simulation under the influence of different noises. Moreover, we identify two key transcription factors (TFs), CREB1 and CREB3, that regulate downstream signaling genes to coordinate cell fate commitment. The dark genes in the single-cell expression datasets are mined to reveal the potential pathway regulation mechanism.

BACKGROUND: In the field of computational personalized medicine, drug response prediction (DRP) is a critical issue. However, existing studies often characterize drugs as strings, a representation that does not align with the natural description of molecules. Additionally, they ignore gene pathway-specific combinatorial implication. RESULTS: In this study, we propose drug Graph and gene Pathway based Drug response prediction method (GPDRP), a new multimodal deep learning model for predicting drug responses based on drug molecular graphs and gene pathway activity. In GPDRP, drugs are represented by molecular graphs, while cell lines are described by gene pathway activity scores. The model separately learns these two types of data using Graph Neural Networks (GNN) with Graph Transformers and deep neural networks. Predictions are subsequently made through fully connected layers. CONCLUSIONS: Our results indicate that Graph Transformer-based model delivers superior performance. We apply GPDRP on hundreds of cancer cell lines' bulk RNA-sequencing data, and it outperforms some recently published models. Furthermore, the generalizability and applicability of GPDRP are demonstrated through its predictions on unknown drug-cell line pairs and xenografts. This underscores the interpretability achieved by incorporating gene pathways.

Complex diseases progression can be generally divided into three states, which are normal state, predisease/critical state and disease state. The sudden deterioration of diseases can be viewed as a bifurcation or a critical transition. Therefore, hunting for the tipping point or critical state is of great importance to prevent the disease deterioration. However, it is still a challenging task to detect the critical states of complex diseases with high-dimensional data, especially based on an individual. In this study, we develop a new method based on network fluctuation of molecules, temporal network flow entropy (TNFE) or temporal differential network flow entropy, to detect the critical states of complex diseases on the basis of each individual. By applying this method to a simulated dataset and six real diseases, including respiratory viral infections and tumors with four time-course and two stage-course high-dimensional omics datasets, the critical states before deterioration were detected and their dynamic network biomarkers were identified successfully. The results on the simulated dataset indicate that the TNFE method is robust under different noise strengths, and is also superior to the existing methods on detecting the critical states. Moreover, the analysis on the real datasets demonstrated the effectiveness of TNFE for providing early-warning signals on various diseases. In addition, we also predicted disease deterioration risk and identified drug targets for cancers based on stage-wise data.

In this letter, we deal with a class of memristor-based neural networks with distributed leakage delays. By applying a new Lyapunov function method, we obtain some sufficient conditions that ensure the existence, uniqueness, and global exponential stability of almost periodic solutions of neural networks. We apply the results of this solution to prove the existence and stability of periodic solutions for this delayed neural network with periodic coefficients. We then provide an example to illustrate the effectiveness of the theoretical results. Our results are completely new and complement the previous studies Chen, Zeng, and Jiang ( 2014 ) and Jiang, Zeng, and Chen ( 2015 ).