Enzyme A stands in for Phospholipase A2, turning phospholipids in the cell membrane into arachidonic acids. From there, Enzyme B stands for Cyclooxygenase (COX), while Enzyme C stands for 5-Lipoxygenase (5-LO). 5-LO produces 5-HPETE, which is turned into leukotrienes. Leukotrienes play a double role in events like asthma, because they cause bronchoconstriction (seen in this patient) and by acting as chemical attractants for inflammatory cells, whose activity will close the airway further. By blocking COX (presumably with an NSAID), we leave more of the substrate arachidonic acid for 5-LOX to ultimately convert into leukotrienes. This is the basis for aspirin-intolerant-asthma, which is frequently associated with nasal polyps. (Samter's triad).
D) Prostacyclin synthase . E) Thromboxane synthase