Skip to main content
SpringerLink
Log in

Krylov and steady-state techniques for the solution of the chemical master equation for the mitogen-activated protein kinase cascade

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Models of chemical kinetics in which some reactions are much faster than others are often treated by a type of quasi-steady-state approximation (QSSA). The total QSSA (tQSSA) was introduced for models of Michaelis-Menten enzyme kinetics and shown to be valid over a wider parameter regime than the usual QSSA. Here, we extend the tQSSA to the Mitogen-Activated Protein Kinase Cascade, an important signaling system in cell biochemistry. These approximations were first developed in a deterministic setting, but here we also describe how to incorporate this approximation into the discrete and stochastic framework of the Chemical Master Equation (CME). The CME gives rise to a large-scale matrix exponential that can be solved by Krylov methods in combination with operator splitting and the tQSSA.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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 (Aug 1998)

    Google Scholar 

  2. Blake, W.J., Kaern, M., Cantor, C.R., Collins, J.J.: Noise in eukaryotic gene expression. Nature 422(6932), 633–637 (Apr 2003)

    Article  Google Scholar 

  3. Booth, B., Zemmel, R.: Prospects for productivity. Nat. Rev. Drug Discov. 3(5), 451–456 (May 2004)

    Article  Google Scholar 

  4. Borghans, J.A., de Boer, R.J., Segel, L.A.: Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58(1), 43–63 (Jan 1996)

    Article  MATH  Google Scholar 

  5. Brent, R.P.: Reducing the retrieval time of scatter storage techniques. Commun. ACM 16(2), 105–109 (February 1973)

    Article  MATH  MathSciNet  Google Scholar 

  6. Briggs, G.E., Haldane, J.B.S.: A note on the kinetics of enzyme action. Biochem. J. 19, 338–339 (1925)

    Google Scholar 

  7. Burrage, K.: Parallel and Sequential Methods for Ordinary Differential Equations. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  8. Burrage, K., Hegland, M., MacNamara, S., Sidje, R.B.: A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems. In: Langville, A.N., Stewart, W.J. (eds.) 150th Markov Anniversary Meeting, Charleston, SC, USA, pp. 21–38. Boson Books, Raleigh (2006)

    Google Scholar 

  9. Burrage, K., Tian, T., Burrage, P.: A multi-scaled approach for simulating chemical reaction systems. Prog. Biophys. Mol. Biol. 85(2–3), 217–234 (2004)

    Article  Google Scholar 

  10. Cao, Y., Gillespie, D.T., Petzold, L.R.: The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122, 014116–1–18 (2005)

    Google Scholar 

  11. Chan, R.H.: Iterative methods for overflow queueing models I and II. Numer. Math. 51(2), 143–180 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ciliberto, A., Capuani, F., Tyson, J.J.: Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation. PLoS Comput. Biol. 3(3), e45 (Mar 2007)

    Article  MathSciNet  Google Scholar 

  13. Dahlquist, G., Björk, Å.: Numerical Methods. Prentice-Hall, New York (1974)

    Google Scholar 

  14. Liu, W.E.D., Vanden-Eijnden, E.: Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123(19), 194107 (Nov 2005)

    Article  Google Scholar 

  15. Ehrenberg, M., Elf, J., Aurell, E., Sandberg, R., Tegnér, J.: Systems biology is taking off. Genome Res. 13(11), 2377–2380 (Nov 2003)

    Article  Google Scholar 

  16. Elowitz, M.B., Surette, M.G., Wolf, P.E., Stock, J.B., Leibler, S.: Protein mobility in the cytoplasm of escherichia coli. J. Bacteriol. 181(1), 197–203 (January 1999)

    Google Scholar 

  17. Fedoroff, N., Fontana, W.: Small numbers of big molecules. Science 297(5584), 1129–1131 (August 2002)

    Article  Google Scholar 

  18. Gillespie, D.: Markov Processes: An Introduction for Physical Scientists. Academic, London (1992)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  20. Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–1733 (2001)

    Article  Google Scholar 

  21. Golub, G., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. J. Soc. Ind. Appl. Math. Ser. B Numer. Anal. 2, 205–224 (1965)

    Article  MathSciNet  Google Scholar 

  22. Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins, Baltimore (1996)

    MATH  Google Scholar 

  23. Golub, G.H., Vanderstraeten, D.: On the preconditioning of matrices with skew-symmetric splittings. Numer. Algorithm 25, 223–239 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hegland, M., Burden, C., Santoso, L., MacNamara, S., Booth, H.: A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math. 205(2), 708–724 (2006) (Special issue on evolutionary problems)

    Article  MathSciNet  Google Scholar 

  25. Henri, V.: Ueber das gesetz der wirkung des invertins. Z. Phys. Chem. 39, 194–216 (1901)

    Google Scholar 

  26. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  27. Huang, C.Y., Ferrell, J.E.: Ultrasensitivity in the mitogen-activated protein kinase cascade. Proc. Natl. Acad. Sci. U. S. A. 93(19), 10078–10083 (Sep 1996)

    Article  Google Scholar 

  28. Hutchison, C.A.: DNA sequencing: bench to bedside and beyond. Nucleic Acids Res. 35(18), 6227–6237 (2007)

    Article  Google Scholar 

  29. Kamvar, S., Haveliwala, T., Manning, C., Golub, G.: Exploiting the block structure of the web for computing pagerank. Technical report, Stanford University (2003)

  30. Langville, A.N., Meyer, C.D.: Google’s PageRank and Beyond: the Science of Search Engine Rankings. Princeton University Press, London (2006)

