Symbolic regression enables coarse-grained model discovery of intracellular signalling dynamics

Abstract

Cells respond to their environment through protein networks often dysregulated in cancer, making dynamical modelling crucial. Limitations in experimental data and computational resources motivate coarse-graining methods to build low-dimensional descriptions. Yet classical approaches to coarse-grained modelling rely on strong assumptions, leaving it unclear when partial experimental observations support reduced descriptions of system dynamics. Here we show that symbolic regression (SR) provides a data-driven way to test whether, and how compactly, the dynamics of a signalling system coarse-grain over the measured variables, and, when they do, infers mechanistically interpretable models. In synthetic enzyme systems, SR recovers Michaelis–Menten kinetics for the two-step mechanism and under three-step extensions. As data quality is degraded, SR simplifies toward effective kinetic laws while preserving correct theoretical limits. Applied to published time-resolved ERK phosphorylation data, SR identifies compact phospho-ERK rate laws in selected cancer-relevant gene overexpression contexts, yielding interpretable kinetic effects. A sparse neural ODE baseline requires few inputs where SR succeeds, but on average more where it fails, indicating that, where a reduced model is learnable at all, SR failure is associated with more complex dynamics that a simple mathematical model cannot describe. Together, these findings establish symbolic regression as a way to test when a compact coarse-grained description is warranted, generating hypotheses where one holds and motivating potential new measurements where it does not. Significance statement Cells process information through complex biochemical networks whose governing equations are usually unknown. Automated equation discovery has succeeded in physics and well-sampled biology, but whether it extends to sparse, noisy, partial data typical of these networks remains unclear. Here, from single-cell measurements of a growth-signalling pathway across dozens of cancer-relevant perturbations, we show it recovers compact models in many of them, some reproducing known regulatory biology and others proposing novel, testable hypotheses. A more expressive deep-learning baseline succeeds more often, but equation discovery’s failures cluster where that baseline uses more measured variables than a compact law can hold. Reducibility to a compact law becomes a measurable property, not a modelling assumption, indicating when dynamics compress and when they do not.