Background: The successful identification of breast cancer (BRCA) prognostic biomarkers is essential for the strategic interference of BRCA patients. Recently, various methods have been proposed for exploring a small prognostic gene set that can distinguish the high-risk group from the low-risk group.Methods: Regularized Cox proportional hazards (RCPH) models were proposed to discover prognostic biomarkers of BRCA from gene expression data. Firstly, the maximum connected network with 1142 genes by mapping 956 differentially expressed genes (DEGs) and 677 previously BRCA-related genes into the gene regulatory network (GRN) was constructed. Then, the 72 union genes of the four feature gene sets identified by Lasso-RCPH, Enet-RCPH, $$L_{0}$$L0 -RCPH and SCAD-RCPH models were recognized as the robust prognostic biomarkers. These biomarkers were validated by literature checks, BRCA-specific GRN and functional enrichment analysis. Finally, an index of prognostic risk score (PRS) for BRCA was established based on univariate and multivariate Cox regression analysis. Survival analysis was performed to investigate the PRS on 1080 BRCA patients from the internal validation. Particularly, the nomogram was constructed to express the relationship between PRS and other clinical information on the discovery dataset. The PRS was also verified on 1848 BRCA patients of ten external validation datasets or collected cohorts.Results: The nomogram highlighted that the importance of PRS in guiding significance for the prognosis of BRCA patients. In addition, the PRS of 301 normal samples and 306 tumor samples from five independent datasets showed that it is significantly higher in tumors than in normal tissues ( $$P<0.05$$P<0.05 ). The protein expression profiles of the three genes, i.e.,ADRB1,SAV1andTSPAN14, involved in the PRS model demonstrated that the latter two genes are more strongly stained in tumor specimens. More importantly, external validation illustrated that the high-risk group has worse survival than the low-risk group ( $$P<0.05$$P<0.05 ) in both internal and external validations.Conclusions: The proposed pipelines of detecting and validating prognostic biomarker genes for BRCA are effective and efficient. Moreover, the proposed PRS is very promising as an important indicator for judging the prognosis of BRCA patients.

Motivation: From a systematic perspective, it is crucial to infer and analyze gene regulatory network (GRN) from high-throughput single-cell RNA sequencing data. However, most existing GRN inference methods mainly focus on the network topology, only few of them consider how to explicitly describe the updated logic rules of regulation in GRNs to obtain their dynamics. Moreover, some inference methods also fail to deal with the over-fitting problem caused by the noise in time series data.Results: In this article, we propose a novel embedded Boolean threshold network method called LogBTF, which effectively infers GRN by integrating regularized logistic regression and Boolean threshold function. First, the continuous gene expression values are converted into Boolean values and the elastic net regression model is adopted to fit the binarized time series data. Then, the estimated regression coefficients are applied to represent the unknown Boolean threshold function of the candidate Boolean threshold network as the dynamical equations. To overcome the multi-collinearity and over-fitting problems, a new and effective approach is designed to optimize the network topology by adding a perturbation design matrix to the input data and thereafter setting sufficiently small elements of the output coefficient vector to zeros. In addition, the cross-validation procedure is implemented into the Boolean threshold network model framework to strengthen the inference capability. Finally, extensive experiments on one simulated Boolean value dataset, dozens of simulation datasets, and three real single-cell RNA sequencing datasets demonstrate that the LogBTF method can infer GRNs from time series data more accurately than some other alternative methods for GRN inference.Availability and implementation: The source data and code are available at https://github.com/zpliulab/LogBTF.

Background: In recent years, the identification of biomarkers or signatures based on gene expression profiling data has attracted much attention in bioinformatics. The successful discovery of breast cancer (BRCA) biomarkers will be beneficial in reducing the risk of BRCA among patients for early detection. Methods: This paper proposes an Ensemble Feature Selection method to screen biomarkers (abbreviat-ed as EFSmarker) for BRCA from publically available gene expression data. Firstly, we employ twelve filter feature selection methods, namely median, variance, Chi-square, Relief, Pearson and Spearman correlation, mutual information, minimal-redundancy-maximal-relevance criterion, ridge regression, decision tree and random forest with Gini index and accuracy index, to calculate the importance (weights or coefficients) of all features on the training dataset. Secondly, we apply the logistic regres-sion classifier on the test dataset to calculate the classification AUC value of each feature subset indi-vidually selected by twelve methods. Thirdly, we provide an ensemble feature selection method by ag-gregating feature importance with classification AUC value. In particular, we establish a feature im-portance score (FIS) to evaluate the importance of each feature underlying all feature selection methods. Finally, the features with higher FIS are taken as identified biomarkers. Results: With the direction of the FIS index induced by the EFSmarker method, 12 genes (COL10A1, COL11A1, MMP11, LOC728264, FIGF, GJB2, INHBA, CD300LG, IGFBP6, PAMR1, CXCL2 and FXYD1) are regarded as diagnostic biomarkers for BRCA. Especially, COL10A1, ranked first with a FIS value of 0.663, is identified as the most credible biomarker. The findings justified via gene and protein expression validation, functional enrichment analysis, literature checking and independent dataset validation verify the effectiveness and efficiency of these selected biomarkers. Conclusion: Our proposed biomarker discovery strategy not only utilizes the feature contribution but also considers the prediction accuracy simultaneously, which may also serve as a model for identifying unknown biomarkers for other diseases from high-throughput gene expression data. The source code and data are available at https://github.com/zpliulab/EFSmarker.