Biomedical Engineering Reference
In-Depth Information
In collagenous soft tissues, collagen molecules are mainly of type I and they
exhibit hydroxyproline-deficient sequences characterized by 60 residues (about
20 nm long), referred to as labile domains and indicated as molecular kinks [
2
].
Their measure of length
'
kinks
is comparable with the value of the persistence
length
1
'
p
for collagen (about 14 nm). This evidence confirms that molecular kinks
are activated by thermal fluctuations [
2
,
9
] and can be extended by forces at
molecular ends that counteract thermal undulations. In this case entropic mecha-
nisms are activated and a transition regime is experienced, from less ordered
molecular states (thermally-activated kinks) to more ordered ones (nearly straight
macromolecule). In this regime, usually referred to as entropic elasticity [
9
], the
mechanical response of collagen macromolecules is mainly dominated by the
flexural behavior of the polypeptide helices rather than by the extensibility of
intra-molecular covalent bonds. Accordingly, neglecting any stretching effect of
the intra-molecular bonds and in agreement with the Worm-Like Chain (WLC)
model [
10
,
11
], the pair of equilibrated forces F
s
to be applied at molecular ends
for obtaining the end-to-end molecular length
'
m
results in:
"
#
1
41
'
m
='
c
Þ
2
1
4
þ
'
m
F
s
ð'
m
Þ¼
q
;
ð
1
Þ
'
c
ð
where
'
c
is the molecular contour length and q
¼
k
B
T
='
p
;
T being the absolute
temperature and k
B
the Boltzmann constant. Equation (
1
) exhibits a pole for
'
m
¼ '
c
, highlighting that the WLC-model is not able to capture an extension of
the end-to-end molecular length over
'
c
involving entropic mechanisms only.
Nevertheless, well-established evidences on collagen show a significant level of
molecular extensibility beyond
'
c
[
12
]. Accordingly, when
'
m
approaches
'
c
the
applied force contributes to activate the stretch of molecular covalent bonds,
inducing the onset of energetic mechanisms [
9
,
11
].
Theoretical models accounting for the extensibility of biopolymer macromol-
ecules over
'
c
have been recently proposed in [
13
-
15
], and the transition regime
from entropic towards energetic elasticity for collagen has been numerically
investigated by using Molecular Dynamical Simulations (MDSs) [
9
]. MDS-based
results proposed in [
9
] have been successfully recovered by the theoretical
approach developed in [
15
], wherein the transition mechanisms were consistently
described by a physically-based lumped-parameter equilibrium formulation,
avoiding phenomenological transition parameters as made in [
13
,
14
].
It is worth pointing out that, as experimental [
12
] and MDS-based [
9
,
16
] results
suggest, the mechanical response of collagen molecules due to energetic effects is
highly non-linear in the first stage, following a pseudo-exponential law, and then
tends asymptotically towards a linearly elastic behavior for
'
m
'
c
. Such an
evidence can be justified by observing that the non-linearities are essentially due to
1
The persistence length is the maximum contour length over which the corresponding molecular
segment appears as straight under thermal fluctuations.
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