The shortcut method is to fit each curve as if it were for a single-substrate enzyme, using the Michaelis-Menten equation. This will give you an apparent Km (Km,app) for the varied substrate, which will be equal to the actual Km only if the enzyme is truly saturated with the other substrate and if there is no substrate inhibition by the other substrate.

The more elaborate method is to measure initial rates for a matrix of concentrations of both substrates, i.e. vary both substrate concentrations. Then fit the entire matrix of rates by nonlinear least squares to the suitable bisubstrate rate equation to obtain estimates of all the relevant kinetic constants. Taking this approach avoids the necessity of achieving thorough saturation and also reveals whether any substrate inhibition is occurring. If substrate inhibition is observed, a modification to the equation is needed.

There are actually several bisubstrate rate equations. Consult Segel's massive book, chapter 9, or other advanced texts on rapid equilibrium and steady-state kinetics to find them. Once you have all the data, you can try fitting various equations to see which fits the best. Examples are Ordered (which you wrote in your question), Rapid Equilibrium Random (mathematically equivalent to Ordered, but mechanistically distinct), and Ping Pong. If you already know that the kinetic mechanism is Ordered, and the order of substrate binding, then go ahead and use that equation to fit the matrix of rates. This will give you the actual values of the Kms and Kia, assuming there is no substrate inhibition.

The modern way:

New insights and tools for the elucidation of lipase catalyzed esterification reaction mechanism in n-hexane: The synthesis of ethyl butyrate (DOI: 10.1016/j.mcat.2018.06.004)

Yes, you need to measure both substrate A at various fixed concentrations of B and vice versa. Since you need at least 12 substrate concentrations each evenly spaced from 0.1 to at least 5 Km (better 10 Km if no substrate inhibition occurs), that classically makes 122 = 144 rates to be determined (all with several time points and the necessary repetitions). However, you could use the sparse matrix approach of experimental design developed by Taguchi to reduce this significantly: for k factors at n values each, the number of required experiments is the next larger figure to E = 2k+1 that is dividable by n, in your case E=5 and the next larger number that can be divided by 12 is the 12 itself. I have described the process on pp 330 of ISBN 978-1-4419-7250-7.

Thank you all for the answers. I have calculated the kinetic parameters using data from matrix of concentrations for both substrates. However, the Km values for both substrates are somewhat higher than what is usually reported in literature for this enzyme family (0.4 mM vs 0.02-0.2 mM reported). How can I validate my data? I am confident that I have measured proper initial rates for all substrate concentrations.

I have used the equation mentioned in the question, but fitted the data using non-linear regression using 1/v vs 1/[A] at different [B], and a secondary plot with the slopes and intercepts from first plot against 1/[B].

@Dr. Shapiro: How can I make an equation in SigmaPlot for non-linear regression with both substrate concentrations?

Dear Avinash

You do not need Sigma Plot, you need the numbers. Here there is a page where you can perform non-linear regression of data using a user-introduced equation:

http://statpages.info/nonlin.html

The only limitation is the variables have to be x1, x2, and so on and parameters names have to be a, b, c, d, e, f, g and h. Upto 8 parameters can be handled.

Then it does run one iteration at a click, so you need to click many times the "iterate" until there is no improvement in the parameter values.

You can use standard errors (weights), or enter your repetitions as several lines with the same variables... i.e.

1 1 2 24

1 1 2 23

1 1 5 35

1 1 5 36.5

In this example you have two observation for the value 2 and two for the value 5 of the third variable (x3). Variables x1 and x2 are 1.

It takes undergraduates a few minutes to learn how to use it, so it should take you 3 seconds.

It does not make a plot, but you can do that in Sigma Plot, once you have the estimates for all parameters.

I have tested it extensively with undergraduates and performs fine.

Best wishes

Rogelio

I have often used Grafit for nonlinear regression of enzyme kinetics rate data. Try to avoid using reciprocal plots, which distort error distributions. Use the original rates.

