Landscape and flux govern cellular mode-hopping between oscillations

Recently, a "mode-hopping" phenomenon has been observed in a NF-κB gene regulatory network with oscillatory tumor necrosis factor (TNF) inputs. It was suggested that noise facilitates the switch between different oscillation modes. However, the underlying mechanism of this noise-induced "cellular mode-hopping" behavior remains elusive. We employed a landscape and flux approach to study the stochastic dynamics and global stability of the NF-κB regulatory system. We used a truncated moment equation approach to calculate the probability distribution and potential landscape for gene regulatory systems. The potential landscape of the NF-κB system exhibits a "double ring valley" shape. Barrier heights from landscape topography provide quantitative measures of the global stability and transition feasibility of the double oscillation system. We found that the landscape and flux jointly govern the dynamical "mode-hopping" behavior of the NF-κB regulatory system. The landscape attracts the system into a "double ring valley," and the flux drives the system to move cyclically. As the external noise increases, relevant barrier heights decrease, and the flux increases. As the amplitude of the TNF input increases, the flux contribution, from the total driving force, increases and the system behavior changes from one to two cycles and ultimately to chaotic dynamics. Therefore, the probabilistic flux may provide an origin of chaotic behavior. We found that the height of the peak of the power spectrum of autocorrelation functions and phase coherence is correlated with barrier heights of the landscape and provides quantitative measures of global stability of the system under intrinsic fluctuations.