Chapter 9 Graph Theory Properties of Cellular Networks Baruch Barzel, 1, 2 Amitabh Sharma 1, 2 and Albert-La ´ szlo ´ Baraba ´si 1, 2 1 Center for Complex Network Research, Department of Physics, Northeastern University, 360 Huntington avenue, Boston, Massachusetts 02115, USA, 2 Center for Cancer System Biology (CCSB) and Department of Cancer Biology, the Dana-Farber Cancer Institute and Department of Genetics, Harvard Medical School, 44 Binney street, Boston, Massachusetts, USA Chapter Outline Introduction 177 Biological Systems As Graphs 178 The Tools of Graph Theory 178 ErdTseRe ´nyi e The Benchmark Network 178 Degrees and Degree Distribution 178 Network Paths and the Small World Phenomena 179 Clustering Coefficient 180 Successes and Failures of the ErdTseRe ´nyi Model 180 Biological Small Worlds 180 Deviations from the ErdTseRe ´nyi Model 180 Scale-Free Nature of Cellular Networks 181 The Scale-Free Property 181 Network Integrity and the Role of Hubs 182 The Origins of the Scale-Free Topology 183 Preferential Attachment in Biological Networks 183 Hierarchy and Modularity 184 Party vs. Date Hubs 184 Degree Correlations 184 Human Disease Network 185 The Building Blocks of Cellular Networks 186 Sub-graphs and Motifs 186 Randomized Networks 186 Autoregulation and the Feedforward Loop 187 Going Beyond Topology 187 Assigning the Weights 188 Characterizing the Weighted Topology 188 Topology Correlated Weights 188 Controllability 188 Differential Networks 189 From Structure to Dynamics 190 References 191 INTRODUCTION From a conceptual point of view, the rise of systems biology can be described as the adoption of a broad-based perspective on biological systems. In that sense, the classical detailed biological analysis is com- plemented by a macroscopic description of the cell as a holistic unit [1e4]. This approach, aiming at a system- level understanding of biology, mandates a crude simplifi- cation of biological processes. In this light, the graph theoretic approach to biological systems focuses on the structural aspects of the interaction patterns, where the interacting species, be them genes, proteins or other bio- logical components, are signified by nodes, and their interactions by the edges drawn between them. These network systems express the underlying architecture which enables the cellular functions to be carried out [5e9]. Undoubtedly, the functionality of the cell cannot be attributed to just one network, but rather to a set of inter- dependent networks ranging from the level of transcription to the processes of metabolism. It is common to divide the cellular functions into three distinct networks, the tran- scriptional network, the protein interaction network and the metabolic network [2]. Although we follow this division throughout this chapter, it should be acknowledged that the true functionality of the cellular unit is a result of the interdependence between these networks, and not merely the interactions in each of them alone. At the current state, the topology of these three fundamental cellular networks has been thoroughly mapped using high-throughput tech- niques. As a result, we now have reliable data on the interaction maps of many organisms. Some examples are proteineprotein interaction networks, which have been Handbook of Systems Biology Concepts and Insights. http://dx.doi.org/10.1016/B978-0-12-385944-0.00009-5 177 Copyright Ó 2013 Elsevier Inc. All rights reserved.
Transcript
Chapter 9
Graph Theory Properties of CellularNetworks
Baruch Barzel,1,2 Amitabh Sharma1,2 and Albert-Laszlo Barabasi1,2
1Center for Complex Network Research, Department of Physics, Northeastern University, 360 Huntington avenue, Boston, Massachusetts 02115, USA,2Center for Cancer System Biology (CCSB) and Department of Cancer Biology, the Dana-Farber Cancer Institute and Department of Genetics, Harvard
Medical School, 44 Binney street, Boston, Massachusetts, USA
Ha
Co
Chapter Outline
Introduction 177
Biological Systems As Graphs 178
The Tools of Graph Theory 178
ndbo
pyrig
ErdTseRenyi e The Benchmark Network 1
78
Degrees and Degree Distribution 1
78
Network Paths and the Small World Phenomena 1
79
Clustering Coefficient 1
80
Successes and Failures of the ErdTseRenyi Model 180
Biological Small Worlds 1
80
Deviations from the ErdTseRenyi Model 1
80
Scale-Free Nature of Cellular Networks 181
The Scale-Free Property 1
81
Network Integrity and the Role of Hubs 1
82
The Origins of the Scale-Free Topology 1
83
Preferential Attachment in Biological Networks 1
83
Hierarchy and Modularity 184
ok of Systems Biology Concepts and Insights. http://dx.doi.org/10.1016/B978-0-12-38594
ht � 2013 Elsevier Inc. All rights reserved.
Party vs. Date Hubs 184
Degree Correlations 184
Human Disease Network 185
The Building Blocks of Cellular Networks 186
4-0.0
Sub-graphs and Motifs 1
86
Randomized Networks 1
86
Autoregulation and the Feedforward Loop 1
87
Going Beyond Topology 187
Assigning the Weights 1
88
Characterizing the Weighted Topology 1
88
Topology Correlated Weights 1
88
Controllability 1
88
Differential Networks 1
89
From Structure to Dynamics 190
References 191
INTRODUCTION
From a conceptual point of view,the rise of systems biology can be described as the adoptionof a broad-based perspective on biological systems. In thatsense, the classical detailed biological analysis is com-plemented by a macroscopic description of the cell asa holistic unit [1e4]. This approach, aiming at a system-level understanding of biology, mandates a crude simplifi-cation of biological processes. In this light, the graphtheoretic approach to biological systems focuses on thestructural aspects of the interaction patterns, where theinteracting species, be them genes, proteins or other bio-logical components, are signified by nodes, and theirinteractions by the edges drawn between them. Thesenetwork systems express the underlying architecture whichenables the cellular functions to be carried out [5e9].
Undoubtedly, the functionality of the cell cannot beattributed to just one network, but rather to a set of inter-dependent networks ranging from the level of transcriptionto the processes of metabolism. It is common to divide thecellular functions into three distinct networks, the tran-scriptional network, the protein interaction network and themetabolic network [2]. Although we follow this divisionthroughout this chapter, it should be acknowledged that thetrue functionality of the cellular unit is a result of theinterdependence between these networks, and not merelythe interactions in each of them alone. At the current state,the topology of these three fundamental cellular networkshas been thoroughly mapped using high-throughput tech-niques. As a result, we now have reliable data on theinteraction maps of many organisms. Some examples areproteineprotein interaction networks, which have been
178 SECTION | II Network Properties of Biological Systems
constructed for organisms such as Homo sapiens, Saccha-romyces cerevisiae (S. cerevisiae) Helicobacter pylori andothers [10e20]. Regulatory and metabolic networks havealso been successfully mapped for yeast, Escherichia coliand various other organisms [21e23]. However, we stilllack data regarding other networks taking part in thecellular processes, such as sRNA and RNAi-mediatednetworks, about which we currently know little.
BIOLOGICAL SYSTEMS AS GRAPHS
Although the foundations of graph theory were laid purelyout of mathematical curiosity, its applicability as a tool forthe characterization of complex systems has long beenappreciated. The notion is that the behavior of complexsystems arises from the coordinated actions of manyinteracting components. The network abstraction can thenbe used to reveal the underlying structure of these inter-actions. The interacting components are signified bya series of nodes, and the interactions between selectedpairs of these nodes are represented by the links (or edges)drawn between them. This abstract description eliminatessome of the details associated with the specific nature of thesystem at hand. However, it allows one to utilize the well-established formalisms of graph theory, thus providinga powerful tool for the analysis and understanding of thesecomplex systems. Moreover, this categorical representationapplied to various systems provides the grounds forcomparison between seemingly distinct networks. Thisprocess has proven highly beneficial, as one of the mostimportant discoveries of recent years was that despite thediversity of cellular networks, several important universalproperties are shared by them all [5].
