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# Insights into the Organization of Biochemical Regulatory Networks Using
Graph Theory Analyses^{*}

^{1}

^{1}To whom correspondence should be addressed. E-mail: ude.mssm@nayaam.iva.

## Abstract

Graph theory has been a valuable mathematical modeling tool to gain insights into the topological organization of biochemical networks. There are two types of insights that may be obtained by graph theory analyses. The first provides an overview of the global organization of biochemical networks; the second uses prior knowledge to place results from multivariate experiments, such as microarray data sets, in the context of known pathways and networks to infer regulation. Using graph analyses, biochemical networks are found to be scale-free and small-world, indicating that these networks contain hubs, which are proteins that interact with many other molecules. These hubs may interact with many different types of proteins at the same time and location or at different times and locations, resulting in diverse biological responses. Groups of components in networks are organized in recurring patterns termed network motifs such as feedback and feed-forward loops. Graph analysis revealed that negative feedback loops are less common and are present mostly in proximity to the membrane, whereas positive feedback loops are highly nested in an architecture that promotes dynamical stability. Cell signaling networks have multiple pathways from some input receptors and few from others. Such topology is reminiscent of a classification system. Signaling networks display a bow-tie structure indicative of funneling information from extracellular signals and then dispatching information from a few specific central intracellular signaling nexuses. These insights show that graph theory is a valuable tool for gaining an understanding of global regulatory features of biochemical networks.

Progress in biochemistry over the past 40 years has allowed us to develop an impressive parts list of cellular components and their interactions. Such interactions give rise to functional subcellular machines such as metabolic circuits, signaling networks, and cytoskeletal structures. Each of these systems contains several hundreds to thousands of different types of components. For example, a recent comprehensive study of mitochondria in the mouse identified over 1000 different types of proteins (1). Understanding the global topological organization of such complex systems is a first step toward a holistic yet detailed functional map of the entire cell. Graph theory, a subfield of mathematics, has been a valuable tool in the past decade to gain insights into the global organization of regulatory biochemical networks as well as to develop more informed hypotheses for new experiments.

Euler's famous publication from 1736 on the Seven Bridges of Königsberg problem (2) initiated graph theory. Over 225 years later, in the late 1950s, a relevant historical development in graph theory was the analysis of random networks by Erdös and Rényi (ER graphs). In the late 1990s, it was recognized that real networks are different from ER graphs. Real-world complex systems abstracted to networks across disciplines, including biochemical networks, have a common global architecture termed small-world (3) and scale-free (4). Small-world indicates a relatively short distance from any node to any other node and a relatively high level of clustering. Clustering means that groups of nodes have many interactions with one another. Scale-free denotes a connectivity distribution that fits a power law. These two seminal observations initiated a new approach to modeling systems of biochemical reactions in a cell. Instead of viewing reactions in pathways as substrates acted upon by enzymes to produce products or as mass-action binding reactions, biochemical interactions in biochemical networks can be abstracted to nodes and links forming a graph (5). Graphs are mathematical structures that have been successfully applied to model complex systems from computer science, electrical engineering, physics, and social sciences and in recent years to represent biological networks.

There are two fundamental approaches in applying graph theory to analyze biochemical regulatory networks. The first is an attempt to understand the global organization of these networks. For this, properties and attributes computed for individual nodes, links, and/or groups of nodes and links are averaged, or the distribution of such properties is analyzed and compared with the distributions found in shuffled networks. The second approach is more practical. By using prior knowledge about biomolecules and their interactions, it is possible to place results from multivariate experiments that produce lists of genes, shown to be altered under different experimental conditions, in the context of known pathways and networks. Here, I describe a few applications and insights from graph theory analyses applied to study biochemical networks in combination with an introduction to concepts and definitions from graph theory.

