Biological systems are spatially organized. This microscopic heterogeneity has been shown to produce emergent complex behaviors such as bistability. Even though the connection between spatiality and dynamic response is essential to understand biological output, its robustness and extent has not been sufficiently explored. This work focuses on a previously described system which is composed of two monostable modules acting on different cellular compartments and sharing species through linear shuttling reactions. One of the two main purposes of this paper is to quantify the frequency of occurrence of bistability throughout the parameter space and to identify which parameters and in which value ranges control the emergence and the properties of bistability. We found that a very small fraction of the sampled parameter space produced a bistable response. Most importantly, shuttling parameters were among the most influential ones to control this property. The other goal of this paper is to simplify the same system as much as possible without losing compartment-induced bistability. This procedure provided a simplified model that still connects two monostable systems by a reduced set of linear shuttling reactions that circulates all the species around the two compartments. Bistable systems are one of the main building blocks of more complex behaviors such as oscillations, memory, and digitalization. Therefore, we expect that the proposed minimal system provides insight into how these behaviors can arise from compartmentalization.

In this article, we consider a double phosphorylation cycle, a ubiquitous signaling component, having the ability to display bistability, a behavior strongly related to the existence of positive feedback loops. If this component is connected to other signaling elements, it very likely undergoes some sort of protein–protein interaction. In several cases, these interactions result in a non-explicit negative feedback effect, leading to interlinked positive and negative feedbacks. This combination was studied in the literature as a way to generate relaxation-type oscillations. Here, we show that the two feedbacks together ensure two types of oscillations, the relaxation-type ones and a smoother type of oscillations functioning in a very narrow range of frequencies, in such a way that outside that range, the amplitude of the oscillations is severely compromised. Even more, we show that the two feedbacks are essential for both oscillatory types to emerge, and it is their hierarchy what determines the type of oscillation at work. We used bifurcation analyses and amplitude vs. frequency curves to characterize and classify the oscillations. We also applied the same ideas to another simple model, with the goal of generalizing what we learned from signaling models. The results obtained display the wealth of oscillatory dynamics that exists in a system with a bistable module nested within a negative feedback loop, showing how to transition between different types of oscillations and other dynamical behaviors such as excitability. Our work provides a framework for the study of other oscillatory systems based on bistable modules, from simple two-component models to more complex examples like the MAPK cascade and experimental cases like cell cycle oscillators.