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Derivation and Use of Mathematical Models in Systems Biology

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Abstract

Systems Biology has brought together researchers from biology, mathematics, physics and computer science to illuminate our understanding of biological mechanisms. In this chapter, we provide an overview of numerical techniques and considerations required to construct useful models describing natural phenomena. Initially, we show how the dynamics of single molecules up to the development of tissues can be described mathematically over both temporal and spatial scales. Importantly, we discuss the issue of model selection whereby multiple models can describe the same phenomena. We then illustrate how reaction rates can be estimated from datasets and experimental observations as well as highlighting the “parameter identifiability problem”. Finally, we suggest ways in which mathematical models can be used to generate new hypotheses and aid researchers in uncovering the design principles regulating specific biological mechanisms. We hope that this chapter will provide an introduction to the ideas of mathematical modelling for those that wish to incorporate it into their research.

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References

  • Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control AC-19:716–723

    Article  Google Scholar 

  • Alon U (2007) An introduction to systems biology: design principles of biological circuits. Chapman and Hall, London

    Google Scholar 

  • von Bertalanffy L (1968) General system theory: foundations, development, applications. George Braziller, New York

    Google Scholar 

  • Bintu L, Buchler NE, Garcia HG, Gerland U, Hwa T, Kondev J, Phillips R (2005) Transcriptional regulation by the numbers: models. Curr Opin Genet Dev 15:116–124

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Blumel M, Dally N, Jung C (2015) Flowering time regulation in crops – what did we learn from arabidopsis? Curr Opin Biotechnol 32:121–129

    Article  PubMed  Google Scholar 

  • Brady SM, Benfey PN (2009) Development and systems biology: riding the genomics wave towards a systems understanding of root development. In: Annual plant reviews: plant systems biology. Wiley-Blackwell, Chichester, chap 11, pp 304–331

    Google Scholar 

  • Brigandt I (2013) Systems biology and the integration of mechanistic explanation and mathematical explanation. Stud Hist Philos Sci C 44:477–492

    Google Scholar 

  • De Caluwé J, Xiao Q, Hermans C, Verbruggen N, Leloup JC, Gonze D (2016) A compact model for the complex plant circadian clock. Front Plant Sci 7:74

    Article  PubMed  PubMed Central  Google Scholar 

  • Dror RO, Dirks RM, Grossman JP, Xu H, Shaw DE (2012) Biomolecular Simulation: A Computational Microscope for Molecular Biology. Annu Rev Biophys 41:429–452

    Article  CAS  PubMed  Google Scholar 

  • Edelstein-Keshet L (1988) Mathematical models in biology, vol 46. SIAM, Philadelphia

    Google Scholar 

  • Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335–338

    Article  CAS  PubMed  Google Scholar 

  • Fayant P, Girlanda O, Chebli Y, Aubin CE, Villemure I, Geitmann A (2010) Finite element model of polar growth in pollen tubes. Plant Cell 22:2579–2593

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Flamholz A, Noor E, Bar-Even A, Milo R (2012) equilibrator – the biochemical thermodynamics calculator. Nucleic Acids Res 40:D770–D775

    Article  CAS  PubMed  Google Scholar 

  • Friel N, Pettitt AN (2008) Marginal likelihood estimation via power posteriors. J R Stat Soc Ser B Stat Methodol 70:589–607

    Article  Google Scholar 

  • Gabor A, Banga JR (2015) Robust and efficient parameter estimation in dynamic models of biological systems. BMC Syst Biol 9:74

    Article  PubMed  PubMed Central  Google Scholar 

  • Gardiner CW (2004) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, Heidelberg

    Book  Google Scholar 

  • Gillespie DT (2000) The chemical Langevin equation. J Chem Phys 113:297–306

    Article  CAS  Google Scholar 

  • Gillespie DT (2007) Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 58:35–55

    Article  CAS  PubMed  Google Scholar 

  • Gilson MK, Zhou HX (2007) Calculation of protein-ligand binding affinities. Annu Rev Biophys Biomol Struct 36:21–42

    Article  CAS  PubMed  Google Scholar 

  • Goodwin BC (1965) Oscillatory behavior in enzymatic control processes. Adv Enzyme Regul 3:425–438

    Article  CAS  PubMed  Google Scholar 

  • Goriely A, Tabor M (2008) Mathematical modeling of hyphal tip growth. Fungal Biol Rev 22:77–83

    Article  Google Scholar 

  • Grima R, Thomas P, Straube AV (2011) How accurate are the nonlinear chemical Fokker-Planck and chemical Langevin equations? J Chem Phys 135:084,103

    Article  Google Scholar 

  • Guerriero ML, Pokhilko A, Fernandez AP, Halliday KJ, Millar AJ, Hillston J (2012) Stochastic properties of the plant circadian clock. J R Soc Interface 9:744–756

