Introduction

In this chapter we present studies of the optimal distance in networks, lopt, defined as the length of the path minimizing the total weight, in the presence of disorder [BBC+03]. Disorder is introduced by assigning random weights to the links or nodes. These weights may represent properties of the links and the transport in them. These properties may include the bandwidth of links in communication networks, delays in the transport of data or material, or the cost of traversing the link. Many real-world networks may be better described by introducing link weights, an example is the airline network where the frequency of flights is an important property (see, e.g., [BBPV04]).

An important quantity characterizing networks is the average distance (minimal hopping) lmin between two nodes in a network containing N nodes. For the Erdős- Rényi networks [ER59, ER60], and the related, more realistic Watts–Strogatz (WS) network [WS98], lmin scales as ln N [Bol85], which leads to the concepts of small world and “six degrees of separation,” while for scale-free networks lmin scales as ln ln N. See Chapter 6 for details.

In most studies, all links in the network are regarded as identical and thus the relevant parameter for information flow including efficient routing, searching, and transport is lmin. In practice, however, the weights (for example, the quality or cost) of links are usually not equal, and thus the length of the optimal path, lopt, minimizing the sum of weights is usually longer than the distance lmin.

One of the most important classes of network studied in the literature is biological networks. This class contains a large variety of naturally occurring networks, formed by the long course of evolution. The human (and animal) body contains a large number of networks, some of which occur in real space, such as the networks of blood vessels, the bronchi, and the nerve system. These networks have been studied for a long time, and many of them are known to be fractal objects, see, for example [BH94, BH96, WBE97]. Another class of network, on which we will focus in this chapter, is logical. These are the networks of gene–gene, gene–protein, and protein–protein interactions. A survey of networks in biology can be found, for example, in [BO04].

In the cell, the genetic information is stored in DNA strands sitting in the cell's nucleus. To pronounce the genetic information DNA is transcribed (copied) to messenger RNA by a protein called RNA polymerase. The messenger RNA is then translated by ribosomes, which form amino acids according to the information in the RNA. Every three letter sequence represents a single amino acid from the twenty amino acids existing in humans and most other known organisms. A chain of amino acids is then created from a sequence of letters and forms a protein. The protein folds into a minimal energy configuration, whose shape determines most of its biological function. The created proteins then interact with the genetic information in several ways.

Introduction

In this appendix we provide basic concepts aimed at introducing the formalism of networks. We first introduce graphs and make simple examples, and then discuss the topological properties of networks. Other general presentations of network structure can be found in review articles and books.

A network (or graph in a more mathematical language) is defined as a set of Nvertices (or nodes) connected by links (or edges). Links, and consequently the whole graph, can be either directed (oriented), if a direction is specified as in Fig. A.1a, or undirected (not oriented), if no direction is specified as in Fig. A.1b. More precisely, undirected links are rather bidirectional ones, since they can be traversed in both directions. For this reason an undirected graph can always be thought of as a directed one where each undirected link is replaced by two directed links pointing in opposite directions (see Fig. A.1c). A link in a directed network is said to be reciprocated if another link between the same pair of vertices, but with opposite direction, is there. Therefore, an undirected network can be regarded as a special case of a directed network where all links are reciprocated. The links of a network may also carry a number, referred to as the weight of the edge, representing the strength of the corresponding interaction. In such a case one speaks of a weighted network. In the present appendix we do not consider weighted networks explicitly.

Introduction

Protein interactions can be identified by a multitude of experimental methods. In fact, the IntAct database of molecular interactions currently lists about 170 different experimental methods and variations thereof that can be used to detect and characterize protein–protein interactions (the main classes are listed in Table 4.1). While we present the commonly used methods in this chapter we will focus on the few technologies which are used in high-throughput studies and thus generated the vast majority of interaction data available today: the yeast two-hybrid (Y2H) assay and protein complex purification and identification by mass spectrometry (MS) (Table 4.2). These two methods represent two fundamentally different sources of interaction data and thus it is important to understand how they work and what strengths and weaknesses each of them has. This is especially important for theoretical analyses which often draw conclusions from datasets which may not be adequate for certain studies. For example, membrane proteins are underrepresented in both yeast two-hybrid and complex purification studies.

Complex versus binary interactions

It is important to note that most methods detect either direct binary interactions or indirect interactions without knowing which proteins are interacting. The yeast two-hybrid system usually detects direct binary interactions while complex purification detects the components of complexes (Fig. 4.1). Complex data are often interpreted as if the proteins that co-purify are interacting in a particular manner, consistent with either a spoke or matrix model.

