Skip to main content

A Stochastic Hybrid Approximation for Chemical Kinetics Based on the Linear Noise Approximation

  • Conference paper
  • First Online:
Computational Methods in Systems Biology (CMSB 2016)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 9859))

Included in the following conference series:

Abstract

The Linear Noise Approximation (LNA) is a continuous approximation of the CME, which improves scalability and is accurate for those reactions satisfying the leap conditions. We formulate a novel stochastic hybrid approximation method for chemical reaction networks based on adaptive partitioning of the species and reactions according to leap conditions into two classes, one solved numerically via the CME and the other using the LNA. The leap criteria are more general than partitioning based on population thresholds, and the method can be combined with any numerical solution of the CME. We then use the hybrid model to derive a fast approximate model checking algorithm for Stochastic Evolution Logic (SEL). Experimental evaluation on several case studies demonstrates that the techniques are able to provide an accurate stochastic characterisation for a large class of systems, especially those presenting dynamical stiffness, resulting in significant improvement of computation time while still maintaining scalability.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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.) CAV 2015. LNCS, vol. 9206, pp. 195–213. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  2. Adler, R.J.: An introduction to continuity, extrema, related topics for general Gaussian processes. Lect. Notes-Monogr. Ser. 12, i-155 (1990)

    MathSciNet  Google Scholar 

  3. Anderson, D.F., Kurtz, T.G.: Stochastic Analysis of Biochemical Systems. Springer, Heidelberg (2015)

    Google Scholar 

  4. Arkin, A., Ross, J., McAdams, H.H.: Stochastic kinetic analysis of developmental pathway bifurcation in phage \(\lambda \)-infected escherichia coli cells. Genetics 149(4), 1633–1648 (1998)

    Google Scholar 

  5. 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)

    Chapter  Google Scholar 

  6. Cao, Y., Gillespie, D.T., Petzold, L.R.: The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122(1), 014116 (2005)

    Article  Google Scholar 

  7. Cardelli, L.: On process rate semantics. Theoret. Comput. Sci. 391(3), 190–215 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cardelli, L., Kwiatkowska, M., Laurenti, L.: Stochastic analysis of chemical reaction networks using linear noise approximation. In: Roux, O., Bourdon, J. (eds.) CMSB 2015. LNCS, vol. 9308, pp. 64–76. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  9. Cardelli, L., Kwiatkowska, M., Laurenti, L.: Programming discrete distributions with chemical reaction networks. In: Rondelez, Y., Woods, D. (eds.) DNA 2016. LNCS, vol. 9818, pp. 35–51. Springer, Heidelberg (2016). doi:10.1007/978-3-319-43994-5_3

    Chapter  Google Scholar 

  10. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence, vol. 282. Wiley, Hoboken (2009)

    MATH  Google Scholar 

  11. Ganguly, A., Altintan, D., Koeppl, H.: Jump-diffusion approximation of stochastic reaction dynamics: error bounds and algorithms. Multiscale Model. Simul. 13(4), 1390–1419 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)

    Article  Google Scholar 

  13. Gillespie, D.T.: The chemical Langevin equation. J. Chem. Phys. 113(1), 297–306 (2000)

    Article  Google Scholar 

  14. Gillespie, D.T.: Simulation methods in systems biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 125–167. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  15. Goutsias, J.: Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122(18), 184102 (2005)

    Article  Google Scholar 

  16. Goutsias, J., Jenkinson, G.: Markovian dynamics on complex reaction networks. Phys. Rep. 529(2), 199–264 (2013)

    Article  MathSciNet  Google Scholar 

  17. Hasenauer, J., Wolf, V., Kazeroonian, A., Theis, F.: Method of conditional moments (mcm) for the chemical master equation. J. Math. Biol. 69(3), 687–735 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Henzinger, T.A., Mikeev, L., Mateescu, M., Wolf, V.: Hybrid numerical solution of the chemical master equation. In: Proceedings of the 8th International Conference on Computational Methods in Systems Biology, pp. 55–65. ACM (2010)

