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- Abstract
- 1. Introduction
- 2. Relevance of the reaction rate constants optimized by progress curve analysis
- 3. Relevance of the Michaelis constant optimized by progress curve analysis
- 4. Monte Carlo simulation as a diagnostic tool in progress curve analysis
- 5. Concluding remarks
- References

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Cent Eur J Biol. Author manuscript; available in PMC 2009 June 25.

Published in final edited form as:

PMCID: PMC2701647

EMSID: UKMS2826

The publisher's final edited version of this article is available at Cent Eur J Biol

See other articles in PMC that cite the published article.

Progress curve analysis is a convenient tool for the characterization of enzyme action: a single reaction mixture provides multiple experimental measured points for continuously varying amounts of substrates and products with exactly the same enzyme and modulator concentrations. The determination of kinetic parameters from the progress curves, however, requires complex mathematical evaluation of the time-course data. Some freely available programs (e.g. FITSIM, DYNAFIT) are widely applied to fit kinetic parameters to user-defined enzymatic mechanisms, but users often overlook the stringent requirements of the analytic procedures for appropriate design of the input experiments. Flaws in the experimental setup result in unreliable parameters with consequent misinterpretation of the biological phenomenon under study. The present commentary suggests some helpful mathematical tools to improve the analytic procedure in order to diagnose major errors in concept and design of kinetic experiments.

Enzyme activity is characterized in terms of rates [1], but the experimental enzyme assays measure substrate or product concentrations and not directly rates. The derivation of reaction rates from the changing concentrations is always an approximation with implicit error, but it is widely used in the determination of kinetic parameters because of the simplicity of its mathematical description with differential rate equations. Thus, in the simplest case the scheme $E+S\underset{{k}_{-1}}{\overset{{k}_{1}}{\rightleftarrows}}ES\stackrel{{k}_{2}}{\to}E+P$ is assumed, where *E* is enzyme, *S* is its substrate, *P* is the measured product, *k _{1}, k_{2}* and

$$\frac{dP}{dt}=\frac{{k}_{2}.E.({S}_{0}-P)}{{K}_{M}+{S}_{0}-P}$$

(1)

where the subscript 0 indicates the initial value of concentration, the ${K}_{M}=\frac{{k}_{-1}+{k}_{2}}{{k}_{1}}$ is the so called Michaelis constant [1]. If *P* is measured over time, typically a linear increase in *P* is approximated from the initial phase of the reaction to define the rate $\frac{dP}{dt}$ and using a range of *S*_{0} a simple optimization procedure can identify the *K _{M}* and

$$t=\frac{1}{{k}_{2}.E}P+\frac{{K}_{M}}{{k}_{2}.E}\mathrm{ln}\frac{{S}_{0}}{{S}_{0}-P}$$

(2)

The problem of using Eq.(2) for optimization stems from the fact that its inverse function *P*=*P ^{M}* (

In order to achieve versatility of the application, the available programs operate with reaction rate constants, e.g. in the scheme shown in the Introduction the values of *k _{1}, k_{-1}* and

For illustration of the issue mentioned above we will take an example from the recent literature. Progress curve analysis is applied in a study on the interactions of various mutants of pancreatic secretory trypsin inhibitor and trypsin [6]. The authors of this report worked with the scheme shown in the Introduction and claim that they have identified the *k _{1}, k_{-1}* and

To further analyze the basic scheme, one can use the data of the control curve from Figure 4B of [6] as mean values of experimental progress curves (this is the control curve, in which the final product concentration approaches best the initial substrate concentration). Because the authors do not provide any information about the variance of their assay, one can assume that the deviation of the final *P* from the initial *S* in Figures 4A and 4C of [6] indicates the scale of the experimental uncertainties and accordingly we model the error of the experiment with flat standard deviation (SD) at each data point as shown in our Figure 1. Using numerical integration of the differential rate equations for the scheme $E+S\underset{{k}_{-1}}{\overset{{k}_{1}}{\rightleftarrows}}ES\stackrel{{k}_{2}}{\to}E+P$, progress curves are simulated with different rate coefficients *k _{1}, k_{-1}* and

