Bistability in One Equation or Fewer

When several genes or proteins modulate one another’s activity as part of a network, they sometimes produce behaviors that no protein could accomplish on its own. Intuition for these emergent behaviors often cannot be obtained simply by tracing causality through the network in discreet steps. Specifically, when a network contains a feedback loop, biologists need specialized tools to understand the network’s behaviors and their necessary conditions. This analysis is grounded in the mathematics of ordinary differential equations. We, however, will demonstrate the use of purely graphical methods to determine, for experimental data, the plausibility of two network behaviors, bistability and irreversibility. We use the Xenopus laevis oocyte maturation network as our example, and we make special use of iterative stability analysis, a graphical tool for determining stability in two dimensions.