Compatibility of competing biochemical modules: capacity, recovery, and finite operation
Abstract
When two biochemical modules draw on one replenished resource and keep private chemical states, “compatible” is not one property. We separate three questions and answer each for an explicit eight-state model in which a glutathione peroxidase branch and a peroxiredoxin–thioredoxin branch share NADPH. Capacity: stationary branch responses decide whether a joint operating point exists; this sharp criterion is inherited from a companion paper. Recovery: we prove that every state of an explicit preparation region has a unique physical solution that meets both service quotas permanently after ms, with exact bounds on shortfall and regeneration expenditure; this statement is verified in Lean 4. An exact-arithmetic tube certificate extends it to every preparation within of a service-deficient state in each concentration independently. In contrast, a one-parameter family of repair kinetics has identical stationary states at every source strength but a necessary recovery delay growing like , so no deadline can be read from stationary data. Duration: an exact storage account shows that a finite regeneration donor cannot support positive service forever, and that finite missions are decided by the actually depleted trajectory. For a one-second mission under a linear donor law, a conservative fixed-band argument needs a M stock; a moving-tube certificate proves that suffices and that fails, from the same preparations. During the successful mission the source falls below the stationary compatibility threshold for more than half the time, while both quotas remain met: stored carrier, not instantaneous capacity, pays the difference. Stationary thresholds therefore can both underestimate readiness time and overestimate finite-mission supply. All conclusions concern the declared maintained model; they assert no global attraction, biological calibration, or clinical meaning.