Biologists now have access to a virtually complete map of all the genes in the human genome, and in the genomes of many other species. They are aggressively assembling a similarly detailed knowledge of the proteome, the full collection of proteins encoded by those genes, and the transcriptome, the diverse set of mRNA molecules that serve as templates for protein manufacture. We increasingly know the “parts list” of molecular biology. Yet we still lack a deep understanding of how all these parts work together to support the complex and coherent activity of the living cell; how cells and organisms manage the concurrent tasks of production and re-production, signalling and regulation, in fluctuating and often hostile environments.

Building a more holistic understanding of cell biology is the aim of the new discipline of systems biology, which views the living cell as a network of interacting processes and gives concrete form to the vision of François Jacob, one of the pioneers in the study of genetic regulatory mechanisms, who spoke in the 1960s of the “logic of life.” Put simply, systems biologists regard the cell as a vastly complex biological “circuit board,” which orchestrates diverse components and modules to achieve robust, reliable and predictable operation. Systems biology suggests that the mechanisms of cell biology can be related to the information sciences, to ideas about information flow and processing in de-centralized networks.

This view, of course, has long been implicit in the study of cell signalling and other key pathways of molecular bio-chemistry.

A primer on statistics and probability theory

This section provides an overview of some of the statistical tools and concepts which are useful for data analysis and the study of complex networks. Our emphasis will be on the practical application of probability theory, rather than its mathematical foundations, which is why we will confine ourselves to self-consistent definitions of the basic ingredients of applied statistics, rather than their derivation from first principles. For those who desire a more rigorous and more detailed treatment of the material, a celebrated introduction to probability theory can be found in, which discusses the contents of this chapter in much greater detail.

Events and probabilities

Tossing a coin – with an outcome of ‘heads’ or ‘tails’ – is one of the simplest examples of a probabilistic event. More complicated examples could be to obtain ‘five’ and ‘two’ when throwing a pair of dice, the ball landing on a red number in a game of roulette, or the spreading of an infection from an infected individual to a healthy one. In all these cases the set of all possible outcomes of an experiment is the sample space. An event can be defined as any member (or subset) of the sample space.

Introduction

Complex systems that describe a wide range of interactions in nature and society can be represented as networks. In general terms, such networks are made of nodes, which represent the objects in a system, and connections that link the nodes, which represent interactions between the objects. In mathematical terms, a network is a graph which comprises of vertices and edges (undirected links) or arcs (directed links). Examples of complex networks include the World Wide Webs, social network of acquaintances between individuals, food webs, metabolic networks, transcriptional networks, signaling networks, neuronal networks and several others. Although the study of networks in the form of graphs is one of the fundamental areas of discrete mathematics, much of our understanding about the underlying organizational principles of complex real-world networks has come to light only recently. While traditionally most complex networks have been modeled as random graphs, it is becoming increasingly clear that the topology and evolution of real networks are not random but are governed by robust design principles.

A number of biological systems ranging from physical interaction between biomolecules to neuronal connections can be represented as networks. Perhaps the classic example of a biological network is the network of metabolic pathways, which is a representation of all the enzymatic inter-conversions between small molecules in a cell. In such a network, nodes represent small molecules, which are either substrates or products of an enzymatic reaction, and directed edges represent an enzymatic reaction that converts a substrate into a product.

Introduction

Complexity in biological systems is the rule rather than the exception. Life seems to depend on structures that not only perform a wide variety of functions, but are adaptable and robust at the same time. For a long time the only scientific approach available to study these complex biological systems has been a purely descriptive one. In the second half of the twentieth century molecular biology emerged, along with the development of a variety of experimental methods that allowed an ever-deeper exploration of the constituent parts of a cell, as well as the ways in which these parts assemble. Our chromosomes were shown to be made of tightly packed DNA double helixes that store our genetic code. RNA polymerase, the protein complex responsible for transcription of the genetic code to messenger RNA, has been identified, along with many major constituents of the fascinating machinery between genetic code and cellular phenotype. As mRNA molecules leave the nucleus they are met by ribosomes, protein complexes that read the genetic code using groups of three mRNA letters to identify the corresponding amino-acid sequence, and thus translate the genetic code into proteins. Proteins are responsible for the majority of biological functions driving a living cell: they orchestrate metabolic reactions, form structural elements like the cytoskeleton, keep track of the extra- and intracellular environment and transmit the signals that constantly reshape gene transcription so that the cell can express precisely the proteins it needs.

