From Individual Trajectories to Population Densities: Weak-Form Inference of Heterogeneous Growth Laws

Single-cell time-lapse microscopy now provides high-resolution size trajectories for thousands of individual cells, yet principled methods for simultaneously learning the functional form of growth laws and the distribution of cell-to-cell physiological variability remain limited. We present a data-driven framework based on the recent Weak-form Estimation of Nonlinear Dynamics (WENDy) algorithm to (i) select the best supported growth law from a set of candidate laws, and (ii) recover individual-level growth parameters via empirical Bayes shrinkage within a random-effects model. We demonstrate the approach on both synthetic individual size data incorporating commonly found growth laws and recorded real size trajectories, showing that the method accurately recovers both growth-law structure and parameter heterogeneity at reasonable noise levels. We finally end with a discussion on how the results obtained by the method here can be upscaled to the population level.