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The simplest case of heterotropic allosterism, the allosteric effector just activates the enzyme activity from a low activity state to a high activity state by binding at a single site separate from the active site. The concentration of the effector that gives a half-maximal activity increase is the dissociation constant of the effector, Kd.

Thanks a lot, then I know. Is it okay to use Vmax and kcat in this case although this is a situation where it is the effector that is at saturating concentration and not the substrate?

You can measure Vmax and kcat for the substrate at various concentrations of the effector. If the allosteric effector is not also a substrate, then these constants are not relevant to the effector.

I agree with Adam B Shapiro, but I would point out that rigorously the concentration of effector for half-maximal activity increase in an activity vs. effector concentration plot is not the dissociation constant of the enzime-effector complex. The following relationship holds:

[A]1/2 = Kd,A x (KM + [S])/(KM + [S]/alpha)

where [A]1/2 is the effector concentration eliciting half-maximal enzyme activity increase, Kd,A is the dissociation constant for the enzime-effector complex, KM is the Michaelis Menten constant, [S] is the substrate concentration (fixed in the activity vs. effector plot), and alpha is the cooperativity constant for substrate and effector binding to the enzyme (i.e., the increase/decrease in KM as a result of the presence of effector at very high concentration).

Therefore:

If [S] >> KM, then, [A]1/2 = alpha x Kd,A

If [S]

Interesting and thanks a lot for both of your answers. . So in my case when I have not studied the Km of the substrate, I guess then the most proper term is just [A]1/2. This leads to some follow-up questions:

-Is it still okay to use the formula I have done: "one site specific binding" in GraphPad Prism to just fit the curve? The fit looks at least very good.

-Is the value I have obtained of [A]1/2 meaningful at all (at least as a rough indicator)? I guess that depends on the size of alpha. What is usually the size of alpha?

-What is the plateau called? As I understood from the previous answer, it is not okay to call it Vmax or kcat.

You can determine [A]1/2, even not knowing KM or alpha, but then it will be a substrate-dependent parameter, since it will be a function of substrate concentration, Michaelis constant, and cooperativity constant.

It will be a similar case when determining inhibition constant (Ki) in enzymatic inhibition assays or effective concentration (EC50) in cellular assays, parameters that are very often dependent on enzyme and substrate concentration and KM, in the case of Ki, or onthe cell number and other hidden factors (e.g., protein/receptor expression level), in the case of EC50.

The "one site specific binding" should be fine, but remember that [A]1/2 will not be the true dissociation constant.

Using [A]1/2 for comparing effectors with the same substrate may be tricky, since the KM would be equal in all cases, but each effector may show a different alpha value with the substrate.

And using [A]1/2 for comparing substrates with the same effector may also be tricky, since each substrate may show a different KM and a different alpha with each effector.

Alpha, the cooperativity constant for substrate and effector binding to the enzyme, may show a wide range of values:

alpha > 1, negative cooperativity between substrate and effector (i.e. effector would be an inhibitor)

alpha =1, no cooperativity and independent binding of effector and substrate

alpha < 1, positive cooperativity between substrate and effector (i.e. effector would be an activator).

Of course, the global effect of an effector must be establish considering its effect on the apparent KM abd apparent Vmax.

When you construct an activity vs. effector concentration plot, you are employing a given [S], not necessarily a very large [S]. Then, the maximal activity in that plot does not correspond to Vmax, which by definition corresponds to very large [S].

Good explanation, thanks a lot. Would you think that it is okay to also call it Kd,app instead of [A]1/2. I don´t see that [A]1/2 is used in the field so I am afraid to only cause confusion by using this term.

Yes, that would be the correct name: apparent dissociation constant, Kd,Aapp, where A is the effector

Perfect, thanks a lot, now it is crystal clear.

Adrian Velazquez-Campoy, Thank you for providing the thorough explanation. I didn't think about cooperativity between substrate and effector.

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I agree strongly with Adrian Velazquez-Campoy . The best term would be Kd,Aapp, instead of [A]1/2, or Km,Aapp. The plateau at the top when you plot Km (or the true Kd!) versus effector concentration is referred to in some articles as Km/A. You will find it mentioned this way, for example in Johnson and Reinhart, 1992 :

( Johnson, J. L., & Reinhart, G. D. (1992). Magnesium-ATP and fructose 6-phosphate interactions with phosphofructokinase from Escherichia coli. Biochemistry, 31(46), 11510-11518.)

and it is used this way in Gregorio Weber's lengthy work on the topic:

( Weber, G. (1975). Energetics of ligand binding to proteins. In Advances in protein chemistry (Vol. 29, pp. 1-83). Academic Press. )

Even this terminology does not describe the full energetic nature of this term, however, since it is temperature sensitive in many cases, due to the changing contribution of entropy. I know that Reinhart and Kimmel found that an inhibitor of PFK from B. stearothermophilus could become an activator if you shifted the temperature. Sadly, I cannot remember that reference off the top of my head so I cannot give it to you here. Kimmel's work is excellent, however.

On your plot, this would mean that Km,Aapp shifts downwards instead of upwards as you increase the effector concentration. Using the generic terms this way allows you to have a meaningful discussion about the energetics of these effectors, and the complexity of the behavior they are inducing. The terms [A]1/2 and [I]1/2 are less useful, since they don't allow you to acknowledge that protein behavior with these effectors depends upon the temperature, and also any other substrate effects that are occuring. An example of these complex substrate effects is in the Johnson and Reinhart paper: the "activator" ADP may be activating the binding of the substrate Fructose-6-P in their enzyme, but it is simultaneously inhibiting the other substrate, ATP, in a non-competitive fashion. Furthermore, the substrate ATP is inhibiting the binding of Fructose-6-P, thus skewing Km pretty wildly away from the true value of Kd. That is too complicated to describe with the simplistic [A]1/2 and [I]1/2 terminology

Thanks a lot for the additional information. I am just a bit confused that you call the plateau Km/A. Do you really mean the plateau (corresponding to Vmax) or do you mean Kd,Aapp?

Km/A was the terminology used to indicate an apparent Km in the saturating presence of the effector A, and it shows up that way in some of the papers on enzyme energetics. Not everyone uses this terminology, though. It is as if different authors write about enzyme energetics in different languages. When you see the term used this way, the authors mean "in the saturating presence of A". I can take no responsibility for the terminology... they coined it when I was maybe four years old. :)

Yes, it is really confusing, so you mean that Km/A is really a velocity term (with a concentration/time type of unit) although it sounds like a binding term (with a concentration unit)?

No, it is not a velocity term. Benjamin is saying that Km/A represents the Km at saturating extremely high concentration of ligand A (not at a certain finite concentration)

Exactly. Adrian Velazquez-Campoy has it correct. Gregorio Weber does go into an equivalent term for kinetics if you read his tome on the topic, but Km/A is the equilibrium term.