In some cases the network description of a cellularsystem is straightforward and natural. Consider, forinstance, the set of physical binding reactions between pairsof proteins or between proteins and other molecules, suchas nucleic acids or metabolites. Here it seems natural to usea node for the representation of each molecular type, and anedge to denote each potential binding reaction. However, inother cases the network description is not unique, and maydiffer according to the motivation of the study. A simpleexample regards the transcriptional regulatory network. Inthis network the edges link between transcription factorsand the genes that they regulate. Here the information flowsfrom the regulating gene to the regulated one, so that thelinks are not symmetrical. The network is thus a directednetwork. Moreover, the relationship between a pair ofinteracting genes can be of an activating or of an inhibitorynature. Thus two different types of directed edges exist,which can be denoted by positive versus negative, orgraphically by /versus dj. Nevertheless, in manycontexts the directed nature of the interactions, or theirsign, is not important, and it is sufficient to model the
system using a regular undirected network. As a morecomplicated system we refer to metabolic networks. Thesesystems can be conceptualized as networks on many levelsof abstraction. For instance, one can visualize the molecularsubstrates as nodes, and the reactions transformingsubstrates to products as links. In this case the links areattributed by the enzyme catalyzing the reaction, and thegraph is directed. However, there are contexts in which it issufficient to use a simpler description, where the enzymesare ignored and the links are undirected. In this case thegraph simply describes the interconnections betweenmetabolites, leaving out the detailed chemistry thatunderlies these connections.
THE TOOLS OF GRAPH THEORY
In the basis of graph theory lies the insight that a complexsystem could be reduced into a series of abstract componentstied together by a set of connections. The spark of this idea iscommonly attributed to the 18th-century mathematicianLeonhard Euler, who in 1735 used it to solve the problem ofthe Seven Bridges of Konigsberg, a problem which backthen confused the residents of the Prussian town. To showthat one cannot visit all of the city’s islands without crossingat least one of the city’s seven bridges twice, Euler con-structed an abstract map of the city in which the islands wererepresented by nodes and the bridges by edges. In doing so,Euler mapped a realistic problem, in all of its complexities,into a clean abstract mathematical representation, whichallowed him to focus strictly on the structural crux of theproblem. However, this spark remained dim and only re-emerged as an elaborate, formalized mathematical theorysome 200 years later, in the 20th century, following the workof Paul Erdos and Alfred Renyi.
ErdoseRenyi e The Benchmark Network
The most elementary network considered in graph theory, isthe ErdoseRenyi random network, where each pair of nodesis connected with equal probability [24e26]. The propertiesof this prototypic network serve as a benchmark to which welater compare the more realistic networks of cellular biology.To construct anErdoseRenyi network,we consider a set ofN
nodes. For each of theNðN � 1Þ
2pairs of nodes in the network
we assign an edgewith probability p, typically chosen so thatp � 1. Simple as it may be, the ErdoseRenyi networkfeatures some surprising characteristics, commonly observedin many real-world biological networks.
Degrees and Degree Distribution
To analyze the components of the network we introducesome elementary network measures. For concreteness, we
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FIGURE 9.1 (A) An undirected network which includes 10 nodes. The
network path between nodes 1 and 8 is emphasized (red). (B) In the
directed version of this network, the path must advance in accordance with
the direction of the edges.
179Chapter | 9 Graph Theory Properties of Cellular Networks
use the ErdoseRenyi network to exemplify them. Themost basic characteristic of a node is its degree, k, definedas the number of links it has to other nodes in the network.In the ErdoseRenyi graph every node can potentially belinked to any of its N � 1 counterparts with independentprobability p. The average degree will thus behki ¼ pðN � 1ÞzpN. The random nature of this networkinvokes some variability in the degree of the nodes, so thatseveral nodes will have more links than the average, whileothers will have less. This variability can be described bythe degree distribution of the graph. Denoted by PðkÞ, it isdefined as the probability that a randomly selected nodewill have exactly k links. As we will see later, the degreedistribution is one of the most fundamental characteristicsof a network, carrying crucial information about itsevolution and formation process. In the ErdoseRenyinetwork PðkÞ follows a Poisson distribution, which indi-cates that most nodes are characterized by roughly thesame degree, the probability to encounter a node witha degree significantly different than hki being vanishinglysmall. The average degree is thus the characteristic scaleof the degree distribution.
In directed networks, we distinguish between thein-degree of a node, denoting the number of incominglinks, and the out-degree, denoting the number of outgoinglinks. For instance, if gene x regulates n other genes, it willhave an out-degree of kout ¼ n, whereas if it is beingregulated by m other genes, its in-degree will be kin ¼ m.Accordingly, the degree distribution in such networks issplit into the incoming distribution PðkinÞ and the outgoingdistribution PðkoutÞ. As an example, consider node numberfour in Figure 9.1, which in the undirected version of thegraph (a) has a total degree of k ¼ 5, while in the directedversion (b) it is characterized by kin ¼ 2 and kout ¼ 3.
Network Paths and the Small WorldPhenomena
A crucial feature of any biological network is its ability tomaintain a flow of information, mass or energy, betweenall of its nodes. From the graph theoretical perspectivethis requirement translates into the existence of a networkpath connecting all (or most) of the nodes in the network.By network path we refer to a route leading from onenode to another by passing solely over existing links(Figure 9.1). Such a group of interconnected nodesconstitutes a connected component, and if indeed a largefraction of the nodes in the graph can be reached from oneanother via these network paths, the graph is said to havea giant connected component. This seemingly remotefeature appears rather frequently and does not requiremuch high-level organization for it to be observed. Infact, for an ErdoseRenyi graph, a giant component will
emerge as long as hki � 1. Moreover, in the case where1 � hki � lnN, this giant component is likely toencompass almost all of the nodes in the network [6,27].
Networks impose a unique metric by which the distancebetween nodes can be measured. Consider the network pathbetween a pair of nodes. Its length is defined as the numberof links it crosses. The length of the shortest path betweena pair of nodes, x and y, is the network distance, lxy.Figure 9.1(a) displays the shortest path between nodes 1and 8 . The network distance between these two nodes isl1;8 ¼ 4. If no such path exists, then the pair of nodescannot communicate with one another and their distance isset to be infinite. The distance between a pair of nodesparameterizes the potential to propagate information fromone to the other, such that close nodes are more likely toaffect one another than far-away nodes. Averaging over allpairs of nodes in the network gives the average path lengthof the network, hli, which offers a measure of the network’soverall connectivity. If the network includes infinite paths,i.e., has isolated components, the averaging is commonlycarried out over the nodes belonging to the giant connectedcomponent. For an ErdoseRenyi network (as well asalmost all other random networks) we find that the averagepath length is strikingly small compared to the networksize. To understand this we focus on the surrounding of
180 SECTION | II Network Properties of Biological Systems
a typical node in the network. It has hki nearest neighbors,all at a distance of l ¼ 1 from it; each of these neighborshas on average hki neighbors of its own, so that there areroughly hki2 nodes at a distance of l ¼ 2. Following thislogic, we find that the number of nodes within a givendistance from any typical node inflates exponentially,leading to a logarithmic dependence of the average pathlength on the network size, N [27]
�l�z
lnN
lnhki: (1)
This simplified argument overlooks the possibilitythat some of the edges might be redundant, i.e., that someof the links emerging from nodes at a distance l mightlink to other nodes which are at the same distance. Still,for large N it captures the behavior of the network at thevicinity of a node, where the probability of linking to analready used node, and thereby creating a loop, is verysmall.
