# Full Analytic Progress Curves of Enzymic Reactions in Vitro

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}=[X]

_{i}(t), depends on the particular parameters a

_{i}specific for the particular processes considered.

_{i}in (1) has to be non-linear in variable [X]

_{i}due to the complexity of the structure of the bio-systems and of the biochemical kinetics.

_{1}and k

_{–1}, which then irreversibly breaks down into the product and the enzyme [6,7,8,9]:

## 2. Method

_{0}],[E

_{0}],0,0) at the time t=0.

_{0}]

_{0}]

_{max}= k

_{2}[E

_{0}]

_{M}ln[S] = [S

_{0}] − V

_{max}t + K

_{M}ln[S

_{0}]

_{max}and a slope of K

_{M}/V

_{max}provide the kinetic parameters V

_{max}and K

_{M}, respectively. Still, this approach has been criticized [5,18], and it is worthwhile investigating whether the exact solution of (14) can be obtained for fitting a non-linear progress curve.

_{W}(τ)/[S

_{0}] and [S]

_{L}(τ)/[S

_{0}], against the scaled time [25]

_{-1}= k

_{2}= 10

^{2}s

^{-1}, k

_{1}= 10

^{6}M

^{-1}s

^{-1}, while the initial conditions are set to [S

_{0}] = 10

^{-4}M and [E

_{0}] = 10

^{-6}M, respectively. As seen in Figure 1, the qualitative and quantitative behaviors of the substrate concentration in both W-Lambert and logistic cases are striking similar.

**Figure 1.**Time-dependent behavior of the substrate scaled concentration for the paradigmatic enzyme-substrate reaction (2) when the W-Lambert (dashed line) and logistic (solid line), (19) and (20) versions of the Michaelis-Menten kinetics, are employed, respectively, with the parametric values k

_{-1}= k

_{2}= 10

^{2}s

^{-1}, k

_{1}= 10

^{6}M

^{-1}s

^{-1}, [S

_{0}] = 10

^{-4}M, and [E

_{0}] = 10

^{-6}M, against the scaled time (22).

_{1}, f

_{2}, and f

_{3}.

## 3. Multiple Alternative Enzyme-Substrate Reactions

_{i}),[E],[ES]

_{i},[P]

_{i})

_{t}

_{=0}= ([S

_{0}]

_{i},[E

_{0}],0,0).

_{i}(t) = [S

_{0}]

_{i}− [S]

_{i}(t) − [ES]

_{i}(t).

_{ij}≅ 1. In this frame, the system (28) can be integrated and the result rearranged so that the proper comparison with the W-Lambert equation (18) to be employed. This causes the W-Lambert transcendent solutions for the system (28) to take the closed forms [12]:

_{ij}<<1. However, in this case the first order of the Taylor expansion of (28b) in (28a) can be retained and, by repeating the previous integration and rearrangement procedure the W-Lambert closed form solution can be cast as [12]:

## 4. Application on Competitive Inhibition

- the respective Michaelis-Menten constants for the substrate and inhibitor branches of (31):
- the respective maximum velocities for the substrate and inhibitor branches of (31):

_{S}]

_{W,L}(t) = [S

_{0}]−[S]

_{W},

_{L}(t)−[ES]

_{W,L}(t),

_{I}]

_{W,L}(t) = [I

_{0}]−[I]

_{W,L}(t)−[EI]

_{W,L}(t),

_{W,L}(t) = [E

_{0}]−[ES]

_{W,L}(t)−[EI]

_{W,L}(t),

**Figure 2.**The scaled progress curves (39) of the species concentrations involved in competitive reaction (31) for the pilot test with k

_{-1}= k

_{2}= 10

^{2}s

^{-1}, k

_{1 }= 10

^{6}M

^{-1}s

^{-1}, k

_{-3 }= k

_{4}= 10s

^{-1}, k

_{3 }= 10

^{5}M

^{-1}s

^{-1}, [S

_{0}] = 10

^{-4}M, [I

_{0}] = 10

^{-5}M, and [E

_{0}] = 10

^{-6}M, arranged as follows: (a) for the leading and inhibitory substrates concentrations, according to (39a), (b) for the substrate-enzyme and inhibitor-enzyme complexes concentrations, according to (39b), (c) for the products of the leading and inhibitory substrates concentrations, according to (39c), and (d) for the enzyme concentration, according to (39d), within the W-Lambert (dashed lines) and logistic (solid lines) Michaelis-Menten kinetics against the scaled time (22), respectively.

_{-1}= k

_{2}= 10

^{2}s

^{-1}, k

_{1}= 10

^{6}M

^{-1}s

^{-1}, k

_{-3}= k

_{4}= 10s

^{-1}, k

_{3}= 10

^{5}M

^{-1}s

^{-1}, while the initial condition have been set to [S

_{0}] = 10

^{-4}M, [I

_{0}] = 10

^{-5}M, and [E

_{0}] = 10

^{-6}M, respectively.

