Home Piecewise synergetic systems and applications in biochemical systems theory
Article
Licensed
Unlicensed Requires Authentication

Piecewise synergetic systems and applications in biochemical systems theory

  • Arcady Ponosov EMAIL logo , Anna Machina and Valeria Tafintseva
Published/Copyright: December 14, 2016
Become an author with De Gruyter Brill

Abstract

We study piecewise synergetic systems originating from Biochemical Systems Theory. In the first part of the paper, the emphasis is put on practical calculations with such systems. We consider four examples: calculation of trajectories and steady states, solution of an optimization problem and a method of estimation of parameters (kinetic orders), all examples being biologically motivated. In the second part of the paper, we study convergence of solutions, in particularly, steady states, of a sequence of piecewise synergetic systems approximating an arbitrary compartment model. This convergence analysis is then applied to the optimization problem and the method of estimating sensitivities (kinetic orders) in a generic compartment model. In this paper we put forward arguments for the importance of the theoretical and numerical analysis of piecewise synergetic systems.


Dedicated to Professor Ivan Kiguradze on the occasion of his 80th birthday


Award Identifier / Grant number: EEA

Funding source: Norges Forskningsråd

Award Identifier / Grant number: 239070

Funding statement: The work of the first author was partially supported by a EEA grant coordinated by Universidad Complutense de Madrid, Spain, and by the grant #239070 of the Norwegian Research Council.

References

[1] Chou I.-C., Martens H. and Voit E. O., Parameter estimation in biochemical systems models with alternating regression, Theoret. Biol. Med. Modell. 3 (2006), Paper No. 25. 10.1186/1742-4682-3-25Search in Google Scholar PubMed PubMed Central

[2] Chou I.-C. and Voit E. O., Recent developments in parameter estimation and structure identification of biochemical and genomic systems, Math. Biosci. 219 (2009), no. 2, 57–83. 10.1016/j.mbs.2009.03.002Search in Google Scholar PubMed PubMed Central

[3] Coelho P. M. B. M., Salvador A. and Savageau M. A., Relating mutant genotype to phenotype via quantitative behavior of the NADPH redox cycle in human Erythrocytes, PloS ONE 5 (2010), no. 9, Article ID e13031. 10.1371/journal.pone.0013031Search in Google Scholar PubMed PubMed Central

[4] de Jong H., Modeling and simulation of genetic regulatory systems: A literature review, J. Comput. Biol. 9 (2002), 67–103. 10.1089/10665270252833208Search in Google Scholar PubMed

[5] Ferrari-Trecate G. and Muselli M., A new learning method for piecewise linear regression, Artificial Neural Networks (ICAAN 2002), Lecture Notes in Comput. Sci. 2415, Springer, Berlin (2002), 444–449. 10.1007/3-540-46084-5_72Search in Google Scholar

[6] Filippov A. F., Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic Publishers, Dordrecht, 1988. 10.1007/978-94-015-7793-9Search in Google Scholar

[7] Floudas C. A., Deterministic Global Optimization. Theory, Methods and Applications, Nonconvex Optim. Appl. 37, Kluwer Academic Publishers, Dordrecht, 2000. 10.1007/978-1-4757-4949-6Search in Google Scholar

[8] Guillén-Gosálbez G. and Sorribas A., Identifying quantitative operation principles in metabolic pathways: A systematic method for searching feasible enzyme activity patterns leading to cellular adaptive responses, BMC Bioinformatics 10 (2009), no. 1, Article ID 386. 10.1186/1471-2105-10-386Search in Google Scholar PubMed PubMed Central

[9] Machina A., Ponosov A. and Voit E. O., Automated piecewise power-law modeling of biological systems, J. Biotechnol. 149 (2010), no. 3, 154–165. 10.1016/j.jbiotec.2009.12.016Search in Google Scholar PubMed

[10] Plahte E. and Kjøglum S., Analysis and generic properties of gene regulatory networks with graded response functions, Phys. D 201 (2005), no. 1–2, 150–176. 10.1016/j.physd.2004.11.014Search in Google Scholar

[11] Plahte E., Mestl T. and Omholt S. W., A methodological basis for description and analysis of systems with complex switch-like interactions, J. Math. Biol. 36 (1998), no. 4, 321–348. 10.1007/s002850050103Search in Google Scholar

[12] Ponosov A., Machina A. and Tafintseva V., Convergence properties of piecewise power approximations, Appl. Math. 7 (2016), no. 13, 1440–1445. 10.4236/am.2016.713124Search in Google Scholar

[13] Savageau M. A., Biochemical systems analysis. I: Some mathematical properties of the rate law for the component enzymatic reactions, J. Theor. Biol. 25 (1969), no. 3, 365–369. 10.1016/S0022-5193(69)80026-3Search in Google Scholar

[14] 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 (1969), no. 3, 370–379. 10.1016/S0022-5193(69)80027-5Search in Google Scholar

[15] Savageau M. A., Biochemical systems analysis. III: Dynamic solutions using a power-law approximation, J. Theor. Biol. 26 (1970), no. 2, 215–226.10.1016/S0022-5193(70)80013-3Search in Google Scholar

[16] Savageau M. A., Alternative designs for a genetic switch: Analysis of switching times using the piecewise power-law representation, Math. Biosci. 180 (2002), 237–253. 10.1016/S0025-5564(02)00113-XSearch in Google Scholar

[17] Sorribas A., Pozo C., Vilaprinyo E., Guillén-Gosálbez G., Jimenez L. and Alves R., Optimization and evolution in metabolic pathways: Global optimization techniques in generalized mass action models, J. Biotechnology 149 (2010), no. 3, 141–153. 10.1016/j.jbiotec.2010.01.026Search in Google Scholar

[18] Sorribas A. and Savageau M. A., A comparison of variant theories of intact biochemical systems. I: Enzyme-enzyme interactions and biochemical systems theory, Math. Biosci. 94 (1989), no. 2, 161–193. 10.1016/0025-5564(89)90064-3Search in Google Scholar

[19] Sorribas A. and Savageau M. A., A comparison of variant theories of intact biochemical systems. II: Flux-oriented and metabolic control theories, Math. Biosci. 94 (1989), no. 2, 195–238. 10.1016/0025-5564(89)90065-5Search in Google Scholar

[20] Sorribas A. and Savageau M. A., Strategies for representing metabolic pathways within biochemical systems theory: Reversible pathways, Math. Biosci. 94 (1989), no. 2, 239–269. 10.1016/0025-5564(89)90066-7Search in Google Scholar

[21] Tafintseva V., Machina A. and Ponosov A., Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks, Nonlinear Anal. Real World Appl. 14 (2013), no. 3, 1732–1754. 10.1016/j.nonrwa.2012.11.008Search in Google Scholar

[22] Tafintseva V., Tøndel K., Ponosov A. and Martens H., Global structure of sloppiness in a nonlinear model, J. Chemometrics 28 (2014), no. 8, 645–655. 10.1002/cem.2651Search in Google Scholar

[23] Voit E. O., Canonical Nonlinear Modeling. S-System Approach to Understanding Complexity, Van Nostrand Reinhold, New York, 1991. Search in Google Scholar

[24] Voit E. O., Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists, Cambridge University Press, Cambridge, 2000. Search in Google Scholar

Received: 2016-9-22
Revised: 2016-9-29
Accepted: 2016-10-2
Published Online: 2016-12-14
Published in Print: 2017-3-1

© 2017 by De Gruyter

Downloaded on 24.7.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2016-0065/html
Scroll to top button