The reaction has 2 substrates, 2'-deoxynucleoside and ATP (other substrates are possible). The equation for the initial rate depends on the kinetic mechanism.

As an example, one likely kinetic mechanism is rapid equilibrium random, for which the initial rate equation is:

v = Vmax[A][B]/{KiaKmB + KmA[B] + KmB[A] + [A][B]}

in which [A] and [B] are the concentrations of the two substrate, KmA and KmB are the Kms of substrates A and B, respectively, and Kia is the dissociation constant of substrate A.

By measuring the initial rate v for a matrix of [A]s and [B]s, you can determine values of the constants Kia, KmA and KmB using a global nonlinear regression procedure. For the rapid equilibrium random kinetic mechanism, you can choose dATP to be substrate A and thereby measure Kia for it.

There are other possibilities for the kinetic mechanism. They do not all have the same initial rate equation. The literature on the enzyme may reveal the right one and give the appropriate rate equation.

Thanks a lot for the answer, that sounds reasonable that it could be a rapid equilibrium random (although I have not proven it). However, since the reviewer suggested me to use curves where I plot how the apparent Vmax changes with inhibitor concentration and how the apparent Km changes with inhibitor concentration, I think I need to plot it that way unless I have a very good argument not to do that. So what do you think about plotting the inhibitor concentration vs Vmax and inhibitor concentration vs 1/Km and just take the inhibitor concentration needed for half inhibition? Is the IC50 I then get the same as Ki?

The reviewer wanted me to check the Ki for the competitive part and the non-competitive part of the inhibition is the same in order to verify that it is a mixed inhibitor with respect to deoxyadenosine as I claimed. That is the reason why I was asked to make the curve. I have already made the data, I just want to check if I use it in the correct way. A similar experiment is also made where ATP is the varied substrate but in this case only the Km is affected.

Sorry, I misunderstood the question.

Intuitively to me, inhibition of this enzyme by dATP seems like it could be somewhat complicated. dATP may be an alternative substrate with respect to ATP if it binds in the ATP binding site, and a competitive inhibitor with respect to both ATP and 2'-deoxyadenosine when it binds in the 2'-deoxyadenosine binding site, as you mentioned. Depending on the relative strengths of these effects, the analysis may be more or less complicated.

If the alternative substrate effect of dATP is substantial, it might result in the inability to achieve complete inhibition of the ATP + 2'deoxyadenosine reaction. The analysis would be different than for a just competitive inhibitor. (Refer to the book mentioned below for an explanation of how to deal with this situation.)

For competitive inhibition, you could use the textbook graphical methods allowing linear regression, such as the double-reciprocal plot (1/v versus 1/[S]) to get Km,apparent at each inhibitor concentration, followed by linear regression of a replot of Km,apparent versus [I] to to read out the Ki (negative value of x-intercept). There are various types of plots and replots for getting Ki. I recommend the book "Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems" by Irwin H. Segel (John Wiley & Sons, 1975) for a thorough description of graphical methods for measuring kinetic parameters.

Those methods are kind of old-fashioned now when it is possible to easily perform nonlinear regression of the original data to rate equations to obtain the values of parameters. This avoids the distortion of error distributions that are introduced by taking reciprocals. With a full matrix of concentrations of both substrates and inhibitor, plus the appropriate kinetic rate equations, you can compare various modes of inhibition and measure their Kis in a few minutes using a suitable computer program. This might even help you narrow down the kinetic mechanism of the uninhibited enzyme reaction.

You can also use IC50 measurements to obtain Ki if you employ the equations described by Cheng and Prusoff (reference below). These were derived by simple algebra from the kinetic rate equations for the inhibition mechanisms and the uninhibited reaction.

Cheng Y, Prusoff WH (December 1973). "Relationship between the inhibition constant (Ki) and the concentration of inhibitor which causes 50 per cent inhibition (I50) of an enzymatic reaction". Biochemical Pharmacology. 22 (23): 3099–108. doi:10.1016/0006-2952(73)90196-2. PMID 4202581.

As far a measuring the value of IC50, you can use nonlinear regression and the Hill equation. For the simplest type of inhibition without cooperativity, and assuming that inhibition goes to 100%, the equation is

% inhibition = 100[I]/(IC50 + [I]), where [I] is the inhibitor concentration.

