Constraints on signaling network logic reveal functional subgraphs on Multiple Myeloma OMIC data
Abstract
Background
The integration of gene expression profiles (GEPs) and large-scale biological networks derived from pathways databases is a subject which is being widely explored. Existing methods are based on network distance measures among significantly measured species. Only a small number of them include the directionality and underlying logic existing in biological networks. In this study we approach the GEP-networks integration problem by considering the network logic, however our approach does not require a prior species selection according to their gene expression level.
Results
We start by modeling the biological network representing its underlying logic using Logic Programming. This model points to reachable network discrete states that maximize a notion of harmony between the molecular species active or inactive possible states and the directionality of the pathways reactions according to their activator or inhibitor control role. Only then, we confront these network states with the GEP. From this confrontation independent graph components are derived, each of them related to a fixed and optimal assignment of active or inactive states. These components allow us to decompose a large-scale network into subgraphs and their molecular species state assignments have different degrees of similarity when compared to the same GEP.
We apply our method to study the set of possible states derived from a subgraph from the NCI-PID Pathway Interaction Database. This graph links Multiple Myeloma (MM) genes to known receptors for this blood cancer.
Conclusion
We discover that the NCI-PID MM graph had 15 independent components, and when confronted to 611 MM GEPs, we find 1 component as being more specific to represent the difference between cancer and healthy profiles.
Keywords
Answer set programming Regulatory network modeling Omic data integrationBackground
The exponential increase of biological data (genomic, transcriptomic, proteomic) [1] and of biological interaction knowledge in Pathway Databases allows modeling cellular regulatory mechanisms. Modeling biological mechanisms is done, most of the time, using boolean or ordinary differential equation representations. Those approaches have shown their efficiency in cellular phenomena study [2], disease research [3, 4], and bio-production optimization [5]. However, those modeling approaches cannot take into account the large amount of OMIC data. This limitation requires that the researcher preselects the OMIC data and network, adding bias to the analysis [6]. A classical way to perform OMIC data preselection is to use differentially expressed genes [7], this leads to select genes by imposing common fixed thresholds while their activation threshold may be specific for each gene. As a consequence the selected pathways may not be specific for the biological problematic. A common way to perform network preselection consists on choosing specific pathways according to the type of data and the biological problematic. Moreover, several regulatory databases such as KEGG, CBN, and Reactome [8, 9, 10] allow to select specific (e.g. apoptosis) pathways directly. Nevertheless, this network preselection approach can hide unsuspected pathways, reducing the possibility to discover new ones.
Some of the methods that identify subnetworks or network components, recognize specific pathways based on differentially expressed genes [11]. However, this kind of approaches considers pathways independently, and does not take into account the interactions between biological compounds. Other methods were developed to find involved pathways by identifying subgraphs or network clusters [12] from a regulatory network using topological informations and then use the gene expression profiles (GEPs) to identify a specific cluster. The majority of such methods uses protein-protein interaction (PPI) networks and GEPs to identify subgraphs [13, 14]. Those methods consider the interactions between biological compounds but infer protein states based on the associated GEP. That is, the built subgraph contains expressed proteins (obtained from associated genes expression) and their interactions [14]. These methods assume that a correleation between gene expression and protein activity exists, which is not necessarily true since an increase on gene expression can account of an increase of protein quantity, however in order to increase the activity of a protein another (e.g. phosphorylation) mechanism may need to be included. Methods using PPI networks are limited since they do not consider causality logic and different interaction roles. While the notion of causality is used by methods such as [15] to find a subgraph which maximizes the genes expression variation information; to our knowledge few subgraph identification methods based on GEPs consider direct interactions in regulatory networks, and much less include the different kind of interaction role (activation or inhibition) [16]. Moreover, the majority of those methods study protein interactions based on GEPs and without taking into account the difference between transcriptional and post-translational regulation. Finally, approaches that include the interaction role in their integrative analysis to link regulatory networks with GEPs [16, 17] use a local strategy, that is, they analyze sequentially each node in the graph with respect to its predecessors.