    Google Scholar 

  19. Hepp, B., Gupta, A., Khammash, M.: Adaptive hybrid simulations for multiscale stochastic reaction networks. J. Chem. Phys. 142(3), 034118 (2015)

    Article  Google Scholar 

  20. Kwiatkowska, M., Norman, G., Parker, D.: Stochastic model checking, pp. 220–270 (2007)

    Google Scholar 

  21. Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Qadeer, S., Gopalakrishnan, G. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  22. Mateescu, M., Wolf, V., Didier, F., Henzinger, T., et al.: Fast adaptive uniformisation of the chemical master equation. IET 4, 441–452 (2010)

    Google Scholar 

  23. McAdams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. Proc. Nat. Acad. Sci. 94(3), 814–819 (1997)

    Article  Google Scholar 

  24. Munsky, B., Khammash, M.: The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys. 124(4), 044104 (2006)

    Article  MATH  Google Scholar 

  25. 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)

    Article  Google Scholar 

  26. Salis, H., Kaznessis, Y.: Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122(5), 054103 (2005)

    Article  Google Scholar 

  27. Srivastava, R., You, L., Summers, J., Yin, J.: Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theoret. Biol. 218(3), 309–321 (2002)

    Article  MathSciNet  Google Scholar 

  28. Thattai, M., Van Oudenaarden, A.: Intrinsic noise in gene regulatory networks. Proc. Nat. Acad. Sci. 98(15), 8614–8619 (2001)

    Article  Google Scholar 

  29. Thomas, P., Popović, N., Grima, R.: Phenotypic switching in gene regulatory networks. Proc. Nat. Acad. Sci. 111(19), 6994–6999 (2014)

    Article  Google Scholar 

  30. Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, vol. 1. Elsevier, Amsterdam (1992)

    MATH  Google Scholar 

  31. 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)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Luca Laurenti .

Editor information

Editors and Affiliations

Appendices

A Proofs

Proposition 1

Let \(x^s \in S^s\) and \(x^f \in S^f\). Then, for \(t \in \mathbb {R}_{\ge 0}\)

$$\begin{aligned} \frac{d P(x^s|t)}{dt}=\sum _{\tau \in R}\beta _{\tau }(x^s-\upsilon _{\tau },t)P(x^s-\upsilon _{\tau },t)- \beta _{\tau }(x^s,t)P(x^s,t) \end{aligned}$$

where \(\beta _{\tau }(x^s,t)= \sum _{x^f \in S^f} \alpha _{\tau }(x^f,x^s)P(x^f|x^s,t)\).

Proof

By using the law of total probability we have

$$\begin{aligned} \frac{d P(x^s|t)}{dt}=\sum _{x^f \in S^f}\frac{d P(x^s,x^f,t)}{dt} \end{aligned}$$

Then, using Eq. (2), and rearranging terms we have

$$\begin{aligned} \sum _{x^f \in S^f}\frac{d P(x^s,x^f,t)}{dt}= \end{aligned}$$
$$\begin{aligned} \sum _{x^f \in S^f}\sum _{\tau \in R^f}\alpha _{\tau }(x^f-\upsilon _{\tau },x^s-\upsilon _{\tau })P(x^f-\upsilon _{\tau },x^s-\upsilon _{\tau },t)- \alpha _{\tau }(x^f,x^s)P(x^f,x^s,t)= \end{aligned}$$
$$\begin{aligned} \sum _{\tau \in R}\beta _{\tau }(x^s-\upsilon _{\tau },t)P(x^s-\upsilon _{\tau },t)- \beta _{\tau }(x^s,t)P(x^s,t) \end{aligned}$$

where \(\beta _{\tau }(x^s,t)= \sum _{x^f \in S^f} \alpha _{\tau }(x^f,x^s)P(x^f|x^s,t)\), that is, the conditional expectation of the propensity rate of \(\tau \) at time t given \(X^s(t)=x^s\).

