AI-driven constrained mathematical models of biological systems

Abstract

Biological systems are inherently complex, governed by nonlinear dynamics, feedback loops, multi-scale interactions, and hidden regulatory mechanisms. Accurately modeling such systems is essential for understanding processes like cellular growth, differentiation, signaling, and immune response. Traditional systems biology approaches rely on prior mechanistic knowledge to construct models using ordinary differential equations (ODEs). While powerful in well-characterized scenarios, these models are often difficult to apply in data-rich but poorly understood systems, as they require manual specification of the governing equations and assume complete knowledge of interactions. To overcome these limitations, we propose a hybrid data-driven framework that integrates classical dynamical systems theory with modern sparse regression techniques. Our approach begins with Takens’ embedding to reconstruct the system’s dynamics from time-series data using delay coordinates. We then apply Sparse Identification of Nonlinear Dynamical Systems (SINDy) to infer minimal and interpretable ODEs that govern the observed dynamics. Recognizing the prevalence of rational nonlinearities in biological systems, such as Michaelis–Menten and Hill kinetics, we extend our method using Implicit SINDy (SINDy-PI), which can identify models involving algebraic constraints and rational terms. We validate our framework on synthetic time-series data generated from a range of well-known biological models, including the FitzHugh–Nagumo, Goodwin, Oregonator, glycolytic oscillators, mass-action systems, Michaelis–Menten kinetics, and microbial growth dynamics. The results demonstrate accurate recovery of the underlying dynamical structure, phase space features, and network topologies. To assess real-world applicability, we apply the framework to two biologically relevant systems: (1) the eukaryotic cell cycle and (2) tumor–immune interactions relevant to cancer immunotherapy. From synthetic data emulating experimental observations, we successfully rediscover governing equations that predict system behavior and reveal key regulatory motifs. Sensitivity analysis and bifurcation techniques further confirm that the inferred models exhibit known transitions and critical points consistent with biological expectations. In summary, our method bridges the gap between mechanistic modeling and data-driven inference. It offers a unified, interpretable, and scalable approach to uncovering meaningful biological dynamics from time-series data. This work advances the field of systems biology by enabling dynamic modeling in settings with limited mechanistic knowledge, ultimately supporting improved understanding, hypothesis generation, and therapeutic design.