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# Mathematical modeling of intracellular signaling pathways

The intercellular communication occurs between biomolecules (mainly proteins), through the fixed set of reaction channels which create highly complex network. Mathematical modeling of such signal processing pathways enables system level understanding of molecular, dynamical mechanisms of a cell.

The biochemical reaction network is the most general model of a signaling pathway. As an underlying mathematical model one can choose the deterministic framework in form of differential equations (ODE, DDE, PDE) or stochastic framework in form of a Markov process or the Chemical Master Equation. The model parameters include the amounts of reacting species and the kinetic rates constants of the reactions.

Obtained mathematical model is in vast majority of cases analytically intractable, thus, the basis for most analyses are the numerical simulations. The common tasks are selection of the correct model (e.g. Bayesian Model Selection) or analysis of the sensitivity of the model with respect to its parameters (e.g. Multi-Parameter Sensitivity Analysis). Moreover, the partially validated models can be useful in designing the most informative experiments (Optimal Experiment Design).

Among the throughly investigated models of the signaling pathways are models of the MAPK cascade, the JAK-STAT or the NF-kappaB pathways, the T Cell receptor or the EGF receptor signaling, or the heat shock response mechanism.

Alongside the development of analysis methods and models of signaling pathways there is an significant progress in development of accompanying technologies. Standards and repositories not only include the model and analysis services (e.g. SBML, BioModels, BioCatalogue), but also the in silico experiments expressed as workflows (e.g. myExperiment). Moreover, the computational cost of the analysis methods, in particular stochastic methods like the Probabilistic Model Checking, stimulates the expansion of grid technologies and the publicly available hardware base.

**Fig. 1. **Thermotolerance in the heat shock response model: the substrate activity (solid) during the two consecutive heat shocks (dotted) of 5 C deg. over the homeostatis level of 37 C deg. The strength of the intoxication by the substrate (colored area) depends on the time gap between heat shocks. Interestingly, activity of the substrate in the second shock can be even higher than activity in the first shock, as shown for the time gap of 800.

**Fig. 2. **The multi-parameter sensitivity analysis (MPSA) workflow with an ODE-based error function (pink and brown boxes represent Essentials steps of the procedure), and the MPSA error surfaces for the enzymatic reaction model. Error surfaces were calculated using the deterministic model with the sum squared error (SSE) of product trajectories (left column) and using the stochastic model with the absolute difference of a value of the reward-based Continuous Stochastic Logic (CSL) formula.