@Dr Juarez: thanks for the explanation, I had modelled the mechanism using Hill system without accounting for the substrate inhibition, which had given me confusing values for the kinetic parameters even with non-linear fitting (although Hill coefficient was 1.8-2.2 and my enzyme is a tetramer, which probably shows real cooperativity).  Nevertheless, I'll check for linearity at low substrate concentrations too. 

Reg the  substrate inhibition, I dissolve the substrate in buffer so there is no question of solvent interference or precipitation. The pH could be a potential problem since the product is acidic but I wonder how large a pH difference could cause pronounced substrate inhibition..

Are you using a coupled assay? I mean are you using ancillary system? If this is the case you should check your assay condition, let me know and may be I can help you

The substrate inhibition you see could be authentic substrate inhibition, which could be modeled using a suitable equation (see the Segel book recommended by Oscar Juarez, the full title of which is "Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems"). There are also many potential reasons why it could be an artifact, some of which were already mentioned. Maria Grazia Tozzi asked whether you are using a coupled assay for detecting product. The reason is that your substrate could be interfering with the activity of the coupling enzyme(s). This could be alleviated by increasing the concentration of the coupling enzyme. Some more details of your system would be helpful. What is the substrate and how are you detecting the product? How pure is the substrate?

I think that trying to include both substrate inhibition and cooperativity into a kinetic mechanism would be very complicated, so it will be important to find out whether the apparent substrate inhibition is authentic or an artifact.

I agree with Adam Shapiro's comment that "trying to include both substrate inhibition and cooperativity into a kinetic mechanism would be very complicated" -- if one were to use the traditional approach using algebraic rate equations, such as those that are published in Segel's book.

However, there is also an entirely different approach to enzyme kinetics, where we dispose of the often complicated algebra, and rely instead on symbolic description of the kinetic mechanism.  Then we let the machine derive the appropriate mathematical formalisms internally.   Here is a review of of the academic freeware package "DynaFit" that does this:

https://www.researchgate.net/publication/38074341_DynaFit--a_software_package_for_enzymology

You can download DynaFit from here:

http://www.biokin.com/dynafit/download.html

I just used DynaFit to fit your data (quickly digitized with the "Engauger" sofware).  Here is the DynaFit input script I used:

[task]

    task = fit

    data = equilibria

[mechanism]

    E + S ES : Ks dissociation

    ES + S ES2 : Ks2 dissociation

    ES2 + S ES3 : Ks3 dissociation

[constants]

    Ks = 100 ?, Ks2 = 1 * Ks, Ks3 = 10 ?

[concentrations]

    E = 0.001

[responses]

    ES2 = 10000000

[data]

    variable S

    set rates

[output]

    directory ./project/researchgate/001/output/fit-004

[settings]

   {Output}

       XAxisLabel = [Substrate]

       YAxisLabel = initial rate

[set:rates]

S v

10 19.3

20 36.5

40 91.2

60 152.9

80 253.1

100 358.2

120 461.4

140 491.9

160 516.2

180 487.1

200 466.0

400 359.2

[end]

The output is shown graphically in the attached picture.  Looks like from this particular data it is indeed possible to extract only two out of the three equilibrium constants that must be involved in the mechanism.  So the difficult part is not so much in how to come up with the fitting model.  You just type ...

    E + S ES : Ks dissociation

    ES + S ES2 : Ks2 dissociation

    ES2 + S ES3 : Ks3 dissociation

... into DynaFit that that will do the trick.  The difficult part instead is how to design the experiment (if possible) so that all features of the model can be anchored down using suitable experimental data.

Incidentally I should clarify that DynaFit does not have some sort of a built-in library of specific mechanisms.  Instead you can type up any arbitrary reaction mechanism, including any number of molecular species interacting in any number of ways.  The names of of molecular species are completely arbitrary, as are the labels you choose for rate constants and equilibrium constants.

I should also add that, in my opinion based on experience, probably the most frequent source of unusual substrate saturation behavior, involving substrate inhibition and occasionally also sigmoid onset, is that the substrate is not chemically pure.   Many years ago I have seen exactly this type of "strange" curve for porcine pepsin (a simple monomeric protease enzyme) with a synthetic peptide substrate that was (a) prepared by an ancient solution-phase method causing a lot of racemization; and (b) not sufficiently purified.   After the same substrate was synthesized using gentle solid-phase technology and sufficiently purified, we obtained a perfect rectangular hyperbola for presumably the "same" substrate.

Hope this helps...

Article DynaFit—A Software Package for Enzymology

Dear all, thanks a lot for your very informative answers. I'm doing a simple discontinuous assay with enzyme action followed by a chemical reaction to detect the product. I did get good results ( better saturation region) with a higher strength buffer, so I think it might be partially due to the pH effect. However, the substrate inhibition was still present from a slightly higher substrate concentration value. @Dr Kuzmic: thanks for introducing the  DynaFit package.I'll go through the  paper and try again with the modified equation.. 

