This is somewhat outside my field, but basically it seems they're defining the independence of the three hypothetical metabolic pathways in two different ways:
* Mathematically the pathways are not independent since P₃ is a combination of P₁ and P₂.
* Biologically the pathways are independent since each requires a different combination of enzymes and therefore genes (the genotype). This is the method used in MPS.
Does that answer your question?