{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T19:18:39Z","timestamp":1786130319323,"version":"3.56.0"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T00:00:00Z","timestamp":1678665600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T00:00:00Z","timestamp":1678665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Northeastern University USA"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Control Signals Syst."],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Previous studies have inferred robust stability of reaction networks by utilizing linear programs or iterative algorithms. Such algorithms become tedious or computationally infeasible for large networks. In addition, they operate like black boxes without offering intuition for the structures that are necessary to maintain stability. In this work, we provide several graphical criteria for constructing robust stability certificates, checking robust non-degeneracy, verifying persistence, and establishing global stability. By characterizing a set of stability-preserving graph modifications that includes the enzymatic modification motif, we show that the stability of arbitrarily large nonlinear networks can be examined by simple visual inspection. We show applications of this technique to ubiquitous motifs in systems biology such as post-translational modification (PTM) cycles, the ribosome flow model (RFM),<jats:italic>T<\/jats:italic>-cell kinetic proofreading, and others. The results of this paper are dedicated in honor of Eduardo D. Sontag\u2019s seventieth birthday and his pioneering work in nonlinear dynamical systems and mathematical systems biology.<\/jats:p>","DOI":"10.1007\/s00498-023-00350-9","type":"journal-article","created":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T18:02:48Z","timestamp":1678730568000},"page":"585-617","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Graphical characterizations of robust stability in biological interaction networks"],"prefix":"10.1007","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6761-4520","authenticated-orcid":false,"given":"M. Ali","family":"Al-Radhawi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,3,13]]},"reference":[{"issue":"1","key":"350_CR1","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1006\/jtbi.2002.2537","volume":"216","author":"M Morohashi","year":"2002","unstructured":"Morohashi M, Winn AE, Borisuk MT, Bolouri H, Doyle J, Kitano H (2002) Robustness as a measure of plausibility in models of biochemical networks. J Theor Biol 216(1):19\u201330","journal-title":"J Theor Biol"},{"issue":"5560","key":"350_CR2","doi-asserted-by":"crossref","first-page":"1662","DOI":"10.1126\/science.1069492","volume":"295","author":"H Kitano","year":"2002","unstructured":"Kitano H (2002) Systems biology: a brief overview. Science 295(5560):1662\u20131664","journal-title":"Science"},{"issue":"11","key":"350_CR3","doi-asserted-by":"crossref","first-page":"1420","DOI":"10.1242\/bio.013714","volume":"4","author":"AL MacLean","year":"2015","unstructured":"MacLean AL, Kirk PD, Stumpf MP (2015) Cellular population dynamics control the robustness of the stem cell niche. Biol Open 4(11):1420\u20131426","journal-title":"Biol Open"},{"issue":"6\u20137","key":"350_CR4","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1007\/s00285-017-1125-6","volume":"75","author":"GP Langlois","year":"2017","unstructured":"Langlois GP, Craig M, Humphries AR, Mackey MC, Mahaffy JM, B\u00e9lair J, Moulin T, Sinclair SR, Wang L (2017) Normal and pathological dynamics of platelets in humans. J Math Biol 75(6\u20137):1411\u20131462","journal-title":"J Math Biol"},{"issue":"6","key":"350_CR5","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1038\/89204","volume":"19","author":"JE Bailey","year":"2001","unstructured":"Bailey JE (2001) Complex biology with no parameters. Nat Biotechnol 19(6):503\u2013504","journal-title":"Nat Biotechnol"},{"issue":"2","key":"350_CR6","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF00251225","volume":"47","author":"F