SPINE: a framework for signaling-regulatory pathway inference from cause-effect experiments
1School of Computer Science, 2School of Medicine, Tel-Aviv University, Tel-Aviv, Israel and 3Department of Bioengineering, University of California, San Diego, CA 92093, USA
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Motivation: The complex program of gene expression allows the cell to cope with changing genetic, developmental and environmental conditions. The accumulating large-scale measurements of gene knockout effects and molecular interactions allow us to begin to uncover regulatory and signaling pathways within the cell that connect causal to affected genes on a network of physical interactions.
Results: We present a novel framework, SPINE, for Signaling-regulatory Pathway INferencE. The framework aims at explaining gene expression experiments in which a gene is knocked out and as a result multiple genes change their expression levels. To this end, an integrated network of protein–protein and protein-DNA interactions is constructed, and signaling pathways connecting the causal gene to the affected genes are searched for in this network. The reconstruction problem is translated into that of assigning an activation/repression attribute with each protein so as to explain (in expectation) a maximum number of the knockout effects observed. We provide an integer programming formulation for the latter problem and solve it using a commercial solver.
We validate the method by applying it to a yeast subnetwork that is involved in mating. In cross-validation tests, SPINE obtains very high accuracy in predicting knockout effects (99%). Next, we apply SPINE to the entire yeast network to predict protein effects and reconstruct signaling and regulatory pathways. Overall, we are able to infer 861 paths with confidence and assign effects to 183 genes. The predicted effects are found to be in high agreement with current biological knowledge.
Availability: The algorithm and data are available at http://cs.tau.ac.il/~roded/SPINE.html
Contact: roded{at}post.tau.ac.il
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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High-throughput technologies are routinely used to map molecular interaction within the cell. These include chromatin immuno-precipitation experiments (Lee et al., 2002) for measuring protein–DNA interactions, and yeast two-hybrid assays (Fields and Song, 1989) and co-immunoprecipitation screens (Gavin et al., 2002) for measuring protein–protein interactions (PPIs). The resulting maps promise to shed light on the regulatory mechanisms that allow the propagation of certain signals within the cell. In particular, we aim at explaining gene expression experiments in which a gene is knocked out and as a result multiple genes change their expression levels.
A series of works concerned the inference of functional interactions in small signaling-regulatory networks from perturbation experiments (Andree et al., 2005; Kholodenko et al., 2002; Yalamanchili et al., 2006). The problem of explaining knockout experiments using a physical network was introduced by Yeang et al. (2004). The authors looked at a specific setting of the problem where the objective is to infer for each edge of the network a direction, specifying the direction in which information flows through that interaction, and a sign, representing the regulatory effect of the interaction (activation or repression). The annotated network can then be used for pathway inference, although this inference problem was not explicitly treated by Yeang et al.
To tackle the network annotation problem, Yeang et al. assumed that explanatory pathways should satisfy several constraints. In particular, they should be directed from the knockout gene to the affected gene, and the aggregate sign along the pathway's edges should complement the sign of the knockout effect. They devised a probabilistic framework for the annotation task, in which the different constraints are translated into potential functions (for edges, paths and knockouts), and the objective is to maximize the product of all potentials. A follow up work (Yeang et al., 2005) devised methods for validation and refinement of the network model. A similar model was used by Yeang and Vingron (2006) to integrate metabolic data with regulatory interactions and gene-knockout data. Workman et al. (2006) used the same model to infer the regulatory pathways that control the response to DNA damage.
While Yeang et al. provided an elegant formulation of the annotation problem, their solution method suffers from several shortcomings: (i) The objective function does not distinguish between situations in which few knockout pairs are explained by many pathways each, and those in which many pairs are explained by few pathways each. Clearly, the latter should be preferable. (ii) The components in the objective function are multiplied, although they are dependent on one another, as also mentioned in (Yeang et al., 2004). (iii) The maximization procedure does not guarantee reaching a global optimum.
