Multi-stage physics-informed neural networks for JAK--STAT5 signaling and ultradian insulin--glucose dynamics: latent-species identifiability and suppression of parameter-induced divergence

Abstract

Coupled diffusion-reaction partial differential equations (PDEs) describe biochemical network dynamics but are difficult to solve for realistic multi-species systems without combining mechanism and data. We present a multi-stage physics-informed neural network (PINN) for multi-species diffusion-reaction PDEs and apply it to two ordinary-differential-equation (ODE) reference systems: the Boehm et al. JAK-STAT5 signaling pathway and the Sturis ultradian insulin-glucose model. For STAT5 we pose a latent-species identifiability test: given sparse observations of eight species, a ten-species model that retains two deliberately withheld but mechanistically standard components--an active receptor-JAK complex and the SOCS negative-feedback inhibitor--recovers the reference trajectory and reduces mean root-mean-square error 3.1-fold relative to an eight-species model that omits them, whereas a PDE-only solution without data anchoring diverges. Because the reference is itself ODE-generated, this demonstrates identifiability against synthetic data, not the discovery of new biology. For the insulin-glucose model the same framework reproduces the [~]120-minute oscillation to 1.0% mean relative error as a benchmark on a stiff, multi-timescale oscillator; its spatial dimension is treated as a numerical construct, not a physical transport setting. A Lyapunov analysis of the STAT5 ODE returns a maximal exponent statistically indistinguishable from zero ({lambda}max {approx} 3.61 x 10-5 min-1, 5/8 trials positive; Lyapunov time [~]1.9 x 104 min, far exceeding the 240-720 min horizon), so the system is effectively non-chaotic and the relevant instability is a bounded, parameter-induced trajectory divergence. Anchoring the solution to baseline data suppresses this divergence, with the reduction growing monotonically with sampling density--from [~]15-19% at eight time points to [~]88-97% at sixty-four, depending on perturbation magnitude. The framework thus offers a data-anchored route to latent-species identifiability and divergence suppression in biochemical ODE/PDE systems, demonstrated here against synthetic reference data. Inside cells, a three-dimensional chemistry of diffusing, reacting molecules drives signaling and rhythm--dynamics that, for realistic networks, strain conventional solvers. Here a multi-stage physics-informed neural network--machine learning constrained by the governing equations--solves stiff, multi-species reaction systems from sparse data. In the JAK-STAT5 signaling pathway, a model that retains two standard but unobserved components (an active receptor complex and a negative-feedback brake) recovers a reference trajectory that a reduced model cannot--a controlled test of whether sparse data can pin down withheld pecies, not a claim of new biology. The same framework reproduces the roughly two-hour insulin-glucose rhythm to within 1% as a benchmark on a stiff oscillator. And anchoring the solution to a few dozen baseline measurements collapses parameter-induced trajectory divergence, turning a parametrically sensitive simulation into a stable one. Where mechanism and data meet, sparse measurements can constrain the structure a model would otherwise leave undetermined.