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Algebraic Systems Biology: A Case Study for the Wnt Pathway

Bulletin of Mathematical Biology volume 78, pages 21–51 (2016)Cite this 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.

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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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Affiliations

  1. San José State University, San José, CA, USA

    Elizabeth Gross

  2. University of Oxford, Oxford, England

    Heather A. Harrington

  3. Pennsylvania State University, State College, PA, USA

    Zvi Rosen

  4. University of California at Berkeley, Berkeley, CA, USA

    Bernd Sturmfels

Authors
  1. Elizabeth Gross
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  2. Heather A. Harrington
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  3. Zvi Rosen
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  4. Bernd Sturmfels
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Correspondence to Bernd Sturmfels.

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Gross, E., Harrington, H.A., Rosen, Z. et al. Algebraic Systems Biology: A Case Study for the Wnt Pathway. Bull Math Biol 78, 21–51 (2016). https://doi.org/10.1007/s11538-015-0125-1

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Keywords

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