. Author manuscript; available in PMC: 2022 Dec 15.

Published in final edited form as: Arch Biochem Biophys. 2021 Oct 26;714:109064. doi: 10.1016/j.abb.2021.109064

Table 1.

Kinetic and thermodynamic signatures of the IF binding model. The K_D,app and K_D have units of concentration. The equilibrium concentrations, [C_eq], are expressed as a ratio of [E_total] where [E_total] = [E] + [EL] + [E*L]. The two observed rates, k_alow and k_fast, as well as their lower ([L] → 0) and upper ([L] →∞) limits have units of s⁻¹ Three unique expressions describe each enzyme state as a function of ligand concentration and time: [E](L,t) = [E_eq] + A_E,slowe^−kslow*t + A_E,faste^−kfast*t, [EL](L,t) = [EL_eq] + A_EL,slowe^−kslow*t + A_EL,faste^−kfast*t, and [E*L](L,t) = [E*L_eq] + A_E*_L,,slowe^−kslow*t + A_E*L,faste^−kfast*t

Induced fit: E⇄E⇄EL (solution: [C] = [C_eq]+A_slowe^−kslowt+A_faste^−kfastt
K_D,app:	$K_{D} (\frac{k_{r}}{k_{r} + k_{f}}) where K_{D} = \frac{k_{o f f}}{k_{o n}}$
[C _eq ]:	$\begin{matrix} [E_{e q}] = \frac{K_{D, a p p} [E_{m a x}]}{K_{D, a p p} + [L]} & [E L_{e q}] = \frac{[L] [E L_{m a x}]}{K_{D, a p p} + [L]} & [E^{} L_{e q}] = \frac{[L] [E^{} L_{m a x}]}{K_{D, a p p} + [L]} \\ where E_{m a x} = [E_{total}], & [E L_{m a x}] = \frac{k_{r}}{k_{f} + k_{r}} [E_{total}] & [E^{*} L_{m a x}] = \frac{k_{f}}{k_{f} + k_{r}} [E_{total}] \end{matrix}$
A_slow and A_fast:	$A_{E, i} = \frac{- k_{o n} [L] (k_{f} + k_{r} - k_{i}) [E_{0}] + k_{o f f} (k_{r} - k_{i}) [E L_{0}] + k_{o f f} k_{r} [E^{} L_{0}]}{k_{i} (k_{i} - k_{j})} A_{E L, i} = - A_{E, i} - A_{E^{} L, i} A_{E^{} L, i} = \frac{k_{o n} [L] k_{f} [E_{0}] - k_{f} (k_{o n} [L] - k_{i}) [E L_{0}] - k_{r} (k_{o n} [L] + k_{o f f} - k_{i}) [E^{} L_{0}]}{k_{i} (k_{i} - k_{j})} when i = slow, j = fast a n d vice versa$
k_slow and k_fast:	$k_{slow} = \frac{b - \sqrt{b^{2} - 4 a c}}{2 a} k_{fast} = \frac{b + \sqrt{b^{2} - 4 a c}}{2 a} where a = 1, b = k_{o n} [L] + k_{o f f} + k_{f} + k_{r}, c = k_{o n} [L] k_{f} + k_{o n} [L] k_{r} + k_{o f f} k_{r} a n d \sum k_{o b s} = b, and \prod k_{o b s} = c$
$\underset{[L] \to 0}{l i m} k_{obs 1, 2} :$	$k_{slow} = \frac{b - \sqrt{b^{2} - 4 a c}}{2 a} k_{fast} = \frac{b + \sqrt{b^{2} - 4 a c}}{2 a} where a = 1, b = k_{o f f} + k_{f} + k_{r}, c = k_{o f f} k_{r}$
$\underset{[L] \to \infty}{l i m} k_{obs 1, 2} :$	$k_{slow} = k_{f} + k_{r} k_{fast} = k_{o n} \infty + k_{o f f} = \infty$