Inferring biological networks by sparse identification of nonlinear dynamics

Inferring the structure and dynamics of network models is critical to understanding the functionality and control of complex systems, such as metabolic and regulatory biological networks. The increasing quality and quantity of experimental data enable statistical approaches based on information theory for model selection and goodness-of-fit metrics. We propose an alternative method to infer networked nonlinear dynamical systems by using sparsity-promoting $\ell_1$ optimization to select a subset of nonlinear interactions representing dynamics on a fully connected network. Our method generalizes the sparse identification of nonlinear dynamics (SINDy) algorithm to dynamical systems with rational function nonlinearities, such as biological networks. We show that dynamical systems with rational nonlinearities may be cast in an implicit form, where the equations may be identified in the null-space of a library of mixed nonlinearities including the state and derivative terms; this approach applies more generally to implicit dynamical systems beyond those containing rational nonlinearities. This method, implicit-SINDy, succeeds in inferring three canonical biological models: Michaelis-Menten enzyme kinetics, the regulatory network for competence in bacteria, and the metabolic network for yeast glycolysis.