Advertisement

Model Identifiability

  • Paola LeccaEmail author
Chapter
  • 46 Downloads
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

Once we have built a model to describe the dynamics of a network, in order to simulate this dynamic, that is, the evolution in the time of the network, we need to know the parameters of the model. Very often the values of the kinetic constants in a network of biochemical interactions, or more generally the arcs’ weights on the network define the force and direction of the interaction between nodes, are obtained from experimental data through various regression and inference techniques. In this chapter, we will tackle a problem that is upstream of the parameter estimation, that is, the possibility to infer them from the data. The problem is known as identifiability. Identifiability is a fundamental prerequisite for model identification. It concerns uniqueness of the model parameters determined from experimental observations. This paper specifically deals with structural or a priori identifiability: whether or not parameters can be identified from a given model structure and experimental measurements. Since experimental data are usually affected by uncertainties, this question is known as practical identifiability. Non-identifiability of parameters induces non-observability of trajectories, reducing the predictive power of the model. We will discuss here a method of parameter identifiability based on the observability rank test and how much it is suitable to handle noisy observations.

References

  1. 1.
    Chou I-C, Voit EO. Recent developments in parameter estimation and structure identification of biochemical and genomic systems. Math Biosci. 2009;219(2):57–83.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Goel G, Chou I-C, Voit EO. System estimation from metabolic time-series data. Bioinformatics. 2008;24(21):2505–11.CrossRefGoogle Scholar
  3. 3.
    Voit EO, Almeida J, Marino S, Lall R, Goel G, Neves AR, Santos H. Regulation of glycolysis in lactococcus lactis: an unfinisched systems biological case study. IEE Proc-Syst Biol. 2006;153(4):286–98.CrossRefGoogle Scholar
  4. 4.
    Polisetty PK, Voit EO. Identification of metabolic system parameters using global optimization methods. Theor Biol Med Model. 2006;3(4):1–15.Google Scholar
  5. 5.
    Rodrigez-Fernandez M, Mendes P, Banga J. A hybrid approach for efficient and robust parameter estimation in biochemical pathways. BioSystems. 2006;83:248–65.CrossRefGoogle Scholar
  6. 6.
    Moles GC, Mendes P, Banga JR. Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Res. 2003;13:2467–74.CrossRefGoogle Scholar
  7. 7.
    Tian T, Xu S, Burrage K. Simulated maximum likelihood method for estimating kinetic rates in gene expression. Bioinformatics. 2007;23(1):84–91.CrossRefGoogle Scholar
  8. 8.
    Chou IC, Martens H, Voit EO. Parameter estimation in biochemical systems models with alternating regression. Theor Biol Med Model. 2006;3:25.CrossRefGoogle Scholar
  9. 9.
    Sugimoto M, Kikuchi S, Tomita M. Reverse engineering of biochemical equations from time-course data by means of genetic programming. BioSystems. 2005;80:155–64.CrossRefGoogle Scholar
  10. 10.
    Reinker S, Altman RM, Timmer J. Parameter estimation in stochastic biochemical reactions. IEEE Proc Syst Biol. 2006;153:168–78.CrossRefGoogle Scholar
  11. 11.
    Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N. Copasi - a complex pathway simulator. Bioinformatics. 2006;22:3067–74.CrossRefGoogle Scholar
  12. 12.
    Zwolak JW, Tyson JJ, Watson LT. Estimating rate constants in cell cycle models. In: Tentner A, editor. Proceedings of high performance constants in cell cycle models, San Diego. 2001. p. 53–7.Google Scholar
  13. 13.
    Vyshemirsky V, Girolami MA. Biobayes: Bayesian inference for systems biology; 2008.Google Scholar
  14. 14.
    Vyshemirsky V, Girolami MA. Bayesian ranking of biochemical system models. Bioinformatics. 2008;24(6):833–9.CrossRefGoogle Scholar
  15. 15.
    Vyshemirsky V, Girolami MA. Biobayes: a software package for bayesian inference in systems biology. Bioinformatics. 2008;24(17):1933–4.CrossRefGoogle Scholar
  16. 16.
    Boys RJ, Wilkinson DJ, Kirkwood TB. Statistics and Computing., Bayesian inference for a discretely observed stochastic kinetic modelNetherlands: Springer; 2008.Google Scholar
