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Monotone Systems in Biology

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Encyclopedia of Systems and Control

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Abstract

Mathematical models arising in biology might sometime exhibit the remarkable feature of preserving ordering of their solutions with respect to initial data: in words, the “more” of x (the state variable) at time 0, the more of it at all subsequent times. Similar monotonicity properties are possibly exhibited also with respect to input levels. When this is the case, important features of the system’s dynamics can be inferred on the basis of purely qualitative or relatively basic quantitative knowledge of the system’s characteristics. We will discuss how monotonicity-related tools can be used to analyze and design biological systems with prescribed dynamical behaviors such as global stability, multistability, or periodic oscillations.

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Bibliography

  • Angeli D, Sontag ED (2003) Monotone control systems. IEEE Trans Aut Cont 48:1684–1698

    Article  MathSciNet  Google Scholar 

  • Angeli D, Sontag ED (2004a) Interconnections of monotone systems with steady-state characteristics. In: Optimal control, stabilization and nonsmooth analysis. Lecture notes in control and information sciences, vol 301. Springer, Berlin, pp 135–154

    Chapter  Google Scholar 

  • Angeli D, Sontag ED (2004b) Multi-stability in monotone input/output systems. Sys Cont Lett 51:185–202

    Article  MathSciNet  Google Scholar 

  • Angeli D, Sontag ED (2008a) Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles. Nonlin Anal Ser B: Real World Appl 9:128–140

    Article  MathSciNet  Google Scholar 

  • Angeli D, Sontag ED (2008b) Oscillations in I/O monotone systems. IEEE Trans on Circ and Sys 55:166–176

    MATH  Google Scholar 

  • Angeli D, Sontag ED (2011) A small-gain result for orthant-monotone systems in feedback: the non sign-definite case. Paper appeared in the 50th IEEE conference on decision and control, Orlando FL, 12–15 Dec 2011

    Google Scholar 

  • Angeli D, Sontag ED (2013) Behavior of responses of monotone and sign-definite systems. In Hüper K, Trumpf J (eds) Mathematical system theory – Festschrift in honor of Uwe Helmke on the occasion of his sixtieth birthday, pp 51–64

    Google Scholar 

  • Angeli D, Ferrell JE, Sontag ED (2004) Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. Proc Natl Acad Sci USA 101:1822–1827

    Article  Google Scholar 

  • Angeli D, de Leenheer P, Sontag ED (2010) Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J Math Bio 61:581–616

    Article  MathSciNet  Google Scholar 

  • Ascensao JA, Datta P, Hancioglu B, Sontag ED, Gennaro ML, Igoshin OA (2016) Non-monotonic response dynamics of glyoxylate shunt genes in Mycobacterium tuberculosis. PLoS Comput Biol 12:e1004741

    Article  Google Scholar 

  • Aswani A, Tomlin C (2009) Monotone piecewise affine systems. IEEE Trans Aut Cont 54:1913–1918

    Article  MathSciNet  Google Scholar 

  • Banaji M (2009) Monotonicity in chemical reaction systems. Dyn Syst 24:1–30

    Article  MathSciNet  Google Scholar 

  • Banaji M, Angeli D (2010) Convergence in strongly monotone systems with an increasing first integral. SIAM J Math Anal 42:334–353

    Article  MathSciNet  Google Scholar 

  • Banaji M, Angeli D (2012) Addendum to “Convergence in strongly monotone systems with an increasing first integral”. SIAM J Math Anal 44:536–537

    Article  MathSciNet  Google Scholar 

  • Blanchini F, Franco E, Giordano G (2015) Structural conditions for oscillations and multistationarity in aggregate monotone systems. In: Proceedings of the IEEE conference on decision and control, pp 609–614

    Google Scholar 

  • Enciso GA, Sontag ED (2005) Monotone systems under positive feedback: multistability and a reduction theorem. Sys Cont Lett 54:159–168

