Download - Network Wrap-Up
26. Lecture WS 2003/04
Bioinformatics III 1
Network Wrap-Up
No lecture Tuesday next week (10.2.)
Questions about lectures 1-12 on 12.2.
Questions about lectures 13-25 on 17.2.
Questions about assignments on 17.2.
today:- review network topologies (Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004))
- review some findings of previous lectures on networks
Jansen et al. Science 302, 449 (2003)
26. Lecture WS 2003/04
Bioinformatics III 2
Characterising metabolic networks
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
To study the network characteristics of the metabolism a graph theoretic description needs to
be established.
(a) Here, the graph theoretic description for a simple pathway (catalysed by Mg2+-dependant
enzymes) is illustrated.
(b) In the most abstract approach all interacting metabolites are considered equally. The
links between nodes represent reactions that interconvert one substrate into another. For
many biological applications it is useful to ignore co-factors, such as the high-energy-
phosphate donor ATP, which results
(c) in a second type of mapping that connects only the main source metabolites to the main
products.
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Bioinformatics III 3
Characterising metabolic networks
(d) The degree distribution, P(k) of the metabolic network illustrates its scale-free topology.
(e) The scaling of the clustering coefficient C(k) with the degree k illustrates the hierarchical
architecture of metabolism (The data shown in d and e represent an average over 43
organisms).
(f) The flux distribution in the central metabolism of Escherichia coli follows a power law,
which indicates that most reactions have small metabolic flux, whereas a few reactions, with
high fluxes, carry most of the metabolic activity. It should be noted that on all three plots the
axis is logarithmic and a straight line on such log–log plots indicates a power-law scaling.
CTP, cytidine triphosphate; GLC, aldo-hexose glucose; UDP, uridine diphosphate; UMP,
uridine monophosphate; UTP, uridine triphosphate.Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
26. Lecture WS 2003/04
Bioinformatics III 4
Yeast protein interaction network
A map of protein–protein interactions in
Saccharomyces cerevisiae, which is
based on early yeast two-hybrid
measurements, illustrates that a few
highly connected nodes (which are also
known as hubs) hold the network
together.
The largest cluster, which contains
78% of all proteins, is shown. The colour
of a node indicates the phenotypic effect
of removing the corresponding protein
(red = lethal, green = non-lethal, orange
= slow growth, yellow = unknown).
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
26. Lecture WS 2003/04
Bioinformatics III 5
Degree
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
The most elementary characteristic of a node is its
degree (or connectivity), k, which tells us how many links
the node has to other nodes. For example, in the
undirected network shown in part a of the figure, node A
has degree k = 5. In networks in which each link has a
selected direction (see figure, part b) there is an
incoming degree, kin, which denotes the number of links
that point to a node, and an outgoing degree, kout, which
denotes the number of links that start from it. For
example, node A in part b of the figure has kin = 4 and
kout = 1. An undirected network with N nodes and L links
is characterized by an average degree <k> = 2L/N
(where <> denotes the average).
26. Lecture WS 2003/04
Bioinformatics III 6
Degree distribution
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
The degree distribution, P(k), gives the probability that a selected node has exactly k links. P(k) is obtained by counting the number o f nodes N(k) with k = 1,2... links and dividing by the total number of nodes N. The degree distribution allows us to distinguish between different classes of networks. For example, a peaked degree distribution, as seen in a random network, indicates that the system has a characteristic degree and that there are no highly connected nodes (which are also known as hubs). By contrast, a power-law degree distribution indicates that a few hubs hold together numerous small nodes.
26. Lecture WS 2003/04
Bioinformatics III 7
Network measures
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
Scale-free networks and the degree exponent
Most biological networks are scale-free, which means that their
degree distribution approximates a power law, P(k) k- , where
is the degree exponent and ~ indicates 'proportional to'. The
value of determines many properties of the system. The
smaller the value of , the more important the role of the hubs
is in the network. Whereas for >3 the hubs are not relevant, for
2> >3 there is a hierarchy of hubs, with the most connected
hub being in contact with a small fraction of all nodes, and for
= 2 a hub-and-spoke network emerges, with the largest hub
being in contact with a large fraction of all nodes. In general, the
unusual properties of scale-free networks are valid only for <
3, when the dispersion of the P(k) distribution, which is defined
as 2 = <k2> - <k>2, increases with the number of nodes (that
is, diverges), resulting in a series of unexpected features,
such as a high degree of robustness against accidental node
failures. For >3, however, most unusual features are absent,
and in many respects the scale-free network behaves like a
random one.
