Calculation of Possible Oligosaccharide Isomers Yields 1.05x10^12 Structures for a Hexasaccharide

1. A Calculation of all Possible Oligosaccharide Isomers Both Branched and Linear Yields 1.05 x 1012 Structures for a Reducing Hexasaccharide The Isomer Barrier to Development of Single-Method Saccharide Sequencing or Synthesis Systems

2. Carbohydrate IT • Carbohydrates, by their unique branching structure, contain an evolutionary potential of information content several orders of magnitude higher in a short sequence than in any other biological oligomer. • This study addresses informational potential inherent in biological recognition systems comprised of complex carbohydrate ligands recognized for targeted activities by specifically binding cognate protein receptors, such as lectins. • Evolution of receptor/ligand cognate pairs in carbohydrates is complex and probably very slow. • Single point mutations in glycosyl transferase proteins are not likely to alter sugar structures, except in cases where a minor amino acid change could alter recognition among closely related sugars comprising otherwise the same structure1 . • 1Yamamoto, F.; Hakomori, S.-i. J Biol Chem 1990, 265:19257-19262.

3. Carbohydrate IT • The polypeptide-based carbohydrate recognition information is carried in one or more genes. • Evolution of biological recognition of just one additional sugar on an existing structure may require a combination of the following: • 1) mutation of the peptide sequence of an existing glycosyl transferase (more likely), or evolution of a novel glycosyl transferase, to make a novel carbohydrate structure, and • 2) evolution of a new lectin binding site or new lectin to contain the new binding/recognition site.

4. Carbohydrate IT • The complex carbohydrate cognate is coded into a specifically ordered set of glycosyl transferase genes where each N-1 precursor is part of the recognition system in the binding site of the “next in-order” glycosyl transferase for acceptance of the next sugar in the sequence from a high energy donor. • Understanding the evolution, genetic control and organization of these newly discovered carbohydrate-protein recognition systems will be a significant research challenge.

5. Carbohydrate IT • In all biological heteropolymers, the linear sequence of monomers comprises, in some manner, a biological code • The ability of proteins to conform in a concave or convex manner to recognize all other biological molecules includes recognition of complex carbohydrates. • Proteins such as lectins, enzymes and antibodies can exhibit exquisite binding specificities for the shape, charge, epimers, anomers, linkage positions, ring size, branching and monosaccharide sequence of carbohydrate ligand molecules where the maximum recognized size is usually hexamer or smaller2. • 2Cisar, J.; Kabat, E.A.; Dorner, M.M.; Liao, J. J Exp Med 1975 42: 435-459. • Takeo, K.; Kabat, E.A. J Immunol 1978, 121: 2305-2310. • Smith-Gill, S.J.; Rupley, J.A.; Pincus, M.R.; Carty, R.P.; Scheraga, H.A. Biochemistry 1984,23: 993-997.

6. Carbohydrate IT • Carbohydrate sequences possess unique solution structures which, although dynamic, are shown by Nuclear Overhauser Effect NMR and molecular modeling to be populated mainly by minimum energy 3-dimensional conformations3 . • Miller, K.E.; Mukhopadhyay, C.; Cagas, P.; Bush, C.A.Biochemistry 1992, 31:6703-6709. • Oligosaccharide haptens, being rather more rigid than short peptides because of steric crowding4 , must be envisioned in 3 dimensional space for specific recognition by proteins. 4Cumming, D.A.; Carver, J.P. Biochemistry 1987, 26: 6664-6676. • French, AD; Mouhous-Riou, N.; Perez, S. Carbohydr Res 1993, 247:51-62. • Poppe, L.; Dabrowski, J.; von der Lieth, C.W.; Koike, K.; Ogawa, T. . Eur J Biochem. 1990, 189, 313-325.