    Article  MathSciNet  Google Scholar 

  • Enciso GA, Sontag ED (2006) Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Disc Contin Dyn Syst 14:549–578

    MathSciNet  MATH  Google Scholar 

  • Enciso GA, Sontag ED (2008) Monotone bifurcation graphs. J Biol Dyn 2:121–139

    Article  MathSciNet  Google Scholar 

  • Farina L, Rinaldi S (2000) Positive linear systems: theory and applications. Wiley, New York

    Book  Google Scholar 

  • Gedeon T, Sontag ED (2007) Oscillations in multi-stable monotone systems with slowly varying feedback. J Differ Equ 239:273–295

    Article  MathSciNet  Google Scholar 

  • Hirsch MW (1985) Systems of differential equations that are competitive or cooperative II: convergence almost everywhere. SIAM J Math Anal 16:423–439

    Article  MathSciNet  Google Scholar 

  • Hirsch MW (1988) Stability and convergence in strongly monotone dynamical systems. Reine Angew Math 383:1–53

    MathSciNet  MATH  Google Scholar 

  • Hirsch MW, Smith HL (2003) Competitive and cooperative systems: a mini-review. In: Positive systems. Lecture notes in control and information science, vol 294. Springer, pp 183–190

    Google Scholar 

  • Maayan A, Iyengar R, Sontag ED (2008) Intracellular regulatory networks are close to monotone systems. IET Syst Biol 2:103–112

    Article  Google Scholar 

  • Mierczynski J (1987) Strictly cooperative systems with a first integral. SIAM J Math Anal 18:642–646

    Article  MathSciNet  Google Scholar 

  • Nikolaev EV, Sontag ED (2016) Quorum-sensing synchronization of synthetic toggle switches: a design based on monotone dynamical systems theory. PLoS Comput Biol 12(4):e1004881. https://doi.org/10.1371/journal.pcbi.1004881

    Article  Google Scholar 

  • Smale S (1976) On the differential equations of species in competition. J Math Biol 3:5–7

    Article  MathSciNet  Google Scholar 

  • Smillie J (1984) Competitive and cooperative tridiagonal systems of differential equations. SIAM J Math Anal 15:530–534

    Article  MathSciNet  Google Scholar 

  • Sootla A, Oyarzún D, Angeli D, Stan GB (2016) Shaping pulses to control bistable systems: analysis, computation and counterexamples. Automatica 63:254–264

    Article  MathSciNet  Google Scholar 

  • Volpert AI, Volpert VA, Volpert VA (1994) Traveling wave solutions of parabolic systems. AMS translations of mathematical monographs, vol 140. American Mathematical Society, Providence

    Book  Google Scholar 

  • Wang L, Sontag ED (2008) Singularly perturbed monotone systems and an application to double phosphorylation cycles. J Nonlin Sci 18:527–550

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, Imperial College London, London, UK

    David Angeli

  2. Dipartimento di Ingegneria dell’Informazione, University of Florence, Florence, Italy

    David Angeli

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  1. David Angeli
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Correspondence to David Angeli .

Editor information

Editors and Affiliations

  1. Electrical and Computer Engineering, Boston University, Boston, MA, USA

    John Baillieul

  2. Automation and Control Solutions, Honeywell, Golden Valley, MN, USA

    Tariq Samad

Section Editor information

  1. Mechanical and Aerospace Engineering, University of California, Los Angeles, CA, USA

    Elisa Franco

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Angeli, D. (2019). Monotone Systems in Biology. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_90-2

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  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_90-2

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-5102-9

  • Online ISBN: 978-1-4471-5102-9

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Chapter history

  1. Latest

    Monotone Systems in Biology
    Published:
    26 September 2019

    DOI: https://doi.org/10.1007/978-1-4471-5102-9_90-2

  2. Original

    Monotone Systems in Biology
    Published:
    28 August 2014

    DOI: https://doi.org/10.1007/978-1-4471-5102-9_90-1

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