Originally proposed as a conceptual metaphor, the Waddington landscape was used to illustrate the directional nature of embryonic development and the relative stability of distinct developmental states. While the Waddington landscape offers a valuable conceptual framework for understanding cellular dynamics, its quantitative reconstruction remains a significant challenge in systems biology and biophysics. Recent methodological advances in single-cell omics technologies, computational modeling approaches, and nonlinear dynamical systems theory have facilitated progress toward quantitative reconstruction of the Waddington landscape, thereby transforming this heuristic metaphor into a predictive theoretical framework. In this review, we summarize the theoretical foundations of the Waddington landscape, categorize current computational and mathematical approaches for the Waddington landscape reconstruction. Meanwhile, we highlight the potential applications and inherent limitations of these approaches in characterizing cellular behaviors, predicting cell fate decisions, and modulating developmental trajectories.

Breast carcinoma (BRCA) is a leading cause of mortality in women worldwide. Understanding the driving factors behind BRCA initiation, progression, and evolution is crucial. This study proposes a novel method to identify stage-related driving factors in BRCA. By utilizing stage-specific functional interaction networks, the multi-omics features and PPI were integrated. A novel rumor-mongering model is introduced to smooth the stage-specific networks and the gravity model is used to balance gene interactions. The top 100 gravity interactions are identified as driving factors. Through biomolecular and enrichment analyses, these driving factors are shown to play a crucial role in BRCA progression. Furthermore, a hybrid hierarchical evolution network illustrates the stage-evolutionary role of driving factors, while a biological functional evolution network demonstrates functional changes in BRCA progression. The proposed method exhibits superior enrichment performance, particularly within targeted pathways, providing valuable insights into the underlying mechanisms of BRCA.

Lung adenocarcinoma, a prevalent subtype of lung cancer, represents one of the most lethal human malignancies. Despite substantial efforts to elucidate its biological underpinnings, the underlying mechanisms governing lung adenocarcinoma remain enigmatic. Modeling and comprehending the dynamics of gene regulatory networks are crucial for unraveling the fundamental mechanisms of lung adenocarcinoma. Conventionally, the cancer is modeled as an equilibrium process based on a time-invariant gene regulatory network to investigate stable cell states. However, the cancer is a nonequilibrium process and the gene regulatory network should be regarded as time-varying in actual. Therefore, a feasible framework was developed to explore the formation and progression of lung adenocarcinoma. On the one hand, to delve into the underlying mechanisms of lung adenocarcinoma formation, the time-invariant gene regulatory network for lung adenocarcinoma was initially undertaken, and the composition of stable cell states was elucidated based on landscape theory. Furthermore, the plasticity of different states was quantified using energy landscape decomposition theory by incorporating cell proliferation. And transition probabilities between different states were defined to elucidate the transition between stable cell states. Additionally, the global sensitivity analysis was performed and a total of three genes and three regulations were identified to be more critical for the formation lung adenocarcinoma, offering a novel strategy for designing network-based therapies for its treatment. On the other hand, the time-invariant gene regulatory network is extended as time-varying to delve into the underlying mechanisms of lung adenocarcinoma progression. The lung adenocarcinoma progression was characterized as four different disease stages based on the mixed states of cell population and the evolutionary direction. And the progressionary mechanism of transition between stages was expounded by evaluating their dynamical transport, with the dynamical transport cost between different stages quantified using Wasserstein metrics.

The evolution of lung adenocarcinoma is accompanied by a multitude of gene mutations and dysfunctions, rendering its phenotypic state and evolutionary direction highly complex. To interpret the evolution of lung adenocarcinoma, various methods have been developed to elucidate the molecular pathogenesis and functional evolution processes. However, most of these methods are constrained by the absence of cancerous temporal information, and the challenges of heterogeneous characteristics. To handle these problems, in this study, a patient quasi-potential landscape method was proposed to estimate the cancerous time of phenotypic states' emergence during the evolutionary process. Subsequently, a total of 39 different oncogenetic paths were identified based on cancerous time and mutations, reflecting the molecular pathogenesis of the evolutionary process of lung adenocarcinoma. To interpret the evolution patterns of lung adenocarcinoma, three oncogenetic graphs were obtained as the common evolutionary patterns by merging the oncogenetic paths. Moreover, patients were evenly re-divided into early, middle, and late evolutionary stages according to cancerous time, and a feasible framework was developed to construct the functional evolution network of lung adenocarcinoma. A total of six significant functional evolution processes were identified from the functional evolution network based on the pathway enrichment analysis, which plays critical roles in understanding the development of lung adenocarcinoma.