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Algebra and Geometry in the Study of Enzymatic Cascades

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World Women in Mathematics 2018

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 20))

Abstract

In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need a priori determination of the parameters, which can be theoretically or practically impossible. This chapter gives a brief introduction to general results based on the network structure. In particular, we describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates and include many post-translational modification networks. We also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and we point out some of the mathematical questions that arise from this application.

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References

  1. Angeli D., De Leenher P., and Sontag. E: A Petri net approach to the study of persistence in chemical reaction networks, Mathematical Biosciences 210, 598–618 (2007).

    Article  MathSciNet  Google Scholar 

  2. Angeli D., De Leenher, P., and Sontag, E.: Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J. Math. Biol. 61, 581–616 (2010).

    Article  MathSciNet  Google Scholar 

  3. Banaji, M.: Inheritance of oscillation in chemical reaction networks. Applied Mathematics and Computation 325, 191–209 (2018).

    Article  MathSciNet  Google Scholar 

  4. Banaji, M., and Pantea, C.: The inheritance of nondegenerate multistationarity in chemical reaction networks. SIAM J Appl Math 78(2), 1105–1130 (2018).

    Article  MathSciNet  Google Scholar 

  5. Bihan F., and Dickenstein, A.: Descartes’ Rule of Signs for Polynomial Systems supported on Circuits, Int. Math. Res. Notices 22, 6867–6893 (2017).

    MathSciNet  MATH  Google Scholar 

  6. Bihan F., Dickenstein, A., and Giaroli, M.: Lower bounds for positive roots and regions of multistationarity in chemical reaction networks. arXiv:1807.05157 (2018).

    Google Scholar 

  7. Bihan, F., Santos, F., and Spaenlehauer, P-J.: A polyhedral method for sparse systems with many positive solutions. SIAM J. Appl. Algebra Geometry 2(4), 620–645 (2018).

    Article  MathSciNet  Google Scholar 

  8. Conradi, C., Feliu, E., Mincheva, M., and Wiuf, C.: Identifying parameter regions for multistationarity. PLoS Comput. Biol. 13(10), e1005751 (2017).

    Article  Google Scholar 

  9. Conradi, C., Flockerzi, D., Raisch, J., and Stelling, J.: Subnetwork analysis reveals dynamic features of complex (bio)chemical networks PNAS 104 (49), 19175–19180 (2007).

    Google Scholar 

  10. Conradi, C., Iosif, A., Kahle, T.: Multistationarity in the space of total concentrations for systems that admit a monomial parametrization. To appear: Bull. Math. Biol. (2019).

    Google Scholar 

  11. Conradi, C., Mincheva, M., Shiu, A: Emergence of oscillations in a mixed-mechanism phosphorylation system. To appear: Bull. Math. Bio. (2019).

    Google Scholar 

  12. Conradi, C., and Shiu, A.: A global convergence result for processive multisite phosphorylation systems. Bull. Math. Biol. 77(1), 126–155 (2015).

    Article  MathSciNet  Google Scholar 

  13. Conradi C., and Pantea C.: Multistationarity in Biochemical Networks: Results, Analysis, and Examples. Algebraic and Combinatorial Computational Biology, Ch. 9, Eds. Robeva R. and Macaulay M., Mathematics in Science and Computation, Academic Press (2019).

    Google Scholar 

  14. Craciun, G., and Feinberg. M., Multiple equilibria in complex chemical reaction networks. I. The injectivity property. SIAM J. Appl. Math. 65(5), 1526–1546 (2005).

    Article  MathSciNet  Google Scholar 

  15. Craciun, G., Helton, J. W., and and Williams, R.: Homotopy methods for counting reaction network equilibria. Math. Biosci. 216(2), 140–149 (2008).

    Article  MathSciNet  Google Scholar 

  16. Dickenstein, A.: Biochemical reaction networks: an invitation for algebraic geometers. MCA 2013, Contemporary Mathematics 656, 65–83 (2016).

    Article  Google Scholar 

  17. Dickenstein, A., Giaroli, M., Pérez Millán, M., and Rischter, R.: Parameter regions that give rise to \(2\left [\tfrac {n}{2}\right ]+1\) positive steady states in the n-site phosphorylation system. arXiv: 1904.11633 (2019).

    Google Scholar 

  18. Dickenstein, A., Pérez Millán, M., Shiu, A., and Tang, X.: Multistationarity in Structured Reaction Networks. Bull. Math. Biol. 81(5), 1527–1581 (2019).