## Nodes and Links

*Graphs* are mathematical structures made of a few related sets. The
first set represents entities. When modeling biochemical networks, these
entities typically represent genes, proteins, or other type of biomolecules.
These entities are formally called *vertices* and less formally
*nodes* or *components*. The second set in a graph describes the
relations between the entities. The elements of this set are formally termed
*edges* for *undirected graphs* and *arcs* for directed
graphs. Directed graphs, termed *digraphs*, can be used to represent
systems in which the causal relationship between vertices is known. For
example, if *A* is upstream of *B*, then there is an arc (arrow)
pointing from *A* to *B*. Other less formal names to describe
edges and arcs are *links* or *interactions*. In biochemical
regulatory networks, these can be direct physical interactions between
proteins, transcription factors binding to promoter sites, other indirect gene
regulation effects, or enzymatic reactions in which enzymes are linked to
their substrates. Throughout the text, the terms graph/network,
vertices/nodes/components, and edges/links/interactions are used
interchangeably but mostly have the same meaning.

There are different types of graphs used to represent different types of
biochemical networks (6). For
example, *mixed graphs* are graphs that are both directed and
undirected. These graphs have two or more sets of relations. Typically, edges
are separated from arcs. Cell signaling pathways are commonly represented
using mixed graphs in which arcs represent activation or inhibition relations,
whereas edges represent physical protein-protein interactions without a
clear-cut directionality such as binding to anchors and scaffolds
(7). Other sets in cell
signaling graphs can represent other properties of edges such as interaction
weights. Weights of arcs can be used to represent the kinetics of biochemical
reactions (8).

Having two types of arcs, such as activation *versus* inhibition
relations, is an example of *edge coloring*. Coloring is the assignment
of labels to vertices or edges with some defined constraints. For example,
*vertex coloring* can be used to distinguish transcription factors from
other proteins in a protein-protein interaction graph. The Gene Ontology
Consortium can be considered a graph-coloring undertaking for labeling genes
and proteins based on their function, location in the cell, and involvement in
biological processes (9). The
Gene Ontology data set itself is stored in a hierarchical tree graph data
structure in which different levels represent the detailed specific
description of terms recounting properties of genes. The Gene Ontology
hierarchical tree is an example of a specialized type of graph in which
specific rules are used to connect vertices.

## Types of Graphs

Another example of a specialized graph in which rules are used to restrict
possible connections between vertices is a *bipartite* graph. These
graphs have two sets of vertices where edges can connect only nodes in
different sets, not nodes within a set. Bipartite networks are used, for
example, to represent metabolic networks separating enzymes from their
substrates and products, disease gene networks connecting diseases with
disease genes (10), and drug
networks connecting drugs with their known biomolecular targets
(11,
12) or to integrate different
“omics” data sets
(13). Another type of graph,
the *planar graph*, can be drawn on a plane with no edge crossings.
Planar graphs are important for visualization. *Acyclic graphs* are
graphs with no cycles. Bayesian networks reconstructed from time-series or
perturbation high-throughput microarrays
(14) or proteomics studies
(15) are typically represented
as acyclic graphs. An acyclic graph is also called a *forest* because
it comprises a collection (*union*) of *trees*. A tree is a
graph in which any two vertices are connected by only one possible
*path*. A graph can be *partitioned* or *cut* into
*subgraphs* or *subnetworks* based on different rules.
Subnetworks of biochemical networks are often used to represent pathways,
modules, or protein complexes. One example of a subgraph is a *spanning
tree*. A spanning tree is a subgraph tree that connects all nodes in a
network without using all links. A *minimum spanning tree* is a
spanning tree that is formed with a minimum cost, where the “cost”
is typically the total number of edges. *Steiner trees* are similar to
minimum spanning trees but extra intermediate vertices and edges may be used
to reduce the overall length/cost of the minimum spanning tree. Steiner trees
can be used to connect lists of “seed” genes that were found to be
altered under different experimental conditions using known protein-protein,
cell signaling, and gene regulatory networks
(16).