Here, we propose a novel combinatorial model for the inference problem. Our optimization goal is to maximize the expected number of explained cause-effect pairs. We reformulate the problem as an integer programming problem and apply a commercial solver to it. Moreover, we provide a measure of confidence in the predictions made, which quantifies the change in the function being optimized when the prediction is flipped.
We validate the method by applying it to a yeast subnetwork involved in mating. In cross-validation tests, SPINE obtains very high accuracy in predicting knockout effects (99%). Next, we apply SPINE to the entire yeast network to predict protein effects and reconstruct signaling and regulatory pathways. Overall, we are able to infer 861 paths with confidence and assign effects to 183 genes. The predicted effects are found to be in high agreement with current biological knowledge. Our results compare favorably to those attained by Yeang et al. (2004) in both the small-scale and the large-scale applications.
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2 METHODS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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2.1 Data description
The input data consist of a directed network of physical interactions
and a collection X of cause-effect pairs of genes. The latter is commonly obtained using knockout experiments in which a gene is perturbed (cause) and the expression levels of all other genes are monitored. Those genes that significantly change their expression levels in response to the knockout are said to be affected by it. All other genes are assumed to be non-affected. In the context of knockout experiments, we refer to a cause-effect pair also as a knockout pair.
The nodes in the network represent genes and their protein products (no distinction is made between those two). The network has two types of edges: protein–DNA interactions and PPIs. Protein–DNA interactions are naturally represented as edges directed from a transcription factor to a regulated gene. Each PPI is represented using two oppositely directed edges. In addition to a direction, every interaction in the network is assigned a positive reliability value, representing the probability that the interaction is true; and a binary sign, representing its effect (0-activation or 1-repression). The former is computed in a pre-processing step (see below), while the latter is to be inferred by the model. In the following, we denote the sign and reliability of each interaction 
by sgn(e) and r(e), respectively. We denote the set of protein–DNA interactions by 
. Finally, we denote by e(s, t) the complement of the observed effect on gene t when knocking out gene s. e(s, t) is assumed to represent the effect of s on t in wild type.
2.2 Explanatory pathways
Yeang et al. (2004) defined certain basic rules for paths that explain knockout pairs. We use the same rules, and define a k-explanatory of a knockout pair 
as an ordered set of vertices 
that satisfies the following conditions:
is a path: 
.
- is a simple path:
.
- The last edge in the path is a protein–DNA edge:
.
- If an intermediate gene along the path has been knocked out, then it should also exhibit an effect on the target protein:
.
For a path , let s() and t() denote its source and target, respectively. Let 
be an indicator variable denoting whether explains the pair 
. An explanatory path is said to be consistent if in addition:
- The aggregate sign of is equal to
. Formally: 
, where 
is the xor operator (addition modulo 2).
- Every proper suffix
of that connects another knockout pair is consistent with respect to it.
Note that we focus here on simple paths, as a protein is likely to play a single role in a given pathway. Further note that the consistency requirement, imposed here on suffixes, is adequate only for sub-paths that end with a protein–DNA edge, as the effect is observed at the transcription level.
These definitions can be easily generalized to accommodate for already known edge signs. For ease of presentation, we concentrate in the following on the case where all edge signs are unknown. In particular, if we denote by 
an indicator variable specifying whether an explanatory path is consistent, then
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| (1) |
Yeang et al. (2004) considered in their work the inference of edge signs only. Another possibility is to assign signs to nodes rather than edges. In this case, all the edges emanating from a node carry its sign. This may be less accurate but would yield many fewer variables and hence avoid overfitting problems and computational bottlenecks. Indeed, most proteins in current databases are classified as either activators or repressors solely, so this relaxation seems to be in line with current biological knowledge. In the rest of the article, we focus on this latter variant, and use the edge variant mainly for comparison purposes.
