Protein Modelling and Surface Folding by Limiting the Degrees of Freedom - Computational Modeling in Tissue Engineering - page 23

Area 123 ¼ Z Area S ð dS 1 þ Z Area S ð dS 1 þ Z

K M L

Area S ð dS 1 and 0\K\M

L K M

We obtain the minimal free energy by minimizing the volume which we have
accounted for by the surface areas as calculated above. Note that more than three
overlapping spheres are improbable for atoms in molecules [ 20 ]. The objective
function for the minimization is given by:

Objective ¼ volume(volumeweighing)
X surfacearea 12 U ; H ð Þ hydrophobicity 1 hydrophobicity 2

The volume weights are proportional to the amount of energy needed to move
the R from cyclohexane to water (0 is neutral, -1 is hydrophilic and 1 is
hydrophobic), using the surface of the whole amino acid, rather than just the R
group [ 65 ]. Each residue has a surface area with a known hydrophobicity [ 12 ]. The
summation is over each set of residues that are touching (i.e. adjacent to each
other). The surface area is the common surface area between the residues. This
term will tend towards having hydrophobic residues together and hydrophilic ones
together, but will avoid having hydrophilic next to hydrophobic residues.
There is also a volume term that minimizes the size of the molecule [ 11 , 58 ].
This volume is given as the volume enclosed by the surface wetted by a solvent
molecule with a *1.4 Å radius. The model is based on changing the distance and
the angles and are adjusted to the distances. If we do not know the structure, we
can calculate in parallel the a-coil and b-sheet simultaneously forming the braid
used to obtain a globular or complex structure.

2.1 Constraints in the Standard Braid Theory

Prohibiting the braids from incidental intersection with themselves or with other
braids is properly observed (rule of ''no intersection'') in this application to keep
the modeled peptide chains from overlapping each other (Appendix 2). The simple
arc length model has been expanded to address the finite volume occupied by each
amino acid residue [ 48 ]. While keeping the length and direction of the arc lengths
constant, each segment is expanded into a bead enveloping the remainder of its
amino acid residues. Each bead interacts with at most two other beads, and the
intersection of any two sequential beads is a single point [ 39 ]. A braid now
represents the peptide chain, which is a collection of beads (Appendix 3). Various
peptide conformations can now be treated as changes in the relative orientation
between pairs of beads. For large, single-chain proteins this is a significantly
simplified approach to molecular modelling (Appendix 4; Appendix 5). The power
of this approach is seen when considering protein structures composed of multiple