Upon the recognition of extracellular growth factors by suitable receptors, plasma membrane-tethered Ras activates the cytoplasmic Raf–MEK–ERK kinase cascade. Activity of Ras is regulated directly by other membrane-tethered proteins: GEFs (activators) and GAPs (inhibitors). The system contains three essential feedback loops: 1) a GEF-mediated fast positive feedback which allows for switch-like activation of Ras; 2) negative feedback from activated ERK to MEK which acts upstream the cascade; and 3) slow negative feedback from activated ERK to Sos1 (a GEF) which allows the system to globally switch off its activity. In our study, we focus on the importance of feedback loops and on the separation of timescales in conjunction with spatial extension of the system. The mathematical model is expressed by reaction–diffusion PDEs with Robin boundary conditions which account for reactions on the membrane.

The model system can be turned on by a locally increased level of extracellular growth factors which activate Ras locally. The patch of active Ras can spread on the membrane through heteroclinic travelling wave propagation, which eventually may lead to the activation of ERK. Cytoplasmic active ERK, which diffuses faster than membrane proteins, leads to global inhibition of membrane-localized Ras. We showed that the system which contains first two feedback loops exhibits bistability, which, in the presence of sufficiently strong ERK–Sos1 feedback, turns into spatially-extended relaxation oscillations. We were able, therefore, to reproduce recurrent activations of ERK observed recently in the experiment by Albeck et al. (Mol Cell 49(2):249–261, 2013). The frequency of the oscillatory response of active ERK depends on the level of extracellular stimulation with growth factors, which is in qualitative agreement with experimentally obtained time courses.

cell signaling; feedback loop; bistability; heteroclinic travelling wave; relaxation oscillations; kinase cascade; cell polarity