Phase Responses of Oscillating Components in a Signaling Pathway

Signal transduction pathways control various events in mammalian cells such as growth, proliferation, differentiation, apoptosis, or migration in response to environmental stimuli. Because of their importance, the activity of signaling pathways is controlled by multiple modes of positive and negative feedback regulation. Although negative feedback regulation primarily functions to stabilize a system, it also becomes a source of emerging oscillations. For example, the oscillatory behavior of mitogen-activated protein kinase (MAPK) activity has been theoretically proposed earlier and experimentally verified recently. However, the physiological function of such oscillatory behavior in biological systems remains unclear. To understand the functional aspects of this behavior, one should analyze the oscillation dynamics from a mathematical point of view. In this study, we applied the phase reduction method to two simple, structurally similar phosphorylation-dephosphorylation cycle models with negative feedback loops (Models A and B) and a MAPK cascade model, whose dynamics all show oscillation. We found that all three models we tested have a Type II phase response. In addition, we found that when a pair of each models A and B coupled through a weak diffusion interaction, they could synchronize with a zero phase difference. A pair of MAPK cascade models also showed synchronous oscillation, however, when a time delay was introduced into the coupling, it showed an asynchronous response. These results imply that structurally similar or even identical biological oscillators can produce differentiated dynamics in response to external perturbations when the cellular environment is altered. Synchronous or asynchronous oscillation may add strength to or dampen the efficiency of signal propagation, depending on subcellular distances and cell density. Phase response analysis allows prediction of dynamics changes in oscillations in multiple cellular environments.