Structural analysis to determine the core of hypoxia response network

Abstract

The advent of sophisticated molecular biology techniques allows to deduce the structure of complex biological networks. However, networks tend to be huge and impose computational challenges on traditional mathematical analysis due to their high dimension and lack of reliable kinetic data. To overcome this problem, complex biological networks are decomposed into modules that are assumed to capture essential aspects of the full network's dynamics. The question that begs for an answer is how to identify the core that is representative of a network's dynamics, its function and robustness. One of the powerful methods to probe into the structure of a network is Petri net analysis. Petri nets support network visualization and execution. They are also equipped with sound mathematical and formal reasoning based on which a network can be decomposed into modules. The structural analysis provides insight into the robustness and facilitates the identification of fragile nodes. The application of these techniques to a previously proposed hypoxia control network reveals three functional modules responsible for degrading the hypoxia-inducible factor (HIF). Interestingly, the structural analysis identifies superfluous network parts and suggests that the reversibility of the reactions are not important for the essential functionality. The core network is determined to be the union of the three reduced individual modules. The structural analysis results are confirmed by numerical integration of the differential equations induced by the individual modules as well as their composition. The structural analysis leads also to a coarse network structure highlighting the structural principles inherent in the three functional modules. Importantly, our analysis identifies the fragile node in this robust network without which the switch-like behavior is shown to be completely absent.

Figure 1. Schematic diagram of hypoxia response network.

This scheme is based on references and . Three pathways, given in green (oxygen-independent pathway), blue and red (oxygen-dependent pathways) can degrade HIF transcription factor. The HRE activation pathway, shown in brown dotted lines, is – following the discussion in – not considered in our analysis.

Figure 2. Petri net representation of the hypoxia response network.

It defines equations (1–13), when read as continuous Petri net with mass action kinetics. The labels are taken as transition identifiers in the structural analysis of the qualitative Petri net, and as kinetic parameters (see Table 2) in the simulative analysis of the continuous Petri net. There are two logical places for , connected to the remaining net by read arcs. See Table 1 for the biological meaning of the other place identifiers (dynamic variables). Each color characterizes an ADT set, compare Table 5. Reduction candidates, as revealed by invariant analysis, are uncolored.

Figure 3. Hypoxia core network.

All reduction candidates were automatically identified by structural analysis and got approved by the numerical simulation experiments of the corresponding ODEs. See Figure 4 for an hierarchical version of this Petri net.

Figure 4. Coarse Petri net structure of the hypoxia core network.

This is the top level of an hierarchical representation of the Petri net given in Figure 3. It reveals the structuring principle inherent in the minimal T-invariants. Each macro transition (drawn as two centric squares) stands for a connected subnet defined by a set of dependent transitions, i.e., transitions occurring together in all non-trivial T-invariants. A, B stand for 1-elementary sets, compare Table 4. The places shown in the coarse net structure are the interface places between the subnets. The coarse net structure clearly identifies the central role of PHD (). Its knock-down would switch off and . See also Figure 5.

Figure 5. Pathways of the hypoxia core network.

In this example, each macro transition path in the coarse net structure corresponds to a non-trivial T-invariant (pathway) of the flat network. There are three such sequences: – direct degradation: (, ), – degradation not requiring ARNT (S4): (, , ), and – degradation requiring ARNT (S4): (, , ).

Figure 6. Experiment 1.

Comparison of the dependency of the steady state value (SSV) of HIF () on the oxygen concentration by systematic silencing network parts as determined by T-invariant analysis by setting sequentially kinetic parameters to zero. (a) full model according to ; (b) kinetic parameters set to zero: ; (c) +(b); (d) +(c); (e) +(d); (f) +(e). The experiment has been done for all three parameter sets as given in . The dark vertical black line indicates the critical oxygen concentration (0.65) for which HIF is completely degraded. It separates hypoxia (left) from normoxia (right). The SSV and the critical oxygen concentration are the same in (a)–(f). Therefore, (f) is considered as the core module for further analysis.

Figure 7. Experiment 2.

Contribution of the individual pathways as induced by the T-invariants , and , see Table 4, determined by numerical integration of the ODEs defined by each pathway. The efficiency of degradation of HIF by oxygen can be ordered as . is the oxygen-independent pathway. degrades more efficiently due to stronger binding of PHD to HIF than ARNT.

Figure 8. Experiment 3.

Pathway is considered separately while varying ARNT initial concentration (). Degradation of HIF through pathway P3 is feasible for low (5) and medium (10) initial concentration of ARNT. For extremely high concentration (100), saturation of HIF by ARNT takes place and thereby the critical oxygen concentration (complete degradation of HIF) does not change.

Figure 9. Experiment 4.

Varying the initial concentration ARNT () in the reduced model to a wide range (0, 5, 1000 … 4500). SSV and critical oxygen concentration are not affected for a very low value of ARNT initial concentration. Extreme concentration of ARNT (1000, 2000 …) which is 100 or more times than that of PHD initial concentration (10) changes both the SSV and the critical oxygen concentration to a lower value and as a result the curves are shifted to the left. We also give the simulation of the full model for varying concentration of ARNT. The concentration change is indicated by the direction of the arrow. The qualitative behavior is the same as for the reduced core model. Even for very high ARNT concentration the critical oxygen concentration is retained, but the HIF steady state values are lowered.

Figure 10. Experiment 5.

Demonstration of fragile node PHD () in the core module of the network. Increase in PHD concentration sHIFts the critical oxygen concentration to a lower value delineating the importance of binding affinity of PHD-HIF complex for degradation. The loss of PHD knocks off the oxygen-dependent degradation pathways , and results in inefficient degradation by the oxygen-independent pathway only, suggesting that PHD is the fragile node in the core module.

Figure 11. Petri net example.

The Petri net for the well known chemical reaction r: and three of its markings (states), connected each by a firing of the transition r. The transition is not enabled anymore in the marking reached after these two single firing steps.

Figure 12. Hierarchical structuring by use of macro transitions.

The three nets are identical and model a single enzymatic reaction following the mass-action kinetics . The transitions and model a reversible reaction. Macro transitions (drawn as two centric squares) hide net details (in this case transition-bordered subnetworks) on the next lower hierarchy level. Macro transitions can be arbitrarily nested resulting into hierarchically structured Petri nets.