If the inhibition does not go to 100% because dATP acts as an alternate substrate, then you can include the parameter MAX, meaning the maximal % inhibition:

% inhibition = MAX[I]/(IC50 + [I])

If there is cooperativity between binding sites, you can include that as the Hill coefficient n:

% inhibition = MAX[I]n/(IC50n + [I]n)

With respect to cooperativity, however, I don't think it is taken into consideration when using the Cheng-Prusoff approach because the kinetic rate equations don't include it.

Thanks a lot for the answer, actually I am already using non-linear regression to obtain Km,app and Vmax values. It is no problem to do that with the software that I am using (GraphPad Prism). It is the next step that I am uncertain about. I have not been able to look in the paper you refer to though (I have ordered it now). The reviewer asks me about making new curves where I plot the obtained Km,app and Vmax,app values as a function of the inhibitor concentrations, which I have done in the attached file. I think this was actually an informative way to do it since it clearly shows that the inhibitor affects both Km and Vmax with dAdo as limiting substrate and only affects Km with ATP as the limiting substrate. However, it raised a lot of new questions as I have shown in the attached file. I am also uncertain why only the Km was affected when ATP was limiting and both Km and Vmax are affected when dAdo was limiting. I have included a model how the inhibitor acts based on previous crystallographic studies of related enzymes. As I mentioned above, it binds in a way where it covers the dAdo site completely and partially the ATP site.

I would prefer to take a different approach to analyzing the kinetics of this system, by globally fitting the {[dAdo], [ATP], [dATP]} data matrix to a plausible kinetic rate equation (e.g. rapid equilibrium bisubstrate) including dATP binding to E, E.dAdo and E.ATP forms of the enzyme. A general equation for the rapid equilibrium bisubstrate system with nonexclusive inhibitor can be found on p. 294 of Segel. (Some of the terms could be dropped.) There are other plausible kinetic mechanisms as well, including the Rapid Equilibrium Random and Ordered Bisubstrate steady-state systems.

You should not confuse the IC50 (the concentration of inhibitor that causes 50% inhibition under whatever arbitrary set of assay conditions is chosen) with the concentration of inhibitor that reduces the value of 1/Km or Vmax by half. I don't think there's a name for that.

Since the structural similarity between dATP and ATP is so close, I would have expected dATP to be an alternative substrate. Maybe this is true only at a higher range of dATP concentrations. Have you tried using dATP as the substrate in place of ATP to see if it is an alternative substrate?

Thanks a lot for the answer. Then I know also that I cannot use the term IC50. Perhaps I can just use more loose terms and say that the effect on Km and Vmax is roughly equal and that the concentration of dATP needed to reduce Vmax and 1/Km by half was around 1 µM in both cases. I do not think I can skip the graphs altogether since the reviewer asked for them.

Thanks also for the suggestion about the the rapid equilibrium two-substrate system. I will look it up and see what it gives.

Regarding the competition, deoxyadenosine binds very tightly to the enzyme compared to ATP (~100 times lower Km). That is the reason why dATP preferably binds with the deoxyadenosine moiety into the deoxyadenosine pocket and not into the pocket where the adenosine moiety of ATP binds. So I think I can rule out that option (that is also established in the field).

I saw this paper that mentioned that some deoxynucleoside kinases go by the Rapid Equilibrium Random steady-state mechanism, and some by the Ordered steady-state mechanism.

https://www.jbc.org/article/S0021-9258(18)66639-6/pdf

By the way, besides kinetic studies, which can be complicated to perform and interpret, you could also try making a direct measurement of the dissociation constant of dATP for your enzyme. Generally speaking, the dissociation constant Kd and the Ki determined from kinetics are the same thing. Isothermal titration calorimetry would probably be a good choice, considering the high potency of the inhibitor.

Perfect, that was really great that you found that paper. Now everything is much clearer to me. The main conclusion of the paper was that some deoxynucleoside kinases are randomly ordered for the two substrates (deoxynucleoside and ATP) and then follow competitive mechanisms whereas some of them bind ATP first and in that case, it is competitive with respect to ATP and non-competitive (or mixed) with respect to the deoxynucleoside. In my case, it is then more similar to the second type. However, I am not sure why it is only partially non-competitive (i.e. mixed) with respect to the deoxynucleoside. Does it mean that it preferentially binds ATP first but not always?

I think it is also a good idea to measure the binding of dATP directly and I can try that.