2.3 Optimization criterion
Intuitively, the goal of the algorithm is to infer regulatory pathways in the network that provide a consistent explanation for the input set of knockout pairs. In practice, the pathways are dependent and there could be more than one pathway explaining a given pair. Hence, rather than trying to pinpoint a single pathway per pair, we aim at assigning signs to the interactions in the network so that the data can be best explained. There are several definitions to what a ‘best’ explanation is. A parsimonious definition would seek a sign assignment so that a maximum number of pairs have at least one consistent path. Note that an implicit assumption here is that the effect propagates through a single pathway, and the existence of other, possibly inconsistent pathways does not contradict the existence of the effect. However, the parsimonious criterion ignores the information on edge reliabilities and could, e.g. prefer solutions in which many pairs are explained by very low probability paths. Thus, we define our optimization problem as that of finding an assignment that will maximize the expected number of pairs that have at least one consistent path. We give a formal definition of this criterion below.
Define the probability of a path as the product of the reliabilities of its edges:
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| (2) |
2.3.1 Expectation calculation
We wish to find a setting of the node or edge signs that maximizes the expected number of satisfied knockout pairs, which is given by:
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| (3) |
Given a collection 
of explanatory paths for a knockout pair (s, t), we denote by 
an indicator variable specifying whether all paths in are consistent with this pair. Clearly, 
. The probability that all the paths exist, p(), is equal to the probability that all the edges in exist. Denoting the latter set of edges by 
we have:
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| (4) |
For a knockout pair 
, denote by the collection of explanatory paths for it, with 
. To compute the probability that at least one of the paths exist, we must take into account the dependencies among them due to edge intersections. Let 
denote the collection of all sets 
of cardinality i. Using the inclusion–exclusion principle, the probability that at least one of the paths in is consistent is:
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| (5) |
As mentioned earlier, in the objective function proposed by Yeang et al. the components are multiplied, although they are dependent on one another. Here, path dependencies are explicitly treated by the model Equation (5). Notably, there are two additional differences, not mentioned above, between our model and that of Yeang et al.: (i) no requirement is placed in Yeang et al. (2004) on the consistency of suffixes of an explanatory path. (ii) SPINE models PPI edges as potentially bi-directional, while in Yeang et al. (2004) they are forced to have a single direction. As we shall see below, in practice, most of the edges attain a single direction also under our model.
2.4 Algorithmic approach
We tackle the optimization problem using an integer program reformulation of it, to which we apply the commercial 
solver. Below, we give the integer programming formulation and the full SPINE algorithm. A diagram that describes the SPINE's input, main phases and output, appears in Figure 1.
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2.4.1 Integer programming formulation
We reformulate the network annotation problem as an integer linear program. The objective function is simply
. The constraints are of two types:- Path constraints: determining the consistency of explanatory paths.
- Knockout pair constraints: determining the expectation value of knockout pairs.
To obtain a linear program, we convert the Boolean equations defining the variables of the model, as well as the objective function, into linear forms.
Denote by 
the longest suffix of which is an explanatory path. Thus, we express 
linearly using two constraints as follows:
- Consistent path sign:
where
(6) 
is a dummy variable, used to express the parity of the repressor signs.
- Consistent suffix:
Note that this constraint is used only in case there is indeed a suffix which is an explanatory path.
(7)
As for Equation (5), the challenge is to express 
, an indicator for a set of paths, in a linear form. This can be done using the following constraint:
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| (8) |
2.4.2 Confidence assignment
SPINE tries to optimize the expected number of explained knockout pairs. For a given instance of the problem, let Omax denote the optimum value of the objective function. Clearly, there may be more than one configuration of protein signs that attains this value. Hence, for each protein v, we provide a confidence value Cv measuring how confident we are in its sign assignment. This value is calculated by running SPINE on a modified instance, in which the sign of the protein is flipped, and recording the difference between the new value of the objective function and Omax. We declare v to be confident if 
, for a pre-specified 
0. We consider a knockout pair to be confidently explained if there is a consistent explanatory path for it whose proteins are all confident.
Yeang et al. (2004) did not define a confidence measure. However, they distinguished between variable assignments that uniquely maximize the objective function, and those that do not. In their model, once all the unique assignments (e.g. of signs) have been determined, one variable is fixed arbitrarily, and the inference procedure is iteratively applied until all the variables are determined. Hence, in contrast to SPINE, some of the inferred assignments may be arbitrary.
