As an example of metabolic pathway, by using GenomicObjectNet, we model the glycolytic pathway with HFPN that was first modeled by Reddy [1] with the discrete Petri net model. Figure 1 and Table 1 show the pathway and the enzymes related to the reactions to model. Together with modeling of the glycolytic pathway, we will model the lac operon gene regulatory mechanism of E. coli with HFPN (Figure 2 (a)), which includes the dual control of the lac operon participating lac repressor, allolactose, catabolite gene activator protein (CAP), and cyclic AMP (cAMP). The modeling is simply done by mapping the information represented in Figure 1 and 2 (a) to places and transitions in the following way (see Figure 3):

• Metabolic pathway: For each object in Figure 1 (ATP, ADP, enzyme for each reaction, Gluc, G6P,Ž¼, Lac), a continuous place is created. The content of the place represents the concentration of the object. For each reaction in Figure 1, we define a continuous transition whose firing speed is given by the Michaelis-Menten equation, where the parameters are roughly provided by setting Vmax=1 and Km=1. The natural degradation of compound is realized by a continuous transition with no outgoing arc which only comsumes but does not produce any. See the details for the left part of Figure 3.
• Gene regulation of the lac operon: Figure 2 (b) shows how places and transitions are created for representing the gene regulation mechanism. The switch mechanism is realized as shown in Figure 2 (c) by using discrete places, discrete transitions and test arcs. The places represent objets such as CAP, cAMP, cAMP/CAP-complex, lac repressor, lactose, galactose, allolactose, b-galactosidase, glucose, mRNAs, etc. whose contents represent the concentrations of the objects. The functions defined for the transitions are again simply given by manual tuning according to logical structure and biochemical knowledge. Some details are given in Figure 2 (b).

Although the necessary parameters were hand-tuned, there was practically no difficulty in obtaining the representation shown in Figure 3. Figure 4 shows the simulation result of concentration behaviors of glucose, lactose, lacZ protein, ATP, and pyruvic acid, which would prove that this model works reasonably at least with respect to the observation.

Figure 1: A part of the glycolytic pathway.

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Table 1: Enzyme reactions: Index numbers correspond to the numbers described in Figure 1.

Figure 2(a): LacZ, the first gene of the lac operon, encodes the enzyme b-galactosidase which breaks down lactose to galactose and glucose. The initiation of transcription of lac operon is controlled by the following mechanisms participating lac repressor protein and CAP: (1) lactose addition increases the concentration of allolactose which binds to the repressor protein and removes it from the DNA, and (2) glucose addition decreases the concentration of cAMP; because cAMP no longer binds to CAP, this gene activator dissociates from the DNA, turning of the operon.

Figure 2(b): The regulatory mechanism is realized as a HFPN model.

Figure 2(c): If the concentrations of CAP and cAMP exceed the levels w_c and w_A, respectively, the complex of CAP and cAMP is produced and binds to the CAP-binding site. In HFPN modeling, we set the test arc weights by w_c=1 and w_A=1 and discrete transition CAP/cAMP produces one token at each time while the contents of continuous places CAP and cAMP are not consumed because the arcs are test arcs except for the transitions representing degradation. Discrete place P_comp represents the CAP-binding site and its capacity (the number of maximum tokens held by the place) is is set to be two. Discrete transition D_comp removes CAP/cAMP-complex from P_comp by one token at each time. In the same way, if the concentration of lac repressor exceeds the level w_l while the concentration of allolactose does not exceed the level w_a, lac repressor binds to the operator site (represented by discrete place P_rep) which is removed from the site at the rate assigned to transition D_rep. Under the condition that the complex of CAP and cAMP binds at promoter site, i.e. P_comp holds at least one token, and lac repressor does not bind to the operator site, i.e. P_rep does not hold any token, the switch of lac operon transcription (represented by discrete place P_Cc) will be turned on.

Figure 3: A HFPN modeling of the glycolytic pathway and lac operon gene regulatory mechanism of E. coli.

Figure 4: This figure shows the behaviors of concentrations of glucose, lactose, lacZ, ATP, and pyruvic acid. As soon as the glucose is almost consumed (around time 20), the production of lacZ protein is begun. When the concentration of lacZ protein reaches at some level (around time 35), the concentration of glucose begins to increase again resulting from breaking down the lactose to the glucose and galactose. At the time when the lactose almost disappears (around time 55), the transcription of lacZ gene is stopped. Growths of the concentrations of ATP and pyruvic acid are also observed in the figure.

Reference

[1] Reddy, V.N. (1994) Modeling biological pathways: A discrete event systems approach. Master's Thesis, The University of Maryland, M.S.94-4. http://www.isr.umd.edu/TechReports/ISR/1994/MS_94-4/MS_94-4.phtml