Acta Informatica

, Volume 54, Issue 6, pp 589–623 | Cite as

Precise parameter synthesis for stochastic biochemical systems

  • Milan Češka
  • Frits Dannenberg
  • Nicola Paoletti
  • Marta Kwiatkowska
  • Luboš Brim
Original Article

Abstract

We consider the problem of synthesising rate parameters for stochastic biochemical networks so that a given time-bounded CSL property is guaranteed to hold, or, in the case of quantitative properties, the probability of satisfying the property is maximised or minimised. Our method is based on extending CSL model checking and standard uniformisation to parametric models, in order to compute safe bounds on the satisfaction probability of the property. We develop synthesis algorithms that yield answers that are precise to within an arbitrarily small tolerance value. The algorithms combine the computation of probability bounds with the refinement and sampling of the parameter space. Our methods are precise and efficient, and improve on existing approximate techniques that employ discretisation and refinement. We evaluate the usefulness of the methods by synthesising rates for three biologically motivated case studies: infection control for a SIR epidemic model; reliability analysis of molecular computation by a DNA walker; and bistability in the gene regulation of the mammalian cell cycle.

Notes

Acknowledgments

We thank David Šafránek for useful discussions about the stochastic models of gene regulation of mammalian cell cycle and Nicolas Basset for helping with the termination of the threshold synthesis algorithm. We also thank Andrej Tokarčík and Petr Pilař for implementing the prototype version of the synthesis algorithms. We finally thank the anonymous reviewers for their insightful feedback.

