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Graph theory has been applied to many aspects of protein research (for a review see
[11]). Applications to protein folding followed two broad approaches. First, protein
structure itself can be considered as a graph consisting of various interactions (such as
covalent bonds, hydrogen bonds, spatial vicinities, contacts etc.) as edges, the nodes being
atoms or residues of the protein. It was found, among others, that the so-called contact
order, i.e. the average sequence distance between residues in atomic contact, seems to be a
key determinant of folding speed [12]. Another line of research concentrates on
characteristic networks of interatomic contacts that may form stabilization centres in
protein structures and can be the reason of the stability of various proteins [13,14]. It was
found that populated conformations seen in molecular dynamics simulations contain
characteristic networks of residues [15,16].
Another line of research was triggered by the finding that the robustness and
stability of networks may be the result of simple topological properties that are invariant
throughout various technical as well as biological systems including social organization,
electrical networks, road networks and the Internet [17]. In the following years the network
topology of a large number of systems have been described, and it was found that some
topology classes, like those characterized by a scale-free distribution of the degree (number
of links at each node), or the so called “small world models” that are characterized by
densely connected subnetworks loosely linked between each other, are indeed found in
various systems within and without biology (for a review see e.g. [18]). The various
network types were described in terms of a number of simple measures borrowed from
graph theory, such as the clustering coefficient, the diameter of the graph etc. This
approach was later extended to descriptions of the entire folding space, using the folding
states as nodes, and transitions as links between them. As the folding states of native
systems cannot be readily studied by physical methods, the investigations were first
directed to model systems. Scala and associates [19] described the folding states of short
peptides using Monte Carlo simulation on lattice models. They found that that the
geometric properties of this network are similar to those of small-world networks, i.e. the
diameter of the conformation space increases for large networks as the logarithm of the
number of conformations, while locally the network appears to have low dimensionality.
Shahnovitch and co-workers analysed the folding states of proteins during molecular
dynamics simulations. It was found that the folding space is reminiscent of scale-free
network, characterized by a majority of less populated states as well as some highly
populated states reminiscent of “hubs” seen in other systems [20].
Our purpose is to describe the folding space of the oxidative folding process using
graph theory. This is an intriguing task since the number of folding states defined in terms
of disulfide links is relatively small, as compared to “ordinary” folding. We will approach
the problem in two steps: i) using graph theory to describe the disulfide intermediates, and
to enumerate the states of the folding space. ii) using a graph-like representation of the
folding space to visualize the experimentally studied folding pathways.
1. Graph representation of oxidative folding intermediates
In proteins containing disulfide bonds, usually all cysteines form part of disulfide bridges,
and the disulfide topology can be unequivocally described by defining which cysteines are
connected. For example, a topology 1-3, 2-4 means that a protein with 4 cysteines has two
disulfide bridges that connect cysteines (1,3) and cysteines (2,4) respectively. Cysteines
can be labelled by their sequence position, or - as in the previous example - in a serial
order from the N-terminus (Figure 3).
