Consider a network of molecular species which can exist in a signal transmitting (“active”) and a signal blocking (“inactive”) form. Only in its active form is a species able to activate other molecules. In principle, the adopted framework allows for the description of a large class of signaling systems [15] and more complex processes such as multiple phosphorylations or the effects of scaffold proteins can be included. For simplicity, the analysis is restricted to simple interactions and single activation steps. Assuming mass action kinetics the non-autonomous ordinary differential equations for the concentration of active species, , read(1)where denotes the strength of an external signal, denotes the rate of activation of the -th molecular species by the signal, and the concentration of the inactive form is denoted . The rate constants specify the rate of activation of by . The zero-nonzero structure of the matrix reflects the circuitry of the elements in the signaling network. The constants characterize the rate of constitutive deactivation of the species , for example by dephosphorylation.

Assuming that production and degradation of each species in the network is always balanced, or a time scale is considered for which changes in total levels are negligible, the conservation relation can be used to uncouple the dynamics of inactive and active forms. Furthermore, because the focus is on the impact of signaling topology and the dynamics in the signal-off case, we set and drastically reduce the kinetic degrees of freedom by imposing for all . With this we can rescale time () and concentrations () resulting in the corresponding dimensionless and autonomous system(2)where .

Due to nonlinearity, multiple stationary states are possible in the system (2). The off-state is always a solution characteristic for signal absence. Any stationary state where the concentrations of some or all active species, , are non-zero is referred to as an autoactived state of (2). A bifurcation of the dynamical system occurs if some stationary state looses its stability. We are specifically interested in transcritical bifurcations where at some point in the parameter space stability switches from one state, which is defined as the ground state, to an autoactivated state. For the sake of simplicity the off-state is assumed to be the ground state of the signaling system.

In terms of graph theory, the quadratic system's matrix is the adjacency matrix of the associated directed graph (or digraph) . The non-zero entries define the edges, i. e. the positive regulatory pattern in the network, whereas the nodes symbolize the species. The digraph is weighted because the non-zero entries can deviate from unity.

The number of incoming edges of a node is its in-degree and accordingly the out-degree is defined as the number of outgoing edges. A source is a node with in-degree zero, a sink has out-degree zero. A series of unidirectional edges is generally called a directed walk or a path if and only if each node is visited only once.

The total number of nodes is denoted order of the graph and the number of edges specifies its size. If the graph is connected, which is a reasonable assumption for signaling networks, a useful measure is the connectivity, . For loop-less graphs (without edges originating and terminating at the same node), it varies in the range of and characterizes the average unspecificity of species in the network.

Any digraph formed by subsets of the original nodes and edges is a subdigraph. A spanning subdigraph has the same node set as the original digraph. A subdigraph in which there exists a path between any two nodes is called a strongly connected component (SCC). A cycle motif is thus a special type of SCC with in-degree and out-degree one for every node. As a closed path it constitutes a positive feedback in the signaling system. An -cycle is defined as a cycle of length , where is the number of nodes or edges in the cycle.

Spectral graph theory is concerned with the eigenvalues of a graph. The ordinary spectrum of a graph is the spectrum of the corresponding adjacency matrix. The adjacency matrices of isospectral graphs have the same principal eigenvalue which is real and in terms of graph theory is denoted the index, , of the graph [18]. Isospectral graphs with distinct topologies are said to belong to the same index class.

All calculations were carried out in Matlab (R2008b, The MathWorks Inc., MI, USA), using additional functions from the Boost Graph Library precompiled in MatlabBGL [19]. The paths in the networks were calculated via the random acyclic subdigraph method [20], [21].

The Jacobian of (2) evaluated at the off-state reads(3)where is the identity matrix. It can be shown that the off-state Jacobian has a similar form for Michaelis-Menten type activation kinetics. We can apply Schur's unitary triangularization theorem to (3) and, subsequently, the Perron-Frobenius theorem for non-negative matrices [22] in order to show that the spectral radius of exactly determines the eigenvalue with maximal real part. Let be the eigenvalues of the off-state Jacobian and the set comprising the corresponding real parts then(4)holds. Therefore, a necessary and sufficient condition for off-state stability in the signal-off case is that the index of the signaling network fulfills . This holds true irrespective of the exact model topology and the precise values of the kinetic parameters in (2). The bifurcation occurs at .

With result (4) it is possible to discuss which topological features of the network affect the relevant qualitative change in the dynamic behavior via the digraph index. As long as we consider weighted digraphs a straightforward answer is not possible, because there are still as many kinetic parameters as edges in the network which of course influence the dynamics. Definitely, the matrix entries (i.e. edge weights) impose lower and upper bounds on the index [18], [22],(5)where and are the th row and column sums, respectively, and and () denote the corresponding sets.

A critical result can be derived if the nodes are permuted such that the adjacency matrix is brought into the so-called irreducible normal form denoted [22]. Then the square matrices aligned in the diagonal with , are either irreducible or 1-by-1 zero matrices. By permutation similarity the eigenvalues of are equal to those of and hence for the spectrum the following holds true (see also [23]):(6)

In fact, the irreducible matrices are the adjacency matrices of the SCCs whereas the acyclic subdigraph of has only zeros in the diagonal of . Hence, for acyclic graphs in general, as is well-known in graph theory [18], [20]. According to (4) a zero index means that acyclic networks always have a stable off-state irrespective of the values of the activation constants. Thus we focus in the following on the influence of the SCCs.

