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simply as K
R
dropping the dependence on the direction. Another feature of K
R
is its very simple analytical expression which simplies both analytical calculation
and numerical estimates. It is also worth noticing that expression (6.16) is a very
natural and intuitive measure of the curvature of the energy landscape, as it can
be seen as a naive generalization of the curvature f
00
(x) of the graph of a one-
variable function to the graph of the N-dimensional function V (q
1
;:::;q
N
): the
Laplacian of the function. However, the previous discussion shows that it is much
more than a naive measure of curvature and that it contains information on the
local neighborhood of the dynamical trajectories.
The Ricci curvature dened in Eq. (6.16) will be used to characterize the geom-
etry of the energy landscape.
6.3.1. Curvature of the energy landscape of simple model proteins
Let us now describe the model whose energy landscape geometry was studied. We
considered a simple model able to describe protein-like polymers as well as polymers
with no tendency to fold; the dierent behaviors being selected upon the choice of
the amino acidic sequence. The model we chose is a minimalistic model originally
introduced by Thirumalai and coworkers [29]. In order to characterize its energy
landscape geometry, we sampled the value of the Ricci curvature K
R
dened in
Eq. (6.16) along its dynamical trajectories using Langevin simulations [25].
The Thirumalai model is a three-dimensional o-lattice model of a polypeptide
which has only three dierent kinds of amino acids: polar (P), hydrophobic (H) and
neutral (N). The potential energy is
V (r
1
;:::;r
N
) = V
bond
(jr
i
r
i1
j) + V
angular
(j#
i
#
i1
j)
+ V
dihedral
(
i
) + V
non-bonded
(r
1
;:::;r
N
)
(6.17)
where
N1
X
k
r
2
(jr
i
r
i1
ja)
2
;
V
bond
=
(6.18)
i=1
N2
X
k
#
2
(j#
i
#
i1
j#
0
)
2
;
V
angular
=
(6.19)
i=1
N3
X
V
dihedral
=
fA
i
[1 + cos
i
] + B
i
[1 + cos(3
i
)]g;
(6.20)
i=1
N3
N
X
X
V
non-bonded
=
V
ij
(jr
i;j
j) ;
(6.21)
i=1
j=i+3
where r
i
is the position vector of the i-th monomer, r
i;j
= r
i
r
j
, #
i
is the i-th
bond angle, i.e., the angle between r
i+1
and r
i
,
i
the i-th dihedral angle, that is
