= Solution
Replace only the free-substrate <equations> by
$$
\boxed{\dot s_i=-B_i-J,\qquad\dot s_e=-B_e+J.}
$$
The 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.
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