The Arrhenius plot (or its modern alternatives, the plots based on Eyring's

https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Kinetics/Modeling_Reaction_Kinetics/Transition_State_Theory/Eyring_equation

and Kramer's theories

http://www.acmm.nl/ensing/thesis/node9.html

) attempts to provide an estimate of the energy barrier of the rate-limiting chemical step in the reaction mechanism of a chemical reaction.

When you add an inhibitor, the kinetic constants determined may become apparent values, and which constant is affected and how, would depend on the kinetic mechanism underlying the inhibitor's action.

For instance an classic uncompetitive or non-competitive inhibitor will lead to an apparent value for the catalytic constant estimate, describe, in the simplest case by:

kCATREAL = kCATApparent · (1+ [I]/KIU)

So if you can measure the value of KIU you should be able to estimate the real kCAT value and this is the one you should use.

In the case of activators, the mechanisms of activation may or may not involve two different values for kCAT. You should actually use the value of the upper boundary for this estimate, since this in the constant reflecting the chemical barriers of the rate-limiting step, since the low apparent k'CAT values rarely do represent a true kinetic constant.

This reference should help you to understand the situation better:

Article Convex Arrhenius plots and their interpretation

you may also refer to this previous discussion thread, related to your question:

https://www.researchgate.net/post/Is_it_right_to_plot_logproduct_formation_rate_against_1_T_instead_of_the_more_popular_logk_v_1_T_in_the_evaluation_of_apparent_activation_energy

Best wishes,

Rogelio

So does the slope change? I am thinking that in the presence of inhibitors the energetics don't actually change. So does the line move up instead of a change in slope?

Hi there,

The presence of an inhibitor or an activator doesn't change the energy of activation of the catalyzed reaction as whatever mode of action of the inhibitor/activator Ea is still the energy required to get tranformation of ES into EP. Ea/R being the slope of the plot, the slope should remain the same.

Dear Dominique,

While that is true most of the time, most enzymes, may be all, have mechanisms with several steps and it is possible to have two or more high energy barriers in between. The Arrhenius plot reveals the height of the transition state (TS) barrier at the rate limiting step. Changes in the chemical environment of a transition state may lower its energy and this is the theoretical ground of what catalysis is about. An activator or an inhibitor may induce a change in conformation, and/or influence the chemical environment at the active site, modifying the height of some barrier relative to another, making a different step rate-limiting.

This has been found to occur frequently with temperature, and Arrhenius plots exhibiting a change in the slope over two ranges of temperature are not that frequent, but a number of cases have been reported. In this case, the rate constants of two steps have a significantly different sensitivity to temperature in a low temperature range one is rate-limiting, but its increase with temperature is steeper than a second constant. At a certain temperature the second constant becomes the rate-limiting and the Arrhenius plot's slope reflect the change.

Finally, It appear to exist a tendency in natural selection to favour K-type allosteric transitions, but V-type allosteric enzymes do exist, specially for non-essential activators. These cases imply a change in kCAT value upon activator addition and so, the Arrhenius plot's slope is likely to change change.

Best wishes,

Rogelio

Thank you Rogelio and Dominique for your remarks. So if the rate limiting step is modulated in some way the slope may change? My interest is in lipid transfer. Where the rate limiting step is capture of the lipid molecule from the membrane. Here of course there is no transformation taking place. The protein simply shuttles lipid between membranes.

Nevertheless it does not occur in one step. First your protein should diffuse to the membrane (3D diffusion), then if lucky, it can stick to the membrane surface and remain bound to scan the surface for its ligand (2D diffusion). Eventually it might bind a suitable ligand or dissociate (abortive hopping).

The mass transfer phenomena at the 2D surface make a singular phenomena. Have a look at this paper:

Article The kinetics of interfacial catalysis by phospholipase A2 an...

In this case, the energy barriers may be, like in protein folding, in the scanning of a lipid surface, in the sticking to the surface, in the conformational change to bind the lipid, or in the dissociation from the membrane once the lipid is bound . Temperature can shift the mechanism from scooting to hopping, which could alter the rate limiting step.

Best wishes

Rogelio