Identifying the Parametric Occurrence of Multiple Steady States for some Biological Networks

Russell Bradford, James Davenport, Matthew England, Hassan Errami, Vladimir P. Gerdt, Dima Grigoriev, Charles Hoyt, Marek Kosta, Ovidiu Radulescu, Thomas Sturm, and Andreas Weber
In: Journal of Symbolic Computation, Elsevier (Jan. 2019)
 

Abstract

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 results to 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 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.

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Bibtex

@ARTICLE{BDEEGGHKRSW19,
    author = {Bradford, Russell and Davenport, James and England, Matthew and Errami, Hassan and Gerdt, Vladimir
              P. and Grigoriev, Dima and Hoyt, Charles and Kosta, Marek and Radulescu, Ovidiu and Sturm, Thomas
              and Weber, Andreas},
     title = {Identifying the Parametric Occurrence of Multiple Steady States for some Biological Networks},
   journal = {Journal of Symbolic Computation, Elsevier},
      year = {2019},
     month = jan,
  abstract = {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 results to 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 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.},
       url = {https://arxiv.org/abs/1902.04882}
}