Handbook of Solvents - George Wypych - ChemTech - Ventech!

12.2 Chain conformations of polysaccharides 723
are also observed. For example, those of (1-4)-linked α-D-glucan and of (1-3)-linked
β-D-glucan seem fairly restricted to some pseudo-helical character compared to the more
disordered set of trajectories that would be obtained if rotations about the glycosidic bonds
were completely unrestricted. Possible interactions between residues of the polysaccharide
chain that are not nearest-neighbors in the primary sequence of the polymer can sometimes
be ignored. In this case, a computer-based polysaccharide chain can be constructed from the
conformational energy map of the dimeric units. The Monte Carlo method 64 and the Flory
matrix methods 65 are commonly used in the so-called “nearest-neighbors approximation” to
mimic the polymer chains in the pure amorphous state or in dilute solution.
The Monte Carlo sample of chains reflects the range of conformations experienced by
any single chain as a function of time or, equivalently, the range of conformations in a large
sample of chemically identical polymer molecules at any instant in time. In either sense, the
sample can be analyzed to deduce both the characteristics of individual chain conformations
and the mean properties of the sample as a whole, which correspond to those in the equilibrium
state of the chain. Results refer, however, to an “unperturbed” chain model that ignores
the consequences of the long range excluded volume effect, because only nearest-neighbor
interactions are accounted for.
Given a sufficient Monte Carlo sample of chains in equally probable representative
conformations, it is possible to assess many mean properties of the polymer in question simply
by computing numerical (unweighted arithmetic) averages over the number of chains in
the sample. For example, the mean square end-to-end distance, the mean square radius of
gyration, or the angular dependence of scattered light (particle scattering factor) are all
mean geometric properties readily computed from a knowledge of the coordinates of the atoms
or atomic groups which are generated in the Monte Carlo sample. The average topological
properties are described through the chain-length dependence of quantities such as the
characteristic ratio, C n, the persistence length, P n, and the correlation function, F n, defined
as:
C
r
L n
= P = L F uu
nL
L i
⎛
2
�
⎞ �
⎜ ⎟
��
0
1
, =
2 ⎜ ⎟∑
,
⎝ 1 ⎠ = 1
n n i
0
n n
0
[12.2.6]
where:
n number of saccharide units
L average virtual bond length
0 mean square end-to-end distance
The virtual bond vector is often defined for each monosaccharidic unit as connecting
oxygen atoms involved in glycosidic linkages. Figure 12.2.11 shows the calculated properties
of equations [12.2.6] for two homoglucan chains which differ only in the configuration
of the anomeric carbon, i.e., (a) the [α-D-(1-3)-glc] n (pseudonigeran) and (b) the
[β-D-(1-3)-glc] n (curdlan). Whatever the chemical features, provided that the molecular
weight is very large (that is for a degree of polymerization n which approaches infinity), the
distribution of unperturbed polymer end-to-end length is Gaussian and Cn equals an asymptotic
characteristic ratio C∞, i.e.: