NETWORKS BASICS Danail Bonchev Center for the Study of Biological Complexity Virginia Commonwealth University Singapore, July 9-17, 2007

Recommended Literature 1. Linked: The New Science of Networks. Albert-László Barabási. Perseus Publisher, ISBN: , 304 pp., Price: $ The Structure and Dynamics of Networks. Mark Newman, Albert-László Barabási, and Duncan J. Watts, Princeton University Press, 2006 | $49.50 / ISBN: ; 624 pp. 3. Evolution of Networks. From Biological Nets to the Internet and WWW. Serguei N Dorogovtsev, Jose Fernando Ferreira Mendes, and A F Ioffe Oxford University Press, 2003, ISBN: , $95.00, 344 pp. 4. An introduction to Systems Biology: Design Principles of Biological Circuits, Uri Alon, Chapman & Hall/CRC, Taylor and Francis Group, 2006, ISBN:

 A branch of science that seeks to integrate different levels of information to understand how biological systems function.  It is not the number and properties of system elements but their relations!!  L. Hood: “Systems biology defines and analyses the interrelationships of all of the elements in a functioning system in order to understand how the system works.” Systems Biology. What Is It?

More on Systems Biology Essence of living systems is flow of mass, energy, and information in space and time. The flow occurs along specific networks  Flow of mass and energy (metabolic networks)  Flow of information involving DNA (transcriptional regulation networks)  Flow of information not involving DNA (signaling networks) The Goal of Systems Biology: To understand the flow of mass, energy, and information in living systems.

Networks and the Core Concepts of Systems Biology (i) Complexity emerges at all levels of the hierarchy of life (ii) System properties emerge from interactions of components (iii) The whole is more than the sum of the parts. (iv) Applied mathematics provides approaches to modeling biological systems.

How to Describe a System As a Whole? Networks - The Language of Complex Systems

What is a Network? Network is a mathematical structure composed of points connected by lines Network Theory Graph Theory Network  Graph Nodes  Vertices (points) Links  Edges (Lines) A network can be build for any functional system F. Harary, Graph Theory, Addison Wesley, Reading, MA, 1969 Gross & Yellen, Handbook of Graph Theory, CRC Press, Boca Raton, FL, 2004 System vs. Parts = Networks vs. Nodes

Networks As Graphs  Networks can be undirected or directed, depending on whether the interaction between two neighboring nodes proceeds in both directions or in only one of them, respectively.  The specificity of network nodes and links can be quantitatively characterized by weights Vertex-WeightedEdge-Weighted 

Networks As Graphs - 2 Networks having no cycles are termed trees. The more cycles the network has, the more complex it is. A network can be connected (presented by a single component) or disconnected (presented by several disjoint components). connecteddisconnected trees cyclic graphs

Networks As Graphs - 3 Some Basic Types of Graphs Paths Stars Cycles Complete Graphs Bipartite Graphs

Air Transportation Network

The World Wide Web

Fragment of a Social Network (Melburn, 2004)

Biological Networks A. Intra-Cellular Networks Protein interaction networks Metabolic Networks Signaling Networks Gene Regulatory Networks Composite networks Networks of Modules, Functional Networks Disease networks B. Inter-Cellular Networks Neural Networks C. Organ and Tissue Networks D. Ecological Networks E. Evolution Network

protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map L-A Barabasi miRNA regulation? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

The Protein Network of Drosophila CuraGen Corporation Science, 2003

Source: ExPASy Metabolic Networks

FAS-L FAS-R FADD CASP10 CASP8 CASP6 CASP3 CASP7 DFF45 DFF40 Death activator DISC Death-Inducing Signaling Complex Heterodimer DFF Initiator Caspases Executor Caspases Start DNA Fragmentation Cleavage of Caspase Substrates Membrane protein Apoptosis Pathway - 1 D. Bonchev, L.B. Kier, C. Cheng, Lecture Series on Computer and Computational Sciences 6, (2006). Apoptosis is a mechanism of controlled cell death critically important in many biological processes

Gene Regulation Networks

The Longevity Gene-Protein Network (LGPN) T. Witten, D. Bonchev, in press C. elegans

Network of Interacting Pathways (NIP) A.Mazurie D.Bonchev G.A. Buck, organisms

Functional Networks Number of shared proteins Cell Cycle Cell Polarity & Structure Intermediate and Energy Metabolism Protein Synthesis and Turnover Protein RNA / Transport RNA Metabolism Signaling Transcription/DNA Maintenance/Chromatin Structure Number of protein complexes Number of proteins Membrane Biogenesis & Turnover Yeast: 1400 proteins, 232 complexes, nine functional groups of complexes (Data A.-M. Gavin et al. (2002) Nature 415, ) D. Bonchev, Chemistry & Biodiversity 1(2004)

Summary  All complex networks in nature and technology have common features.  They differ considerably from random networks of the same size  By studying network structure and dynamics, and by using comparative network analysis, one can get answers of important biological questions.