    Article  MathSciNet  Google Scholar 

  19. Eithun, M., and Shiu, A.: An all-encompassing global convergence result for processive multisite phosphorylation systems. Math. Biosci. 291, 1–9 (2017).

    Article  MathSciNet  Google Scholar 

  20. Érdi, P., and Tóth, J.: Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models. Manchester University Press (1989).

    Google Scholar 

  21. Faugère, J.-C., Moroz, G., Rouillier, F., Safey El Din, M.: Classification of the Perspective-Three-Point problem, Discriminant variety and Real solving polynomial systems of inequalities. ISSAC 2008 Proceedings, D. Jeffrey (eds), Hagenberg (2008).

    Google Scholar 

  22. Feinberg, M.: Foundations of Chemical Reaction Network Theory, Applied Mathematical Series, Vol. 202, Springer Nature Switzerland (2019).

    Book  Google Scholar 

  23. Feliu, E., and Wiuf, C.: Enzyme-sharing as a cause of multi-stationarity in signalling systems. J. R. Soc. Interface 9 (71), 1224–1232 (2012).

    Article  Google Scholar 

  24. Feliu, E., and Wiuf, C.: Simplifying biochemical models with intermediate species. J. R. Soc. Interface 10: 20130484 (2013).

    Article  Google Scholar 

  25. Gatermann, K., Eiswirth, M., and Sensse, A.: Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems, J. Symb. Comput. 40(6), 1361–1382 (2005).

    Article  MathSciNet  Google Scholar 

  26. Gelfand, I., Kapranov, M., and Zelevinsky,A.: Discriminants, resultants and multidimensional determinants. Birkhäuser Boston (1994).

    Google Scholar 

  27. Giaroli, M., Bihan, F., and Dickenstein, A .: Regions of multistationarity in cascades of Goldbeter-Koshland loops. J. Math. Biol. 78(4), 1115–1145 (2019).

    Article  MathSciNet  Google Scholar 

  28. Gnacadja, G.: Reachability, persistence, and constructive chemical reaction networks (part II): a formalism for species composition in chemical reaction network theory and application to persistence, J. Math. Chem. 49(10) , 2137–2157 (2011).

    Article  MathSciNet  Google Scholar 

  29. Gnacadja, G.: Reachability, persistence, and constructive chemical reaction networks (part III): a mathematical formalism for binary enzymatic networks and application to persistence, J. Math. Chem. 49(10), 2158–2176 (2011).

    Article  MathSciNet  Google Scholar 

  30. Gross, E., Harrington, H. A., Rosen, Z., and Sturmfels, B.: Algebraic Systems Biology: A Case Study for the Wnt Pathway Bull Math Biol. 78(1), 21–51 (2016).

    Google Scholar 

  31. Gunawardena, J.: A linear framework for time-scale separation in nonlinear biochemical systems, PLoS ONE 7:e36321 (2012).

    Article  Google Scholar 

  32. Harrington, H. A, Mehta, D., Byrne, H., and Hauenstein, J.: Decomposing the parameter space of biological networks via a numerical discriminant approach. arXiv:1604.02623 (2016).

    Google Scholar 

  33. Hell, J., and Rendall, A. D.: A proof of bistability for the dual futile cycle. Nonlin. Anal. RWA 24, 175–189 (2015).

    Article  MathSciNet  Google Scholar 

  34. Hell, J., and Rendall, A. D.: Sustained oscillations in the MAP kinase cascade. Math.. Biosci. 282, 162–173 (2016).

    Article  MathSciNet  Google Scholar 

  35. Hell, J., and Rendall, A. D.: Dynamical Features of the MAP Kinase Cascade. In: Graw F., Matth’́aus F., Pahle J. (eds) Modeling Cellular Systems. Contributions in Mathematical and Computational Sciences, vol 11. Springer, Cham (2017).

    Chapter  Google Scholar 

  36. Holstein, K., Flockerzi, D., and Conradi, C.: Multistationarity in sequential distributed multisite phosphorylation networks. Bull. Math. Biol. 75(11), 2028–2058 (2013).

    Article  MathSciNet  Google Scholar 

  37. Joshi,B., and Shiu, A.: Which small reaction networks are multistationary? SIAM J. Appl. Dyn. Syst. 16(2), 802–833 (2017).

    Article  MathSciNet  Google Scholar 

  38. Mirzaev I., and Gunawardena J. : Laplacian dynamics on general graphs, Bull. Math. Biol. 75, 2118–2149 (2013).

    Article  MathSciNet  Google Scholar 

  39. Müller, S., Feliu, E., Regensburger, G., Conradi, C., Shiu, A., and Dickenstein, A.: Sign conditions for injectivity of generalized polynomial maps with appl. to chemical reaction networks and real algebraic geometry, Found. Comput. Math. 6(1), 69–97 (2016).

