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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Bates D, Hauenstein J, Sommese A, Wampler C (2013) Numerically solving polynomial systems with Bertini. Software, Environments, and Tools, vol 25. SIAM, Philadelphia
Bates D, Gross E, Leykin A, Rodriguez J (2013) Bertini for Macaulay2. arXiv:1310.3297
Chen C, Davenport J, Moreno Maza M, Xia B, Xiao R (2013) Computing with semi-algebraic sets: relaxation techniques and effective boundaries. J Symb Comput 52:72–96
Craciun G, Feinberg M (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J Appl Math 65:1526–1546
Draisma J, Horobet E, Ottaviani G, Sturmfels B, Thomas RR (2015) The Euclidean distance degree of an algebraic variety. Found Comput Math (to appear). arXiv:1309.0049
Faugère J-C, Moroz G, Rouillier F, Safey El Din M (2008) Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities. In: ISSAC 2008. ACM, New York, pp 79–86
Feliu E, Wiuf C (2012) Variable elimination in chemical reaction networks with mass-action kinetics. SIAM J Appl Math 72:959–981
Garcia-Puente L, Petrovic S, Sullivant S (2013) Graphical models. J Softw Algebra Geom 5:1–7
Grayson D, Stillman M (2002) Macaulay2, a software system for research in algebraic geometry. www.math.uiuc.edu/Macaulay2/
Gross E, Davis B, Ho K, Bates D, Harrington H (2015) Model selection using numerical algebraic geometry. arXiv:1507.04331
Harrington H, Ho K, Thorne T, Stumpf M (2012) Parameter-free model discrimination criterion based on steady state coplanarity. Proc Natl Acad Sci 109:15746–15751
Karp R, Pérez Millán M, Desgupta T, Dickenstein A, Gunawardena J (2012) Complex-linear invariants of biochemical networks. J Theor Biol 311:130–138
Klipp E, Liebermeister W, Wierling C, Kowald A, Lehrach H, Herwig R (2009) Systems biology. Wiley, New York
Király F, Rosen Z, Theran L (2013) Algebraic matroids with graph symmetry. arXiv:1312.3777
Király F, Theran L, Tomioka R (2015) The algebraic combinatorial approach for low-rank matrix completion. J Mach Learn Res 16:1391–1436
MacLean A, Harrington H, Stumpf M, Byrne H (2015) Mathematical and statistical techniques for systems medicine: the Wnt signaling pathway as a case study. In: Schmitz, ULF, Wolkenhauer, OLAF (eds) Systems Medicine. Methods Molecular Biology, vol 1386. Springer, New York
MacLean A, Rosen Z, Byrne H, Harrington H (2015) Parameter-free methods distinguish Wnt pathway models and guide design of experiments. Proc Natl Acad Sci 112:2652–2657
Meshkat N, Sullivant S (2014) Identifiable reparametrizations of linear compartment models. J Symb Comput 63:46–67
Morgan A, Sommese A (1989) Coefficient-parameter polynomial continuation. Appl Math Comput 29:123–160
Ottaviani G, Spaenlehauer P-J, Sturmfels B (2014) Exact solutions in structured low-rank approximation. SIAM J Matrix Anal Appl 35:1521–1542
Oxley J (2011) Matroid theory. Oxford University Press, Oxford
Pérez Millán M, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74:1027–1065
Rodriguez J, Tang X (2015) Data-discriminants of likelihood equations. In: Proceedings of the 2015 ACM on international symposium on symbolic and algebraic computation, pp 307–314
Rosen Z (2014) Computing algebraic matroids. arXiv:1403.8148
Shiu A, Sturmfels B (2010) Siphons in chemical reaction networks. Bull Math Biol 72:1448–1463
Tan CW, Gardiner BS, Hirokawa Y, Layton MJ, Smith DW, Burgess AW (2012) Wnt Signalling pathway parameters for mammalian cells. PLoS ONE 7:e31882
Verschelde J (1999) Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans Math Softw 25:251–276
Voit E (2012) A first course in systems biology. Garland Science, New York
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.
About this article
Cite this article
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
- Biochemical reaction networks
- Nonlinear algebra
- \(\beta \)-catenin/Wnt signaling
- Steady-state variety
- Algebraic matroids