Box 2.

Cooperativity

Assumption: The cooperative complex contains N copies of identical protomers, and each of them is in a distinct transition-state (or phase) of the functional cycle

(i) Energy conservation

$\sum _i^N{W}_{\text{C}i}=0$

where W Ci is the cooperative work that is output by the protomer in the i-th step. A negative W Ci would indicate that the i-th step absorbs energy from the cooperative complex

(ii) Arrhenius theorem

$k_i{\text{e}}^{-W\text{c}i/RT}=k_j{\text{e}}^{-W\text{c}j/RT}$

where k i is the (pseudo) first-order rate constant of the i-th step in a hypothetical non-cooperative protomer

(iii) Cooperative work

$W_{Ci}=\frac{-RT}{N}ln\frac{{\prod }_j^N{k}_j}{k_i^N}=RTln\frac{k_i}{k_{\text{c}}}$

$k_{\text{c}}={\left(\prod _j^N{k}_j\right)}^{\frac{1}{N}}$

where k c is the pseudo first-order rate constant of the cooperative complex. For instance, for the AcrB trimer, k C = (k 1· k 2· k 3)1/3