Conceived and designed the experiments: LR MR BA MS. Performed the experiments: LR MR. Analyzed the data: LR MR MS. Contributed reagents/materials/analysis tools: BA. Wrote the paper: LR MR MS.
The authors have declared that no competing interests exist.
Networks play a crucial role in computational biology, yet their analysis and representation is still an open problem. Power Graph Analysis is a lossless transformation of biological networks into a compact, less redundant representation, exploiting the abundance of cliques and bicliques as elementary topological motifs. We demonstrate with five examples the advantages of Power Graph Analysis. Investigating protein-protein interaction networks, we show how the catalytic subunits of the casein kinase II complex are distinguishable from the regulatory subunits, how interaction profiles and sequence phylogeny of SH3 domains correlate, and how false positive interactions among high-throughput interactions are spotted. Additionally, we demonstrate the generality of Power Graph Analysis by applying it to two other types of networks. We show how power graphs induce a clustering of both transcription factors and target genes in bipartite transcription networks, and how the erosion of a phosphatase domain in type 22 non-receptor tyrosine phosphatases is detected. We apply Power Graph Analysis to high-throughput protein interaction networks and show that up to 85% (56% on average) of the information is redundant. Experimental networks are more compressible than rewired ones of same degree distribution, indicating that experimental networks are rich in cliques and bicliques. Power Graphs are a novel representation of networks, which reduces network complexity by explicitly representing re-occurring network motifs. Power Graphs compress up to 85% of the edges in protein interaction networks and are applicable to all types of networks such as protein interactions, regulatory networks, or homology networks.
Networks play a crucial role in biology and are often used as a way to represent experimental results. Yet, their analysis and representation is still an open problem. Recent experimental and computational progress yields networks of increased size and complexity. There are, for example, small- and large-scale interaction networks, regulatory networks, genetic networks, protein-ligand interaction networks, and homology networks analyzed and published regularly. A common way to access the information in a network is though direct visualization, but this fails as it often just results in “fur balls” from which little insight can be gathered. On the other hand, clustering techniques manage to avoid the problems caused by the large number of nodes and even larger number of edges by coarse-graining the networks and thus abstracting details. But these also fail, since, in fact, much of the biology lies in the details. This work presents a novel methodology for analyzing and representing networks. Power Graphs are a lossless representation of networks, which reduces network complexity by explicitly representing re-occurring network motifs. Moreover, power graphs can be clearly visualized: they compress up to 90% of the edges in biological networks and are applicable to all types of networks such as protein interaction, regulatory networks, or homology networks.
In recent years, novel high-throughput methods, such as yeast two-hybrid assays
In the case of protein interaction networks, their topology has been explored through the clustering of proteins into groups that share the same biological function, are similarly localized in the cell, or are part of a complex. To this end, several algorithms have been developed, such as socio-affinity clustering
How does the underlying biology manifest itself in protein interaction networks?
Stars often occur because of hub proteins or when affinity purification complexes are interpreted using the spoke model. Bicliques often arise because of domain-domain or domain-motif interactions inducing protein interactions
The abundance of stars, cliques, and bicliques suggests that modeling protein interaction networks as a collection of binary interactions is an obstacle toward a detailed analysis of the wealth of information contained in high-throughput networks. These networks have many edges that redundantly diffuse the information instead of highlighting it. In this study we introduce a new network representation and analysis paradigm that not only groups proteins into biologically relevant modules but also conveys in all detail–without loss of information–and with fewer symbols, the subtle connection patterns within and between groups of proteins.
As
Some recent large-scale experiments
This point is crucial for the interpretation of results from pull-down assays where whole complexes are identified rather than binary interactions
The problem with this perspective is that the spoke model underestimates, and the matrix model overestimates the number of true physical interactions between the members of a complex. For both models the use of binary interactions does not convey succinctly an otherwise simple connection pattern. Let
A recent survey of the yeast proteome investigated the modularity of the yeast cell machinery
Two catalytic alpha subunits (CKA1, CKA2) and two regulatory beta subunits (CKB1, CKB2) interacting with the FACT complex, with sub-complex NIP1-RPG-PRT1, and with the PAF1 complex. The graph representation (A) consists of 80 edges whereas the power graph representation (B) has 30 power edges, thus an edge reduction of 62%. This simplification of the representation makes the separation of the regulatory subunits from the catalytic subunits immediately apparent without loss of information on individual interactions.
Other complexes are visible in the power graph representation. For example, the proteins POB3 and SPT16 are grouped together in one power node. They form a complex known as the heterodimeric FACT complex SPT16/POB3, a complex involved in the transcription elongation on chromatin templates. It is known that the casein kinase II complex activates the FACT complex
Overall we see that the power graph representation manages to give an insightful picture of the underlying biology. It should be stressed that these representations are obtained without the addition of biological background knowledge but instead based on the network topology alone. Power Graphs thus provide useful hints into the existence of complexes, their internal organization, and their relationships.
Importantly, the power graph representation is a lossless representation, meaning that all and only interactions from the original network are represented faithfully, which is usually not the case for most clustering methods.
Similarly to the survey of the yeast proteome by Gavin et al.
(A) Standard graph representation. (B) power graph representation. The ORC complex is visible with a power node of proteins–ORC1/ORC4/ORC5–carrying a nucleotide binding P-loop domain [SCOP:52540]. Histones subtypes HTA1/2, HTB1/2, HHT1/2, and HHF1/2 share the same color. Histones HTA2, HTB2 and HHF1 are segregated from their twin subtypes HTA1, HTB1 and HHF2. The FACT complex SPT16/POB3 is again delineated.
Interacting with histones is the ORC Complex (Origin Recognition Complex) responsible for marking origin regions prior to DNA replication. On
Surprisingly, histones HTA2, HTB2 and HHF1 are segregated from their twin subtypes HTA1, HTB1 and HHF2, as subunits ORC2 and ORC6 interact with HTA2, HTB2 and HHF1 and not with the HTA1, HTB1, and HHF2. This is contradictory to the identity/near identity of these pairs of histones. The power graphs shows the separation between these two types of histones.
Why have these mostly identical proteins different interaction partners? In the case of H2A histones, each subtype has been shown to be sufficient for cell viability, and no clear functional difference were reported apart from homozygous strains for hta1− exhibiting a slower growth
Yet, this hypothesis does not explain that co-regulated HTA2 and HTB2 are both seen interacting with ORC2 and ORC6, whereas the differently co-regulated HTA1 and HTB1 do not
In reference
(A) Protein interaction network showing the 105 interaction partners of the SH3 domain carrying proteins: SHO1, ABP1, MYO5, BOI1, BOI2, RVS167, YHR016C and YFR024. The underlying network consists of 182 interactions represented here as 36 power edges–a reduction of 80%–leaving all but only the core information. Class 1 motif (RxxPxxP) proteins are shown in black. Class 2 motif (PxxPxR) proteins are shown in light grey
We investigated how the interaction profiles of these eight SH3 carrying proteins relate to the domain sequences.
The pair of SH3-carrying proteins YHR016C/YFR024 that are grouped in one power node in
As we have seen previously on specific examples, power graph analysis can help disentangle complex protein interaction networks. A quantitative analysis requires the definition of measures. Here we introduce the edge reduction measure:
From a visual complexity standpoint, trading edges for a hierarchy of sets of nodes is advantageous since the edges of a clique or biclique necessarily cross in two dimensions, whereas the circles delineating power nodes–by definition–do not.
The edge reduction of the rewired networks is represented using a a box-plot. 50% of edge reduction values are inside the box. Most networks exhibit a significant deviation from the null model as indicated by high z-scores between 2.2 and 242.
Protein Interaction Network | # Nodes | # Edges | Avg. Degree | e.r. | c.r |
Lim et al. (2006) | 571 | 701 | 2.45 | 85% | 12.1 |
Hazbun et al. (2003) | 2243 | 3130 | 2.79 | 79% | 13 |
Kim et al. (2006) | 577 | 1090 | 3.78 | 67% | 4.1 |
Gunsalus et al. (2004) | 281 | 514 | 3.6 | 65% | 4.6 |
Gavin et al. (2006) | 1462 | 6942 | 9.4 | 64% | 7.2 |
Ewing et al. (2007) | 2294 | 6449 | 5.62 | 54% | 6.6 |
Ito et al. (2001) | 3243 | 4367 | 2.69 | 53% | 5.3 |
Rual et al. (2005) | 1527 | 2529 | 3.31 | 50% | 4.5 |
Krogan et al. (2006) | 2708 | 7123 | 5.26 | 49% | 4.5 |
Stanyon et al. (2004) | 478 | 1778 | 7.43 | 48% | 5.3 |
Stanyon et al. (2004) | 478 | 1778 | 7.43 | 48% | 5.3 |
Butland et al. (2005) | 1277 | 5324 | 8.33 | 43% | 6.0 |
Arifuzzaman et al. (2006) | 2457 | 8663 | 7.05 | 39% | 5.4 |
Lacount et al. (2005) | 1272 | 2643 | 4.16 | 38% | 3.8 |
Average degree, edge reduction (e.r.), and edge to power node conversion rate (c.r.).