In a directed network, network paths are commonlydefined to propagate only in the direction of the edges. Thisway the paths reflect the flow of information between thenodes. The network distance between a pair of nodes is nolonger symmetrical, as displayed in Figure 9.1(b). Thedistance from node 1 to node 8 in the directed network isl1;8 ¼ 6 (as opposed to l1;8 ¼ 4 in the undirected version).However, no path exists in the opposite direction, renderingl8;1 infinite.
Clustering Coefficient
Certain networks show a tendency to form clusters ofinterconnected nodes [28]. In such networks, if the nodesy and z are both connected to some other node, x, it is likelythat there is also a direct link between y and z themselves.To quantify this we denote the number of links connectingx ‘s nearest neighbors to one another by cx,. This numbercan take values ranging from zero, in the case that no such
links exist, tokxðkx � 1Þ
2in the case that all of the pairs
among x ‘s kx nearest neighbors are connected. The clus-
tering coefficient is thus defined as Cx ¼ 2cxkxðkx � 1Þ,
taking values which are between zero and unity [29e30].For instance, the clustering of node four in Figure 9.1 is
C4 ¼ 2� 2
5� 4¼ 0:2. Averaging over all the nodes in the
network gives the average clustering coefficient hC i. Foran ErdoseRenyi graph the probability of all pairs of nodesbeing linked is uniform, regardless of whether they sharea mutual neighbor or not. The average clustering coefficientis thus hCiER ¼ p, which, for a sparse graph, is typicallyvery small.
SUCCESSES AND FAILURES OF THEERDOSeRENYI MODEL
Biological Small Worlds
The ErdoseRenyi network model is greatly oversimplifiedand therefore overlooks many important features observedin real biological networks. Nevertheless, this model doesprove successful in predicting the overall connectivityobserved is practically all analyzed biological network.These networks all feature a giant connected component,such that almost all pairs of nodes are connected by finitenetwork paths.Moreover, the average path length is found tobe consistent with Eq. (1), so that the interacting nodes in thenetwork are typically just a few steps away fromone another,meaning that cellular networks, like many other networks innature, feature the small world effect. For instance, inmetabolism it was found that most pairs of metabolites canbe linked by paths averaging approximately three edges inlength. These extremely short average path lengths are notunique to any specific species: theywere found in asmany as43 different species, ranging from the evolutionary reducedmetabolic network of parasitic bacteria to the highlydeveloped networks of large multicellular organisms [31].Similar, albeit less dramatic, results apply for protein andgenetic interaction networks, where the average path lengthranges from about four to eight edges [32e33].
Deviations from the ErdoseRenyi Model
In most cellular networks a tendency to form cliques isobserved, where the neighbors of one node tend to bethemselves connected. Thus the average clustering coeffi-cient, hCi, of most cellular networks is significantly largerthan that of an equivalent ErdoseRenyi network. By anequivalent network, we refer to an ErdoseRenyi networkwith the same size and average degree. For instance, pro-teineprotein interaction networks feature a clusteringcoefficient which is typically about an order of magnitudehigher than that observed in their randomly rewiredequivalents [32]. Similar findings also characterize meta-bolic networks [34e35].
The emergence of high clustering provides the first hinttowards the recognition that the ErdoseRenyi model cannotaccount for the topological properties of realistic networks.However, the most significant indication in that directioncomes from the degree distributions observed in actualnetworks. In contrast to the Poisson degree distribution,which is the fingerprint of the ErdoseRenyi graph, cellularnetworks consistently follow a power-law degree distribu-tion [2,5,31,36e45], predicting that the probability fora randomly chosen node to have exactly k links is given by
PðkÞwk�g; (2)
181Chapter | 9 Graph Theory Properties of Cellular Networks
where g takes values which are typically between 2 and 3.This finding has profound implications for the architecture ofbiological networks, as well as their evolution and func-tionality. These implications are discussed in the nextsection.
SCALE-FREE NATURE OF CELLULARNETWORKS
As the structure of cellular networks was elucidated, itbecame evident that their topology does not obey thetypical narrow distribution observed for many otherquantities in nature. Instead of the commonly foundPoisson, Gaussian and exponential distributions, cellularnetworks feature a power-law degree distribution. Thefirst evidence for this came from metabolic networks,where we take the nodes to represent the metabolites andthe directed links to represent the enzyme-catalyzedchemical reactions between them. The analysis of meta-bolic networks from as many as 43 different organismsrevealed that they are all characterized by a power-lawdegree distribution [31]. Similar findings followed fromthe study of proteineprotein interaction networks [32],and transcriptional regulatory networks [38,44].
The Scale-Free Property
In contrast to the Poisson (and other narrow) distributions,the power-law distribution is not concentrated around itsmean. Networks characterized by such a degree distributionare thus highly non-uniform e most of the nodes have onlya few links, whereas a few nodes have a disproportionatelylarge number of links. These highly connected nodes, oftencalled hubs, are the glue that binds the majority of low-degree nodes together. The presence of these hubs, which is
FIGURE 9.2 (A) The yeast proteineprotein interaction network. The size o
sizes serves as a visual expression of the scale-free nature of the degree distribu
that hubs are typically surrounded by low-degree nodes.
strictly banned in narrow degree distributions, is observed inpractically all the analyzed cellular networks, ranging fromthe ultra-reactive pyruvate and coenzyme A in metabolicnetworks, to the insulin receptor in proteineprotein inter-action networks [2,5]. This can be seen in Figure 9.2(a),where we display the proteineprotein interaction network ofS. cerevisiae. In this representation the node size is propor-tional to its degree, so that the clearly visible variability inthe node sizes illustrates the heterogeneity in their degrees.While most proteins participate in one, two or three inter-actions, a few hubs participate in well above 10, and thedegree of some even exceeds 100.
These highly heterogeneous topologies differ essen-tially from the classic ErdoseRenyi networks, in that theydo not have a typical node. In an ErdoseRenyi network, thedegrees of most nodes are in the vicinity of the averagedegree. The scarcity of nodes with any given degree can beestimated by comparing with the average degree of thenetwork. In that sense the average degree provides a char-acteristic scale by which the rest of the nodes should bemeasured. In contrast, a power-law degree distribution, ofthe form of Eq. (2), allows for the coexistence of nodes withan extremely broad range of degrees, freeing the network ofany typical scale. The cellular networks are thus scale-free(SF) [36]. Graphically, the power-law degree distributionforms a straight line when plotted on logarithmic axes, withthe slope providing the scaling exponent, g (Figure 9.3).This yields an intuitive illustration for the concept of the SFtopology. It shows, graphically, that one cannot assigna typical scaling, since the graph duplicates itself regardlessof the scaling used in the horizontal axis (denoting thedegrees).
