# Assigning probabilities to qualitative dynamics of gene regulatory networks

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## Abstract

Mathematical and computational modeling frameworks play the leading role in the analysis and prediction of the dynamics of gene regulatory networks. The literature abounds in various approaches, all of which characterized by strengths and weaknesses. Among the others, Ordinary Differential Equations (ODE) models give a more general and detailed description of the network structure. But, analytical computations might be prohibitive as soon as the network dimension increases, and numerical simulation could be nontrivial, time-consuming and very often impracticable as precise and quantitative information on model parameters are frequently unknown and hard to estimate from experimental data. Last but not least, they do not account for the intrinsic stochasticity of regulation. In the present paper we consider a class of ODE models with stochastic parameters. The proposed method separates the deterministic aspects from the stochastic ones. Under specific assumptions and conditions, all possible trajectories of an ODE model, where incomplete knowledge of parameter values is symbolically and qualitatively expressed by initial inequalities only, are simulated in a single run from an initial state. Then, the occurrence probability of each trajectory, characterized by a sequence of parameter inequalities, is computed by associating probability density functions with network parameters. As demonstrated by its application to the gene repressilator system, the method seems particularly promising in the design of synthetic networks.

## Keywords

Gene regulatory networks Nonlinear ODE models Stochastic parameters Qualitative simulation## Mathematics Subject Classification (2010)

34C60 34E15 60H10 92B99## Notes

### Acknowledgments

We gratefully thank Diana X. Tran for useful general discussions on Synthetic Biology, and, in particular, on the repressilator system. This work is carried out within the FLAGSHIP “InterOmics” project (PB.P05) that is funded and supported by the Italian Ministry of Education, University and Research (MIUR) and National Research Council (CNR) organizations.

## References

- Alon U (2007) Introduction to systems biology: design principles of biological circuits. Chapman & Hall, Boca RatonGoogle Scholar
- Bokes P, King JR, Wood AT, Losse M (2012) Multiscale stochastic modelling of gene expression. J Math Biol 65(3):493–520CrossRefzbMATHMathSciNetGoogle Scholar
- Bruggeman FJ, Westerhoff HV (2007) The nature of systems biology. Tren Microbiol 15(1):45–50CrossRefGoogle Scholar
- Cao Y, Samuels DC (2009) Discrete stochastic simulation methods for chemically reacting systems. Meth Enzymol 454:115–140CrossRefGoogle Scholar
- de Jong H (2002) Modeling and simulation of genetic regulatory systems: a literature review. J Comput Biol 9(1):67–104CrossRefGoogle Scholar
- de Jong H, Geiselmann J, Hernadez C, Page M (2003) Genetic network analyzer: qualitative simulation of genetic regulatory networks. Bioinformatics 19(3):336–344CrossRefGoogle Scholar
- Drulhe S, Ferrari-Trecate G, de Jong H (2008) The switching threshold reconstruction problem for piecewise affine models of genetic regulatory networks. IEEE Trans Autom Cont 53:153–165CrossRefGoogle Scholar
- Edwards R, Ironi L (2013) Periodic solutions of gene networks with steep sigmoidal regulatory functions. Technical report, 1PV13/1/0 IMATI-CNRGoogle Scholar
- Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403(6767):335–337CrossRefGoogle Scholar
- Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434CrossRefMathSciNetGoogle Scholar
- Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361CrossRefGoogle Scholar
- Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys 115:1716–1733CrossRefGoogle Scholar
- Hartwell LH, Hopfield JJ, Leibler S, Murray AW (1999) From molecular to modular cell biology. Nature 402(6761):C47–C52CrossRefGoogle Scholar
- Heinemann M, Panke S (2006) Synthetic biology—putting engineering into biology. Bioinformatics 22(22):2790–2799CrossRefGoogle Scholar
- Ironi L, Panzeri L (2009) A computational framework for qualitative simulation of nonlinear dynamical models of gene-regulatory networks. BMC Bioinfo 10(Suppl 12):S14CrossRefGoogle Scholar
- Ironi L, Panzeri L, Plahte E, Simoncini V (2011) Dynamics of actively regulated gene networks. Physica D 240:779–794CrossRefzbMATHMathSciNetGoogle Scholar
- Ironi L, Tran DX (2014) A simulator of the nonlinear dynamics of gene-regulatory networks. Technical report, IMATI-CNR (in preparation)Google Scholar
- Karlebach G, Shamir R (2008) Modelling and analysis of gene regulatory networks. Nat Rev Mol Cell Biol 9(10):770–780CrossRefGoogle Scholar
- Kell DB (2006) Metabolomics, modelling and machine learning in systems biology-towards an understanding of the languages of cells. Febs J 273(5):873–894CrossRefGoogle Scholar
- Khalil AS, Collins JJ (2010) Synthetic biology: applications come of age. Nat Rev Gen 11(5):367–379CrossRefGoogle Scholar
- Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Kroese DP, Taimre T, Botev ZI (2011) Handbook of Monte Carlo methods. Wiley Series in Probability and Statistics, Wiley, New YorkGoogle Scholar
- Kulasiri D, Nguyen LK, Samarasinghe S, Xie Z (2008) A review of systems biology perspective on genetic regulatory networks with examples. Curr Bioinfo 3:197–225CrossRefGoogle Scholar
- Machina A, Edwards R, van den Driessche P (2013) Singular dynamics in gene network models. SIAM J Appl Dyn Sys 12(1):95–125CrossRefzbMATHGoogle Scholar
- McAdams HH, Arkin A (1976) Stochastic mechanisms in gene expression. Proc Natl Acad Sci USA 94:814–819CrossRefGoogle Scholar
- Øksendal B (2003) Stochastic differential equations: an introduction with applications, 6th edn. Springer, BerlinCrossRefGoogle Scholar
- Paszek P (2007) Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function. Bull Math Biol 69:1567–1601CrossRefzbMATHMathSciNetGoogle Scholar
- Plahte E, Kjøglum S (2005) Analysis and generic properties of gene regulatory networks with graded response functions. Physica D 201(1–2):150–176CrossRefzbMATHMathSciNetGoogle Scholar
- Purcell O, Savery NJ, Grierson CS, di Bernardo M (2010) A comparative analysis of synthetic genetic oscillators. J Roy Soc Interf 7:1503–1524CrossRefGoogle Scholar
- Rubino G, Tuffin B (eds) (2009) Rare event simulation using Monte Carlo methods. John Wiley & Sons Ltd, ChichesterGoogle Scholar
- Veflingstad SR, Plahte E (2007) Analysis of gene regulatory network models with graded and binary transcriptional responses. Biosystems 90:323–339CrossRefGoogle Scholar