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Bulletin of Mathematical Biology

, Volume 78, Issue 1, pp 21–51 | Cite as

Algebraic Systems Biology: A Case Study for the Wnt Pathway

  • Elizabeth Gross
  • Heather A. Harrington
  • Zvi Rosen
  • Bernd Sturmfels
Original Article

Abstract

Steady-state analysis of dynamical systems for biological networks gives rise to algebraic varieties in high-dimensional spaces whose study is of interest in their own right. We demonstrate this for the shuttle model of the Wnt signaling pathway. Here, the variety is described by a polynomial system in 19 unknowns and 36 parameters. It has degree 9 over the parameter space. This case study explores multistationarity, model comparison, dynamics within regions of the state space, identifiability, and parameter estimation, from a geometric point of view. We employ current methods from computational algebraic geometry, polyhedral geometry, and combinatorics.

Keywords

Biochemical reaction networks Nonlinear algebra \(\beta \)-catenin/Wnt signaling Steady-state variety  Polyhedra Algebraic matroids 

Notes

Acknowledgments

This project was supported by UK Royal Society International Exchange Award 2014/R1 IE140219. EG, BS and HAH initiated discussions at an American Institute of Mathematics workshop in Palo Alto. Part of the work was carried out at the Simons Institute for Theory of Computing in Berkeley. HAH gratefully acknowledges EPSRC Fellowship EP/K041096/1. EG, ZR, and BS were also supported by the US National Science Foundation, through Grants DMS-1304167, DMS-0943745, and DMS-1419018, respectively. Thanks to Helen Byrne and Reinhard Laubenbacher for comments on early drafts of the paper.

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Copyright information

© Society for Mathematical Biology 2015

Authors and Affiliations

  • Elizabeth Gross
    • 1
  • Heather A. Harrington
    • 2
  • Zvi Rosen
    • 3
  • Bernd Sturmfels
    • 4
  1. 1.San José State UniversitySan JoséUSA
  2. 2.University of OxfordOxfordEngland
  3. 3.Pennsylvania State UniversityState CollegeUSA
  4. 4.University of California at BerkeleyBerkeleyUSA

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