Horn","year":"1972","unstructured":"Horn F, Jackson R (1972) General mass action kinetics. Arch Ration Mech Anal 47(2):81\u2013116","journal-title":"Arch Ration Mech Anal"},{"issue":"10","key":"350_CR7","doi-asserted-by":"crossref","first-page":"2229","DOI":"10.1016\/0009-2509(87)80099-4","volume":"42","author":"M Feinberg","year":"1987","unstructured":"Feinberg M (1987) Chemical reaction network structure and the stability of complex isothermal reactors\u2013I. The deficiency zero and deficiency one theorems. Chem Eng Sci 42(10):2229\u20132268","journal-title":"Chem Eng Sci"},{"issue":"7","key":"350_CR8","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1109\/9.935056","volume":"46","author":"ED Sontag","year":"2001","unstructured":"Sontag ED (2001) Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans Autom Control 46(7):1028\u20131047","journal-title":"IEEE Trans Autom Control"},{"issue":"4","key":"350_CR9","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1007\/s00285-009-0309-0","volume":"61","author":"D Angeli","year":"2010","unstructured":"Angeli D, De Leenheer P, Sontag E (2010) Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J Math Biol 61(4):581\u2013616","journal-title":"J Math Biol"},{"key":"350_CR10","doi-asserted-by":"crossref","unstructured":"Ali\u00a0Al-Radhawi M, Angeli D (2013) Piecewise linear in rates Lyapunov functions for complex reaction networks. In: Proceedings of the 52nd IEEE control and decision conference (CDC), pp 4595\u20134600","DOI":"10.1109\/CDC.2013.6760606"},{"issue":"1","key":"350_CR11","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1109\/TAC.2015.2427691","volume":"61","author":"M Ali Al-Radhawi","year":"2016","unstructured":"Ali Al-Radhawi M, Angeli D (2016) New approach to the stability of chemical reaction networks: piecewise linear in rates Lyapunov functions. IEEE Trans Automatic Control 61(1):76\u201389","journal-title":"IEEE Trans Automatic Control"},{"issue":"2","key":"350_CR12","doi-asserted-by":"crossref","first-page":"1007681","DOI":"10.1371\/journal.pcbi.1007681","volume":"16","author":"M Ali Al-Radhawi","year":"2020","unstructured":"Ali Al-Radhawi M, Angeli D, Sontag ED (2020) A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks. PLoS Comput Biol 16(2):1007681","journal-title":"PLoS Comput Biol"},{"key":"350_CR13","doi-asserted-by":"crossref","unstructured":"Ali\u00a0Al-Radhawi M, Angeli D (2014) Robust Lyapunov functions for complex reaction networks: an uncertain system framework. In: Proceedings of the IEEE 53rd conference on decision and control (CDC), pp 3101\u20133106","DOI":"10.1109\/CDC.2014.7039867"},{"issue":"10","key":"350_CR14","doi-asserted-by":"crossref","first-page":"2482","DOI":"10.1016\/j.automatica.2014.08.012","volume":"50","author":"F Blanchini","year":"2014","unstructured":"Blanchini F, Giordano G (2014) Piecewise-linear Lyapunov functions for structural stability of biochemical networks. Automatica 50(10):2482\u20132493","journal-title":"Automatica"},{"key":"350_CR15","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.automatica.2017.08.022","volume":"86","author":"F Blanchini","year":"2017","unstructured":"Blanchini F, Giordano G (2017) Polyhedral Lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the jacobian is non-singular. Automatica 86:183\u2013191","journal-title":"Automatica"},{"key":"350_CR16","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1109\/LCSYS.2021.3081369","volume":"6","author":"F Blanchini","year":"2021","unstructured":"Blanchini F, Giordano G (2021) Dual chemical reaction networks and implications for Lyapunov-based structural stability. IEEE Control Syst Lett 6:488\u2013493","journal-title":"IEEE Control Syst Lett"},{"key":"350_CR17","doi-asserted-by":"crossref","DOI":"10.1201\/9781420011432","volume-title":"An introduction to systems biology: design principles of biological circuits","author":"U Alon","year":"2006","unstructured":"Alon U (2006) An introduction to systems biology: design principles of biological circuits. Chapman and Hall\/CRC, London"},{"issue":"1","key":"350_CR18","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.nonrwa.2006.09.006","volume":"9","author":"D Angeli","year":"2008","unstructured":"Angeli D, Sontag ED (2008) Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles. Nonlinear Anal Real World Appl 9(1):128\u2013140","journal-title":"Nonlinear Anal Real World Appl"},{"key":"350_CR19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.mbs.2017.05.006","volume":"291","author":"M Eithun","year":"2017","unstructured":"Eithun M, Shiu A (2017) An all-encompassing global convergence result for processive multisite phosphorylation systems. Math Biosci 291:1\u20139","journal-title":"Math Biosci"},{"issue":"5","key":"350_CR20","doi-asserted-by":"crossref","first-page":"1545","DOI":"10.1109\/TCBB.2012.88","volume":"9","author":"M Margaliot","year":"2012","unstructured":"Margaliot M, Tuller T (2012) Stability analysis of the ribosome flow model. IEEE\/ACM Trans Comput Biol Bioinf 9(5):1545\u20131552","journal-title":"IEEE\/ACM Trans Comput Biol Bioinf"},{"issue":"116","key":"350_CR21","doi-asserted-by":"crossref","first-page":"20151062","DOI":"10.1098\/rsif.2015.1062","volume":"13","author":"A Raveh","year":"2016","unstructured":"Raveh A, Margaliot M, Sontag ED, Tuller T (2016) A model for competition for ribosomes in the cell. J R Soc Interface 13(116):20151062","journal-title":"J R Soc Interface"},{"issue":"5","key":"350_CR22","doi-asserted-by":"crossref","first-page":"1555","DOI":"10.1109\/LCSYS.2020.3041213","volume":"5","author":"J Miller","year":"2021","unstructured":"Miller J, Al-Radhawi MA, Sontag ED (2021) Mediating ribosomal competition by splitting pools. IEEE Control Syst Lett 5(5):1555\u20131560","journal-title":"IEEE Control Syst Lett"},{"key":"350_CR23","doi-asserted-by":"crossref","unstructured":"Craciun G, Feinberg M (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J Appl Math 1526\u20131546","DOI":"10.1137\/S0036139904440278"},{"issue":"2","key":"350_CR24","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1137\/15M1034441","volume":"15","author":"M Banaji","year":"2016","unstructured":"Banaji M, Pantea C (2016) Some results on injectivity and multistationarity in chemical reaction networks. SIAM J Appl Dyn Syst 15(2):807\u2013869","journal-title":"SIAM J Appl Dyn Syst"},{"issue":"7","key":"350_CR25","doi-asserted-by":"crossref","first-page":"1662","DOI":"10.1007\/s11538-017-0303-4","volume":"79","author":"MM de Freitas","year":"2017","unstructured":"de Freitas MM, Wiuf C, Feliu E (2017) Intermediates and generic convergence to equilibria. Bull Math Biol 79(7):1662\u20131686","journal-title":"Bull Math Biol"},{"issue":"2","key":"350_CR26","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1016\/j.mbs.2007.07.003","volume":"210","author":"D Angeli","year":"2007","unstructured":"Angeli D, De Leenheer P, Sontag ED (2007) A Petri net approach to the study of persistence in chemical reaction networks. Math Biosci 210(2):598\u2013618","journal-title":"Math Biosci"},{"issue":"6","key":"350_CR27","doi-asserted-by":"crossref","first-page":"1683","DOI":"10.1007\/s00285-020-01477-y","volume":"80","author":"E Gross","year":"2020","unstructured":"Gross E, Harrington H, Meshkat N, Shiu A (2020) Joining and decomposing reaction networks. J Math Biol 80(6):1683\u20131731","journal-title":"J Math Biol"},{"key":"350_CR28","doi-asserted-by":"crossref","DOI":"10.1016\/j.amc.2022.127109","volume":"426","author":"M Banaji","year":"2022","unstructured":"Banaji M, Boros B, Hofbauer J (2022) Adding species to chemical reaction networks: Preserving rank preserves nondegenerate behaviours. Appl Math Comput 426:127109","journal-title":"Appl Math Comput"},{"key":"350_CR29","volume-title":"Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models","author":"P \u00c9rdi","year":"1989","unstructured":"\u00c9rdi P, T\u00f3th J (1989) Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models. Manchester University