algebraic systems biology a case study for the wnt pathway

Algebraic Systems Biology: A Case Study for the Wnt Pathway

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• Published: 08 December 2015
• Volume 78 , pages 21–51, ( 2016 )

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• Elizabeth Gross 1 ,
• Heather A. Harrington 2 ,
• Zvi Rosen 3 &
• Bernd Sturmfels 4  

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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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San José State University, San José, CA, USA

Elizabeth Gross

University of Oxford, Oxford, England

Heather A. Harrington

Pennsylvania State University, State College, PA, USA

University of California at Berkeley, Berkeley, CA, USA

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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Received : 11 February 2015

Accepted : 12 November 2015

Published : 08 December 2015

Issue Date : January 2016

DOI : https://doi.org/10.1007/s11538-015-0125-1

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Bulletin of Mathematical Biology > 2016 > 78 > 1 > 21-51

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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Elizabeth Gross

• San José State University, San José, USA

Heather A. Harrington

• University of Oxford, Oxford, England
• Pennsylvania State University, State College, USA

Bernd Sturmfels

• University of California at Berkeley, Berkeley, USA

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

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Algebraic Systems Biology: A Case Study for the Wnt Pathway

Affiliations.

• 1 San José State University, San José, CA, USA. [email protected].
• 2 University of Oxford, Oxford, England. [email protected].
• 3 Pennsylvania State University, State College, PA, USA. [email protected].
• 4 University of California at Berkeley, Berkeley, CA, USA. [email protected].
• PMID: 26645985
• DOI: 10.1007/s11538-015-0125-1

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: -catenin/Wnt signaling; Algebraic matroids; Biochemical reaction networks; Nonlinear algebra; Polyhedra; Steady-state variety.

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Algebraic Systems Biology: A Case Study for the Wnt Pathway

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Algebraic systems biology: Wnt signaling as a case study Seminar

Event details.

Applied Mathematics Seminar

In this talk I will focus on a specific signaling pathway of gene regulation and present different approaches for analysing models from systems biology. The focus of these techniques will be a set of models of the Wnt signaling pathway, which is involved in development, adult tissue cells and cancer.

We propose a new mechanistic model that includes spatial localization, compare this model and existing models from the literature to data. We apply statistical and algebraic approaches-- focusing on both parameter-dependent and parameter-independent methods to compare and reject models. We use ideas from numerical algebraic geometry, combinatorics and statistics to better understand the Wnt signaling system and we also inform design of experiments. The techniques are applicable to other problems in systems biology.

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Heather Harrington , University of Oxford. EPSRC Postdoctoral Research Fellow

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BJ Stolz, J Tanner, HA Harrington, V Nanda Geometric anomaly detection in data Proceedings of the national academy of sciences 117 (33), 19664-19669

E Gross, H Harrington, N Meshkat, A Shiu Joining and decomposing reaction networks Journal of mathematical biology 80, 1683-1731

MF Adamer, HA Harrington, EA Gaffney, TE Woolley Coloured noise from stochastic inflows in reaction–diffusion systems Bulletin of Mathematical Biology 82, 1-28

E Yeung, S McFann, L Marsh, E Dufresne, S Filippi, HA Harrington, ... Inference of multisite phosphorylation rate constants and their modulation by pathogenic mutations Current Biology 30 (5), 877-882. e6

A Barbensi, D Celoria, HA Harrington, A Stasiak, D Buck Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology Science advances 6 (9), eaay1458

HA Harrington, D Mehta, HM Byrne, JD Hauenstein Decomposing the parameter space of biological networks via a numerical discriminant approach Maple in Mathematics Education and Research: Third Maple Conference, MC 2019 …

Dufresne E, Edwards PB, Harrington HA, Hauenstein JD Sampling real algebraic varieties for topological data analysis 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA) 2020 1531-1536. arXiv:1802.07716.

Byrne HM, Harrington HA,Muschel R, Reinert G, Stolz BJ, Tillmann U Topology characterises spatial networks of tumour vasculature Mathematics Today, 2019 55(5):206-210. arXiv:1907.08711.