Of particular significance in the characterization of thedegree distribution is the value of the scaling exponent, g.The broadness of the distribution becomes larger as the
f the nodes is proportional to their degree. The heterogeneity in the node
tion. (B) A small portion of the network reveals that it is disassortative, i.e.,
10−1
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100 101 102 100
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FIGURE 9.3 The degree distribution of the ErdTseRenyi network (A1) follows a Poisson (A2) as opposed to the cellular networks (B1) and (C1) for
which it follows a power-law (B2) e (C2). While the Poisson distribution is concentrated around the mean, the degrees observed in the cellular networks
range over several orders of magnitude. This non-uniformity of the cellular networks is expressed in the coexistence of dense and sparse patches visible in
the network graph. For the metabolic network, where the edges are directed, both the in-degree and the out-degree distributions are plotted. In all graphs
the dots represent the data, while the solid lines are fits to Poisson and to power-law distributions accordingly.
182 SECTION | II Network Properties of Biological Systems
value of g becomes smaller. This means that smaller g
values characterize more degree heterogeneous networks.More specifically, Eq. (2) features three different regimesas the value of g is changed. To observe this we considerthe value of the n th moment, hkni, of the distribution, givenby
RknPðkÞdk. Note that for an infinite network, where k
ranges from zero to infinity, this integral diverges if n� g
� �1. For g � 2 the divergence is observed already at thelevel of the first moment. In such cases, the distribution isso broad that the average is undefined. In practice, when thenetwork is finite, this will take form in a topology wherealmost all the nodes have degrees significantly lower thanthe average, and a small minority of nodes will have sucha high degree that they connect directly to a significantfraction of the nodes in the network, i.e., their degree is ofthe order N. For 2 < g � 3 (typical of most cellularnetworks) the distribution has a finite average, but thesecond moment diverges. This means that the variance,s2 ¼ hk2i � hki2, becomes undefined, capturing the highvariability of the distribution. These mathematical
pathologies are removed once g � 3, as when the scalingexponent is above this threshold, for many practicalpurposes the scale-free nature of the distribution is nolonger relevant. These three regimes are also expressed inthe average path length of the network, as discussed inmore detail below.
Network Integrity and the Role of Hubs
The SF topology allows for a disproportionate number ofhighly connected nodes. These nodes play a crucial role inthe structural integrity of the network. To understand thiswe consider the majority of nodes in the network, whichhave only a few links. They are all likely to be connected tothe hubs by very short paths (of one or two edges). Inaddition, any selected pair of hubs is also likely to be veryclose to one another, due to the large number of links thatthey have. The result is that in SF networks the pathbetween nodes becomes even shorter than in ErdoseRenyinetworks, the hubs playing the role of network short-cuts
183Chapter | 9 Graph Theory Properties of Cellular Networks
[46e48]. In fact, in an SF network, where the scalingexponent is 2 < g < 3 the average path length satisfieshliwln lnN, adding an additional logarithmic correction tothe average path length characteristic of ErdoseRenyinetworks (Eq. (1)). For g ¼ 3 it is found that
hliw lnN
ln lnN, and for g > 3 the result is like that of the
ErdoseRenyi network. Cellular networks, for which thescaling exponent is usually between 2 and 3, are thus ultra-small worlds [49].
The analysis above, regarding the importance of thehubs as the structural backbone of SF networks, has somesurprising implications on the robustness of cellularnetworks to random perturbations. Our intuition leads us toview complex systems as highly intricate structures, whichdepend strongly on the proper functionality of all of theircomponents. When a significant fraction of their nodes fail,these systems are expected to become dysfunctional. Incontrast, biological networks prove to be astoundinglyresilient against component failure [50e54]. From thetopological perspective this can be attributed to their SFtopology and its hub-based backbone. Scale-free networkshave been shown to maintain their structural integrity evenunder the deletion of as many as 80% of their nodes. Theremaining 20% will still form a connected component[55e57]. This is although in an ErdoseRenyi network theremoval of nodes beyond a certain fraction inevitablyresults in the network disintegrating into small isolatedcomponents [27]. The source of this topological resilienceof SF networks is rooted in their inherent non-uniformity.The vast majority of nodes in SF networks have merely oneor two links, playing a marginal role in maintaining theintegrity of the network. Most random failures will occuron these unimportant nodes and thus will not significantlydisrupt the network’s functionality. The relative scarcenessof the hubs, and, on the other hand, their central role inmaintaining the network’s structural integrity, ensures thatrandom failures will rarely break down the network.
The robustness of cellular networks, which reliesstrongly on the hub nodes, is, however, a double-edgedsword. Despite allowing the networks to withstand a largenumber of random failures, it makes them extremelyvulnerable to intentional interventions. The removal of justa small number of key hubs will cause the SF network tobreak down into isolated dysfunctional clusters [56e57].Supporting evidence for this comes from the small numberof lethal genes found in many organisms, and, on the otherhand, by the relatively large number of hubs found amongthese genes [39,41,58e65].
The Origins of the Scale-Free Topology
The SF topology is a universal feature of many realnetworks, both in the context of biology and in social and
technological systems [5]. This ubiquitous topologicalfeature not only characterizes the architecture of a givennetwork, it also serves as an indicator for its formationprocess. This idea is captured by the BarabasieAlbertmodel, which attributes the emergence of an SF topologyto the presence of two fundamental formation processes:network growth and preferential attachment [36]. Bygrowth we refer to the fact that networks are not static:they evolve in time by constantly adding new nodes andnew links. By preferential attachment we mean that nodesare more likely to link to already highly connected nodes.For a more accurate definition, consider an evolvingnetwork, where at each time step a single new node isintroduced, drawing m new links to any of the existingnodes. According to the preferential attachment mecha-nism the new node will choose to connect to an existingnode, x, with a probability proportional to x’s currentdegree, namely
PðxÞ ¼ kxPiki; (3)
where the sum in the denominator is over all nodes in thecurrent state of the network. These two processes, growthand preferential attachment, give rise to the observedpower-law degree distributions. It can be shown that anyone of these processes alone is insufficient and does notyield the desired SF topology. Network growth is required,as otherwise the network reaches saturation and the degreedistribution becomes nearly Gaussian. The preferentialattachment mechanism is needed to support the formationof hubs [36]. By this mechanism, if a node has many links itis more likely to acquire new links, creating a state wherethe rich get richer. The result is that the more connectednodes gain new links at a higher rate, and eventuallyemerge as hubs. Eliminating the preferential attachmentmechanism leads to an exponential distribution, much lessbroad than a power-law.
Preferential Attachment in BiologicalNetworks
The realization of the BarabasieAlbert model in theformation of cellular networks is rooted in the process ofgene duplication [66e71]. This process is clearly respon-sible for network growth, as duplicated genes produceduplicate proteins and thus introduce new nodes into thenetwork. The more delicate point is that gene duplicationalso adheres to the rules of preferential attachment. Tounderstand this, consider an interaction network whichgrows via node duplication. At each time step a randomnode is chosen, say x, and an identical node, ~x, is created.This newly created duplicate node will have exactly thesame interactions as the original node. This means that each
184 SECTION | II Network Properties of Biological Systems
of x’s nearest neighbors will receive a new edge. Therefore,the distribution of new links in the network is biasedtowards the more connected nodes. Indeed, a node withmany nearest neighbors is more likely to have one of itsneighbors chosen for duplication. In fact, for a given nodewith degree k, the probability for a randomly chosen nodeto be linked to it is directly proportional to k. Thus itsprobability to gain a link in the growth process is alsoproportional to k, consistently with Eq. (3).
