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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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
Bertsimas D, Weismantel R (2005) Optimization over integers. Dynamic Ideas. Belmont
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
Carathéodory C (1911) Über den Variablilitätsbereich der fourierschen Konstanten von positiven harmonischen Funktionen. Rendiconto del Circolo Matematico di Paleromo 32:193–217
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
Cook W, Fonlupt J, Schrijver A (1986) An integer analogue of Carathéodory’s theorem. J Comb Theory B 40:63–70
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
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
Farkas J (1894) On the applications of the mechanical principle of fourier. Mathematikai és Thermészettudományi Értesitö 12:457–472
Farkas J (1898) A parametric method for the mechanical principle of fourier. Mathematikai és Physikai Lapok 7:63–71
Firla RT, Ziegler GM (1999) Hilbert bases, unimodular triangulations, and binary covers of rational polyhedral cones. Discret Comput Geom 21:205–216
Giles FR, Pulleyblank WR (1979) Total dual integrality and integer polyhedra. Linear Algebra Appl 25:191–196
Gordan PA (1873) Über die Auflösung linearer Gleichungen mit reellen Coefficienten. Mathematische Annalen 6:23–28
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
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
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
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
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
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
Marwan W, Wagler A, Weismantel R (2008) A mathematical approach to solve the network reconstruction problem. Math Methods Oper Res 67:117–132
Minkowski H (1896) Geometrie der Zahlen. Teubner, Leipzig
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
Oda T (1988) Convex bodies and algebraic geometry. Springer, Berlin
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
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
Schrijver A (1986) Theory of linear and integer programming. Wiley, New York
Schuster R, Hilgetag C (1994) On elementary flux modes in biochemical reaction systems at steady state. J Biol Syst 2:165–182
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
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
Sturmfels B, Weismantel R, Ziegler GM (1995) Gröbner bases of lattices, corner polyhedra and integer programming. Contrib Algebra Geom 36:281–298
Thomas RR (1995) A geometric Buchberger algorithm for integer programming. Math Oper Res 20:864–884
Thomas RR, Weismantel R (1997) Truncated Gröbner bases for integer programming. Appl Algebra Eng Commun Comput 8:241–257
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
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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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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DOI: https://doi.org/10.1007/s11047-009-9165-5