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The combinatorics of modeling and analyzing biological systems

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Abstract

The purpose of this paper is to present a strictly mathematical model for interaction networks, to address the question of steady-state analysis, and to outline an approach for reconstructing models from experimental data. Our expositions require notations and basic results from discrete mathematics. Therefore, we also introduce some elementary background material from this field.

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References

  • Andrec M, Kholodenko B, Levy R, Sontag E (2005) Interference of signaling and gene regulatory networks by steady-state perturbation experiments: structure and accuracy. J Theor Biol 232:427–441

    MathSciNet  Google Scholar 

  • Bertsimas D, Weismantel R (2005) Optimization over integers. Dynamic Ideas. Belmont

    Google Scholar 

  • Bruns W, Gubaladze J, Henk M, Martin A, Weismantel R (1999) A counterexample to an integer analogue of Carathéodory’s theorem. J Pure Appl Math 510:179–185

    MATH  Google Scholar 

  • Carathéodory C (1911) Über den Variablilitätsbereich der fourierschen Konstanten von positiven harmonischen Funktionen. Rendiconto del Circolo Matematico di Paleromo 32:193–217

    Article  MATH  Google Scholar 

  • Conti P, Traverso C (1991) Buchberger algorithm and integer programming. Lecture notes in computer science 539. Springer, Berlin, pp 130–139

  • Cook W (1983) Operations that preserve total dual integrality. Oper Res Lett 2:31–35

    Article  MathSciNet  MATH  Google Scholar 

  • Cook W, Fonlupt J, Schrijver A (1986) An integer analogue of Carathéodory’s theorem. J Comb Theory B 40:63–70

    Article  MathSciNet  MATH  Google Scholar 

  • Cornuéjols G, Urbaniak R, Weismantel R, Wolsey LA (1997) Decomposition of integer programs and of generating sets. In: Burkard R, Woeginger GJ (eds) Proceedings of the 5th European symposium on algorithms, pp 92–103

  • Durzinsky M, Marwan W, Wagler A, Weismantel R (2008a) Automatic reconstruction of molecular and genetic networks from experimental time series data. Biosystems 93:181–190

    Article  Google Scholar 

  • Durzinsky M, Wagler A, Weismantel R (2008b) A combinatorial approach to reconstruct Petri nets from experimental data. In: Heiner M, Uhrmacher AM (eds) Proceedings of CMSB 2008. LNBI 5307, pp 328–346

  • Durzinsky M, Wagler A, Weismantel R (2009) An algorithmic framework for network reconstruction. In: Heiner M, Uhrmacher AM (eds) Special issue foundations of formal reconstruction of biochemical networks. J Theor Comput Sci (in press)

  • Ewald G (1996) Combinatorial convexity and algebraic geometry. Springer, Berlin

    MATH  Google Scholar 

  • Farkas J (1894) On the applications of the mechanical principle of fourier. Mathematikai és Thermészettudományi Értesitö 12:457–472

    Google Scholar 

  • Farkas J (1898) A parametric method for the mechanical principle of fourier. Mathematikai és Physikai Lapok 7:63–71

    Google Scholar 

  • Firla RT, Ziegler GM (1999) Hilbert bases, unimodular triangulations, and binary covers of rational polyhedral cones. Discret Comput Geom 21:205–216

    Article  MathSciNet  MATH  Google Scholar 

  • Giles FR, Pulleyblank WR (1979) Total dual integrality and integer polyhedra. Linear Algebra Appl 25:191–196

    Article  MathSciNet  MATH  Google Scholar 

  • Gordan PA (1873) Über die Auflösung linearer Gleichungen mit reellen Coefficienten. Mathematische Annalen 6:23–28

    Article  MathSciNet  Google Scholar 

  • Heiner M, Gilbert D, Donaldson R (2008) Petri nets for systems and synthetic biology. In: Bernardo M, Degano P, Zavattaro G (eds) SFM 2008. Springer LNCS 5016, pp 215–264

  • Henk M, Weismantel R (2000) On minimal solutions of Diophantine equations. Contrib Algebra Geom 41:49–55

    MathSciNet  MATH  Google Scholar 

  • Koch I, Heiner M (2008) Petri nets. In: Junker BH, Schreiber F (eds) Biological network analysis. Wiley Book Series on Bioinformatics, pp 139–179

  • Lamparter T, Marwan W (2001) Spectroscopic detection of a phytochrome-like photoreceptor in the Myxomycete Physarum polycephalum and the kinetic mechanism for the photocontrol of sporulation by Pfr. Photochem Photobiol 73:697–702

    Article  Google Scholar 

  • Larhlimi A, Bockmayr A (2005) Minimal metabolic behaviors and the reversible metabolic space. Matheon Preprint Nr. 299, FU Berlin

  • Laubenbacher R, Stigler B (2005) A computational algebra approach to reverse engineering of gene regulatory networks. J Theor Biol 229:523–537

