Protein Modelling and Surface Folding by Limiting the Degrees of Freedom - Computational Modeling in Tissue Engineering - page 35

b

a

x ( t )

x ( i t )

i 1

Fig. 15 The sequence [ a ; ðÞ is length of a single coil] gives an approximation to the polygon arc
P; the length between two points (a, b), where D is segments of arc-length

Appendix 2

Boundary Determination to Prevent Overlap
A path is a one-dimensional sub-manifold M of R 3 , so that, for any point x 2 M
there is a local parameterization near x. C k k ð Þ denotes the curvature of the path
and D denotes the coordinates identifying the path. The output of each iteration is
a set of coordinates in three dimensions, D ¼ x 1 ; x 2 ; ... ; x ð Þ identifying a path.
We denote by length bond is the polygonal arc around the path (Fig. 15 ). The
curvature C k and the arc-length are non-regular. Let x ¼ x(t), with a t b and
consider a partition [ 15 ]:
a ¼ t 0 \t 1 \ \t n ¼ b, of an interval (a, b).
The sequence (a, b) are the boundaries of a single coil) gives an approximation
to the polygon arc C. As illustrated the length between two points (a, b), where D
are segments of arc-length given by:

k ð D) ¼ X D j ¼ X k ¼ X

n n n

x i x i 1 xt ðÞ xt i ðÞ

ð 2 : 1 Þ

k k k

j ¼ 1 i ¼ 1 i ¼ 1

The arc-length can be bounded from above and from below. The upper bound is
given by:

k ðÞ X
K D j

Þ¼ 1

q þ K ; D ð k K D j ð 2 : 2 Þ

ð Þ \ D 6¼;

And the lower bound is:

k ðÞ X
K D j

Þ¼ 1

q K ; D ð k K D j ð 2 : 3 Þ

ð Þ D

where q þ ð K,D) is the ratio of the total measure of the set in the system K (is the
volume minimization) so that the transformation (projection) of the segments
and the curve C give the lower and the upper bound a, ð .

b ¼ q þ ¼ lim

k ð Þ!1 sup q þ K ; D

ð Þ¼ lim

k !1 Supq

ð K ; D Þ ð 2 : 4 Þ

þ
k ðÞ k

a ¼ q ¼ lim

k ð Þ!1 inf q K ; D

ð Þ¼ lim

k !1

inf
k ðÞ k

q K ; D ð Þ ð 2 : 5 Þ