Identifying the parametric occurrence of multiple steady states for some biological networks

Russell Bradford, James H. Davenport, Matthew England, Hassan Errami, Vladimir Gerdt, Dima Grigoriev, Charles Hoyt, Marek Košta, Ovidiu Radulescu, Thomas Sturm, Andreas Weber

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We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic and numerical methods. The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters. We apply combinations of symbolic computation methods designed for mixed equality / inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory. We are able to determine semi-algebraic conditions for multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.

Original languageEnglish
Pages (from-to)84-119
Number of pages36
JournalJournal of Symbolic Computation
Early online date15 Jul 2019
Publication statusPublished - 1 May 2020


  • Biological networks
  • MAPK
  • Mixed equation/inequality solving
  • Real quantifier elimination
  • Signalling pathways

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics


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