The edge reduction and conversion rate are dependent on the abundance of stars, cliques and bicliques in the network–as these motifs require just one power edge to represent arbitrarily many edges. In particular, from the example previously discussed (casein kinase II complex, nucleosome) we would expect cliques and bicliques to be the culprit. To ascertain that their abundance is indeed the explanation for the higher edge reductions, we examine the count of power edges of different sizes.
The area of each disc is proportional to the logarithm of the number of corresponding cliques (diagonal) and bicliques (non-diagonal). Stars are found along the first column or row. For example, there are 11 bicliques between two nodes and 4 nodes, and 34 bicliques of 6 nodes. The diagram is symmetric along the diagonal. Protein interaction networks from Gavin et al. (red) compared to corresponding rewired networks (blue). The high z-score (242) can be explained by significant abundance of cliques and bicliques compared to a random null-model obtained through rewiring. Note that despite the fact that the number of edges is constant, the total count of cliques, bicliques, and stars, is not necessarily constant.
Having observed an abundance of cliques and bicliques, there remains the possibility that this is solely caused by experimental or methodological artifacts. However, we know of at least one case for which this cannot be the explanation: the Structural Interaction Network (SIN) by Kim et al. is a set of interactions carefully curated using structural information: all interactions reported are direct physical interactions explained by a known structural binding
(A) Close-up of a
These results corroborate studies that looked at network motifs identified as functional units in the context of biological networks
It has been argued recently that other distributions than the power-law are a better fit to the observed degree distributions of protein interaction networks
To further support the idea that power nodes are not artifacts of the networks topology but have in fact a biological interpretation, we analyzed the enrichment of power nodes in InterPro domains
Our null hypothesis is that “annotations are randomly distributed” following an hyper-geometric distribution. In order to take into account missing domain annotations, only power nodes for which more than two thirds of the proteins are annotated with at least one term or domain are considered. Moreover we use the Bonferroni correction since we do multiple hypothesis testing.
Network | n.s.a. | ||
Kim et al. (SIN)(2006) | 90% | 96% | 0% |
Krogan et al. (2006) | 78% | 88% | 6% |
Gavin et al. (2006) | 70% | 90% | 3% |
Rual et al. (2005) | 65% | 80% | 1% |
Ewing et al. (2007) | 54% | 80% | 8% |
Ito et al. (2001) | 51% | 86% | 7% |
Arifuzzaman et al. (2006) | 46% | 73% | 0% |
Hazbun et al. (2003) | 43% | 69% | 17% |
Butland et al. (2005) | 41% | 76% | 0% |
Lim et al. (2006) | 39% | 56% | 10% |
Lacount et al. (2005) | 20% | 54% | 29% |
Stanyon et al. (2004) | 15% | 47% | 13% |
See
Network | n.s.a. | ||
Kim et al. (SIN)(2006) | 63% | 89% | 0% |
Gavin et al. (2006) | 58% | 73% | 0% |
Krogan et al. (2006) | 51% | 60% | 1% |
Hazbun et al. (2003) | 21% | 33% | 1% |
Rual et al. (2005) | 19% | 35% | 1% |
Ito et al. (2001) | 16% | 29% | 0% |
Ewing et al. (2007) | 15% | 28% | 5% |
Butland et al. (2005) | 15% | 35% | 1% |
Lim et al. (2006) | 11% | 29% | 0% |
Arifuzzaman et al. (2006) | 7% | 22% | 1% |
Stanyon et al. (2004) | 7% | 21% | 9% |
Lacount et al. (2005) | 5% | 39% | 59% |
See
Other biological networks benefit from Power Graph Analysis, too. Examples are protein homology networks
Bicliques can occur in regulatory networks due to two reasons: some transcription factors operate within complexes–combinatorial regulation–and regulatory motifs in promoter regions can be shared and repeated for different genes. In the case of homology networks, proteins sharing a sequence region of high similarity–i.e. a domain–induce cliques. Bicliques are similarly induced between sub-groups of similar proteins due additional region of sequence similarity.
