LECTURE 2 1.Complex Network Models 2.Properties of Protein-Protein Interaction Networks

Complex Network Models: Average Path length L, Clustering coefficient C, Degree Distribution P(k) help understand the structure of the network. Some well-known types of Network Models are as follows: Regular Coupled Networks Random Graphs Small world Model Scale-free Model Hierarchical Networks

Regular networks

Diamond Crystal Graphite Crystal Regular networks Both diamond and graphite are carbon

Regular network (A ring lattice) Average path length L is high Clustering coefficient C is high Degree distribution is delta type.

Random Graph Erdos and Renyi introduced the concept of random graph around 40 years ago.

Random Graph p=0 p=0.1 p=0.15 p=0.25 N=10 E max = N(N-1)/2 =45

p=0.25 Average path length L is Low Clustering coefficient C is low Degree distribution is exponential type. Random Graph

Usually to compare a real network with a random network we first generate a random network of the same size i.e. with the same number of nodes and edges. Other than Erdos Reyini random graphs there are other type of random graphs A Random graph can be constructed such that it matches the degree distribution or some other topological properties of a given graph Geometric random graphs Random Graph

Small world model ( Watts and Strogatz) Oftentimes,soon after meeting a stranger, one is surprised to find that they have a common friend in between; so they both cheer: “ What a small world! ” What a small world!!

Small world model ( Watts and Strogatz) Begin with a nearest-neighbor coupled network Randomly rewire each edge of the network with some probability p

Small world model ( Watts and Strogatz) Average path length L is Low Clustering coefficient C is high Degree distribution is exponential type.

Start with a small number of nodes; at every time step, a new node is introduced and is connected to already- existing nodes following Preferential Attachment (probability is high that a new node be connected to high degree nodes) Scale-free model ( Barab á si and Albert)

Average path length L is Low Clustering coefficient C is not clearly known. Degree distribution is power-law type. P(k) ~ k -γ

Scale-free networks exhibit robustness Robustness – The ability of complex systems to maintain their function even when the structure of the system changes significantly Tolerant to random removal of nodes (mutations) Vulnerable to targeted attack of hubs (mutations) – Drug targets

The term “scale-free” refers to any functional form f(x) that remains unchanged to within a multiplicative factor under a rescaling of the independent variable x i.e. f(ax) = bf(x). This means power-law forms (P(k) ~ k -γ ), since these are the only solutions to f(ax) = bf(x), and hence “power-law” is referred to as “scale-free”. Scale-free model ( Barab á si and Albert)

Hierarchical Graphs The starting point of this construction is a small cluster of four densely linked nodes (see the four central nodes in figure).Next, three replicas of this module are generated and the three external nodes of the replicated clusters connected to the central node of the old cluster, which produces a large 16-node module. Three replicas of this 16-node module are then generated and the 12 peripheral nodes connected to the central node of the old module, which produces a new module of 64 nodes. NETWORK BIOLOGY: UNDERSTANDING THE CELL’S FUNCTIONAL ORGANIZATION Albert-László Barabási & Zoltán N. Oltvai NATURE REVIEWS | GENETICS VOLUME 5 | FEBRUARY 2004 | 101

Hierarchical Graphs The hierarchical network model seamlessly integrates a scale-free topology with an inherent modular structure by generating a network that has a power-law degree distribution with degree exponent γ = 1 +ln4/ln3 = 2.26 and a large, system-size independent average clustering coefficient ~ 0.6. The most important signature of hierarchical modularity is the scaling of the clustering coefficient, which follows C(k) ~ k –1 a straight line of slope –1 on a log–log plot NETWORK BIOLOGY: UNDERSTANDING THE CELL’S FUNCTIONAL ORGANIZATION Albert-László Barabási & Zoltán N. Oltvai NATURE REVIEWS | GENETICS VOLUME 5 | FEBRUARY 2004 | 101

NETWORK BIOLOGY: UNDERSTANDING THE CELL’S FUNCTIONAL ORGANIZATION Albert-László Barabási & Zoltán N. Oltvai NATURE REVIEWS | GENETICS VOLUME 5 | FEBRUARY 2004 | 101 Comparison of random, scale- free and hierarchical networks

Typical protein-protein interaction A protein binds with another or several other proteins in order to perform different biological functions---they are called protein complexes. protein-protein interaction

PROTEIN- PROTEIN INTERACTIONS by Catherine Royer Biophysics Textbook Online protein-protein interaction This complex transport oxygen from lungs to cells all over the body through blood circulation

PROTEIN- PROTEIN INTERACTIONS by Catherine Royer Biophysics Textbook Online protein-protein interaction

Usually protein-protein interaction data are produced by Laboratory experiments (Yeast two-hybrid, pull-down assay etc.) Network of interactions and complexes The results of the experiments are converted to binary interactions. The binary interactions can be represented as a network/graph where a node represents a protein and an edge represents an interaction. A D F B C E Bait protein Interacted protein A BFDEC Spoke approach C B D F E A Matrix approach detected complex data

AtpBAtpA AtpGAtpE AtpAAtpH AtpBAtpH AtpGAtpH AtpEAtpH List of interactions Corresponding network Adjacency matrix Network of interactions

The yeast protein interaction network evolves rapidly and contain few redundant duplicate genes by A. Wagner. Mol. Biology and Evolution proteins and 899 interactions S. Cerevisiae giant component consists of 466 proteins

The yeast protein interaction network evolves rapidly and contain few redundant duplicate genes by A. Wagner. Mol. Biol. Evol Average degree ~ 2 Clustering co- efficient = Degree distribution is scale free

An E. coli interaction network from DIP ( 300 proteins and 287 interactions E. coli Giant component 93 proteins Components of this graph has been determined by applying Depth First Search Algorithm There are total 62 components

An E. coli interaction network from DIP ( Average degree ~ Clustering co-efficient = 0.29 Degree distribution ~ scale free

Lethality and Centrality in protein networks by H. Jeong, S. P. Mason, A.-L. Barabasi, Z. N. Oltvai Nature, May proteins and 2240 interactions S. Cerevisiae Almost all proteins are connected Degree distribution is scale free

Average degree 5.42 Clustering co- efficient = 0.18 Giant component consists of 4385 proteins PPI network based on MIPS database consisting of 4546 proteins interactions

Degree distribution ~ scale free PPI network based on MIPS database consisting of 4546 proteins interactions

# of protein s # of Interac. Average degree Clusterin g Coeffi. Giant Compo. Degree Distribu ~ Exist 47.3% Power law Exist 31% Almost Power law ______ Exist ~100% Power law Exist ~96% Not exactly Power law A complete PPI network tends to be a connected graph And tends to have Power law distribution

We learnt 1.Properties of some complex network models 2.Properties of Protein-Protein Interaction Networks