Observation fields and sharp reconstruction for mixtures of saturating compartments

We ask what the total load in r noninteracting saturating compartments determines about the individual compartments. For Michaelis-Menten kinetics with unknown rates, saturation constants and states, we determine the generic observation field of the pooled total. It consists of all rational functions of the rate, saturation constant and state triples that are invariant under simultaneous relabeling. We give generators that recover the pairing of each rate with its saturation constant and load. The total and its first 3r derivatives generate the field. For two compartments, derivatives through order five leave thirteen generic complex possibilities for the unordered triples. The sixth derivative resolves them. With known parameters the observation field is the fixed field of a group of compartment permutations. This description also holds for Haldane kinetics with a common known time scale and a common dimensionless inhibition parameter. For Michaelis-Menten compartments with a common known rate and saturation constant, an explicit linear system reconstructs the unordered states from the total and its first r-1 derivatives. This order is generically sharp.