Most biochemical networks are not fully characterized. In many of them,
there are interactions and components that are not connected with the rest of
the network. Such networks typically have a *giant connected
component*. It is important to consider that graphs can be alternately
represented as a symmetric adjacency matrix where vertices are represented as
identical row and column labels, and the matrix contents consist of the
presence or absence of edges (0s and 1s) and/or the strength and/or direction
between interacting biochemical entities. The matrix formulation of graphs
allows manipulation and analysis using powerful tools from linear algebra.

## Properties of Nodes

Vertices and edges in networks can have an assortment of attributes or
properties. Two vertices are considered *adjacent* or
*connected* if there is an edge that links them. Such vertices are also
called *neighbors*. An important attribute/property of vertices is
their *vertex degree* (also called *valence*), which is commonly
denoted with *k*. This means that *k* is also the number of
neighbors a vertex has. In digraphs, it is important to distinguish between
*in-degree* and *out-degree*. Different types of biochemical
networks across different species were found to have a connectivity degree
distribution that fits a power-law function
(4,
17,
18). This means that most
nodes have few neighbors but that a substantial number of nodes have high
degree (Fig. 1*A*). The
power-law connectivity distribution observation can be explained by the fact
the proteins in the cell are heterogeneous, serving many and different
functions. Power-law distributions are commonly observed in highly
heterogeneous complex systems. Vertices with high degree are informally called
*hubs*. Analysis of protein-protein interaction networks demonstrated
that hubs can be classified into “party” hubs and
“date” hubs (Fig.
1*B*) (19).
Party hubs are proteins that interact with their neighbors in the same place
at the same time, whereas date hubs are proteins that interact at different
times in different places within the cell. Another classification of hubs
showed that hubs can be divided into single-domain or multidomain hubs
(20)
(Fig. 1*C*). Some
examples of single-domain date hubs are protein kinases A and C and the
phosphatase PP2A, which have many known substrates. CASK is an example of a
party hub with multiple domains. *Assortative mixing* is when the
probability for interactions between nodes is biased due to nodes' properties.
For example, assortative mixing by valence is when hubs are frequently
connected to one another (21).
Biochemical networks in general were found not to display assortative mixing
by valence as compared with other networks, for example, brain networks
constructed from functional magnetic resonance images
(22). On the other hand,
assortative mixing by function, location, or biological process is obviously
highly pervasive in regulatory biochemical networks.

## Paths in Biochemical Networks

A *path* in a graph represents a sequence of alternating neighboring
nodes and links with no repeating nodes. Some of graph theory's most famed
algorithms are those developed by Dijkstra
(23) and Floyd
(24) to find the shortest path
(*geodesic path*) between two vertices in a network. Finding the
shortest path between a cell-surface receptor and downstream transcription
factors in a cell signaling network can be used to identify important new
signaling pathways. Such an approach was useful to hypothesize potential
signaling mechanisms in Neuro2A cells downstream of CB1R receptors. Cells were
stimulated with a CB1R agonist, and assessment of activity for hundreds of
canonical transcription factors was performed. It was found that after 20 min,
CB1R activation modulates the activity of 23 transcription factors
(25). Using known cell
signaling and protein-protein interactions extracted from published
experimental studies, new biological roles for pathways and co-regulators were
identified. In another study, a global analysis of paths from receptors to
effectors in a literature-based mammalian cell signaling network showed that
from some receptors, *e.g.* the
*N*-methyl-d-aspartate receptor, there are many paths to
effectors, *e.g.* the transcription factor cAMP-responsive
element-binding protein (CREB), whereas from other receptors, there are only a
few (Fig. 1*D*)
(26). This topological feature
can be due to biased research (most data from popular proteins and pathways)
but can also indicate a design that is commonly observed in learning
classifier systems implemented in computer programs.

