Synthetic paradoxical control circuits have been used to regulate cell population density. However, paradoxical interactions and inherent time delays in these circuits complicate stable population control due to their effects on system dynamics. This article proposes a strategy to achieve robust population control by analyzing the bistable switching and oscillatory behavior in a delayed paradoxical cell population control circuit. First, five distinct population outcomes are identified, along with stabilization criteria for these states, highlighting the presence of bistability. Specifically, we reveal two bistable switching mechanisms: one driven by the initial cell population density and the other by the time delay for blasticidin-induced cell death. Furthermore, we demonstrate that the time delay can destabilize the effective control state via a Hopf bifurcation, leading to periodic oscillations, which can be mitigated by adjusting the blasticidin concentration. Based on these findings, we propose a robust control strategy by regulating initial cell population density and blasticidin concentration. These insights advance the design of robust population control circuits, with broad implications for synthetic biology and cell therapy.

Circadian rhythm disruptions are linked to increased cancer risk and unfavorable prognosis in patients with cancer, highlighting the critical role of the interplay between the circadian rhythm factor Per2 and the tumor suppressor p53. This brief presents, for the first time, a mathematical model to capture the dynamics of the p53-Per2 network in DNA-damaged cells. The model accurately describes the different stages of the process from unstressed cells to cellular repair and finally to apoptosis as the degree of DNA damage increases. Furthermore, it is found that increasing the inhibition of Per2 by p53 leads to the phase advance of Per2 oscillations, whereas by modulating the inhibition of Mdm2 by Per2, an independent amplitude modulation of active p53 can be achieved, with the range of modulation increasing with the strength of the inhibition. Moreover, the effects of time delays inherent in the transcription, translation, and nuclear translocation of Per2 on the circadian rhythm of DNA-damaged cells are quantitatively investigated by theoretical analyses. It is found that time delays can induce stable oscillations through a supercritical Hopf bifurcation, thereby maintaining the circadian function of DNA-damaged cells and enhancing their DNA-damage repair capacity. This study proposes new insights into cancer prevention and treatment strategies.

This article addresses the disturbance decoupling problem (DDP) of probabilistic Boolean networks (PBNs). Existing research on the DDP of Boolean networks (BNs) has predominantly concentrated on deterministic BNs (DBNs), which inadequately model the complexities of real-world systems, particularly those exhibiting stochastic behavior and uncertainty. We introduce two definitions tailored to PBNs: disturbance decoupling in distribution and disturbance decoupling with probability one. Utilizing an algebraic representation framework, we derive algebraic conditions for both types of disturbance decoupling. Additionally, we employ a vertex partitioning approach to establish graphical conditions pertinent to these disturbance decoupling types. Our investigation further reveals the connections between DBNs and PBNs concerning disturbance decoupling. To demonstrate the effectiveness of our theoretical results, we present three examples.

This article proposes a nonaugmented method for investigating the minimal observability problem of Boolean networks (BNs). This method can be applied to more general BNs and reduce the computational and space complexity of existing results. First, unobservable states concerning an unobservable BN are classified into three categories using the vertex-colored state transition graph, each accompanied by a necessary and sufficient condition for determining additional measurements to make them distinguishable. Then, an algorithm is designed to identify the additional measurements that would render an unobservable BN observable using the conditions. Next, to determine the minimum added measurements, a necessary and sufficient condition and an algorithm based on a constructed matrix are presented. Finally, the results obtained are compared with existing literature and illustrated with examples.