Theoretically, our method is closely related to the minimal precursor sets of a metabolic network. The minimal precursor sets describe all minimal subsets of metabolites that can be used to synthesize a particular metabolite or set of metabolites. A combinatorial formula was developed to calculate the value of Pout from the set of minimal precursor sets (S) and the value of Pin. Using this formula, an exact producibility curve can be plotted and theoretical values of the producibility metric (PM) can be calculated. This calculation is shown for increasingly complicated and general minimal precursor set structures. (A) In the simplest case, a single metabolite serves as the minimal precursor set for the target. This leads to a straight line producibility curve (Pout = Pin) and an expected PM value of 0.5. (B) If the target metabolite can be produced from one minimal precursor set of size m Pout is equal to Pin raised to the power of m due to the assumption that all metabolites are added independently. As m increases the producibility curve bows outward towards Pin = 1, and the PM decreases below 0.5. (C) If the target metabolite can be produced from n different minimal precursor sets of size 1 Pout can be defined based on the probabilistic rule for an or relationship, or it can be represented with a combinatorial formula. represents the binomial coefficients for n choose i. As n increases the producibility curve bows inward towards Pin = 0, and the PM increases above 0.5. (D) In the most general case, minimal precursor sets can be overlapping and any size/number. The producibility curve and PM vary depending on these properties. A combinatorial formula can be used to represent this general case. is the cardinality of the set of minimal precursor sets (the total number of minimal precursor sets). is the cardinality of the union of the minimal precursor sets being enumerated by the recursive sum (the number of unique metabolites in the combination of minimal precursor sets being enumerated). The recursive sum over ji enumerates all possible combinations of i minimal precursor sets by summing over the indices ji where ji is between ji-1 and , thus enumerating binomial combinations. The outer sum implements the inclusion exclusion principle.