Switching behavior (bistability) resulting from interacting signaling pathways. (**a**) By binding to α1 and α2 adrenoceptors, norepinephrine can activate both the G_{q} and G_{i/o} pathways. Interactions between these pathways can result in a positive-feedback loop []. A key question regarding the behavior of this system is whether the positive-feedback loop can lead to bistability. (**b**) To answer this question, we can develop a model of this system to determine whether bistability occurs. The first two ordinary differential equations (ODEs) describe the formation of receptor–ligand complexes for α1 and α2 adrenoceptors. The equations state that the rates of change of the receptor–ligand complexes are equal to the rate of binding of the ligand to the receptor (first term), minus the rate of dissociation of the ligand (second term), minus the rate of binding of the G proteins to the complexes, plus the rate of activation of the G proteins. In these equations, the rate of change in concentration is represented as *d*[*C*]/*dt*, where [*C*] represents the concentration of species *C*. The parameter *k*_{i} represents the kinetic constant (forward or reverse rate) for reaction *i*. The last equation of this set states that the rate of change of phosphorylated mitogen-activated protein kinase (MAPK) is equal to the rate of phosphorylation by MEK minus the rate of dephosphorylation. Here, the phosphorylation of MAPK is assumed to follow Michaelis–Menten kinetics, with the rate specified by the affinity constant *K*_{m} and the turnover number *k*_{cat} of MEK. Similar equations can be written for the remaining components of the system. After the equations have been written for all the components, and values assigned for concentration and rate terms, the system can be analyzed for many properties. For instance, bistability can be explored by examining the effect of kinetic parameters and initial concentrations on the steady-state solutions, which are calculated by setting the rate of change of all molecules to zero. Abbreviations: AA, arachidonic acid; AR, adrenoceptor; cPLA_{2}, cytoplasmic phospholipase A_{2}; DAG, diacylglycerol; IP_{3}, inositol (1,4,5)-trisphosphate; NE, norepinephrine; PKC, protein kinase C; PLCβ, phospholipase Cβ. Image in (a) reproduced, with permission, from .

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