Stability and bifurcation analysis of a model for the signal transduction process with a signal amplification delay

Abstract

All living cells need to sense and respond to their environment. Cells communicate with each other through extracellular signaling molecules [1]. Signal transduction is the process by which information from an extracellular signal is transmitted from the plasma membrane into the cell and along an intracellular chain of signaling molecules to stimulate a cellular response. Many situations have been reported where altered signaling pathways produce dramatic changes in cell survival, cell proliferation, morphology, angiogenesis, longevity, or other properties that characterize cancer cells. Signal transduction abnormalities have been linked to the development of many serious disorders, such as chronic myelogenous leukemia and Alzheimer's disease [2, 3] . In this study, a model with delay of the signal transduction process is analyzed. After showing that the model admits positive solutions, we derive conditions on the system parameters which give rise to different dynamical behaviors which could be expected in the signaling pathway under the impact of delays. Numerical simulations are carried out and discussed in support of the theoretical analysis. We found that the system changes its dynamic behavior from stable to unstable around the system's steady state when the delay increases in value so that it crosses a critical value via a Hopf bifurcation and bifurcation of a family of periodic solutions can be expected if the delay is in the vicinity of the critical value. Numerical simulations are carried out to support the theoretical predictions concerning various dynamical behaviours permitted by different values of the amplification effect delay.