    Article  MathSciNet  Google Scholar 

  • Liu J (1991) Hilbert bases with the Carathéodory property. PhD thesis, Cornell University

  • Marwan W (2003) Theory of time-resolved somatic complementation and its use for the analysis of the sporulation control network of Physarum polycephalum. Genetics 164:105–115

    Google Scholar 

  • Marwan W, Starostzik C (2002) The sequence of regulatory events in the sporulation control network of Physarum polycephalum analysed by time-resolved somatic complementation of mutants. Protist 153:391–400

    Article  Google Scholar 

  • Marwan W, Sujatha A, Starostzik C (2005) Reconstructing the regulatory network controlling commitment and sporulation in Physarum polycephalum based on hierarchical Petri Net modeling and simulation. J Theor Biol 236:349–365

    Article  Google Scholar 

  • Marwan W, Wagler A, Weismantel R (2008) A mathematical approach to solve the network reconstruction problem. Math Methods Oper Res 67:117–132

    Article  MathSciNet  MATH  Google Scholar 

  • Minkowski H (1896) Geometrie der Zahlen. Teubner, Leipzig

    Google Scholar 

  • Nutsch T, Marwan W, Oesterhelt D, Gilles ED (2003) Signal processing and flagellar motor switching during phototaxis of Halobacterium salinarum. Genome Res 13:2406–2412

    Article  Google Scholar 

  • Oda T (1988) Convex bodies and algebraic geometry. Springer, Berlin

    MATH  Google Scholar 

  • Pottier L (1991) Minimal solutions of linear diophantine systems: bounds and algorithms. In: Book RV (ed) Rewriting techniques and applications. Lecture notes in computer science 488. Springer, Berlin, pp 162–173

  • Runge T (2004) Methodik zur Modellierung und Validierung von biochemischen Netzwerken mit gefärbten Petri Netzen. Diplomarbeit, Technische Universität Brandenburgs, Cottbus

  • Scarf HE (1981) Production sets with indivisibilities I: generalities. Econometrica 49:1–32

    Article  MathSciNet  MATH  Google Scholar 

  • Schilling CH, Letscher D, Palsson B (2000) Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective. J Theor Biol 203:229–248

    Article  Google Scholar 

  • Schrijver A (1986) Theory of linear and integer programming. Wiley, New York

    MATH  Google Scholar 

  • Schuster R, Hilgetag C (1994) On elementary flux modes in biochemical reaction systems at steady state. J Biol Syst 2:165–182

    Article  Google Scholar 

  • Sebö A (1990) Hilbert bases, Carathéodory’s theorem and combinatorial optimization. In: Kannan R, Pulleyblank WR (eds) Proceedings of the 1st conference on integer programming and combinatorial optimization, Waterloo, pp 431–455

  • Starostzik C, Marwan W (1995) Functional mapping of the branched signal transduction pathway that controls sporulation in Physarum polycephalum. Photochem Photobiol 62:930–933

    Article  Google Scholar 

  • Sturmfels B (1996) Gröbner bases and convex polytopes. American Mathematical Society, Providence

  • Sturmfels B, Thomas RR (1997) Variation of cost functions in integer programming. Math Program 77:357–388

    MathSciNet  MATH  Google Scholar 

  • Sturmfels B, Weismantel R, Ziegler GM (1995) Gröbner bases of lattices, corner polyhedra and integer programming. Contrib Algebra Geom 36:281–298

    MathSciNet  MATH  Google Scholar 

  • Thomas RR (1995) A geometric Buchberger algorithm for integer programming. Math Oper Res 20:864–884

    Article  MathSciNet  MATH  Google Scholar 

  • Thomas RR, Weismantel R (1997) Truncated Gröbner bases for integer programming. Appl Algebra Eng Commun Comput 8:241–257

    Article  MathSciNet  MATH  Google Scholar 

  • Torres LM, Wagler A (2009) Encoding the dynamics of deterministic systems. Otto-von-Guericke University Magdeburg, Preprint 09–29, Math Methods Oper Res (submitted)

  • Torres LM, Wagler A, Weismantel R (2008) Modeling the dynamic behavior of deterministic biological systems. In: Proceedings of ALIO/EURO workshop on applied combinatorial optimization, Buenos Aires, 2008. ISBN 978-950-29-1116-8

  • Urbaniak R, Weismantel R, Ziegler GM (1997) A variant of Buchberger’s algorithm for integer programming. SIAM J Discret Math 1:96–108

    Article  MathSciNet  Google Scholar 

  • van der Corput JG (1931) Über Systeme von linear-homogenen Gleichungen und Ungleichungen. Proceedings Koninklijke Akademie van Wetenschappen te Amsterdam 34:368–371

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Correspondence to Annegret K. Wagler.

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Wagler, A.K., Weismantel, R. The combinatorics of modeling and analyzing biological systems. Nat Comput 10, 655–681 (2011). https://doi.org/10.1007/s11047-009-9165-5

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