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Bioinformatics III 8
Shortest path and mean path length
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
Distance in networks is measured with the path length, which
tells us how many links we need to pass through to travel
between two nodes. As there are many alternative paths
between two nodes, the shortest path — the path with the
smallest number of links between the selected nodes — has a
special role. In directed networks, the distance ℓAB from node A
to node B is often different from the distance ℓBA from B to A. For
example, in part b of the figure, ℓBA = 1, whereas ℓAB = 3. Often
there is no direct path between two nodes. As shown in part b of
the figure, although there is a path from C to A, there is no path
from A to C. The mean path length, <ℓ>, represents the average
over the shortest paths between all pairs of nodes and offers a
measure of a network's overall navigability.
26. Lecture WS 2003/04
Bioinformatics III 9
Clustering coefficient
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
In many networks, if node A is connected to B, and B is connected to C,
then it is highly probable that A also has a direct link to C. This
phenomenon can be quantified using the clustering coefficient33 CI =
2nI/k(k-1), where nI is the number of links connecting the kI neighbours of
node I to each other. In other words, CI gives the number of 'triangles'
that go through node I, whereas kI (kI -1)/2 is the total number of triangles
that could pass through node I, should all of node I's neighbours be
connected to each other. For example, only one pair of node A's five
neighbours in part a of the figure are linked together (B and C), which
gives nA = 1 and CA = 2/20. By contrast, none of node F's neighbours link
to each other, giving CF = 0. The average clustering coefficient, <C >,
characterizes the overall tendency of nodes to form clusters or groups.
An important measure of the network's structure is the function C(k),
which is defined as the average clustering coefficient of all nodes with k
links. For many real networks C(k) k-1, which is an indication of a
network's hierarchical character.
The average degree <k>, average path length <ℓ> and average
clustering coefficient <C> depend on the number of nodes and links (N
and L) in the network. By contrast, the P(k) and C(k ) functions are
independent of the network's size and they therefore capture a network's
generic features, which allows them to be used to classify various
networks.
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Bioinformatics III 10
Origin of scale-free topology and hubs in biological networks
The origin of the scale-free topology in complex networks can be reduced to two basic mechanisms: growth and preferential attachment. Growth means that the network emerges through the subsequent addition of new nodes, such as the new red node that is added to the network that is shown in part a . Preferential attachment means that new nodes prefer to link to more connected nodes. For example, the probability that the red node will connect to node 1 is twice as large as connecting to node 2, as the degree of node 1 (k1=4) is twice the degree of node 2 (k2 =2). Growth
and preferential attachment generate hubs through a 'rich-gets-richer' mechanism: the more connected a node is, the more likely it is that new nodes will link to it, which allows the highly connected nodes to acquire new links faster than their less connected peers. In protein interaction networks, scale-free topology seems to have its origin in gene duplication. Part b shows a small protein interaction network (blue) and the genes that encode the proteins (green). When cells divide, occasionally one or several genes are copied twice into the offspring's genome (illustrated by the green and red circles). This induces growth in the protein interaction network because now we have an extra gene that encodes a new protein (red circle). The new protein has the same structure as the old one, so they both interact with the same proteins. Ultimately, the proteins that interacted with the original duplicated protein will each gain a new interaction to the new protein. Therefore proteins with a large number of interactions tend to gain links more often, as it is more likely that they interact with the protein that has been duplicated. This is a mechanism that generates preferential attachment in cellular networks. Indeed, in the example that is shown in part b it does not matter which gene is duplicated, the most connected central protein (hub) gains one interaction. In contrast, the square, which has only one link, gains a new link only if the hub is duplicated.
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
26. Lecture WS 2003/04
Bioinformatics III 11
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
Aa
The Erdös–Rényi (ER) model of a random network starts with N nodes
and connects each pair of nodes with probability p, which creates a
graph with approximately pN (N-1)/2 randomly placed links.
Ab
The node degrees follow a Poisson distribution, which indicates that
most nodes have approximately the same number of links (close to the
average degree <k>). The tail (high k region) of the degree distribution
P(k ) decreases exponentially, which indicates that nodes that
significantly deviate from the average are extremely rare.
Ac
The clustering coefficient is independent of a node's degree, so C(k)
appears as a horizontal line if plotted as a function of k. The mean path
length is proportional to the logarithm of the network size, l log N, which
indicates that it is characterized by the small-world property.
