Topological estimation of signal flow in complex signaling networks

Abstract

In a cell, any information about extra- or intra-cellular changes is transferred and processed through a signaling network and dysregulation of signal flow often leads to disease such as cancer. So, understanding of signal flow in the signaling network is critical to identify drug targets. Owing to the development of high-throughput measurement technologies, the structure of a signaling network is becoming more available, but detailed kinetic parameter information about molecular interactions is still very limited. A question then arises as to whether we can estimate the signal flow based only on the structure information of a signaling network. To answer this question, we develop a novel algorithm that can estimate the signal flow using only the topological information and apply it to predict the direction of activity change in various signaling networks. Interestingly, we find that the average accuracy of the estimation algorithm is about 60-80% even though we only use the topological information. We also find that this predictive power gets collapsed if we randomly alter the network topology, showing the importance of network topology. Our study provides a basis for utilizing the topological information of signaling networks in precision medicine or drug target discovery.

Figure 1

Illustration of a complex signaling network. (a) The topology of an exemplary signaling network. (b) The outcome of signaling. Red and blue links represent activating and inhibiting signals, respectively. Red and blue nodes indicate up-regulated and down-regulated nodes, respectively.

Figure 2

Topological information and signal flow. (a) A biochemical reaction such as phosphorylation of a protein in signaling networks can be represented by a directed link with a sign (i.e., a signed edge of a digraph). Activation and inhibition are denoted as plus (+) and minus (−) signs, respectively. (b) Signal flow is estimated by calculating the multiplication of the link weight and the activity of source node. (c) There are four types of signal flow. The sign of link and the sign of signal flow can be same or opposite depending on the source node activity and link weight where pointed arrow (→) indicates a positive weight and blunt arrow (⊣) denotes a negative weight.

Figure 3

A toy example to explain how signal propagation algorithm works. (a) Toy example network and its link weight normalization. (b–d) The temporal evolution of activities and signal flows, (b) when node A is activated, or (c) when node A is activated and node E is inhibited. (d) The net effect of inhibiting node E calculated by comparing the results of (b) and (c). The real numbers in (b) and (c) denote the log-activity, x. The real numbers in (d) denote the difference between the log-activities of (b) and (c). The colors of nodes and links denote the relative quantity of the activities and signal flows. Red and blue colors of circles represent positive (up-regulated) and negative (down-regulated) activities, respectively. Red and blue colors of arrows denote positive (activating) and negative (inhibiting) signal flows, respectively. The values for basal activities of nodes A and B were assigned +1 and −1 (i.e., b_A =+1 and b_B = −1), respectively. The initial state, x(t = 1), was equal to the basal activity, b in each case, and the hyperparameter, α, is set to 0.5.

Figure 4

Workflow for testing the algorithm. A single dataset includes the topological information of a signaling network and perturbation conditions. Signal propagation algorithm estimates the signal flow using the network topology, and predicts the DAC. Accuracy is calculated as evaluation statistics by comparing the prediction results of the algorithm with the actual DAC of biomolecules. The hyperparameter, α = 0.5 was used and link weight normalization was applied, unless it is explicitly denoted.

Figure 5

Overall accuracy of the algorithm. (a) The accuracies of predicting the DAC for the six datasets. (b) The accuracies of predicting the DAC under the degree-preserving randomization of network topology. In the randomization, the sign of a link is flipped or the targets of two links are swapped. (c) The accuracies of randomizing link weights based on the sampling policy. D: decay links only, weight ~ (0.001, 1); D + A: both decay and amplification links, weight ~ (0.001, 1000); N: link weight normalization is applied.

Figure 6

Network topology and hierarchical clustering result of B2009. (a) The network topology of B2009 where EGFR and IR signaling pathways are interconnected. Red nodes are the perturbation targets. (b) The hierarchical clustering of the average accuracies for the 200 sub-datasets of B2009. A single data element of the table represents the average accuracy for predicting the DAC of the readout across the 200 sub-datasets.

Figure 7

In-depth analysis of B2009. Perturbation conditions that SP algorithm failed to accurately predict GS-RAS-RAF-MEK-ERK cascade (a,c,e). The modification of network topology (removal of the link between AKT and RAF) and the adjustment of the four link weights improved the results (b,d,f). Signal flows and node activities are visualized for the perturbation of IRS and PIP3 (c,d) and the perturbation of RAF and IRS (e,f) before and after modifying the network topology and adjusting the link weights. In the adjustment of weights, the original weight values of four links (ERK to GS, ERK to GAB1, PI3K to PIP3, and PIP3 to IRS) were multiplied by 20, 20, 1.2, and 2, respectively after the weight normalization.