Figure 1: On the left, a diagram of the macromolecules involved in gene expression is illustrated. Solid lines represent the transcription of mRNA, while dashed lines depict the translation of the corresponding protein. On the right is a picture of an interaction graph of a three-gene regulatory network that describes how genes influence one another. A wiring diagram, where nodes are genes and edges define the interactions among the genes, characterizes a gene regulatory network. In the Boolean model, genes are recognized as either ON (1) or OFF (0). This model reduces the quantitative complexity of the gene interactions to the qualitative, logical structure of the network. Figure 2: Example interaction graph for a three-gene regulatory network. Figure 3: Time series for the continuous model of a three-gene regulatory network with a positive feedback circuit, where r = 1 and, from the left to right, n = 2,5, 10. Figure 4: Time series for the continuous model of a three-gene regulatory network with a positive feedback circuit, where r = 50 and, from the left to right, n = 2,5, 10. Figure 5: Time series for the continuous model of a three-gene regulatory network with ¢ negative feedback circuit, where r = 1 and, from the left to right, n = 2,5, 10. The Jacobian of the system at the fixed point can be written as To begin the process of using generalized models, we assume that there is at least one positive equilibrium We now define the normalized variables We obtain the equations The Jacobian of the normalized model becomes By substituting r for the timescale ratio and noting that the A, B, C, and D represent the only possible nonzero partial derivatives, the Jacobian becomes In this generalized model, a;, 6;, and 7; are measures of the (inverse) time scales of the mRNA and protein concentrations. We assume that the time scale ratio is Figure 7: The Hopf bifurcation surface