References

  1. 1.
    Abate, A., Brim, L., Češka, M., Kwiatkowska, M.: Adaptive aggregation of markov chains: quantitative analysis of chemical reaction networks. In: Kroening, D., Păsăreanu, C.S. (eds.) Computer Aided Verification (CAV), LNCS, vol. 9206, pp. 195–213. Springer (2015)Google Scholar
  2. 2.
    Andreychenko, A., Mikeev, L., Spieler, D., Wolf, V.: Parameter identification for Markov models of biochemical reactions. In: Gopalakrishnan, G., Qadeer, S. (eds.) Computer Aided Verification (CAV), LNCS, pp. 83–98. Springer (2011)Google Scholar
  3. 3.
    Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Verifying continuous time Markov chains. In: Alur, R., Henzinger, T.A. (eds.) Computer Aided Verification (CAV), LNCS, vol. 1102, pp. 269–276. Springer (1996)Google Scholar
  4. 4.
    Baier, C., Haverkort, B., Hermanns, H., Katoen, J.: Model-checking algorithms for continuous-time Markov chains. IEEE Trans Softw Eng 29(6), 524–541 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Bartocci, E., Bortolussi, L., Nenzi, L.: A temporal logic approach to modular design of synthetic biological circuits. In: Gupta, A., Henzinger, T.A. (eds.) Computational Methods in Systems Biology (CMSB), pp. 164–177. Springer (2013)Google Scholar
  6. 6.
    Batt, G., Yordanov, B., Weiss, R., Belta, C.: Robustness analysis and tuning of synthetic gene networks. Bioinformatics 23(18), 2415–2422 (2007)CrossRefGoogle Scholar
  7. 7.
    Belta, C., Habets, L.: Controlling a class of nonlinear systems on rectangles. IEEE Trans Autom Control 51(11), 1749–1759 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Billingsley, P.: Probability and Measure. Wiley, Hoboken (2008)MATHGoogle Scholar
  9. 9.
    Bortolussi, L., Milios, D., Sanguinetti, G.: Smoothed model checking for uncertain continuous time markov chains. CoRR. arXiv:1402.1450 (2014)
  10. 10.
    Bortolussi, L., Sanguinetti, G.: Learning and designing stochastic processes from logical constraints. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) Quantitative Evaluation of Systems (QEST), LNCS, vol. 8054, pp. 89–105. Springer (2013)Google Scholar
  11. 11.
    Brim, L., Češka, M., Dražan, S., Šafránek, D.: Exploring parameter space of stochastic biochemical systems using quantitative model checking. In: Sharygina, N., Veith, H. (eds.) Computer Aided Verification (CAV), LNCS, vol. 8044, pp. 107–123. Springer (2013)Google Scholar
  12. 12.
    Caron, R., Traynor, T.: The zero set of a polynomial. Technical report, University of Windsor (2005)Google Scholar
  13. 13.
    Češka, M., Dannenberg, F., Kwiatkowska, M., Paoletti, N.: Precise parameter synthesis for stochastic biochemical systems. In: Mendes, P., Dada, J.O., Smallbone, K. (eds.) Computational Methods in Systems Biology (CMSB), pp. 86–98. Springer (2014)Google Scholar
  14. 14.
    Chen, T., Hahn, E.M., Han, T., Kwiatkowska, M., Qu, H., Zhang, L.: Model repair for Markov decision processes. In: Theoretical Aspects of Software Engineering (TASE), pp. 85–92. IEEE (2013)Google Scholar
  15. 15.
    Courant, R., John, F.: Introduction to Calculus and Analysis, vol. 2. Springer, Berlin (2012)MATHGoogle Scholar
  16. 16.
    Dannenberg, F., Hahn, E.M., Kwiatkowska, M.: Computing cumulative rewards using fast adaptive uniformisation. In: Gupta, A., Henzinger, T.A. (eds.) Computational Methods in Systems Biology (CMSB), LNCS, vol. 8130, pp. 33–49. Springer (2013)Google Scholar
  17. 17.
    Dannenberg, F., Kwiatkowska, M., Thachuk, C., Turberfield, A.: DNA walker circuits: computational potential, design, and verification. Nat. Comput. 14, 195–211 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Donzé, A.: Breach, a toolbox for verification and parameter synthesis of hybrid systems. In: Touili, T., Cook, B., Jackson, P. (eds.) Computer Aided Verification (CAV), LNCS, vol. 6174, pp. 167–170. Springer (2010)Google Scholar
  19. 19.
    Fox, B.L., Glynn, P.W.: Computing Poisson probabilities. CACM 31(4), 440–445 (1988)Google Scholar
  20. 20.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2381 (1977)CrossRefGoogle Scholar
  21. 21.
    Grassmann, W.: Transient solutions in Markovian queueing systems. Comput. Oper. Res. 4(1), 47–53 (1977)CrossRefGoogle Scholar
  22. 22.
    Hahn, E.M., Hermanns, H., Zhang, L.: Probabilistic reachability for parametric Markov models. Int. J. Softw. Tools Technol. Transf. (STTT) 13(1), 3–19 (2011)CrossRefGoogle Scholar
  23. 23.
    Han, T., Katoen, J., Mereacre, A.: Approximate parameter synthesis for probabilistic time-bounded reachability. In: Real-Time Systems Symposium (RTSS), pp. 173–182. IEEE (2008)Google Scholar
