A scalable method for parameter-free simulation and validation of mechanistic cellular signal transduction network models

The metabolic modelling community has established the gold standard for bottom-up systems biology with reconstruction, validation and simulation of mechanistic genome-scale models. Similar methods have not been established for signal transduction networks, where the representation of complexes and internal states leads to scalability issues in both model formulation and execution. While rule- and agent-based methods allow efficient model definition and execution, respectively, model parametrisation introduces an additional layer of uncertainty due to the sparsity of reliably measured parameters. Here, we present a scalable method for parameter-free simulation of mechanistic signal transduction networks. It is based on rxncon and uses a bipartite Boolean logic with separate update rules for reactions and states. Using two generic update rules, we enable translation of any rxncon model into a unique Boolean model, which can be used for network validation and simulation—allowing the prediction of system-level function directly from molecular mechanistic data. Through scalable model definition and simulation, and the independence of quantitative parameters, it opens up for simulation and validation of mechanistic genome-scale models of signal transduction networks.


Supplementary Discussion
This section contains an annotated version of the state target updates for the minimal modification motif (Figure 2), and a detailed illustration of the equations from the results section using the HOG pathway model as an example.

States
The remaining four states (S2-S5) correspond to the catalysts of the four reactions and are constantly true.
The reaction updates are straightforward as there are no contingencies: They only require the reacting components to be true: I.e., the catalysts and, except in the case of synthesis, component A -which is present if either the phosphorylated or unphosphorylated state is true.
However, we do not use these reaction update rules in the test: We replace the reaction updates with a constant true/false value in the model to generate 16 (2 4 ) permutations, which, in combination with the four permutations of initial values for S A-{0} and S A-{P} , generates 64 model variants.
The initial values are changed in the _initial_vals.csv file, which by default reads: The interaction motif analysis (Figure 3) is performed the same way, except that we get 128 permutations as there are four reactions and three elemental states involved. The minimal motif models are attached as Supplementary Models 3 (covalent modification motif) and 4 (interaction motif).

Smoothed expressions
With smoothing, the model generation creates the following .boolnet file: The only difference is in the S0/S1 lines, where the smoothing terms appear in the source state expression. Compare the smoothed (lower) with the non-smoothed (upper) expression. For S0, dephosphorylation (R1) now triggers under the following conditions: The reaction (R1) is true and the source state (S1) is (1) indirectly synthesised, (2) produced by phosphorylation in the absence of degradation, or (3) present and not consumed or degraded. 3b is the original term from the unsmoothed export and is redundant with the smoothed expression.
In the following, we exemplify the application of the equations on the Hog pathway model. For improved readability, components, reactions, states and inputs are prefixed with C, R, S and I, with their actual names written in subscript.

Eq.1: Components = ⋂ ⋃
Most components (in this particular model) have a single site (residue or domain) that can be in the neutral or modified/bound state. Their component expressions consist of a simple OR statement between the neutral and modified/bound states: This means that at a protein is considered degraded as soon as all states for a single site have disappeared.
Finally, there is one component (PPT) in the system that does not carry any states. This is included in the model with a component state: Eq. 5. Synthesis:

⋂ ⋃ ′ for non-neutral states
The Hog pathway model does not include synthesis or degradation of components.
Eq. 6. Contingencies: There are five regulated reactions in the Hog pathway model. The first depend on the model input Eq. 8. State updates: As no synthesis or degradation reactions are considered in the Hog model, the components are constitutive and equation 8 simplifies to: I.e., all components in the state must be true and the state must either be produced (a producing reaction must be true AND the source state for that reaction must be true) OR the state must be present and not consumed (the state is true AND either the consuming reaction OR one of its source states must be false).