In this study we propose a method based on exhaustive and global graph coloring approaches [18]. These approaches are able to predict the graph coloring configurations, in terms of discrete states (e.g. active or inactive) of the molecular species of a biological network with respect to a set of experimental observations. In this work we extend those approaches by looking for harmonious or perfect colorations. The intuition behind the harmonious or perfectness notion is to point to reachable network discrete states that maximize the agreement between the molecular species active or inactive states and the directionality of the pathways reactions according to their activator or inhibitor control role. This can be expressed in natural language as follows: “for a given node in the graph we impose that its discrete active or inactive state is explained by a maximal number of regulators”. This statement is inspired from a hypothesis of redundancy in biological networks control, and we use Logic Programming to express this statement and search for coloring models where it holds for every node in the graph. Afterwards, we correlate the graph coloring models that maximize the perfectness notion and in this way build correlated graph components. After adding experimental data, our method is able to identify components of interest. We present an application of this method with transcriptomic data from myeloma cells (MC) of 602 MM patients and from normal plasma cells (NPC) of 9 healthy donors. Multiple myeloma is a hematologic malignancy representing 1% of all cancer [19] with a survival rate of 49.6% after 5 years. Our method of perfect graph colorings identification allowed us to identify 15 components. One of these components was statistically specific to MC in comparison to NPC. Using gene ontology enrichment analysis with the PANTHER tool we were able to associate this component to oncogenic phenomena.
Methods
Answer Set Programming (ASP)
The perfect colorations identification is implemented in Answer Set Programming (ASP) [20]. This declarative programming approach allows us to express a problem in the form of a logic program (LP). The syntax of ASP is close to Prolog syntax because the grammatical structure of both LPs rules expresses a logical implication from the right terms of the rule towards the left terms of the rule. However, ASP semantics, which stands for the meaning of the vocabulary symbols used in each rule, allows a different type of solving mechanism. While in Prolog there is an inference process to search for an answer to a query, ASP programs allow to find all (Herbrand stable) models satisfying all the LP rules.
An ASP program consists of a set of predicates and first order logic rules of the form :
where Ai are atoms, i.e elements of the Herbrand base, which is composed of all the possible relations or predicates in first order logic of the LP. The Herbrand base is built by instantiating the LP predicates with the LP terms (constants or elements of the Herbrand universe). Basically, the line 1 explicits that A0 will be trueif A1,..., An are true and An+1,..., An+k cannot be proven to be true (not in the Herbrand base). In ASP, a solution or answer set is a stable Herbrand model, that is, a minimal set of true atoms without variables (grounded atoms) where all the logical rules are satisfied. We give now a brief description of the ASP rules used in this study; for deeper ASP understanding, please refer to [20, 21]. Variables in ASP start with uppercase letter whereas variables starting with lowercase letters denote constants. We use the following rule to generate candidate solutions:
This rule is satisfied when n predicates a(X,Z) are true, where X ranges over the domain of true predicates b(X) and Z is fixed by predicate c(Z). Another rule we use is expressed as:
This rule generates a predicate sum(X) where X is the number of predicates a(Z) which are true and ranged by the domain of true predicates b(Z). Finally, we used the following rule for optimization:
This rule expresses the selection of the answer sets with the minimal value of X, where predicate sum(X) is true. The “@p” indicates the optimization priority. The higher the value of p, the higher the priority.
Modeling perfect coloring with ASP
Instantiation
Graph: a graph G(V,E) is composed of a set of nodes V and edges E. Edge: an edge is a tuple with 2 nodes (source and target), a sign (1 for activation, -1 for inhibition) and a weight.
Node: nodes are identified by the union of all sources and targets in the edges.
Target: a target is a node with at least one predecessor. We can identify those targets by looking for the union of all targets in the edges (line 12)
Candidate solutions generation
A colored graph is a graph in which all nodes are associated to a sign: up standing for “+” and down for “-”. These signs refer to the qualitative variation that one may experimentally measure in a molecular species (component of the graph) when comparing 2 cellular states, for example after v.s. before a stress condition. In this work we are interested on modeling sets of possible state variations of the components of the graph (line 16).
Definitions
Local consistent node coloring.
