Exploring Parameter Space of Stochastic Biochemical Systems Using Quantitative Model Checking

  • Luboš Brim
  • Milan Češka
  • Sven Dražan
  • David Šafránek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8044)


We propose an automated method for exploring kinetic parameters of stochastic biochemical systems. The main question addressed is how the validity of an a priori given hypothesis expressed as a temporal logic property depends on kinetic parameters. Our aim is to compute a landscape function that, for each parameter point from the inspected parameter space, returns the quantitative model checking result for the respective continuous time Markov chain. Since the parameter space is in principle dense, it is infeasible to compute the landscape function directly. Hence, we design an effective method that iteratively approximates the lower and upper bounds of the landscape function with respect to a given accuracy. To this end, we modify the standard uniformization technique and introduce an iterative parameter space decomposition. We also demonstrate our approach on two biologically motivated case studies.


  1. 1.
    Andreychenko, A., Mikeev, L., Spieler, D., Wolf, V.: Parameter Identification for Markov Models of Biochemical Reactions. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 83–98. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Verifying Continuous Time Markov Chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  3. 3.
    Baier, C., Haverkort, B., Hermanns, H., Katoen, J.P.: Model Checking Continuous-Time Markov Chains by Transient Analysis. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 358–372. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Ballarini, P., Forlin, M., Mazza, T., Prandi, D.: Efficient Parallel Statistical Model Checking of Biochemical Networks. In: PDMC 2009. EPTCS, vol. 14, pp. 47–61 (2009)Google Scholar
  5. 5.
    Barbuti, R., Levi, F., Milazzo, P., Scatena, G.: Probabilistic Model Checking of Biological Systems with Uncertain Kinetic Rates. Theor. Comput. Sci. 419, 2–16 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bernardini, F., Biggs, C., Derrick, J., Gheorghe, M., Niranjan, M., Sanguinetti, G.: Parameter Estimation and Model Checking in a Model of Prokaryotic Autoregulation. Tech. rep., University of Sheffield (2007)Google Scholar
  7. 7.
    Daigle, B., Roh, M., Petzold, L., Niemi, J.: Accelerated Maximum Likelihood Parameter Estimation for Stochastic Biochemical Systems. BMC Bioinformatics 13(1), 68–71 (2012)CrossRefGoogle Scholar
  8. 8.
    Degasperi, A., Gilmore, S.: Sensitivity Analysis of Stochastic Models of Bistable Biochemical Reactions. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 1–20. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Didier, F., Henzinger, T.A., Mateescu, M., Wolf, V.: Fast Adaptive Uniformization of the Chemical Master Equation. In: HIBI 2009, pp. 118–127. IEEE Computer Society (2009)Google Scholar
  10. 10.
    El Samad, H., Khammash, M., Petzold, L., Gillespie, D.: Stochastic Modelling of Gene Regulatory Networks. Int. J. of Robust and Nonlinear Control 15(15), 691–711 (2005)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fox, B.L., Glynn, P.W.: Computing Poisson Probabilities. CACM 31(4), 440–445 (1988)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gillespie, D.T.: Exact Stochastic Simulation of Coupled Chemical Reactions. Journal of Physical Chemistry 81(25), 2340–2381 (1977)CrossRefGoogle Scholar
  13. 13.
    Golightly, A., Wilkinson, D.J.: Bayesian Parameter Inference for Stochastic Biochemical Network Models Using Particle Markov Chain Monte Carlo. Interface Focus 1(6), 807–820 (2011)CrossRefGoogle Scholar
  14. 14.
    Grassmann, W.: Transient Solutions in Markovian Queueing Systems. Computers & Operations Research 4(1), 47–53 (1977)CrossRefGoogle Scholar
  15. 15.
    Hahn, E.M., Han, T., Zhang, L.: Synthesis for PCTL in Parametric Markov Decision Processes. In: NASA Formal Methods, pp. 146–161 (2011)Google Scholar
  16. 16.
    Henzinger, T.A., Mateescu, M., Wolf, V.: Sliding Window Abstraction for Infinite Markov Chains. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 337–352. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Jha, S.K., Clarke, E.M., Langmead, C.J., Legay, A., Platzer, A., Zuliani, P.: A Bayesian Approach to Model Checking Biological Systems. In: Degano, P., Gorrieri, R. (eds.) CMSB 2009. LNCS, vol. 5688, pp. 218–234. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Koh, C.H., Palaniappan, S., Thiagarajan, P., Wong, L.: Improved Statistical Model Checking Methods for Pathway Analysis. BMC Bioinformatics 13(suppl. 17), S15 (2012)Google Scholar
  19. 19.
    Kwiatkowska, M., Norman, G., Pacheco, A.: Model Checking Expected Time and Expected Reward Formulae with Random Time Bounds. Compu. Math. Appl. 51(2), 305–316 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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)CrossRefGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Mikeev, L., Neuhäußer, M., Spieler, D., Wolf, V.: On-the-fly Verification and Optimization of DTA-properties for Large Markov Chains. Form. Method. Syst. Des., 1–25 (2012)Google Scholar
  23. 23.
    Reinker, S., Altman, R., Timmer, J.: Parameter Estimation in Stochastic Biochemical Reactions. IEEE Proc. Syst. Biol. 153(4), 168–178 (2006)CrossRefGoogle Scholar
  24. 24.
    Schlögl, F.: Chemical Reaction Models for Non-Equilibrium Phase Transitions. Zeitschrift fur Physik 253, 147–161 (1972)CrossRefGoogle Scholar
  25. 25.
    Swat, M., Kel, A., Herzel, H.: Bifurcation Analysis of the Regulatory Modules of the Mammalian G1/S transition. Bioinformatics 20(10), 1506–1511 (2004)CrossRefGoogle Scholar
  26. 26.
    Vellela, M., Qian, H.: Stochastic Dynamics and Non-Equilibrium Thermodynamics of a Bistable Chemical System: the Schlögl Model Revisited. Journal of The Royal Society Interface 6(39), 925–940 (2009)CrossRefGoogle Scholar
  27. 27.
    Yang, E., van Nimwegen, E., Zavolan, M., Rajewsky, N., Schroeder, M.K., Magnasco, M., Darnell, J.E.: Decay Rates of Human mRNAs: Correlation With Functional Characteristics and Sequence Attributes. Genome Research 13(8), 1863–1872 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Luboš Brim
    • 1
  • Milan Češka
    • 1
  • Sven Dražan
    • 1
  • David Šafránek
    • 1
  1. 1.Systems Biology Laboratory at Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations