Protein Modelling and Surface Folding by Limiting the Degrees of Freedom - Computational Modeling in Tissue Engineering - page 36

Fig. 16 The radii are chosen
so that the intersection of the
closure of any two beads S i

p

ij

v v

and S j is a single point p ij ; .

The point p ij ; is the origin of
a right and a left vector
v iR ; v jL

iR jL

S

S

i j

Hence the boundaries of C are given in ( 2.4 ) and in ( 2.5 ).

Appendix 3

The geometry structure of the protein is defined by a braid (see B2for a
description of a chain as a collection of beads forming a braid). The jth molecule
of the chain is fitted to a conveniently shaped open bead Sj (see B3) with is 0
center located at the center of the bead and the radius r i has size such that the ith
bead does not overlap with the jth bead when i 6¼ j :
The radii in Fig. 16 r i are chosen so that the intersection of the closure of any
two beads S i is a single point p ij ; (see B3). The point p ij ; is the origin of a

right and a left vector v iR ; v jL . In this process it is important to translate (projec-
tion) and rotate these vectors. The mathematics of this construction justifies
geometry of the bead construction.

and S j

Appendix 4

With our model of collagen in mind, we next introduced the concept of the braid
group. The braid was defined as the union of the backbones creating a string
representing the amino acids. The collagen has three strands (as a group) or coils
and each strand has a back bone, represented as the union of all points x (t i - 1 ,t i )
that are generated:

Bonds ¼ N
n ¼ 1

f xt i 1 ; t i ð Þ g ð 4 : 1 Þ

A braid is a collection of beads for which two operators ; ð Þ can be defined. The
bead in the collection can be projected using least of the squares. Let B denote this
collection of beads,soB ¼ braid ð Þ , and B, ðÞ is a group. We are checking the
segments of the radius of bead of a single braid. The enclosed volume shrinks
driven by minimization and through the homoeopathy is guaranty [ 2 ] (see B5).
We are modelling three coils, and their geometrical configuration has an equiva-
lence class denoted by r i and r i .Abraid is equivalent and it is called isotope if
the three coils cannot pass each other or themselves without intersecting [ 8 ]
Fig. 17 .