Skip to main content
SpringerLink
Log in

\(H_\infty \) Feedback Control Theory in Biochemical Systems

  • Chapter
  • First Online:
Advances in H∞ Control Theory

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 481))

  • 489 Accesses

Abstract

A new approach is presented for the study of \(H_\infty \) optimal control of biochemical pathways. Starting with various linear models of single enzymatic reaction systems, a simple unbranched four-block enzyme system that contains a negative feedback loop is analyzed in the open- and the closed-loop configurations, where it is shown that the closed-loop configuration is one of the static output-feedback control systems. The original nonlinear four-block system is modeled as a polytopic-type uncertain linear system, where the extent of the nonlinearity can be “tuned” by a fictitious uncertainty interval, thus better capturing the nonlinear nature of the systems under study. The sensitivity of the latter enzymatic system to variations in certain variables is explored via the optimal \(H_\infty \) control approach. Based on this approach, the threonine synthesis pathway that contains three negative feedback loops is analyzed and is shown to be optimal in the \(H_\infty \) sense.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover 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. Gershon, E., Hiler, R., Shaked, U.: Classical control theory approach to enzymatic reactions. In: Proceedings of the European Control Conference (ECC03), Cambridge, England (2003)

    Google Scholar 

  2. Gershon, E., Shaked, U.: \(H_\infty \) feedback control of biochemical pathways via system uncertainty. In: Proceedings of the 5rd IFAC Symposium on Robust Control Design (ROCOND), Toulouse, France, July 2006

    Google Scholar 

  3. Gershon, E., Shaked, U.: \(H_\infty \) feedback-control theory in biochemical systems. Int. J. Robust Nonlinear Control 18, 18–50 (2008)

    Article  MathSciNet  Google Scholar 

  4. Gershon, E., Yokev, O., Shaked, U.: \(H_\infty \) feedback-control of the Threonine Synthesis pathway via system uncertainty. In: Proceedings of the European Control Conference (ECC07), Kos, Greece, June 2007

    Google Scholar 

  5. Gershon, E., Shaked, U.: Robust polytopic analysis of the feedback-control of Glycolysis in Yeasts via some system norms. In: Proceedings of the 20th Mediterranean Conference on Control and Automation (MED12), Barcelona, Spain, July 2012

    Google Scholar 

  6. Gershon, E., Navon, M.: Robust feedback-control analysis of the Threonine Synthesis Pathway via various system norms. In: Proceedings of the 22nd Mediterranean Conference on Control and Automation (MED14), Palermo, Sicily, June 2014

    Google Scholar 

  7. Gershon, E., Navon, M., Shaked, U.: Robust peak-to-peak and \(H_\infty \) static output-feedback control of the Threonine Synthesis Pathway. In: Proceedings of the European Control Conference (ECC15), Linz, Austria, July 2015

    Google Scholar 

  8. Lehninger, A.L.: Principles of Biochemistry. Worth Publishers (1982)

    Google Scholar 

  9. Segel, I.R.: Enzyme Kinetics—Behavior and Analysis of Rapid Equilibrium and Steady-state Enzyme Systems. Wiley (1975)

    Google Scholar 

  10. Wilkinson, F.: Chemical Kinetics and Reaction Mechanisms. Van Nostrand Reinhold Company (1980)

    Google Scholar 

  11. Chassagnole, C., Rais, B., Quentin, E., Fell, D., Mazat, J.P.: An integrated study of Threonine-pathway enzyme kinetics in Echerichia coli. Biochem. J. 356, 415–423 (2001)

    Article  Google Scholar 

  12. Chassagnole, C., Rais, B., Quentin, E., Fell, D., Mazat, J.P.: Threonine synthesis from aspartate in Echerichia coli cell-free extract: pathway dynamics. Biochem. J. 356, 425–432 (2001)

    Article  Google Scholar 

  13. Chassagnole, C., Rais, B., Quentin, E., Fell, D., Mazat, J.P.: Control of Threonine-synthesis pathway in Echerichia coli: theoretical and experimental approach. Biochem. J. 356, 433–444 (2001)

    Article  Google Scholar 

  14. Franklin, G.F., Fowell, J.D., Emami-Naeini, A.: Feedback Control of Dynamic Sysytems, 2nd edn. Addison-Wiesley, New York (1991)

    Google Scholar 

  15. Atkinson, D.E.: Enzymes as control elements in metabolic regulation. In: Boyer, P. (ed.) The Enzymes, vol. 1, 3rd edn., p. 461. Academic Press (1970)

    Google Scholar 

  16. Ellison, W.R., Lueck, J.D., Fromm, H.J.: Studies on the mechanism of orthophosphate regulation of Bovine Brain Hexokinase. J. Biol. Chem. 250(5), 1864–1871 (1975)

    Google Scholar 

  17. Ardehali, H., Printz, R.L., Whitesell, R.R., May, J.M., Granner, D.K.: Functional interaction between the N- and C-terminal halves of Human Hexokinase II. J. Biol. Chem. 274, 15986–15989 (1999)

    Article  Google Scholar 

  18. Aleshin, A.E., Zeng, C., Bourenkov, G.P., Bartunik, H.D., Fromm, H.J., Honzatko, R.B.: The mechanism of regulation of Hexokinase: new insight from the crystal structure of recombinant human brain hexokinase comlexed with glucose and glucose-6-phospahte. Structure 6(1), 39–50 (1998)

