# Modelling genetic regulatory networks from specified behaviours

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## Keywords

Cystic Fibrosis Linear Temporal Logic Path Condition Symbolic Execution Genetic Regulatory Network## Introduction

Modelling and simulation are needed to understand genetic regulatory networks. But parameters of the models are usually difficult to determine. To deal with this problem we propose a methodology in which the qualitative approach developed by R. Thomas [1] is used. The parameters of the model, which are related to the kinetic parameters of a differential description, may be unknown. We translate the set of possible models into one Symbolic Transition System, and the known behaviours into temporal logic formulas; then the constraints on the parameters corresponding to all models having the specified behaviours can be determined.

## Models

In the asynchronous discrete modelling of regulatory networks [1], each variable *x* represents the concentration of a constituent of the network. In each state, the value of *x* is an integer bounded by the number of variables that *x* can regulate. Each state and each variable is associated with a parameter that has an integer value. This parameter is the value toward which the variable tends in the associated state.

## Examples

*Pseudomonas aeruginosa*are bacteria that secrete mucus in lungs affected by cystic fibrosis, but not in common environment. As it increases respiratory deficiency, this is a major cause of mortality in this disease. The regulatory network proposed in [2], contains the protein AlgU, and an inhibitor complex anti-AlgU (Fig. 1A). Bacteriophage lambda is a virus whose DNA can integrate into bacterial chromosome. After infection, most of the bacteria display a lytic response, but some display a lysogenic response, i.e. survive and carry lambda genome, becoming immune to infection. The graph of interactions described in [3] involves four genes, cI, cro, cII and N (Fig. 1B). The lytic (resp. lysogenic) response leads to the states where cro (resp. cI) is fully expressed. In these two cases the parameters are unknown.

## Methods

The set of all discrete models associated with a graph of interactions are translated into a Symbolic Transition System. Then we apply symbolic execution techniques [4], to construct a tree of states sequences, such that each path is associated with the constraint on parameters necessary to its existence: this constraint is called *path condition*. To search a specific path in the symbolic execution tree, we have adapted model-checking techniques for Linear Temporal Logic (LTL): all paths verifying the LTL formula are selected, and the disjunction of the associated path conditions is synthesised. The resulting constraint represents all parameters compatible with the specified behaviour.

## Results

It has been observed that mucoid *P. aeruginosa* can continue to produce mucus isolated from infected lungs. It is commonly thought that the mucoidy is due to a mutation which cancels the inhibition of algU gene; an alternative hypothesis is that it is an epigenetic modification, occurring without mutation [2]. With the method described here it is possible to find the constraints such that the resulting models have two stable behaviours, one mucoid and one non-mucoid: the 8 selected models are compatible with the epigenetic hypothesis. In the case of lambda-phage, there are 2156 coherent models with pathways from initial state to lysis and to lysogeny. But in all these models, there is a common path to lysis, and at least one of two precise paths to lysogeny.

## Conclusion

Because of the partial knowledge of systems, even with a qualitative formalism, different models can fit with experimental results. Our method allows manipulating not only one model, but a set of coherent models. Then we can efficiently respond to two kinds of questions: is there any selected model coherent with a hypothetic behaviour (as the epigenetic modification in *P. aeruginosa*)? Are there common behaviours in selected models (as pathways to lysis or lysogeny in lambda-phage)? Moreover, by keeping this set, new experimental results can be added incrementally to restrict and refine the models.

## References

- 1.Thomas R, D'Ari R: Biological feedback. 1990, Boca Raton: CRC PressGoogle Scholar
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- 3.Thieffry D, Thomas R: Dynamical behaviour of biological regulatory networks–II. Immunity control in bacteriophage lambda. Bull Math Biol. 1995, 57 (2): 277-297.PubMedGoogle Scholar
- 4.Gaston C, Le Gall P, Rapin N, Touil A: Symbolic execution techniques for test purpose definition. TestCom, Lecture Notes in Computer Science. 2006, 3964: 1-18. 10.1007/11754008_1.CrossRefGoogle Scholar

## Copyright information

This article is published under license to BioMed Central Ltd.