Constants in Enzyme Kinetics under the Rapid Equilibrium Assumptions

I am new to enzyme kinetics, and I am trying to write an rate equation for a rapid equilibrium random terreactant system. I have been consulting the books *Enzyme Kinetics and Mechanism* by Cook and Cleland and *Enzyme Kinetics* by Segel. One thing that have been confusing to me is that, sometimes they refer Ka (Michaelis constant for reactant A) and Kia (inhibition constant for reactant A) as dissociation constant in their books, and I am not sure which one to use.


From what I understand, Ki is dissociation constant except two systems, namely Ordered Bi Bi and Ordered Uni Bi (under **Haldane Relationships** In Cook and Cleland's book, as well as illustrated clearly through a lot of other cases in Segel's book). I feel like dissociation constant and inhibition constant should be two different concepts but it seems like they use both interchangeably.


For Ka, Ka is the dissociation constant when the system is rapid equilibrium Uni-Uni (specifically, when they discuss the Michaelis–Menten/Briggs–Haldane equation, as MM equation is derived with assumption of rapid equilibrium). In the case of random Bi-Bi, Cook and Cleland defined Ka as the dissociation of A from the ternary complex, and Kia as the dissociation of A from the binary complex; whereas Segel used dissociation constant KA for $\ce{E + A <=> EA}$ and $\alpha$KA for $\ce{EB + A <=> EAB}$ (i.e. the dissociation constant is modified by another substrate that is already bound to the enzyme with a factor of $\alpha$. What is the difference between them?


I guess what ultimately confused me is in the case of terreactant: Segel used the same method for defining the constants for each reaction step; Cook and Cleland just used "const" "coef A" etc. to describe the rate equations without further details. The paper they referenced, [*Methods in Enzymology* 87, 353](https://doi.org/10.1016/S0076-6879(82)87021-3), only provides the deduction of possible mechanism based on the missing terms in the denominator without further defining them since they depend on the mechanism. The paper *Rapid-Equilibrium Enzyme Kinetics* by Alberty (2008) provided an explicit expression for random terreactant system, following Cleland's nomenclature and used a mixture of Michaelis and inhibition constants in the equation, but I do not understand how six constants (KA, KB, KC, KIA, KIB, KIC) cover all 12 edges in the system. I do not need the exact expression of the constants in terms of the rate constants (the small letter *k*s), but I would very much like to know what are the terms (Michaelis, inhibition or dissociation) should be used when describing the rate equation of a rapid equilibrium random terreactant system.




I do not have access to either books of *Enzyme Kinetics and Mechanism* by Cook and Cleland and *Enzyme Kinetics* by Segel. However, I read some of Cleland and coworkers' papers (e.g., Ref.1 & 2). In those and other peer-viewed papers by Alberty, my understanding of the nomenclature of rapid equilibrium of enzyme kinetics as follows (I avoid use of $K\_\mathrm{a}$ and use $K\_\ce{A}$ instead to avoid the confusion with acid dissociation constants):


1. As you have mentioned, $K\_\ce{A}$ is the dissociation of $\ce{A}$ from the ternary complex such as $\ce{EAB}$ such that $\ce{EAB <=> EB + A}$. Thus, $K\_\ce{A} = \dfrac{[\ce{EB}][\ce{A}]}{[\ce{EAB}]}$ (Ref.3).
2. You have mentioned $K\_\ce{IA}$ is the inhibition constant. But I don't think it has anything to do with inhibition here. As Ref.2 and other paper mentioned, it is the dissociation of $\ce{A}$ from the binary complex such as $\ce{EA}$ such that $\ce{EA <=> E + A}$. Thus, $K\_\ce{IA} = \dfrac{[\ce{E}][\ce{A}]}{[\ce{EA}]}$ (Ref.3).


As Ref.1 mentioned, before you are trying to write an rate equation for a rapid equilibrium random terreactant system or any other system, you may need to have an idea of the reaction mechanism (e.g., like Ping Pong type of mechanism). Accordingly you have to do a lot of assumptions (e.g., steady state steps) to get calculated some of those unknown contributions such as $coef\mathrm{A}$ and $coef\mathrm{B}$ in final equation.


**Note:** A good reference to read is Ref.4 and references therein to get some help for derive the equation with appropriate assumptions.




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**References:**


1. Ronald E. Viola and W. Wallace Cleland, "[19] Initial velocity analysis for terreactant mechanisms," *Methods in Enzymology* **1982**, *87*, 353-366 (DOI: <https://doi.org/10.1016/S0076-6879(82)87021-3>).
2. Gerald Litwack and W. W. Cleland, "Studies on the Tyrosine Aminotransferase Mechanism," *Biochemistry* **1968**, *7(6)*, 2072–2079 (DOI: <https://doi.org/10.1021/bi00846a008>).
3. Robert A. Alberty, "Determination of Rapid-Equilibrium Kinetic Parameters of Ordered and Random Enzyme-Catalyzed Reaction A + B = P + Q," *J. Phys. Chem. B* **2009**, *113(29)*, 10043–10048 (DOI: <https://doi.org/10.1021/jp9021097>).
4. S. Cha, "A Simple Method for Derivation of Rate Equations for Enzyme-catalyzed Reactions under the Rapid Equilibrium Assumption or Combined Assumptions of Equilibrium and Steady State," *J. Biol. Chem.* **1968**, *243(4)*, 820-825 (DOI: <https://doi.org/10.1016/S0021-9258(19)81739-8>).


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