7. Carbohydrate IT • Carbohydrate polymers themselves often contain a complex multifaceted sequence, and specific proteins can bind to relatively short subsets or haptens within longer saccharide sequences, such as in heparin5 • Atha, D.H.; Lormeau, J.C.; Petitou, M.; Rosenberg, R.D.; Choay, J. Biochemistry 198726: 6454-6461. • Riensenfeld, J.; Hook, M.; Bjork, I.; Lindahl, U.; Ajaxon, B. Fed Proc1977, 36, 39-43. • vanBoeckel, C.A.A.; Petitou, M. Angewandte Chemie Int. Ed. 1993, 32, 1671-1818.

8. Carbohydrate IT • A lectin or other carbohydrate binding protein can act • in control mechanisms, such as selectins in inflammation • as signals for polypeptide location within the cell, such as lysosomal protein markers6, • In single celled organisms as recognition markers for predation or adhesion • In viruses to target cell-surface structures for adhesion and invasion. • in the metazoan, for specific cell surface recognition of one cell by another. • 6Reitman, M.L.; Kornfeld, S. J Biol Chem 1981, 256:11977-11980.

9. Carbohydrate IT • A large collective of low avidity interactions may take place to dramatically increase binding strength where multimeric intercellular binding occurs7 • A monomer may have a millimolar binding constant • Dimer -> micromolar • Trimer-> nanomolar (multivalency of adhesive sites) • 7Lee, Y.C. Ciba Found Symp 1990, 145:80-95. • Higher orders -> velcro effect • Specific spacing of carbohydrate moieties within a structure may confer several orders of magnitude tighter binding.

10. Carbohydrate IT • Possible higher complexity might occur where low avidity binding of patterns of sets of carbohydrates by sets of binding proteins may form recognition systems which may play a powerful role in intercellular sociology • during development8 • Feizi, T. Nature 1985, 314, 53-57. • Feizi, T. Adv Exp Med Biol 1988, 228,317-329. • in the immune system9 • Brandley, B.K.; Swiedler,S.J. ; Robbins, P.W. Cell 1990, 63, 861-870. • Aruffo, A.; Stamenkovic,I.; Melnick, M.; Underhill,C.B.; Seed, B. Cell 1990, 61, 1303-1310. • Polley, M.J.; Phillips,M.L.; Wayner,E.; Nudelman, E.; Singhal, A.K.; Hakomori, S.-i.; Paulson, J.C. Proc. Natl. Acad. Sci USA 1991, 88: 6224-6229. • Yuen, C.T.; Lawson, A.M.; Chai, W.; Larkin, M.; Stoll, M.S.; Stuart, A.C.; Sullivan, F.X.; Ahern, T.J.; Feizi, T. Biochemistry 1992, 31:9126-9131. • in parasitology10 • Friedman, M.J.; Fukuda, M.; Laine, R.A. Science 1985, 228: 75-77. • and other microbial pathogenesis11 • Srnka, C.A.;Tiemeyer, M .; Gilbert, J.H.; Moreland, M.; Schweingruber,H.; de Lappe,B.W.; James, P.G.; Gant, T .;Willoughby, R.E.; Yolken, R.H.; Nashed, M.A.; Abbas, S.A.; Laine, R.A. Virology 1992, 190: 794-805. .

11. Carbohydrate IT • Numerous reviews and recent papers have been written regarding new discoveries in carbohydrate-based recognition systems such as the • "Selectins"8, • glycosaminoglycan clotting factors12 , • tumor markers13, • parasite recognition systems9, • rhizobium nodulation systems14, • plant pathogen recognition15 and • others16 • 8(op.cit) • 9(op.cit) • 12vanBoeckel, C.A.A.; Petitou, M. Angewandte Chemie Int. Ed. 1993, 32: 1671-1718. • 13Hakomori, S. Am J Clin Pathol 1984, 82:635-648. • Hoff, S.D.; Irimura, T.; Matsushita, Y.; Ota, D.M.; Cleary, K.R.; Hakomori, S.-i. Arch Surg 1990, 125, 206-209. • 14Truchet, G.; Roche, P.; Lerouge, P.; Vasse,J.; Camut, S.; de Billy,F.; Prome,J.-C.; Denarie ,J . Nature 1991, 351: 670-673. • Fisher, R..F; Long, S.R.Nature 1992, 357: 655-660. • 15Maniara, G.; Laine, R.A. ; Kuc, J. Physiol. Plant. Pathol. 1984,24: 177-186. • 16Karlsson, K.A . Chem. Phys. Lipids 1986,42: ,153-175.

