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RJP, PAI, and AL conceived and designed the experiments, performed the experiments, and analyzed the data. RJP, PAI and AL contributed reagents/materials/analysis tools. RJP, PAI, and AL wrote the paper.

The authors have declared that no competing interests exist.

Biological networks, such as those describing gene regulation, signal transduction, and neural synapses, are representations of large-scale dynamic systems. Discovery of organizing principles of biological networks can be enhanced by embracing the notion that there is a deep interplay between network structure and system dynamics. Recently, many structural characteristics of these non-random networks have been identified, but dynamical implications of the features have not been explored comprehensively. We demonstrate by exhaustive computational analysis that a dynamical property—stability or robustness to small perturbations—is highly correlated with the relative abundance of small subnetworks (network motifs) in several previously determined biological networks. We propose that robust dynamical stability is an influential property that can determine the non-random structure of biological networks.

The authors model how network motifs respond to small-scale perturbations and find a strong correlation between motif stability and abundance in a network, suggesting that dynamic properties of network motifs may play a role in overall network structure.

Life is a dynamical process. As the intricate connectedness of biological systems is revealed, it is essential to keep in mind that network maps are graphical representations of dynamic systems. A network with fixed structure is dynamic in the sense that the nodes take on values corresponding to

Example biological networks, such as those analyzed here, are artificially separated from the highly integrated and complex whole, which includes interconnected metabolic, signal transduction, transcriptional, cytoskeletal, and other types of interlocked networks and pathways. Even with this limitation, consideration of the structure of isolated networks has profoundly influenced contemporary biology, which is increasingly focused on systems-level concepts. The topological structures of the networks studied here: the transcriptional regulatory networks of

Presently, it is not clear what determines the particular frequencies of all possible network motifs in a specific network. At least two alternative explanations can be offered. One can hypothesize that certain constraints on development of a network as a whole determine which motifs become abundant. Conversely, some network motifs may possess properties important enough to become overrepresented and thereby drive the network evolution and its ultimate structure. The latter explanation is consistent with the frequently made assumption that the function of a small biological network or pathway is largely determined by the connections among the constituting genes, proteins, metabolites, or cells [

A comprehensive analysis of the dynamics of networks, large or small, is considerably more complicated than the corresponding analysis of their structure. For instance, structures of all possible three- or four-node network motifs can be generated and enumerated easily. However, their dynamics may not be completely determined owing to unknown and potentially complex functional dependencies between nodes, the lack of knowledge of parameters defining specific instances of motifs in real networks, and “unmodeled” interactions that may be absent from the network representation yet relevant to dynamics. In addition, motifs with the same topology may give rise to different dynamic behaviors, as was recently demonstrated experimentally using small synthetic genetic circuits [

Although the particular parametric values can confer unique properties onto individual instances of motifs, we argue that all instances of a particular motif display characteristics that can be studied comprehensively. Here we present the analysis of a specific robust property that can be displayed by network motifs: robust stability to small-scale perturbations of the activities of the biological entities. Small perturbations include intrinsic stochastic fluctuations (noise) and transient up- or down-regulation of activity. “Small” implies that a linear approximation of the potentially nonlinear relations is still valid (see

Intuitively, one expects that for complex biological networks to function robustly, it is necessary that they display stability and resist both noise and stressful small-scale perturbations. These basic homeostatic properties, however, are not inherent to any biological or biochemical system, as small stimuli can, in principle, trigger large-scale sustained responses, especially if feedback interactions between members of a biological network are involved. For example, relatively short stimulations by tumor necrosis factor α (5 min or less) can trigger sustained (60 min) activation of the NF-κB pathway leading to expression of a battery of genes [

The dynamic behavior of a circuit is determined by the direction, sign, and strength of the connections. To develop some intuition, consider the case of a two-node feedback loop (_{ij}_{ii}

(Top) Schematic of a simple feedback circuit with the nodes and edges labeled.

(Middle) We illustrate the dependence of stability on the values of _{12} and _{21} as they are varied from −1 to 1, with constant self-degradation terms _{11} = −0.3 and _{22} = −0.7. The system can be stable (green), oscillatory (blue), or unstable (red). These regions are determined by gain k = _{12}_{21}. Shown in white are contours of constant gain k. The quadrants labeled (+) correspond to positive feedback (_{12} and _{11} have the same sign), while the quadrants labeled (−) correspond to negative feedback (_{11} and _{21} have opposite signs).

(Bottom) The stability regions vary as the values of self-degradation terms _{11} and _{12} change. The more stable the open-loop nodes (more negative _{11} and _{22}), the greater the regions of closed-loop stability. However, if _{11} and _{22} are close in sign and magnitude, the size of the oscillatory regions increases.

