A Real-life Application of Barabasi’s Scale-Free Power-Law Presentation for ENGS 112 Doug Madory Wed, 1 JUN 05 Fri, 27 MAY 05

Background Common property of many large networks is vertex connectivities follow a scale-free power- law distribution. Consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected.

So what? The objective of network theory is not network diagrams, but insight! Application of Barabasi’s theory to bioinformatics has produced several significant biological discoveries

Determining Roles of Proteins Within Metabolism Proteins are traditionally identified on the basis of their individual actions Modern research is trying to determine contextual or cellular function of proteins  Requires analysis of 1000’s of simultaneous protein-protein interactions – unworkable!  Must analyze as a complex network

Yeast proteome

Protein-Protein Interaction Map of protein-protein interactions forms a scale- free power-law network  Few highly-connected proteins play central role in mediating interactions among numerous, less connected proteins  Consequence is tolerance to random errors  Removal of highly-connected proteins rapidly increases network diameter computationally

Highly-Connected Proteins When highly-connected proteins are removed in order of connectivity, mortality of cell increases  Highly-connected proteins paramount to survival  93% of proteins have <5 links, 21% essential  0.7% of proteins have >15 links, 62% essential Conversely when proteins are removed at random, effect is negligible

More Characteristics of Highly-Connected Proteins Most hub proteins same across species 4% of all proteins were found in all organisms of experiment  These were also the most highly connected proteins Species-specific differences expressed in least connected proteins

Small-World in Organisms Connectivity characterized by network diameter  Shortest biochemical pathway averaged over all pairs of substrates For all known non-biological networks average node connectivity is fixed  Implies increased diameter as new nodes added  Therefore more complex organisms should have greater network diameters – but they don’t!!!

Conservation of Diameter All metabolic networks share same diameter! As organism complexity increases individual proteins are increasingly connected to maintain constant metabolic network diameter Larger diameter would attenuate organism’s ability to respond efficiently to external changes

Conservation of Diameter Minitab analysis of Barabasi’s data for diameter

Conservation of Gamma All metabolic networks share power-law  a. A. Fulgidus (archae)  b. E. coli (bacterium)  c. C. Elegans (eukaryote)  d. All 43 organisms (avg)  for all life about 2.2

Conservation of  Minitab analysis of Barabasi’s data for 

Conclusions Barabasi’s network theory offers insights into metabolic networks in cellular biology Correlation between connectivity and indispensability of a protein confirms that robustness against lethal mutations is derived from organization of protein interactions Metabolic networks within all living things have almost same diameter and 