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keeps the bond distance almost constant, a three-body potential V
2
, which measures
the energetic cost of local bending, and a long{range Lennard-Jones potential V
3
acting between all pairs of monomers i and j such thatjijj> 1. For the sake of
simplicity, the monomers are assumed to have the same unitary mass. The space
coordinates of the i-th monomer are (x
i
;y
i
) and their conjugated momenta are
(p
x;i
;p
y;i
) = ( x
i
; y
i
) . The variable r
i;j
=
p
(x
i
x
j
)
2
+ (y
i
y
j
)
2
is the distance
between i-th and j-th monomer and
i
is the bond angle at the i-th monomer. V
3
is the only contribution that depends on the nature of the monomers. The coe-
cients c
i;j
=
8
(1 +
i
+
j
+ 5
i
j
) are dened in such a way that the interaction
is attractive if both residues are either hydrophobic or polar (with c
i;j
= 1 and
1=2, respectively), while it is repulsive if the residues belong to dierent species
(c
ij
=1=2).
Here, we focus our investigation on three sequences of twenty monomers that
represent the three classes of dierent folding behaviors observed in this model:
[S0] a homo-polymer composed of 20 H residues;
[S1]=[HHHP HHHP HHHP PHHP PHHH] a sequence that has been iden-
tied as a fast-folder in [19];
[S2]=[PPPH HPHH HHHH HHHP HHPH] a randomly generated sequence,
that has been identied as a slow-folder in [17].
The three characteristic temperatures T
(collapse temperature), T
f
(folding
temperature), T
g
(\glassy" temperature) determined by means of Langevin simula-
tions in [10] are reported in Table 6.1 for the three analyzed sequences.
Table 6.1.
Characteristic temperatures for the analyzed sequences.
homopolymer S0
fast-folder S1
slow-folder S2
T
0.16
0.11
0.13
T
f
0.044
0.061
0.044
T
g
0.022
0.048
0.025
A thorough discussion of numerical techniques that can be used to eectively
reconstructing the connectivity graph for the models above and and to test the
reliability of the graph in reproducing the observed equilibrium and nonequilibrium
properties of the models can be found in [6].
6.2.2.1. Topological properties of the renormalized connectivity graph
We now consider the topological properties of the connectivity graph and in par-
ticular we focus on how the renormalization procedure aects the topology. We
perform this analysis on all the three sequences previously introduced with the aim
to describe how the topology of the renormalized connectivity graph inuences the
folding propensity of the system (see [6] for the details).
