Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems.
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.
Angeli
David
D
Dipartimento di Sistemi e Informatica, University of Florence, 50139 Florence, Italy.
Ferrell
James E
JE
Sontag
Eduardo D
ED
D016203:CDC2 Protein Kinase / Q000378:metabolism*; D018797:Cell Cycle Proteins; D019926:Cyclin B / Q000378:metabolism; D025461:Feedback, Physiological; D048369:MAP Kinase Kinase 1; D020935:MAP Kinase Signaling System; D019950:Mitogen-Activated Protein Kinase 1 / Q000378:metabolism; D020929:Mitogen-Activated Protein Kinase Kinases / Q000378:metabolism; D008954:Models, Biological; D009687:Nuclear Proteins; D011505:Protein-Tyrosine Kinases / Q000378:metabolism*; D017231:Proto-Oncogene Proteins c-mos / Q000378:metabolism; D013997:Time Factors
D018797:Cell Cycle Proteins; D019926:Cyclin B; D009687:Nuclear Proteins; D011505:Protein-Tyrosine Kinases; D017231:Proto-Oncogene Proteins c-mos; D016203:CDC2 Protein Kinase; D019950:Mitogen-Activated Protein Kinase 1; D048369:MAP Kinase Kinase 1; D020929:Mitogen-Activated Protein Kinase Kinases