We emphasized earlier that positive feedback typically generates a bistable system with two stable steady states. Why does the system modeled in Figure 8–81D have three? This conundrum can be explained by solving the reaction rate equations (Equation set 8–9, Figure 8–81B) for various different starting conditions of [X] and [Y], determining all values of [X] and [Y] as a function of time. Starting Figure 8–82 Analysis of the stability of a system’s steady states. (A) The dotted with each set of initial concentrations of [X] and [Y], these calculations produce lines are the nullclines for the system shown a so-called trajectory of points, each indicated by a curved green line on Figure in Figure 8–81. Also shown are dynamic 8–82A. A fascinating pattern emerges: each trajectory moves across the plot and trajectories (green) that show the changes settles in one of two steady states, but never in the third (middle steady state). over time in [X] and [Y], starting at a variety We conclude that the middle steady state is unstable because it cannot “attract” of different initial concentrations (determined by solution of Equation set 8–9; see Figure any trajectories. The system therefore has only two stable steady states. Thus, the 8–81B). By plotting [X] versus [Y] at each number of stable steady states in a system need not be equal to the total number time point, we find that, although there are of its theoretically possible steady states. In fact, stable steady states are usually three possible steady states in this system, separated by unstable ones, as in our example. the dynamic trajectories converge on only Once this system adopts a fate by settling in one of the two steady states, does two of them. The middle steady state is avoided: it is unstable, being unable to it have the ability to switch to the other state? The numerical solution of Equation attract any trajectories. (B) Imagine that the system is at the upper-left steady state and experiences a perturbation (black arrows), such as a random fluctuation in the production rates of X and/or Y. If the perturbation is small (arrow 1), the system will return to the same steady state. On the other hand, a perturbation that drives the system beyond the unstable (middle) steady state (arrow 2) causes it to switch to the lower-right steady state. The set of perturbations that a system can withstand without switching from one steady state to the other is known as the region of concentration of X concentration of X attraction of that steady state.
set 8–9 can again provide an answer. In Figure 8–82B, we show the solution of this equation set for two perturbations from the upper-left steady state. For a small perturbation, the system returns to its original steady state. But the larger perturbation causes the system to switch to the alternate steady state. Thus, this system can be switched from one stable steady state to the other by subjecting it to an input (or a perturbation) that is large enough to make the other steady state more attractive. More generally, every stable steady state has a corresponding region of attraction, which can be intuitively thought of as the range of perturbations (of [X] or [Y] in this example) for which the dynamic trajectories converge back to that particular steady state, rather than switch to the other one.
The concept of a region of attraction has interesting implications for the heritability of transcriptional states and the transition rates between them. If the region of attraction around one steady state is large, for example, then most cells in the population will assume this particular state. Furthermore, this state is likely to be inherited by daughter cells, since minor perturbations, like those ensuing from an asymmetric distribution of molecules during cell division, will rarely be sufficient to induce switching to the other steady state. We should expect that the use of positive feedback, coupled to cooperativity, will quite often be associated with systems requiring stable cell memory.
Robustness Is an Important Characteristic of Biological Networks