A Graph-Based Toy Model of Chemistry

Large scale chemical reaction networks are a ubiquitous phenomenon, from the metabolism of living cells to processes in planetary atmospheres and chemical technology. At least some of these networks exhibit distinctive global features such as the "small world" behavior. The systematic study of such properties, however, suffers from substantial sampling biases in the few networks that are known in detail. A computational model for generating them is therefore required. Here we present a Toy Model that provides a consistent framework in which generic properties of extensive chemical reaction networks can be explored in detail and that at the same time preserves the "look-and-feel" of chemistry: Molecules are represented as labeled graphs, i.e., by their structural formulas; their basic properties are derived by a caricature version of the Extended Hückel MO theory that operates directly on the graphs; chemical reaction mechanisms are implemented as graph rewriting rules acting on the structural formulas; reactivities and selectivities are modeled by a variant of the Frontier Molecular Orbital Theory based on the Extended Hückel scheme. The approach is illustrated for two types of reaction networks: Diels-Alder reactions and the formose reaction implicated in prebiotic sugar synthesis.

[1]  Marcel Thürk.  Ein Modell zur Selbstorganisation von Automatenalgorithmen zum Studium molekularer Evolution , 1993 .

[2]  Giuseppe Del Re,et al.  812. A simple MO–LCAO method for the calculation of charge distributions in saturated organic molecules , 1958 .

[3]  L W Buss,et al.  What would be conserved if "the tape were played twice"? , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[4]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[5]  Otto Diels,et al.  Synthesen in der hydroaromatischen Reihe , 1928 .

[6]  Carson C. Chow,et al.  Small Worlds , 2000 .

[7]  W. Banzhaf,et al.  Self-organization in a system of binary strings with spatial interactions , 1999 .

[8]  Eric Fontain,et al.  The generation of reaction networks with RAIN. 1. The reaction generator , 1991, J. Chem. Inf. Comput. Sci..

[9]  Ante Graovac,et al.  The Interplay Between Graph Theory and Molecular Orbital Theory , 1994 .

[10]  C. Sandorfy,et al.  LCAO MO CALCULATIONS ON SATURATED HYDROCARBONS AND THEIR SUBSTITUTED DERIVATIVES , 1955 .

[11]  Peter F. Stadler,et al.  Relevant cycles in Chemical reaction Networks , 2001, Adv. Complex Syst..

[12]  Gérard Berry,et al.  The chemical abstract machine , 1989, POPL '90.

[13]  C. Cramer,et al.  Implicit Solvation Models: Equilibria, Structure, Spectra, and Dynamics. , 1999, Chemical reviews.

[14]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[15]  R. Read.  Every one a winner , 1978 .

[16]  Andrew V. Zeigarnik,et al.  On hypercycles and hypercircuits in hypergraphs , 1998, Discrete Mathematical Chemistry.

[17]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[18]  John S. McCaskill,et al.  Graph Replacement Chemistry for DNA Processing , 2000, DNA Computing.

[19]  Gesine Reinert,et al.  Small worlds , 2001, Random Struct. Algorithms.

[20]  J. Köbler,et al.  The Graph Isomorphism Problem: Its Structural Complexity , 1993 .

[21]  M. Nagl.  Graph-Grammatiken: Theorie, Anwendungen, Implementierung , 1979 .

[22]  E. Hückel,et al.  Quantentheoretische Beiträge zum Benzolproblem , 1931 .

[23]  R. Read.  Every one a Winner or how to Avoid Isomorphism Search when Cataloguing Combinatorial Configurations , 1978 .

[24]  S. Cannizzaro,et al.  Ueber den der Benzoësäure entsprechenden Alkohol , 1854 .

[25]  S. Schuster,et al.  The modelling of metabolic systems. Structure, control and optimality. , 1998, Bio Systems.

[26]  J. Sauer,et al.  Diels‐Alder reactions II: The Reaction Mechanism , 1967 .

[27]  David Weininger,et al.  SMILES. 2. Algorithm for generation of unique SMILES notation , 1989, J. Chem. Inf. Comput. Sci..

[28]  Lionel Goodman,et al.  Hyperconjugation not steric repulsion leads to the staggered structure of ethane , 2001, Nature.

[29]  Gerhard Goos,et al.  Efficient Graph Rewriting and Its Implementation , 1995 .

[30]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[31]  Swarna M. Patra,et al.  Graph-theoretic study of certain interstellar reactions , 1997 .

[32]  Jean-Loup Faulon,et al.  Stochastic Generator of Chemical Structure. 3. Reaction Network Generation , 2000, J. Chem. Inf. Comput. Sci..

[33]  Emanuel Pfeil,et al.  Die Bildung von Zuckern aus Formaldehyd unter der Einwirkung von Laugen , 1961 .

[34]  R. J. Bagley,et al.  Spontaneous emergence of a metabolism , 1990 .

[35]  G. Klopman,et al.  Chemical reactivity and the concept of charge- and frontier-controlled reactions , 1968 .

[36]  Johann Gasteiger,et al.  Simulation of Organic Reactions: From the Degradation of Chemicals to Combinatorial Synthesis , 2000, J. Chem. Inf. Comput. Sci..

[37]  Masahiro Aoyama,et al.  Olefin Metathesis , 2020, Name Reactions.

[38]  Lionel Salem,et al.  Intermolecular orbital theory of the interaction between conjugated systems , 1968 .

[39]  Ian Fleming,et al.  Frontier Orbitals and Organic Chemical Reactions , 1977 .

[40]  Klaus Müllen,et al.  Dendritic and hyperbranched polyphenylenes via a simple Diels-Alder route , 1997 .

[41]  Ivar Ugi,et al.  New elements in the representation of the logical structure of chemistry by qualitative mathematical models and corresponding data structures , 1993 .

[42]  Alain Devaquet,et al.  Self-consistent field intermolecular orbital theory of the interaction between conjugated systems , 1970 .

[43]  Kenichi Fukui,et al.  A Molecular Orbital Theory of Reactivity in Aromatic Hydrocarbons , 1952 .

[44]  J. J. Sylvester,et al.  On an Application of the New Atomic Theory to the Graphical Representation of the Invariants and Covariants of Binary Quantics, With Three Appendices, [Continued] , 1878 .

[45]  Ronald Breslow,et al.  On the mechanism of the formose reaction , 1959 .

[46]  Michael Frenklach,et al.  Modeling of Large Reaction Systems , 1987 .

[47]  Cayley.  LVII. On the mathematical theory of isomers , 1874 .

[48]  Eugene M. Luks.  Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time , 1980, FOCS.

[49]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[50]  Wolfgang Banzhaf,et al.  Artificial ChemistriesA Review , 2001, Artificial Life.

[51]  C. Ponnamperuma,et al.  Model for Origin of Monosaccharides , 1967, Nature.

[52]  W. Demore,et al.  Photochemistry of Planetary Atmospheres , 1998 .

[53]  R. Hoffmann.  An Extended Hückel Theory. I. Hydrocarbons , 1963 .

[54]  Max Wolfsberg,et al.  The Spectra and Electronic Structure of the Tetrahedral Ions MnO4−, CrO4−−, and ClO4− , 1952 .

[55]  Peter Decker,et al.  Bioids : X. Identification of formose sugars, presumable prebiotic metabolites, using capillary gas chromatography/gas chromatography—mas spectrometry of n-butoxime trifluoroacetates on OV-225 , 1982 .

[56]  D. Heidrich,et al.  Properties of Chemically Interesting Potential Energy Surfaces , 1991 .

[57]  D. Fell.  Understanding the Control of Metabolism , 1996 .