2.5 A speedup heuristic
To deal with large integer programs, we infer the model variables using multi-level runs. At each level 
, we start from the confident variables inferred on previous levels, and search for a setting of the remaining variables of the model that maximizes the expected number of explained knockout pairs using explanatory paths of length 
. This heuristic is based on the assumption that short, confident pathways are more likely than long ones.
We use = 0 and consider only confident predictions, as non-confident predictions can be arbitrarily flipped without changing the overall value of the objective function. The full SPINE algorithm is summarized in Figure 2. Note that this algorithm solves both the node- and the edge-variant of the problem.
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2.6 A toy example
Figure 3a gives an example of a simple input network. The nodes
represent proteins, the solid edges represent protein–DNA interactions (for convenience, no PPIs are used) and the dashed edges represent knockout pairs along with the required knockout effect. The reliability of each edge appears to its side. The explanatory paths in this example (with their probabilities in parentheses) are:
(0.9).
(1).
(0.45).
(0.8).
The knockout pair indicators are:
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Using Equation (3), we get the total expectation:
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When looking for an assignment of signs to edges, there are many possible configurations that satisfy all those paths. In all those configurations 
must have a positive sign, since both knockout pairs (A, D) and (B, D) have the same effect. All the other edges remain unsigned, since there is more than one possibility to set their signs. Hence, there is a single confident edge in this variant, and no consistent path is confidently inferred (Fig. 3b).
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In contrast, in the node-based variant, many more variables are determined (Fig. 3c). Indeed, A will be signed positive for the same argument as above. Thus, E has to be signed negative, and G has to be signed positive. The only unsigned nodes in this example are B and C: signing both of them as either positive or negative will make
2 a consistent path. |
3 RESULTS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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We tested SPINE both in a small-scale and a large-scale setting on a yeast (Saccharomyces cerevisiae) data set. First, we applied SPINE to part of a yeast subnetwork involved in mating in order to test the different variants of the algorithm and for purposes of comparison to a previous analysis made in Yeang et al. (2004). Next, we applied SPINE to genome-wide yeast data and evaluated its prediction performance.
3.1 Data acquisition
We constructed an integrated yeast network of 15 147 protein–protein and 5568 protein–DNA interactions, spanning a total of 5313 genes/proteins.
PPIs were taken from DIP (Salwinski et al., 2004, April 2005 download) and were assigned confidence scores using a previously described logistic regression method (Shlomi et al., 2006). Each PPI was represented by two oppositely directed edges. Protein–DNA (directed) interactions were taken from (Macisaac et al., 2006) and were assigned a confidence of 0.96 based on the false positive rate estimations in (Lee et al., 2002).
Gene expression data were taken from (Hughes et al., 2000) and included genome-wide gene expression measurements in yeast under 210 different single-gene knockouts. A previous analysis by Yeang et al. (2004) yielded a collection X of 24 457 pairs of a knocked out gene and an affected gene, all with P-value 
.
3.2 Yeast mating subnetwork
Yeang et al. (2004) have defined a yeast subnetwork involved in mating, which consists of 33 protein–DNA interactions, and 25 PPIs, covering a set of 149 knockout pairs. Applying our algorithm to this network with path length bound of 5 (as in Yeang et al., 2004) resulted in inferring consistent explanatory paths for all 103 reachable knockout pairs.
To evaluate the performance of our algorithm, we tested it in a cross-validation setting on this network, and compared our results to those attained by Yeang et al. (2004). For purposes of comparison, we first used the edge variant of our algorithm with the same evaluation criterion used in Yeang et al. (2004). In detail, at each iteration one or five knockout pairs were hidden, and each was considered to be successfully predicted if all explanatory paths for it were consistent with its true effect. The results are shown in Table 1.