12. Carbohydrate IT • Banausic motives and research by new start-up companies have recently driven science to many new discoveries in the immune cell recognition systems (Selectins and others) • Current molecular understanding of this system alone augurs a giant breakthrough in immunochemistry. • Taken together, these interesting findings give bold introduction to a new excitement in carbohydrate biochemistry.

13. Carbohydrate IT • A growing specialty area of biochemistry concerns itself with the biology of protein recognition of specific carbohydrates. • This field has been coined "Glycobiology" by Raymond Dwek17 , • Opdenakker, G.; Rudd, P.M.; Ponting, C.P.; Dwek, R.A. FASEB J 1993, 7:1330-7 • Rademacher, T.W.; Parekh, R.B.; Dwek, R.A. Annu Rev Biochem198857, 785-838. • The name Glycobiology has also been adopted by a Journal and a North American Scientific Society of some 1000+ members (formerly the Society for Complex Carbohydrates).

14. Carbohydrate IT • What, therefore, are the structural components that make carbohydrates so complex • and what is the magnitude of potential information content for which it is apparent that higher organisms have exploited.

15. Carbohydrate IT • Usually, with some exceptions, saccharide binding proteins recognize a 6 sugar oligomer or smaller. • Within a hexasaccharide sequence comprised of a set of 6 different sugars (hexoses in this example) which may be repeated or used more than once in a structure, more than 1.05 x 1012 possible carbohydrate structures exist. • In contrast a set of 6 different amino acids which can be repeated in permuted structures can only generate 46656 different molecules, more than 7 orders of magnitude lower than possible for carbohydrates.

16. Carbohydrate IT • Carbohydrates have 8 major structural features comprising • 1) epimers; including D and L forms; • 2) linear sequence of core and of linear branches; • 3) ring size, usually 5 or 6 membered; • 4) anomeric configuration; a and ß • 5) linkage position; (i.e., 1->2, 1->3, 1->4, 1->5, 1->6 etc) • 6) branching positions and arborization; • 7) reducing terminal attachment, (glycoside, acetal, ketal) • 8) derivatives (ester, ether, phosphate, sulfate, lactyl, etc.) • all of the above contribute to large numbers of equal-mass isomers in a short sequence potentially each recognizable by specifically binding proteins.

17. Carbohydrate IT • Calculation of the isomers of an oligosaccharide was mentioned in Nathan Sharon's collected lectures18 as originating with John Clamp (of Britain) who estimated 1056 isomers for a trisaccharide comprised of 3 different hexoses. • 18Sharon, N. , Complex Carbohydrates, Their Chemistry, Biosynthesis and Functions, Addison-Wesley Publishing Company, Advanced Book Program, Reading, Mass., 1975; p . 7. • This calculation was based on • 6 sequence permutations of 3 different monomers (3!), • 8 permutations of alpha and beta anomeric configurations at each of three sugars (23) • 16 possibilities of attachment of the reducing terminal and internal sugar (to the 2,3,4 or 6 hydroxyl of their respective aglycones) (42). • This number, 6 x 8 x 16 = 768, • It is not clear how the number 1056 was calculated by Clamp. • 3 amino acids in a row or 3 nucleic acids would give 6 isomers as 3!

