A statistical model for amylopectin and glycogen. The condensation of A‐R‐Bf−1 units

The authors have developed a statistical model for a branched polymer built from a monomeric unit (such as glucose) which has two types of reactive groups (aldehydic and alcoholic). The treatment differs from that of Flory's in that the different B groups in the A‐R‐B f‐1 monomer have differing reactivities instead of equal reactivities. The mole fraction of a particular x ‐mer having b 1 , b 2 ,… b f‐1 linkages at their respective B 1 , B 2 ,…B f−1 functional groups is: where {article}{empty}{document}x = 1 + ∑i = 1f - 1 bᵢ{document} is the total number of monomer units per molecule and p 1 , p 2 ,… p f‐1 are the probabilities that any monomer unit is linked at the B 1 , B 2 ,…B f‐1 functional group. From the above, x̄ n , the number‐average DP, x̄ w , the weight‐average DP, x̄ z , the Z‐average DP, and w x , the weight fraction of an x ‐mer, have been obtained. It is shown that the limit of x̄ z / x̄ w is 3 and the limit of x̄ w / x̄ n is infinity as the degree of polymerization increases for all cases where “f” is greater than two. These results were used for the randomly branched polymers, amylopectin and glycogen, where the functionality of the monomer in both cases is three. p 4 and p 6 were used to represent the probabilities that any glucose unit is linked at the 4 or 6 position. Using values of p 6 = 0.05 and p 4 = 0.94933, x̄ n = 1500, M̄ n = 243,000, M̄ w =35,100,000, and M̄ z = 104,500,000, giving M̄ w / M̄ n = 140 and M̄ z / M̄ w = 2.99. Literature values for molecular weights of amylopectin and glycogen are compared with the above statistical model and with less random statistical models.