Inferring biological networks by sparse identification of nonlinear dynamics

Abstract: 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.

• The paper introduces implicit-SINDy, an extension of the SINDy algorithm using sparsity-promoting optimization to infer biological networks, specifically addressing rational function nonlinearities prevalent in these systems.
• Validated on models including enzyme kinetics, bacterial competence regulation, and yeast glycolysis, implicit-SINDy accurately reconstructs network dynamics and identifies correct parameters.
• This method offers a scalable, computationally efficient approach for biological network inference, reducing the need for prior knowledge and accelerating model discovery in systems biology.

Inferring Biological Networks by Sparse Identification of Nonlinear Dynamics

The paper "Inferring Biological Networks by Sparse Identification of Nonlinear Dynamics" introduces a novel methodological advancement tailored for the inference of complex biological networks. The proposed approach, termed implicit-SINDy, extends the framework of the sparse identification of nonlinear dynamics (SINDy) algorithm, accommodating rational function nonlinearities to model the dynamics of biological systems. This enhancement addresses the prevalent challenge of rational nonlinearities in the modeling of metabolic and regulatory networks, which traditional polynomial-based models fall short of encapsulating adequately.

Methodological Framework

The implicit-SINDy method diverges from classical information-theoretic approaches for model selection, which involve evaluating a combinatorial space of possible network configurations. Instead, it employs sparsity-promoting $\ell_1$ optimization to discern the minimal subset of nonlinear interactions within a fully connected network framework. The methodology is particularly adept in systems characterized by underlying rational functions, commonly encountered in biological networks due to steady-state assumptions or time-scale separation.

To operationalize this methodology, the paper delineates a workflow wherein implicit dynamic equations are formulated by constructing an augmented library of non-linear terms derived from both state variables and their derivatives. Specifically, the augmented library facilitates the incorporation of potential rational functions in the form of implicit expressions. Subsequently, the research details the application of an alternating directions method (ADM) to extract the sparsest vector in the null space of the library, thereby identifying active terms that drive the network dynamics.

Validation on Biological Models

The robustness and applicability of the implicit-SINDy method are validated against three fundamental biological models representative of different classes of network dynamics:

1. Michaelis-Menten Enzyme Kinetics: By deriving implicit forms for the kinetics, the method accurately infers the rational expressions integral to the enzymatic reaction processes.
2. Regulatory Networks for Bacterial Competence: The framework is applied to infer bi-stable gene regulatory circuits in bacterial competence, successfully identifying both positive and negative feedback loops crucial to the regulation of these systems.
3. Metabolic Networks for Yeast Glycolysis: The method extends to a seven-node metabolic network illustrating glycolytic oscillations in yeast, reconstructing complex models comprising both rational and polynomial dynamics accurately.

In all cases, implicit-SINDy exhibits proficiency in extracting the correct model parameters with high precision, thus endorsing its potential as an automated inference strategy for biological networks.

Implications and Future Directions

The theoretical implications of this research are substantial, promising a scalable and computationally viable approach to model discovery in systems biology, where classical brute-force methods become untenable. Implicit-SINDy provides an attractive alternative, especially given the limited need for prior knowledge of the network's nonlinearities. The method elegantly side-steps the computational burdens associated with projecting traditional information criteria across expansive combinatorial landscapes, instead positioning itself as an efficient machine-learning inspired tool for biological model inference.

From a practical standpoint, the advancement equips researchers with a streamlined technique to harness the wealth of experimental data emerging from modern biological studies, potentially accelerating discoveries in metabolic engineering and disease modeling. However, addressing challenges such as increased robustness against noise and optimizing sampling requirements remain meaningful avenues for ongoing research.

Ultimately, implicit-SINDy augments the repertoire of tools available for discerning intricate biological networks. As biological data continues to proliferate, the method holds promise for significant contributions to the understanding of cellular functionalities and control mechanisms, paving the way for refined models and improved therapeutic interventions across the biological sciences.