    Article  CAS  PubMed  Google Scholar 

  • Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N, Singhal M, Xu L, Mendes P, Kummer U (2006) Copasi – a complex pathway simulator. Bioinformatics 22:3067–3074

    Article  CAS  PubMed  Google Scholar 

  • Hucka M, Finney A, Sauro HM, Bolouri H, Doyle JC, Kitano H (2003) The systems biology markup language (sbml): a medium for representation and exchange of biochemical network models. Bioinformatics 19:524–531

    Article  CAS  PubMed  Google Scholar 

  • Jones GW, Chapman SJ (2012) Modeling growth in biological materials. SIAM Rev 54:52–118

    Article  Google Scholar 

  • van Kampen NG (1981) Ito versus stratonovich. J Stat Phys 24:175–187

    Article  Google Scholar 

  • Kato N, He H, Steger AP (2010) A systems model of vesicle trafficking in arabidopsis pollen tubes. Plant Physiol 152:590–601

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Kholodenko BN (2006) Cell-signalling dynamics in time and space. Nat Rev Mol Cell Biol 7:165–176

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Kitano H (2002a) Computational systems biology. Nature 420:206–210

    Article  CAS  PubMed  Google Scholar 

  • Kitano H (2002b) Systems biology: a brief overview. Science 295:1662–1664

    Article  CAS  PubMed  Google Scholar 

  • Klipp E, Herwig R, Kowald A, Wierling C, Lehrach H (2005) Systems biology in practice. Wiley-VCH, Weinheim

    Book  Google Scholar 

  • Kondo S, Miura T (2010) Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329:1616–1620

    Article  CAS  PubMed  Google Scholar 

  • Kroeger JH, Geitmann A, Grant M (2008) Model for calcium dependent oscillatory growth in pollen tubes. J Theor Biol 253:363–374

    Article  CAS  PubMed  Google Scholar 

  • Locke JCW, Southern MM, Kozma-Bognar L, Hibberd V, Brown PE, Turner MS, Millar AJ (2005) Extension of a genetic network model by iterative experimentation and mathematical analysis. Mol Syst Biol 2005.0013

    Google Scholar 

  • Locke JCW, Kozma-Bognar L, Gould PD, Feher B, Kevei E, Nagy F, Turner MS, Hall A, Millar AJ (2006) Experimental validation of a predicted feedback loop in the multi-oscillator clock of arabidopsis thaliana. Mol Syst Biol 2:59

    Article  PubMed  PubMed Central  Google Scholar 

  • Lotka AJ (1920) Analytical note on certain rhythmic relations in organic systems. Proc Natl Acad Sci USA 6:410–415

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • MacDonald JT, Barnes C, Kitney RI, Freemont PS, Stan GBV (2011) Computational design approaches and tools for synthetic biology. Integr Biol 3:97–108

    Article  Google Scholar 

  • McClung CR (2014) Wheels within wheels: new transcriptional feedback loops in the arabidopsis circadian clock. F1000Prime Rep 6:2

    Google Scholar 

  • McKay MD, Beckman RJ, Conover WJ (2000) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42:55–61

    Article  Google Scholar 

  • Mogilner A, Allard J, Wollman R (2012) Cell polarity: quantitative modeling as a tool in cell biology. Science 336:175–179

    Article  CAS  PubMed  Google Scholar 

  • Murray JD (2002a) Mathematical biology I: an introduction. Springer, New York

    Google Scholar 

  • Murray JD (2002b) Mathematical biology II: spatial models and biomedical applications. Springer, New York

    Google Scholar 

  • Orth JD, Thiele I, Palsson BO (2010) What is flux balance analysis? Nat Biotechnol 28:245–248

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Paulsson J (2004) Summing up the noise in gene networks. Nature 427:415–418

    Article  CAS  PubMed  Google Scholar 

  • Phillips R, Kondev J, Theriot J, Garcia HG (2013) Physical biology of the cell, 2nd edn. Garland Science, New York

    Google Scholar 

  • Pokhilko A, Hodge SK, Stratford K, Knox K, Edwards KD, Thomson AW, Mizuno T, Millar AJ (2010) Data assimilation constrains new connections and components in a complex, eukaryotic circadian clock model. Mol Syst Biol 6:416

    Article  PubMed  PubMed Central  Google Scholar 

  • Pokhilko A, Fernandez AP, Edwards KD, Southern MM, Halliday KJ, Millar AJ (2012) The clock gene circuit in arabidopsis includes a repressilator with additional feedback loops. Mol Syst Biol 8:574

    Article  PubMed  PubMed Central  Google Scholar 

  • Rao YVC (2004) An introduction to thermodynamics. Universities Press, Telangana

    Google Scholar 

  • Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmüller U, Timmer J (2009) Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25:1923–1929

