Context-Specific Nested Effects Models

Abstract

Advances in systems biology have made clear the importance of network models for capturing knowledge about complex relationships in gene regulation, metabolism, and cellular signaling. A common approach to uncovering biological networks involves performing perturbations on elements of the network, such as gene knockdown experiments, and measuring how the perturbation affects some reporter of the process under study. In this article, we develop context-specific nested effects models (CSNEMs), an approach to inferring such networks that generalizes nested effects models (NEMs). The main contribution of this work is that CSNEMs explicitly model the participation of a gene in multiple contexts, meaning that a gene can appear in multiple places in the network. Biologically, the representation of regulators in multiple contexts may indicate that these regulators have distinct roles in different cellular compartments or cell cycle phases. We present an evaluation of the method on simulated data as well as on data from a study of the sodium chloride stress response in Saccharomyces cerevisiae.

Keywords: context specific; graph; inference; nested effects models; network.

FIG. 1.

(A) An example of effect nesting in an NEM and (B) a partial intersection of effects as captured by a CSNEM. The table underneath each graph represents the differential expressions of the high-dimensional phenotypes observed in the screen, with rows corresponding to single-gene knockouts and each column corresponding to an effect, one dimension of a phenotype, where a “1” indicates that a perturbation changed the response of the effect, and a “0” indicates that it did not. CSNEM, context-specific nested effects model; NEM, nested effects model.

FIG. 2.

(A) A graph G, (B) the accessibility matrix $\Gamma$, (C) an attachment matrix $\Theta$, and (D) the resultant effect matrix $F=\Gamma \Theta$.

FIG. 3.

Building a CSNEM from a mixture of NEMs. (A) Three NEMs that compose a mixture. (B) A single graph obtained by an edge-preserving merge of the three NEMs. (C) The corresponding action-set graph.

FIG. 4.

The action-set graph of a nontransitive NEM cycle structure. (A) A nontransitive cycle NEM and its corresponding effect matrix. (B) The reversed cycle and its corresponding effect matrix. (C) The action-set graph that corresponds to both graphs.

FIG. 5.

Box plots of simulation F-measures. Each plot represents an aggregate of results from 30 random simulation replicates. Grid rows correspond to the number of contexts in the generating model, the x-axis in each of the grid cells indicates the number of contexts in the learned model, and the y-axis represents: (A) the F-measure of recovering the generating model's effect matrix from the learned model across different sizes of action sets (grid columns) from log-odds matrices generated with $\beta =10$, (B) the F-measure of recovering ancestry relationships, (C) the F-measure of learning the effect matrix of a 20-action network from log-odds matrices generated with varying settings of β (grid columns), and (D) the F-measure of learning the effect matrix from 10-action networks of varying density (grid columns) with log-odds generated using $\beta =10$.

FIG. 6.

The 3-CSNEM network learned from Saccharomyces cerevisiae NaCl stress knockout microarray data. Action nodes and action-action edges are colored according to the NEM member in the mixture from which they came, in cyan, magenta, or yellow. Nodes that were merged because of identical ancestors in multiple mixture members are colored according to subtractive color mixing (cyan and magenta make blue, cyan and yellow make green, magenta and yellow make red, and all three make black). Effects are colored and grouped according to the actions to which they are attached. Where the number of effects in a group is less than 10, the effects are listed. Where it is 10 or more, the number of effects in the group is shown. Action-action edges are solid and action-effect edges are dashed.

FIG. 7.

The NEM network learned from S. cerevisiae NaCl stress knockout microarray data. Action nodes are in black and effect nodes are in blue. Effects are grouped according to the actions to which they are attached. Where the number of effects in a group is less than 10, the effects are listed. Where it is 10 or more, the number of effects in the group is shown.

FIG. 8.

Comparison of effect group GO term enrichments. Columns correspond to GO terms and rows correspond to actions in the NEM and CSNEM, and to possible combinations of contexts of the 3-CSNEM. A point indicates that a GO term was found to be significantly enriched. Points are colored by knockout in the NEM and CSNEM plots and by context in the context-membership plot. GO, Gene Ontology.