However, there are still a few things that confuse me from the referee´s response. It is not only asking me to plot the Vmax,app and Km,app values as a function of inhibitor concentration, it also asks me to derive the apparent Ki´s for the non-competitive (Kinc) and competitive (Kic) parts of the inhibition from these curves and it claims that Kic=Kinc means that dATP binds to a single site. I think the proposed curves are illustrative to show the type of inhibition, but the second part of the request raises a lot of questions. Are Kinc and Kic values possible to calculate from the curves? Is it customary at all to divide Ki into competitive and non-competitive fractions? Is it really true that Kic=Kinc means single-site binding and is that reasoning valid for a bisubstrate enzyme with overlapping active sites (if so, is it for ordered or random reactions)?

In my opinion, the terminology is a problem. The terms competitive, noncompetitive, uncompetitive and mixed inhibition are confusing rather than clarifying. They work OK with single-substrate enzymes if you can remember what they mean, but they get in the way of understanding what is going on in multi-substrate enzymes. I prefer a different approach:

During the reaction cycle, the enzyme goes through multiple substrate-bound states. In a random mechanism, these are E, EA, EB and EAB. In an ordered mechanism, they are E, EA and EAB.

An inhibitor that interacts with a substrate binding site (directly or indirectly, but its usually directly) can have different affinities for each of these states, but it may not be able to bind to some of them at all. For dATP with your enzyme, the inhibitor definitely can't bind to the state in which dAdo is bound, since they would have to occupy the same binding site simultaneously. dATP probably also can't bind to the state in which ATP is bound for the same reason, namely overlapping phosphate binding sites). Therefore, the only state to which dATP can bind is E.

Since there is only one state to which dATP can bind, there is only one Ki, which is the dissociation constant of dATP binding to E. There's no need to talk about a competitive Ki for one substrate and a noncompetitive Ki for the other. (An exception would be if dATP binds to an allosteric site, which would most likely affect Vmax. (And there's only one Vmax.))

It should be possible for you to write kinetic rate equations for the case of dATP binding only to E for the various plausible kinetic mechanisms of the enzyme, then use a global analysis of the data to obtain a value for the parameter Ki, along with the other kinetic constants. You should be able to dig these equations out of Segel. They essentially amount to attaching the multiplicative factor (1+[I]/Ki) to the term(s) in the denominator of the rate equation that correspond(s) to the E state, when the equation is written in the form v = Vmax[A][B]/denominator, where all the other rate constants are in the denominator. For example, for the Ordered Bi steady-state system, the initial rate equation is:

v = Vmax[A][B]/{KiaKmB + KmA[B] + KmB[A] + [A][B]}

and for an inhibitor binding only to E, it's:

v = Vmax[A][B]/{(KiaKmB + KmA[B])(1+[I]/Ki) + KmB[A] + [A][B]}

because both KiaKmB and KmA[B] refer to the E state, whereas KmB[A] refers to the EA state.

In contrast, in the Rapid Equilibrium Random system, the initial rate equation has the same form, mathematically, as the Ordered Bi steady state equation, but the denominator terms and kinetic constants have different meanings:

v = Vmax[A][B]/{alphaKAKB + alphaKA[B] + alphaKB[A] + [A][B]}

Here, an inhibitor binding only to E would have the equation:

v = Vmax[A][B]/{alphaKAKB(1+[I]/Ki) + alphaKA[B] + alphaKB[A] + [A][B]}

This approach to the analysis shows that you can potentially use inhibition by dATP to distinguish between Steady State Ordered and Rapid Equilibrium Random kinetic mechanisms, if you know from the start that dATP can only bind to the E state. (There are other kinetic mechanisms, of course, but these two are impossible to distinguish between by initial rate mechanism alone without the use of inhibitors.)

This approach can also be used to test other possibilities by doing the global fit to other cases, such as by allowing the inhibitor to bind to additional enzyme-ligand states. You can compare the results of different fits using the chi-squared goodness-of-fit values (lowest is best).

For the sake of satisfying the reviewer, however, you can use the old-fashioned reciprocal plot and replot method to calculate Ki, based on its effect on the Km for each substrate, as you have done. I think the reviewer's statement about Kic=Kinc meaning there is only one inhibitor binding site is probably true, but I haven't thought about things this way. I've always used the method I described above. I guess the idea is just to measure the Ki when varying each substrate independently, and comparing the values.

You could also try a direct dATP binding measurement in the presence and absence of a saturating concentration of each substrate, as a direct means of discerning whether it's possible for the inhibitor to bind in the presence of each substrate. This could be a technically challenging experiment, however.

