Number of isomers in a 3-sugar oligosaccharide

Question

I recently read that 12288 different oligosaccharides can be formed from the D-isomers of glucose, mannose and galactose. I wanted to try to calculate this, but I keep falling very short of this number.

Each molecule has the potential to form 6 unique bonds:

The disaccharide can form 13 unique bonds:

(Note, bonds formed by anomeric carbons shown with an arrow)

So, if a monosaccharide can form 6 different bonds, this means that:

6 possible bonds x 5 bonds on 2nd sugar x 2 anomers for seconds sugar = 60 possible disaccharides

Each of these disaccharides has 13 possible bonds it can make to a third saccharide:

13 bonds x 5 bonds of 3rd molecule x 2 anomers = 130 combinations for a single dissacharide.

130 combinations per disaccharide x 60 possible disaccharides =7800 isomers.

This is probably a counting error. But maybe I am missing a stereocenter?

Answer 1

This question is interesting and neglected. Bob made a sensible attempt at solving the problem. I took a different approach. [The number of isomers 12,288 was found in Stryer’s Biochemistry (Google: stryer 12288 isomers)]. Of course, oligo- means a few. I presume this means three in this case. But can D-glucose, D-galactose and D-mannose be used as many times as you want? That would mean 27 (3³ = 27) permutations which would eventually make the total much greater than 12,288. However, if each hexose is to be used once, then six (6! = 6 factorial) permutations are possible for the location of the hexoses in the oligosaccharide. Division of 12,288 by 6 removes this issue from the question and leaves 2,048 as the number to justify. Now 2048 = 2¹¹. The problem reduces to finding 11 stereocenters, if possible!

Let us assume as a point of departure that diagramatic structure ABC (top of the illustration; The connectivity C₁-C₄ is arbitrary.) shows where bonds are formed between the C₁ anomeric carbons and the hydroxyls at C₂, C₃, C₄ and C₆ of another hexose. Another pair of assumptions is that all rings are 6-membered (pyranose) and that all hexoses, as defined, are in the D-series. Finally, assume for the time being that there are no C₁-O-C₁ connections as typified in figs. D and E.

Rather than define C₁ of hexose A as having two stereocenters (α and β anomers), this assignment will be considered in a different way. Hexose A is given a value of zero because no attachments may be made at its hydroxyl groups at C₂, C₃, C₄ and C₆. Hexose B has four hydroxyl groups (blue dots) that can form two anomeric, glycosidic bonds at C₁ with hexose A. The AB connection creates 2⁴ (16) stereoisomers. In a similar fashion the hydroxyls at C₂, C₃, C₄ and C₆ of hexose C (red dots) form another 16 stereoisomers with C₁ of hexose B. The total count thus far is 3! x 2⁸ = 6 x 256 = 1536. This result is only one-eightth of the goal of 12,288 or short by a factor of 2³. To this point the eight hydroxyl groups considered can only form two configurations each.

Now the anomeric hydroxyl of hexose C when bonded to the anomeric carbon of hexose B will give not two stereoisomers but rather four (αα, αβ, βα, ββ)(see fig. E). Similarly, the same relationship exists between hexoses B and A (see fig. D). These configurations add a factor of 16 (2⁴) to the total. This reasoning leads to a value of (3!) x (2¹²) = 6 x 4096 = 24,576.

If the original analysis considered a structure such as ABC having a mutarotating C₁-OH, then the total might be 12,288. However, ABC is a connectivity map, not a chemical structure.

What may be concluded from all of this? The premises set down here may not be those applied by the proposers of 12,288. There may be a flaw in this, or their, analysis. The answer may be 12,288, 24,576 or something else.