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.
Bibliographical noteThe version of the paper on Arxiv is one that was created after the first and main peer review. Please note that the Appendix in the Arxiv version will appear as supplementary data in the final version of record.
FunderD. Grigoriev, O. Radulescu, T. Sturm, and A. Weber are grateful to ANR - 17-CE40-0036 / DFG - 391322026 SYMBIONT. J.H. Davenport, M. England and T. Sturm are grateful to the European Union's Horizon 2020 Research and Innovation programme, under grant agreement No 712689 (SC 2 ). H. Errami, O. Radulescu, and A. Weber thanks the French-German Procope-DAAD program for partial support of this research. V. Gerdt was partially supported by the RUDN University Program 5-100 . D. Grigoriev is grateful to the grant RSF 16-11-10075 and to MCCME for wonderful working conditions and an inspiring atmosphere. M. Košta has been supported by the DFG/ANR Project STU 483/2-1 SMArT.
- Mixed Equation / Inequality Solving
- Real Quantifier Elimination
- Biological Networks
- Signaling Pathways