    Article  Google Scholar 

  19. Aleshin, A.E., Kirby, C., Xiaofeng, L., Bourenkov, G.P., Bartunik, H.D., Fromm, H.J., Honzatko, R.B.: Crystal structures of motant monomeric Hexokinase I reveal multiple ADP binding sites and conformational changes relevant to allosteric regulation. J. Mol. Biol. 296, 1001–1015 (2000)

    Article  Google Scholar 

  20. Liu, X., Sup Kim, C., Kurbanov, F.T., Honzatko, R.B., Fromm, H.J.: Dual mechanism for Glucose 6-Phosphate inhibition of human Hexokinase. J. Biol. Chem. 274, 31155–31159 (1999)

    Article  Google Scholar 

  21. de Cerqueira Cesar, M., Wilson, J.E.: Functional characteristics of hexokinase bound to the type a and type B sites of bovine brain mitochondria. Arch. Biochem. Biophys. 397(1), 106–112 (2002)

    Google Scholar 

  22. Arora, K.K., Filburn, C.R., Pederson, P.I.: Structure/function relationships in hexokinase. Site-directed mutational analyses and characterization of overexpressed fragments implicate different functions for the N- and C-terminal halves of the enzyme. J. Biol. Chem. 268, 18259–18266 (1993)

    Google Scholar 

  23. Wolkenhauer, O., Ghosh, B.K., Cho, K.H.: Control and coordination in biochemical networks. IEEE Control Syst. 24(4), 30–34 (2004)

    Article  Google Scholar 

  24. Fell, D.: Metabolic control analysis: a survey of it’s theoretical and experimental development. Biochem. J. 286, 313–330 (1992)

    Article  Google Scholar 

  25. Fell, D.: Frontiers in Metabolism: Understanding the Control of Metabolism. Portland Press (1997)

    Google Scholar 

  26. Voit, E.O.: Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists. Cambridge University Press, Cambridge, UK (2000)

    Google Scholar 

  27. Goel, G.: Reconstructing Biochemical Systems. Systems Modeling and Analysis Tools for Decoding Biological Designs. VDM Verlag Dr. Mller, Saarbrcken, Germany (2008)

    Google Scholar 

  28. Torres, N.V., Voit, E.O.: Pathway Analysis and Optimization in Metabolic Engineering. Cambridge University Press, Cambridge, UK (2005)

    Google Scholar 

  29. Voit, E.O.: Biochemical systems theory: a review. ISRN Biomath. 2013(Article ID 897658), 53 (2013). https://doi.org/10.1155/2013/897658

    Article  Google Scholar 

  30. Baker, G.A., Graves-Morris, P.: Pade Approximation. Cambridge University Press (2009)

    Google Scholar 

  31. Xie, L., Fu, M., de Souza, C.E.: \(H_\infty \) control and quadratic stabilization of systems with parameter uncertainty via output feedback. IEEE Trans. Autom. Control 37, 1253–1256 (1992)

    Article  MathSciNet  Google Scholar 

  32. Alves, R., Savageau, M.A.: Effect of overall feedback inhibition in unbranched biosynthetic pathways. Biophys. J. 79, 2290–2304 (2000)

    Article  Google Scholar 

  33. Schuster, S., Heinrich, R.: Minimization of intermediate concentrations as a suggested optimality principle for biochemical networks, I. Theoretical analysis. J. Math. Biol. 29, 425–442 (1991)

    Article  Google Scholar 

  34. Savinell, J.M., Palsson, B.O.: Network analysis of intermediary metabolizm using linear optimization, I. Developmenmt of mathematical formalism. J. Theor. Biol. 154, 421–454 (1992)

    Article  Google Scholar 

  35. Mends, P., Kell, D.B.: Non-linear optimaization of biochemical pathways: applications to metabolic engineering and parameter estimation. Bioinform. 14(10), 869–883 (1998)

    Google Scholar 

  36. Morohashi, M., Winn, A.E., Borisuk, M.T., Bolouri, H., Doyle, L., Kitano, H.: Robustness as a measure of plausibilty in models of biochemical netwroks. J. Theor. Biol. 216, 19–30 (2002)

    Article  Google Scholar 

  37. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  38. Petersen, I.R.: Quadratic stabilizability of uncertain linear systems containing both constant and time varying uncertain parameters. J. Optim. Theory Appl. 57(3), 439–461 (1988)

    Article  MathSciNet  Google Scholar 

  39. Savageau, M.A.: Biochemical systems analysis. I. Some mathematical properties of the rate law for the component enzymatic reactions. J. Theor. Biol. 25, 365–369 (1969)

    Article  Google Scholar 

  40. Savageau, M.A.: Biochemical systems analysis. II. The steady-state solutions for an n-pool system using a power-law approximation. J. Theor. Biol. 25, 370–379 (1969)

    Article  Google Scholar 

  41. Savageau, M.A.: Biochemical systems analysis. 3. Dynamic solutions using a power-law approximation. J. Theor. Biol. 26, 215–226 (1970)

    Article  Google Scholar 

  42. Savageau, M.: Biochemical Systems Analysis: A Study of Function and Design in Molecular Biology. Addison Wesley, Reading, MA (1976)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, Holon Institute of Technology, Holon, Israel

    Eli Gershon

  2. School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel

    Uri Shaked

Authors
  1. Eli Gershon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uri Shaked
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eli Gershon .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gershon, E., Shaked, U. (2019). \(H_\infty \) Feedback Control Theory in Biochemical Systems. In: Advances in H∞ Control Theory. Lecture Notes in Control and Information Sciences, vol 481. Springer, Cham. https://doi.org/10.1007/978-3-030-16008-1_16

Download citation

Publish with us

Policies and ethics

Search

Navigation