Global Dynamical Structure Reconstruction from Reconstructed Dynamical Structure Subnetworks: Applications to Biochemical Reaction Networks

Two key variables that often determine the behavior of a dynamical system are its network structure and parametric realization. The structure of the network generally is determined by how states in the system causally depend on each other; edges in the network are determined by causal dependence while nodes are determined by the states of the system. Network structure alone does not determine dynamical behavior, though, parametric information is also important in determining what dynamical behaviors a system can achieve. Rather, network structure, or topology, often defines or narrows the possible behaviors a system can achieve. Without any structural constraints, a dynamical system can have arbitrary input-output behavior. Once network structure is imposed, the set of realizable input-output trajectories can be reduced. This is particularly evident in biological networks; certain network topologies are referred to as network motifs [3]. In systems and synthetic biology, these network motifs are broadly accepted as enabling useful dynamical behavior. For example, an incoherent feed forward loop can be used for fold-change detection or adaptation, a cyclic network of repressors is associated with either oscillations or multistability, and a dual negative feedback network of two nodes is used as memory module or toggle switch. Network structure is thus an important aspect of designing synthetic biological circuits. By selecting an appropriate network motif and validating its functionality in practice, synthetic biologists are able to guide the phenotype of biological systems to match desired performance specifications. It thus seems that the choice of network structure between engineered systems or even fundamental physical components is an important design variable to be considered. How components are interconnected implicitly defines network structure, which in turn constrains dynamical behavior of the system. Certain network structures can give rise to undesirable dynamic behavior [4]. Choosing the right network structure is thus an important problem in the synthesis of robust engineered dynamical systems. Similarly, once a dynamical system has been designed and implemented, verifying that the network structure of a dynamical system is operating as designed is an equally important problem. This is especially critical, when the engineered system does not behave as expected (a pervasive challenge in current efforts to implement synthetic biocircuits) [5]. The problem of verifying or reverse-engineering a system’s network structure from measurement data is called a network reconstruction problem. Network reconstruction problems are a specific class of system identification problems, where the model class of interest not only encodes parametric but structural information. In the next section we motivate and formulate the network reconstruction problem for different network representation models and argue that one particular representation is well suited for biochemical reaction networks: the dynamical structure function.