  17. 17.
    Golightly A, Wilkinson DJ. Bayesian inference for nonlinear multivariate diffusion models observed with error computational statistics and data analysis. Computational Statistics and Data Analysis. 2008;52(3):1674–93.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wilkinson D. Stochastic Modelling for Systems Biology. Boca Raton: Chapman and Hall/CRC; 2006.zbMATHGoogle Scholar
  19. 19.
    Wilkinson DJ. Bayesian methods in bioinformatics and computational systems biology. Briefings in Bioinformatics. 2007;1(8):109–16. CrossRefGoogle Scholar
  20. 20.
    Lecca P, Palmisano A, Ihekwaba A, Priami C. Calibration of dynamic models of biological systems with kinfer. Eur Biophys J.Google Scholar
  21. 21.
    Geffen D, Findeise R, Schliemann M, Allgoever F, Guay M. Observability based paramter identifiability for biochemial reaction network. In: 2008 American control conference; 2008. p. 2130–34, June 11–13 2008.Google Scholar
  22. 22.
    Anguelova M. Nonlinear observability and identifiability: general theory and a case study of a kinetic model of s. cerevisiae; 2004.Google Scholar
  23. 23.
    Denis-Vidal L, Joly-Blanchard G, Noiret C. Some effective approaches to check the identifiability of uncontrolled nonlinear systems. Math Comput Simul. 2001;57(2):35–44.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Geffen D. Parameter identifiability of biochemical reaction networks in systems biology. PhD thesis, Department of Chemical Engineering of Queen’s University, Kingston, Ontario, Canada; 2008.Google Scholar
  25. 25.
    Ljung L, Glad T. On global identifiability for arbitrary model parametrization. Automica. 1994;30(2):265–76.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pohjanpalo H. System identifiability based on the power series expansion of the solution. Mathematical Biosciences. 1978;41:21–33.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Liu Y-Y, Slotine J-J, Barabasi A-L. Observability of complex systems. Proc Natl Acad Sci. 2013;110(7):2460–5.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Villaverde Alejandro F. Observability and structural identifiability of nonlinear biological systems. Complexity. 2019;2019:1–12.zbMATHGoogle Scholar
  29. 29.
    Stigter JD, Joubert D, Molenaar J. Observability of complex systems: finding the gap. Sci Rep. 2017;7(1).Google Scholar
  30. 30.
    Lin Wu, Li Min, Wang Jian-Xin, Fang-Xiang Wu. Controllability and its applications to biological networks. J Comput Sci Technol. 2019;34(1):16–34.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Anguelova M. Observability and identifiability of nonlinear systems with applications in biology. PhD thesis, Chalmers University of Technology and Gẗeborg University; 2007.Google Scholar
  32. 32.
    Lecca Paola, Re Angela. Identifying necessary and sufficient conditions for the observability of models of biochemical processes. Biophys Chem. 2019;254:106257.CrossRefGoogle Scholar
  33. 33.
    Aguirre LA, Portes LL, Letellier C. Structural, dynamical and symbolic observability: from dynamical systems to networks. PLOS ONE. 2018;13(10):e0206180.CrossRefGoogle Scholar
  34. 34.
    Hong wei Lou and Rong Yang. Necessary and sufficient conditions for distinguishability of linear control systems. Acta Math Appl Sinica Engl Ser. 2014;30(2):473–82.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Krener J. \((ad_{f, g})\), \((ad_{f, g})\) and locally \((ad_{f, g})\) invariant and controllability distributions. SIAM J Control Optim. 1985;23(4):523–49.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kawano Yu, Ohtsuka Toshiyuki. Observability analysis of nonlinear systems using pseudo-linear transformation. IFAC Proc Vol. 2013;46(23):606–11.CrossRefGoogle Scholar
  37. 37.
    Halás M, Kawano Y, Moog CH, Ohtsuka T. Realization of a nonlinear system in the feedforward form: a polynomial approach. IFAC Proc Vol. 2014;47(3):9480–5.CrossRefGoogle Scholar
  38. 38.
    Stephanopoulos Gregory. Metabolic fluxes and metabolic engineering. Metab Eng. 1999;1(1):1–11.CrossRefGoogle Scholar
  39. 39.
    Gaspard PP. Rössler systems.Google Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Faculty of Computer ScienceFree University of Bozen-BolzanoBozen-BolzanoItaly

Personalised recommendations