Introduction

Specific sensory and signaling systems allow living cells to gather and transmit information from the environment. All perceived signals are used in order to adjust cellular metabolism, growth, and development to environmental conditions. At the same time the cell is able to sense the intracellular milieu, e.g. energy and nutrient availability, redox state and so on and it accordingly adapts its physiological state. The importance of such changes in cellular processes is underlined by the presence of multiple regulatory systems (see Table 3.1), the most important of which controls the rate of transcription of a gene. The extremely different cell types in higher eukaryotes are a consequence of expression pattern differences, as well as cellular proliferation and differentiation, which are controlled by complex regulatory circuits originating space- and time-dependent transcriptional patterns. Thus, understanding the dynamic link between genotype and phenotype remains a central issue in biology.

Signals sensed by the cell are translated into changes in the rate of transcription of well-defined groups of genes through the activation of specific proteins (transcription factors, TF). TFs have high affinity for specific short sequences located upstream of genes and regulate transcription either positively or negatively. The binding of a TF to its target on the gene's promoter controls when expression occurs, at what level, under what conditions, and in which cells or tissues [662]. Interactions with other proteins, chromatin remodeling, modification complexes and the general transcription machinery affect the DNA-binding characteristics of a TF thereby influencing the rate of transcription.

The network hypothesis

A cell is an enormously complex entity made up by myriad interacting molecular components that perform the biochemical reactions that maintain life. This book is about the network hypothesis, according to which it is possible to describe a cell through the set of interconnections between its component molecules. Hence, it becomes convenient to focus on these interactions rather than on the molecules themselves to describe the functioning of the cell.

The central dogma in molecular biology describes the way in which a cell processes the information required to produce the molecules necessary to sustain its existence and reproduction. It is also becoming increasingly clear, however, that in order to establish a more complete description of the manner in which a cell works we require a deeper understanding of the manner in which the sets of interconnections between these molecules are defining the identity of the cell itself. It is therefore important to investigate whether the genetic makeup of an organism does not only specify the rules for generating proteins, but also the way in which these proteins interact among themselves and with the other molecules in a cell.

Introduction

Most of biology is interpreted via macromolecular interactions. Protein interaction networks represent the first genome-wide drafts of those interactions and have been explored as models for understanding cellular processes. Given the constant flow of new experimental data on the correlation of genes and proteins within an organism or even between different species, there is the need to rationalize in a solid framework the thousands of possible protein interactions inferred by experiments. Computational models can help in this task investigating the microscopic mechanisms responsible for the behaviors observed in experiments as different as yeast two-hybrid, mass spectroscopy, gene co-expression, synthetic lethality, just to mention the most popular.

In protein interaction networks, nodes and links represent the proteins and the interactions between them, respectively (Fig. 5.1). However, depending on the approach that has been used to generate the map, a link does not always indicate a direct physical interaction. It can also represent correlated expression in the cell, performance of successive steps in a metabolic pathway, similar genomic context and so on. Observation of direct binding is a good indication that two interacting proteins cooperate in the same biological pathway and whenever possible this chapter will restrict to this type of interactions.

Modeling can focus on different network properties. Some approaches use evolutionary arguments. Others take into account genomic information as well as the physico-chemical properties of proteins in a statistical way.

Introduction

This chapter deals with the complex network of biochemical reactions known as cellular metabolism. Understanding how the different components of this network coordinate their action towards generating coherent pipelines of chemical transformations, how the pipelines themselves are promptly assembled, disassembled and controlled as a function of changing environmental conditions, and how evolutionary adaptation shapes this whole system, constitute fundamental ongoing challenges. These questions are not only intellectually fascinating, but also practically important for many biomedical, engineering, and environmental problems. Because of the complexity of these networks, mathematical models and computer simulations are an essential component of this challenge. This chapter aims at providing a concise and elementary introduction to some basic concepts on mathematical modeling of metabolic networks, with a few examples of recent research applications. Those interested in serious background should refer to classical biochemistry textbooks and recent books on metabolic engineering and computational models of biological networks.

Cellular metabolism and its regulation

In the busy economy of a cell, the balance of resources is essential for survival and reproduction. The main currencies, free energy stored in chemical bonds, and molecular building blocks, can be used for a variety of purposes, from the synthesis of new molecules, to the maintenance of gradients across the membrane; from the capacity to move and find more food, to the production of all components necessary for self-reproduction. This economy involves several hundred to thousands of types of small molecules and biochemical reactions (Fig. 6.1).