Fundamental biological questions to answer: (i) Which interactions and groups of interactions are likely to have equivalent functions across species? (ii) Based on these similarities, can we predict new functional information about proteins and interactions that are poorly characterized? (iii) What do these relationships tell us about the evolution of proteins, networks and whole species? (iv) How to reduce the noise in biological data: Which interactions represent true binding events? False-positive interaction is unlikely to be reproduced across the interaction maps of multiple species. Fundamental Biological Questions to Answer (i) Which interactions and groups of interactions are likely to have equivalent functions across species? (ii) Based on these similarities, can we predict new functional information about proteins and interactions that are poorly characterized? (iii) What do these relationships tell us about the evolution of proteins, networks and whole species? (iv) How to reduce the noise in biological data: Which interactions represent true binding events? False-positive interaction is unlikely to be reproduced across the interaction maps of multiple species.

All Complex Dynamic Networks Have Similar Structure and Common Properties  Scale-Freeness  Small-Worldness  Centrality  Robustness/Fragility  Hubs

How To Characterize a Network?

Quantifying Networks A1. Connectivity-based: A. Graph-Theoretical (Topological) Descriptors A2. Distance-based B. Information-Theoretic Descriptors B2. Structural C. Complexity Measures C3. Walk Count B1, Compositional C1. Subgraph Count C2. Overall Connectivity C4. Small-World Connectivity

Connectivity-Based Topological Descriptors Adjacency Matrix Adjacency relation, a ij a ij = 1 ( neighbors ) a ij = 0 ( otherwise ) Adjacency Matrix a i A(G) = a i – node degree V = 5 E = 4 G random node numbering node degrees

Connectivity Descriptors Local (node) descriptors: vertex (node) degrees, a i Global (Network) descriptors: total adjacency, A

Average and Normalized Descriptors Average vertex (node) degree: Network connectedness (density) Connectivity Descriptors = 8/5 = 1.6 A = = 8 Conn = 8/5.4 = 0.4 = 40% Example

Adjacency in Directed Graphs Adjacency relation, a ij a ij = -1 (incoming edge(arc)) a ij = +1 (outgoing edge(arc)) a ij = 0 (otherwise) In-degree = -1 Out-degree = a i (in) a i (out) A(DG) = -1,1 -3,1 0,1-1,1 0,1

Distance-Based Topological Descriptors Distance Matrix Distance relation : d ij = 1 for i,j - neighbors The distance between two non-neighboring nodes is equal to the number of edges along the shortest path that connects them d i D(G) = d i – node distance (node distance degree) d 26 = ? d 57 =?

Distance Descriptors Node descriptors : Network descriptors: Network distance, D(G) Network diameter, Diam(G ) Node eccentricity, e i e i = Max(d ij ) node distance, d i Network radius, Rad(G)

Distance Descriptors-2 Average and Normalized Descriptors Average node distance, Average network distance, (average degree of separation, average path length) = 36/5 = 7.2 D = = 36 = 36/5.4 = 1.8 Example

Distances in Directed Networks  Some distances in directed graphs are equal to infinity !! How to calculate D and ?  In-distances and out-distances 0, 2 -4, 0 0,3 -2, d 21 =? d 13 =? D(in) = D(out) = 6 = 6/(4x3) = 0.5 ???