We analyze the response of this circuit to a small perturbation from a steady-state under different assumptions on the parameters (_{11}_{22}_{12}_{21}_{11}_{22}), the greater the regions of closed-loop stability. Consequently, positive feedback produces stability if constitutive degradation dominates the feedback gain. Both positive and negative feedback can produce oscillations in general (only negative feedback can produce oscillations in the two-node case). Finally, upstream input nodes and downstream output nodes can be added to the system without altering the stability, provided that the number or size of feedback loops is unchanged.

Similar analysis can be extended to all possible three- or four-node network motifs by defining a metric, the structural stability score (SSS), as the probability that the dynamical system corresponding to a given motif relaxes monotonically to steady-state following a small perturbation. An SSS value of 1 indicates that non-oscillatory relaxation to a steady-state (henceforth simply termed stability) is guaranteed by connectivity and does not depend on the specific parameter values that define the functional interactions, thus making the system

Determination of the SSS values (fully described in

Next, we contrasted the abundance of motifs in several real biological networks with the corresponding motif SSS values (

The number of instances of all three-node (left) and four-node (right) motifs in biological networks (log scale), and SSS (black dashed line). The motifs are sorted on the

Above, we chose a conservative definition of stability, classifying any system with oscillatory behavior as unstable, even if the oscillations are damped. However, our results still hold if we adopt the more traditional, and less conservative, definition of stability in which damped oscillations are classified as stable dynamics, irrespective of the potential presence of oscillation (see

We also noted that, for the same SSS values, the motif abundance was correlated with the number of edges in the motif, as can be seen from the particular ordering of the three-node motifs used (

Is the non-random character of network organization driven, at least to some extent, by the structural stability of network motifs? If this is the case, one would expect that motifs of relatively higher stability would be overrepresented compared to their relatively less stable counterparts when compared to random networks of the same size. To test this hypothesis, we generated 100 Erdös-Renyi (ER)–type random graphs with the same number of nodes and edges as each of the real networks. Lacking any organizing principle, the distribution of motifs in this type of random graph is determined by the density of edges [

(Top panel, left) All 13 network motifs are sorted on the

(Bottom five panels, left) Normalized Z scores (green bars) for all 13 motifs of the transcriptional networks of

(Right) Z scores of network motifs in the three-edge and four-edge density groups are plotted in columns labeled I, II, and III, corresponding to the three stability classes, SSS = 1, SSS ≈ 0.4, and SSS < 0.2, respectively. Note that the four-edge density group does not contain any motifs with SSS = 1 since four edges among three nodes dictates at least one feedback loop.

(Top panel) All 199 motifs are sorted on the

(Bottom five panels) Normalized Z scores (green bars) for all 199 network motifs of the indicated biological networks, shown with outlines of the SSS from the top panel (dotted black outline provided as a guide to the eye). Each vertical red line indicates a change in the number of motif edges, indicating boundaries of density groups. The composition of the four-node density groups is specified in

The top panels of

Four-node network motifs are combinatoric elaborations of three-node motifs [

On average, the most structurally stable motifs have higher Z scores than those with lower SSS within each density group (

Density groups consisting of four, five, and six-edge network motifs contain at least one motif in each stability class. Box and whisker plots indicate a difference in the average Z score between stability classes (I, II, and III correspond to SSS = 1, SSS ≈ 0.4, and SSS < 0.2, respectively). The networks examined here are the transcriptional networks of (A)

In addition to statistically significant differences in average Z scores among stability classes, there is similarity in stability properties among highly overrepresented network motifs.

In the preceding investigation, we extended a structural analysis technique (decomposition of a network into subgraphs) to a dynamical analysis. Our characterization of motif dynamics implicitly assumed that subsystems consisting of a few nodes could behave relatively autonomously despite being embedded in a larger network. Conceptually, the assumption is valid when the activity of a node does not propagate a long distance through the network, creating the situation in which small subgraphs are functionally relevant. This situation can arise when (i) only a small fraction of the nodes are activated, or (ii) only a small fraction of the potential regulatory interactions are activated. Under these conditions, the “active network” is a fragmented version of the full potential network. Note that the preceding dynamical analysis was appropriate for only a small perturbation of the activities of the nodes. Now we utilize the concept of a small perturbation to demonstrate that the active network can be a fragmented version of the full potential network, with true functional regulatory units consisting of three or four nodes.

Regulation of gene expression is dependent on the particular demands of a cell with respect to its environment. For example, many transcription factors are active only in the presence of additional signaling molecules. cAMP receptor protein, the most connected node in the bacterial transcription regulation network, is active only when bound to co-regulator cAMP, which, in turn, is present under only special circumstances, such as absence of glucose in the medium. In the absence of a cAMP signal, cAMP receptor protein is inactive, and its links are inactive. We define a context-dependent network to be the subset of the nodes and links that are activated in a certain context, such as glucose deprivation of a bacterial culture. With this principle in mind, we used high-throughput gene expression datasets to infer the context-dependent gene regulation networks, with specific contexts defined by mild environmental stress perturbations. These contexts are relevant to the preceding dynamical analysis.

