Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2009 / B / 4 / c / Solution

Replace only the free-substrate equations byThe other four equations remain as above. The carrier invariant remains unchanged because no protein is added or removed. The two external substrate terms cancel in the total-substrate balance, so the second invariant also remains unchanged. This cancellation describes a balanced source and sink, even though there is a nonzero throughput. A constant sink is physically meaningful only while substrate concentrations remain nonnegative; it is not a state-dependent removal law valid after an interior reservoir is exhausted.