Press, Manchester"},{"issue":"3\u20134","key":"350_CR30","doi-asserted-by":"crossref","first-page":"398","DOI":"10.3166\/ejc.15.398-406","volume":"15","author":"D Angeli","year":"2009","unstructured":"Angeli D (2009) A tutorial on chemical reaction network dynamics. Eur J Control 15(3\u20134):398\u2013406","journal-title":"Eur J Control"},{"issue":"1","key":"350_CR31","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1186\/1741-7007-12-29","volume":"12","author":"J Gunawardena","year":"2014","unstructured":"Gunawardena J (2014) Models in biology:\u2018accurate descriptions of our pathetic thinking\u2019. BMC Biol 12(1):29","journal-title":"BMC Biol"},{"issue":"4","key":"350_CR32","doi-asserted-by":"crossref","first-page":"6477","DOI":"10.4249\/scholarpedia.6477","volume":"3","author":"CA Petri","year":"2008","unstructured":"Petri CA, Reisig W (2008) Petri net. Scholarpedia 3(4):6477","journal-title":"Scholarpedia"},{"issue":"4","key":"350_CR33","doi-asserted-by":"crossref","first-page":"1321","DOI":"10.1137\/050634177","volume":"66","author":"G Craciun","year":"2006","unstructured":"Craciun G, Feinberg M (2006) Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph. SIAM J Appl Math 66(4):1321\u20131338","journal-title":"SIAM J Appl Math"},{"issue":"4","key":"350_CR34","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1109\/5.24143","volume":"77","author":"T Murata","year":"1989","unstructured":"Murata T (1989) Petri nets: properties, analysis and applications. Proc IEEE 77(4):541\u2013580","journal-title":"Proc IEEE"},{"key":"350_CR35","volume-title":"Stability theory by Liapunov\u2019s second method","author":"T Yoshizawa","year":"1966","unstructured":"Yoshizawa T (1966) Stability theory by Liapunov\u2019s second method. Mathematical Society of Japan, Tokyo"},{"key":"350_CR36","volume-title":"Introduction to dynamic systems; theory, models, and applications","author":"DG Luenberger","year":"1979","unstructured":"Luenberger DG (1979) Introduction to dynamic systems; theory, models, and applications. Wiley, New York, NY"},{"issue":"4","key":"350_CR37","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1109\/MCS.2009.932926","volume":"29","author":"V Chellaboina","year":"2009","unstructured":"Chellaboina V, Bhat S, Haddad WM, Bernstein DS (2009) Modeling and analysis of mass-action kinetics. IEEE Control Syst Mag 29(4):60\u201378","journal-title":"IEEE Control Syst Mag"},{"key":"350_CR38","doi-asserted-by":"crossref","unstructured":"Marinescu D, Beaven M, Stansifer R (1991) A parallel algorithm for computing invariants of Petri net models. In: Proceedings of the fourth international workshop on petri nets and performance models, pp 136\u2013137","DOI":"10.1109\/PNPM.1991.238774"},{"issue":"6","key":"350_CR39","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1109\/TCS.1978.1084490","volume":"25","author":"H Maeda","year":"1978","unstructured":"Maeda H, Kodama S, Ohta Y (1978) Asymptotic behavior of nonlinear compartmental systems: nonoscillation and stability. IEEE Trans Circuit Syst 25(6):372\u2013378","journal-title":"IEEE Trans Circuit Syst"},{"issue":"1","key":"350_CR40","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1137\/1035003","volume":"35","author":"JA Jacquez","year":"1993","unstructured":"Jacquez JA, Simon CP (1993) Qualitative theory of compartmental systems. SIAM Rev 35(1):43\u201379","journal-title":"SIAM Rev"},{"issue":"11","key":"350_CR41","doi-asserted-by":"crossref","first-page":"3828","DOI":"10.1529\/biophysj.107.110866","volume":"93","author":"J Gunawardena","year":"2007","unstructured":"Gunawardena J (2007) Distributivity and processivity in multisite phosphorylation can be distinguished through steady-state invariants. Biophys J 93(11):3828\u20133834","journal-title":"Biophys J"},{"key":"350_CR42","unstructured":"Ali Al-Radhawi M (Dec 2015) New approach to the stability and control of reaction networks. PhD thesis, PhD Dissertation, Imperial College