research-article

A Case Study on the Parametric Occurrence of Multiple Steady States

Authors:

Russell Bradford,

James H. Davenport,

Matthew England,

Vladimir Gerdt,

Dima Grigoriev,

Ovidiu Radulescu,

Andreas WeberAuthors Info & Claims

ISSAC '17: Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

Pages 45 - 52

https://doi.org/10.1145/3087604.3087622

Published: 23 July 2017 Publication History

Abstract

We consider the problem of determining multiple steady states for positive real values in models of biological networks. Investigating the potential for these in models of the mitogen-activated protein kinases (MAPK) network has consumed considerable effort using special insights into the structure of corresponding models. Here we apply combinations of symbolic computation methods for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition. We determine multistationarity of an 11-dimensional MAPK network when numeric values are known for all but potentially one parameter. More precisely, our considered model has 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment, and furthermore positivity conditions on all variables and parameters.

References

[1]

D. S. Arnon, G. E. Collins, and S. McCallum. Cylindrical algebraic decomposition I: The basic algorithm. SIAM J. Comput., 13(4):865--877, 1984.

Digital Library

[2]

D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler. Bertini: Software for numerical algebraic geometry.

[3]

U. S. Bhalla and R. Iyengar. Emergent properties of networks of biological signaling pathways. Science, 283(5400):381--387, 1999.

[4]

C. Chen, J. Davenport, J. May, M. Moreno Maza, B. Xia, and R. Xiao. Triangular decomposition of semi-algebraic systems. J. Symb. Comput., 49:3--26, 2013.

Digital Library

[5]

C. Chen, M. Moreno Maza, B. Xia, and L. Yang. Computing cylindrical algebraic decomposition via triangular decomposition. In Proceedings of the ISSAC 2009, pages 95--102. ACM, 2009.

Digital Library

[6]

C. Conradi, E. Feliu, M. Mincheva, and C. Wiuf. Identifying parameter regions for multistationarity. Preprint: arXiv:1608.03993, 2017.

[7]

C. Conradi, D. Flockerzi, and J. Raisch. Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space. Math. Biosci., 211(1):105--31, 2008.

[8]

C. Conradi and M. Mincheva. Catalytic constants enable the emergence of bistability in dual phosphorylation. Journal of The Royal Society Interface, 11(95), 2014.

[9]

G. Craciun, A. Dickenstein, A. Shiu, and B. Sturmfels. Toric dynamical systems. J. Symb. Comput., 44(11):1551--1565, 2009.

Digital Library

[10]

A. Dolzmann and T. Sturm. Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin, 31(2):2--9, 1997.

Digital Library

[11]

M. England, R. Bradford, and J. Davenport. Improving the use of equational constraints in cylindrical algebraic decomposition. In Proceedings of the ISSAC 2015, pages 165--172. ACM, 2015.

Digital Library

[12]

M. England and J. Davenport. The complexity of cylindrical algebraic decomposition with respect to polynomial degre. In Proceedings of the CASC 2016, volume 9890 of LNCS, pages 172--192. Springer, 2016.

[13]

M. Feinberg. Stability of complex isothermal reactors--I. the deficiency zero and deficiency one theorems. Chem. Eng. Sci., 42(10):2229--2268, 1987.

[14]

D. Grigoriev and N. N. Vorobjov. Solving systems of polynomial inequalities in subexponential time. J. Symb. Comput., 5:37--64, 1988.

Digital Library

[15]

E. Gross, H. A. Harrington, Z. Rosen, and B. Sturmfels. Algebraic systems biology: A case study for the Wnt pathway. Bull. Math. Biol., 78(1):21--51, 2016.

[16]

H. Hong, R. Liska, and S. Steinberg. Testing stability by quantifier elimination. J. Symb. Comput., 24(2):161--187, 1997.

Digital Library

[17]

M. D. Johnston. A note on "MAPK networks and their capacity for multistationarity due to toric steady states". arXiv:1407.5651, 2014.

