The Mitogen Activated Protein Kinase (MAPK) cascade, found in all eucaryotic cells, is one of the most studied pathways in systems biology. Consisting of three sequential levels of proteins, it is activated by growth factors in cell surface receptors (input to the system), leading to the phosphorylation of the last-level kinase (output). This kinase is then involved in further phosphorylation of transcription factors, in processes like cell proliferation and differentiation [1,2]. Huang and Ferrell developed a model of 22 equations to describe the cascade [3].

It has been found that adding to the model an explicit negative feedback loop, from the output to the input, was necessary to match oscillations found in experimental results [4]. On the other hand, computational work has shown the possibility of reverberations, seen as a transition between steady states involving intermediate oscillations, upon a switch-on or switch-off in the input [5]. When combined with dynamical systems analysis, we show that it is possible to take advantage of the underlying behavior and lead the input back and forth across a bifurcation to obtain bursting, a complex phenomenon well studied in neuroscience where variables exhibit rapid oscillations followed by periods of quiescence. This pattern of high-frequency activity and resting states can be sustained in time.

In this work, we simulate the Huang-Ferrell MAPK cascade with the addition of one differential equation for the usually fixed input, born from a feedback loop regulation of the input by the output of the cascade. With a constant positive rate and a negative rate determined by the output (a negative feedback loop) and the same input (regulating its own production), we present how the output can develop bursting by properly setting the input parameters.

Depending on the bifurcations, further additions to the system of equations can help to develop bursting. We study this in both the MAPK cascade and a version of the Goodwin model, which serves as a relatively simple cascade motif. We show that in these systems the delay from input to output, characteristic of these motifs, is an essential ingredient not only for oscillations, but also for the onset of bursting. A further delay when going from output to input will prove beneficial for achieving trains of oscillations. At the same time, a transition between sustained bursting and damped bursting can be achieved, in an analogue to sustained and damped oscillations. This phenomenon of damped bursting is, to our knowledge, reported here for the first time. Moreover, we discover the existence of a nonlinear resonance when the cascade is periodically stimulated at some preferred frequency.

Our goal is to uncover new dynamics in well-known pathways which could be then linked to different biological functions, as well as explore the mechanisms behind bursting, involving ingredients such as delay, positive feedback, multiple negative feedbacks, and coexistence of stable steady states and limit cycles.

1. Lewis, T. S., Shapiro, P. S., and Ahn., N. G. (1998). Signal transduction through MAP kinase cascades. Adv. Cancer Res. 74, 49–139. doi:10.1016/s0065-230x(08)60765-4
2. Kochańczyk, M., Kocieniewski, P., Kozłowska, E., Jaruszewicz-Błońska, J., Sparta, B., Pargett, M., et al. (2017). Relaxation oscillations and hierarchy of feedbacks in MAPK signaling. Sci. Rep. 7, 38244. doi:10.1038/srep38244
3. Huang, C. Y. F., and Ferrell, J. E. (1996). Ultrasensitivity in the mitogen-activated protein kinase cascade. Proc. Natl. Acad. Sci. U. S. A. 93 (19), 10078–10083. doi:10.1073/pnas.93.19.10078
4. Shankaran, H. et al. Rapid and sustained nuclear-cytoplasmic ERK oscillations induced by epidermal growth factor. Mol. Sys. Biol. 5, 332 (2009). doi:10.1038/msb.2009.90
5. Mitra, T., Menon, S.N. & Sinha, S. Emergent memory in cell signaling: Persistent adaptive dynamics in cascades can arise from the diversity of relaxation time-scales. Sci Rep 8, 13230 (2018). doi.org/10.1038/s41598-018-31626-9