The mathematical exploration for the mechanism of lung adenocarcinoma formation and progression.

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.