    Article  CAS  PubMed  Google Scholar 

  • Raue A, Karlsson J, Saccomani MP, Jirstrand M, Timmer J (2014) Comparison of approaches for parameter identifiability analysis of biological systems. Bioinformatics p doi:10.1093/bioinformatics/btu006

    Google Scholar 

  • Rausenberger J, Tscheuschler A, Nordmeier W, Wüst F, Timmer J, Schäfer E, Fleck C, Hiltbrunner A (2011) Photoconversion and nuclear trafficking cycles determine phytochrome A’s response profile to far-red light. Cell 146:813–825

    Article  CAS  PubMed  Google Scholar 

  • Rybel BD, Adibi M, Breda AS, Wendrich JR, Smit ME, Novak O, Yamaguchi N, Yoshida S, van Isterdael G, Palovaara J, Nijsse B, Boekschoten MV, Hooiveld G, Beeckman T, Wagner D, Ljung K, Fleck C, Weijers D (2014) Integration of growth and patterning during vascular tissue formation in arabidopsis. Science 345:1255,215

    Article  Google Scholar 

  • Schomburg I, Chang A, Ebeling C, Gremse M, Heldt C, Huhn G, Schomburg D (2004) Brenda, the enzyme database: updates and major developments. Nucleic Acids Res 32:D431–D433

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Seaton DD, Smith RW, Song YH, MacGregor DR, Stewart K, Steel G, Foreman J, Penfield S, Imaizumi T, Millar AJ, Halliday KJ (2015) Linked circadian outputs control elongation growth and flowering in response to photoperiod and temperature. Mol Syst Biol 11:776

    Article  PubMed  PubMed Central  Google Scholar 

  • Shoval O, Sheftel H, Shinar G, Hart Y, Ramote O, Mayo A, Dekel E, Kavanagh K, Alon U (2012) Evolutionary trade-offs, pareto optimality, and the geometry of phenotype space. Science 336:1157–1160

    Article  CAS  PubMed  Google Scholar 

  • Simon D (2013) Evolutionary optimization algorithms: biologically inspired and population-based approaches to computer intelligence. Wiley, New Jersey

    Google Scholar 

  • Song YH, Smith RW, To BJ, Millar AJ, Imaizumi T (2012) Fkf1 conveys timing information for constans stabilization in photoperiodic flowering. Science 336:1045–1049

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Thomas P, Matuschek H, Grima R (2013) How reliable is the linear noise approximation of gene regulatory networks? BMC Genomics 14((Suppl 4):S5)

    Google Scholar 

  • Vyshemirsky V, Girolami MA (2008) Bayesian ranking of biochemical system models. Bioinformatics 24:833–839

    Article  CAS  PubMed  Google Scholar 

  • Westerhoff HV, Kell DB (2007) The methodologies of systems biology. Elsevier, Amsterdam

    Book  Google Scholar 

  • Wittig U, Kania R, Golebiewski M, Rey M, Shi L, Jong L, Algaa E, Weidemann A, Sauer-Danzwith H, Mir S, Krebs O, Bittkowski M, Wetsch E, Rojas I, Muller W (2012) Sabio-rk: the database for biochemical reaction kinetics. Nucleic Acids Res 40:D790–D796

    Article  CAS  PubMed  Google Scholar 

  • Wolpert L (1996) One hundred years of positional information. Trends Genet 12:359–364

    Article  CAS  PubMed  Google Scholar 

  • Zeilinger MN, Farre EM, Taylor SR, Kay SA, III FJD (2006) A novel computational model of the circadian clock in arabidopsis that incorporates PRR7 and PRR9. Mol Syst Biol 2:58

    Article  PubMed  PubMed Central  Google Scholar 

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Acknowledgements

Given the large field of mathematical modelling in biological systems we would like to apologise to any readers who feel that we have neglected important references. The references contained herein are those that the authors believe would provide a useful introduction to interested readers. RWS is funded by FP7 Marie Curie Initial Training Network grant agreement number 316723. CF is funded by HFSP Research grant RGP0025/2013.

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Authors and Affiliations

  1. Laboratory of Systems and Synthetic Biology, Wageningen UR, 6700 EJ, Wageningen, The Netherlands

    Robert W. Smith & Christian Fleck

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  1. Robert W. Smith
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  2. Christian Fleck
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Correspondence to Christian Fleck .

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Editors and Affiliations

  1. Department of Molecular Biology, University of Salzburg, Salzburg, Austria

    Gerhard Obermeyer

  2. Cell Biology Molecular Genetics, University of Maryland, College Park, Maryland, USA

    José Feijó

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Smith, R.W., Fleck, C. (2017). Derivation and Use of Mathematical Models in Systems Biology. In: Obermeyer, G., Feijó, J. (eds) Pollen Tip Growth. Springer, Cham. https://doi.org/10.1007/978-3-319-56645-0_13

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