A node colored in a consistent way will be a node where its color is explained by at least one of its direct predecessor in the graph [18]. There are two possibilities for the coloring of a node n so that it will be explained by one of its predecessors p. This will depend on the sign of the edge from p to n. If the edge is an activation (line 17), p has to be associated with the same sign, otherwise if it is an inhibition (line 18), p has to be associated with the opposite sign. Because a node needs a predecessor to have a consistent color, this rule is only relevant for graph targets.
Imperfect target coloring.
An imperfect node coloring happens when a node is colored with a sign not explained by at least one of its direct predecessors in the graph.
Imperfect weighted regulator.
An imperfect weighted regulator p is a direct predecessor of a node n that does not explain consistently the color of n. The weight of this rule will be the weight of the edge from p to n.
Optimization constraints
Our method identifies graph colorings which minimize conflicts between target and predecessors, that is, it finds perfect graph colorings with minimal conflicts. In order to do this we apply 3 minimizations.
Inconsistency minimization
The first optimization will select the colored graphs with the minimal number of inconsistent targets. For this, we will first identify the inconsistent targets (line 23), then count the sum of those inconsistent targets (line 24). Finally, we will minimize this sum (line 25).
Imperfect target coloring minimization
The second optimization aims to reduce the solutions space to the graph with the minimal number of imperfect targets. In the same way as previously, the sum of imperfect target colorings is computed for each solution (line 26), then the solutions with the minimal number of imperfect colorations will be selected (line 27).
Imperfect weighted regulator minimization
The last optimization will minimize the sum of imperfect weighted regulators. First, for each target we compute the sum of the weights from the imperfect weighted regulators (line 28). Then we can compute the sum of weights for a colored graph (line 30). Finally, we can select the colored graph with the minimal sum of the weights associated to imperfect regulators (line 31).
Component identification
Correlation matrix informing about the dependence between two nodes colorations among perfect colorations. a and b inform for each coloring combination occurrence
Coloring | up | down |
---|---|---|
up | a | b |
down | b | a |
Maximal similarity
Space solution reduction
Due to our candidate solution generation, the space of solutions for a graph of n nodes will have a size of 2^{n}. Because our graph coloring method is based on 2 signs with symmetric rules, we can observe that a coloring model and its reverse represents the same coloring perfectness. Therefore, it is possible to instantiate a node with a fixed color to reduce to half the solution space size. For example with line 32, we fixed the node node0 in the graph to down.
The first and second reduction methods identify subcomponents. Aggregating molecular-species nodes within subcomponent nodes reduces the number of nodes in the graph. The third method reduces the number of edges and detects components which are isolated of the rest of the graph.
Reduction based on the consistency (Fig. 2a)
This reduction method first identifies nodes which are candidates to have a sign correlation in consistent solutions, then it merges those nodes into a subcomponent-node. For that purpose we look for a specific pattern: a node with only one predecessor and a single incoming edge. This pattern will be merged into a component that will be composed of both elements and the sign of their correlation in a consistent solution (“+” if positive correlation, “-” for negative correlation). This process of pattern identification and merging of nodes into a subcomponent will be repeated until no new pattern is detected. Notice that the assembling of a subcomponent-node with a new molecular-species or subcomponent node generates a new subcomponent-node.
Reduction based on the co-regulators (Fig. 2b)
The second reduction identifies nodes candidates to have a sign correlation in candidate coloring solutions with minimized imperfect coloring. For this, we look for another pattern: two nodes without predecessors which share the same and unique successor (Fig. 2b). Those nodes can be merged into a subcomponent-node. In the same way as previously, the process of pattern recognition and then merging of nodes into a subcomponent will be repeated until no new pattern is detected.
Reduction based on the edges balance (Fig. 2c)
From both previous reduction methods we obtain a new graph composed of subcomponents. We consider here a non-merged molecular-species node as a subcomponent composed of one node. Then, we compute the edges weight between nodes of the graph by adding the weight of all the edges of the same sign that go from the molecular-species nodes of the source subcomponent to the molecular-species nodes of the target subcomponent. By merging together the edges of the same sign between two subcomponents, we may obtain subcomponents sharing at most 2 edges, e_{1} and e_{2}, which are opposite signed and weighted respectively w_{1} and w_{2}. In this case, we will compute new weights: \(w_{1}\prime = w_{1} - min(w_{1},w_{2})\) and \(w_{2}\prime = w_{2} - min(w_{1},w_{2})\). In case a new weight is equal to zero (Fig. 2c), we can delete the associated edge. After this edge reduction we may obtain disconnected subcomponents that are isolated from the graph. These subcomponents are color-independent of the rest of the graph and constitute a component as defined in “Component identification” section However, our method stores the information that targets of these components will be always consistent since they receive positive and negative interactions coming from the component. Also, on these targets, the perfectness constraint will not be verified.
Implementation
To identify perfect graph colorings we used Answer Set Programming (ASP), namely clingo 4.5.4. The graph extraction from PID and the reduction algorithms were implemented with python 2.7 using the package NetworkX [23]. The components identification from perfect graph colorings were implemented in R [24] and python 2.7. All the computation (graph extraction, perfect coloration identification, components identification and MS computing) were made on a standard machine.
Toy example
Graph reduction
Perfect coloring and components identification
Perfect colorations for the toy example graph and the space solution reduction
A + | B +, C + | D +, E +, F + | G + | H + | |
---|---|---|---|---|---|
Coloration 1 | down | down | down | up | up |
Coloration 2 | up | up | up | down | down |
Maximal similarity computing
For a component, there are two possible colorings (component configurations) due to the symmetric property. For example, the component “A +,B +, C+, D +, E +, F +, G -, H -” (Fig. 7) has two possible configurations: C^{1}={(A,up),(B,up),(C,up),(D,up),(E,up),(F,up),(G,down),(H,down)} and C^{2}={(A,down),(B,down),(C,down),(D,down),(E,down),(F,down),(G,up),(H,up)}. Let us suppose a gene expression profile {D=up,E=up,G=up}. We can compute the similarity, Sim, between the expression profile and each coloring configuration as Sim_{C}^{1}=2 and Sim_{C}^{2}=1. The maximal similarity (MS) will be the maximal value between these two values divided by the number of observations in the profile, that is, MS=max(Sim_{C}^{1},Sim_{C}^{2})/3=2/3.
Application
Results and discussions
Perfect colorations
The graph reduction based on the consistency then co-regulators allowed to reduce the graph to 194 subcomponents and 408 edges. The edge weight computing and balance reduced the graph to 194 subcomponents and 389 edges. That is a reduction to 8% and 14% of the original number of nodes (2269) and edges (2683) respectively.
Perfect coloration results for initial and reduced graph
Graph | # Nodes | #Targets | # Edges | Solution space | Number of inconsistent targets | Number of imperfect colorations | Number of imperfect weighted regulator | Computation time |
---|---|---|---|---|---|---|---|---|
Original | 2269 | 2267 | 2683 | 2^{2269} | 0 | 35 | 36 | 4332 sec |
Reduced | 193 | 183 | 389 | 2^{193} | 0 | 35 | 36 | 14 sec |
Components identification
From those 16834 perfect colorations we identified 15 components (Fig. 8-b). 11 components were composed of 1 node (1 gene for each component), 2 were composed of 2 nodes (1 gene for each component), one was composed of 422 nodes (with 167 genes) and the last component was composed of 1832 nodes (with 349 genes).
Components validation
Results for the components analysis. The “Validation p-value” refers to the comparison between real and randomized data. The “Specificity p-value” refers to the comparison between MC and NPC data
Component | # Nodes | # Genes | Validation p-value | Specificity p-value |
---|---|---|---|---|
C ^{2} | 422 | 167 | 8,904e-03 | 0.019 |
C ^{6} | 1832 | 349 | 7.91e-05 | 0.573 |
In order to validate the similarity computing, we generated for each dataset, 5 randomized datasets by scrambling observed signs. As previously, for each randomized dataset, we computed the MS with the components configuration. Then, for each component, we compared the MS between real data and randomized data with a Welch’s t-test (Table 4, Validation p-value). Both components have a p-value lower than 0.05, allowing us to conclude to a statistical significance.
