A Real-life Application of

Barabasi’s Scale-Free

Power-LawPresentation 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 proteinsRequires 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

3.53.43.33.23.13.0

Median

Mean

3.323.303.283.263.243.223.20

Anderson-Darling Normality Test

Variance 0.0151Skewness 0.104338Kurtosis -0.311452N 43

Minimum 3.0000

A-Squared

1st Quartile 3.2000Median 3.30003rd Quartile 3.4000Maximum 3.5000

95% Confidence I nterval for Mean

3.2528

1.48

3.3286

95% Confidence I nterval for Median

3.2000 3.3000

95% Confidence I nterval for StDev

0.1015 0.1564

P-Value < 0.005

Mean 3.2907StDev 0.1231

95% Confidence I ntervals

Summary 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

2.42.32.22.12.0

Median

Mean

2.202.192.182.172.16

Anderson-Darling Normality Test

Variance 0.0080Skewness 0.024458Kurtosis 0.234872N 86

Minimum 2.0000

A-Squared

1st Quartile 2.1000Median 2.20003rd Quartile 2.2000Maximum 2.4000

95% Confidence Interval for Mean

2.1646

4.99

2.2029

95% Confidence Interval for Median

2.2000 2.2000

95% Confidence Interval for StDev

0.0776 0.1050

P-Value < 0.005

Mean 2.1837StDev 0.0893

95% Confidence I ntervals

Summary for g

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