LNA++: Linear Noise Approximation with First and Second Order Sensitivities

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11095)


The linear noise approximation (LNA) provides an approximate description of the statistical moments of stochastic chemical reaction networks (CRNs). LNA is a commonly used modeling paradigm describing the probability distribution of systems of biochemical species in the intracellular environment. Unlike exact formulations, the LNA remains computationally feasible even for CRNs with many reactions. The tractability of the LNA makes it a common choice for inference of unknown chemical reaction parameters. However, this task is impeded by a lack of suitable inference tools for arbitrary CRN models. In particular, no available tool provides temporal cross-correlations, parameter sensitivities and efficient numerical integration. In this manuscript we present LNA++, which allows for fast derivation and simulation of the LNA including the computation of means, covariances, and temporal cross-covariances. For efficient parameter estimation and uncertainty analysis, LNA++ implements first and second order sensitivity equations. Interfaces are provided for easy integration with Matlab and Python.

Implementation and availability: LNA++ is implemented as a combination of C/C++, Matlab and Python scripts. Code base and the release used for this publication are available on GitHub ( and Zenodo (


Linear noise approximation Automatic construction Numerical simulation Sensitivity analysis MATLAB Python 


  1. 1.
    Engblom, S.: Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput. 180(2), 498–515 (2006). Scholar
  2. 2.
    Fröhlich, F., Thomas, P., Kazeroonian, A., Theis, F.J., Grima, R., Hasenauer, J.: Inference for stochastic chemical kinetics using moment equations and system size expansion. PLoS Comput. Biol. 12(7), e1005030 (2016). Scholar
  3. 3.
    Giagos, V.: Inference for stochastic kinetic genetic networks using the linear noise approximation, May 2011.
  4. 4.
    Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188(1), 404–425 (1992). Scholar
  5. 5.
    Hasenauer, J., Wolf, V., Kazeroonian, A., Theis, F.J.: Method of conditional moments (MCM) for the chemical master equation. J. Math. Biol. 69(3), 687–735 (2014). Scholar
  6. 6.
    Hindmarsh, A.C., et al.: SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31(3), 363–396 (2005). Scholar
  7. 7.
    Hucka, M., et al.: The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 19(4), 524–531 (2003). Scholar
  8. 8.
    Kazeroonian, A., Fröhlich, F., Raue, A., Theis, F.J., Hasenauer, J.: CERENA: Chemical REaction network analyzer - a toolbox for the simulation and analysis of stochastic chemical kinetics. PLoS One 11(1), e0146732 (2016). Scholar
  9. 9.
    Komorowski, M., Costa, M.J., Rand, D.A., Stumpf, M.P.H.: Sensitivity, robustness, and identifiability in stochastic chemical kinetics models. Proc. Natl. Acad. Sci. U.S.A. 108(21), 8645–8650 (2011). Scholar
  10. 10.
    Komorowski, M., Finkenstädt, B., Harper, C.V., Rand, D.A.: Bayesian inference of biochemical kinetic parameters using the linear noise approximation. BMC Bioinform. 10(1), 343 (2009). Scholar
  11. 11.
    Komorowski, M., Zurauskiene, J., Stumpf, M.P.H.: StochSens-MATLAB package for sensitivity analysis of stochastic chemical systems. Bioinformatics 28(5), 731–733 (2012). Scholar
  12. 12.
    Munsky, B., Khammash, M.: The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys. 124(4), 044104 (2006). Scholar
  13. 13.
    Sanft, K.R., Wu, S., Roh, M., Fu, J., Lim, R.K., Petzold, L.R.: StochKit2: software for discrete stochastic simulation of biochemical systems with events. Bioinformatics 27(17), 2457–2458 (2011)CrossRefGoogle Scholar
  14. 14.
    Stathopoulos, V., Girolami, M.A.: Markov chain Monte Carlo inference for Markov jump processes via the linear noise approximation. Philos. Trans. Ser. A 371(1984), 20110541 (2013). Scholar
  15. 15.
    Thomas, P., Matuschek, H., Grima, R.: Intrinsic noise analyzer: a software package for the exploration of stochastic biochemical kinetics using the system size expansion. PLoS One 7(6), e38518 (2013). Scholar
  16. 16.
    van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, 3rd edn. North-Holland, Amsterdam (2007)zbMATHGoogle Scholar
  17. 17.
    Veldhuizen, T.: Blitz++ User’s Guide, March 2006.

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational BiologyOberschleißheimGermany
  2. 2.Center for MathematicsTechnische Universität MünchenMunichGermany

Personalised recommendations