Helical structures from neutral biopolymers - Physical Gels from Biological and Synthetic Polymers - page 204

which indicates that the contact number (average number of neighbours in contact with
a certain particle) at the threshold c critical is of the order of 1 (actually 1.4). Networks of
rod-like macromolecules show thresholds for the gel

- sol transition at a low polymer

concentration, c critical ≈ 0.05 wt%, in agreement with the excluded volume of the

molecules.
Gels of (albeit structurally quite different) colloidal rods were also investigated
experimentally by Philipse and Wierenga ( 1998 ). They also considered the case of
heterogeneous networks, which leads to very low critical volume fractions at the thresh-
old, when particles form fractal clusters.
The gelation threshold in gelatins is compatible with the formation of a loose network
of rods with a large aspect ratio. As the diameter of the triple helix is around 1 nm, the
length of the equivalent rods at the threshold could be around 200

- 300 nm. However, the

gels are not made of rods with a fixed length; the helices grow step by step, increasing

the helix fraction with time, but the distribution of the helical sequences is probably not
identical in all circumstances. However, the elasticity behaves as if the volume fraction of
helices or the total length of helices per unit volume is the only parameter which has an
in uence on the elasticity.

>> 2c critical
helix
Here the percolating system changes towards a more homogeneous network. However,
even then the helix concentration determines the elasticity of the network. The helix
concentration (g cm − 3 ) can be converted to the total length of triple helices per unit
volume, L v , as if all the helical sequences were assembled end to end in one single ribbon.
The ribbon creates a

(b) At higher helical concentrations , c

filamentous structure with tightly entangled strands. Transmission
electron microscopy (TEM) imaging of network replicas indeed shows a

filamentous

structure.
In statistical analysis, if

filaments adopt all orientation angles in a given volume, the
nearest-neighbour distance d is related to the total length L v of

filaments by a very simple

relation:

2L v Þ - 1 = 2

¼ð : ð 7 : 9 Þ

d

10
gcm − 3 , and knowing from the collagen structure that the triple helix has an average mass
per mole per unit length of 1000 g mol − 1 nm − 1 ,we

Considering, for instance, a helix fraction of χ = 0.40 for a gel with concentration c ≈

2.4×10 4 μm/μm 3 , which is a

nd L v ≈

large value. The distance d derived from ( 7.9 )is ≈ 4 nm, in agreement with TEM and

neutron scattering investigations. Therefore, filaments are tightly entangled, so that in the

range 0.01 g cm − 3 < c helix < 0.05 g cm − 3 , the calculated distance between the strands
(mesh size of the network of helices) is 7 nm > d > 3 nm, which is much smaller than the
persistence length of collagen, l p ≈

170 nm. The network of helices forms an intricate
structure at scales such that they have to be considered as rigid objects (d << l p ), a
situation comparable to that of actin gels ( Chapter 9 ), for example.
We can then consider the gel network to consist of semi- exible strands of persistence

length l p , of length L and linear density 1/d 2 , but further assumptions on the deformation