18. Carbohydrate IT However, due to not considering repeating sugars, ring size or branching, both Clamp and Sharon underestimated the number of isomers by nearly 2 orders of magnitude. Richard Schmidt in 1986 published a table showing a calculation of 720 isomers for a trisaccharide, 34,560 for a tetrasaccharide and 2,144,640 for a pentasaccharide19 19Schmidt, R. R. (1986) Angew. Chem. Int. Ed. Engl. 1986, 25, 212-235. In 1988, Laine, et al. published a formula including a ring size term, and estimated the resulting number for a linear, reducing pentasaccharide with non-repeating units as follows20 : n! x 2na x 2nr x 4n-1 Laine, R.A.; Pamidimukkala, K.M.; French, A.L.; Hall, R.W.; Abbas, S.A.; Jain, R.K. ; Matta, K.L. J. Am. Chem. Soc. 1988, 110: 6931-6939.

19. Carbohydrate IT • n! x 2na x 2nr x 4n-1 • where "n" is the number of monosaccharides connected to each other in an “oligosaccharide”, • 2nsubscript "a" is the anomeric term, • 2nsubscript "r" the ring size term • linkage position is represented by 4n-1 • Employed in a specific calculation for a linear pentasaccharide comprised of 5 different non-repeating hexoses this resulted in 31,457,280 isomers, all having the same mass.

20. Carbohydrate IT • However, the number of possible isomers is actually much larger due to branching and the natural possiblity of repeated monomers. • Carl G. Hellerqvist, in 1990, estimated 2.72 billion possible structures for a hexasaccharide containing aminosugars, fucose and hexoses21. • Hellerqvist, C.G. Methods in Enzymology 1990,193, 554-573. • Hellerqvist’s theme was to show how these numbers are lowered by successive analytical steps

21. Carbohydrate IT • Sugar monomers are often repeated in natural carbohydrates just as in peptides. • Repeating saccharides, for example, were considered in a separate calculation by Richard Schmidt (op. cit.). • Therefore, in the Clamp/Sharon formula 3!x23x42 for the number of possible trisaccharides from a set of 3 hexoses, the first term should have been 33 = 27 instead of 3! = 6. • The total should have been multiplied by another term for ring size, since most hexoses can occur in either pyranose (6-membered) or furanose (5-membered) forms.

22. Carbohydrate IT • Considering the 5 membered ring would have increased the result for a trisaccharide by a factor of 23 possibilities or x8. • The furanose form presents the possibility that in a trisaccharide of sequence ABC, sugar A could have been connected through the 5 position of sugar B, for example, possibly increasing the number of potential linkage positions to 5 instead of 4 in a hexose. • However this factor is taken into account by the ring size term keeping the number of possibilities of linkage positions at 42=16. • Thus, the correct number for linear trisaccharides made up from a set of 3 hexoses is 27 x 8 x 8 x 16 = 27,648. • Remember, a tripeptide, if aa’s repeated would be only 33 = 27

23. Branching oligosaccharides • Oligosaccharides can be made up of 2 sugars or more attached to the same moiety: Consider Sugars A and B attached in different ways to sugar C, for example: • A(1->6) B(1->6 • \ \ • C(1->R)* or C(1->R) • / / • B(1->3) A(1->3) • *R = reducing end attachment site (protein, lipid, other aglycon)

24. Carbohydrate IT • Another possibility for the configuration of a trisaccharide is a branched structure where sugars A and B are both glycosidically attached to sugar C by a 2,3; 2,4; 2,6; 3,4; 3,6 or 4,6 branching pattern (six possibilities). • Where sugar C is in the furanose form, however, additional branching possibilities include 2,3; 2,5; 2,6; 3,5; 3,6; 5,6 for a total of 12 different branched structures. • The ring size term 2nr, however, when applied to the branching sugar, takes into account the additional 6 structures.