Beyer et al. presented an integrative approach for assigning transcription factors to target genes in
(A) Power node hierarchy of the complete bipartite network between
The transcription factors MSN2, MSN4, and SKN7 are known to regulate the expression of genes in response to stresses, such as heat and osmotic shock, oxidative stress, low pH, glucose starvation, sorbic acid and high ethanol concentrations
Power Graph Analysis is useful for its ability to decompose a bipartite network into an union of bicliques. This decomposition leads naturally to a hierarchy of clusters of transcription factors linked to a hierarchy of clusters of target genes.
The protein tyrosine phosphatase (PTP) family
The power graph of the protein tyrosine phosphatase homology network is shown in
(A) The original homology network has 279 nodes and 4849 edges. The power graph has 209 power edges - with the addition of 95 non-singleton power nodes. Each node represents a human protein tyrosine phosphatase, with an edge between two proteins corresponding to highly significant alignments with E-values of at most 10−46. The network is obtained by an all against all BLASTP scan using the NCBI BLASTP tool
The choice of a threshold for the E-value has an impact on the representation. We observe that for the value of 10−46 the power graph reveals the most details. In this case, the lossless reduction in complexity achieved by the power graph representation reaches 95% edge reduction–from 4849 edges to 209 with 95 power nodes. The clustering of proteins in the power graph corresponds to the known classification of PTPs: 82% of leaf power nodes (that do not contain power nodes) have all of their proteins belonging to exactly the same sub-family. While the previous results could have been obtained through the hierarchical clustering of the sequences, Power Graph Analysis reveals additional details.
The cross-links between different regions of the hierarchy constitute a new insight with respect to traditional clustering methods. For example, a group of 6 type B receptor PTPs are linked by a power edge to two type 2 non-receptor PTPs.
The detection of similarity cross-links in the hierarchy is the contribution of Power Graph Analysis to the analysis of homology networks. These cross-links constitute a weak signal in networks and are difficult to detect. In this case the evidence for this domain erosion is carried by only eight similarity links between four and two proteins whereas the original network has 4849 edges. In the power graph representation it is one power edge among only 209.
Protein networks, and in particular protein interaction networks from high-throughput measurements are known to suffer from many false positives and negatives. To investigate the robustness of power graph analysis, we compare a network's power graph to the power graphs with increasing levels of noise modelled with the addition, removal or rewiring of edges.
Noise level is defined as the number of edges different from the original networks. Random rewiring leaves the total number of edges unchanged, thus a noise level of 100% means that all edges have changed. (A) Comparison of the power node hierarchies. The F1-measure of the precision and recall is computed between the power nodes found for the original network, and power nodes found for the rewired networks. (B) Comparison of the proximity of nodes in the power node hierarchies. Recall is obtained by comparing pairs of nodes together in a power node with the corresponding pairs of nodes in the power graph after random rewiring, the more distant in the power node hierarchy the lower the recall. Precision is obtained by starting from pairs of nodes together in power nodes found in the rewired networks and looking how far–in the power node hierarchy–are the corresponding nodes in the original network. The F1-measure of precision and recall is reported.
Power Graph Analysis lies at the crossing point of clustering, network motif analysis, information compression, and visualisation. In the previous results, we showed that Power Graph Analysis reveals known underlying biology when applied to protein interaction networks, regulatory and homology networks. It also leads to new insights and new hypotheses. In particular, we presented evidence that the similarity of interaction profiles for peptide-binding SH3 domains correlates with the sequence similarity of these domains. We also discussed how the difference of interaction profiles of otherwise near-identical histone subtypes–visible in the power graph representation–suggests that the TAP methodology interfered with the histone regulatory mechanisms and led to low expression levels of histones subtypes HTA1 and HTB1. Examining other types of networks, we showed that Power Graph Analysis of predicted transcription factors for target genes by Beyer et al.
The main reason behind the usefulness of Power Graph Analysis is the observation that experimental protein interaction networks, bipartite regulatory networks, protein homology networks, and other biological networks have an abundance of cliques and bicliques. Moreover, for small-scale interaction networks and some high quality networks, such as SIN
Cliques and bicliques in biological networks have been noticed in the past
With Power Graph Analysis it is possible to decompose and represent biological networks as combinations of two simple elements: cliques and bicliques. New analysis methodologies and algorithms can be developed to leverage the information compression made possible by Power Graphs. These directly operate on Power Graphs instead of traditional node-and-edge-graphs. Indeed, one important finding is that the information contained in diverse biological networks, such as protein interaction networks, regulatory networks, and homology networks is highly compressible–even up to 95% for some homology networks. We argue that avoiding this excess of redundant information is possible and desirable.
The advantages and uses of Power Graph Analysis are:
The simpler representation of complex networks without loss of information.
