Modeling integrated cellular machinery using hybrid Petri-Boolean networks

Abstract

The behavior and phenotypic changes of cells are governed by a cellular circuitry that represents a set of biochemical reactions. Based on biological functions, this circuitry is divided into three types of networks, each encoding for a major biological process: signal transduction, transcription regulation, and metabolism. This division has generally enabled taming computational complexity dealing with the entire system, allowed for using modeling techniques that are specific to each of the components, and achieved separation of the different time scales at which reactions in each of the three networks occur. Nonetheless, with this division comes loss of information and power needed to elucidate certain cellular phenomena. Within the cell, these three types of networks work in tandem, and each produces signals and/or substances that are used by the others to process information and operate normally. Therefore, computational techniques for modeling integrated cellular machinery are needed. In this work, we propose an integrated hybrid model (IHM) that combines Petri nets and Boolean networks to model integrated cellular networks. Coupled with a stochastic simulation mechanism, the model simulates the dynamics of the integrated network, and can be perturbed to generate testable hypotheses. Our model is qualitative and is mostly built upon knowledge from the literature and requires fine-tuning of very few parameters. We validated our model on two systems: the transcriptional regulation of glucose metabolism in human cells, and cellular osmoregulation in S. cerevisiae. The model produced results that are in very good agreement with experimental data, and produces valid hypotheses. The abstract nature of our model and the ease of its construction makes it a very good candidate for modeling integrated networks from qualitative data. The results it produces can guide the practitioner to zoom into components and interconnections and investigate them using such more detailed mathematical models.

Figure 1. Illustration of Petri nets and Boolean networks.

Consider a cellular network that involves three molecular species , and , where is self-regulatory (activating), inhibits , and both and activate in a cooperative manner. (Left) A Petri net representation, with three places corresponding to the molecular species, and two transitions corresponding to the reactions. A read arc (line with arrows on both ends) connecting place to transition means that when transition fires, the number of tokens in place does not change. Notice that the inhibition of is represented by transition which consumes tokens from . (Right) A Boolean network representation, with three Boolean variables corresponding to the molecular species. The primed version of a variable indicated the next-state of that variable. In other words, these Boolean formulas can be interpreted a , , and .

Figure 2. Graphical representation of glucose system.

Red shapes are Petri net places (signaling and metabolism), and small black squares on the arrows represent Petri net transitions (dashed lines correspond to enzymatic interactions). Green squares are Boolean network elements for regulatory components. Blue ovals are also Petri net places and correspond to interconnection elements. The Petri-to-Boolean arithmetic conditions are noted on/through red arrows (specific values are defined in the section of parametrizing the model). The Boolean-to-Petri connections are indicated with green arrows. The initial condition defined by vector , is set as follows: all Petri net places have 0 tokens except ADP (10 tokens) and Glucose (20 tokens); all Boolean network elements are set to 0, except HNF3beta and HNF1beta, which are set to 1. The ‘’ connections into Boolean variables correspond to the negation functions. For the Petri net component, the ‘’ connection from transition to place is a schematic representation of inhibition, which is implemented using the standard Petri net definition as being an input place to transition . Transitions without inputs or outputs represent sources and sinks, respectively.

Figure 3. Comparison of blood glucose (left) and insulin (right) dynamics in a single fast/feed cycle as simulated by our model with the experiential data by Korach-André et al (top) and ODE-based model by Liu et al (bottom).

Our IHM is shown in solid black line. Experimental data and ODE model results are reconstructed from Figure 7 in . The results of IHM dynamics qualitatively match to both experimental data and ODE-generated data.

Figure 4. The dynamics of all components in the IHM for feed/fast cycle in for the transcriptional regulation of glucose metabolism.

The plots show selected species from different components — component interconnections (top left), selected species from liver metabolism (top right), selected species from pancreatic beta-cell and liver signaling (bottom left), and selected species pancreatic beta-cell regulation (bottom right). X-axis for Petri net components are expressed in tokens, and for Boolean component is an average of Boolean values, 0 and 1.

Figure 5. The dynamics of IHM under normal Akt and reduced Akt (kdAkt) as compared to the experimental data in ( Figures 2B and 2D in [62]).

The kdAkt experiment was modeled by IHM by reducing the rate at which Akt suppresses FOXO and increasing the rate of the source transition into FOXO. In all images, yellow background indicates feeding stage, and red corresponds to fasting. The experimental data measures the glucose levels at the feeding stage and insulin secretory response during fasting. IHM shows the entire cycle. We observe the glucose of kdAkt model being higher than normal condition, as well as lower insulin secretion in reduced Akt scenario. These results correspond to the observations in the experimental data.

Figure 6. Glucose response from PI3K inhibition.

The comparison between IHM model (left) and experiment (right). Inhibiting PI3K was modeled by setting rate between Secreted insulin and PI3K to 0. In IHM model, glucose is higher with PI3K inhibition which is consistent with experimental data. The experimental data is reconstructed from Figure 3C in .

Figure 7. Diagram of the S. cerevisiae HOG pathway.

Graphical representation of glucose system. Red shapes are Petri net places (signaling and metabolism), and small black squares on the arrows represent Petri net transitions (dashed lines correspond to enzymatic interactions). Green squares are Boolean network elements for regulatory components. Blue ovals are also Petri net places and correspond to interconnection elements. The Petri-to-Boolean arithmetic conditions are noted on/through red arrows (specific values are defined in the section of parametrizing the model). The Boolean-to-Petri connections are indicated with green arrows. The initial condition defined by vector , is set as follows: all Petri net places have 0 tokens except ADP, which has 10 tokens; all Boolean network elements are set to 0. See caption of Figure 2 for more details about the representation.

Figure 8. Validation of our model against idFBA and ODE-based model as generated by Lee et al (contrast to Figure 9 in [10]).

The plots show the dynamics under osmotic stress (solid lines), and under no osmotic stress (dashed lines). The colors on all plots are indicated in the top left panel. The correspondence in qualitative behavior for all solid lines indicate similar results for all models under osmotic stress; for all dashed lines indicate similar results for all models under no osmotic stress.

Figure 9. Dynamics of IHM for all components of HOG pathway under osmotic stress.

The plots show selected species from different components — component interconnections (top left), species from metabolism (top right), selected species from regulation (bottom left), and selected species signaling (bottom right). X-axis for Petri net components are expressed in tokens, and for Boolean component is an average of Boolean values, 0 and 1.