How it works

The chemistry that feels most like chemistry. Molecules are structural formulas - labelled graphs of atoms and bonds - and reactions are graph rewriting rules that must conserve atoms and bond order. What makes it distinctive is that energies are not supplied: the VSEPR rules turn each structure into an orbital graph and an extended-Hückel calculation gives its energy, so activation energies and hence rates are *computed from the structure itself*. Networks built this way have the look and feel of real reaction chemistry.

S

a molecule is its structural formula: a vertex- and edge-labelled graph of atoms and bonds. The VSEPR rules turn that graph unambiguously into the orbital graph of Polansky, whose vertices are the outer orbitals (1s for H, sp3/sp2/sp hybrids plus the leftover p orbitals for C, N and O) and whose edges are orbital overlaps: sigma along a bond, pi between p orbitals, and the weaker semi-direct, hyperconjugation and banana-bond overlaps. Species id and structure are the canonical SMILES of the structure formula, so molecules are identified up to graph isomorphism. · implicit

R

graph rewriting rules on the structure formula, each a left graph, a right graph and a context; a bimolecular mechanism is split into one half rule per educt plus a join rule that forms the intermolecular bond. Every rule must conserve vertex labels (the atoms) and total bond order (the valence electrons). Implemented: the Diels-Alder rule transcribed verbatim from appendix B of the 2003 paper, the aldol condensation of its figure 5, and keto-enol tautomerism. · arity 1, 2

A

ode, ssa, graph-rewrite · none

Parameters

What to look for

• reaction networks with the look-and-feel of real chemistry: the Diels-Alder rule builds cyclohexene from butadiene and ethene and 4-vinylcyclohexene from two butadienes, and the aldol rule builds glyceraldehyde, erythrose and dihydroxyacetone from formaldehyde and glycol aldehyde
• built-in thermodynamics: every species carries a total atomization energy computed from its graph alone, so reaction energies come out of the model rather than being assigned (the Diels-Alder cycloadditions are exothermic, as published)
• the total atomization energy of a homologous series is linear in chain length, the straight line of figure 3, and the C6H10 isomers fall in a band of about 90 kcal/mol as that figure shows
• an activation-energy threshold decides how large the network grows: lowering barrier_cutoff removes channels and shrinks the closure, which is the threshold series of the ECAL figures 3 and 4
• generic network statistics: the substrate graph of the Diels-Alder network is clustered and short-pathed (small-world), while the formose network is not, because its many non-reactive species must first be toggled by a keto-enol isomerisation (ECAL Table 1)

Implementation decisions

• The book gives ToyChem one paragraph and one figure, so everything is reconstructed from the three original papers. The ToyChem package technical report (book [105], http://www.bioinf.uni-leipzig.de/Publications/PREPRINTS/05-002.pdf) is dead at that URL and at every variant tried, and the Internet Archive was offline throughout this session, so the released code was not consulted; the Haskell rewrite server and its client/server protocol are not reproduced.
• Book reference [300] (Fujita's imaginary transition structures) belongs to the RNA artificial chemistry of section 18.1.1, which reuses ToyChem's rewriting scheme, not to ToyChem itself; it is kept in refs because the catalog entry cross-references that section, but nothing here implements it. Section 18.1.1 is about RNA folding and contributes no ToyChem specification.
• Energy: Table 1 of the 2003 paper is implemented exactly - the Coulomb integrals, the sigma and pi overlap tables, kappa = 1.75, semi-direct 0.1 (0.0 with hydrogen), indirect 0.0, hyperconjugation 0.8, banana bonds 0.7 (3-ring) and 0.8 (4-ring). Two things the table leaves open were fixed by reading: the hyperconjugation factor scales the pi-table entry of the two elements (it is listed under 'sigma-pi coupling' and no direct p-sp3 value is tabulated), and the atomization reference is the valence state, one electron per valence orbital, since the model's orbitals are the hybridised ones.
• Absolute energies are not reproducible from the published parametrisation: the 2005 tables were computed with the later C/H/N/O/P/S parameter set, whose values were only ever distributed with the unavailable technical report. On the four species of its Table 2 that are unambiguous, this reconstruction reproduces the published total atomization energies up to a single scale factor with correlation r = 0.9997 and a ratio spread of 0.011 (computed/published ratio 0.379-0.390), so relative energetics - orderings, reaction energies and barriers - carry over while the kcal/mol scale does not. The tests assert the relative agreement, never the absolute numbers.
• Reaction rates: the papers give the Klopman-Salem energy increment (equations 10-14) and its frontier-orbital simplification Delta E = xi/(E_zeta - E_alpha), and say the reactivities 'can be translated into reaction rate constants e.g. using Arrhenius' law', without printing xi or a prefactor. Chemart therefore uses the frontier gap E_LUMO - E_HOMO of the partners as the Arrhenius activation energy Ea, keeps xi/gap alongside as the paper's reactivity index, and leaves A at 1: no invented number enters a rate. The full four-point Klopman-Salem expansion over atom quadruples is not implemented, only the frontier-orbital limit the paper itself used for figures 7 and 8.
• Rules: the Diels-Alder rule is appendix B transcribed node for node (context six carbons; left 1=2, 2-3, 3=4, 5=6; right 1-2, 2=3, 3-4, 4-5, 5-6, 6-1), applied to the disjoint union of the educts so that it covers the intermolecular and the intramolecular case alike. The aldol condensation is figure 5's two half rules plus the join rule, read as the standard mechanism the figure draws: the enol's nucleophilic carbon attacks the carbonyl carbon, the enol oxygen becomes a carbonyl, and its hydrogen moves to the former carbonyl oxygen, giving the beta-hydroxy carbonyl the figure names. Keto-enol tautomerism is added because the formose network needs it - the ECAL paper says a species must be toggled from non-reactive to reactive by a keto-enol isomerisation - and the paper's 'dismutations' (Cannizzaro) are not implemented.
• Aromatic bonds are excluded from the diene and dienophile patterns. The rules act on Kekule structure formulae, where a benzene ring would match the diene pattern and a Diels-Alder addition to it would be a pure artifact; the paper's own seed set contains phthalic anhydride, so the guard is needed to reproduce figure 7 at all.
• The species budget replaces the iteration count. The paper builds figure 7 with 'three iterations of the orderly generation algorithm'; chemart.expand runs the same recursion (equation 16: only combinations involving a species found in the previous round are tried) to closure or to a species budget, so max_species is the knob and the round structure is not exposed.
• max_carbons reproduces the cut stated in the caption of figure 8 ('we do not permit carbon chains with more than four members to undergo further aldol condensation'); it is applied to the products of the aldol rule. With it the formose closure is complete and small, which is why formose is the default network.
• The second butadiene-dimer regioisomer of the 2005 Table 2 (the fused bicyclic, -1588.74 kcal/mol) is not produced here: it comes from that paper's explicit collision simulation, which builds products bond by bond, not from the prescribed Diels-Alder rewrite rule. The collision simulation itself is not implemented.
• The caption of figure 3 of the 2003 paper calls its isomer series 'C4H10 isomers' and then lists hexynes, hexadienes and methylcyclopentenes, which are C6H10; the total atomization energies plotted (1520-1610 kcal/mol) confirm C6H10. The catalog and the tests follow the data, not the caption.
• No initial state and no flow: the papers give the initial mixtures of figures 7 and 8 but no concentrations and no reactor, so the seed set is recorded in extras.seed and nothing is invented. Reverse reactions are not generated as pairs, so thermodynamic consistency is not claimed even though every species carries an energy.

Sources

• Banzhaf, W. & Yamamoto, L. (2015). Artificial Chemistries, section 11.3.1 and figure 11.12: one paragraph on the Benko et al. toy model - molecules as graphs, hybrid orbitals from the VSEPR rules abstracted into an orbital graph carrying bonds and energy, graph rewriting, conservation of mass and atom types, parameters for HCNOPS, and automatic generation of the differential equations of the resulting network.
• Benko, G., Flamm, C. & Stadler, P. F. (2003). A graph-based toy model of chemistry. J. Chem. Inf. Comput. Sci. 43(4):1085-1093 (book [103]), doi:10.1021/ci0200570. Open copy: https://ul.qucosa.de/api/qucosa%3A32097/attachment/ATT-0/ - section 2 (the orbital graph, the Wolfsberg-Helmholtz parametrisation, equations 1-9), section 3 (graph rewriting, the two conservation laws, half rules and join rules, the Klopman-Salem formula equations 10-14 and its frontier-orbital simplification Delta E = xi/(E_zeta - E_alpha)), section 4 (orderly generation equation 16, canonical SMILES as the isomorphism test, the Diels-Alder network of figure 7 and the formose network of figure 8), appendix A with Table 1 (the complete parametrisation reproduced in the module) and appendix B (the Diels-Alder rule in GML, transcribed verbatim).
• Benko, G., Flamm, C. & Stadler, P. F. (2003). Generic properties of chemical networks: artificial chemistry based on graph rewriting. Advances in Artificial Life (ECAL), LNCS 2801:10-19 (book [102]), doi:10.1007/978-3-540-39432-7_2. Open copy: https://ul.qucosa.de/api/qucosa%3A32096/attachment/ATT-0/ - the substrate graph as the one-mode projection of the reaction hypergraph and its measures <k>, <L>, <C> with the Erdos-Renyi references of equation (5); Table 1 (Formose n=48 <k>=3.25 <L>=3.55 <L_rand>=3.28 <C>=0.15 <C_rand>=0.068; Diels-Alder n=40 <k>=4.65 <L>=2.15 <L_rand>=2.40 <C>=0.72 <C_rand>=0.110; E. coli n=282); figures 3-4 (the network grows as the reactivity threshold is lowered) and figure 5 (degree rank statistics with power-law slope -1.19).
• Benko, G., Flamm, C. & Stadler, P. F. (2005). Explicit collision simulation of chemical reactions in a graph based artificial chemistry. ECAL, LNCS 3630:725-733 (book [104]), doi:10.1007/11553090_73. Open copy: https://ul.qucosa.de/api/qucosa%3A32092/attachment/ATT-0/ - the later parametrisation (C, H, N, O, P, S, backbonding, banana overlaps, indirect overlaps), the elementary bond formation/fission/shift moves, the table of allowed orbital interactions, and Tables 2 and 3 of total atomization energies in kcal/mol used here as the energy benchmark.
• Gillespie, R. J. & Nyholm, R. S. (1957). Inorganic stereochemistry. Quart. Rev. Chem. Soc. 11:339-380 (book [333]): the VSEPR rules that map the structure formula onto the hybrid orbitals, cited by the book for exactly that step.

Banzhaf & Yamamoto, Artificial Chemistries (2015), §11.3.1; used in the RNA AC of 18.1.1

Preview at default parameters

11 species · 16 reactions · status complete · seed 1

O=CCO -> OC=CO  [arrhenius A=1.0 Ea=131.11036505407327 reactivity=0.17588653643431806 rules=keto-enol]
OC=CO -> O=CCO  [arrhenius A=1.0 Ea=137.97957985078907 reactivity=0.16713015088854197 rules=keto-enol]
C=O + OC=CO -> O=CC(O)CO  [arrhenius A=1.0 Ea=106.95649437718302 reactivity=0.215606804750694 rules=aldol]
O=CCO + OC=CO -> O=CC(O)C(O)CO  [arrhenius A=1.0 Ea=88.27936923241182 reactivity=0.2612223920550319 rules=aldol]
O=CC(O)CO -> OC=C(O)CO  [arrhenius A=1.0 Ea=130.82968441387322 reactivity=0.1762638815748351 rules=keto-enol]
O=CC(O)C(O)CO -> OC=C(O)C(O)CO  [arrhenius A=1.0 Ea=130.82082709909778 reactivity=0.1762758156431121 rules=keto-enol]
OC=C(O)CO -> O=C(CO)CO  [arrhenius A=1.0 Ea=124.60595143283525 reactivity=0.185067789578494 rules=keto-enol]
OC=C(O)CO -> O=CC(O)CO  [arrhenius A=1.0 Ea=124.60595143283525 reactivity=0.185067789578494 rules=keto-enol]
OC=C(O)C(O)CO -> O=C(CO)C(O)CO  [arrhenius A=1.0 Ea=124.33340531322227 reactivity=0.18547346903195952 rules=keto-enol]
OC=C(O)C(O)CO -> O=CC(O)C(O)CO  [arrhenius A=1.0 Ea=124.33340531322227 reactivity=0.18547346903195952 rules=keto-enol]
C=O + OC=C(O)CO -> O=C(CO)C(O)CO  [arrhenius A=1.0 Ea=107.4525194253242 reactivity=0.21461151514484766 rules=aldol]
C=O + OC=C(O)CO -> O=CC(O)(CO)CO  [arrhenius A=1.0 Ea=107.4525194253242 reactivity=0.21461151514484766 rules=aldol]
O=C(CO)CO -> OC=C(O)CO  [arrhenius A=1.0 Ea=122.67942342810865 reactivity=0.1879740494013139 rules=keto-enol]
O=C(CO)C(O)CO -> OC=C(O)C(O)CO  [arrhenius A=1.0 Ea=122.52753234286943 reactivity=0.18820707117049865 rules=keto-enol]
O=C(CO)C(O)CO -> OCC(O)=C(O)CO  [arrhenius A=1.0 Ea=122.52753234286943 reactivity=0.18820707117049865 rules=keto-enol]
OCC(O)=C(O)CO -> O=C(CO)C(O)CO  [arrhenius A=1.0 Ea=112.30982558220896 reactivity=0.20532974635527376 rules=keto-enol]

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