  In this paper, we model the mechanics of a collagen pair in the connective
tissue extracellular matrix that exists in abundance throughout animals,
including the human body. This connective tissue comprises repeated units of
two main structures, namely collagens as well as axial, parallel and regular
anionic glycosaminoglycan between collagens. The collagen fibril can be modeled
by Hooke's law whereas anionic glycosaminoglycan behaves more like a
rubber-band rod and as such can be better modeled by the worm-like chain model.
While both computer simulations and continuum mechanics models have been
investigated the behavior of this connective tissue typically, authors either
assume a simple form of the molecular potential energy or entirely ignore the
microscopic structure of the connective tissue. Here, we apply basic physical
methodologies and simple applied mathematical modeling techniques to describe
the collagen pair quantitatively. We find that the growth of fibrils is
intimately related to the maximum length of the anionic glycosaminoglycan and
the relative displacement of two adjacent fibrils, which in return is closely
related to the effectiveness of anionic glycosaminoglycan in transmitting
forces between fibrils. These reveal the importance of the anionic
glycosaminoglycan in maintaining the structural shape of the connective tissue
extracellular matrix and eventually the shape modulus of human tissues. We also
find that some macroscopic properties, like the maximum molecular energy and
the breaking fraction of the collagen, are also related to the microscopic
characteristics of the anionic glycosaminoglycan.
