Bistability in one equation or fewer

Abstract

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.

Fig. 1.

A schematic of the Xenopus laevis MAP kinase pathway (a) as connected in vivo and (b) separated into a cascade and a positive feedback, for purposes of theoretical analysis.

Fig. 2.

Cross section of a landscape containing a perfectly spherical rock. The rock can come to rest at three positions: the two valley floors, and at the very top of the small hill. If the rock is perched at the hilltop, and it is pushed, it will clearly accelerate in the direction it was pushed and roll down the hillside. Thus, the hilltop position is unstable. A rock perched at a valley bottom, however, if pushed, will stop rolling, then reverse direction and eventually settle again at the valley bottom. The two valley bottom positions are stable. You can map the mathematical concept of stability onto an intuitive understanding of stability you already possess.

Fig. 3.

Open-loop dose–response curves for the separated legs of the MAPK system, assuming Michaelis–Menten-shaped curves. Note the two intersection points (a), indicating two steady states. Suppose the system comes to rest at the nonzero steady state (b). Following a small perturbation with iterative stability analysis, we end up back at the steady-state; the nonzero steady-state is stable. If the system is at rest at the graph’s origin (c), then a small perturbation will grow with each iteration until the system comes to rest at the stable steady state. Thus, the steady state at the origin is unstable.

Fig. 4.

The in vivo p42 MAPK → Mos curve displays a sigmoidal, not a Michaelis–Menten shape. A hill coefficient of 4.9 causes p42 MAPK activity to markedly accelerate with respect to Mos concentration before saturating. For reference, a Michaelis–Menten-shaped curve contains no such acceleration. The difference in shape can be seen on both semi-log (a) and linear (b) axes. Data reprinted from (1).

Fig. 5.

We induce some sigmoidal curvature in the Mos → p42 MAPK open-loop curve (a). Note there are now three intersections of the two open-loop curves and three steady-states. A perturbation to a system at rest at the origin now results in a return to the origin—the origin has become a stable steady state (b). Perturbations from the middle steady state will grow, coming to rest at either the origin or the upper steady state (c). In total, there are now two stable steady states and one unstable.

Fig. 6.

We incrementally increase the basal activity of p42 MAPK, which represents the effect of progesterone in vivo. This shifts the Mos → p42 MAPK open-loop curve in the vertical axis (a). Each shift has the effect of changing the positions of the three steady states in the Mos axis. Now, we can plot the location of these steady states while varying basal p42 MAPK activity over a wide range (b). As basal p42 MAPK activity increases, the two stable steady states move up the Mos axis (now displayed as the vertical axis), while the unstable steady state moves down the Mos axis. At a high-enough basal p42 MAPK activity, the lower stable steady state and the unstable steady state meet and annihilate, leaving only the upper stable steady state. A system at rest at the lower stable steady state, when experiencing this annihilation event, will explosively jump from a low Mos concentration to a high Mos concentration without coming to rest at any intermediate levels. Even if the basal p42 MAPK level is lowered after this explosive event, the system will forever remain at the higher of the two stable steady states. The transition from low to high Mos was irreversible.