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2. Description of the oxidative folding space as graphs
The graph descriptions introduced above can be applied both to fully and to partially
oxidized intermediates, and the transitions between them can be conveniently described by
comparing the adjacency matrices of the two states. The sum of the elements in the i-th
column plus the i-th row (S
i
=6
j
A
ji
6
j
A
ij
) shows if the i-th cysteine forms a bridge. The
sum of the differences calculated between these measures of two adjacency matrices
describing two intermediates, (SD =6
i
'S
i
) shows how many cyesteins gained or lost a
pair. If two states are connected by a disulfide interchange reaction, the number of disulfide
bridges NB remains the same by definition, and it is easy to show that SD will differ
exactly by 2. For redox steps in which one disulfide bridge is established or lost, NB and
SD will increase or decrease by one and two, respectively. On the above basis one can
easily enumerate, for a protein with any number of cysteine residues, a) the oxidative
folding states and b) the possible transition steps between them. In other words one can
draw a network of all possible oxidative folding pathways. The characteristics of a few
systems are summarized in Table 1.
Table 1.
Number of possible intermediates in and graph parameters of oxidative
folding networks.
N of
cysteine
s
N of
intermediat
es (nodes)
Redox
transition
s
Shufflin
g
transitio
ns
Total no
of
transitions
(edges)
Clustering
coefficient
C
Average
path
length
1
1
0
0
0
1.000
0.000
2
2
1
0
1
1.000
1.000
3
4
3
3
6
1.000
1.000
4
10
12
12
24
0.400
1.467
5
26
40
60
100
0.410
1.810
6
76
150
240
390
0.247
2.293
7
232
546
1050
1596
0.253
2.640
8
764
2128
8736
10864
0.181
3.149
9
2620
8352
19152
27504
0.182
3.550
10
9496
34380
83520
117900
0.142
3.977
The results show that on one hand, the clustering coefficient of the system decreases
while on the other, the average path length increases with the number of cysteines. Both
findings are consistent with the view that the folding space of peptides with many cysteines
may be too complex and thus the systems may be unable to fold fast enough.
The pathways can also be graphically represented, and in order to simplify the
resulting picture, we chose a 3D representation wherein the states having the same number
of disulfide bridges are placed on separate planes. In this representation, the shuffling
transitions are within the planes, and the redox edges connect adjacent planes.
It is noted that the experimental methods do not reveal all possible intermediates;
some of them may be too short-lived or not abundant enough so as to be noticed an
isolated. In spite of these limitations, the folding pathways appear as connected subgraphs
within the network of all possible intermediates, showing that the experimental techniques
actually identified states that can interconvert into one another. Only in EGF do we see an
“isolated” intermediate which suggest that some intermediates of the pathway were not
observed experimentally.