E Gross, H Harrington, N Meshkat, A Shiu Linear compartmental models: input-output equations and operations that preserve identifiability SIAM Journal on Applied Mathematics 79 (4), 1423-1447

HA Harrington, N Otter, H Schenck, U Tillmann Stratifying multiparameter persistent homology SIAM Journal on Applied Algebra and Geometry 3 (3), 439-471

A Barbensi, D Buck, HA Harrington, M Lackenby Double branched covers of knotoids arXiv preprint arXiv:1811.09121

L Speidel, HA Harrington, SJ Chapman, MA Porter Topological data analysis of continuum percolation with disks Physical Review E 98 (1), 012318

MF Adamer, TE Woolley, HA Harrington Graph-facilitated resonant mode counting in stochastic interaction networks Journal of the Royal Society Interface 14 (137), 20170447

N Otter, MA Porter, U Tillmann, P Grindrod, HA Harrington A roadmap for the computation of persistent homology EPJ Data Science 6, 1-38

Smith RCG, Stumpf PS, Ridden SJ, Sim A, Filippi S, Harrington HA, MacArthur BD Nanog fluctuations in embryonic stem cells highlight the problem of measurement in cell biology Biophysical journal 112 (12), 2641-2652 DOI

BJ Stolz, HA Harrington, MA Porter Persistent homology of time-dependent functional networks constructed from coupled time series Chaos: An Interdisciplinary Journal of Nonlinear Science 27 (4) DOI

Drellich E, Gainer-Dewar A, Harrington HA, He Q, Heitsch C, Poznanovic S. Geometric combinatorics and computational molecular biology: Branching polytopes for RNA sequences Accepted (to appear AMS Contemporary Mathematics Volume). arXiv:1509.04090. 

Kay SK, Harrington HA, Shepherd S, Brennan K, Dale T, Osborne JM, Gavaghan DJ, Byrne HM The role of the Hes1 crosstalk hub in Notch-Wnt interactions of the intestinal crypt PLoS computational biology 13 (2), e1005400 DOI

R Van Gorder, H Harrington Reduction of dimension for nonlinear dynamical systems Nonlinear Dynamics 88 (1)

E Gross, B Davis, KL Ho, DJ Bates, HA Harrington Numerical algebraic geometry for model selection and its application to the life sciences Journal of The Royal Society Interface 13 (123), 20160256 DOI

BJ Stolz, HA Harrington, MA Porter The topological" shape" of Brexit arXiv preprint arXiv:1610.00752

E Dufresne, HA Harrington, DV Raman The geometry of sloppiness arXiv preprint arXiv:1608.05679

D Levy, HA Harrington, RA Van Gorder Role of seasonality on predator–prey–subsidy population dynamics Journal of theoretical biology 396, 163-181

HA Harrington, KL Ho, N Meshkat Differential algebra for model comparison arXiv preprint arXiv:1603.09730

A Seigal, M Beguerisse-Diaz, B Schoeberl, M Niepel, HA Harrington Tensors and algebra give interpretable groups for crosstalk mechanisms in breast cancer arXiv preprint arXiv:1612.08116 1

E Gross, HA Harrington, Z Rosen, B Sturmfels Algebraic systems biology: a case study for the Wnt pathway Bulletin of mathematical biology 78, 21-51 DOI

AL MacLean, HA Harrington, MPH Stumpf, HM Byrne Mathematical and statistical techniques for systems medicine: the Wnt signaling pathway as a case study Systems Medicine, 405-439

*Taylor D, *Klimm F, Harrington HA, Kramar M, Mischaikow K, Porter MA, Mucha PJ. Topological data analysis of contagion maps for examining spreading processes on networks Nature communications 6 (1), 7723 

AL MacLean, Z Rosen, HM Byrne, HA Harrington Parameter-free methods distinguish Wnt pathway models and guide design of experiments Proceedings of the National Academy of Sciences 112 (9), 2652-2657

Jonanovic G, Sheng X, Ale A, Feliu E, Harrington HA, Kirk P, Wiuf C, Buck M, Stumpf MPH. Phosphorelay of non-orthodox two component systems functions through a bi-molecular mechanism in vivo: the case of ArcB Molecular BioSystems 11 (5), 1348-1359

AL MacLean, HA Harrington, MPH Stumpf, MDH Hansen Epithelial-mesenchymal transition in metastatic cancer cell populations affects tumor dormancy in a simple mathematical model Biomedicines 2 (4), 384-402

Michailovici I, Harrington H, Azogui HH, Yahalom-Ronen Y, Plotnikov A, Ching S, Stumpf MPH, Klein OD, Seger R, Tzahor E. Nuclear to cytoplasmic shuttling of ERK promotes differentiation of muscle stem/progenitor cells Development 141 (13), 2611-2620

HA Harrington, E Feliu, C Wiuf, MPH Stumpf Cellular compartments cause multistability and allow cells to process more information Biophysical journal 104 (8), 1824-1831

HA Harrington, MB Díaz, MP Rombach, LM Keating, MA Porter Teach network science to teenagers arXiv preprint arXiv:1302.6567

Chaidos A, Barnes C, Cowan G, May P, Melo V, Hatjiharissi E, Papaioannou M, Harrington H, Doolittle H, Terpos E, Abdalla S, Yarranton H, Naresh K, Foroni L, Reid A, Rahemtulla A, Stumpf M, Roberts I, Karadimitris A. Clinical drug resistance linked to interconvertible phenotypic and functional states of tumor-propagating cells in multiple myeloma Blood, The Journal of the American Society of Hematology 121 (2), 318-328

