Abstract
Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analysed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on three case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN.
This research is supported by a Royal Society Research Professorship and ERC AdG VERIWARE.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bortolussi, L., Lanciani, R.: Model checking Markov population models by central limit approximation. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 123–138. Springer, Heidelberg (2013)
Cardelli, L.: Two-domain DNA strand displacement. Math. Struct. Comput. Sci. 23(02), 247–271 (2013)
Cardelli, L.: Morphisms of reaction networks that couple structure to function. BMC Syst. Biol. 8(1), 84 (2014)
Cardelli, L., Kwiatkowska, M., Laurenti, L.: Stochastic analysis of chemical reaction networks using linear noise approximation. arXiv preprint (2015). arXiv:1506.07861
Csikász-Nagy, A., Cardelli, L., Soyer, O.S.: Response dynamics of phosphorelays suggest their potential utility in cell signalling. J. R. Soc. Interface 8(57), 480–488 (2011)
Elf, J., Ehrenberg, M.: Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res. 13(11), 2475–2484 (2003)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence, vol. 282. Wiley, New York (2009)
Gillespie, D.T.: Deterministic limit of stochastic chemical kinetics. J. Phys. Chem. B 113(6), 1640–1644 (2009)
Gillespie, D.T., Hellander, A., Petzold, L.R.: Perspective: stochastic algorithms for chemical kinetics. J. Chem. Phys. 138(17), 170901 (2013)
Itō, I.: Essentials of Stochastic Processes, vol. 231. American Mathematical Soc., Providence (2006)
Kwiatkowska, M., Norman, G., Parker, D.: Stochastic model checking. In: Bernardo, M., Hillston, J. (eds.) SFM 2007. LNCS, vol. 4486, pp. 220–270. Springer, Heidelberg (2007)
Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)
Patel, J.K., Read, C.B.: Handbook of the Normal Distribution, vol. 150. CRC Press, Boca Raton (1996)
Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, vol. 1. Elsevier, Amsterdam (1992)
Wallace, E., Gillespie, D., Sanft, K., Petzold, L.: Linear noise approximation is valid over limited times for any chemical system that is sufficiently large. IET Syst. Biol. 6(4), 102–115 (2012)
Wolf, V., Goel, R., Mateescu, M., Henzinger, T.A.: Solving the chemical master equation using sliding windows. BMC Syst. Biol. 4(1), 42 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Cardelli, L., Kwiatkowska, M., Laurenti, L. (2015). Stochastic Analysis of Chemical Reaction Networks Using Linear Noise Approximation. In: Roux, O., Bourdon, J. (eds) Computational Methods in Systems Biology. CMSB 2015. Lecture Notes in Computer Science(), vol 9308. Springer, Cham. https://doi.org/10.1007/978-3-319-23401-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-23401-4_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23400-7
Online ISBN: 978-3-319-23401-4
eBook Packages: Computer ScienceComputer Science (R0)