[18]

B. Joshi and A. Shiu. A Survey of Methods for Deciding Whether a Reaction Network is Multistationary. Math. Model. Nat. Phenom., 10(5):47--67, 2015.

[19]

C. Li, M. Donizelli, N. Rodriguez, H. Dharuri, L. Endler, V. Chelliah, L. Li, E. He, A. Henry, M. I. Stefan, J. L. Snoep, M. Hucka, N. Le Novère, and C. Laibe. BioModels database: An enhanced, curated and annotated resource for published quantitative kinetic models. BMC Systems Biology, 4:92, 2010.

[20]

N. I. Markevich, J. B. Hoek, and B. N. Kholodenko. Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades. J. Cell Biol., 164(3):353--359, 2004.

[21]

M. Pérez Millán and A. Dickenstein. The structure of MESSI biological systems. Preprint: arXiv:1612.08763, 2016.

[22]

M. Pérez Millán and A. G. Turjanski. MAPK's networks and their capacity for multistationarity due to toric steady states. Math. Biosci., 262:125--37, 2015.

[23]

S. Schuster and T. Höfer. Determining all extreme semi-positive conservation relations in chemical reaction systems: a test criterion for conservativity. J. Chem. Soc. Faraday T., 87(16):2561--2566, 1991.

[24]

A. J. Sommese, J. Verschelde, and C. W. Wampler. Introduction to numerical algebraic geometry. In Solving Polynomial Equations: Foundations, Algorithms, and Applications, pages 301--337. Springer, 2005.

[25]

D. Wang. Elimination Methods. Springer, 2000.

[26]

D. Wang and B. Xia. Stability analysis of biological systems with real solution classification. In Proceedings of the ISSAC 2005, pages 354--361. ACM, 2005.

Digital Library

[27]

V. Weispfenning. Quantifier elimination for real algebra--the quadratic case and beyond. Appl. Algebr. Eng. Comm., 8(2):85--101, 1997.

[28]

G. Weng, U. S. Bhalla, and R. Iyengar. Complexity in biological signaling systems. Science, 284(5411):92--6, 1999.

Cited By

Reyes BOtero-Muras IPetyuk V(2022)A numerical approach for detecting switch-like bistability in mass action chemical reaction networks with conservation lawsBMC Bioinformatics10.1186/s12859-021-04477-x23:1Online publication date: 4-Jan-2022
https://doi.org/10.1186/s12859-021-04477-x
Röst GSadeghimanesh A(2021)Exotic Bifurcations in Three Connected Populations with Allee EffectInternational Journal of Bifurcation and Chaos10.1142/S021812742150202331:13Online publication date: 25-Oct-2021
https://doi.org/10.1142/S0218127421502023
Kruff NLüders CRadulescu OSturm TWalcher S(2021)Algorithmic Reduction of Biological Networks with Multiple Time ScalesMathematics in Computer Science10.1007/s11786-021-00515-215:3(499-534)Online publication date: 8-Jul-2021
https://doi.org/10.1007/s11786-021-00515-2
Show More Cited By

Index Terms

A Case Study on the Parametric Occurrence of Multiple Steady States
1. Applied computing
  1. Life and medical sciences
    1. Computational biology
      1. Biological networks
2. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Equation and inequality solving algorithms

Recommendations

Case study: Molecular dynamics study on the ligand recognition by tandem SH3 domains of p47phox, regulating NADPH oxidase activity

The phagocyte NADPH oxidase complex plays a crucial role in host defense against microbial infection through the production of superoxides. Chronic granulomatous disease (CGD) is an inherited immune deficiency caused by the absence of certain components ...
Steady states of constrained reaction systems

Reaction systems, a mathematical formalism inspired by the mechanisms within a biological cell, focuses on an abstract set-based representation of chemical reactions via facilitation and inhibition. The simple yet elegant nature of reaction systems ...
Characterizing the protonation states of the catalytic residues in apo and substrate-bound human T-cell leukemia virus type 1 protease

Display Omitted Protonation states of the catalytic residues in HTLV-1 protease were investigated.In apo HTLV-1 protease, the two catalytic residues are both unprotonated.In HTLV-1 protease-substrate complex, Asp32 is protonated, Asp32' is ...