If there is "only" substrate inhibition but not a sigmoid offset, then you would use this notation in DynaFit:

[task]

    task = fit

    data = equilibria

[mechanism]

    E + S ES : Ks dissociation

    ES + S ES2 : Ks2 dissociation

[constants]

    Ks = 100 ?, Ks2 = 1000 ?

[responses]

    ES = ...

[data]

   ...

Note that now there are only two equilibrium constants.  Ks2 would the the "substrate inhibition" constant -- if it is real and not an artifact due to an impurity (always possible...).  There is only one "response" because only ES leads to the formation of the final product (which does not be shown explicitly).

Feel free to email me the new data if you get stuck with DynaFit.  To setup a DynaFit "script" file for something like this takes literally just few seconds, so it's not a bother at all.

Hope this helps...

Enzyme kinetic parameters (Vmax and Km) can be estimated using the Lineweaver–Burk plot (1/v vs 1/s). The y-intercept of the graph is equivalent to the inverse of Vmax; the x-intercept of the graph represents −1/Km. The Michaelis constant Km is the substrate concentration at which the reaction rate is at half-maximum, and is an inverse measure of the substrate's affinity for the enzyme—as a small Km indicates high affinity, meaning that the rate will approach Vmax more quickly. The value of Km is dependent on both the enzyme and the substrate, as well as conditions such as temperature and pH.

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@Dr Kuzmic: I tried modifying the experiment, by using a higher strength buffer and also using a pure substrate (Sigma). There doesnt seem to be much difference in the results, there is still a sigmoid plot combined with substrate inhibition. I've also checked the linearity of the reaction with both time course and enzyme concentration. So I'm pretty sure now that this strange kinetic plot is a result of the enzyme mechanism, and the values for enzyme activity (v) are reproducible. However, the Hill's plot  with GraphPad doesnt give satisfactory results for K(0.5). I've tried using DynaFit as you had suggested, but I'm not clear about which reaction mechanism or constants to use and also how to interpret the results. How do I write a reaction mechanism for say the MWC model or sequential model of cooperativity and include substrate inhibition in it? Also, I noticed that there is no mention of vmax in the DynaFit results. Could you pl explain me the details so I can plan the experiment better and interpret the results to understand its biological significance?

@Avinash: In my experience is is much more effective to do a DynaFit analysis on an actual data set, rather than tring to explain the theoretical principles in the abstract.

That said, please feel free to send me the raw data (http://www.biokin.com/contact.html).  You could just copy and paste the data into the email as plain text (first column = [Substrate], second column = initial rate).

I'd be glad to do the analysis and then I and explain very much more easily why there is no "Vmax" in the DynaFit formalism, etc.  For now, just briefly, please see the attached reprint.

Hope this helps...

Dear Peter Kumzic,

I am also interested in your program and I obtained the license from my email, but when I installed Dynafit, there is no file called 'install license.'  If I open the file, I am told to get a license and if I continue I get a blank page with File and Output. If I open the file section and insert the license information, I get no further and if I restart the program, I am still asked for the license.

Please advise

Martin Levine

@Martin Levine:

Sorry about the hassle Martin.  Perhaps the documentation is not as clear as it needs to be.  There should be no file called "install license", as you mention in your message above.  Instead there is supposed to be a file called "license.txt", which should be attached to the auto-generated email.  Then there is menu item called "File ... Install License".  In any case, the details of license installation are explained here:

DynaFit - "Getting Started Manual":

HTML -- http://www.biokin.com/dynafit/html-dfitstart/dfitstart.html

Section "Installing the License File"

PDF -- http://www.biokin.com/dynafit/pdf/dfitstart.pdf

Page 13

Hope this helps.  If you still get stuck please do not post here but instead contact me via .

Cheers,

-P.K.

OK - Done - thanks ML

Hello! I have a curve that looks really similar to yours. Do you mind letting me know how you fit your curve eventually?? Thanks a lot!

From my experience this type of sigmoidal profile was the result of substrate hydrophobicity. I have seen a very similar profile when I used an enzyme with a hydrophobic peptide substrates. At low concentration the peptide had a tendency to stick to plastic and at high concentration it used to aggregate and formed micelles unacessible for enzyme. Addition of DMSO (10%) and shifting to low-binding plates resolved the problem immediately. I considered a number of potential and fancy models including some slow conformational changesArticle Cooperativity in Monomeric Enzymes with Single Ligand-Binding Sites

Substrate impurity (rather then inhibition) is yet another reason for deviations from MM profiles. When I studied phosphatases, the presence of inorganic phosphate in phosphate-monoester substrates was the reason. For what appeared to be a clear substrate inhibition, turned out a classic product inhibition that was gone immediately after I figured how to synthesize and isolate the molecules nearly free of inorganic phosphate.

Commercial coupled assays applied without proper validation frequently yield ridiculous kinetic results that have nothing to do with the target enzyme behavior.

The bottom line - before I resort to complex kinetic modeling trying to explain some bizarre enzymatic behavior, I now always look for simple explanations first. . A quick NMR or LC/MS of substrate, the order of substrate addition, additives (BSA,DMSO, osmolytes, salts), even proper calibration and validation of multichannel pipeters can help a lot in understanding what is really going on.