25. Branched Carbohydrates • Since each branch can occur in two different ways, such as A6,B3 or B6,A3 there are again 12 different ways to branch these three sugars. • The permutation term, En, however takes care of this A6,B3 and B6,A3 branching duplex. • Possible branched trisaccharides from a set of 3 hexoses, each one unique and different from the linear structures are 27 x 8 x 8 x 6 = 10,368. • En * 2nr * 2na * x 6n-2 (branched forms)

26. Carbohydrate IT • The total structures from a trisaccharide comprised of 3 hexoses, choosing among a set of only 3 different hexoses is 27,648 (linear forms) plus 10,368 (branched forms) =38,016, • This number is about 40 fold higher than Clamp's, Sharon's or Schmidt's estimate of 720 - 1050. The formula for isomers of a trisaccharide having a reducing end is thus: • En * 2nr * 2na * 4n-1 (linear forms) + • En * 2nr * 2na * x 6n-2 (branched forms)

27. Analytical Challenge • Use of NMR as a single spectroscopic method: • Each trisaccharide would contain 15 ring protons including the anomeric, • thus the proton NMR spectrum would need to resolve 38,016 x 15 = 570,240 "different" proton environments within 0.5 ppm. (the natural dispersion for the ring protons, the anomeric protons are downfield) • It is doubtful that a tenth of this number of lines could be resolved using multi-dimension proton NMR, (requiring a terahertz instrument). • Today, A Gigahertz NMR is about the limiting practical application. • In fact, the carbon-13 spectrum, thirty times more dispersed, would need to resolve 38,016 x 18 carbons = 684,288 lines if they happened all to be different, an impossibility. • NMR by itself, therefore cannot be used to absolutely identify trisaccharides or higher oligomers by virtue of chemical shift values.

28. Analytical Challenge • As for mass spectrometry, • All 38,016 trisaccharide isomers have the same mass. • Partial fragmentation in collisional activated mass spectrometry might provide the combination of partial degradation and spectral patterns to resolve such parameters as position of linkage22, or anomeric config.22 • But this approach may not be sufficient without other sensitive chemical manipulations. • Mendonca, S. Richard B. Cole, Junhua Zhu, Yang Cai, Alfred D. French, Glenn P. Johnson, and Roger A. Laine, 2003,Incremented Alkyl Derivatives Enhance Collision Induced Glycosidic Bond Cleavage in Mass Spectrometry of Disaccharides J Am Soc Mass Spectrom.14:63-78. • Yoon, E.; Laine, R.A. Biological Mass Spectrom. 1992,21, 479-485. • Laine, R.A. ; Yoon, E.; Mahier,T.J.; Abbas,S.A.; deLappe, B.W.; Jain, R.K.; Matta, K.L. Biological Mass Spectrometry 1991,20: 505-514. • Laine, RA (1990) Glycoconjugates: Overview and Strategy in Mass Spectrometry, Methods Enzymol. 193: 539-553. (ed: JA McCloskey) • Laine, R.A.. Methods in Enzymology. 1989, 179: 157-164. • Laine, R.A.; Pamidimukkala, K.M.; French, A.L.; Hall, R.W.; Abbas, S.A.; Jain, R.K. ; Matta, K.L. J. Am. Chem. Soc. 1988,110:6931-6939.

29. Carbohydrate IT • NON-reducing oligosaccharides: Trisaccharides can also be configured with the trehalose-type aldose-1->1-aldose or the sucrose/raffinose non-reducing aldose-1->2 ketose internal linkage structure, • Larger oligosaccharides can be linked in a head-to-tail cyclodextrin fashion. • These kinds of permutations would add a large number to this calculation. • At first blush, for the set of "cyclodextric" hexasaccharides, the linear permutations number calculated below would be multiplied by 4 due to the linkage term added by the extra head-to-tail linkage, making the clyclodextrics alone close to 0.8 trillion. • However, since there would be no reducing and non-reducing terminals, many of the cyclic "isomers" might be identical depending on the chosen starting position. This will require some additional noodling. • To simplify, the scope of this lecture will be limited to the much more common reducing-end saccharides. • There have been no reported estimations of all isomers resulting from oligosaccharide branching, therefore this is a new approach.