*Harrington HA, *Ho K, Thorne T, Stumpf MPH. A parameter-free model selection criterion based on steady-state coplanarity Proc Nat Acad Sci. 2012 Sept;109(39):15746-15751.arXiv:1109.3670. DOI

Harrington HA, Komorowski M, Beguerisse-Díaz M, Ratto GM, Stumpf MPH. Mathematical modeling reveals the functional implications of the different nuclear shuttling rates of Erk1 and Erk2 Physical biology 9 (3), 036001

RJ Tanaka, M Ono, HA Harrington Skin barrier homeostasis in atopic dermatitis: feedback regulation of kallikrein activity PLoS One 6 (5), e19895

KL Ho, HA Harrington Bistability in apoptosis by receptor clustering PLoS computational biology 6 (10), e1000956

Boyle J, Harrington HA, Piper E, Elderfield K, Stark J, Landis RC, Haskard DO. Coronary intraplaque hemorrhage evokes a novel atheroprotective macrophage phenotype The American journal of pathology 174 (3), 1097-1108

HA Harrington, KL Ho, S Ghosh, KC Tung Construction and analysis of a modular model of caspase activation in apoptosis Theoretical Biology and Medical Modelling 5, 1-15

HA Harrington, M Maier, L Naidoo, N Whitaker, PG Kevrekidis A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors Mathematical and Computer Modelling 46 (3-4), 513-524

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• DOI: 10.1007/s11538-015-0125-1
• Corpus ID: 6403736

Algebraic Systems Biology: A Case Study for the Wnt Pathway

• Elizabeth Gross , H. Harrington , +1 author B. Sturmfels
• Published in Bulletin of Mathematical… 11 February 2015
• Mathematics, Biology

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Supporting statements, 64 citations, algebra and geometry in the study of enzymatic cascades, decomposing the parameter space of biological networks via a numerical discriminant approach, numerical algebraic geometry for model selection and its application to the life sciences, symbolic analysis of multiple steady states in a mapk chemical reaction network, a case study on the parametric occurrence of multiple steady states, algebra, geometry and topology of erk kinetics, identifying the parametric occurrence of multiple steady states for some biological networks, complexity of model testing for dynamical systems with toric steady states.

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A Deficiency-Based Approach to Parametrizing Positive Equilibria of Biochemical Reaction Systems

Newton-okounkov bodies of chemical reaction systems.

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Mathematical and statistical techniques for systems medicine: the wnt signaling pathway as a case study., complex-linear invariants of biochemical networks..

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Parameter-free methods distinguish Wnt pathway models and guide design of experiments

Variable elimination in chemical reaction networks with mass-action kinetics, identifiable reparametrizations of linear compartment models, chemical reaction systems with toric steady states, coefficient-parameter polynomial continuation, multiple equilibria in complex chemical reaction networks: i. the injectivity property, computing algebraic matroids, wnt signalling pathway parameters for mammalian cells, related papers.

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Glioma stem cells as promoter of glioma progression: a systematic review of molecular pathways and targeted therapies.

1. Introduction

2. materials and methods, 2.1. literature review, 2.2. data extraction, 2.3. outcomes, 2.4. risk of bias assessment, 2.5. statistical analysis, 3.1. literature review, 3.2. data analysis, 4. discussion, 4.1. gcss cell lines, 4.2. molecular pathways involved in gcs-mediated glioma progression, 4.3. therapeutics targets and agents, 4.4. limitations and future directions, 5. conclusions, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Agosti, E.; Antonietti, S.; Ius, T.; Fontanella, M.M.; Zeppieri, M.; Panciani, P.P. Glioma Stem Cells as Promoter of Glioma Progression: A Systematic Review of Molecular Pathways and Targeted Therapies. Int. J. Mol. Sci. 2024 , 25 , 7979. https://doi.org/10.3390/ijms25147979

Agosti E, Antonietti S, Ius T, Fontanella MM, Zeppieri M, Panciani PP. Glioma Stem Cells as Promoter of Glioma Progression: A Systematic Review of Molecular Pathways and Targeted Therapies. International Journal of Molecular Sciences . 2024; 25(14):7979. https://doi.org/10.3390/ijms25147979

Agosti, Edoardo, Sara Antonietti, Tamara Ius, Marco Maria Fontanella, Marco Zeppieri, and Pier Paolo Panciani. 2024. "Glioma Stem Cells as Promoter of Glioma Progression: A Systematic Review of Molecular Pathways and Targeted Therapies" International Journal of Molecular Sciences 25, no. 14: 7979. https://doi.org/10.3390/ijms25147979

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