Random networks
26. Lecture WS 2003/04
Bioinformatics III 12
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
Scale-free networks Scale-free networks are characterized by a power-law degree
distribution; the probability that a node has k links follows P(k) ~ k- ,
where is the degree exponent. The probability that a node is highly
connected is statistically more significant than in a random graph, the
network's properties often being determined by a relatively small number
of highly connected nodes that are known as hubs (see figure, part Ba;
blue nodes). In the Barabási–Albert model of a scale-free network, at
each time point a node with M links is added to the network, which
connects to an already existing node I with probability I = kI/JkJ,
where kI is the degree of node I and J is the index denoting the sum over
network nodes. The network that is generated by this growth process has
a power-law degree distribution that is characterized by the degree
exponent = 3.
Bb Such distributions are seen as a straight line on a log–log plot. The
network that is created by the Barabási–Albert model does not have an
inherent modularity, so C(k) is independent of k (Bc). Scale-free
networks with degree exponents 2< <3, a range that is observed in
most biological and non-biological networks, are ultra-small, with the
average path length following ℓ ~ log log N, which is significantly shorter
than log N that characterizes random small-world networks.
26. Lecture WS 2003/04
Bioinformatics III 13
Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)
Hierarchical networks To account for the coexistence of modularity, local clustering and scale-
free topology in many real systems it has to be assumed that clusters
combine in an iterative manner, generating a hierarchical network.
The starting point of this construction is a small cluster of four densely
linked nodes (see the four central nodes in Ca). 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 16 peripheral nodes connected to the central node of
the old module, which produces a new module of 64 nodes. 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 = 1 + ln4/ln3 =
2.26 (see Cb) and a large, system-size independent average clustering
coefficient <C> ~ 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 (see Cc). A hierarchical architecture implies that
sparsely connected nodes are part of highly clustered areas, with
communication between the different highly clustered neighbourhoods
being maintained by a few hubs (see Ca).
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Bioinformatics III 14
Reminder A few remarks on the past lectures ...
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Bioinformatics III 15
V14: Prediction of P-P interaction from correlated mutationsResults obtained by i2h in a set of 14 two domain proteins of known structure = proteins with two interacting domains. Treat the 2 domains as different proteins.
A: Interaction index for the 133 pairs with 11 or more sequences in common. The true positive hits are highlighted with filled squares.
B: Representation of i2h results, reminiscent of those obtained in the experimental yeast two-hybrid system. The diameter of the black circles is proportional to the interaction index; true pairs are highlighted with gray squares. Empty spaces correspond to those cases in which the i2h system could not be applied, because they contained <11 sequences from different species in common for the two domains.
In most cases, i2h scored the correct pair of protein domains above all other possible interactions.
Pazos, Valencia, Proteins 47, 219 (2002)
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Bioinformatics III 16
V14: Co-localization of interaction partners
Use localization data to assess the
quality of prediction because two
predicted interacting partners
sharing the same subcellular
location are more likely to form a
true interaction.
Comparison of colocalization index
(defined as the ratio of the number
of protein pairs in which both
partners have the same subcellular
localization to the number of
protein pairs where both partners
have any sub-cellular localization
annotation).
Lu, ..., Skolnick, Genome Res 13, 1146 (2003)
Multithreading predictions (MTA) are
less reliable than high-confidence inter-
actions, but score quite well amongst
predictions + HTS screens.
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Bioinformatics III 17
V14:Do partners have the same function?
Proteins from different groups of
biological functions may interact with
each other.
However, the degree to which interacting
proteins are annotated to the same
functional category is a measure of
quality for predicted interactions.
Here, the predictions cluster fairly well
along the diagonal.
Lu, ..., Skolnick, Genome Res 13, 1146 (2003)
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V15: Statistical significance of complexes and modules
Number of complete cliques (Q = 1) as
a function of clique size enumerated in
the network of protein interactions
(red) and in randomly rewired graphs
(blue, averaged >1,000 graphs where
number of interactions for each protein
is preserved).
Inset shows the same plot in log-
normal scale. Note the dramatic
enrichment in the number of cliques in
the protein-interaction graph
compared with the random graphs.
Most of these cliques are parts of
bigger complexes and modules.
Spirin, Mirny, PNAS 100, 12123 (2003)
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Bioinformatics III 19
V15: Architecture of protein network
Fragment of the protein network. Nodes
and interactions in discovered clusters
are shown in bold. Nodes are colored by
functional categories in MIPS:
red, transcription regulation;
blue, cell-cycle/cell-fate control;
green, RNA processing; and
yellow, protein transport.
Complexes shown are the SAGA/TFIID
complex (red), the anaphase-promoting
complex (blue), and the TRAPP complex
(yellow).
Spirin, Mirny, PNAS 100, 12123 (2003)
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Bioinformatics III 20
V15: Evolution of the yeast protein interaction network
Isotemporal categories are designed
through a binary (b) coding scheme.
The b code represents the
distribution of each yeast protein's
orthologs in the universal tree of life.
Bit value 1 indicates the presence of
at least one orthologous hit for a
yeast protein in a corresponding
group of genomes, and bit value 0
indicates the absence of any
orthologous hit. The presented
example is 110011 in the b format
and 51 in the d format. Orthologous
identifications are based on COGs at
NCBI and in von Mering et al. (2002).
Qin et al. PNAS 100, 12820 (2003)
Previously, phylogenetic profileswere used to detect proteininteraction partners.Here, use phylogenetic profiles to detect modules.
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Bioinformatics III 21
V15: Evolution of the yeast protein interaction network
Interaction patterns.
Z scores for all possible interactions
of the isotemporal categories in the
protein interaction network.
For categories i and j,
Zi,j = (Fi,jobs – Fi,j
mean)/i,j
where Fi,jobs is the observed number
of interactions, and Fi,jmean and i,j are
the average number of interactions
and the SD, respectively, in 10,000
MS02 null models.
Qin et al. PNAS 100, 12820 (2003)
The diagonal distribution of large positive Z scores indicates that yeast proteins
tend to interact with proteins from the same or closely related isotemperal
categories.
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V19: Flux balancingmass conservation.
Therefore one may analyze metabolic systems by requiring mass conservation.
Only required: knowledge about stoichiometry of metabolic pathways and
metabolic demands
For each metabolite:
Under steady-state conditions, the mass balance constraints in a metabolic
network can be represented mathematically by the matrix equation:
S · v = 0
where the matrix S is the m n stoichiometric matrix,
m = the number of metabolites and n = the number of reactions in the network.
The vector v represents all fluxes in the metabolic network, including the internal
fluxes, transport fluxes and the growth flux.
)( dtransporteuseddegradeddsynthesizei
i VVVVdt
dXv
Any chemical reaction requires
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V19: Flux balance analysis
Since the number of metabolites is generally smaller than the number of reactions
(m < n) the flux-balance equation is typically underdetermined.
Therefore there are generally multiple feasible flux distributions that satisfy the mass
balance constraints.
The set of solutions are confined to the nullspace of matrix S.
To find the „true“ biological flux in cells ( e.g. Heinzle, Huber, UdS) one needs
additional (experimental) information,
or one may impose constraints
on the magnitude of each individual metabolic flux.
The intersection of the nullspace and the region defined by those linear inequalities
defines a region in flux space = the feasible set of fluxes.
iii v
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V19: Rerouting of metabolic fluxes
(Black) Flux distribution for the wild-type.
(Red) zwf- mutant. Biomass yield is 99% of
wild-type result.
(Blue) zwf- pnt- mutant. Biomass yield is 92% of
wildtype result. The solid lines represent
enzymes that are being used, with the
corresponding flux value noted.
Note how E.coli in silico circumvents removal of
one critical reaction (red arrow) by increasing
the flux through the alternative G6P P6P
reaction.
Edwards & Palsson PNAS 97, 5528 (2000)
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V20: Extreme Pathwaysintroduced into metabolic analysis by the lab of Bernard Palsson
(Dept. of Bioengineering, UC San Diego). The publications of this lab
are available at http://gcrg.ucsd.edu/publications/index.html
Extreme pathway
technique is based
on the stoichiometric
matrix representation
of metabolic networks.
All external fluxes are
defined as pointing outwards.
Schilling, Letscher, Palsson,
J. theor. Biol. 203, 229 (2000)
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V20: Feasible solution set for a metabolic reaction network
(A) The steady-state operation of the metabolic network is restricted to the region
within a cone, defined as the feasible set. The feasible set contains all flux vectors
that satisfy the physicochemical constrains. Thus, the feasible set defines the
capabilities of the metabolic network. All feasible metabolic flux distributions lie
within the feasible set, and
(B) in the limiting case, where all constraints on the metabolic network are known,
such as the enzyme kinetics and gene regulation, the feasible set may be reduced
to a single point. This single point must lie within the feasible set.
Edwards & Palsson PNAS 97, 5528 (2000)
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V21: Reconfigured Network
Klamt & Stelling Trends Biotech 21, 64 (2003)
A C P
B
D
A(ext) B(ext) C(ext)R1 R2 R3
R5
R4 R8
R9
R6
R7bR7f
3 EFMs are not systemically independent:EFM1 = EP4 + EP5EFM2 = EP3 + EP5EFM4 = EP2 + EP3
26. Lecture WS 2003/04
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V21: relation between EP and EFMsWhat is the consequence of when all exchange fluxes (and hence all
reactions in the network) are irreversible?
Klamt & Stelling Trends Biotech 21, 64 (2003)
EFMs and EPs always co-incide!
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V22: Correlation between genes of same metabolic pathway
Ihmels, Levy, Barkai, Nat. Biotech 22, 86 (2004)
Genes of the glycolysis pathway
(according KEGG) were clustered
and ordered based on the correlation
in their expression profiles.
Shown here is the matrix of their
pair-wise correlations.
The cluster of highly correlated
genes (orange frame) corresponds
to genes that encode the central
glycolysis enzymes.
The linear arrangement of these
genes along the pathway is shown at
right.
Of the 46 genes assigned to the
glycolysis pathway in the KEGG
database, only 24 show a correlated
expression pattern.
In general, the coregulated genes
belong to the central pieces of
pathways.
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The connectivity of a given metabolite
is defined as the number of reactions
connecting it to other metabolites.
Shown are the distributions of
connectivity between metabolites in an
unrestricted network () and in a
network where only correlated
reactions are considered ().
In accordance with previous results
(Jeong et al. 2000) , the connectivity
distribution between metabolites
follows a power law (log-log plot).
Ihmels, Levy, Barkai, Nat. Biotech 22, 86 (2004)
V22: Connectivity of metabolites
In contrast, when coexpression is
used as a criterion to distinguish
functional links, the connectivity
distribution becomes exponential
(log-linear plot).
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V22: Co-expression of transporters
Transporter genes are
co-expressed with the relevant
metabolic pathways providing
the pathways with its metabolites.
Co-expression is marked in green.
Ihmels, Levy, Barkai, Nat. Biotech 22, 86 (2004)
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V25: Combination of data sets into probabilistic interactomes
(B) Combination of data sets into
probabilistic interactomes.
The 4 interaction data sets
from HT experiments were
combined into 1 PIE.
The PIE represents a
transformation of the
individual binary-valued
interaction sets into a data
set where every protein pair
is weighed according to the
likelihood that it exists in a
complex. A „naïve” Bayesian network is used to model
the PIP data. These information sets hardly
overlap.
Jansen et al. Science 302, 449 (2003)
Because the 4 experimental
interaction data sets contain
correlated evidence, a fully
connected Bayesian network
is used.
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V25: Static naive Bayesian Networks
The likelihood ratio L defined as
relates prior and posterior odds according to Bayes‘ rule:
negffP
posffPffL
N
NN ...
......
1
11
priorNpost OffLO ...1
In the special case that the N features are conditionally independent
(i.e. they provide uncorrelated evidence) the Bayesian network is a so-called
„naïve” network, and L can be simplified to:
N
i
N
i i
iiN negfP
posfPfLffL
1 11...
Jansen et al. Science 302, 449 (2003)
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V25: PIP vs. the information sources
Ratio of true to false positives (TP/FP) increases
monotonically with Lcut, confirming L as an
appropriate measure of the odds of a real
interaction.
The ratio is computed as:
Protein pairs with Lcut > 600 have a > 50%
chance of being in the same complex.Jansen et al. Science 302, 449 (2003)
cut
cut
LL
LL
cut
cut
Lneg
Lpos
LFP
LTP
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V25: PIE vs. the information sources
9897 interactions are predicted from PIP and
163 from PIE.
In contrast, likelihood ratios derived from single
genomic factors (e.g. mRNA coexpression) or
from individual interaction experiments (e.g. the
Ho data set) did no exceed the cutoff when used
alone.
This demonstrates that information sources that,
taken alone, are only weak predictors of
interactions can yield reliable predictions when
combined.
Jansen et al. Science 302, 449 (2003)