 Network Node Accessibility Acc (G) = N d (DG)/N d Distances in Directed Networks - 2  Adjusted Average Network Distance Example: G D = 52, = 52/(6x5) = DG D = 34, = 34/20 = 1.70 < D(G)?? AC = 20/30 = 0.667AD = 1.70/0.667 = 2.55 > D(G)

Shannon’s Information Theory References 1. Shannon, C.; Weaver, W. Mathematical Theory of Communications.University of Illinois Press: Urbana, MI, Bonchev, D. Information ‑ Theoretic Indices for Characterization of Chemical Structures. Research Studies Press: Chichester, UK,1983. The more diverse the distribution of system elements, the larger its information content. Information is a measure of system’s diversity How to Measure Information? What Is Information? Wiener: Information is neither matter, nor energy. Forget about meaning! Information is contained in any system, the elements of which can be grouped according to one or more criteria. The more complex the system, the larger its information content. Information is a measure of system complexity

Shannon’s Information Theory Basic Equations Mean Information:, bits/element Total Information:, bits Finite Probability Scheme: System of N elements and k equivalence classes with equivalence criterion α: class number of elements probability 1 N 1 p 1 2 N 2 p 2 ………………………………………………………………... k N k p k where p i = N i / N, and Σ p i = 1. Normalized Information:

Network Information Descriptors Information on the system elements equivalence, e I distances degrees e I tot (deg) = 6log 2 6 – 3log 2 3 – 3x1log1 = bits Vertex degree equivalence distribution: 6{3, 1, 1, 1} e I(deg) = -(3/6) 2 log(3/6) – 3x(1/6)log 2 (1/6) = 1.79 bits/node e I norm (deg) = 10.75/(6log 2 6) = e I tot (dist) = 6log 2 6 – 2x2log 2 2 – 2x1log1 = bits e I(deg) = - 2x(2/6)log 2 (2/6) – 2x(1/6)log 2 (1/6) = 1.92 bits/node e I (deg) = 11.51/(6log 2 6) = Vertex distance equivalence distribution: 6{ 2, 2, 1, 1} Composition distribution: 6{2,2,1,1}

Information on the system elements weight (or magnitude), m I weighted information descriptors (indices) distances D = 52 degrees A = 14 Network Information Descriptors - 2 m I(dist) = 52log 2 52–12log 2 12–2x9log 2 9–2x8log 2 8-6log 2 6 = bits m I(deg) = -(12/52)log 2 (12/52) – 2x(9/52)log 2 (9/52) – 2x(8/52)log 2 (8/52) – (6/52)log 2 (6/52) = 2.55 bits/node m I norm (deg) = /(52log 2 52) = m I tot (deg) = 14log 2 14 – 4log 2 4 – 3log x2log log 2 1 = bits m I(deg) = -(4/14)log 2 (4/14) –(3/14)log 2 (3/14) – 3x(2/14)log 2 (2/14) – (1/14)log 2 (1/14) = 2.47 bits/node m I norm (deg) = 34.55/(14log 2 14) = Distance magnitude distribution: 52 {12, 2x9, 2x8, 6} Vertex degree distribution: 14 {4, 3, 3x2, 1}

Network Complexity Descriptors - 1 Subgraph Count, e SC SC = 17 (5, 4, 4, 3, 1) OC = 76 ( 8, 16, 23, 21, 8) V = 5, E = 4 e=0 e= e= e= e= SC = 5 0 OC = 8 1 SC = 4 1 OC = 16 2 SC = 4 2 OC = 23 3 SC = 3 3 OC = 21 4 SC=1 4 OC = 8 Example e = number of edges Overall Connectivity, e OC

Network Complexity Descriptors - 2 Walk Count, WC Example WC = 106 ( 8, 16, 28, 54) 1 3 l = 1 3 l=2 1 The three complexity measures, SC, OC, and WC, can discriminate very subtle complexity features. 1 2 SC 28(5,8,9,5,1) 30(5,9,10,5,1) OC(in) 111(12,28,41,25,5) 135(16,40,49,25,5) WC 15(5,5,5) 21(5,7,9) l=2 For networks use only complexity measures with e = 1, 2, and 3!!

Small-World Connectivity Network Complexity Descriptors - 3 Network complexity increases with connectivity Network complexity increases with the decrease in its radius Can one unite the two patterns into a single complexity measure? D. Bonchev and G. A. Buck, Quantitative Measures of Network Complexity. In: Complexity in Chemistry, Biology and Ecology, D. Bonchev and D. H. Rouvray, Eds., Springer, New York, 2005, p b i is a measure for node centrality

SC = OC = WC = B1 = B2 = SC = OC = WC = B1 = B2 = SC = OC = WC = A/D = B2 = Examples of Increasing Complexity: N = 5

Thank You for Your Attention!!!