30. Carbohydrate IT • To simplify and address the issue of carbohydrate isomers in a biologically relevant size more thoroughly, we will estimate all of the possible isomers for a reducing hexasaccharide comprised from a set of 6 hexoses in the D -configuration. • Since both D - and L- configurations of hexoses appear in nature, especially in plants, fungi and microbes, the possible isomers are even higher than we are considering here (by a factor of 26).

31. Carbohydrate IT • Although in this calculation we will only consider the possible D - isomers, we must consider that the pure L- forms generate an equal number. and • The mixed D,L forms would add a multiple of 64 to the total number.

32. Carbohydrate IT • LINEAR STRUCTURES: • The total number of possible structures, S*, of a D hexose-containing hexasaccharide begins with the value for a linear chain of 6 different non-repeating sugars ABCDEF, whose general formula is as follows: • A: S*=n!*2na*2nr*(4n-1) • Where • n is the number of different hexoses in a string. • n! is the linear permutation term, no sugar monomers repeated (6! = 120). • 2na is the term for anomeric isomers • (26) = 64. • 2nr is the term for ring size (pyranose or furanose) • 26=64 • 4n-1 is the linkage position term (45 = 1024)

33. Carbohydrate IT • While all 5 of the carbons 2 - 6 hydroxyls can participate in the linkage position when considering pyranose and furanose forms, pyranose excludes the 5 linkage and furanose excludes the 4 linkage, therefore this part of the linkage is taken into account by ring size, above. • This number for linear non-repeating structures of a hexasaccharide considering only D stereochemistry would be: • A: S* = 6! * 26 * 26 * 45 = 3,019,898,880 (three billion!)

34. Carbohydrate IT • Table 1. • Linear Isomers of D -Hexoses, each hexose used once. • Oligosaccharide size: Hexose Set Linear Isomers • _____________________________________________________________ • Monosaccharide 1 4 • Disaccharide 2 128 • Trisaccharide 3 6144 • Tetrasaccharide 4 393,216 • Pentasaccharide 5 31,457,280 • Hexasaccharide 6 3,019,898,880 • _____________________________________________________________

35. Carbohydrate IT • if each or any of the members of the 6 sugar set could be repeated, equation A becomes A' as follows: • A': S* = En * 2na * 2nr * (4n-1) • where n is the length of the chain in monomers, and E is the number of different kinds of monomers (epimers) in the set. • En is the linear permutation term where individual sugar types can be repeated within the chain. • The remaining terms are the same as in equation A.

36. Carbohydrate IT • In this case, the number of permutations for a linear hexasaccharide would be as follows: • A': S* = 66 * 26 * 26 * 45 = 46656 * 64 * 64 * 1024 = 195,689,447,424 • Nearly 200 billion, an astonishing number!

37. Carbohydrate IT • Table 2: • Linear Isomers from a set of 1-6 D-Hexoses • _____________________________________________________________ • Oligosaccharide size: Hexose Set Linear Isomers • _____________________________________________________________ • Monosaccharide 1 4 • Disaccharide 2 256 • Trisaccharide 3 27,648 • Tetrasaccharide 4 4,194,304 • Pentasaccharide 5 819,200,000 • Hexasaccharide 6 195,689,447,424 • ____________________________________________________________ • Note that all of the mono- to pentasaccharides added together comprise less than 0.5% of the number for the total hexasaccharide isomers.

