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b
a
N
C
1
2
3
4
1-3, 2-4 or abab topology
Figure 3. Nomenclature for disulfide topologies. Disulfides can be labeled by the
sequence positions, or simply by the sequential number of the cysteine residues they
connect ( 1-3, 2-4 topology). Alternatively, it is customary to alphabetically label the
disulfide bridges, and describe the topology by assigning the bridge label to the
cysteines, starting from the N terminus ( abab topology).
The number of fully connected (disulfide bonded) isomers in a protein chain with n
disulfide bonds (2 n cysteines) can be deduced from simple combinatorial considerations as
(2 n )!/( n !*2 n ). According to this formula proteins with two disulfide bridges have 3 fully
oxidized isomers, 3-disulfide proteins have 15 and 4-disulfide proteins have 105. In other
words, the number of intermediates increases very fast as a function of the number of
constituent cysteines, and it has been hypothesized that the reason why the number of
cysteines in autonomously folding protein domains is not very large is because the too high
number of possible intermediates would slow down the folding process.
1
2
3
4
1
2
3
4
1
1
1
1
2
1
2
3
3
1
4
4
1-3, 2-4 1-2, 3-4
Figure 4 Adjacency matrices of two disulfide topologies of a peptide with two
disulfide bridges
For a complete description of the folding process we have to consider both fully
oxidized intermediates and the ones with free cysteine residues. For this purpose we will
use a formal description of the intermediates as (undirected) graphs, with cysteines as nodes
and disulfide bridges as edges (the main chain will not be represented). For the majority of
naturally occurring protein structures the resulting graphs will be extremely simple
especially if described as an adjacency matrix. Such an adjacency matrix is symmetrical,
and contains 1 if two cysteines form a disulfide bonds and zero otherwise. As one cysteine
can form only one disufide bridge, each column and each row of the resulting matrix will
have atmost one value of 1. The adjacency matrix of two disulfide topologies of a 2-
disulfide proteins are shown in Figure 4.
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