Theorem 2

Assume \(\varLambda ^s_t\) is non-empty and \(S^s\) is the state space of \(X^s(t)\). Then, the stochastic process \(Z^h:\varOmega \times \mathcal {R}_{\ge 0}\rightarrow \mathcal {S}\), with \(\varOmega \) its sample space and \((\mathcal {S},\mathcal {B})\) a measurable space, is such that for \(A\in \mathcal {B}\) and \(t\in \mathbb {R}_{\ge 0}\)

$$\begin{aligned} P(Z^h(t)\in A|X(0)=x_0)=\sum _{x^s \in S^s} P(Z_{x^s}(t)\in A)P(X^s(t)=x_s) \end{aligned}$$

where \(Z_{x^s}(t)\) is a Gaussian random variable with expected value and variance

$$\begin{aligned} E[Z_{x^s}(t)]=B\cdot \begin{pmatrix} E[\bar{X}^f(t)]\\ x^s \end{pmatrix} \quad C[Z_{x^s}(t)]=B\cdot \begin{pmatrix} C[\bar{X}^f(t)]&{}0\\ 0&{}0 \end{pmatrix} \cdot B^T \end{aligned}$$

where \(\bar{X}^f\) is the virtual fast process introduced in Sect. 3.

Proof

By the law of total probability we have

$$ P(Z(t)\in A|X(0)=x_0)=\sum _{x^s \in S^s}P(Z(t)\in A|X^s(t)\!=\!x^s,X(0)=x_0)P(X^s(t)\!=\!x^s|X(0)=x_0). $$

By application of the LNA it follows that \(X^f(t)\) conditioned on the event \(X^s(t)=x^s\) is a Gaussian random variable with expected value and variance

$$ E[X^f(t)|X^s(t)=x^s]=\begin{pmatrix} E[\bar{X}^f(t)]\\ x^s \end{pmatrix} $$

and covariance matrix

$$ C[X^f(t)|X^s(t)=x^s]=\begin{pmatrix} C[\bar{X}^f(t)]&{}0\\ 0&{}0 \end{pmatrix} $$

Given a multidimensional Gaussian distribution, each linear combination of its components is still Gaussian. As a consequence, \(E[Z^h(t)|X^s(t)=x^s]=B\cdot E[X^f(t)|X^s(t)=x^s]\) and \(C[Z^h(t)|X^s(t)=x^s]=B\cdot C[X^f(t)|X^s(t)=x^s]\cdot B^T\).

Theorem 3

Assume \(\varLambda ^s_t\) is non-empty. Then, for \(t\in \mathbb {R}_{\ge 0}\)

$$\begin{aligned} E[Z^h(t)|X(0)=x_0]=\sum _{x^s \in S^s} E[Z_{x^s}(t)\in A]P(X^s(t)=x_s) \end{aligned}$$
$$\begin{aligned} C[Z^h(t)|X(0)=x_0]=\sum _{x^s \in S^s} C[Z_{x^s}(t)\in A]P(X^s(t)=x_s) \end{aligned}$$

Proof

The proof follows from the application of the law of total expectation for random variables with mutually exclusive and exhaustive events.

B Figures

Fig. 2.
figure 2

Comparison of expected value and variance of mRNA in Example 2 in interval [0, 200] as calculated by direct solution of the CME (Fig. 2a) and by our algorithm (Fig. 2b).

Fig. 3.
figure 3

Comparison of the probability distribution of RNA at time \(t=200\) as calculated by numerical hybrid algorithm (Fig. 3a) and by the LNA (Fig. 3b).

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Cardelli, L., Kwiatkowska, M., Laurenti, L. (2016). A Stochastic Hybrid Approximation for Chemical Kinetics Based on the Linear Noise Approximation. In: Bartocci, E., Lio, P., Paoletti, N. (eds) Computational Methods in Systems Biology. CMSB 2016. Lecture Notes in Computer Science(), vol 9859. Springer, Cham. https://doi.org/10.1007/978-3-319-45177-0_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45177-0_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45176-3

  • Online ISBN: 978-3-319-45177-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics