Inverse scattering theory describes the conditions necessary and sufficient to determine an unknown potential from known scattering data. No similar theory exists for when and how one may deduce the kinetics of an unknown chemical reaction from quantitative information about its final state and its dependence on initial conditions-except it is known to be impossible for equilibrium reactions. This article presents a case study of a far-from-equilibrium reaction: it presents a systematic phenomenological analysis of experimental time series for the amount of final product, a biopolymer, formed from various initial concentrations of monomers. Distinct mathematical properties of the kinetics of the unknown reaction pathway are found. These properties are shown to restrict the kinetics to a single model that generalizes Oosawa's classical nucleation-polymerization model. The methods used here to analyze the self-assembly of microtubules from tubulin are general, and many other reactions and processes may be studied as inverse problems with these methods when enough experimental data are available.