One of the predictions of the BarabasieAlbert model isthat nodes can become well connected by virtue of beingolder. A node that was introduced early in the history of thenetwork will have more time to accumulate links, and, bythe ‘rich get richer’ mechanism, enhance its chances ofbecoming a hub [36]. In metabolic networks, we find thatthe hubs do, indeed, tend to be older. Some examples arecoenzyme A, NAD and GTP, remnants of the RNA world,which are among the most connected substrates of themetabolic network [34]. Similar findings rise from theanalysis of proteineprotein interaction networks, where, onaverage, the evolutionary ancient proteins are characterizedby higher degrees [72e73]. This offers direct empiricalevidence for the preferential attachment hypothesis.
HIERARCHY AND MODULARITY
The ability of complex systems to function properly andcarry out vital tasks requires the cooperation of manyindependent components. In many artificial networks this iscommonly achieved by relying on a hierarchical design.The network is layered, and nodes at one level orchestratethe behavior of their subordinates belonging to a levelbelow. In that sense we tend to picture network hierarchy asa tree-like topology. However, the idea of having distincthierarchical layers of nodes stands in sharp contrast to thescale-free nature of the cellular networks. The presence ofhubs, which connect directly to a large fraction of the nodesin the network, will inevitably break down the layeredtopology. We therefore have to adopt a different notion ofhierarchy to account for the functional design of biologicalnetworks.
The conceptual idea is that the functionality of theseelaborate networks can be broken into distinct, relativelyisolated tasks [35,74e78]. From a structural point of view,this will be expressed in networks composed of highlyinterconnected sub-graphs, or modules. The hierarchicaldisposition of a given node can be characterized by thenumber of such sub-graphs to which it belongs. This way,a node which is placed low in the hierarchy will participatein just one functional task, and hence belong to just onemodule. Higher in the hierarchy we find nodes that bridgebetween two or three different modules. Eventually, at thehighest level of the hierarchy will reside the hubs, which donot belong to any specific sub-graph but rather connect
many sub-graphs that would otherwise be isolated. Thequantifiable fingerprint of such a hierarchical design can befound in CðkÞ, which describes the dependence of theclustering coefficient on the degree [35,45,79e80]. Low-degree nodes will tend to belong to a specific module andthus feature a high clustering coefficient e indeed, almostall their neighbors will themselves be part of the samemodule. The hubs, on the other hand, will be connected tomany nodes from different modules, and accordingly willtend to have a low clustering coefficient.
The analysis of cellular networks shows clear evidenceof hierarchical topology. The dependence of the clusteringcoefficient on the degree features a power-law scaling,CðkÞwk�b. This has been observed for metabolic networks[35], proteineprotein interaction networks [32] and regu-latory networks, with b taking values typically between1 and 2.
PARTY VS. DATE HUBS
We have already acknowledged the crucial role that thehubs play in the integration of the network. In the abovediscussion we further emphasized their importance whenthe network has a modular structure, as the mediatorsbetween separate modules. In this context, an interestingdistinction between two types of hubs has been proposed[81]. The first type, named party hubs, corresponds to ourusual perception of hubs as nodes that interact with manyother nodes simultaneously. The second type, date hubs,bind to their partners at different times or at differentcellular locations. While the party hubs tend to interactwithin a module, it is the date hubs that typically connectbetween separate modules. So that it is mainly the latterthat serve as the integrators of the network. In the analysisof the yeast proteineprotein interaction network these twotypes of hubs were indeed identified [81]. When the datehubs were systematically removed, the network split intosmall disconnected modules. In contrast, the removal ofparty hubs, despite diluting the modules themselves,harmed the overall integrity of the network to a much lesserextent.
DEGREE CORRELATIONS
It is commonly observed in networks that similar nodestend to connect to one another. This feature, termedassortative mixing, can be related to any characteristic ofa node, and in particular to the node’s degree. For instance,in social networks individuals with many friends tend tolink to others who too have a high degree. However, asshown in Figure 9.2(b), in the featured proteineproteininteraction network the opposite is true: the network isdisassortative, which means that the hubs tend to avoideach other, leading to a network where highly connected
185Chapter | 9 Graph Theory Properties of Cellular Networks
nodes are surrounded by low-degree nodes [82]. This dis-assortativity is observed in most biological networks,including the metabolic and regulatory networks, and is, infact, a property shared by technological networks, such asthe power grid or the internet [83e84].
To classify a network as assortative or disassortative wefirst need to define our expectations of a neutral network.What we are aiming at is to characterize the expectedcorrelations between the degrees of nearest neighbors in theabsence of any assortative bias. To do this we consider therandom selection of an edge in the network and calculatethe probability that at one of its ends resides a node withdegree of k and at the other a node with a degree of k0. Letus first calculate the probability for the first node: i.e., weare seeking the probability of finding a node with k links atthe end of a randomly selected edge. This is essentiallydifferent from the direct selection of a random node, since itgives an advantage to nodes with a higher degree. Thereason is simply because such nodes have a larger numberof edges to which they are attached. For instance, to reacha node with a single edge through this procedure, one mustpinpoint the one edge leading to it. On the other hand, thereare k potential edges through which a k degree node can bereached, making this outcome k times more likely. Thus thedesired probability is proportional to the abundance of kdegree nodes, as well as to the degree itself, i.e., it is
qk ¼ kPðkÞhki , where the denominator is used as a normali-
zation constant. In a neutral network, the degree distribu-tion of the nodes that lie at the other end of the selectededge is independent of qk. Thus, in the absence of degreecorrelations, the probability that a randomly selected edgelinks between two nodes with a degree of k and k0 is simply
QNeukk0 ¼ qkqk0 . To evaluate the assortativity of the network
we compare the observed probability Qkk0 to QNeukk0 . For an
assortative network, the observed probability will showa positive bias along the diagonal, where the value of k isclose to that of k0. Disassortativity will be expressed asa negative bias along the diagonal, and a tendency to havemore links where ksk0.
Another, more compact description of the degreecorrelations can be viewed by observing the averagedegree of a node’s nearest neighbors. We denote thisaverage by Knn. We then average over all nodes witha given degree, k, to obtain KnnðkÞ, namely the averagedegree of the neighbors surrounding a typical node with klinks. In a neutral network, Knn should not depend on k,but if degree correlations are present they will beexpressed as a monotonic increase or decrease in KnnðkÞ.In the proteineprotein interaction network displayed inFigure 9.2, this dependency is clearly visible: the averagedegree of the hub’s nearest neighbors is between 1 and 2,and yet the low-degree nodes are almost all connected to
the hubs, so that for them Knn is much greater. Indeed, theanalysis shows that for this network KnnðkÞwk�a, whereaz0:24 [85].
One can obtain an even more compact parameterizationfor a network’s assortativity, by referring to the Pearsoncorrelation coefficient measured between the degrees ofpairs of connected nodes. This can be explicitly done byextracting the correlation coefficient for k and k0 from thedistribution given by Qkk0. The result is [83]
r ¼ hkk0i � hkihk0is2
; (4)
where s2 is the variance obtained from the distribution qk.
The parameter r takes values between 1, for a perfectlyassortative network, and �1, when the network is perfectlydisassortative. For the yeast proteineprotein interactionnetwork shown in Figure 9.2(a), it measures r ¼ �0:156,confirming that the network is, indeed, disassortative.
