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6.3. Geometry of Energy Landscapes
The local strategy to analyze energy landscapes described in Sec. 6.2 requires a
huge computational eort. So the following question arises: is there some global
property of the energy landscape which can be easily computed numerically as an
average along dynamical trajectories and which is able to identify polymers having
a protein-like behavior? We shall show in the following that such a quantity indeed
exists, at least for the minimalistic model we considered, and that it is of a geometric
nature. In particular, the uctuations of a suitably dened curvature of the energy
landscape clearly mark the folding transition while do not show any remarkable
feature when the polymer undergoes a hydrophobic collapse without a preferred
native state. This is at variance with thermodynamic global observables, like the
specic heat, which show a very similar behavior in the case of a folding transition
and of a simple hydrophobic collapse.
The intuitive reason why geometric information on the landscape, and especially
curvature, could be relevant to the problem of folding is that the dynamics on a
landscape would be heavily aected by the local curvature: minima of the energy
landscape are associated to positive curvatures and stable dynamics, while saddles
involve negative curvatures, at least along some direction, thus implying some in-
stability. One can reasonably expect that the arrangement and detailed properties
of minima and saddles might reect in some global feature of the distribution of
curvatures of the landscape, when averaged along a typical trajectory.
The denition of the curvature of a manifold M depends on the choice of a metric
g [23, 24]: once the couple (M;g) is given, a covariant derivative and a curvature
tensor R(e
i
;e
j
) can be dened; the latter measures the noncommutativity of the
covariant derivatives in the coordinate directions e
i
and e
j
. A scalar measure of the
curvature at any given point P2M is the the sectional curvature
K(e
i
;e
j
) =hR(e
i
;e
j
)e
j
;e
i
i;
(6.10)
whereh;istands for the scalar product. At any point of an N-dimensional mani-
fold there are N(N1) sectional curvatures, whose knowledge determines the full
curvature tensor at that point. One can however dene some simpler curvatures
(paying the price of losing some information): the Ricci curvature K
R
(e
i
) is the
sum of the K's over the N1 directions orthogonal to e
i
,
X
N
K
R
(e
i
) =
K(e
i
;e
j
) ;
(6.11)
j=1
and summing also on the N directions e
i
one gets the scalar curvature
X
N
X
N
R=
K
R
(e
i
) =
K(e
i
;e
j
) ;
(6.12)
i=1
i;j=1
K
R
N1
and
R
N(N1)
then,
can be considered as average curvatures at a given point.
