MOTIVATION: Trajectory inference methods are essential for extracting temporal ordering from static single-cell transcriptomic profiles, thus facilitating the accurate delineation of cellular developmental hierarchies and cell-fate transitions. However, numerous existing methods treat trajectory inference as an unsupervised learning task, rendering them susceptible to technical noise and data sparsity, which often lead to unstable reconstructions and ambiguous lineage assignments. RESULTS: Here, we introduce BayesTraj, a semi-supervised Bayesian framework that incorporates prior knowledge of lineage topology and marker-gene expression to robustly reconstruct differentiation trajectories from scRNA-seq data. BayesTraj models cellular differentiation as a probabilistic mixture of latent lineages and captures marker-gene dynamics through parametric functions. Posterior inference is conducted using Hamiltonian Monte Carlo (HMC), yielding estimates of pseudotime, lineage proportions, and gene activation parameters. Evaluations on both simulated and real datasets with diverse branching structures demonstrate that BayesTraj consistently outperforms state-of-the-art methods in pseudotime inference. In addition, it provides per-cell branch-assignment probabilities, enabling the quantification of differentiation potential using Shannon entropy and the detection of lineage-specific gene expression via Bayesian model comparison. AVAILABILITY AND IMPLEMENTATION: BayesTraj is written in R and available at https://github.com/SDU-W-Zhanglab/BayesTraj and has been archived on Zenodo (DOI: 10.5281/zenodo.16758038).

As for a pipe-line plant subjected to complex logical control specifications, an approach is proposed to generate the control codes via Petri net such that the plant is run as concurrently as possible. A PN model, called the plant net, is designed to represent the whole process of a pipe-line system. In this plant net, the tasks, including fluid transportation from one tank to another and cleaning tank, are represented by operational places, and level sensors in tanks are modeled by labels assigned with transitions. Further, conflict structures are used to describe the conflict relations among tasks due to the shared valves and pipes. By introducing a priority order in the transitions, an algorithm is presented to compute the control commands for valves based on this plant net. A beer filtration plant is taken as an example to illustrate the method.