Structure of Self-Assembled Globular Proteins - Food Colloids: Self-Assembly and Material Science - page 52

of independent elastic chains of the network formed by random aggregation of
the globular proteins. 16 This concentration can be calculated using the mean-
field approximation. In this approach there is no direct relationship between the
structure and the elasticity; the contribution of an elastically active chain to the
shear modulus is supposed to be independent of its size and structure.
The only model that involves a direct relationship between structure and
mechanical properties is the so-called fractal gel model. 65-67 The system is
considered as an ensemble of close-packed blobs connected to each other
through a stress-bearing backbone. When all protein is incorporated in the
gel, the size of the elastic blobs is approximately equal to the value of R a
determined with scattering techniques. Each blob acts as an elastic spring with a
spring constant that depends on the size of the blob and the fractal dimension of
the elastic backbone, d b . If the elastic backbone is fully flexible, the elastic energy
per blob in the linear response regime is proportional to the thermal energy, i.e.,
E p kT. If the elasticity is dominated by the bending enthalpy of the elastic
backbone, then E is inversely proportional to the number of elementary units of
the elastic backbone, N b , which is related to the blob size R b byN b / R d b .The
shear modulus is then E multiplied by the number concentration of the blobs (n):

G / nkT ;

ð entropy Þ

ð 7 Þ

G / nR d b
b

: ð bending enthalpy Þ

Here we have assumed that all links between elementary units are the same.
Sometimes a distinction is made between links within blobs and those between
blobs, but in the case of globular protein gelation this distinction does not
apply. In fact, as mentioned above, the blobs are only a conceptual tool to
describe the length-scale below which the system has a fractal structure and
above which the system is homogeneous.
For globular protein gels, the shear modulus is determined by the enthalpy
because G is much larger than expected for purely entropic elasticity. 28 When
all the proteins are part of the gel, we have n p R b and R b / C 1 =ð d f 3 Þ ; then G
is given by

G / C ð 3 þ d b Þ=ð 3 d f Þ :

ð 8 Þ

At concentrations close to the gel point, not all the aggregated protein is part of
the gel and Equation (8) is no longer valid, because only the gel fraction F g
contributes to the elastic modulus. However, the sol fraction does contribute to
the scattering intensity; so R b is not proportional to R a unless F g ¼ 1.
According to the percolation model, 5 the gel fraction (and thus the shear
modulus) reach zero at the gel point concentration C g or the gel time t g
following the power law F g p e 1.2 , where e can be (C - C g )/C g for a given
heating time much larger than t g ,or(t - t g )/t g for a given protein concentration
much larger than C g . However, this limiting power-law dependence is only
valid for e { 1 and it is dicult to observe experimentally.
The model leading to Equation (8) is based on general properties of the gel
structure. It does not give the pre-factor of the power-law increase; that