Modelling Gene Regulatory Networks Using Galerkin Techniques Based on State Space Aggregation and Sparse Grids | SpringerLink
Skip to main content

Modelling Gene Regulatory Networks Using Galerkin Techniques Based on State Space Aggregation and Sparse Grids

  • Conference paper
Modeling, Simulation and Optimization of Complex Processes

Abstract

An important driver of the dynamics of gene regulatory networks is noise generated by transcription and translation processes involving genes and their products. As relatively small numbers of copies of each substrate are involved, such systems are best described by stochastic models. With these models, the stochastic master equations, one can follow the time development of the probability distributions for the states defined by the vectors of copy numbers of each substance. Challenges are posed by the large discrete state spaces, and are mainly due to high dimensionality.

In order to address this challenge we propose effective approximation techniques, and, in particular, numerical techniques to solve the master equations. Two theoretical results show that the numerical methods are optimal. The techniques are combined with sparse grids to give an effective method to solve high-dimensional problems.

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

Access this chapter

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 17159
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 21449
Price includes VAT (Japan)
  • 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. A. Arkin, J. Ross, and H. H. McAdams. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected escherichia coli. Genetics, 149:1633–1648, 1998.

    Google Scholar 

  2. D. Braess. Finite Elements. Cambridge, second edition, 2005.

    Google Scholar 

  3. G. Dahlquist. Stability and error bounds in the numerical integration of ordinary differential equations. Kungl. Tekn. Högsk. Handl. Stockholm. No., 130:87, 1959.

    MathSciNet  Google Scholar 

  4. L. Ferm and P. Lötstedt. Numerical method for coupling the macro and meso scales in stochastic chemical kinetics. Technical Report 2006-001, Uppsala University, January 2006.

    Google Scholar 

  5. J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing, 67(3):225–253, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. T. Gillespie. Markov Processes: an introduction for physical scientists. Academic Press, San Diego, USA, 1992.

    MATH  Google Scholar 

  7. M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In Iterative methods in linear algebra (Brussels, 1991), pages 263–281. North-Holland, Amsterdam, 1992.

    Google Scholar 

  8. M. Hegland. Adaptive sparse grids. ANZIAM J., 44((C)):C335–C353, 2002.

    Google Scholar 

  9. M. Hegland. Additive sparse grid fitting. In Curve and surface fitting (Saint-Malo, 2002), Mod. Methods Math., pages 209–218. Nashboro Press, Brentwood, TN, 2003.

    Google Scholar 

  10. M. Hegland, C. Burden, L. Santoso, S. MacNamara, and H. Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comp. Appl. Math., 205:708–724, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original.

    MATH  Google Scholar 

  12. B. Lewin. Genes VIII. Pearson Prentice Hall, 2004.

    Google Scholar 

  13. C. Pflaum and A. Zhou. Error analysis of the combination technique. Numer. Math., 84(2):327–350, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Ptashne and A. Gann. Genes and Signals. Cold Spring Harbor Laboratory Press, 2002.

    Google Scholar 

  15. M. A. Shea and G. K. Ackers. The o r control system of bacteriophage lambda, a physical-chemical model for gene regulation. Journal of Molecular Biology, 181:211–230, 1985.

    Article  Google Scholar 

  16. T. Ström. On logarithmic norms. SIAM J. Numer. Anal., 12(5):741–753, 1975.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hegland, M., Burden, C., Santoso, L. (2008). Modelling Gene Regulatory Networks Using Galerkin Techniques Based on State Space Aggregation and Sparse Grids. In: Bock, H.G., Kostina, E., Phu, H.X., Rannacher, R. (eds) Modeling, Simulation and Optimization of Complex Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79409-7_17

Download citation

Publish with us

Policies and ethics