Formal modeling and analysis of the MAL-associated biological regulatory network: insight into cerebral malaria

Abstract

The discrete modeling formalism of René Thomas is a well known approach for the modeling and analysis of Biological Regulatory Networks (BRNs). This formalism uses a set of parameters which reflect the dynamics of the BRN under study. These parameters are initially unknown but may be deduced from the appropriately chosen observed dynamics of a BRN. The discrete model can be further enriched by using the model checking tool HyTech along with delay parameters. This paves the way to accurately analyse a BRN and to make predictions about critical trajectories which lead to a normal or diseased response. In this paper, we apply the formal discrete and hybrid (discrete and continuous) modeling approaches to characterize behavior of the BRN associated with MyD88-adapter-like (MAL)--a key protein involved with innate immune response to infections. In order to demonstrate the practical effectiveness of our current work, different trajectories and corresponding conditions that may lead to the development of cerebral malaria (CM) are identified. Our results suggest that the system converges towards hyperinflammation if Bruton's tyrosine kinase (BTK) remains constitutively active along with pre-existing high cytokine levels which may play an important role in CM pathogenesis.

Figure 1. TLR2/4 signalling pathway.

The TLR2/4 signalling pathway starts with recognition (1) of PAMPs by TLRs. This activates (2) BTK which phosphorylates (3) MAL. MyD88 adapter protein and kinases are recruited (4) and activated around MAL. This eventually leads to the activation (5) of NF-B as IB is degraded. The proinflammatory cytokine genes are expressed (6) producing INCY that are secreted (7). INCY are responsible (8) for the production of inflammation and activation of their respective receptors. This again activates NF- B (9a) and through an alternate pathway induces the production of SOCS-1 (9b). SOCS-1 negatively regulates MAL by polyubiquitination (10a) and blocks NF- B mediated expression (10b). Abbreviations: TLR, toll like receptors; PAMPs, pathogen associated molecular patterns; BTK, bruton's tyrosine kinase; MAL, MyD88 adapter like; MyD88, myeloid differentiation response protein; NF-B, nuclear factor kappa-light-chain-enhancer of activated B cells; IB, inhibitor of B; INCY, proinflammatory cytokines; SOCS-1, suppressor of cytokine signaling-1 , , .

Figure 2. MAL associated BRN.

Numerals (1 and 2) represent the threshold levels of interactions; plus (+) signs indicate activation while inactivation is indicated by a minus (−) sign. Arrows indicate the direction of activation/inactivation. The thresholds values are set according to Definition 2.

Figure 3. State graph of the MAL associated BRN.

The complete state graph is obtained by using the GENOTECH tool. Definitions 2 and 4 assist in setting the values of the K-parameters. The K-parameters are set such that they result in a model coherent with the observed steady states behaviors. In the case of MAL-associated BRN these states are (0,0,0,0,0) and (0,0,1,2,1). In the state (0,0,0,0,0) the system does nothing when there is no signal of pathogen or returns to this state after proper response. In the state (0,0,1,2,1), the system produces inflammation continuously. The set of logical parameters is: , , , , , , , , , , , , , , , and .

Figure 4. Normal and Divergent paths.

Each circle represents a particular state (configuration) and inside the circle the values 0,1 and 2 represent qualitative levels of proteins according to the order (BTK, MAL, NF-kB, INCY, SOCS-1). The solid lines show the transition which leads to normal state (0,0,0,0,0) and the dotted lines show the transitions towards the state of hyperinflammation (0,0,1,2,1). The conditions for all the transitions in this figure are given in Table 1.

Figure 5. Activation and degradation of entities.

(A.) Sigmoidal (above) and Step function (below) representation of activation. (B.) Sigmoidal (above) and Step function (below) representation of degradation (inhibition).

Figure 6. A toy BRN.

A toy example of a BRN where , and represent biological entities. The labels ‘+’ and ‘1’ represent the activation and the threshold concentration respectively.

Figure 7. State graph.

The state graph of the BRN in Figure 3. Each node represents a qualitative (discrete) state of the BRN. The values inside a state represent the concentrations of the entities , and .

Figure 8. Snapshot of the MAL associated BRN construction in GENOTECH.

The BRN is constructed as a directed graph by using the Gene’ New and Interaction’New menu options. An edge between two entities shows an interaction which is labeled with threshold and the sign of interaction (+for activation and −for degradation).

Figure 9. Snapshot of the logical parameters.

Each entity of the BRN is assigned a set of logical parameters by using the properties option accessible by right clicking that entity.

Figure 10. Snapshot of the state graph in GENOTECH.

The stable states are highlighted in red. The right panel shows the analysis command which include: show path which highlights paths between two selected states; show cycle which highlights existing cycles (closed path), if any, in the state graph; show neighboring states highlights all the neighboring states of the selected state. The conversion menu contains the commands to export the graph to DOT and HyTech formats.

Figure 11. Timing diagram.

Timing diagram showing the evolution of proteins involved in BRN. Here the concentrations are represented by two levels x and x+1 on the vertical axis. The horizontal axis represent the time of evolution. The dotted lines are the boundaries of the different configurations of the discrete concentrations. Each configuration is according to the order (BTK , MAL, NF-kB, INCY, SOCS-1).

Figure 12. Different representations of the evolution of an entity.

Evolutions of an entity is shown as: sigmoidal representation of the activation (A.), discrete approximation of the activation (B.), piece-wise linear approximation of the activation (C.), sigmoidal representation of the degradation (D.), discrete approximation of the degradation (E.) and piece-wise linear approximation of the degradation (F.).

Figure 13. Bio-LHA.

Partial view of Bio-LHA of the MAL associated BRN. The labels 11000, 01000 and 11100 shows the state (configuration) of BRN consisting of five entities (BTK, MAL, NF-kB, INCY, SOCS-1).