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It can be shown that the reference probability as dened by this renormalization
prescription is still a stationary state of the Laplacian matrix of the renormalized
graph. Moreover this probability distribution ensures that the renormalized transi-
tion rates satisfy the detailed balance.
We stress that connectivity graphs obtained by applying the renormalization
protocol here described depend on temperature in a twofold manner: the obvious
Arrhenius dependency of the jumping rates and the somewhat arbitrary renormal-
ization threshold that dictates which connections to coalesce. In order to realisti-
cally reproduce the metastable states of the system the latter must be of the same
order of magnitude of the temperature. It is however dicult to devise any theo-
retical argument suggesting what should be its precise numerical value. In [6] it has
been shown that the large scale dynamical features of the system do not crucially
depend on the choice of this parameter and that the renormalization procedure does
not alter the long timescales of the dynamics on the graph.
6.2.2. Two-dimensional models
The idea of approximating the thermalized dynamics of a polypeptidic chain by a
stochastic dynamics on a directed graph was tested using a very simple model, rst
introduced in [17]. This is a slight modication of the 2{d o-lattice HP model
originally proposed by Stillinger et al. in [18]. It is dened by the Hamiltonian
X
L
L1
X
p
x;i
+ p
y;i
2
H =
+
V
1
(r
i;i+1
)
i=1
i=1
L1
X
L2
X
X
L
+
V
2
(
i
) +
V
3
(r
ij
;
i
;
j
)
(6.8)
i=2
i=1
j=i+2
which contains the phenomenological potentials
V
1
(r
i;i+1
) = (r
i;i+1
r
0
)
2
;
1cos
i
16
V
2
(
i
) =
;
(6.9)
1
r
12
i;j
c
i;j
r
i;j
V
3
(r
i;j
) =
:
All the parameters are expressed in terms of adimensional arbitrary units: for in-
stance, and r
0
are xed to the values 20 and 1, respectively. The model Hamilo-
nian represents a one-dimensional chain of L point-like monomers corresponding to
the residues of a real protein. Only two types of residues are considered: hydro-
phobic, H, and polar, P . Accordingly, a heteropolymer is identied by a sequence
of discrete variablesf
i
g(with i = 1;:::;L) along the chain:
i
=1 indicates
that the i-th residue is of type H or P, respecively. The intramolecular potential
is composed of three terms: a sti nearest-neighbour harmonic potential V
1
, which