    Google Scholar 

  40. Manrai A. K., and Gunawardena J.: The Geometry of Multisite Phosphorylation, Biophysical Journal (2008), Vol. 95, 5533–5543.

    Google Scholar 

  41. Patel, A.L., and Shvartsman, S.Y.: Outstanding questions in developmental ERK signaling. Development 145(14) (2018).

    Article  Google Scholar 

  42. Pérez Millán, and Dickenstein, A.: The structure of MESSI biological systems. SIAM J. Appl. Dyn. Syst. 17(2), 1650–1682 (2018).

    Article  MathSciNet  Google Scholar 

  43. Pérez Millán, M., Dickenstein, A., Shiu, A., and Conradi,C.: Chemical reaction systems with toric steady states. Bull. Math. Biol. 74(5) 1027–1065 (2012).

    Article  MathSciNet  Google Scholar 

  44. Qiao, L., Nachbar, R. B., Kevrekidis, I. G., and Shvartsman, S. Y.: Bistability and oscillations in the Huang-Ferrell model ofMAPK signalling. PLoS Comp. Biol. 3, 1819–1826 (2007).

    Article  Google Scholar 

  45. Rubinstein, B., Mattingly, H., Berezhkovskii, A., and Shvartsman,S.: Long-term dynamics of multisite phosphorylation. Mol. Biol. Cell 27(14), 2331–2340 (2016).

    Article  Google Scholar 

  46. Sadeghimanesh, AH., and Feliu,E.: The multistationarity structure of networks with intermediates and a binomial core network. Bull. Math. Biol. 81:2428–2462 (2019).

    Article  MathSciNet  Google Scholar 

  47. Safey El Din, M.: RAGlib library, available at: https://www-polsys.lip6.fr/~safey/RAGLib/.

  48. M. Sáez, C. Wiuf, E. Feliu: Graphical reduction of reaction networks by linear elimination of species. J. Math. Biol. 74(1-2), 195–237 (2017).

    Article  MathSciNet  Google Scholar 

  49. Shinar, G., and Feinberg, M.: Structural sources of robustness in biochemical reaction networks, Science 327(5971) 1389–1391 (2010), .

    Article  Google Scholar 

  50. Suwanmajo, T., and Krishnan, J.: Mixed mechanisms of multi-site phosphorylation Journal of The Royal Society Interface 12(107) (2015).

    Article  Google Scholar 

  51. Thomson,T., and Gunawardena, J.: The rational parameterisation theorem for multisite post-translational modification systems. J. Theoret. Biol. 261(4), 626–636 (2009).

    Article  MathSciNet  Google Scholar 

  52. Wang, L., Sontag, E.: On the number of steady states in a multiple futile cycle. J. Math. Biol. 57(1), 29–52 (2008).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I am very grateful to the Committee for Women in Mathematics of the International Mathematical Union for the organization of this meeting and for the many activities they take in charge to ensure development and recognition of female mathematicians around the world. I am particulary grateful to Carolina Araujo for all her work to make (WM)2 a great success. I also thank Magalí Giaroli, Mercedes Pérez Millán and Enrique Tobis for their help with the figures.

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Authors and Affiliations

  1. Department of Mathematics, FCEN, University of Buenos Aires, Ciudad Universitaria, Pab. I, Buenos Aires, Argentina

    Alicia Dickenstein

  2. IMAS, UBA-CONICET, Ciudad Universitaria, Pab. I, Buenos Aires, Argentina

    Alicia Dickenstein

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  1. Alicia Dickenstein
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Correspondence to Alicia Dickenstein .

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Editors and Affiliations

  1. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil

    Carolina Araujo

  2. Department of Mathematics, University of Wisconsin–Madison, Madison, WI, USA

    Georgia Benkart

  3. Department of Mathematics, University of Western Australia, Crawley, WA, Australia

    Cheryl E. Praeger

  4. Department of Mathematics, Boğaziçi University, Bebek, Istanbul, Turkey

    Betül Tanbay

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Dickenstein, A. (2019). Algebra and Geometry in the Study of Enzymatic Cascades. In: Araujo, C., Benkart, G., Praeger, C., Tanbay, B. (eds) World Women in Mathematics 2018. Association for Women in Mathematics Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-030-21170-7_2

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