Comments

Information & Contributors

Information

Published In

cover image ACM Other conferences

ISSAC '17: Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

July 2017

466 pages

ISBN:9781450350648

DOI:10.1145/3087604

Editor:
Michael Burr
Clemson University, USA
,
General Chair:
Chee K. Yap
New York University, USA
,
Program Chair:
Mohab Safey El Din
University Pierre et Marie Curie, France

Copyright © 2017 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

In-Cooperation

SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 23 July 2017

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Research-article

Funding Sources

Conference

ISSAC '17

ISSAC '17: International Symposium on Symbolic and Algebraic Computation

July 23 - 28, 2017

Kaiserslautern, Germany

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

17
Total Citations
View Citations
259
Total Downloads

Downloads (Last 12 months)12
Downloads (Last 6 weeks)1

Reflects downloads up to 21 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

Reyes BOtero-Muras IPetyuk V(2022)A numerical approach for detecting switch-like bistability in mass action chemical reaction networks with conservation lawsBMC Bioinformatics10.1186/s12859-021-04477-x23:1Online publication date: 4-Jan-2022
https://doi.org/10.1186/s12859-021-04477-x
Röst GSadeghimanesh A(2021)Exotic Bifurcations in Three Connected Populations with Allee EffectInternational Journal of Bifurcation and Chaos10.1142/S021812742150202331:13Online publication date: 25-Oct-2021
https://doi.org/10.1142/S0218127421502023
Kruff NLüders CRadulescu OSturm TWalcher S(2021)Algorithmic Reduction of Biological Networks with Multiple Time ScalesMathematics in Computer Science10.1007/s11786-021-00515-215:3(499-534)Online publication date: 8-Jul-2021
https://doi.org/10.1007/s11786-021-00515-2
Rahkooy HSturm T(2021)Parametric Toricity of Steady State Varieties of Reaction NetworksComputer Algebra in Scientific Computing10.1007/978-3-030-85165-1_18(314-333)Online publication date: 13-Sep-2021
https://dl.acm.org/doi/10.1007/978-3-030-85165-1_18
Nam KGyori BAmethyst SBates DGunawardena J(2020)Robustness and parameter geography in post-translational modification systemsPLOS Computational Biology10.1371/journal.pcbi.100757316:5(e1007573)Online publication date: 4-May-2020
https://doi.org/10.1371/journal.pcbi.1007573
England MBradford RDavenport J(2020)Cylindrical algebraic decomposition with equational constraintsJournal of Symbolic Computation10.1016/j.jsc.2019.07.019100:C(38-71)Online publication date: 1-Sep-2020
https://dl.acm.org/doi/10.1016/j.jsc.2019.07.019
Florescu DEngland M(2020)Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output CorrectnessMathematical Aspects of Computer and Information Sciences10.1007/978-3-030-43120-4_27(341-356)Online publication date: 18-Mar-2020
https://doi.org/10.1007/978-3-030-43120-4_27
Huang ZEngland MWilson DBridge JDavenport JPaulson L(2019)Using Machine Learning to Improve Cylindrical Algebraic DecompositionMathematics in Computer Science10.1007/s11786-019-00394-813:4(461-488)Online publication date: 3-Apr-2019
https://doi.org/10.1007/s11786-019-00394-8
Conradi CIosif AKahle T(2019)Multistationarity in the Space of Total Concentrations for Systems that Admit a Monomial ParametrizationBulletin of Mathematical Biology10.1007/s11538-019-00639-481:10(4174-4209)Online publication date: 22-Jul-2019
https://doi.org/10.1007/s11538-019-00639-4
Sadeghimanesh AFeliu E(2019)The Multistationarity Structure of Networks with Intermediates and a Binomial Core NetworkBulletin of Mathematical Biology10.1007/s11538-019-00612-181:7(2428-2462)Online publication date: 17-May-2019
https://doi.org/10.1007/s11538-019-00612-1
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents