Abstract
Enumeration of molecules is one of the fundamental problems in bioinformatics and chemoinformatics which is also important from a practical viewpoint. We consider the problem of enumerating the stereo-isomers of tree-like polyinositol molecules (with chemical formula \(\hbox {C}_{6n}\hbox {O}_{5n+6}\hbox {H}_{4n+2}\) where \(n\) is the number of hexagonal oinositol rings) and monosubstituted tree-like polyinositols (with chemical formula \(\hbox {C}_{6n}\hbox {O}_{5n+6}\hbox {H}_{4n+1}\hbox {Z}\)). We establish recursion counting formulas for the numbers of the stereo-isomers for these two classes of molecules, in which chirality is also taken into account. In our study, the generating function, Pólya enumeration theory and ‘Dissimilarity Characteristic Theorem’ play important roles. Compared to some known computer programs such as ISOMERS, MOLGEN, exhaustive construction and Dynamic Programming etc., our method is more efficient to our enumeration problem with larger number of inositol rings. Further more, based on the obtained recursion formulas, we derive the asymptotic values for the numbers of these two stereo-isomers from which we conclude that almost all tree-like and monosubstituted tree-like polyinositols are chiral.
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References
A. Cayley, On the theory of the analytical forms called trees. Phil. Mag. 13, 172–176 (1857)
G. Pólya, R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (Springer, Berlin, 1987)
T. Imada, S. Ota, H. Nagamochi, T. Akutsu, Efficient enumeration of stereoisomers of outerplanar chemical graphs using dynamic programming. J. Chem. Inf. Model. 51, 2788–2807 (2011)
J.L. Faulon, D. Visco, D. Roe, Enumerating molecules, in Reviews in Computational Chemistry, ed. by K.B. Lipkowitz, R. Larter, T.R. cundari (Wiley-VCH, Hoboken, 2005), pp. 209–257
L. Bouveault, De l’isomérie optique dans les corps à chaines fermées. Bull. la Socieété Chimique Paris 11, 44–147 (1894)
T. Posternak, The Cyclitols (Hermann, Paris, 1965)
K. Ma, L.A. Thomason, J. McLaurin, Scyllo-inositol, preclinical, and clinical data for Alzheimers disease. Adv. Pharmacol. 64, 177–212 (2012)
L.E. Brammer Jr, T. Hudlicky, Inositol synthesis: concise preparation of L-chiro-inositol and muco-inositol from a common intermediate. Tetrahedron: Asymmetry 9, 2011–2014 (1998)
M. Desjardins, L.E. Brammer Jr, T. Hudlicky, A practical multigram-scale synthesis of allo-inositol. Carbohydr. Res. 304, 39–42 (1997)
M. Mandel, T. Hudlicky, L.D. Kwart, G.M. Whited, Unusual oxidation of 1-halo-1, 3-dienes with parmanganate. Expedient syntheses of (+)-D-chiro-3-inosose and (+)-D-chiro-inositol from chlorobenzene. J. Org. Chem. 58, 2331–2333 (1993)
H.A.J. Carless, K. Busia, Synthesis of racemic 6-deoxy-6-fluoro-chiro-inositol 2, 3, 5-trisphosphate. Carbohydr. Res. 234, 207–215 (1992)
H.A.J. Carless, S.S. Malik, Enantiospecific synthesis of C-methyl azidoinositols and aminocyclitols from toluene. Tetrahedron: Asymmetry 3, 1135–1138 (1992)
H.A.J. Carless, O.Z. Oak, Total synthesis of (-)-laminitol (1D–4C-methyl-myo-inositol) via microbial oxidation of toluene. Tetrahedron Lett. 32, 1671–1674 (1991)
T. Hudlicky, K.A. Abboud, J. Bolonick, R. Maurya, M.L. Stantonga, A.J. Thorpe, Concise syntheses of 1, 2-L-chiro-inositol conjugates and oligomers-a novel class of saccharide mimics with promising molecular properties. Chem. Commun. 15, 1717–1718 (1996)
S. Freeman, T. Hudlicky, New oligomers of conduritol-F and muco-inositol. Synthesis and biological evaluation as glycosidase inhibitors. Bio. Med. Chem. Lett. 14, 1209–1212 (2004)
B.J. Paul, J. Willis, T.A. Martinot, I. Ghiviriga, K.A. Abboud, T. Hudlicky, Synthesis, structure, and biological evaluation of novel N-and O-linked diinositols. J. Am. Chem. Soc. 124, 10416–10426 (2002)
H.R. Henze, C.M. Blair, The number of structurally isomeric alcohols of the methanol series. J. Am. Chem. Soc. 53, 3042–3046 (1931)
R. Otter, The number of trees. Ann. Math. 49, 583–599 (1948)
F. Harary, R.W. Robinson, A.J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of varous species. J. Austral. Math. Soc. 20, 483–503 (1975)
R.W. Robinson, F. Harary, A.T. Balaban, The numbers of chiral and achiral alkanes and monosubstituded alkanes. Tetrahedron 32, 355–361 (1976)
F. Harary, R.C. Read, The enumeration of tree-like polyhexes. Proc. Edinb. Math. Soc. 17, 1–13 (1970)
I. Gutman, S.J. Cyvin, J. Brunvoll, Enumeration of the isomers of phenylenes. Monatsh. Chem. 125, 887–894 (1994)
M.R.P.F.N. Costa, R.C.S. Dias, Prediction of mean square radius of gyration of tree-Like polymers by a general kinetic approach. Polymer 48, 1785–1801 (2007)
R.C. Read, The enumeration of acyclic chemical compoundsl, in Chemical Applications of Graph Theory, ed. by A.T. Balaban (Academic Press, New York, 1976), pp. 24–61
C. Yeh, Isomer enumeration of alkenes and aliphatic cyclopropane derivatives. J. Chem. Inf. Comput. Sci. 36, 854–856 (1996)
F. Harary, E.M. Palmer, Graphical Enumeration (Academic Press, New York, 1973)
L. Bytautas, D.J. Klein, Chemical combinatorics for alkane-isomer enumeration and more. J. Chem. Inf. Comput. Sci. 38, 1063–1078 (1998)
J. Brunvoll, S.J. Cyvin, B.N. Cyvin, Enumeration of tree-like octagonal systems. J. Math. Chem. 21, 193–196 (1997)
F. Bergeron, G. Labelle, P. Leroux, Combinatorial Species and Tree-Like Structures (Cambridge University Press, Cambridge, 1998)
A.T. Balaban, J.W. Kennedy, L. Quintas, The number of alkanes having \(n\) carbons and a longest chain of length \(d\): an application of a theorem of Pólya. J. Chem. Educ. 65, 304–313 (1988)
M. Gordon, J.W. Kennedy, The counting and coding of trees of fixed diameter. SIAM J. Appl. Math. 28, 376–398 (1975)
J.V. Knop, W.R. Müller, K. Szymanski, N. Trinajstić, Computer Generation of Certain Classes of Molecules (Association of Chemists and Technologists of Croatia, Zagreb, 1985)
Y. Ishida, Y. Kato, L. Zhao, H. Nagamochi, T. Akutsu, Branch-and-bound algorithms for enumerating treelike chemical graphs with given path frequency using detachment-cut. J. Chem. Inf. Model. 50, 934–946 (2010)
K.C. Deng, J.G. Qian, F.J. Zhang, Enumerating tree-like polyphenyl isomers. J. Stat. Mech.: Theory Exp. P12018 (2012). http://iopscience.iop.org/1742-5468/2012/12
R. Friedberg, Dual trees and resummation theorems. J. Math. Phys. 16, 20–30 (1975)
A. Miličević, N. Trinajstić, Combinatorial enumeration in chemistry, in Chemical Modelling: Applications and Theory, vol. 4, ed. by A. Hinchliffe (The Royal Society of Chemistry, Cambridge, 2006), pp. 405–469
T. Hudlicky, K.A. Abboud, D.A. Entwistle, R. Fan, R. Maurya, A.J. Thorpe, J. Bolonick, B. Myers, Toluene-dioxygenase-mediated cis-dihydroxylation of aromatics in enantioselective synthesis. iterative glycoconjugate coupling strategy and combinatorial design for the synthesis of oligomers of nor-saccharides, inositols and pseudosugars with interesting molecular properties. Synthesis 7, 897–911 (1996)
H. Dolhaine, H. Hönig, M. Van Almsick, Sample applications of an algorithm for the calculation of the number of isomers with more than one type of achiral substituent. MATCH Commun. Math. Comput. Chem. 39, 21–37 (1999)
H. Dolhaine, H. Hönig, Full isomer-tables of inositol-oligomers up to tetramers. MATCH Commun. Math. Comput. Chem. 46, 91–119 (2002)
H. Dolhaine, H. Hönig, On the theorical number of some inositol tetramers. MATCH Commun. Math. Comput. Chem. 46, 71–89 (2002)
C. Rücker, R. Gugisch, A. Kerber, Manual construction and mathematics-and computer-aided counting of stereoisomers. The example of oligoinositols. J. Chem. Inf. Comput. Sci. 44, 1654–1665 (2004)
E.F. Beckenbach, Applied Combinatorial Mathematics (Wiley, New York, 1964)
G. Pólya, G. Szegö, Problems and Theorems in analysis (Springer, Berlin, 1972)
F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969)
A. Meir, J.W. Moon, On an asymptotic method in enumeration. J. Comb. Theory Ser. A. 51, 77–89 (1989)
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Research supported by NSFC (No. 11271307).
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Deng, K., Qian, J. Enumerating stereo-isomers of tree-like polyinositols. J Math Chem 52, 1581–1598 (2014). https://doi.org/10.1007/s10910-014-0338-9
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DOI: https://doi.org/10.1007/s10910-014-0338-9