Components specification
Biological results
Five first results of the Gene Ontology Enrichment Analysis for the component C^{2}
GO biological process | found | expected | Fold enrichment | P-value |
---|---|---|---|---|
regulation of cell death | 75 | 11.98 | 6.26 | 6.46E-37 |
regulation of programmed cell death | 73 | 11.21 | 6.51 | 8.33E-37 |
regulation of apoptotic process | 72 | 11.11 | 6.48 | 4.90E-36 |
single-organism cellular process | 149 | 77.70 | 1.92 | 9.90E-28 |
positive regulation of metabolic process | 87 | 24.50 | 3.55 | 7.81E-26 |
Five first results of the Gene Ontology Enrichment Analysis for the component C^{6}
GO biological process | found | expected | Fold Enrichment | P-value |
---|---|---|---|---|
response to organic substance | 182 | 42.74 | 4.26 | 1.02E-68 |
response to chemical | 203 | 64.12 | 3.17 | 2.13E-57 |
response to oxygen-containing compound | 129 | 23.26 | 5.55 | 1.32E-56 |
positive regulation of biological process | 233 | 88.29 | 2.64 | 1.39E-55 |
regulation of cell proliferation | 132 | 25.67 | 5.14 | 1.98E-54 |
The genes included in C^{2} (Table 5) seem strongly associated with cell death pathways: the three first biological processes are linked to cell death. Nonetheless, those pathways are strongly implicated in cancer disease [29]. On the other side, the component C^{6} (Table 6) does not look associated to redundant pathways since we cannot associate genes in the component C^{6} with a specific pathway. We notice however that C^{6} will describe cellular events linked with cell proliferation.
Comparison with other clustering methods
Where N_{i} stands for the number of genes in the cluster i. We compute the CQ with 5 clusterings: 2 obtained using ClusterONE (CO_{1} and CO_{2}), 2 obtained using the fuzzy c-means algorithm (FA_{1} and FA_{2}), and the last based on our component identification algorithm (CI). The parameters used to obtain the clusters and GO enrichment analysis were set as follows. For CO_{1} we used the basic parameters while we imposed to identify 2 clusters for CO_{2}. For FA_{1} and FA_{2} we imposed the cluster search fixing 2 centers. In the case of FA_{1} we used overlapping genes for the GO enrichment analysis. We removed those overlapping genes for FA_{2}. For our method, due to the fact that only the components 2 and 6 had more than one gene we consider the other components as outliers.
Results of the comparison with other clustering methods
Clustering method | #clusters | #enriched clusters | #genes | μ ^{ S E} | σ ^{ S E} | CQ | Loss information ratio |
---|---|---|---|---|---|---|---|
C0_{1} | 105 | 24 | 344 | 0.10 | 0.155 | 37.82 | 35.8% |
C0_{2} | 2 | 2 | 101 | 0.069 | < 0.001 | 6.96 | 88.2% |
F A _{1} | 2 | 2 | 688 | 0.065 | 0.02 | 44.94 | 23.8% |
F A _{2} | 2 | 2 | 380 | 0.089 | 0.008 | 33.66 | 42.9% |
CI | 15 | 2 | 511 | 0.089 | 0.006 | 46.17 | 21.6% |
PID-NCI graph | 1 | 1 | 524 | 0.11 | ∅ | 58.93. | 0% |
Conclusion
In this study, we proposed a method that imposes constraints to model graph coloration on biological signaling and regulatory networks. This method is able to reduce a regulatory network to subparts called components. These components describe network variables that are independent from others in the context of the perfect coloring constraints. Moreover, by using observations, we can select some of those components based on the maximal similarity between components configurations and those observations. The main points where our method is different from other subgraph identification methods are: (i) our method extracts network subcomponents by considering only the network logic (causality and inhibition/activation roles), while other methods consider topological features without logic; (2) the order of the analysis, our method first extracts logic network subcomponents states (harmonious colorings) and then confront these states to gene expression profiles (GEPs), adding less bias to the network v.s. data confrontation; and (3) when in a later step we integrate GEPs, we do it by locating GEPs measurements in the transcriptional layer, without overlapping transcriptional regulation with post-translational regulation. Using our method we were able to represent the species state variations (colorings) of a subgraph of the PID-NCI signaling and regulatory network (2269 nodes and 2683 edges) with 15 components. Each component will aggregate molecular-species having the same state-shift behavior given the PID-NCI graph topology. Only two (C^{2} and C^{6}) of these 15 components include more than two molecular species nodes. From GO enrichment analyses, C^{2} is strongly associated to cell death pathways, this biological process is robustly associated to cancer. The C^{6} component cannot be associated to any specific pathway of cancer. Interestingly, this component specification was done independently of the GEP up-/down-regulation states. We have compared the identification of these 2 components by our method with respect to 4 other clustering results obtained with two different clustering methods on the same data. Our results show that our method retrieves larger and meaningful information, in the context of GO annotations associated to the genes within these components or clusters, than these other approaches.
