Biomedical Engineering Reference
In-Depth Information
interaction energy of native contacts. Therefore, native interactions tend to be
weaker and less optimized in longer proteins. This may be interpreted as yet another
manifestation of the tendency of proteins to have marginal stability.
7
Inverse Folding
Closely related to the question of protein evolution is the inverse folding problem.
The inverse folding problem can be formulated as the study of the statistical
properties of protein sequences that fold into a given target structure, and is of
importance for bioinformatics applications such as structure prediction, as well
as for theoretical modeling. Interestingly, it is possible to analytically solve the
inverse folding problem within the hydrophobic approximation of the energy. This
approximation exploits the fact that the contact energy matrix U.a; b/ is well
approximated by its main eigenvector h.a/,
U.a; b/
"
H
h.a/ h.b/ ;
(9)
where "
H
<0and the eigenvector h.a/ is related to the hydrophobicity of residue
a [
64
,
65
]. Using this approximation, the native energy can be expressed as
E.
A
;
C
nat
/
"
H
X
ij
C
nat
ij
h
i
h
j
;
(10)
where h
i
D
h.A
i
/ is the hydrophobicity profile of sequence
A
. It is immediately
seen that the optimal hydrophobicity profile that minimizes the native energy for a
given value of the mean squared hydrophobicity
h
h
2
coincides with the principal
eigenvector of the contact matrix. If we further impose a condition on the mean
hydrophobicity
i
in order to constrain stability against unfolding, we find that the
optimal hydrophobicity profile is proportional to the so-called effective connectivity
(EC) profile c
i
[
66
], a structural profile that almost coincides with the principal
eigenvector for single domain proteins, and generalizes it for multi-domain proteins.
The EC has large components in the core of the protein, where residues have many
contacts, and small components on the surface. Thus, the optimal hydrophobicity
profile h
opt
h
h
i
i
/
c
i
expresses the well-known fact that buried positions tend to be
hydrophobic and surface positions tend to be hydrophilic, but in a quantitative
fashion. The optimal hydrophobicity profile is in very good agreement both with
the hydrophobicity profile averaged over sequences obtained by simulating protein
evolution with structural conservation, and with the hydrophobicity profile averaged
over positions in the PDB that have similar EC components. We are currently
investigating modifications of this framework that allow for enforcing stability
against misfolding in a more explicit way.
The sequence that best matches the optimal hydrophobicity profile is analogous
to the prototype sequence found by Bornberg-Bauer and Chan in simulations of
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