Dear Adam B Shapiro,

There are many software packages to perform non-linear regression analysis. Some like Origin, Prisma, Graphpad, Sigma Stat, Sigma Plot, maple, matlab and mathematica (along with many others) are commertial and usually user-friendly, but you need to buy them and can be quite expensive.

Other differ in their approach and power but are freeware and have steep learning curves. These category include octave, GNUplot, XMGrace, and R, amongst other. If you need my recommendation on one of these, I would go for GNUplot, may be the easiest to learn of the group, though octave and R can do quite a lot more. For instance, with Octave it is possible to fit data to sets of differential equations, using numerical integration+non-linear regression; though for that you need some programming experience.

The page I quoted offer the best of both worlds. It is simple, can be learned pretty fast, it is robust and can fit data to any equation with several variables and up to 8 parameters.

Best wishes,

Rogelio

With non linear regression, I get the following values:

vmax 733.34 25.63

Ka 0.66 0.05 0.094

Kb 27.12 2.77 6.8

Kia 0.43 0.096 1.06

The second column is standard error, and the third column are values from literature for the same enzyme from another organism. Are the error values acceptable? Also, what could be probable reasons for the higher Ka and Kb values - can they be attributed to the kinetic mechanism, or do I need to generate more data to recalculate the values? In case it is due to kinetic mechanism, what other terms should be introduced in the rate equation to account for the additional factors?

There are several thing you should check.

A) your coefficients of variation (100^*se/value) are 3.49% 7.58% 10.2% and 22.3%. The fit seems acceptable.

B) Check at the residuals. If you used the page I suggested, the residuals are printed below the RMS value. If you used some other program, use your parameter values, and in a spreadsheet you can calculate the theoretical values from the equation, using the concentrations of A and B corresponding to each observation (each measurement you have). Then calculate the difference

residual=(V_observed - V_calculated)

Calculate the correlation coefficient for the residuals against each variable. i.e.

residual vs. [A], residuals vs. [B], residuals vs. V_observed.

B) In my experience the Kia and Kb differences are not so large and can be attributed to differences in the temperature, pH, reagents purity, equipment calibration, enzyme purity and so on. So ( these are not that surprising.

C) The difference in Vmax is most likely to do with either differences in the source of enzyme (organism, way of production, etc.) and/or differences in the amount of enzyme (you did not provide units, so I do not know if the values are specific activity or just velocity).

D) The difference in Ka is large, your enzyme may be different to the one reported in its affinity and reactivity towards substrate A (is it the same substrate?).

Finally, I noticed you mentioned NAD was the second substrate. This is not so common amongst dehydrogenases, most have the NAD as first substrate to enter and NADH last substrate to leave the enzyme.

But also there is something unusual about your values, though you did not provide units, and I can only guess. If all units for the K values are mM. the Kb of 27 mM would be for NAD, that is too large for anything I have seen in other dehydrogenases. If it is 27 µM is pretty low for a Km. Now if the Ka for the reduced substrate is 0.66 mM and its Kia 0.43 mM these values mean strong affinity, but seem possible, however if these are µM, they seem pretty low, it is not common to have such a tight binding for the reduced substrate. Are you sure NAD is the first substrate?

Check at the inhibition pattern by NADH of your enzyme at fixed concentration of substrate A, and fixed-varied concentration of NAD. The inhibition should be non-competitive, sometimes uncompetitive, but should not be competitive. If it is, you may have the wrong order of substrates. The velocity equation would still fit the data, but the parameter values should change.

Best wishes,

Rogelio

One problem that may be encountered is that the data do not have homogeneous absolute, but homogeneous relative errors (common in experiments with radiactive labels). In that case, one needs to minimize chi2. This can be done only by simplex algorithm (Caceci & Cacheris 1984, Nelder & Mead 1965). If you use the Marquardt-Levenberg method to minimise the sum of squared residuals instead, the curves fit the high concentrations perfectly fine, but not the low. For an example, see doi:10.1046/j.1432-1327.1999.00643.x.

All optimization methods have limitations, for most searches are NP-complete problems. That is why careful analysis of the estimates and residuals is important. Yet, at the end of the day, data quality is the most relevant factor.

Best wishes

Rogelio