Network analysis methodologies and algorithms can be reformulated on top of Power Graph Analysis.
Cliques and bicliques–which are abundant and relevant for biological networks–are explicitly represented.
As a side effect of the decomposition, nodes are clustered by connectivity and neighbourhood similarity.
The connectivity information between these clusters is preserved.
Other graph formalisms have been proposed, such as
As we showed, Power Graph Analysis is a novel network analysis paradigm that provides a basis for new methodologies. One immediate example is visualisation. Several tools exist to visualise biological networks, such as Cytoscape
Given a graph
The following two conditions are required for simplifying the representations:
Power node hierarchy condition: Any two power nodes are either disjoint, or one is included in the other.
Power edge disjointness condition: Each edge of the original graph is represented by one and only one power edge.
Relaxing the previous two conditions leads to abstract Power Graphs that are difficult to visualize.
We have developed an algorithm for computing near-minimal power graph representations from graphs. The first phase of the algorithm collects candidate power nodes and the second phase uses these to search and add power edges abstracting a maximum number of edges from
A set of nodes is a candidate power node if its nodes have neighbours in common. We use a hierarchical clustering algorithm
To detect stars and other highly asymmetric bicliques in phase two, additional to the hierarchy of sets of nodes achieved with the clustering we add to the candidate power nodes for each node
First a neighbourhood similarity clustering of the nodes is performed providing candidate power nodes. In a second step power edges are searched between nodes and candidate power nodes. Note that modular decomposition would not consider as a module the set of nodes having similar but non-identical neighbourhoods. The power graph algorithm finds this candidate and uses it to succinctly represent the biclique.
The minimal power graph problem is to be seen as an optimization problem in which the power graph achieving the highest edge reduction is searched. The greedy power edge search follows the heuristic of making the locally optimum decision at each step with the hope of finding the global optimum, or at least a close approximation
Among the candidate power nodes found in phase one each pair that corresponds to a power edge is a candidate power edges. The candidates abstracting the most edges are added successively to the power graph.
The power graph algorithm shares similarities to existing algorithms, such as modular decomposition
Modular decomposition identifies modules as sets of nodes having
We have conducted experiments to understand the behaviour of the edge reduction for two important classes of networks: synthetic random networks generated according to the Erdös-Rényi model
(A) Edge reduction versus edge density. Edge reduction attains a minimum for an edge density between 0.1 and 0.2 and the raises linearly (B) Edge to power node conversion rate versus edge density.
Network rewiring is done by choosing randomly two edges
We evaluate the enrichment of a cluster's proteins with domains using p-values assuming an hyper-geometric distribution
This is the probability that the cluster has
The biological function and complex assignments for the examples where obtained through SGD
Name | Description | Database ID |
CKA1 | Alpha catalytic subunit of casein kinase 2 | [SGD:YIL035C] |
CKA2 | Alpha' catalytic subunit of casein kinase 2 | [SGD:YOR061W] |
CKB1 | Beta regulatory subunit of casein kinase 2 | [SGD:YGL019W] |
CKB2 | Beta' catalytic subunit of casein kinase 2 | [SGD:YOR039W] |
NIP1 | Subcomplex (Prt1p-Rpg1p-Nip1p) of eIF3 | [SGD:YMR309C] |
RPG1 | Subcomplex (Prt1p-Rpg1p-Nip1p) of eIF3 | [SGD:YBR079C] |
PRT1 | Sbcomplex (Prt1p-Rpg1p-Nip1p) of eIF3 | [SGD:YOR361C] |
UTP22 | Possible U3 snoRNP protein | [SGD:YGR090W] |
ROK1 | ATP-dependent RNA helicase of the DEAD box family | [SGD:YGL171W] |
