Hamiltonian representation of generalized ribosome flow models

Authors

Mihaly A. Vaghy, Gabor Szederkenyi

Abstract

In this paper we study a class of compartmental models with bounded capacities, called generalized ribosome flow models and show that they are formally kinetic, i.e. we can assign a chemical reaction network (CRN) to the system based on the compartmental structure which realizes the nonlinear dynamics. We decompose the model into two dual subsystems both having positive linear first integrals representing conservation. Based on the dynamics of the reaction network we construct a port-Hamiltonian representation of the system in the original and also in the reduced state spaces with clear connection between the structure matrices and the compartmental graph topology. We finally demonstrate that the system is non-expansive in the -norm both in the original and in the reduced state spaces. The generality of our approach ensures that the results are valid for a wide class of reaction rate functions used in the CRN representation.

Citation

• Journal: 2022 European Control Conference (ECC)
• Year: 2022
• Volume:
• Issue:
• Pages: 657–662
• Publisher: IEEE
• DOI: 10.23919/ecc55457.2022.9838067

BibTeX

@inproceedings{Vaghy_2022,
  title={{Hamiltonian representation of generalized ribosome flow models}},
  DOI={10.23919/ecc55457.2022.9838067},
  booktitle={{2022 European Control Conference (ECC)}},
  publisher={IEEE},
  author={Vaghy, Mihaly A. and Szederkenyi, Gabor},
  year={2022},
  pages={657--662}
}

Download the bib file

References

• Raveh, A., Zarai, Y., Margaliot, M. & Tuller, T. Ribosome Flow Model on a Ring. IEEE/ACM Trans. Comput. Biol. and Bioinf. 12, 1429–1439 (2015) – 10.1109/tcbb.2015.2418782
• Russo, G., di Bernardo, M. & Sontag, E. D. Global Entrainment of Transcriptional Systems to Periodic Inputs. PLoS Comput Biol 6, e1000739 (2010) – 10.1371/journal.pcbi.1000739
• lipták, Traffic reaction model. ArXiv Preprint (2021)
• Du, Q., Huang, Z. & LeFloch, P. G. Nonlocal Conservation Laws. A New Class of Monotonicity-Preserving Models. SIAM J. Numer. Anal. 55, 2465–2489 (2017) – 10.1137/16m1105372
• Kessels, F. Traffic Flow Modelling. EURO Advanced Tutorials on Operational Research (Springer International Publishing, 2019). doi:10.1007/978-3-319-78695-7 – 10.1007/978-3-319-78695-7
• lighthill, On Kinematic Waves. II. A Theory of Traffic Flow on Long Crowded Roads. Proceedings of the Royal Society of London A Mathematical and Physical Sciences (0)
• Richards, P. I. Shock Waves on the Highway. Operations Research 4, 42–51 (1956) – 10.1287/opre.4.1.42
• erdi, Mathematical Models of Chemical Reactions Theory and Applications of Deterministic and Stochastic Models (1989)
• Anderson, D. F., Brunner, J. D., Craciun, G. & Johnston, M. D. On classes of reaction networks and their associated polynomial dynamical systems. J Math Chem 58, 1895–1925 (2020) – 10.1007/s10910-020-01148-9
• Angeli, D. A Tutorial on Chemical Reaction Network Dynamics. European Journal of Control 15, 398–406 (2009) – 10.3166/ejc.15.398-406
• Feinberg, M. Foundations of Chemical Reaction Network Theory. Applied Mathematical Sciences (Springer International Publishing, 2019). doi:10.1007/978-3-030-03858-8 – 10.1007/978-3-030-03858-8
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• smith, Monotone Dynamical Systems An Introduction to the Theory of Competitive and Cooperative Systems Ser Mathematical Surveys and Monographs (1995)
• Brown, R. F. Compartmental System Analysis: State of the Art. IEEE Trans. Biomed. Eng. BME-27, 1–11 (1980) – 10.1109/tbme.1980.326685
• nijmeijer, Nonlinear Dynamic Control Systems (2016)
• Jacquez, J. A. & Simon, C. P. Qualitative Theory of Compartmental Systems. SIAM Rev. 35, 43–79 (1993) – 10.1137/1035003
• Cobelli, C. & Romanin-Jacur, G. On the structural identifiability of biological compartmental systems in a general input-output configuration. Mathematical Biosciences 30, 139–151 (1976) – 10.1016/0025-5564(76)90021-3
• vidyasagar, Nonlinear Systems Analysis (1978)
• Maeda, H. & Kodama, S. Some results on nonlinear compartmental systems. IEEE Trans. Circuits Syst. 26, 203–204 (1979) – 10.1109/tcs.1979.1084618
• Margaliot, M. & Tuller, T. Stability Analysis of the Ribosome Flow Model. IEEE/ACM Trans. Comput. Biol. and Bioinf. 9, 1545–1552 (2012) – 10.1109/tcbb.2012.88
• Zazworsky, R. & Knudsen, H. Controllability and observability of linear time-invariant compartmental models. IEEE Trans. Automat. Contr. 23, 872–877 (1978) – 10.1109/tac.1978.1101853
• Brauer, F. Compartmental Models in Epidemiology. Lecture Notes in Mathematics 19–79 (2008) doi:10.1007/978-3-540-78911-6_2 – 10.1007/978-3-540-78911-6_2
• Bar-Shalom, E., Ovseevich, A. & Margaliot, M. Ribosome Flow Model with Different Site Sizes. SIAM J. Appl. Dyn. Syst. 19, 541–576 (2020) – 10.1137/19m1250571
• Haddad, W. M., Chellaboina, V. & Hui, Q. Nonnegative and Compartmental Dynamical Systems. (Princeton University Press, 2010). doi:10.1515/9781400832248 – 10.1515/9781400832248
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. FnT in Systems and Control 1, 173–378 (2014) – 10.1561/2600000002
• Wang, L., Maschke, B. & van der Schaft, A. Port-Hamiltonian modeling of non-isothermal chemical reaction networks. J Math Chem 56, 1707–1727 (2018) – 10.1007/s10910-018-0882-9
• Otero-Muras, I., Szederkényi, G., Alonso, A. A. & Hangos, K. M. Local dissipative Hamiltonian description of reversible reaction networks. Systems & Control Letters 57, 554–560 (2008) – 10.1016/j.sysconle.2007.12.003
• Haddad, W. M. & Chellaboina, V. Stability and dissipativity theory for nonnegative dynamical systems: a unified analysis framework for biological and physiological systems. Nonlinear Analysis: Real World Applications 6, 35–65 (2005) – 10.1016/j.nonrwa.2004.01.006
• Ortega, R., Astolfi, A., Bastin, G. & Rodriguez, H. Stabilization of food-chain systems using a port-controlled Hamiltonian description. Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334) (2000) doi:10.1109/acc.2000.878579 – 10.1109/acc.2000.878579
• Craciun, G. et al. Realizations of kinetic differential equations. Mathematical Biosciences and Engineering 17, 862–892 (2020) – 10.3934/mbe.2020046
• Ballesteros, A., Blasco, A. & Gutierrez-Sagredo, I. Hamiltonian structure of compartmental epidemiological models. Physica D: Nonlinear Phenomena 413, 132656 (2020) – 10.1016/j.physd.2020.132656