### Topics

- I guess you already plotted the data since you said you generated it and it was biphasic. I think the question you are asking is how to fit the data or how to analyze the data. To do so, you have to model your enzymatic reaction first and choose or get a right equation if there is one to do the analysis. Make sure the biphasic is not an artifact. If it is real, you got a problem. Good luck!
- My question is how to plot both phases separately to get 2 Kms and 2 Vmaxs. The first phase is straight forward but the second phase is difficult to plot. Any ideas?
- If you insist plotting the two phases separately, you already assumed the reaction scheme is composed of two independent parallel Michaelis Menten reactions that have same product. In this case, you can use sum of two hyperbolas to fit the whole data set and use the fitting parameter to regenerate two phases separately.

I guess I still don't understand your data was from real measurement or from some exercise. If it was from real measurement, it must be more complicated with v depending on substrate concentration biphasically other than two simple reactions. - A reciprocal plot of your data might help you extrapolate the individual parameters. You might want to check out Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems by Irwin H. Segel for a detailed explanation of how to analyze your data.
- Sorry, I don´t understand what you mean with a biphasic enzyme. Could you provide of simple sketch the reaction mechanism ? I search in google, but can´t find it?. It is only to learn.
- I recommend that you read Cleland's Enzyme Kinetics book. Could you provide a little more detail as to what you are doing and seeing? Appropriate analysis with curve fitting equations will depend on many assumptions. Also, if you are using an indirect assay, you could be limited by a component of the coupled reaction.
- You can use sequential Michaelis-Menten curves

I have odeled human butyrylcholinesterase which has substrate activation and shown that you can have inhibitors and activators of the non-substrated activated or the substrate activated form of the enzyme

Use the equation

v=(V1([S]/([S]+Ks1)))-(V1([S]/([S]+Ks2)))+(V2([S]/([S]+Ks2)))

this segregates the fraction of the enzyme population found in each form based on the concentration of substrate.

It also allows for easy insertion of terms that affect the kinetic constants for each form of the enzyme

Check out the paper I wrote where I use this equation to model 2 inhibitors affecting the enzyme activity of of butyrylcholinesterase at the same time

Biochim Biophys Acta. 2011 Dec;1810(12):1230-5. Epub 2011 Aug 19.

Synergistic inhibition of butyrylcholinesterase by galantamine and citalopram.

Walsh R, Rockwood K, Martin E, Darvesh S.

I also don't suggest you use Segel or Cleland books as changes to the kinetic constants for enzyme systems based on traditional modeling methods are have problems associated with them

the whole designation of inhibitors as competitive, mixed non-competitive and non-competitive is most likely erroneous. Non-competitive inhibition when rearranged from

Vmax/(1+[I]/Ki)

to

Vmax-(Vmax [I]/([I]+Ki))

demonstrates that change in Vmax is a result of mass action binding of the inhibitor to the enzyme. This is the same curve observed with the Michaelis-Menten equation or receptor binding

when you try the same thing with the competitive inhibition equation you the Km being divided by 1-the mass action binding term which results in a need for the mix noncompetitive inhibition which has two binding constants for one binding event.

by replacing the competitive term with a term that does produce the same sort of mass binding term associated with the non-competitive equation you can produce an equation that only requires one binding constant to describe changes in Km and Vmax

The other problem with competitive, non-competitive and mixed noncompetitive inhibition is a failure to distinguish between binding constants and effect on the enzyme

taking the non-competitive inhibitor term above

Vmax-(Vmax [I]/([I]+Ki))

while this tern does recognize the mass binding of the inhibitior with

[I]/([I]+Ki)

it assumes that the inhibition is total with the inhibitor binding

"-Vmax"

while there can be all sorts of different levels of activity remaining when the inhibitor binds so this term should be replaces with a delta Vmax term

deriving kientic equations the way segel does in his book ignore the difference between binding constants and effect on the catalytic system.and should be considered carefully before you decide to use them.

Using terms like

V1-(V1 [I]/([I]+Ki))

and

Ks1-(Ks1 [I]/([I]+Ki))

in the equation above should help you model just about any effect you see in your kinetics with the biphasic enzyme

you can check out more in my book chapter on the subject at

http://www.intechopen.com/books/medicinal-chemistry-and-drug-design/alternative-perspectives-of-enzyme-kinetic-modeling

good luck - Thanks a lot Dr Ryan Walsh for your help. I will consult your paper. Later I managed to find two references about similar enzymes (manganese and lignin-peroxidases) which also showed biphasic behavior similar to my enzyme.

