Structural Properties of an S-system Model of Mycobacterium tuberculosis Gene Regulation

Honeylou F. Farinas1,2, Eduardo R. Mendoza2,3,4,5, and Angelyn R. Lao2*

1Department of Mathematics, Mariano Marcos State University
Ilocos Norte 2906 Philippines
2Mathematics and Statistics Department, De La Salle University
Metro Manila 1004 Philippines
3Institute of Mathematical Sciences and Physics, University of the Philippines
Los Banos, Laguna 4031 Philippines
4Max Planck Institute of Biochemistry
Martinsried near Munich, Germany
5Faculty of Physics, Ludwig Maximilian University
Munich 80539 Germany

*Corresponding Author: This email address is being protected from spambots. You need JavaScript enabled to view it.



Magombedze and Mulder (2013) studied the gene regulatory system of Mycobacterium tuberculosis (Mtb) by partitioning this into three subsystems based on putative gene function and role in dormancy/latency development. Each subsystem, in the form of S-system, is represented by an embedded chemical reaction network (CRN), defined by a species subset and a reaction subset induced by the set of digraph vertices of the subsystem. For the embedded networks of S-system, we showed interesting structural properties and proved that all S-system CRNs (with at least two species) are discordant. Analyzing the subsystems as subnetworks, where arcs between vertices belonging to different subsystems are retained, we formed a digraph homomorphism from the corresponding subnetworks to the embedded networks. Lastly, we explored the modularity concept of CRN in the context of the digraph.




ARCEO CPP, JOSE EC, MARIN-SANGUINO A, MENDOZA ER. 2015. Chemical reaction network approaches to biochemical systems theory. Math Biosci 269: 135–152.
ARCEO CPP, JOSE EC, LAO AR, MENDOZA ER. 2017. Reaction networks and kinetics of biochemical systems. Math Biosci 283:13–29.
ARCEO CPP, JOSE EC, LAO AR, MENDOZA ER. 2018. Reactant subspaces and kinetics of chemical reaction networks. J Math Chem 56(2): 395–422.
BOROS B. 2013. On the positive steady states of Deficiency-One Mass Action Systems. [Ph.D. Thesis]. Budapest, Hungary: Eotvos Lorand University.DONNELL P, BANAJI M, MARGINEAN A, PANTEA C. 2014. CoNtRol: an open source framework for the analysis of chemical reaction networks. Bioinformatics 30(11): 1633–1634.
ELLISON P. 1998. The advanced deficiency algorithm and its applications to mechanism discrimination [Ph.D. Thesis]. University of Rochester.
FEINBERG M. 2019. Foundations of chemical reaction network theory. New York: Springer International Publishing.
FEINBERG M. 1987. Chemical reaction network structure and the stability of complex isothermal reactors: I. The deficiency zero and deficiency one theorems. Chem Eng Sci 42: 2229–2268.
FELIU E, WIUF C. 2012. Preclusion of switch behaviour in networks with mass action kinetics. Appl Math Comput 219: 1449–1467.
FORTUN NT, MENDOZA ER, RAZON LF, LAO AR. 2019. A Deficiency Zero Theorem for a class of power law kinetic systems with non-reactant determined interactions. MATCH Commun Math Comput Chem 81: 621–638.
GUIRADO E, SCHLESINGER L. 2013. Modeling the Mycobacterium tuberculosis granuloma–the critical battlefield in host immunity and disease. Frontiers in Immunology 4: 98.
HORN F, JACKSON R. 1972. General mass action kinetics. Arch Ration Mech Anal 47(2): 81–116.
IBARGÜEN-MONDRAGÓN E, ESTEVA L, CHÁVEZ-GALÁN L. 2011. A mathematical model for cellular immunology of tuberculosis. Math Biosci Eng 8(4): 973.
IBARGÜEN-MONDRAGÓN E, ESTEVA L. 2013. On the interactions of sensitive and resistant Mycobacterium tuberculosis to antibiotics. Math Biosci 246(1): 84–93.
JOSHI B, SHIU A. 2012. Simplifying the Jacobian Criterion for precluding multistationarity in chemical reaction networks. SIAM J Appl Math 72: 857–876.
JOSHI B, SHIU A. 2013. Atoms of multistationarity in chemical reaction networks. J Math Chem 51: 153–178.
JOSHI B, SHIU A. 2015. A survey of methods for deciding whether a reaction network is multistationary. Math Model Nat Phenom 10(5): 47–67.
KIM PMS. 2003. Understanding Subsystems in Biology through Dimensionality Reduction, Graph Partitioning and Analytical Modeling.
KIRSCHNER D, PIENAAR E, MARINO S, LINDERMAN J. 2017. A review of computational and mathematical modeling contributions within-host infection and treatment. Current Opinion in Systems Biology 3: 170–185.
LI W, SCHUURMANS D. 2011. Modular Community Detection in Networks. IJCAI Proceedings International Jt Conf Arti Intell. p. 1366–1371.
LORENZ DM, JENG A, DEEM MW. 2011. The emergence of modularity in biological systems. Phys Life Rev 8: 129–160.
MAGOMBEDZE G, MULDER N. 2013. Understanding TB latency using computational and dynamic modelling procedures. Infect Genet Evol 13: 267–283.
PEDRUZZI G, RAO KV, CHATTERJEE S. 2015. Mathematical model of mycobacterium–host interaction describes physiology of persistence. J Theor Bio 376: 105–117.
SHI R, LI Y, TANG S. 2014. A mathematical model with optimal controls for cellular immunology of tuberculosis. Taiwanese J Math 18(2): 575–597.
SHINAR G, FEINBERG M. 2012. Concordant chemical reaction networks. Math Biosci 240: 92–113.
VOIT EO. 2000. Computational Analysis of Biochemical Systems. Cambridge Univ Press.
[WHO] World Health Organization. 2019. Global Tuberculosis Report 2019. Retrieved from
YOUNG D, STARK J, KIRSCHNER D. 2008. Systems Biology of Persistent Infection: Tuberculosis as a Case Study. Nat Rev Microbiol 6: 520–528.