{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:30:59Z","timestamp":1787229059145,"version":"build-2736575974"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Many biochemical processes can successfully be described by dynamical systems allowing some form of switching when, depending on their initial conditions, solutions of the dynamical system end up in different regions of state space (associated with different biochemical functions). Switching is often realized by a bistable system (i.e., a dynamical system allowing two stable steady state solutions) and, in the majority of cases, bistability is established numerically. In our view, this approach is too restrictive. On the one hand, due to predominant parameter uncertainty, numerical methods are generally difficult to apply to realistic models originating in systems biology. On the other hand, switching already arises with the occurrence of a saddle-type steady state (characterized by a Jacobian where exactly one eigenvalue is positive and the remaining eigenvalues have negative real part). Consequently we derive conditions based on linear inequalities that allow the analytic computation of states and parameters where the Jacobian derived from a mass action network has a defective zero eigenvalue so that\u2014under certain genericity conditions\u2014a saddle-node bifurcation occurs. Our conditions are applicable to general mass action networks involving at least one conservation relation; however, they are only sufficient (as infeasibility of linear inequalities does not exclude defective zero eigenvalues).<\/jats:p>","DOI":"10.1137\/10081722x","type":"journal-article","created":{"date-parts":[[2012,1,26]],"date-time":"2012-01-26T18:08:22Z","timestamp":1327601302000},"page":"110-134","source":"Crossref","is-referenced-by-count":17,"title":["Switching in Mass Action Networks Based on Linear Inequalities"],"prefix":"10.1137","volume":"11","author":[{"given":"Carsten","family":"Conradi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dietrich","family":"Flockerzi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,1,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(95)00026-N"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-9822(01)00330-X"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2009.v7.n4.a4"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2009.07.003"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/060673412"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1063\/1.1780011"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"R. Brualdi and B. Shader,\n                      Matrices of Sign-Solvable Linear Systems\n                      , Cambridge University Press, Cambridge, MA, 1995.","DOI":"10.1017\/CBO9780511574733"},{"key":"R8","unstructured":"C. Chicone,\n                      Ordinary Differential Equations with Applications\n                      , Springer, New York, 1999."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"C. Conradi and D. Flockerzi,\n                      Multistationarity in mass action networks with applications to ERK activation\n                      , J. Math. Biol. (2011), to appear; DOI 10.1007\/s00285-011-0453-1.","DOI":"10.1007\/s00285-011-0453-1"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"C. Conradi, D. Flockerzi, and J. Raisch,\n                      Saddle-node bifurcations in biochemical reaction networks with mass action kinetics and application to a double-phosphorylation mechanism\n                      , in Proceedings of the 2007 American Control Conference, New York, 2007, pp. 6103\u20136109 (CD-ROM).","DOI":"10.1109\/ACC.2007.4282717"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2007.10.004"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0705731104"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139904440278"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/050634177"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0602767103"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/S1381-1169(99)00371-4"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF00375614"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF00375615"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/138\/1\/012006"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2004.04.003"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/j.pbiomolbio.2009.06.004"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1186\/1752-0509-1-2"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.13001\/1081-3810.1021"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2008.925474"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-007-0099-1"},{"key":"R26","unstructured":"R. T. Rockafellar,\n                      Convex Analysis\n                      , Princeton University Press, Princeton, NJ, 1970."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/0009-2509(94)80061-8"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"N. Sleumer,\n                      Output-sensitive cell enumeration in hyperplane arrangements\n                      , in Algorithm Theory\u2014SWAT'98, Lecture Notes in Comput. Sci. 1432, Springer, Berlin, Heidelberg, 1998.","DOI":"10.1007\/BFb0054377"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1159\/000076100"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1093\/bioinformatics\/btn401"},{"key":"R31","unstructured":"J. H. 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