Many gene regulatory networks have periodic behavior, for instance the cell cycle or the circadian clock. Therefore, the study of formal methods to analyze limit cycles in mathematical models of gene regulatory networks is of interest. In this work, we study a pre-existing hybrid modeling framework (HGRN) which extends René Thomas’ widespread discrete modeling. We propose a new formal method to find all limit cycles that are simple and deterministic, and analyze their stability, that is, the ability of the model to converge back to the cycle after a small perturbation. Up to now, only limit cycles in two dimensions (with two genes) have been studied; our work fills this gap by proposing a generic approach applicable in higher dimensions. For this, the hybrid states are abstracted to consider only their borders, in order to enumerate all simple abstract cycles containing possible concrete trajectories. Then, a Poincaré map is used, based on the notion of transition matrix of the concrete continuous dynamics inside these abstract paths. We successfully applied this method on existing models: three HGRNs of negative feedback loops with 3 components, and a HGRN of the cell cycle with 5 components.
Sun, H., Folschette, M., & Magnin, M. (2022). Limit Cycle Analysis of a Class of Hybrid Gene Regulatory Networks. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 13447 LNBI, pp. 217–236). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-031-15034-0_11