Use steady or initial-rate conditions with effectively clamped substrate concentrations. For either substrate

X \in {S, R}

, put

U_{X} = [A X]

and

V_{X} = [A X^{*}]

. The intermediate balances are

0 = k_{+}^{X} [A] [X] - (k_{-}^{X} + m) U_{X}, 0 = m U_{X} - (l_{-}^{X *} + V_{A}) V_{X} .

(1)

The product flux is therefore

J_{X} = V_{A} V_{X} = \frac{k _{+}^{X} [ A ] [ X ] m V _{A}}{( k _{-}^{X} + m ) ( l _{-}^{X *} + V _{A} )} .

(2)

The free enzyme concentration is common to the two competing pathways, so it cancels even when enzyme sequestration changes its magnitude. Consequently

F = \frac{J _{R}}{J _{S}} = \frac{[ R ]}{[ S ]} \frac{k _{+}^{R}}{k _{+}^{S}} \frac{k _{-}^{S} + m}{k _{-}^{R} + m} \frac{l _{-}^{S *} + V _{A}}{l _{-}^{R *} + V _{A}} .

(3)

Each post-binding factor is the ratio of probabilities of proceeding rather than dissociating at that stage. The expression assumes positive common forward rates and no significant product back reaction. Without the steady/intermediate-balance or initial-rate assumption, the ratio can depend on intermediate populations and observation time; binding and dissociation constants alone would not determine a unique transient ratio.