  24. 24.
    Jensen, A.: Markoff chains as an aid in the study of Markoff processes. Skand. Aktuarietidskr. 36, 87–91 (1953)MathSciNetMATHGoogle Scholar
  25. 25.
    Jha, S.K., Langmead, C.J.: Synthesis and infeasibility analysis for stochastic models of biochemical systems using statistical model checking and abstraction refinement. Theor. Comput. Sci. 412(21), 2162–2187 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Katoen, J.P., Klink, D., Leucker, M., Wolf, V.: Three-valued abstraction for continuous-time markov chains. In: Damm, W., Hermanns, H. (eds.) Computer Aided Verification (CAV), LNCS, vol. 4590, pp. 311–324. Springer (2007)Google Scholar
  27. 27.
    Kermack, W., McKendrick, A.: Contributions to the mathematical theory of epidemicsii. The problem of endemicity. Bull. Math. Biol. 53(1), 57–87 (1991)Google Scholar
  28. 28.
    Klarner, H., Streck, A., Šafránek, D., Kolčák, J., Siebert, H.: Parameter identification and model ranking of thomas networks. In: Gilbert, D., Heiner, M. (eds.) Computational Methods in Systems Biology (CMSB), pp. 207–226. Springer (2012)Google Scholar
  29. 29.
    Koksal, A.S., Pu, Y., Srivastava, S., Bodik, R., Fisher, J., Piterman, N.: Synthesis of biological models from mutation experiments. SIGPLAN Not. 48(1), 469–482 (2013)CrossRefMATHGoogle Scholar
  30. 30.
    Kwiatkowska, M., Norman, G., Pacheco, A.: Model checking expected time and expected reward formulae with random time bounds. Comput. Math. Appl. 51, 305–316 (2006)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kwiatkowska, M., Norman, G., Parker, D.: Stochastic model checking. In: Bernardo, M., Hillston, J. (eds.) Formal Methods for Performance Evaluation (SFM), LNCS, vol. 4486, pp. 220–270. Springer (2007)Google Scholar
  32. 32.
    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 (2011)Google Scholar
  33. 33.
    Madsen, C., Myers, C., Roehner, N., Winstead, C., Zhang, Z.: Utilizing stochastic model checking to analyze genetic circuits. In: Computational Intelligence in Bioinformatics and Computational Biology (CIBCB), pp. 379–386. IEEE (2012)Google Scholar
  34. 34.
    Mateescu, M., Wolf, V., Didier, F., Henzinger, T.A.: Fast adaptive uniformization of the chemical master equation. IET Syst. Biol. 4(6), 441–452 (2010)CrossRefGoogle Scholar
  35. 35.
    Michaelis, L., Menten, M.L.: Die kinetik der invertinwirkung. Biochem. z 49(333–369), 352 (1913)Google Scholar
  36. 36.
    Paoletti, N., Yordanov, B., Hamadi, Y., Wintersteiger, C.M., Kugler, H.: Analyzing and synthesizing genomic logic functions. In: Biere, A., Bloem, R. (eds.) Computer Aided Verification (CAV), pp. 343–357. Springer (2014)Google Scholar
  37. 37.
    Rao, C.V., Arkin, A.P.: Stochastic chemical kinetics and the quasi-steady-state assumption: application to the gillespie algorithm. J. Chem. Phys. 118(11), 4999–5010 (2003)CrossRefGoogle Scholar
  38. 38.
    Reibman, A.: Numerical transient analysis of Markov models. Comput. Oper. Res. 15(1), 19–36 (1988)CrossRefMATHGoogle Scholar
  39. 39.
    Sanft, K.R., Gillespie, D.T., Petzold, L.R.: Legitimacy of the stochastic Michaelis-Menten approximation. Syst. Biol., IET 5(1), 58–69 (2011)Google Scholar
  40. 40.
    Sassi, B., Amin, M., Girard, A.: Control of polynomial dynamical systems on rectangles. In: European Control Conference (ECC), pp. 658–663. IEEE (2013)Google Scholar
  41. 41.
    Sen, K., Viswanathan, M., Agha, G.: Model-checking markov chains in the presence of uncertainties. In: Hermanns, H., Palsberg, J. (eds.) Tools and Algorithms for the Construction and Analysis of Systems (TACAS), LNCS, vol. 3920, pp. 394–410. Springer (2006)Google Scholar
  42. 42.
    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
  43. 43.
    Wickham, S.F.J., Bath, J., Katsuda, Y., Endo, M., Hidaka, K., Sugiyama, H., Turberfield, A.J.: A DNA-based molecular motor that can navigate a network of tracks. Nat. Nanotechnol. 7, 169–173 (2012)CrossRefGoogle Scholar
  44. 44.
    Zhang, J., Watson, L.T., Cao, Y.: Adaptive aggregation method for the chemical master equation. Int. J. Comput. Biol. Drug Des. 2(2), 134–148 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Milan Češka
    • 2
    • 3
  • Frits Dannenberg
    • 3
  • Nicola Paoletti
    • 3
  • Marta Kwiatkowska
    • 3
  • Luboš Brim
    • 1
  1. 1.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic
  2. 2.Faculty of Information TechnologyBrno University of TechnologyBrnoCzech Republic
  3. 3.Department of Computer ScienceUniversity of OxfordOxfordUK

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