A Case Study on the Parametric Occurrence of Multiple Steady States

Abstract

We consider the problem of determining multiple steady states for positive real values in models of biological networks. Investigating the potential for these in models of the mitogen-activated protein kinases (MAPK) network has consumed considerable effort using special insights into the structure of corresponding models. Here we apply combinations of symbolic computation methods for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition. We determine multistationarity of an 11-dimensional MAPK network when numeric values are known for all but potentially one parameter. More precisely, our considered model has 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment, and furthermore positivity conditions on all variables and parameters.

Accepted into ISSAC 2017. This version has additional page showing all 11 CAD trees discussed in Section 2.1.1

References in corpus (4)

• Identifying parameter regions for multistationarity
• Improving the use of equational constraints in cylindrical algebraic decomposition
• The complexity of cylindrical algebraic decomposition with respect to polynomial degree
• A note on "MAPK networks and their capacity for multistationarity due to toric steady states"

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• The multistationarity structure of networks with intermediates and a binomial core network
• Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition
• Symbolic Versus Numerical Computation and Visualization of Parameter Regions for Multistationarity of Biological Networks
• Machine Learning for Mathematical Software
• Multistationarity in the space of total concentrations for systems that admit a monomial parametrization
• TheoryGuru: A Mathematica Package to apply Quantifier Elimination
• Algorithmically generating new algebraic features of polynomial systems for machine learning
• Algorithmic Reduction of Biological Networks With Multiple Time Scales
• Improved cross-validation for classifiers that make algorithmic choices to minimise runtime without compromising output correctness
• Quantifier Elimination for Reasoning in Economics
• Non-linear Real Arithmetic Benchmarks derived from Automated Reasoning in Economics
• Parametric Toricity of Steady State Varieties of Reaction Networks
• Positive Solutions of Systems of Signed Parametric Polynomial Inequalities
• Groebner bases of reaction networks with intermediate species