A meta-analysis of Boolean network models reveals design principles of gene regulatory networks

TL;DR

A meta-analysis of this largest repository of expert-curated Boolean GRN models reveals several design principles, including more canalization, redundancy, and stable dynamics than expected.

Abstract

Gene regulatory networks (GRNs) play a central role in cellular decision-making. Understanding their structure and how it impacts their dynamics constitutes thus a fundamental biological question. GRNs are frequently modeled as Boolean networks, which are intuitive, simple to describe, and can yield qualitative results even when data is sparse. We assembled the largest repository of expert-curated Boolean GRN models. A meta-analysis of this diverse set of models reveals several design principles. GRNs exhibit more canalization, redundancy and stable dynamics than expected. Moreover, they are enriched for certain recurring network motifs. This raises the important question why evolution favors these design mechanisms.

Figures (19)

• Figure 1: Summary statistics of the analyzed GRN models. (A) Plot of the number of genes and external parameters for each model sorted by number of genes. (B-C) For each model, the number of genes is plotted against (B) the number of external parameters and (C) the average essential in-degree of the genes. The Spearman correlation coefficient and associated p-value are shown in red. (D) In-degree (red circles) and out-degree (black stars) distribution derived from all 5112 update rules. (E) Prevalence of each type of regulation (activation: blue, inhibition: orange, conditional: gray) stratified by the number of regulators (x-axis). Non-essential regulations are excluded.
• Figure 2: High prevalence of canalization. (A) Expected distribution of the canalizing depth for random Boolean functions for different numbers of essential inputs (1-10), based on 1000 random functions each. (B) Stratification of all identified update rules based on the number of essential inputs (rows) and the canalizing depth (columns). Update rules with more than ten inputs were omitted here, see table \ref{['tab:canalization_full']} for a full table. The color gradient in A and B is computed separately for each row. (C-D) The distribution of the (C) canalizing strength and (D) normalized input redundancy of all observed 0-canalizing functions with 3-6 essential inputs (that is, all functions in the blue box in B) is shown (blue), as well as the expected distribution for random Boolean functions (orange), derived from 1000 samples each. Horizontal lines depict the respective mean values.
• Figure 3: High prevalence of redundancy. The empirical distribution of the redundancy, measured by the number of symmetry groups (y-axis), is computed for all identified update rules (blue), stratified by the number of essential inputs (x-axis). For comparison, the expected distribution of the number of symmetry groups for random Boolean functions with 1-10 essential inputs is included (green), as well as the expected distribution for random Boolean functions with the same canalizing depth distribution as observed update rules (orange), as shown in Fig. \ref{['fig:canalization']}A. Each expected distribution was generated using 1000 random functions. fig. \ref{['fig:redundancy_detail']} contains the explicit values of each distribution.
• Figure 4: Abundance of coherent feed-forward loops. (A) Total number of the different types of FFLs in the 122 GRNs (colored bars). Conditional FFLs (gray) contain at least one conditional regulation preventing the determination of their exact type. Black horizontal lines indicate the respective expected number, which is based on null model 1 (see Methods). Type 1-4 FFLs are coherent, while type 5-8 FFLs are incoherent. (B) Proportion (stacked bar, color-coded as in A) and total number (black line) of the different types of FFLs for each network. The 17 networks without any FFLs are omitted. (C) For each target gene in a FFL (green), the edge effectiveness of the master regulator (blue) and the intermediate regulator (orange) is compared, stratified by the essential in-degree of the target gene. Horizontal lines depict the respective mean values. n = number of target genes with given essential in-degree, p = p-value from a two-tailed Wilcoxon signed-rank test.
• Figure 5: Abundance of clusters of feed-forward loops. Total number of the different types of clusters of FFLs in the 122 GRN models. Nodes in the motif graphs are color-coded based on their role in the two clustered FFLs: master regulators (blue), intermediate genes (orange), target genes (green), genes that appear in both FFLs but with a different role (gray).