The effective graph reveals redundancy, canalization, and control pathways in biochemical regulation and signaling

Abstract

The ability to map causal interactions underlying genetic control and cellular signaling has led to increasingly accurate models of the complex biochemical networks that regulate cellular function. These network models provide deep insights into the organization, dynamics, and function of biochemical systems: for example, by revealing genetic control pathways involved in disease. However, the traditional representation of biochemical networks as binary interaction graphs fails to accurately represent an important dynamical feature of these multivariate systems: some pathways propagate control signals much more effectively than do others. Such heterogeneity of interactions reflects canalization-the system is robust to dynamical interventions in redundant pathways but responsive to interventions in effective pathways. Here, we introduce the effective graph, a weighted graph that captures the nonlinear logical redundancy present in biochemical network regulation, signaling, and control. Using 78 experimentally validated models derived from systems biology, we demonstrate that 1) redundant pathways are prevalent in biological models of biochemical regulation, 2) the effective graph provides a probabilistic but precise characterization of multivariate dynamics in a causal graph form, and 3) the effective graph provides an accurate explanation of how dynamical perturbation and control signals, such as those induced by cancer drug therapies, propagate in biochemical pathways. Overall, our results indicate that the effective graph provides an enriched description of the structure and dynamics of networked multivariate causal interactions. We demonstrate that it improves explainability, prediction, and control of complex dynamical systems in general and biochemical regulation in particular.

Keywords: Boolean network; biochemical regulation; canalization; complex networks.

Fig. 1.

Constructing the effective graph. (A, Left) The interaction graph of automaton $x_4$ (green node), with $k=3$ input variables (blue nodes, $x_1,x_2,x_3$) and (A, Right) its corresponding Boolean logic given by the LUT, with bias $\rho \left(x_4\right)=1/4$. (B, Left) The effective graph of automata $x_4$ is built from the wild card redescription of the LUT (B, Right), $F^{\prime }$, which shows that input $x_3$ is always redundant (only wild cards in its column) and that $x_4=x_1\wedge x_2$. Edge thickness denotes edge effectiveness, $e_{ji}$, with the fully redundant edge shown in dashed red. The total input redundancy of automaton $x_4$ is $k_r\left(x_4\right)=1.75$, and therefore, its effective connectivity is $k_e\left(x_4\right)=1.25$.

Fig. 2.

Central tendency, variation, and heterogeneity of edge effectiveness of Boolean automata in biochemical regulation and random ensembles. (A) The distributions of edge effectiveness for ensembles of $10^4$ automata with degree $k=6$ at each bias $\rho$. (B) The distribution of edge effectiveness of the 630 incoming interactions to 105 automata with degree $k=6$ in Cell Collective models (green) compared with a bias-matched sample of random Boolean automata (pink). (C) The distributions of edge effectiveness Gini coefficients for inputs to automata in each of the random ensembles from A. (D) The distribution of edge effectiveness Gini coefficients for inputs to the 105 automata with degree $k=6$ in the Cell Collective models (green) compared with the bias-matched ensemble of random Boolean automata (cyan).

Fig. 3.

The effective graph captures the spread of perturbations. The predictive power of the edge-product approximation using the two models, ${\mathcal{M}}_{\text{IG}}$ (blue) and ${\mathcal{M}}_{\text{EG}}$ (red), measured by the Spearman rank correlation (vertical axis) with the total impact, ${\iota }_{ij}\left(t\right)$, sampled from $10^4$ trajectories, after $t$ steps (horizontal axis). The shaded region denotes one SD for a sample of 100 random networks and 10 perturbed nodes per network.

Fig. 4.

Study of the A. thaliana BN model. (A) The interaction graph for the A. thaliana BN. (B) The effective graph. Edge thickness denotes effectiveness, $e_{ji}$; dashed red indicates fully redundant edges (Table 1 shows parameter values); node color intensity denotes effective out-degree; and green nodes denote cases of null effective out-degree ($k_e^{out}=0$). (C) A threshold effective graph showing only edges with $e_{ji}\ge 0.4$ to enhance visibility of the largest connected component that allows $LFY$ to function as a master regulator and reveals that $WUS$ functions simply as an autoregulator; green nodes denote cases of null effective out-degree at this threshold level. (D) Spearman’s rank correlation (vertical axis) between the true impact of perturbing each node [${\iota }_{ij}\left(t\right)$] and respective path-length approximation predictions using the interaction (blue) and effective (red) graphs after $t$ steps (horizontal axis); $FUL$ cannot be computed (N/A, not available) because it has null impact on other variables (validating our observation of a fully redundant output).

Fig. 5.

Study of the ER+ breast cancer BN model. (A) Hierarchical rendering of the effective graph for the BN model of ER+ breast cancer. Edge thickness denotes its effectiveness, thresholded to $e_{ji}>0.2$; node color denotes constituent pathways (legend is in the top right corner). (B) Conditional effective graph with Alpelisib= ON (pinned state denoted with bold text and blue border), revealing how it renders much of the influence from $RTK$ (receptor tyrosine kinases) pathway redundant (red dashed edges) while fixing the state of several variables in the $PI3K$ pathway, such as the phospholipid $PIP3$ (phospholipid); variables whose state becomes fixed (constants) are denoted by a blue border, and edges that transmit a constant input state are denoted by a dashed blue color. (C) Spreading dynamics of perturbations to each of the seven drugs in the model and the proportion of network effectively reachable. (D) Effectiveness of outgoing edges of drug variables; $k^{out}$ and $k_e^{out}$ denote out-degree and effective out-degree, respectively.