The mechanism responsible for the disassortative natureof biological networks remains unclear. It cannot beaccounted for by the BarabasieAlbert mechanism, whichdoes not yield any degree correlations [83]. From a func-tional point of view, it highlights the modular structure ofbiological networks, possibly strengthening even furtherthe central role of the hubs. It was also shown that dis-assortativity harms the resilience of the network and makesit more vulnerable to the intentional removal of hubs, sincein such networks the majority of low-degree nodes areconnected solely to the hubs. On the other hand, dis-assortativity has a positive contribution to the integrity ofthe network when it is not under attack, as typically a dis-assortative network will feature a larger giant connectedcomponent than an assortative or a neutral one [83]. Thisonce again emphasizes the resilience of cellular networksagainst random failure, compared to their vulnerabilityagainst selected node removal.
HUMAN DISEASE NETWORK
The applications of graph theory to systems biology can gobeyond the mapping of the concrete network systems foundwithin the cell. Graphs could also be used as a means oforganizing biological information in a way that couldpotentially spark new insights. An innovative example isprovided by the network approach to the study of humandiseases [86]. In this approach two networks are con-structed. The first is the human disease network. In thisnetwork the nodes represent genetic disorders and the edgeslink disorders which are associated with mutations in thesame gene. The second network is the disease genenetwork. Here the nodes represent genes, and the edges linkgenes which are associated with the same disorder. Bothnetworks are found to be highly clustered, showing that
186 SECTION | II Network Properties of Biological Systems
diseases, as well as disease genes, tend to divide intomodules, or families, of related disorders. In the diseasegene network genes that contribute to the same disordertend to be correlated in many other ways as well. They havean increased tendency to be expressed together in specifictissues; they typically display high co-expression levels;and in many cases they share common cellular and func-tional characteristics, as annotated in the Gene Ontology[86]. This network is also in close relationship with theproteineprotein interaction network, as disease-relatedgenes are very likely to have their products interact togetherthrough physical binding. A surprising feature revealed bythis analysis is the distinction between lethal genes anddisease genes. In many cases, the products of lethal genesare highly connected nodes in the protein interactionnetwork [41]. This emphasizes the importance of the hubsfor the proper function of the network. In contrast, diseasegenes tend to avoid the hubs, and the vast majority of themare non-essential. It has been suggested that this is drivenby natural selection, enabling the proliferation of mutationsonly if they harmed non-vital genes [86]. Confirmingevidence comes from the fact that somatic mutations,which do not harm the organism’s reproduction, are indeedmore frequently related to hub genes. In a broaderperspective, this network approach to human diseasesoffers a tool for the understanding of general patterns ingenetic disorders, and could potentially reveal connectionswhich are not apparent in the study of individual disorders[87e91].
THE BUILDING BLOCKS OF CELLULARNETWORKS
In the previous section we discussed the macroscopicaspects of the hierarchical topology. We have shown thatthe hierarchy in cellular networks is closely intertwinedwith their modular structure. Indeed, from a functionalpoint of view biology is full of examples of modularity,where a distinct group of proteins, genes or metabolites is
responsible for the execution of some basic biologicaloperations. Topologically, as discussed earlier, this isexpressed in the emergence of various sub-graphscomposed of highly interlinked groups of nodes. The highclustering typical of cellular networks provides the quan-titative evidence for this modular network structure. In thissection we focus on the typical recurring structures of thesesub-graphs, and their meaning. In a sense, we are loweringthe altitude of the bird’s-eye perspective with which weviewed the networks until now, going from the macroscopicanalysis to a more focused look at the building blocks ofour complex systems.
Sub-graphs and Motifs
In order to conduct a fruitful analysis of network modules,we need first to develop a scheme by which we can identifywhat are meaningful modules. For instance, considera tetrahedral sub-graph, which is a fully connected set offour nodes as shown in Figure 9.4(a). We can evaluate theabundance of this sub-graph in our network, but this willnot be sufficient in order to tag it as a significant functionalmodule. The randomness in the network topology makes itprobable that such a module is due to appear in the networkby chance. We thus consider a certain module to bea significant motif if it is over-represented in that network,that is, more abundant than expected by chance alone[92e93]. The idea is that if the network has the tendency toover-represent a certain module, there must be an evolu-tionary or functional need for it. Since natural selectiondiscriminates on the basis of functional criteria, it will bethese motifs that are likely to be capable of carryingimportant biological functions.
Randomized Networks
As stated above, for a certain module to qualify as a motif itmust be more abundant than would be expected by chance.However, we have not accurately defined what we mean by
(C) (D)X
Z
Y
X
Z
Y
e autoregulator; (C) the coherent feedforward loop and (D) the incoherent
187Chapter | 9 Graph Theory Properties of Cellular Networks
this criterion. It might, at first glance, seem intuitive to usethe ErdoseRenyi networks as the grounds for comparison.However, a more careful look shows that this is not suffi-cient. The reason is that the expected frequency of a givensub-graph is dictated by the degree distribution of the graphas a whole. Consider, for instance, the tetrahedral modulediscussed above. This module consists of four nodes and sixlinks. In order for such a module to emerge, first we need tohave a node with a degree of at least 3. Then there must beat least three additional links among this node’s nearestneighbors. The likelihood of the first condition is dictatedby PðkÞ, and the likelihood of the second is determined byCðkÞ. In the broader sense what this means is that themacroscopic features of the network, given by PðkÞ andCðkÞ, are in close relationship with the detailed structure ofits modules. In the context of the current discussion, itstates that the abundance of a given sub-graph is notindependent of PðkÞ. Thus, in order to deem a certainmodule as over-represented in a particular network, wemust compare its abundance to that of a randomizednetwork with the same degree distribution [94]. Sucha randomized network can be constructed by randomlyrewiring all the links in the original network, preservingeach node’s degree, and hence PðkÞ, but deleting finestructure, such as the recurrence of motifs.
Autoregulation and the Feedforward Loop
We now briefly discuss two noted examples of highlyrecurring motifs found in transcriptional regulatorynetworks. The first motif is the negative autoregulator,which is one of the simplest and most abundant networkmotifs found in E. coli [95e96]. It includes a single tran-scription factor, which represses its own transcription.Graphically, this motif, shown in Figure 9.4(b), is simplya single node loop. It was shown to have two importantfunctions. The first function is response acceleration.Compared to alternative regulating processes, such asprotein degradation, the process of autoregulation allowsfor a faster response to signals. This was shown boththeoretically and experimentally by employing syntheticgene circuits in E. coli [97]. The second advantage is thatthe motif increases the stability of the gene productconcentration against stochastic noise. It therefore reducesthe variations in protein levels between different cells[98e99].
Another motif frequently encountered in regulatorynetworks is the feedforward loop [100]. This motif consistsof three nodes, x, y and z, where x is directly linked to both yand z, and in addition y is also directly linked to z(Figure 9.4(c),(d)). The direct links can symbolize theactivation or the inhibition of the target gene, or anycombination thereof. Thus eight different versions of thismotif can be constructed, each with a different biological
function [101]. To demonstrate the functional importanceof this motif, we focus on two different versions of themotif. The first is a coherent feedforward loop, observed inthe arabinose utilization and in the flagella systems of E.coli [102e103], and the second is an incoherent feedfor-ward loop, which appears in the galactose system of E. coli[104].