When comparing the 611 gene expression profiles from myeloma cells, and healthy donors and shuffled data with the the genes present in the 15 components, we observed that C^{2} and C^{6} were the components which were significantly more specific to real data. Also, C^{2} was having a significant statistical specificity when compared unhealthy and healthy expression profiles.
Our method seems efficient to identify and select functional components specific to the gene expression profiles used in our study taking into account the computational complexity that represents analyzing large-scale networks. However in this case study the reduction to 15 components, with two validated ones with respect to shuffled data, does not allow us to provide a deeper understanding, especially with respect to the subtypes of patients based on the overall survival. As a perspective of this work, we wish to improve the subcomponent identification in order to be able to compute larger regulatory networks, and potentially full databases. For this purpose, we would like to implement the components identification in ASP. Another research line will be to apply this method to other data (regulatory network and observations data) as well as to model with other more refined modeling frameworks the subcomponent C^{2} to investigate the patient subtypes overall survival. One last perspective of this study could be to explore those targets wich are perfectly colored in all GEPs. This identification could be another strategy to improve the space solution reduction.
Notes
Acknowledgements
Tis study was supported by Intergroupe Francophone du Myélome and by a French Institute National du Cancer Grant EVACAMM PROG/09/10 (to H.A.L., S.M.), a National Institutes of Health Grant PO1CA155258-01 (to S.M., H.A.L., N.C.M.), and a research grant from Celgene. B.M.’s PhD scholarship was funded by GRIOTE project. We would like to thank Elise Douillard, Magali Devic, Emilie Morenton and Nathalie Roi for excellent technical assistance. We are most grateful to the bioinformatics core facility of Nantes (BiRD - Biogenouest) for its technical support.
Funding
Publication cost of this article was funded by the “Pays de la Loire” Regional GRIOTE project.
Availability of data and materials
Minimum Information About a Microarray Experiment-compliant data has been deposited at: Gene Expression Omnibus with accession number GSE83503. All graphs used in this study are available online using cynetshare. The subgraph extracted from NCI-PID (goo.gl/jGAxhg). The graph illustrated in the Fig. 8b (goo.gl/5V5uOZ) The subgraphs with the nodes in the component 2 (goo.gl/o9Ah7L) and the component 6 (goo.gl/x47OL9). The Cytoscape session containing all graphs is available at goo.gl/9XwBnL. The workflow implementation is available at github.com/BertrandMiannay/Iggy-POC.
About this supplement
This article has been published as part of BMC Systems Biology Volume 12 Supplement 3, 2018: Selected original research articles from the Fourth International Workshop on Computational Network Biology: Modeling, Analysis, and Control (CNB-MAC 2017): systems biology. The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume-12-supplement-3.
Authors’ contributions
BM implemented the perfect coloring model, the components identification and the MS computing, performed the computational analysis and wrote the paper. BM and CG conceived and supervised the study, BM, SM, FM and CG discussed the results of the data analysis, and drafted the manuscript. BM and CG wrote the paper. All authors read and approved the final manuscript.
Ethics approval and consent to participate
The experiments were undertaken with the understanding and written informed consent of each subject.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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