A novel structure-based control method for analyzing nonlinear dynamics in biological networks

Exploring complex biological systems requires adequate knowledge of the system’s underlying wiring diagram but not its specific functional forms. Thus, exploration actually requires the concepts and approaches delivered by structure-based network control, which investigates the controllability of complex networks through a minimum set of input nodes. Traditional structure-based control methods focus on the structure of complex systems with linear dynamics and may not match the meaning of control well in some biological systems. Here we took into consideration the nonlinear dynamics of some biological networks and formalized the nonlinear control problem of undirected dynamical networks (NCU). Then, we designed and implemented a novel and general graphic-theoretic algorithm (NCUA) from the perspective of the feedback vertex set to discover the possible minimum sets of the input nodes in controlling the network state. We applied our NCUA to both synthetic networks and real-world networks to investigate how the network parameters, such as the scaling exponent and the degree heterogeneity, affect the control characteristics of networks with nonlinear dynamics. The NCUA was applied to analyze the patient-specific molecular networks corresponding to patients across multiple datasets from The Cancer Genome Atlas (TCGA), which demonstrates the advantages of the nonlinear control method to characterize and quantify the patient-state change over the other state-of-the-art linear control methods. Thus, our model opens a new way to control the undesired transition of cancer states and provides a powerful tool for theoretical research on network control, especially in biological fields. Author summary Complex biological systems usually have nonlinear dynamics, such as the biological gene (protein) interaction network and gene co-expression networks. However, most of the structure-based network control methods focus on the structure of complex systems with linear dynamics. Thus, the ultimate purpose to control biological networks is still too complicated to be directly solved by such network control methods. We currently lack a framework to control the biological networks with nonlinear and undirected dynamics theoretically and computationally. Here, we discuss the concept of the nonlinear control problem of undirected dynamical networks (NCU) and present the novel graphic-theoretic algorithm from the perspective of a feedback vertex set for identifying the possible sets with minimum input nodes in controlling the networks. The NCUA searches the minimum set of input nodes to drive the network from the undesired attractor to the desired attractor, which is different from conventional linear network control, such as that found in the Maximum Matching Sets (MMS) and Minimum Dominating Sets (MDS) algorithms. In this work, we evaluated the NCUA on multiple synthetic scale-free networks and real complex networks with nonlinear dynamics and found the novel control characteristics of the undirected scale-free networks. We used the NCUA to thoroughly investigate the sample-specific networks and their nonlinear controllability corresponding to cancer samples from TCGA which are enriched with known driver genes and known drug target as controls of pathologic phenotype transitions. We found that our NCUA control method has a better predicted performance for indicating and quantifying the patient biological system changes than that of the state-of-the-art linear control methods. Our approach provides a powerful tool for theoretical research on network control, especially in a range of biological fields.