Analysis of Enzymatic Regulatory Five-Element Model

Given the background of the enzymatic regulatory network, this paper explores important properties of the classical five-element model, such as essentiality, adaptation and oscillation, which play important roles in system biology. Firstly, we prove that the five-element model is an essential network, which indicates the network can not be reduced. Secondly, as negative feedback loops and incoherent feedforward loops show oscillation and adaptation found in three-node and four-node motifs, the five-element model, containing these loops, can oscillate within a certain range of given bifurcation parameters by the Hopf bifurcation. Thirdly, we propose an algebraic condition for perfect adaptation when the five-element model converges to a fixed point. We also consider adaptation in the oscillating system which has not been studied in the literature, and show that the oscillating five-element model can exhibit strong adaptation by simulation. Moreover, we find that the frequency of the oscillation is tunable while its amplitude almost remains constant within a range of inputs by a case study. Therefore, our approach shows that the five-element model exhibits significant advantages in important dynamic properties in the context of system biology.