If the expert re-evaluation of the optimized rate constants from Figure 1 is performed as discussed in the preceding section, the Michaelis constant ${K}_{M}=\frac{{k}_{-1}+{k}_{2}}{{k}_{1}}$ can be calculated from the three sets of parameters. The three combinations of rate constants used in Figure 1 yield remarkably similar *K _{M}* values: 85.06, 83.71 and 82.98 μM. Does this mean that if once all three rate constants cannot be defined, working with a single substrate concentration progress curve analysis still allows at least reliable determination of

Instead of the rate constants, in the alternative analysis the *K _{M}* and

Progress curve of trypsin catalyzed reaction analyzed by optimization of the Michaelis and the catalytic constants. The data are the same as in Figure 1, but the integrated equation
is used for fitting the parameters *K* **...**

$$t=\frac{1}{{k}_{2}.E}P+\frac{{K}_{M}}{{k}_{2}.E}\mathrm{ln}\frac{{S}_{0}}{{S}_{0}-P}$$

When biological processes are formalized in mathematical terms, measured experimental data are used to derive model parameters as general characteristics of the process under study. A legitimate requirement in biological modeling is to provide not only the best estimate of the parameters, but also a statistical measure of their variability. However, even in the case of the simplest models the statistical distribution of the parameters can be described only with complicated and model-specific analytic weighting procedures for conversion of the uncertainties of the measurement into confidence intervals of the parameters. An alternative robust approach for determination of statistical variance of model parameters is the Monte Carlo simulation and its variants [10,11]. The principle of this approach is that the measured data define the distribution of the sampled quantity (e.g. product concentration in different points of the progress curve) and thereafter a computer performs a great number (1000-1500) of virtual experiments: it draws random samples from the modeled distribution of experimental data and estimates the synthetic model parameters on the basis of these quasi-experimental points. The synthetic sets of parameters are flipped around the original model parameters. The mean of the gained 1000-1500 points determines the best guess for the needed model parameters, whereas their distribution, which reflects the uncertainties of the experiment, determines their multidimensional “root” confidence intervals (for further details see [12,13]).

In addition to the typical application of Monte Carlo simulations described above, their inclusion as a routine supplement of progress curve analysis can be of special benefit for screening the adequateness of interpretation of the model parameters. For example, Monte Carlo simulation with the data and the optimization approach of Figure 1 would explicitly show that the experimental setup is not sensitive to the values of the three rate constants. Figure 3 illustrates that with these experimental data the fitting procedure to a model with three parameters converges successfully with ranges of *k _{1}* and

Even if the model turns to be adequate, Monte Carlo simulations can help the identification of additional shortcomings in the experimental design, e.g. insufficiency of measured data. This issue is exemplified by Figure 4. Running the optimization with largely different initial values for *K _{M}* and

The technical development since the introduction of progress curve analytic programs like FITSIM and DYNAFIT [4,5] makes possible the everyday application of computer-intensive procedures for a wide audience of researchers in enzyme kinetics. Thus, Monte Carlo simulations can be performed on desktop computers within reasonable timeframe even for complex reaction schemes. In addition to reliable description of the confidence intervals of the model parameters, their implementation as a routine utility in progress curve analysis would automate tasks, which are available as separate optional tests in the widely used analytic programs. For example, a test for the sensitivity of the experimental data to the fitted kinetic parameters can be performed with FITSIM [4], but some of its users skip it [6], which results in unreliable interpretation. A routine Monte Carlo simulation (Figure 3) would warn the user in similar situations that a re-evaluation of the experimental design and the model is warranted. However, it should be emphasized that a computer cannot substitute completely the expertise of the user, e.g. the insufficiency of data illustrated in Figure 4 can be easily missed, if the optimization is not started from largely different initial values of the model parameters, which should be suggested by the expert user. In summary, Monte Carlo simulations can reduce the risk for misuses of progress curve analysis, but do not eliminate the necessity for careful expert inspection of the final analytic output.

This work was supported by the Wellcome Trust [083174/B/07/Z] and the Hungarian Scientific Research Fund [OTKA K60123].

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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