RRP7 | Involved in rRNA processing and ribosome biogenesis | [SGD:YCL031C] |
YLR003C | Uncharacterized, may participate in DNA replication | [SGD:YLR003C] |
YKL088W | Predicted phosphopantothenoylcysteine decarboxylase | [SGD:YKL088W] |
POB3 | Subunit of the FACT complex (RNA Pol II trans. elong.) | [SGD:YML069W] |
SPT16 | Subunit of the FACT complex (RNA Pol II trans. elong.) | [SGD:YGL207W] |
HHO1 | Histone H1 | [SGD:YPL127C] |
HTA1 | One of two nearly identical histone H2A subtypes | [SGD:YDR225W] |
HTA2 | One of two nearly identical histone H2A subtypes | [SGD:YBL003C] |
HTB1 | One of two nearly identical histone H2B subtypes | [SGD:YDR224C] |
HTB2 | One of two nearly identical histone H2B subtypes | [SGD:YBL002W] |
HHT1 | One of two identical histone H3 proteins | [SGD:YBR010W] |
HHT2 | One of two identical histone H3 proteins | [SGD:YNL031C] |
HHF1 | One of two identical histone H4 proteins | [SGD:YBR009C] |
HHF2 | One of two identical histone H4 proteins | [SGD:YNL030W] |
HTZ1 | Histone variant H2AZ of histone H2A in nucleosomes | [SGD:YOL012C] |
ORC1 | ORC complex subunit 1, binds on replication origins | [SGD:YML065W] |
ORC2 | ORC complex subunit 2, binds on replication origins | [SGD:YBR060C] |
ORC3 | ORC complex subunit 3, binds on replication origins | [SGD:YLL004W] |
ORC4 | ORC complex subunit 3, binds on replication origins | [SGD:YPR162C] |
ORC5 | ORC complex subunit 3, binds on replication origins | [SGD:YNL261W] |
ORC6 | ORC complex subunit 3, binds on replication origins | [SGD:YHR118C] |
RVB1 | Essential protein involved in transcription vregulation | [SGD:YDR190C] |
RVB2 | Essential protein involved in transcription bregulation | [SGD:YPL235W] |
ARP4 | Nuclear actin-related involved in chromatin remodeling | [SGD:YJL081C] |
ARP5 | Nuclear actin-related involved in chromatin remodeling | [SGD:YNL059C] |
SWR1 | Swi2/Snf2-related ATPase, SWR1 complex | [SGD:YDR334W] |
SWC6 | Nucleosome-binding component of the SWR1 complex | [SGD:YML041C] |
PIL1 | Primary component of eisosomes | [SGD:YGR086C] |
SHO1 | Transmembrane osmosensor | [SGD:YER118C] |
ABP1 | Actin-binding protein, cortical actin cytoskeleton | [SGD:YCR088W] |
MYO5 | One of two type I myosins | [SGD:YMR109W] |
BOI1 | Polar growth related, functionally redundant with Boi2 | [SGD:YBL085W] |
BOI2 | Polar growth related, functionally redundant with Boi1 | [SGD:YGL171W] |
RVS167 | Actin-associated protein | [SGD:YGL171W] |
YSC84 | Actin cytoskeleton organization related. | [SGD:Yhr016c] |
LSB3 | ATP-dependent RNA helicase of the DEAD box family | [SGD:YFR024C-A] |
YAP1 | bZIP T.F, mediates resistance to cadmium | [SGD:YML007W] |
YAP2 | AP-1-like bZIP, involved in stress responses | [SGD:YDR423C] |
YAP6 | Putative bZIP T.F, sodium and lithium tolerance | [SGD:YDR259C] |
YAP7 | Putative bZIP T.F | [SGD:YOL028C] |
MSN2 | Transcriptional activator, response to stress | [SGD:YMR037C] |
MSN4 | Transcriptional activator, response to stress | [SGD:YKL062W] |
SKN7 | Nuclear response regulator,response to oxidative stress | [SGD:YHR206W] |
P23470 | Protein-tyrosine phosphatase gamma | [SP:P23470] |
A6NEQ4 | Uncharacterized Protein-tyrosine phosphatase gamma | [SP:A6NEQ4] |
Q9P0U2 | Protein tyrosine phosphatase, non-receptor type 22 | [SP:Q9P0U2] |
Q5TBC0 | Protein tyrosine phosphatase, non-receptor type 22 | [SP:Q5TBC0] |
Q9Y2R2 | Tyrosine-protein phosphatase non-receptor type 22 | [SP:Q9Y2R2] |
A0N0K6 | Protein tyrosine phosphatase, non-receptor type 22 | [SP:A0N0K6] |
Q9Y406 | Protein tyrosine phosphatase, non-receptor type 20 | [SP:Q9Y406] |
Q5SRF2 | Protein tyrosine phosphatase, non-receptor type 20 | [SP:Q5SRF2] |
Q4JDL3 | Tyrosine-protein phosphatase non-receptor type 20 | [SP:Q4JDL3] |
Thanks to Christof Winter for detailed feedback and discussions on the biological relevance of power graphs and to Andreas Henschel, Frank Dressel and Annalisa Marsico for critique and feedback. Thanks also go to Andreas Beyer for critique and for suggesting the analysis of his transcription factor to target genes network