The topology of signaling networks also displays a bow-tie structure, in
which signals from many receptors converge on the same intermediate components
and then are directed to regulate different transcription factor effectors
(Fig. 1*E*). This type
of organization is common for Toll-like receptors sharing adaptor proteins
such as MyD88 (27), G
protein-coupled receptors sharing Gα and Gβγ
(28), and growth factor
receptors sharing adaptor proteins such as SOS1 and GRB2. The shortest path
algorithm can be used to find automatically and display previously
characterized interactions that “connect” genes and proteins
(29) or to compute global
network properties such as *characteristic path length*
(3) or network
*diameter*. Network diameter is simply the longest of the shortest
paths among all possible shortest paths between all pairs of nodes in a
network. The characteristic path length is the average shortest path across
all possible pairs of nodes.

## Network Motifs

Biochemical networks contain many three-node *cliques*. A clique is
a *complete* subgraph in which all possible links between a subset of
nodes are operational. Completing “defective cliques” was used to
predict not yet observed interactions using the known protein-protein
interactions of a yeast network
(30). Small cliques in
biochemical networks are only one kind of a possible set of small biochemical
circuits. The different kinds of small biochemical circuits are collectively
termed *network motifs*. More precisely, network motifs are subgraphs
that are over-represented in real networks relative to the same subgraphs in
*shuffled networks*
(31). Shuffled networks are
networks in which the edges of real networks are systematically randomized
while keeping intact some general properties of the original topology such as
the connectivity degree
(32).

Biochemical networks such as signal transduction networks and gene
regulatory networks show similar patterns of network motifs. For example, the
*bifan* motif (33,
34) is made of two upstream
regulators both regulating the same two downstream effectors
(Fig. 1*F*). This dual
regulation structure was identified statistically as the most over-represented
network motif in gene regulatory networks of yeast
(31) and *Escherichia
coli* (31,
35) as well as in a mammalian
neuronal cell signaling network
(7). One example of a bifan
motif in cell signaling networks is the regulation of transcription factors
ATF2 and Elk by the kinases JNK (c-Jun N-terminal
kinase) and p38
(33). The abundance of bifans
is most likely due to a large number of isoforms generated through gene
duplication-divergence evolution. The bifan motif and other motifs such as
feedback and feed-forward loops were found to act as noise filters
(33,
36,
37). Two types of network
motifs, namely feedback and feed-forward loops, are very important for
characterizing the dynamics of biochemical networks
(38,
39). Graph analysis of a large
cell signaling network suggested that negative feedback loops are more
prevalent than positive feedback loops near the cell surface
(7), a design that could be
helpful for dampening noise while amplifying persistent extracellular signals
(Fig. 1*G*).

A paucity of negative feedback and feed-forward loops in yeast, *E.
coli*, and mammalian cell signaling networks was also observed
(40). This feature of the
topology suggests that negative loops have not been favored through evolution
because of their potential to introduce dynamical instabilities. Hence, it
appears that negative regulators are less regulated outgoing hubs, examples of
which are known in cell signaling networks. For instance, phosphatases such as
PP1 and PP2A are enzymes that deactivate most of their effectors through
dephosphorylation (Fig.
1*H*). On the other hand, positive feedback loops are
highly nested, where the same proteins function in many positive feedback
loops, a topology that also favors dynamical stability
(Fig. 1*I*)
(41). Some regulatory motifs
in biochemical networks have long been known, *e.g.* the negative
feedback loop in the synthesis of branched chain amino acid from threonine to
isoleucine (42). The concept
of network motifs is illustrated by several examples from cell signaling
(Fig. 2).

**Example of network motifs within cell signaling networks.**

*PFBL*, positive feedback loop;

*NFBL*, negative feedback loop;

*PFFL*, positive feed-forward loop;

*NFFL*, negative feed-forward loop;

*CaM*, calmodulin;

*CaN*, calcineurin;

*AC1*, adenylyl cyclase I;

**...**

The presence of network motifs that are dense in links, like the bifan,
points to the fact that biochemical networks typically have high
*clustering coefficients*
(3). A clustering coefficient
measures the level of density in local connectivity around the neighborhood of
a node. High clustering also suggests that biochemical networks are organized
into modules. Such modules can be identified using network clustering
algorithms. A popular measure for identifying clusters in networks is the
*betweenness centrality* measure. Betweenness centrality is computed
for each vertex or edge by counting the number of times the shortest paths
pass through the vertex or the edge
(43). If many short paths go
through a vertex and if the vertex has a relatively low degree, the vertex
must be connecting different modules. Such a vertex can be removed for the
purpose of isolating and identifying modules/clusters.