1. Ruiz-Duenas, FJ et al. 2001 Biochemical Society Transactions, 29(2): 116-122

2. Timofeevski, SL et al 2000 Archives of Biochem. Biophys. 373(1):147-153 - We have discussed biphasic kinetics with relation to an enzyme that exists as an equilibrium of two quaternary structure isoforms. The example is the enzyme porphobilinogen synthase and solution is to treat the protein as a morpheein and fit the data to a double hyperbolic equation...essentially the sum of two MM equations. From this you will get two Km values and two Vmax values. See for instance:

1. Selwood, Trevor; Jaffe, Eileen K. (2012). "Dynamic dissociating homo-oligomers and the control of protein function". Archives of Biochemistry and Biophysics 519 (2): 131–43.

2. Jaffe, Eileen K.; Lawrence, Sarah H. (2012). "Allostery and the dynamic oligomerization of porphobilinogen synthase". Archives of Biochemistry and Biophysics 519 (2): 144–53. doi:10.1016/j.abb.2011.10.010. PMC 3291741.

3. Lawrence, Sarah H.; Jaffe, Eileen K. (2008). "Expanding the concepts in protein structure-function relationships and enzyme kinetics: Teaching using morpheeins". Biochemistry and Molecular Biology Education 36 (4): 274–283. - It all depends how you model the reaction and what your reaction scheme is. Like I said before, you can use a sum of two hyperbolas if you reaction scheme is a two parallel simple Michaelis Menten reactions with the same product. If your biphasic feature resulted from more complicated cases, you cannot use a sum of two hyperbolas to analyze your data. For example, if the substrate and enzyme binding stoichiometry is larger than 2, you end up having more phases in your v vs [S] plot.
- The problem with simply adding two MM equations as suggested by Eileen and Wenxiang above is that adding the two curves does not reflect the change in state induced in the enzyme and does not allow for modeling of modifiers of the different forms of the enzyme.

If you were to assume you had an inhibitor of only the allosterically modified form of the enzyme; in the system where you add the MM equations together the inhibition of V2 could never be more than V2 and V1 would still exist.

While if you use the expanded form

v=(V1([S]/([S]+Ks1)))-(V1([S]/([S]+Ks2)))+(V2([S]/([S]+Ks2)))

as the modified form appears the unmodified disappears and so each form of the enzyme can be analyzed for the effect of inhibitors or activators separately. - According to Dr. Walsh's equation:

v=(V1([S]/([S]+Ks1)))-(V1([S]/([S]+Ks2)))+(V2([S]/([S]+Ks2)))

=(V1([S]/([S]+Ks1)))+((V2-V1)([S]/([S]+Ks2)))

=(V1([S]/([S]+Ks1)))+(V2eff ([S]/([S]+Ks2)))

V2eff=V2-V1

It is basically a sum of two hyperbolas.

Again the analysis should depend on what reaction scheme is. Dr. Walsh assumed there is allosterical inhibition, but the original poster never specified reaction scheme other than biphasic observation. - The equation can be used for allosteric inhibition but also can be used for activation as is demonstrated in the excel figure I attached in the last post

and yes while it can be factored like Dr Wenxiang Cao has pointed out, it is inappropriate as once again it obscures the transition from one state to the other (see the excel file for details) and prevents analysis of other modifiers of enzyme activity on the individual forms of the enzyme. - Thanks Eileen Jaffe, Ryan Walsh and Wenxiang Cao for your excellent suggestions. All your comments were very useful.
- Is there a modeling software which can model sigmoidal biphasic data? For normal plots including Hill, I use Enzyme kinetic module EK2 linked to Sigma plot.
- You can use excel

Kemmer G, Keller S. Nonlinear least-squares data fitting in Excel spreadsheets. Nat Protoc. 2010 Feb;5(2):267-81. - If you use SigmaPlot, you can simply enter any equation you wish into the regression wizard. It is actually much easier to do this if you do NOT use the kinetics module.
- I totally agree with Eileen. It will be much easier and you have much more control just using real data analysis software like Origin, IGOR, Prism, SigmaPlot,, KaleidaGraph etc, or using Matlab with your own code. If you really really want to use kinetics simulation software, try KinTek kinetic explorer or free software Tenua.
- thanks for all valuable suggestions. I managed to fit data perfectly to biphasic equation using Origin.
- Hi! DynaFit has been very useful for me...

## All Answers (21)