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Evidently, the results are very similar to those of Yeang et al. (2004) with SPINE providing a slightly higher accuracy, both in the leave-one-out and leave-5-out tests. Notably, some of the knockout pairs remain undecided in the SPINE application since no explanatory path for them is predicted confidently. For SPINE, the only incorrect prediction was for knockout pair (FUS3, PRY2), which is the only knockout pair involving FUS3. The only explanatory path from FUS3 to PRY2 is FUS3
STE12 PRY2; both interactions were signed as positive, yielding a positive path, in contrast to the needed knockout effect. Interestingly, the interaction FUS3 STE12 is known to be positive, as FUS3 is known to activate STE12. Next, we wished to test the effect of maximizing the expected number of satisfied knockout pairs, rather than simply their number. In these tests, we used a more relaxed criterion for computing prediction accuracy: a knockout pair was considered to be predicted successfully if the expected number of explanatory paths that are consistent with its true effect exceeded the expected number of the non-consistent ones (if the numbers are equal, we consider the pair as undecided). Using this criterion seems more natural under our model and indeed improved the accuracy for the node variant, although the results for the edge variant remained the same (data not shown). We tested the maximization criteria in both the edge- and node-variant. As shown in Table 2, maximizing the mere number of satisfied pairs resulted in markedly smaller numbers of confident sign variables, in both variants, although both criteria attained similar accuracy levels.
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By comparing the cross-validation results for SPINE's edge and node variants, it is evident that the latter attains similar accuracy levels although it has significantly less variables to be optimized and, hence, is less prone to overfitting.
Finally, we conducted a noisy cross-validation test. In each trial, the sign of one knockout pair was flipped, and the sign of a second knockout pair was hidden and inferred. The results show that the model is robust to these perturbations and the accuracy is maintained at very high levels (Table 2).
3.3 Genome-wide application
Next, we applied our method to the genome-wide data. Due to the size of the network, and following Yeang et al. (2004), we restricted the computation to paths of length at most 3. Out of 231 proteins participating in such paths, 183 protein signs were confidently inferred. Among 974 knockout pairs that had explanatory paths, 861 were confidently explained using the inferred signs. The contribution of each level in SPINE's execution is shown in Table 3. The amount of explained knockout pairs and inferred proteins increases in each level, and this demonstrates the importance of looking at paths rather than considering direct edges only. In comparison, when using only protein–DNA interactions the explanatory power of the model is markedly decreased: only 188 knockout pairs are confidently explained and only 30 protein signs are inferred.
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In order to validate our predictions, we gathered information on activator and repressor proteins from four different sources: GO (Ashburner et al., 2000, December 2006 download)—molecular functions ‘transcriptional activator activity’ and ‘transcriptional repressor activity’; KEGG (Kanehisa and Goto, 2000), Guelzim et al. (2002) and Myers and Kornberg (2000). We excluded 19 genes that appeared as both activators and repressors in the different sources. The results are summarized in Table 4, showing a significant overlap between the model's prediction and the known signs. Examining the predictions made in each level of the algorithm we observe that most predictions, and also most errors, are made in level 3: 24/27 activators and 8/15 repressors are correctly predicted.
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The confidence measure can vary from one inferred sign to another. Thus, we checked the accuracy of our predictions when using varying thresholds of the confidence measure for determining the set of predicted signs. The inferred signs were grouped into bins, such as each bin contains the inferred signs above a certain confidence threshold. The results clearly show that increasing the confidence threshold also increases the accuracy of the predictions (Fig. 4).
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Next, we wished to compare our prediction performance to that of Yeang et al. (2005). As we could not readily apply the latter method to our network, nor to the intersection of our network and that in Yeang et al. (2005), we used the results of applying their method to the genome-wide network that appears in Yeang et al. (2005). We note that the two networks differ in
20% of their constituent interactions, although their sizes are about the same; thus, the comparison presented below should be interpreted with caution as the difference in networks may bias the results. Recall that the method of Yeang et al. infers edge signs. To infer node signs, we applied a majority rule to the set of edge signs incident to each node (nodes for which no majority existed remained undecided). The statistics of these predictions are shown in Table 4. It can be seen that the overall performance of the algorithms is similar with respect to activator predictions, but SPINE attains higher accuracy in repressor predictions while the results of Yeang et al. are not significant. To further evaluate the results of both algorithms, we computed their precision and recall. Precision is the percent of correct predictions out of all predictions (for both activators and repressors); recall is the percent of correct predictions out of all known signs. The precision and recall for SPINE were 75% and 24%, respectively. The method of Yeang et al. yielded lower rates in both measures with 68% precision and 20% recall.
