Exploring robustness of bistability in prototypical positive feedback loops

Bistability is a dynamical property of biological systems which have the ability to possess two distinct stable steady states. Bistability is the hallmark of decision-making processes and underlies basic cellular functions such as cell cycle progression, cellular differentiation, and apoptosis. It is crucial for a bistable system to operate robustly, meaning that it has to be able to maintain the bistable behavior in the presence of perturbations in its kinetic parameters. We aim to understand how different parameter configurations and ultrasensitive mechanisms such as molecular cooperativity, homodimerization and titration, organize bistability and its robustness in prototypical feedback loop systems. We in particular show that the coupling between a positive and a negative feedback loop, enclosed under the titration mechanism, can enlarge the bistability range of a single parameter, and therefore contribute to the robustness of bistability. We also develop a method based on the open-loop approach to explore parametric regions inside the bistability area of bifurcation diagrams, in which the sensitivity of unstable steady state to parameters of a system can be minimized. Unstable steady states are key organizers of bistability and minimization of their sensitivity to parameters leads to the persistence of the bistable behavior against parameter perturbations. Our results provide insight into the role of different parameters as well as homodimerization and titration mechanisms in creating robust bistability in positive feedback systems. Additionally, we study the galactose network in {\it Saccharomyces cerevisiae}, in which bistability creates a persistent memory of the carbon source that is available in the environment. We reconstruct the bistable behavior of the network by developing a mathematical model that represents the molecular interactions of the network. Using the experimental data extracted from different layers of the network, we perform nonlinear regression to estimate the parameter values of the model. Our investigations reveal the significance of homodimerization and titration in creating bistability in the galactose network. In summary, our results provide a better understanding of how parameter configurations and different ultrasensitive regulatory motifs contribute to bistability and its robustness. The results can be used to efficiently design and synthesize robust bistable switches.