Combining boolean method with delay times for determining behaviors of biological networks

Boolean functions have been used to analyze the molecular networks of cells. For example, ArarrB represents if A becomes active B will be activated. This method is effective for qualitatively analyzing networks but is not suitable for studies of kinetic behaviors of networks. In the present paper, a dynamic Boolean method was developed by combining Boolean operations with molecular interaction parameters (delay or response times). The Boolean operations characterize the discrete interactions among biological components. The delay times describe the quantitative kinetics. The combination of the two characterizes the discrete biological interactions of networks. For example, A_{t ArarrB}B represents that if A becomes active B will be activated after an activation time t_ArarrB. By using this dynamic logic method, we achieved the following results: we proved the general theorems to determine bistable states and oscillation behaviors of networks, we showed that time delays are essential for oscillation behaviors, we proved that single variable networks are either bistable or oscillatory, and we explained why a signal can have multiply responses from different networks. In addition, we analyzed the mitosis cycle of budding yeast cells. We showed that the mitosis cycle is not only robust against structural changes but also robust against fluctuation in kinetic parameters (e.g. delay times).