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10 6
9
a
b
8
7
10 5
6
N nodes
<j>
5
10 4
4
3
10 3
2
0
0.02
0.04
0.06
0.08
0
0.02
0.04
0.06
0.08
T
T
Fig. 6.1. Order (a) and average connectivity (b) of the renormalized connectivity graph as a
function of temperature. Empty circles correspond to the homopolymer, lled squares and crosses
to the slow{folding and fast{folding heteropolymers, respectively. Dashed lines represent least{
square exponential ts to the data.
Preliminary to the study any topological characteristic we analyze how renormal-
ization aects the size of the graphs under study. In Fig. 6.1(a) we show the order
of the renormalized connectivity graph, i. e., the number of nodes it is composed
of, for a temperatures that encompasses both the glassy and transitions T = 0:00,
0:02, 0:04, 0:06, 0:08 of all the sequences under study. The eect of renormaliza-
tion on graph sizes are signicantly weaker for the homopolymer than for the two
heteropolymers. While in the rst case the order of the connectivity graph reduces
by a mere 30%, in the latter cases it decreases by an order of magnitude or more.
The dierent sensitivity of the sizes of connectivity graphs to temperature
changes is an eect of the dierences in the distribution of the energy barrier heights
W. Figure 6.2 shows the distribution of W for the three analyzed sequences both at
high and low energies (main panel and inset, respectively). A general shift toward
high barriers can be observed in the case of the homopolymer. The consequent
decrease in the fraction of low energy barriers explains the relative insensitivity of
the homopolymer to renormalization: in this temperature range the same increase
in temperature causes the coalescence of many less connections in the homopolymer
than in the heteropolymer.
As far as topology is concerned, Fig. 6.1(b) shows the average node connectivi-
ties of the sequences under study. The homopolymer, that is characterized by the
presence of many more metastable states at all temperatures, has instead a lower
connectivity than heteropolymers, almost 3 against 6 at T = 0. This discrepancy
however attenuates as temperature increases: while the homopolymer connectivity
remains substantially unchanged with temperature, the connectivity of heteropoly-
mers markedly decreases. This drop reects a tendency of the connectivity distri-
butions of the two heteropolymers to peak at low values.
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