We considered five different context-dependent gene regulation networks based on the genomic expression pattern of the yeast

Do common “driving forces” underlie the organization of biological networks? It seems fantastic to suggest that such forces could exist, considering that the biological entities involved are as diverse as genes, enzymes, and whole cells. Nevertheless, even functionally unrelated systems may have evolved under fundamental constraints. The analysis presented here suggests that the dynamic properties of small network motifs contribute to the structural organization of biological networks. In particular, robustness of small regulatory motifs to small perturbations is highly correlated with the non-random organization of these networks. Inspection of highly overrepresented motifs in the four-node significance profile (see

There are two separate trends in the motif significance profiles (see

An evolutionary argument may help explain the overrepresentation of structurally stable motifs in real networks compared to random graphs. Evolutionary pressure may select for network innovations that are structurally stable because these configurations are robust to variations in the strength of the connections. A high SSS indicates that it is likely that randomly assigned connection strengths and signs will result in a stable equilibrium, while a low SSS indicates that stability is possible although it requires precisely weighted connection strengths. Easily parameterized network designs that are predisposed to dynamical stability can be advantageous considering the evolutionary mechanisms of random mutation and natural selection. Of course, stability to small perturbations is by no means the only functional constraint on network performance and structure. For instance, in the developmental transcriptional regulation network in

The relationship between structural stability and overrepresentation of motifs can be illustrated using the example of three-node loops. The three-node feed-forward loop (motif 7) is an acyclic topology. Like all feed-forward architectures, it has an SSS of 1, implying that its relative abundance in a network might be high, as is indeed the case. As suggested above, structural stability is usually necessary but not sufficient for the emergence and preservation of network motifs. In the case of the feed-forward loop, previous modeling has demonstrated that this motif can produce a wealth of computational functions [

An important caveat of our work is that we have assumed that the network operates in isolation of other components of the overall system that may influence stability. Clearly, these example networks are a significant simplification of a considerably more complicated reality of biological regulation. For example, in assigning SSS values in transcriptional regulation networks, protein–protein interactions and other modifications of transcription factors and their targets are ignored. In this case, it is likely that these unmodeled components represent dynamics with faster time scales, and that the transcriptional network represents the slower, dominant dynamic behavior. Moreover, it is reasonable to conjecture that a network with higher SSS would be more tolerant of perturbations in the faster scale components than one with a lower SSS value. However, in some cases, these unmodeled components may exert considerable influence on the system. For example, in a recently published model of the heat-shock response in

Our characterization of motif dynamics implicitly assumed that subsystems consisting of a few nodes could behave relatively autonomously despite being embedded in a larger network. We demonstrated how the yeast gene expression network is only partially activated by mild environmental perturbations (heat shock, osmotic shock, etc.), concluding that nodes and links in large-scale networks can be interpreted as potentially active, rather than “always on.” A context-dependent network, composed of the subset of the nodes and links that are activated at a given moment, can be a fragmented version of the full potential network, where network motifs are on the scale of three or four nodes. We demonstrated context-dependence in yeast gene regulation, but we expect that, in general, it is appropriate to assume that all elements in a network are not “always on.” The more fragmented a network in various functional contexts, the more relevant our analysis of dynamic properties of network motifs.

In summary, our results suggest that robust stability of network motifs is an important determinant of biological network structure. This conclusion favors the intuitively appealing claim that biological networks need to be resistant to small-scale perturbation, including noise, and that this resistance may be structurally embedded in the network organization. While our analysis shows a strong correlation between network motif abundance and structural stability, it leaves us to speculate as to why motifs with various numbers of edges are overrepresented in different networks. Since the bacteria and yeast networks regulate gene expression in relatively simple organisms, many features provided by the presence of less stable, feedback-containing motifs may not be necessary or could potentially be detrimental. In contrast, transcriptional networks of higher organisms or non-transcriptional regulatory networks may benefit from the increased occurrence of feedback-containing motifs and more complex functions potentially provided by them. Stability to small perturbations can be important for robust network performance. Thus, it is reasonable to expect that motif distribution among diverse networks represents a balance of abundant, stable motifs and relatively rare, potentially unstable motifs, allowing greater functional flexibility coupled with predominant dynamic stability.

Given complete knowledge of the functional dependencies of the nodes, a dynamical system corresponding to a particular motif consisting of

(I) An example network motif (10) depicted with nodes and edges labeled. The sign and weight of an edge from node _{ij}_{ii}

(II) A motif is a graphical depiction of a dynamical system that can be modeled using (possibly nonlinear) ordinary differential equations, which are functions of the state of the nodes in the motif.