39. Analysis, synthesis • A technological barrier to simple one-method analytical differentiation among this many structures is even more apparent than with trisaccharides as noted above. • Also, organic synthesis of one pure hexasaccharide among 0.2 x 1012possible structures is a daunting task. • Indeed, synthesis of a trisaccharide is estimated by most oligosaccharide synthesis chemists to take 20 man-weeks compared with 3 hours for a tripeptide. There are few 95% yield reactions in oligosaccharide synthesis. (some new automated machines alter this)

40. Carbohydrate IT • In addition, the above numbers would be increased by a large number of biologically possible compounds with branched chains. • BRANCHED STRUCTURES: • The monosaccharide in position "F" is assigned to be the reducing-end throughout, designated as "FR". • MONOSACCHARIDE BRANCHES: • For the singly branched compounds, examples are as follows: • B->C->D->E->FR B->C->D->E->FR B->C->D->E->FR B->C->D->E->FR • | | | | • A A A A • I II III IV • We will omit the arrows in the structures which are understood as pointing toward the reducing end "FR".

41. Carbohydrate IT • Each of the above represented examples of singly branched species can be considered as a separate saccharide that has a fixed branch point with regard to the location of the branching sugar moiety within the chain, the branch being movable among the hydroxyls on the branch point sugar. • All of the monosaccharides in the hexamer are then considered to contribute to isomers just as the linear form, but with the branch positions movable among carbons on each monomer capable of forming branches within the chain.

42. Carbohydrate IT • The general formula for sets of oligosaccharide isomers branched with a single monosaccharide along the core chain would be: • B: S*= En * 2na * 2nr * (4n-3)*[6*(n-2)]. • - where n-2 is the number of core monosaccharides that can originate monosaccharide branches. • - 4n-3 are the permutations of positions of linkage on unbranched monomers within the chain. • - 6*(n-2) are the possible arrangements of branches on each of the hexopyranoses in the chain capable of producing a branch (n-2 ).

43. Carbohydrate IT • These would be, for example, in I, above, the A,B branches on C inserted as either A,B or B,A, respectively on the 2,3; 2,4; 2,6; 3,4; 3,6; or 4,6 positions of pyranoses and 2,3; 2,5; 2,6; 3,5; 3,6; and 5,6 positions of furanoses. • However, we assume that permutations of the ABC monomers are included in the En term, therefore 12 possibilities remain for each possible branch position. • However, the pyranose/furanose term, 2nr includes the alternate set of 6 structures. Since the 6 possible positions for branching in each ring form account for 12 possibilities by the multiple of 2 in the ring form term, the factor for single branches should be 6*(n-2).

44. Carbohydrate IT • In this case, the number of isomers for each of configurations I-IV, above, would be: • B: S*B1 = 66 * 26 * 26 * (43) * [6*(n-2)] = • 46656 x 64 x 64 x 64 x 24 = 293,534,171,136 • This first branching example gives nearly 300 billion additional possible structures!

45. Carbohydrate IT • DISACCHARIDE BRANCHES: • For hexasaccharides with a single disaccharide branch, the set would appear as follows: • C-D-E-FR C-D-E-FR C-D-E-FR etc. • | | | • AB AB AB • V VI VII

46. Carbohydrate IT • B->C->D->E->FR B->C->D->E->FR B->C->D->E->FR B->C->D->E->FR • | | | | • A A A A • I II III IV • C-D-E-FR C-D-E-FR C-D-E-FR etc. • | | | • AB AB AB • V VI VII

47. Carbohydrate IT • V is the same as II, where ABDEFR can be considered the "core" structure with a single monosaccharide branch on D, however, VI and VII are novel arrangements.

48. Carbohydrate IT • The formula for this set would be • C: S*=En*2na*2nr*(4n-3)*[6*(n-4)]. • where disaccharide branches that generate new compounds beyond single branches already considered can only happen on n-4 of the monomers. Tetrasaccharides and below would not produce novel compounds.

49. Carbohydrate IT • for hexasaccharides the numerical total is • 46556*64*64*64*12= 146,452,512,768 • (novel structures beyond linear and single-branched hexasaccharides made up of 6 different hexoses.)

50. Carbohydrate IT • TRISACCHARIDE BRANCHES: to the core chain, • D-E-FR D-E-FR etc. • | | • ABC ABC • VIII IX