Thanks a lot for all the help. However, although I agree with the reasoning behind choosing the formulas, I am uncertain how to plot the data. In the software I am using (GraphPad Prism), it is possible to plot the data according to fixed models for competitive, non-competitive, uncompetitive and mixed inhibition. The mixed model looks like this:

VmaxApp=Vmax/(1+I/(Alpha*Ki))

KmApp=Km*(1+I/Ki)/(1+I/(Alpha*Ki))

Y=VmaxApp*X/(KmApp + X)

The corresponding competitive model looks like this:

KmObs=Km(1+[I]/Ki)

Y=Vmax*X/(KmObs+X)

In the data table, I then list substrate concentrations (X) and enzyme activities (Y) at different inhibitor concentrations (each Y column is with a different inhibitor concentration). In these experiments I have used a fixed saturating concentration of the other substrate. Since it is possible to change the equations to user-defined versions (by cloning the equation and modifying it), I wonder if I can change the two equations in order to fit your two equations above for random and ordered substrate kinetics.

The plot with enzyme activity as a function of [ATP] at different inhibitor concentrations fits perfectly with the competitive model above. Can I then say that the obtained Ki value is really the Ki value of the reaction or do I need to modify the equation to a two-substrate version? The other substrate (deoxyadenosine) is in 100-fold excess (100-fold). According to the paper about the deoxynucleoside kinases that you recommended to me earlier, my data fits with that ATP is the first substrate to bind. I wonder if it then can be treated as a pseudo-single substrate and that is the reason why it fits so well to the competitive model.

In the corresponding experiment where enzyme activity was plotted as a function of [deoxyadenosine] both Vmax,app and Km,app are affected and in this case none of the models fitted well (but the mixed model was at least the best one). Can the mixed model above then be rewritten with the other substrate concentration treated as a constant)? One uncertainty is also why the inhibitor affects both Vmax,app and Km,app (and not only Vmax). Can this be interpreted as that ATP is not always the first substrate to bind?

Apparently, there is a way of getting Prism to work with multiple independent variables for a global regression analysis, described here. I haven't tried it.

https://www.graphpad.com/guides/prism/latest/curve-fitting/reg_defining_equation_with_two_or_.htm

I've always used a program called Grafit to do this type of analysis, for which it's designed.

I think you are right that you can reduce the kinetics to a pseudo-single substrate system by saturating with deoxyadenosine as the second substrate. This pretty much guarantees that dATP will be competitive with ATP, since it will not be able to compete with deoxyadenosine. The measured Ki should accurately represent the competition with ATP when deoxyadenosine is saturating. The inhibition mechanism and Ki may come out differently if you vary both substrates. Do you get the result that dATP is simply competitive with deoxyadenosine when you fully saturate with 100X Km of ATP?

At a sub-saturating concentration of ATP and varying the deoxyadenosine concentration, if dATP competes with both substrates, I think it makes sense that dATP affects Vmax(apparent). Vmax(apparent) at any given deoxyadenosine concentration depends on the proportion of enzyme that has ATP bound, so the competition with ATP binding by dATP lowers the apparent Vmax(apparent) when varying dATP, unless the ATP concentration is saturating to prevent the competition.

This kind of thing is easier to interpret using the framework I mentioned previously, where you can test different ideas about inhibitor-enzyme complexes by changing the rate equation.

Thanks a lot for the explanation, it is really good that I can treat it as pseudo-single substrate.

In the experiment with variable [dAdo], I cannot use 100x Km because of practical reasons. ATP will then compete out the product from the filters that I am using in the assay. So unfortunately, I am then restricted to maximally 2 mM ATP (10x Km). In the experiments with variable ATP, there is no such restriction since I can then use a less sensitive filter-independent HPLC-dependent assay. However, maybe it is sufficient to use the data I have just to show that both Km,app and Vmax,app are affected with variable [dAdo] and refer to the article you showed me that has previously been reported that the data supports a model where ATP binds first. The only question that remains is why both Km,app and Vmax,app are affected (and not only Vmax). If the inhibitor only can bind to E-free, it should only compete with the first substrate. Is that indicative of that the binding order is not that strict although ATP preferably binds first? On the other hand, the binding of deoxyadenosine must be very poor to the empty enzyme considering that I am using a 100x Km concentration and still it does not seem to affect the E-free concentration considerably in the experiment with variable ATP.

I think you should consider doing substrate and inhibitor binding experiments to supplement the kinetic studies. Various methods are available, such as ITC, SPR , MST, BLI, and NMR. The vacancy peak method of Hummel and Dreyer can be done with just an HPLC and a size exclusion column. Article Macromolecule-ligand binding studied by the Hummel and Dreye...

Thanks a lot for the suggestion, I will take a look at it.