In the coherent feedforward loop all the directed linksrepresent the process of activation. Thus gene x activatesboth genes y and z, and yet gene y itself activates z onceagain. This might seem redundant, but can be shown to haveimportant functional implications. Consider the case wherethe target gene, z, can only be activated if it receives a signalfrom both x and y. Using a computational analogy, we saythat it serves as an AND gate, as it yields a positive outputonly when both of its inputs are positive. The motif willfeature a time lag from when x is activated to when zresponds. This is because z will be activated only aftera sufficient concentration of y products has been produced.The result is that short sporadic expressions of the x genewill die off before z is ever activated. This motif, thereforefunctions as a filter, ignoring stochastic short-term pertur-bations and responding only to persistent ones. Thecomplementary feature rises when the target gene serves asan OR gate. In this case, z is activated by either x or y. Herethe delayed response will appear if x suddenly ceases to beexpressed, in which case z will still remain active for sometime, as long as a sufficient abundance of y’s productpersists. Thus the stability of z’s expression is assuredagainst sudden short-term drops in the production of x. Thistype of behavior is observed in the flagella system of E. coli,where a persistent activation of the flagella is maintainedeven under transient loss of the input signal [105].
A surprising, but nevertheless prevalent, version of thefeedforward motif is the incoherent feedforward loop.Here, while x activates both y and z, the link between y andz is inhibitory. This seemingly contradictory wiring leads toan interesting functional feature. Consider a sudden acti-vation of the gene x, due to, say, an external signal. Asa result both y and z will be activated too. For a short timeafter x’s activation, the expression levels of z will beconstantly rising, owing to its activation by x. However,after a sufficient amount of y products has been produced,the expression of z will be suppressed, due to its inhibitionby y. This version of the motif therefore translatesa persistent signal induced by x into a spike of activation ofthe target gene, z.
GOING BEYOND TOPOLOGY
Despite its success, the purely topological approachpossesses inherent limitations in the race to understandcellular networks. In focusing on topology alone, we haveneglected the fact that not all edges are created equal.
188 SECTION | II Network Properties of Biological Systems
In practice, the dynamical functionality of a complexnetwork is probably affected not just by the binary patternof who is connected to whom, but also by the nature of thisconnection and by its strength. Indeed, in a realistic bio-logical network several reactions are more dominant thanothers e a feature that is overlooked by topology-basedanalyses. To obtain a more effective description we assigndifferent weights to the edges, based on the intensity of theinteraction [106e107]. This gives rise to weightednetworks, where the link between a pair of nodes i, j, is nolonger represented by the discrete state of present versusabsent, but by a continuous number, wij, evaluating itsimportance.
Assigning the Weights
In metabolic networks the most natural measure for theweight of a given reaction is its flux, i.e., the rate by whicha substrate is being converted to its product. The flux-balance approach has proved very successful in retrievingthese fluxes [108e109]. In this approach one writes a set ofequations for the metabolic fluxes, based on the assumptionthat all the metabolic reactions are balanced, that is, theconcentrations of the reactants are at a steady state. Thisamounts to a set of linear algebraic equations for the fluxes.Typically these equations are underdetermined, as theyinclude more variables than equations. To further narrowthe solution space one imposes biological and chemicalconstraints. These constraints may emerge from experi-mental data, for instance, if some of the fluxes can bedirectly measured. Other constraints may be thermody-namic in nature, for instance if a certain reaction is knownto be irreversible because the product has a much lower freeenergy than the substrates. Finally, after characterizing thediminished solution space to which the fluxes are con-strained, the specific solution is chosen to be the one thatoptimizes the predefined biological function (e.g., maximalgrowth rate). For more detailed information regarding flux-balance methods see Chapter 12.
In transcriptional regulatory networks one can rely onmicroarray datasets to express the strength of a connectionbetween a pair of genes. The co-expression of a pair ofgenes can be evaluated by measuring the correlations intheir expression patterns. Alternatively, one can search forlocal similarities in the perturbed transcriptome profiles ofthe genes, and use those to infer the network connectionsand their weights [110e111].
Characterizing the Weighted Topology
Metabolic flux-balance analysis has been applied to themetabolic network of E. coli, and the complete weightednetwork has been obtained. Similarly to the topologicalfindings, the weighted reactions were found to display
strikingly high variability [112]. The reaction weights, basedon the calculated fluxes, range over several orders ofmagnitude. The weight distribution, like the degree distri-bution, follows a power-law, PðwÞww�a, which, for E. coli,has a scaling exponent ofaz1:5.While the specificfluxes forthe different reactions depend on the environmental condi-tions, the aggregate behavior, captured by the weight distri-bution, remains unchanged under various environmentalconditions. The fact that a < 2 emphasizes the broadness ofthe observed weight distribution, as, mathematically, themean value obtained for such a distribution diverges. Inpractice, one finds that almost all fluxes are below theaverage, and a few fluxesmeasure orders ofmagnitude aboveit, dominating the dynamics of the system. This provides aninteresting illustration for the biochemical activity ofmetabolism. It suggests that under any given conditionsmetabolism is dominated by a small set of highly activereactions, embedded in a background ofmostly dim chemicalactivity. A similar pattern of a highly uneven load distributionoccurs in the regulatory network of S. cerevisiae. Also theremost genes have weak correlations, while a few pairs showquite significant correlation coefficients [113].
Topology Correlated Weights
The connectedness of a node, characterized topologically,is captured by its degree. The weighted network analog isthe node’s strength. The strength is defined as the sum of allweights assigned to the node’s set of links, namelysi ¼
P
jwij, where the absence of a link between a pair of
nodes is denoted by setting the corresponding weight tozero. To characterize the relationship between the networktopology and the link weights, we measure the dependenceof the strength on the node degrees, namely sðkÞ. Thisfunction typically takes the form of sðkÞwkb where a lineardependence, namely b ¼ 1, reflects the absence of suchdegree-strength correlations, meaning that the weights areevenly distributed among the edges, so that a node acquiresmore strength simply because it has more links. However,in real systems it is commonly observed that b > 1. Thisimplies that nodes with a higher degree tend to also havelinks with higher weight. This feature was explicitlyobserved in the E. coli metabolic network [114], where itwas found that highly weighted links favored highly con-nected nodes. For each pair of connected nodes the weight,wij, is measured via flux balance analysis, finding that it
features a power-law dependence on the degrees of the two
linked nodes, i.e. hwijiwðkikjÞq, with qz0:5.
Controllability
The functional state of a cellular network at any given timecan be characterized by the concentrations of the reacting
189Chapter | 9 Graph Theory Properties of Cellular Networks
molecules, be them proteins, metabolites or any otherbiomolecules. This defines a vast state space that thenetwork could explore. However, there are cases wheresome parts of this vast state space are restricted owing tothe dynamics of the network. For instance, consider thesimple case of just two interacting metabolites, X and Y , inwhich X produces Y , namely X/Y. The state of thissystem is described by a point in the two-dimensional spacegiven by N
! ¼ ðNX;NYÞ, where NXðYÞ is the concentrationof XðYÞ. However, by assigning a certain value to NX , theresult for NY becomes predetermined via Y ’s production byX. The system is therefore confined to a small subspace ofthe complete two-dimensional state space. If one wished tosteer the system to any desired state, the node X must beexplicitly driven by an external input. We say [115e116]that this system can be controlled (that is, manipulated intoany desired state) with a single driver node, X. In general, ifwe wish to control a complex network, we first need toidentify the set of driver nodes that, if driven by differentsignals, offer full control over the network. We are partic-ularly interested in identifying the minimum number ofdriver nodes, whose control is sufficient to fully control thesystem’s dynamics [117]. An illustration for simple three-node networks is shown in Figure 9.5.