London"},{"issue":"6","key":"350_CR43","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1137\/060673412","volume":"67","author":"M Banaji","year":"2007","unstructured":"Banaji M, Donnell P, Baigent S (2007) P matrix properties, injectivity, and stability in chemical reaction systems. SIAM J Appl Math 67(6):1523\u20131547","journal-title":"SIAM J Appl Math"},{"issue":"1","key":"350_CR44","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0025-5564(80)90096-6","volume":"51","author":"TC Gard","year":"1980","unstructured":"Gard TC (1980) Persistence in food chains with general interactions. Math Biosci 51(1):165\u2013174","journal-title":"Math Biosci"},{"key":"350_CR45","doi-asserted-by":"crossref","unstructured":"Waltman P (1991) A brief survey of persistence in dynamical systems. In: Busenberg S, Martelli M (eds) Delay differential equations and dynamical systems, pp 31\u201340. Springer","DOI":"10.1007\/BFb0083477"},{"key":"350_CR46","doi-asserted-by":"crossref","unstructured":"Angeli D, De\u00a0Leenheer P, Sontag E (2007) A petri net approach to persistence analysis in chemical reaction networks. In: Queinnec I, Tarbouriech S, Garcia G, Niculescu SI (eds) Biology and control theory: current challenges, pp 181\u2013216. Springer","DOI":"10.1007\/978-3-540-71988-5_9"},{"issue":"1","key":"350_CR47","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1137\/090779401","volume":"71","author":"D Angeli","year":"2011","unstructured":"Angeli D, De Leenheer P, Sontag ED (2011) Persistence results for chemical reaction networks with time-dependent kinetics and no global conservation laws. SIAM J Appl Math 71(1):128\u2013146","journal-title":"SIAM J Appl Math"},{"issue":"11","key":"350_CR48","doi-asserted-by":"crossref","first-page":"6840","DOI":"10.1073\/pnas.78.11.6840","volume":"78","author":"A Goldbeter","year":"1981","unstructured":"Goldbeter A, Koshland DE (1981) An amplified sensitivity arising from covalent modification in biological systems. Proc Natl Acad Sci 78(11):6840\u20136844","journal-title":"Proc Natl Acad Sci"},{"issue":"11","key":"350_CR49","doi-asserted-by":"crossref","first-page":"5042","DOI":"10.1073\/pnas.92.11.5042","volume":"92","author":"TW McKeithan","year":"1995","unstructured":"McKeithan TW (1995) Kinetic proofreading in T-cell receptor signal transduction. Proc Natl Acad Sci 92(11):5042\u20135046","journal-title":"Proc Natl Acad Sci"},{"issue":"9","key":"350_CR50","doi-asserted-by":"crossref","first-page":"1002127","DOI":"10.1371\/journal.pcbi.1002127","volume":"7","author":"S Reuveni","year":"2011","unstructured":"Reuveni S, Meilijson I, Kupiec M, Ruppin E, Tuller T (2011) Genome-scale analysis of translation elongation with a ribosome flow model. PLoS Comput Biol 7(9):1002127","journal-title":"PLoS Comput Biol"},{"key":"350_CR51","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511810817","volume-title":"Matrix analysis","author":"RA Horn","year":"1985","unstructured":"Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge"}],"container-title":["Mathematics of Control, Signals, and Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00498-023-00350-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00498-023-00350-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00498-023-00350-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,8]],"date-time":"2023-12-08T21:26:42Z","timestamp":1702070802000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00498-023-00350-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,13]]},"references-count":51,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["350"],"URL":"https:\/\/doi.org\/10.1007\/s00498-023-00350-9","relation":{},"ISSN":["0932-4194","1435-568X"],"issn-type":[{"value":"0932-4194","type":"print"},{"value":"1435-568X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,13]]},"assertion":[{"value":"9 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 March 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declaration"}},{"value":"The author declares no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}