We will call the set whose elements are the SCCs the feedback family of the network and denote their indices by . The union of spectra (6) shows that the index of the whole network is determined by the component with maximal index [23],(7)The SCC with maximal index is called the dominant component of the feedback family. Relation (7) is intriguing because it implicates that the positive feedback cycles outside of the dominant component are neutral with respect to off-state stability. If the -th SCC dominates the network index, condition (4) reduces to . If all or more SCCs have the same maximal index, these components are dominant representatives of an index class called the dominant class of the network. The index of any network equals the index of its dominant class. Note, that the number of representatives in a dominant class is irrelevant for stability. Unlike the assumption raised by means of numerical calculations [16], this analytical result shows that there is no straightforward relation between number of feedback cycles and off-state stability. Thus, dominance over off-state stability is not an intrinsic property of a single strongly connected motif but rather depends on the context given by the specific composition of the feedback family.

Because always holds true, condition (4) can provide a natural definition of the network's off-state robustness. The basic idea is that the extent of the parameter space, for which a key function of a system is maintained, can be seen as a possible measure for robustness of a biochemical network towards this function [4], [8].

In the following we confine the treatment to simple (i.e. unweighted) digraphs by setting and assume that there is no trans-autoactivation. The resulting binary adjacency matrix is denoted and its index has no values between zero and one [22]. The advantage of restricting the analysis to simple loop-less digraphs lies in the fact that in-degree and out-degree of each node directly reflect the number of upstream activators and downstream targets of a signaling compound, respectively. For an accurate kinetic analysis these restrictions are too strong. However, because our focus is on constraints imposed solely by topology the simplification is reasonable. In the more general case different edge weights could partially compensate each other to allow for stability.

Relation (4) now reads and the critical value for the bifurcation parameter results in . Thus the structural design of a signaling system encoded in the index distinguishes parameter regions with the desirable property of stability () and a region in which a dysfunction of the system by autoactivation cannot be avoided (). Apparently, the larger the stable regime the more reliably the signaling system performs its function in the sense that additional combinations of the parameters , and are buffered by the network structure. Therefore, the value provides a measure for what we call the kinetic off-state robustness(8)of a network design. This of course results in for acyclic networks. In the following we discuss networks with non-empty feedback family. It is appropriate to classify the SCCs as either regular or irregular digraphs because the structural characteristics decisive in kinetic off-state robustness differ in both cases.

Natural variation among individual cells in a population is a prerequisite of evolution. Thus, within our framework selection pressure towards off-state robustness in signaling systems can only act as long as there are topologies with distinct indices. However, as the spectrum does not fully determine topology there is considerable structural plasticity within index classes, for example acyclic networks may be in-trees or out-trees. To evaluate the significance of those structural properties which vary among networks with the same index, we further elaborate the concept of off-state robustness.

So far, we analyzed the implications of different topologies while assuming the topology of a particular system to be fixed. In the following, we consider the fact that living cells are steadily subjected to variation even on the structural level of signaling systems due to random genetic events, diseases or other exogenous factors. If to each network topology a dynamical system is attached, structural perturbations may affect dynamic properties like off-state robustness. We assume that during evolution signaling networks have been molded into topological designs which support kinetic off-state robustness despite structural defects.

For illustration and to put our hypothesis to the test five signaling systems have been evaluated: The Transpath [33] kinase network which was already analyzed in [17] for other properties and some recent canonical signaling networks [34]–[37] retrieved from the Science STKE database [38]. The positive interaction maps of the STKE networks were derived as follows. First, the negative interactions and those whose sign is currently unknown were deleted. Any node becoming isolated by this procedure was ignored. Second, because we are considering only the signal-off status, all nodes defined as an external signal are discarded. Because the corresponding edges have to be deleted as well, again some nodes may become isolated. The topological properties shown to be important for our study are summarized in Table 1.

We want to demonstrate that there exist real signaling systems that may indeed exhibit off-state robustness and that the corresponding topological properties are not the product of chance. For this, structural features are compared with random counterparts to assess their statistical significance. Random networks can be created by different means. Here a rewiring method is used which is described in [39]. Starting with the real network, repeatedly, two edges are chosen at random, their target nodes or their nodes of origin are exchanged, provided the resulting edges do not already exist. This procedure ensures that in the rewired network order, size and the in-degrees as well as out-degrees for each node are preserved. As a consequence, of the three properties considered above, only the path sum differs between the random networks.

The results for the five biological networks are depicted in Figure 3. The graphs in the left column depict the distributions of the indices of the perturbed networks, and the graphs in the right column represent the distributions of their structural robustness values. All five networks are acyclic and the index distributions clearly indicate that the maximal kinetic off-state robustnesses can not be expected by chance. For the Transpath network for example, only 35 out of 100,000 random networks are acyclic. Therefore, an acyclic network design is most likely a striking feature. Further, the structural robustness for three of the five networks is the highest observed for all perturbations. Only for the B Cell Activation network and for the IFNγ pathway, perturbed network structures with a slightly higher structural robustness were found. We hypothesize that these findings also reflect evolution towards structural off-state robustness. Table 2 gives an overview of the p-values of all signaling networks analyzed. All analyzed networks have a significantly low index and high structural off-state robustness. The path sums are all lower in the original networks than in the random samples, however, compared to the values of random networks being in the same index class only the TGF and (weakly) the IFN pathways seem to have significantly low p-values. To assess the effect of connectivity and divergence the original networks would have to be tested against random networks generated by different means, for example by generating Erdös-Renyi graphs of the same order.