In contrast to Yeang et al. SPINE does not constrain the directions of the PPIs. Interestingly, most (312 out of 340) PPIs appeared in one direction only in the inferred consistent pathways.
Figure 5 shows an example subnetwork inferred by the model, including both molecular and functional data. The explanatory pathways that appear in the subnetwork go through GAL4. GAL4 is a known activator. It is a DNA-binding transcription factor required for the activation of the GAL genes in response to galactose. Other components in this subnetwork are MED2, GAL11, SRB4, SRB5 and SRB6, which are mediators that function as a bridge between the regulatory proteins and the basal polymerase II transcription machinery (Myers and Kornberg, 2000).
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SPINE found a confident optimal solution, Soptimal, in which GAL4 is a confident activator, i.e. signing GAL4 as repressor reduces the expected number of explained knockout pairs. We checked the effect of signing GAL4 as a repressor, by running SPINE again, while fixing GAL4 as a repressor. Once reached an optimum, we searched for a solution with the same optimal value, but with minimum changes compared to solution Soptimal. As shown in Figure 5a, all the explanatory pathways that go through GAL4 are consistent when it is an activator, whereas when GAL4 is a repressor some of those pathways become non-consistent. Some genes, such as, e.g. HAP5, remain with the same sign even when GAL4 is a repressor, due to their importance in the different pathways, including pathways that does not appear in the network. However, some genes do change their sign, such as, e.g. SUA7. Figure 5b shows the decrease in the expectation of knockout pairs connected in the network.
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4 CONCLUSIONS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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This article addresses the problem of inferring signaling-regulatory pathways explaining cause-effect experiments. We have translated the reconstruction problem into that of assigning an activation/repression attribute with each protein so as to explain (in expectation) a maximum number of the knockout effects observed. We provided an integer programming formulation for the latter problem and solved it using a commercial solver. The resulting framework, SPINE, was tested on both small- and large-scale networks, and provided highly accurate results, comparing favorably to a previous approach by Yeang et al. (2004).
There are several important contributions of the current work: (i) a basic optimization criterion and a biologically motivated scheme to solve it in practice via multi-level runs; (ii) a confidence measure for the signs inferred by the model and (iii) a general approach for assigning both edge and node signs, providing a balance between the flexibility of the assignment and its confidence.
While our algorithm provides a valuable framework for analyzing gene expression data in the context of a physical network, several of its limitations should be acknowledged. First, the algorithm is computationally intensive due to the resulting large integer programs. A practical approach to analyze larger networks could be to limit the accuracy of the solver. Second, it could be beneficial to integrate into the framework information on protein complexes (Gavin et al., 2006; Hollunder et al., 2005), which could be perhaps viewed as unified single nodes in the context of the current analysis. Finally, our algorithm does not consider the strength of the knockout effect (and the associated P-value). A refined optimization criterion that combines this information could improve the accuracy of the predictions.
We have tested our method on cause-effect data gathered from gene knockout experiments. Although gene knockouts are routine in model organisms such as yeast, they are more technically involved in mammalian species such as mouse. For these higher eukaryotes, other ways of genetically perturbing the cell are gaining in practice, such as RNA interference and eQTL analysis. In addition, although DNA microarrays are currently the most widespread technology for measuring global changes in cellular state, a variety of other measurement systems, such as mass spectrometry for monitoring protein abundance, protein post-translational modification or small molecule abundance, are gaining in popularity. The general formulation of our model allows its application to those types of cause-effect data as well.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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R.S. was supported by an Alon Fellowship. T.S. was supported by an Eshkol Fellowship from the Israeli Ministry of Science. This research was supported by the Israel Science Foundation (grant no. 385/06).
Conflict of Interest: none declared.
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REFERENCES |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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