How ScaleFree 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

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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

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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)

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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 networks average
  degree, nodes that have significantly more
  or less links than are very rare.

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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

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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)

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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).

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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/globalguerrill
  as/2004/05/scalefree_terro.html
  
• Scalefree 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)

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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.
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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 Manhattans Upper West Side.

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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

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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

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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.

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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.

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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 prejected.

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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
  

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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
  

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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)
  
•  

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Scale-free or notwhy 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

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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

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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

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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