Applying the above concepts to transcriptional regula-tory networks reveals that they are typically difficult tocontrol. This is expressed by their relatively large numberof driver nodes. As an example, for a typical transcriptionalregulatory network approximately 80% of the nodes aredriver nodes, indicating that in order to steer thesenetworks, the majority of the nodes have to be explicitlycontrolled [117]. A higher extent of controllability isobserved for metabolic networks, where the fraction ofdriver nodes is typically around one-third. From a topo-logical point of view, dense and homogeneous networks arerelatively easy to control, whereas sparse and inhomoge-neous ones are hard to control. This implies that the degree
Nz
Y
Initial state
Final state
(A) (B
Nx
Ny
FIGURE 9.5 (A) The state of a three-node network is given by the conce
dimensional state-space. Controlling the network means steering it from any
nodes must be explicitly controlled in order to manipulate the network. (C) H
distribution plays an important role in determining thecontrollability of a network. More specifically, scale-freenetworks, which are highly non-regular, will feature a largenumber of driver nodes and thus be difficult to control.Interestingly, controllability is not governed by the hubs, asthe driver nodes tend to avoid the high-degree nodes.
The results presented above might seem to defy ourintuitive perception of biological networks as systemswhich are expected to be firmly controlled. However, whenexamined once again, they might offer some deeper insightinto the nature of control in biological systems. The factthat there are many driver nodes, and that they are typicallythe less central nodes in the network, is an expression of thehighly constrained nature of these networks. It shows thatthe cellular networks are not free to explore the entire statespace, but are rather confined to a restricted area of thisspace. Thus the only way in which these networks can bedriven into a predefined final state is by explicitly drivingalmost each one of their nodes by an external signal eespecially the low-degree nodes, whose state is otherwisegoverned by the hubs. In a sense, one can interpret theseresults as the strategy of cellular networks to circumventexternal control and maintain their function, even if a largenumber of nodes are being influenced.
Differential Networks
A given network topology may give rise to a variety ofdynamical behaviors under different dynamical rules orenvironmental conditions. External stimuli may trigger theactivity of different parts of the network, further affecting itsdynamical functionality. However, a much broader range ofdynamical behaviors could be achieved under the rule ofa non-static topology, where the structure of the networkitself can react to environmental changes and externalstimuli. Recently, it has been found that cellular networksindeed take advantage of this source of dynamical diversity,
X X
YZ Z
) (C)
ntration assigned to each of the nodes. This defines a point in the three-
initial state to any desired final state. (B) In this network two of the three
ere it is sufficient to control just one node.
190 SECTION | II Network Properties of Biological Systems
altering the architecture of the network itself under differentconditions or biological states, such as tissue type, diseasestate or the surrounding environment [118].
FROM STRUCTURE TO DYNAMICS
The path taken throughout this chapter outlines, in somesense, the approach of network biology towards its futurechallenges. We begin by describing the network represen-tation of cellular systems, examining their topologicalproperties. We then follow with a discussion regardingmotifs, weighted networks and controllability, whichaddresses the dynamics and function of these networks. Inthis spirit, we end this chapter with what is probably themost pressing challenge of this area of research e thebridging between structure and dynamics. We are currentlyat a stage where the topological aspects of cellular networkshave been thoroughly elucidated and their evolutionaryorigins fairly understood. However, we still lack a completetheory which could interpret the topological findings intoa set of dynamical predictions, from which the actualfunctionality of the networks could be inferred [5]. Belowwe stress, in a very broad fashion, the strategic path thatcould meet this challenge [119e120].
The most fundamental question we must address iswhether the gap between structure and dynamical behaviorcould at all be bridged. We need to take into account thatthe topology is one actor in a highly detailed cast ofnetwork characteristics. In the most detailed description ofthese cellular systems, all the interactions can take ondifferent reaction processes and different strengths. Byreaction processes we refer to the types of interaction, e.g.,chemical, regulatory, etc., and by strengths we mean thatsimilar processes may occur at different rates. So whereasstructurally we denote all the various interactions bynetwork links, one should ask: is the process of geneticregulation really comparable to that of physical binding? Isit guaranteed that two structurally identical networks willexpress similar behavior even if they differ in some otherdetails? Or perhaps these details, which are overlooked bygraph theoretic analysis, are not important.
A more constructive approach towards the abovequestions is to ask how far can one actually progress withstructure alone? It seems clear that a complete time-dependent dynamics of the system would require theincorporation of all of the details mentioned above, and isthus beyond the scope of network biology. On the otherhand, what could be achievable based on a structuralanalysis is a macroscopic understanding of networkdynamics. More specifically, network science is notexpected to be successful in predicting the behavior ofa specific set of nodes. It could, however, provide answersto general questions regarding the network as a whole.Questions such as: which are the most effective nodes in
this network? Does this network support long-range inter-actions, or is the impact of nodes contained locally? Are theconcentrations of the nodes in this network stable or gov-erned by fluctuations? Will a small perturbation causea macroscopic failure? At least some of these characteris-tics, and others like them, might be determined by thetopology of the network, regardless of the other details ofthe interactions. And if that is the case, we should be able toaddress these important questions with the tools of networkanalysis.
In a broader perspective, applying graph theory to thestudy of complex systems is aimed at bringing about anintuitive, visual and mathematical toolkit for their under-standing. In that sense, the challenge of this approach is todevise a set of intuitive dynamical interpretations to thealready defined set of topological features. The idea is toassign a functional meaning to characterizations such asa broad degree distribution, high clustering, small world-ness, etc. Along this path, future research needs to chal-lenge some of the common wisdoms regarding theseattributions between structure and dynamics. For instance,the intuitive notion that in a small world topology all thenodes are affected by one another, since they are just a fewreactions away; or the common perception that the hubs arethe most influential nodes in the network. Once thesestatements, and others like them, are examined, they willbring forth a new intuition on the meaning of differentstructural attributes. Then, by analyzing the structure ofa network, researchers will be able to make generalassessments regarding its expected dynamics.
The rapidly improving experimental techniques inbiology will hopefully enable the dynamical predictionsderived from network analysis to be tested. However, evenwhere the existing experimental procedures are insuffi-cient, help might arrive from unexpected sources. Perhapsthe greatest success of the network approach thus far is inrevealing the universal nature of the topology of networksecellular and others e providing a set of tools and criteria bywhich to classify and characterize the structure of thesediverse systems [5]. A similar degree of universality in thedynamics of networks, if found, will provide us witha parallel set of unifying principles, allowing us to describe,using a common platform, various dynamical processes andmake meaningful predictions on the behavior of networksfrom diverse fields. These universal dynamical aspectscould then be inferred from one system to the other.Metaphorically, this expands the boundaries of the classicbiology laboratories far beyond their traditional walls. Asdata are currently collected in vast amounts from biolog-ical, social and technological systems, the abilities thatnetwork science creates to learn from one system aboutthe other provide a crucial source of empirical strength,a strength that may one day help make complex systemsslightly more simple.
191Chapter | 9 Graph Theory Properties of Cellular Networks
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