= Solution
In <alternating-access carrier transport>, the four reversible reaction pairs are $S_i+C_i\rightleftharpoons P_i$, $S_e+C_e\rightleftharpoons P_e$, $C_i\rightleftharpoons C_e$ and $P_i\rightleftharpoons P_e$. Binding and unbinding use $k_+,k_-$; both conformational pairs use $k$ in each direction. Define the net binding rates $B_i=k_+s_ic_i-k_-p_i$ and $B_e=k_+s_ec_e-k_-p_e$. <law of mass action>[Mass action] and the change of each species in each reaction give
$$
\boxed{\begin{aligned}
\dot s_i&=-B_i,&\dot s_e&=-B_e,\\
\dot p_i&=B_i+k(p_e-p_i),&\dot p_e&=B_e+k(p_i-p_e),\\
\dot c_i&=-B_i+k(c_e-c_i),&\dot c_e&=-B_e+k(c_i-c_e).
\end{aligned}}
$$
For example binding consumes one free interior <substrate> and one free interior carrier, and produces one interior complex; loaded-carrier transformation then moves that bound <substrate> across the <cell membrane>[membrane]. A bare-carrier transformation alone does not transport <substrate>. The <concentration> convention implicitly uses compatible compartment-volume <normalization>, as required for the stated unweighted <conservation laws>.
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