Modular Control of Boolean Network Models

Abstract

The concept of control is crucial for effectively understanding and applying biological network models. Key structural features relate to control functions through gene regulation, signaling, or metabolic mechanisms, and computational models need to encode these. Applications often focus on model-based control, such as in biomedicine or metabolic engineering. In a recent paper, the authors developed a theoretical framework of modularity in Boolean networks, which led to a canonical semidirect product decomposition of these systems. In this paper, we present an approach to model-based control that exploits this modular structure, as well as the canalizing features of the regulatory mechanisms. We show how to identify control strategies from the individual modules, and we present a criterion based on canalizing features of the regulatory rules to identify modules that do not contribute to network control and can be excluded. For even moderately sized networks, finding global control inputs is computationally challenging. Our modular approach leads to an efficient approach to solving this problem. We apply it to a published Boolean network model of blood cancer large granular lymphocyte (T-LGL) leukemia to identify a minimal control set that achieves a desired control objective.

Keywords: Boolean networks; Canalization; Control; Gene regulatory networks; Modularity.

Fig. 1

Control via modularity. First, the network is decomposed into its constituent modules: $F_1,\cdots ,F_m$. Then, controls $u_1,\cdots ,u_m$ are identified for each module. Combining the controls of the modules $u=(u_1,\dots ,u_m)$ yields a set of controls for the whole network.

Fig. 2

Wiring diagram and state space of the Boolean network in Example 2.4-2.11. (a) The wiring diagram encodes the dependency between variables. (b) The state space is a directed graph with edges between all states and their images. This graph therefore encodes all possible trajectories.

Fig. 3

Boolean network decomposition into modules. (a) Wiring diagram of a non-strongly connected Boolean network where the modules are highlighted by amber and green boxes. (b) Directed acyclic graph describing the corresponding connections between the modules.

Fig. 4

Wiring diagram, directed acyclic graph, and state space for the network in Examples 3.13 and 3.17. (a) Wiring diagram where the modules are highlighted by amber and green boxes. (b) Directed acyclic graph describing the corresponding connections between the modules. (c) State space of the network generated with Cyclone (Dimitrova et al. 2023).

Fig. 5

Wiring diagram and directed acyclic graph for the network in Examples 3.15. (a) Wiring diagram where the modules are highlighted by amber and green boxes. (b) Directed acyclic graph describing the corresponding connections between the modules.

Fig. 6

Example of a modular directed acylic graph structure to illustrate Condition (iii) in Theorems 4.11 and 4.13. (a) Module $F_i$ can be removed from the control search as long as conditions (i) and (ii) are satisfied, and the phenotype of interest depends only on any subset of variables that are part of the blue modules. (b) When a node in module $F_m$ is regulated by a node in module $F_k$ (indicated by the red edge in the directed acyclic graph), the phenotype may no longer depend on nodes in $F_m$, in order for module $F_i$ to be removable from the control search.

Fig. 7

Control via modularity and canalization. Once the network is decomposed into modules $F_1,\dots ,F_m$, we can override the effect of module $F_i$ by the using another module ($F_k$ in this case) whose variables are inputs of $f_x$ that are located in more dominant layers than the layers containing the variables of $F_i$.

Fig. 8

(a) Wiring diagram of the T-LGL model, published in Saadatpour et al. (2011), which describes the mechanisms that regulate apoptosis. The non-trivial modules (i.e., the modules containing multiple nodes) are highlighted by amber, green, and gray boxes. (b) The regulatory inputs of the node DISC. (c) Writing the regulatory function corresponding to node DISC in its extended monomial form (Theorem 4.5) reveals its canalizing structure.