Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems

Abstract

It is becoming increasingly clear that bistability (or, more generally, multistability) is an important recurring theme in cell signaling. Bistability may be of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or "remember" transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex. Here, we show that for a class of feedback systems of arbitrary order the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this open-loop, feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2/Wee1 system and a more complicated five-variable mitogen-activated protein kinase cascade.

Fig. 1.

Feedback systems that may exhibit bistability. (a) A two-component positive-feedback loop, which can be analyzed by classical phase plane techniques. (b) A two-component, mutually inhibitory feedback loop, which can also be analyzed by classical phase plane techniques. (c) A longer mutually inhibitory feedback loop, which cannot be analyzed by classical phase plane techniques.

Fig. 2.

Analysis of the Cdc2-cyclin B/Wee1 system by numerical simulation. (a) Schematic depiction of the system. (b–d) Phase plane plots of the Cdc2-cyclin B system. Constants are: α₁ = α₂ = 1, β₁ = 200, β₂ = 10, γ₁ = γ₂ = 4, K₁ = 30, K₂ = 1. Three different feedback gains are shown: v = 1(b), v = 1.9 (c), and v = 0.75 (d).

Fig. 3.

Mathematical analysis of the Cdc2-cyclin B/Wee1 system, by breaking the feedback loop. (a) Schematic view of a feedback system before (Left) and after (Right) breaking the feedback loop. ω is the input of the open-loop system; η is the output. (b) Incidence graph for system 2. (c) Steady-state I/O static characteristic curve (k_η is a function of ω) for system 2 (red), with constants chosen as in Fig. 2 b–d. The solid blue line represents ω as a function of η for unitary feedback. There are three intersection points (I, II, and III), which represent two stable steady states (I and III) and one unstable steady state (II). The dashed blue lines represent ω as a function of η for the values of the feedback gain v above which (v ≳ 1.8) and below which (v ≲ 0.83) the system becomes monostable. (d) Bifurcation diagram for the system, showing bistability when the feedback strength v is between ≈0.83 and ≈1.8. The bifurcation diagram is obtained from the characteristic as the plot of the curve [ω/k(ω),k(ω)], with ω ranging over the allowed range of inputs.

Fig. 4.

Analysis of a system with similar-looking characteristic curves (compare Fig. 3c) but qualitatively different stability behavior. (a) Steady-state I/O static characteristic curve for system 4. Constants are: c = 0.8, b = 500/140, K = 405/14. (b) Phase plane for system 4. (c) Incidence graph for system 4.

Fig. 5.

Bistability in a MAPK cascade. (a) Schematic depiction of the Mos-MEK-p42 MAPK cascade, a linear cascade of protein kinases embedded in a positive-feedback loop (Left), together with the corresponding open-loop system (Right). (b) Incidence graph for a 2D subsystem (a single level) of a kinase cascade. (c) Steady-state I/O characteristic (k_η as a function of ω) for the MAPK cascade (red curve), plotted together with the diagonal, representing ω as a function of η with unity feedback (blue line). (d) Experimental demonstration of a sigmoidal response of MAPK to Mos. Experimental data are taken from ref. and are means ± SD. (e) Bifurcation diagram for the MAPK cascade, showing the stable on-state (red curve), the stable offstate (green curve), and the unstable threshold (black curve) as a function of feedback strength (v). (f) Simulations show that trajectories funnel toward one of two stable steady states, denoted by red and green ticks, as required by our theorem. Shown are calculated concentrations of Mos (x), active MEK (y₃), and active MAPK (z₃) for 11 choices of initial conditions, as a function of time.