Signal transduction is an important and basic mechanism to cell life activities. The stochastic state transition of receptor induces the release of singling molecular, which triggers the state transition of other receptors. It constructs a nonlinear singling network, and leads to robust switchlike properties which are critical to biological function. Network architectures and state transitions of receptor will affect the performance of this biological network. In this work, we perform a study of nonlinear signaling on biological network with multistate by analyzing network dynamics of the Ca$^{2+}$ induced Ca$^{2+}$ release mechanism, where fast and slow processes are involved and the receptor has four conformational states. Three types of networks, Erdos-Renyi network, Watts-Strogatz network and BaraBasi-Albert network, are considered with different parameters. The dynamics of the biological networks exhibit different patterns at different time scales. At short time scale, the second open state is essential to reproduce the quasi-bistable regime, which emerges at a critical strength of connection for all three states involved in the fast processes and disappears at another critical point. The pattern at short time scale is not sensitive to the network architecture. At long time scale, only monostable regime is observed, and difference of network architectures affects the results more seriously. Our finding identifies features of nonlinear signaling networks with multistate that may underlie their biological function.
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