    MATH  Google Scholar 

  31. Lodish, H., Berk, A., Zipursky, L.S., Matsudaira, P., Baltimore, D., Darnell, J.: Molecular Cell Biology. W.H. Freeman, San Francisco (2008)

    Google Scholar 

  32. MacNamara, S., Bersani, A.M., Burrage, K., Sidje, R.B.: Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation. J. Chem. Phys. 129, 95–105 (2008)

    Google Scholar 

  33. MacNamara, S., Burrage, K., Sidje, R.B.: An improved finite state projection algorithm for the numerical solution of the chemical master equation with applications. In: Read, W., Roberts, A.J. (eds.) Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006, vol. 48 of ANZIAM J., pp. C413–C435. James Cook University, Townsville (October 2007)

    Google Scholar 

  34. MacNamara, S., Burrage, K., Sidje, R.B.: Multiscale modeling of chemical kinetics via the master equation. Multiscale Model. Simul. 6(4), 1146–1168 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia (2000)

    MATH  Google Scholar 

  36. Michaelis, L., Menten, M.L.: Die kinetik der invertinwirkung. Biochem. Z. 49, 333–369 (1913)

    Google Scholar 

  37. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20, 801–836 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  38. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, 25 years later. SIAM Rev. 45(1), 3–49 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  40. Nicolau, Jr. D.V., Burrage, K., Parton, R.G., Hancock, J.F.: Identifying optimal lipid raft characteristics required to promote nanoscale protein-protein interactions on the plasma membrane. Mol. Cell Biol. 26(1), 313–323 (2006)

    Article  Google Scholar 

  41. Ortega, J.M., Rheinbolt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, London (1970)

    MATH  Google Scholar 

  42. Pedersen, M.G., Bersani, A.M., Bersani, E.: The total quasi-steady-state approximation for fully competitive enzyme reactions. Bull. Math. Biol. 69(1), 433–457 (Jan 2007)

    Article  Google Scholar 

  43. Rao, C.V., Arkin, A.P.: Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J. Chem. Phys. 118, 4999–5010 (2003)

    Article  Google Scholar 

  44. Rathinam, M., Petzold, L.R., Cao, Y., Gillespie, D.T.: Stiffness in stochastic chemically reacting systems: the implicit tau-leaping method. J. Chem. Phys. 119, 12784–12794 (December 2003)

    Article  Google Scholar 

  45. Rubinow, S.I., Lebowitz, J.L.: Time-dependent Michaelis-Menten kinetics for an enzyme-substrate-inhibitor system. J. Am. Chem. Soc. 92, 3888–3893 (1970)

    Article  Google Scholar 

  46. Saad, Y.: Analysis of some krylov subspace approximaitons to the matrix exponential operator. Siam J. Numer. Anal. 29(1), 209–228 (February 1992)

    Article  MATH  MathSciNet  Google Scholar 

  47. Schnell, S., Maini, P.K.: A century of enzyme kinetics: Reliability of the k m and v max estimates. Comments Theor. Biol. 8, 169–187 (2003)

    Article  Google Scholar 

  48. Segel, L.A., Slemrod, M.: The quasi-steady-state assumption: a case study in perturbation. SIAM Rev. 31(3), 446–447 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  49. Sidje, R.B.: Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1), 130–156 (1998)

    Article  MATH  Google Scholar 

  50. Sidje, R.B.: Parallel algorithms for large sparse matrix exponentials: application to numerical transient analysis of Markov processes. PhD thesis, University of Rennes (1994)

  51. Sidje, R.B., Stewart, W.J.: A numerical study of large sparse matrix exponentials arising in Markov chains. Comput. Stat. Data Anal. 29, 345–368 (1999)

    Article  MATH  Google Scholar 

  52. Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)

    MATH  Google Scholar 

  53. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(2), 506–517 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  54. Strang, G.: Linear Algebra and Its Applications. Thomson, Brooks/Cole, Belmont (2006)

    Google Scholar 

  55. Straus, O.H., Goldstein, A.: Zone behaviour of enzymes. J. Gen. Physiol. 26, 559–585 (1943)

    Article  Google Scholar 

  56. Tian, T., Burrage, K.: Binomial leap methods for simulating stochastic chemical kinetics. J. Chem. Phys. 121(21), 10356–10364 (2004)

    Article  Google Scholar 

  57. Tzafriri, A.R.: Michaelis-menten kinetics at high enzyme concentrations. Bull. Math. Biol. 65(6), 1111–1129 (Nov 2003)

    Article  Google Scholar 

  58. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. Elsevier Science, Amsterdam (2001)

    MATH  Google Scholar 

  59. Voet, D., Voet, J.G.: Biochemistry. Wiley, New York (2004)

    Google Scholar 

  60. Wolinsky, H.: The thousand-dollar genome. Genetic brinkmanship or personalized medicine? EMBO Rep. 8(10), 900–903 (Oct 2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Australian Centre in Bioinformatics and The Institute for Molecular Biosciences, The University of Queensland, Brisbane, Queensland, 4072, Australia

    Shev MacNamara & Kevin Burrage

  2. Computing Laboratory and Oxford Centre for Integrative Systems Biology, University of Oxford, Oxford, UK

    Kevin Burrage

Authors
  1. Shev MacNamara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kevin Burrage
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kevin Burrage.

Rights and permissions

Reprints and permissions

About this article

Cite this article

MacNamara, S., Burrage, K. Krylov and steady-state techniques for the solution of the chemical master equation for the mitogen-activated protein kinase cascade. Numer Algor 51, 281–307 (2009). https://doi.org/10.1007/s11075-008-9239-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-008-9239-y

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation