Fault Tolerance in Protein Interaction Networks:
Stable Bipartite Subgraphs and Redundant Pathways
Lenore Cowen
Tufts University
Protein-protein interaction
Protein-protein interaction
PPI: A simple graph model
vertices ↔ genes/proteins
edges ↔ physical interactions
simplifications:
• undirected
• loses temporal information
• difficult to decompose into separate
processes
• conflates different PPI types into one
class of "physical interactions"
Current data
• High-throughput methods are allowing us to
fill in many edges in our simple model, often
between unannotated proteins.
What we want:
What we have:
Question: Can we
infer anything about
"real" pathways from
the low-resolution
graph model of
pairwise interactions?
Interaction types
• We distinguish here between two types of
interaction:
– physical interactions
– genetic interactions
Genetic interactions (epistasis)
Only 18% of yeast genes are essential (the yeast
dies when they’re removed).
essential gene.
yeast.
yeast dies.
gene deleted.
Genetic interaction:
synthetic lethality
Some pairs of nonessential genes exhibit
interesting correlative relationships.
nonessential gene.
nonessential gene.
yeast lives.
dies.
both genes deleted at once.
gene deleted.
gene deleted.
Nonessential Genes
– Some genes are non-essential because they are
only required under certain conditions (i.e. an
enzyme to metabolize a particular nutrient).
– Other genes are non-essential because the
network has some built-in redundancy.
• One gene (completely or partially) compensates for the
loss of another.
• One functional pathway (completely or partially)
compensates for the loss of another.
Redundant pathways
and synthetic lethality
Kelley and Ideker (2005):
Between-Pathway Model (BPM)
In reality, the data are very incomplete:
Between-Pathway Model (BPM)
Kelley and Ideker (2005)
and Ulitsky and Shamir (2007)
• Goal: detect putative BPMs in yeast interactome
• Method:
1) find densely-connected subsets of the physical
protein-protein interaction (PI) network
(putative pathways)
2) check the genetic interaction (GI) network to see if
patterns in density of genetic interactions correlate
with these putative pathways
3) check resulting structures for overrepresentation of
biological function (gene set enrichment)
Kelley and Ideker (2005)
and Ulitsky and Shamir (2007)
(1)
(2)
enriched for function X
enriched for function Y
(3)
Kelley and Ideker (2005)
and Ulitsky and Shamir (2007)
• Problems:
– Sparse data limits the potential scope of discovery
– independent validation is difficult
Our method
• We show how to systematically search for
stable bipartite subgraphs (putative BPMs)
• We use only synthetic lethality interactions to
search for BPMs:
– allows the use of PIs for independent statistical
validation of putative BPMs
– scope of potential discovery is greater than when
using PIs as seed structures
How should we look for bipartite
subgraphs?
Maximum bipartition
• Definition: Given any graph G, a maximum
bipartition of G is an assignment of each node
of G to one of two sets, A and B, in such a way
that the number of edges that CROSS the
partition is maximized.
Maximum bipartition
• Definition: Given any graph G, a maximum
bipartition of G is an assignment of each node
of G to one of two sets, A and B, in such a way
that the number of edges that CROSS the
partition is maximized.
• Fact: Maximum bipartition is NP-hard.
We don’t want a maximum bipartition
anyway!
We don’t want to force a choice of sides!
Maximal bipartition
• Definition: Given any graph G, a maximal
bipartition of G is an assignment of each node
of G to one of two sets, in such a way that
moving any single node from one set to the
other does not increase the number of edges
of G which cross between the two sets.
Maximal bipartition
Algorithm
• Randomly assign a set-label to each node in G.
• Call a node v “happy” if at least half of its
neighbors are in the opposite set from v, and
“unhappy” otherwise.
• While there exists an unhappy node:
– Pick one such node at random.
– Flip its set label.
Algorithm
(an “unhappy” node flips to “happy.”)
Algorithm
Claim: This procedure terminates in at most |E|
steps, where |E| is the number of edges in G.
Proof: While a particular node may switch its
affiliation many times over the course of the
algorithm, notice that each time a flip is
performed, the number of edges crossing
between the two partitions increases by at
least one. So there can be at most |E| steps.
Algorithm
Claim: On termination, every node is “happy.”
Proof: [This is just the termination condition of
the while-loop.]
Observe that the partition generated in this way
is maximal: flipping any single node cannot
increase the number of edges crossing
between partitions, because all nodes are
happy.
Stable Bipartite Subgraph: Motivation
If a gene exists within a BPM, then we expect the two
pathways of the BPM to fall into opposite sets within
most maximal partitions (because the partitioning
algorithm is looking to maximize the number of edges
crossing between sets).
So in a maximal partition,
genes in the same pathway as a BPM gene g should
tend to be assigned to the same set as g;
those in the opposite pathway should wind up in the
opposite set;
and those in neither pathway should bounce around
with little or no correlation to g’s set-assignment.
Stable Bipartite Subgraph
Definition: For a node m, repeat this procedure
k times to find maximal bipartite subgraphs.
Let A be the set of nodes that occur in the
same partition as m at least r percent of the
time. Let B be the set of nodes that occur in
the opposite partition of m at least r percent
of the time. Return A and B as m’s stable
bipartite subgraph.
Stable Bipartite Subgraph
Definition: For a node m, repeat this procedure
k times to find maximal bipartite subgraphs.
Let A be the set of nodes that occur in the
same partition as m at least r percent of the
time. Let B be the set of nodes that occur in
the opposite partition of m at least r percent
of the time. Return A and B as m’s stable
bipartite subgraph.
The stable bipartite subgraphs are our BPMs!
(k=250; r= 70 percent)
Test Datasets
• original physical + genetic interaction data used in
Kelley + Ideker (2005)
682 genes (nodes)
1,858 edges (SL interactions)
• up-to-date set of physical + genetic interactions
taken from BioGRID database (October 2007)
1,678 genes (nodes)
6,818 edges (SL interactions)
Return Stable BPMs?
Example BPM
How do we know it is meaningful?
Biological validation: Enrichment results. We
find things that are known to be functionally
related in our putative pathways. [GO
Enrichment]
Statistical validation:
- Location of known PI edges
- Prediction of new SL edges
Results
Network
BPMs
SL edges
covered
%Enrich.
pathways
Kelley&
Ideker
G
360
687
251/720
34.9%
Our
Results
G
602
1,526
643/1204
53.4%
Ulitsky&
ShamirA
G’
140
<3,765
100/280
35.7%
Ulitsky&
ShamirB
G’
270
<3,765
177/540
32.8%
Our
Results
G*
1,510
4,949
1528/3020
50.6%
Results
SGD GO-SLIM coverage
Ulitsky + Shamir
Us
46.3%
79.8%
Results: Dually-enriched BPMs
Results: Differentially-enriched BPMs
Example BPM
Example BPM
Website
http://bcb.cs.tufts.edu/.yeast.bpm/
Website
http://bcb.cs.tufts.edu/.yeast.bpm/
Website
http://bcb.cs.tufts.edu/.yeast.bpm/
Results: BPM Validation
In addition to validation based on coherence of
biological function, we can also statisticially
validate our methods directly from the
structure of the network!
Method 1: Examine the distribution of known PIs
within each BPM.
Results: BPM Validation
Goal: estimate the probability of seeing as many
or fewer physical interactions between the
two sets as were actually observed.
Results: BPM Validation
Results: BPM Validation
Method 2: Examine the distribution of new SL
interactions appearing within each BPM in the
Kelley/Ideker network.
Results: BPM Validation
Goal: estimate the probability of seeing as many
or more new synthetic-lethality interactions
appearing between the two sets as were
actually observed.
Results: BPM Validation
• Results: Across the set of 175 candidate BPMs
from G which contained at least 20 new SL edges
in G+, the average probability that the observed
between-pathway bias would occur by chance
was 0.017.
• Since these new edges were not used to
construct candidate BPMs in G, their distribution
bias provides independent support for the
hypothesis that stable subgraphs do indeed
correspond to biologically meaningful structures.
Validation: Microarray Data
• Rosetta compendium (Hughes et al, 2000):
-- contains yeast expression profiles of 276
deletion mutants:
i.e. for each gene in the yeast genome,
measures how its expression levels change
when particular gene g is deleted, as
compared to wildtype yeast.
Delete a gene in pathway 1; see if
changes in pathway 2 coherent
BPM
Deleted Gene
Pathway restriction
log10 ratio
Sort
At step i: N to 1
Calculate weighted percent of
genes in pathway seen so far
and precent of genes not in
pathway:
Score is max difference
How to validate a pathway
Using a permutation test we sample 99
random subsets of genes the same size as the
pathway
We calculate the cluster rank score for each of
these 99 sets
We sort the test plus the pathway score
The p-value is the percentile
A pathway is validated if its p-value is <=0.1
Delete a gene in pathway 1; see if
changes in pathway 2 coherent
We call a pathway “Validated” if its Cluster Rank Score has p-value < .1
Kelley-Ideker Histogram of the Lowest CRS per Pathway per BPM
This histogram displays all the CRS scores from all of the results from Kelley and Ideker’s
BPMs bucketed according to their lowest p value score. The p value scores <= 0.10
indicate a validated BPM.
Ulitskyi Histogram of the Lowest CRS per Pathway per BPM
This histogram displays all the CRS scores from all of the results from Ulitskyi’s BPMs
bucketed according to their lowest p value score. The p value scores <= 0.10 indicate a
validated BPM.
Ma Histogram of the Lowest CRS per Pathway per BPM
This histogram displays all the CRS scores from all of the results from Ma’s BPMs
bucketed according to their lowest p value score. The p value scores <= 0.10 indicate a
validated BPM.
Brady Histogram of the Lowest CRS per BPM
This histogram displays all the CRS scores from all of the results from Brady’s BPMs
bucketed according to their lowest p value score. The p value scores <= 0.10 indicate a
validated BPM. Clearly, Brady’s BPMs are disproportionately represented in the lower p
value range.
Results
BPM dataset # paths hit
knockouts
Kelley-Ideker 160
(05)
Ulitsky36
Shamir (07)
Ma et al.
54
(08)
Our results 959
# validated
pathways
16
% validated
pathways
10%
5
14%
6
11%
230
24%
A Tantalizing Peek of What We can Do
With More Data!
• A heat map of the
differential expression
of yeast genes in
pathway 2 in response
to the deletion of two
different genes (SHE4
and GAS1) from
pathway 1 in a validated
BPM of Ma et al.
A random-gene validation test couples
the two pathways together
Co-authors and collaborators
•
•
•
•
•
•
Arthur Brady
Noah Daniels
Ben Hescott
Max Leiserson
Kyle Maxwell
Donna Slonim
thanks.
A Graph Theory Problem
• Our algorithm samples from the maximal
bipartite subgraphs. With what distribution? Is
it uniform? Proportional to the number of
edges that cross the cut?? ???
• What are the properties of the stable bipartite
subgraphs of the synthetic lethal network?
Are they conserved across species?
Approach
• Run the partitioning algorithm 250 times on
the yeast SL network (G).
• For each gene g in G,
– Construct a set A consisting of g and all nodes in G
which wind up in the same set as g at least 70% of
the time.
– Construct another set B consisting of all nodes in
G which wind up in the opposite set from g at
least 70% of the time.
• We call the subgraph of G defined by A and B
the “stable bipartite subgraph of g”, and
designate it as a candidate BPM.