(III) Assuming that the motif can achieve steady-state dynamics, the stability of the equilibrium may be analyzed by making a linear approximation of the (possibly nonlinear) functions. Linearization involves computation of the Jacobian, a matrix of partial derivatives of the functional dependencies expressed in the equations with respect to the variables. The linearized system is evaluated at the equilibrium point(s), resulting in specific values of _{ij}

(IV) The general form of the Jacobian matrix corresponding to this motif contains zero entries wherever one node exerts no influence on another node and non-zero elements wherever it does. An instance of a Jacobian consistent with the corresponding motif is created by sampling the non-zero terms from some distribution. Eigenvalues of the Jacobian indicate the local stability properties of the equilibrium point. Sampling many instances of structurally equivalent Jacobians allows computation of the

where the variable _{i}_{i}_{i}_{i},

The linearized system in a small neighborhood

Although the precise values of the elements of the Jacobian matrix might not be known, it is clear that the matrix has zero-valued elements whenever one node exerts no influence on another node, and non-zero elements whenever it does. The sign of these elements depends on whether the influence is activating (positive) or inhibitory (negative). We note that the Jacobian can be reduced to the corresponding adjacency matrix by normalizing the entries to ones or zeros. Thus, the adjacency matrix is a particular example of a Jacobian consistent with the potentially nonlinear differential equations. Thus, in our analysis, the general form of the Jacobian _{ij}

We analyzed the stability properties of the 13 three-node and 199 four-node motifs using root locus analysis (see

Rather than calculate an exact solution for all possible three- and four-node motifs, we employed a numerical method to estimate stability under a particular probability distribution for the _{ij}_{ij}

We used the Mfinder1.1 software program distributed by U. Alon's group (

Motif abundances in the real networks were compared to those in ER- type random graphs with the same number of nodes and edges as the corresponding real network. This is a different null model than previously described [

where Nreal_{i} is the abundance of the _{i}> and std(Nrand_{i}) are the mean and standard deviation of its abundance in the corresponding 100 random graphs with the same number of nodes and edges as the real network. We normalized the vector of Z scores to unit length:

The

The SSS is represented as a stacked bar graph (top panel) in which the black portion of the bars represents the component of SSS due to monotonic decay, and the grey portion of the bars represents the component of SSS due to damped oscillations. The ordering of three-node motifs by the more expansive definition (traditional in the field of control systems engineering) produces the identical ordering of three-node motifs as conservative definitions of SSS that exclude oscillations (compare to

(19 KB PDF).

The SSS is represented as a stacked bar graph (top panel) in which the black portion of the bars represents the component of SSS due to monotonic decay, and the grey portion of the bars represents the component of SSS due to damped oscillations. The ordering of four-node motifs by the more expansive definition (traditional in the field of control systems engineering) produces a substantially similar ordering of four-node motifs as the conservative definition of SSS that excludes oscillations (compare to

(168 KB PDF).

Overrepresented motifs (positive Z scores) tend to have higher SSS scores than other motifs with the same number of edges (compare to

(19 KB PDF).

Overrepresented motifs (positive Z scores) tend to have higher SSS scores than other motifs with the same number of edges (compare to

(165 KB PDF).

(A) The full potential network affecting the expression of YHR087W (bottom node) was topologically sorted to reveal the causal structure. Green nodes indicate that the gene was regulated (> 2-fold change in expression) in at least one of the five environmental perturbation experiments examined here (heat shock, H_{2}O_{2}, dithiothrietol, diamide, hyper-osmotic shock). Green links are regulatory interactions that were inferred from the regulated genes, irrespective of whether or not the mRNA expression of the upstream transcription factor was observed. (All inbound links to a regulated node are inferred to be potentially active.)

(B) The context-dependent regulatory network in response to a mild heat shock was obtained by traversing the green links backwards from YHR087W, stopping when no more green links were encountered. The regulatory unit is larger than the average motif size of three nodes, yet dynamical stability is a robust property of the structure.

(C) The context-dependent regulatory networks for hyper-osmotic shock and diamide perturbations are a subgraph of the heat-shock regulatory structure.

(B and C) The context-dependent motifs have SSS = 1, indicating that dynamical stability is a robust property of the structure of the regulatory unit. Images generated with Cytoscape1.1 software [

(56 KB PDF).

ID labels match the output from mfinder1.1 motif-finding software provided by U. Alon.

(215 KB PDF).

Stability analysis using root locus.

(45 KB PDF).

We thank Uri Alon for providing datasets and the mfinder software. We are grateful to Elias Issa, Jef Boeke, members of the Levchenko lab, and three anonymous reviewers for discussion and critical review. This work was supported by the National Institutes of Health (1-U54-RR020839 to AL, and 71920 to PAI) and the National Science Foundation (083500 to PAI).

Erdös-Renyi

structural stability score

signal transduction knowledge environment