Biomedical Engineering Reference
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Fig. 1.2 Extraction procedure for the bending potential of X-X-P combinations. ( a ) Normalized
distribution of the bending angle θ , which is obtained by mapping all the Ramachandran data ( in-
set ) to the pseudo-bending angle θ through Eq. ( 1.1 ). Inset : Ramachandran data for the central
residue of X-X-P combinations extracted form the coil library. ( b ) Obtained bending potential,
U(θ) after applying the Boltzmann-inversion method on the probability distribution P(θ) pre-
sented in ( a )
Four basic types of Ramachandran plots have been reported in the literature depend-
ing on the stereo-chemistry of the amino-acids: Glycine, Proline, 'Generic' (which
refers to the remaining 18 amino acids), and 'Pre-Proline' (which refers to residues
preceding a Proline) (Ho and Brasseur, 2005 ). As a result, different potential func-
tions are expected for Glycine (G), Proline (P), and the rest of the amino acids (X)
depending on their neighboring residues.
1.2.4 Extraction of Potential Functions
Bending potentials for the pseudo-bond angles are obtained by Boltzmann-inversion
of the θ probability distribution. Initially, φ and ψ dihedral angles for the central
residue of different triple combinations of P, G and X are extracted from the 3-
residue-fragments in the coil library. The extraction procedure is depicted schemat-
ically in Fig. 1.2 for X-X-P combinations. In Fig. 1.2 (a-inset) the φ and ψ values
are plotted for all X amino acids (i.e. those amino acids that are not P or G) that have
an X preceding it and a P following it. In the next step, each datapoint in the Ra-
machandran space is mapped to θ using Eq. ( 1.1 ). Collecting all datapoints in data
bins gives the θ probability distribution (Fig. 1.2 (a)) which is then directly converted
to the bending potential by Eq. ( 1.3 )(seeFig. 1.2 (b)).
In order to develop the bending potentials, one can consider different levels of ac-
curacy. This could range from developing 27 bending potentials for all combinations
of G, P and X to just three sets of potentials for our three letter amino-acid alpha-
bet ignoring any neighbor dependence. Studying proteins with different amino-acid
contents shows that including the neighbor-residue effect is important only if the
considered residue is preceding a Proline residue. Therefore, 6 sets of bending po-
tentials are suggested in which we distinguish those central residues that do and do
not precede a Proline.
 
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