How Scale-Free are Biological Networks.

1. How Scale-Free are Biological Networks. Raya Khanin and Ernst Wit Department of Statistics University of Glasgow

2. Biological Networks Protein interaction network: proteins that are (or might be) connected by physical interactions Metabolic network: metabolic products and substrates that participate in one reaction Gene regulatory network: two genes are connected if the expression of one gene modulates expression of another one by either activation or inhibition

3. Biological networks • Node represents a gene, protein or metabolite • Edge represents an association, interaction, co-expression • Directed edge stands for the modulation (regulation) of one node by another: e.g. arrow from gene X to gene Y means gene X affects expression of gene Y

4. Network measures Degree (or connectivity) of a node, k, is the number of links (edges) this node has. The degree distribution, P(k), is the probability that a selected node has exactly k links. Networks are classified by their degree distributions. (Barabasi and Oltvai, Nature, 2004)

5. Random Network • A fixed number of nodes are connected randomly to each other: start with N nodes and connect each pair with probability p creating a graph with pN(N-1)/2 randomly placed links. • The degrees follow a Poisson distribution: most nodes have roughly the same number of links, approximately equal to the network’s average degree, <k>; nodes that have significantly more or less links than <k> are very rare.

6. Scale-Free Network • Scale-free networks have a few nodes with a very large number of links (hubs) and many nodes with with only a few links. • It indicates the absence of a typical node in the network • Scale-free networks are characterized by a power-law distribution:

7. Comparing Random and Scale-free distribution • In the random network, the five nodes with the most links (in red) are connected to only 27% of all nodes (green). In the scale-free network, the five most connected nodes (red) are connected to 60% of all nodes (green) (source: Nature)

8. Examples of scale-free networks Network of citations between scientific papers Network of collaborations, movie actors Electrical power grids, airline traffic routes, railway networks World Wide Web and other communication systems Biological Networks "These laws, applying equally well to the cell and the ecosystem, demonstrate how unavoidable nature's laws are and how deeply self-organization shapes the world around us.“(A.Barabasi, 2002).

9. Google: scale-free networks “Scale-free networks are everywhere. They can be seen in … terrorist networks. What this means for counter-terrorism experts?” Global guerrillas (network organization, infrastructure disruption, and the emerging marketplace of violence) http://globalguerrillas.typepad.com/globalguerrillas/2004/05/scalefree_terro.html Scale|free : http://www.scalefree.info (Social Networking, Communities of Practice and Knowledge Management: publish articles such as “love and knowledge”, “what lovers tell us about persuasion”, “how ‘social contagion’ affects consumer behaviour)

10. Scale-free Networks Properties of scale-free systems are invariant to changes in scale. The ratio of two connectivities in a scale-free network is invariant under rescaling: This implies that scale-free networks are self-similar, i.e. any part of the network is statistically similar to the whole network and parameters are assumed to be independent of the system size.

11. Scale-free Networks Scale-free (self-similarity) properties of a common cauliflower plant: it is virtually impossible to determine whether one is looking at a photograph of a complete vegetable or its part, unless an additional scale-dependent object (a match) is added. (A) Complete vegetable; (B) a small segment of the same vegetable; (C) small part of the segment shown in B. The same match was used in all three photographs to provide a sense of scale for an otherwise scale-free structure (Gomez et al, 2001). Note: vegetable was purchased in Sloan supermarket in Manhattan’s Upper West Side.

12. Biological networks are reported to be scale-free Metabolic networks Protein interaction networks Protein domain networks Gene co-expression networks Distribution of gene expression and spot intensities on microarrays Frequency of occurrence of generalized parts in genomes of different organisms

13. Example of gene network • To find indication of a power-law the data is usually graphically fitted by a straight line on a log-log scale • Log contract data • Fit to a few points is not particularly good Number of regulated genes per regulating protein: Guelzim et al, Nature, 2002

14. Estimating the power exponent by maximum-likelihood • the power-law distribution • Observed values: xi is a connectivity of node i • the likelihood function for N observed connectivities • the log-likelihood is maximized by finding zeros of its derivative using the Newton-Raphston method.

15. Goodness-of-fit testing • E(k) - expected(k) with estimated ; O(k) – observed(k) • Consider a chi-squared statistic, (approximately) under H0: network is scale-free • Pool for connectivity values over k*, for which the expected number of connections is less than 5. As a result, the chi-squared statistic is approximately chi-squared distributed with k*-2 degrees of freedom, if the data come truly from a power-law distribution.

16. Goodness-of-fit testing • The p-value for each of the networks can be calculated by the exceedence probability of a chi-squared distribution: t is calculated (observed) values of T with estimated p-value is the probability a network has such connectivities if they were drawn from the power-law distribution. • If p<0.05, H0 (network is scale-free) is rejected.

17. Exponents of the power-law and scale-free p-values Datasets p-value Uetz 25 2.05 0.00004 Schwikowski 26 1.865 0 Ito 27 2.02 0 Li 28 2.3 0 Rain 29 2.1 0 Giot 21 1.53 0 Tong 9 1.44 0 Lee 20 1.99 0 Guelzim 17 1.49 0 Spellman/Cho 1.27(1.06) 0

18. Truncated power-law is the cut-off, s. t. the number of connections is less than expected for pure scale-free networks for and the behaviour is approximately scale-free within the range

19. Parameters of the truncated power-law and truncated power-law p-values Datasets p-value Uetz 25 1.6 8.67 0.37 Schwikowski 26 1.26 6.2 0.105 Ito 27 1.79 26 0 Li 28 2.1 19.5 0.0178 Rain 1.12 11.5 0.2 Giot 21 1.09 20 0.0013 Tong9 0.96 23.7 0 Lee 20 1.96 294 0 Guelzim17 1.18 15 0.00001 Spellman/Cho 1.07(0.78) 73(99) 0.7(0.1)

20. Scale-free or not:why is it important? • Network architecture • Evolutionary models • Scalability (self-similarity) criterion does not hold; one has to exercise caution while applying the features of an already studied part of the network to investigate properties of the unknown part of the same network, or of other networks that seem similar

21. Example of gene network • Guelzim and co-authors (Nature, 2002) conclude that connectivity distributions of gene transcriptional network for yeast and E-coli are scale-free, and thus “bacterial and fungal genetic networks are free of characteristic scale with respect to the distributions of both regulating and departing connections”. Number of regulated genes per regulating protein We found, however, that the scale-free property does not even hold and therefore such biological conclusions are invalid

22. Qualitative properties of biological networks • Existence of hubs • Many nodes with a few connections • “Lethality and centrality” (the more essential gene/protein is the more connections it has) • Small-world property (short average path) • High clustering of nodes

23. Alternative non-scale free distributions • generalized truncated power-law • stretched exponential distribution • geometric random graph: a geometric graph with n independently and uniformly distributed points in a metric space