## Conclusions

One of the limitations of graph theory applications in analyzing
biochemical networks is the static quality of graphs. Biochemical networks are
dynamical, and the abstraction to graphs can mask temporal aspects of
information flow. The nodes and links of biochemical networks change with
time. Static graph representation of a system is, however, a prerequisite for
building detailed dynamical models
(44). Most dynamical modeling
approaches, *e.g.* Boolean networks
(45), Petri nets
(46), and event ontologies
(INOH Pathway Database), can be used to simulate network dynamics while using
the graph representation as the skeleton of the model. Modeling the dynamics
of biochemical networks provides closer to reality recapitulation of the
system's behavior *in silico*, which can be useful for developing more
quantitative hypotheses.

The challenge with building dynamical models of biochemical networks is
that they require kinetic and quantity parameters, which are difficult to
obtain experimentally. Another obstacle in both graph theory and dynamical
modeling is that most applications are *N*P-hard. This means that time
for execution grows exponentially with *N*, where *N* can be the
number of steps in a path or the number of nodes in a graph. This
computational challenge places practical limitations on calculating static and
dynamical properties of large regulatory biochemical networks. To overcome
this challenge, sampling (47)
and parallelization of algorithms
(48) can be applied.

In summary, graph analysis of biochemical networks has been useful for obtaining an overview of the organizations of different types of biochemical networks across species. In general, most networks have a connectivity distribution that fits a power law, high clustering coefficients, and relatively short average path lengths; the networks are organized in hierarchical modularity, where hubs serve as party or date hubs and can be divided into multisite or single-site hubs, and assortative mixing by valence is not common, whereas assortative mixing by function, location, or biological process is evident. Biochemical network motifs are enriched in dense substructures where the bifan motif is the most over-represented, probably due to duplication-divergence, and where negative feedback and feed-forward loops are less common than positive loops. Cell signaling networks have many paths from some input receptors and few from others, a topology reminiscent of a classification system. Signaling networks also display a bow-tie structure. These are only a handful of topological patterns out of many. Such topological properties are likely to have consequences for the dynamical behavior of a system. Initial dynamical analyses of these properties are consistent with an architecture that supports stability, noise filtering, modularity, redundancy, and robustness to failure as well as variations of kinetic rates and concentrations.

We are just starting to understand the intricate dynamics of large and complex biochemical systems in which graph theory plays an important role in organizing the accumulated knowledge. Graph theory is also useful for the analysis of multivariate data when lists of genes or proteins can be placed in the context of prior knowledge to develop more informed hypotheses about how multiple factors cooperate to produce complex phenotypes. In the new world of Big Data (massively abundant data) and Cloud Computing (data can be accessed from everywhere and processed anywhere), graph theory plays an increasingly important role in the transition from the classical approach of hypothesizing and testing experimentally to hypothesizing, modeling, and testing to measure everything, identify patterns, model, and modify (manipulate) input-output relationships (49).

## Acknowledgments

I thank Professor Iyengar for helpful suggestions.

## Notes

^{*}This work was supported, in whole or in part, by National
Institutes of Health Grant
1P50GM071558-01A27398. This work was also supported by
a startup fund from the Mount Sinai School of
Medicine (to A. M.). This is the fifth article of six in the
Thematic Minireview Series on Computational Biochemistry:
Systems Biology. This minireview will be reprinted in the
2009 Minireview Compendium, which will be
available in January, 2010.

## References

**American Society for Biochemistry and Molecular Biology**

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