## 5. Conclusions

## Acknowledgements

## References

- Mayr, E. What evolution is? The Orion Publishing Group Ltd., “Science Masters” Brockmann Inc., 2001. [Google Scholar]
- Thom, R. Structural Stability and Morphogenesis – An Outline of a General Theory of Models; W. I. Benjamin, Inc.: Reading, Massachusetts , 1975. [Google Scholar]
- Crampin, E. J.; Schnell, S.; McSharry, P. E. Mathematical and Computational Techniques to Deduce Complex Biochemical Reaction Mechanisms. Prog. Biophys. Mol. Biol.
**2004**, 86, 77–112. [Google Scholar] - Mandl, F. Quantum mechanics; John Wiley & Sons: Chichester, 1992. [Google Scholar]
- Schnell, S.; Maini, P. K. A Century of Enzyme Kinetics: Reliability of the KM and Vmax Estimates. Comm. Theor. Biol.
**2003**, 8, 169–187. [Google Scholar] - Brown, A. J. Influence of Oxygen and Concentration on Alcohol Fermentation. J. Chem. Soc. Trans.
**1892**, 61, 369–385. [Google Scholar] - Brown, A. J. Enzyme Action. J. Chem. Soc. Trans.
**1902**, 81, 373–388. [Google Scholar] - Henri, V. Über das gesetz der wirkung des invertins. Z. Phys. Chem.
**1901**, 39, 194–216. [Google Scholar] - Michaelis, L.; Menten, M. L. Die kinetik der invertinwirkung. Biochem. Z.
**1913**, 49, 333–369. [Google Scholar] - Voet, D.; Voet, J. G. Biochemistry, (second edition); John Wiley & Sons, Inc.: New York, 1995; Chapter 13. [Google Scholar]
- Schnell, S.; Mendoza, C. Time-Dependent Closed Form Solution for Fully Competitive Enzyme Reactions. Bull. Math. Biol.
**2000**, 62, 321–336. [Google Scholar] - Schnell, S.; Mendoza, C. Enzyme Kinetics of Multiple Alternative Substrates. J. Math. Chem.
**2000**, 27, 155–170. [Google Scholar] - Cornish-Bowden, A. Fundamentals of Enzyme Kinetics; Butterworths: London, 1979. [Google Scholar]
- Segel, I. H. Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Systems; Wiley: New York, 1975. [Google Scholar]
- Ritchie, R. J.; Prvan, T. A Simulation Study on Designing Experiments to Measure the Km of Michaelis-Menten Kinetics Curves. J. Theor. Biol.
**1996**, 178, 239–254. [Google Scholar] - Zimmerle, C. T.; Frieden, C. Analysis of Progress Curves by Simulations Generated by Numerical Integration. Biochem. J.
**1989**, 258, 381–387. [Google Scholar] - Szedlacsek, S. E.; Ostafe, V.; Duggleby, R. G.; Serban, M.; Vlad, M. O. Progress-Curve Equations for Reversible Enzyme-Catalysed Reactions Inhibited by Tight-Binding Inhibitors. Biochem. J.
**1990**, 265, 647–653. [Google Scholar] - Goudar, C. T.; Sonnad, J. R.; Duggleby, R. G. Parameter Estimation Using a Direct Solution of the Integrated Michaelis-Menten Equation. Biochim. Biophys. Acta
**1999**, 1429, 377–383. [Google Scholar] - Câteau, H.; Tanaka, S. Kinetic Analysis of Multisite Phosphorylation Using Analytic Solutions to Michaelis-Menten Equation. J. Theor. Biol.
**2002**, 217, 1–14. [Google Scholar] - Gray, P; Scott, S. K. Chemical Oscillations and Instabilities. Non-linear Chemical Kinetics; Clarendon Press: Oxford, 1990. [Google Scholar]
- Segel, L. A. On the Validity of the Steady State Assumption of Enzyme Kinetics. Bull. Math. Biol.
**1988**, 50, 579–593. [Google Scholar] - Segel, L. A.; Slemrod, M. The Quasi-Steady-State Assumption: A Case Study in Perturbation. SIAM Rev.
**1989**, 31, 446–477. [Google Scholar] - Duggleby, R. G.; Morrison, J. F. The Analysis of Progress Curves for Enzyme-Catalysed Reactions by Non-Linear Regression. Biochim. Biophys. Acta
**1977**, 481, 297–312. [Google Scholar] - Duggleby, R. G. Quantitative Analysis of the Time Courses of Enzyme-Catalyzed Reactions. Methods
**2001**, 24, 168–174. [Google Scholar] - Schnell, S.; Mendoza, C. Closed Form Solution for Time-Dependent Enzyme Kinetics. J. Theor. Biol.
**1997**, 187, 207–212. [Google Scholar] - Rubinow, S. I. Introduction to Mathematical Biology; Wiley: New York, 1975. [Google Scholar]
- Haldane, J. B. S.; Stern, K. G. Allgemeine Chemie der Enzyme; Dresden: Verlag von Steinkopff, 1932. [Google Scholar]
- Lineweaver, H.; Burk, D. The Determination of the Enzyme Dissociation Constants. J. Am. Chem. Soc.
**1934**, 56, 658–666. [Google Scholar] - Cornish-Bowden, A. The Use of the Direct Linear Plot for Determining Initial Velocities. Biochem. J.
**1975**, 149, 305–312. [Google Scholar] - Barry, D. A.; Parlange, J. –Y; Li, L.; Prommer, H.; Cunningham, C. J.; Stagnitti, F. Analytical Approximations for Real Values of Lambert W-function. Math. Comp. Simulation
**2000**, 53, 95–103. [Google Scholar] - Hayes, B. Why W? American Scientist
**2005**, 93, 104–108. [Google Scholar] - Lacrămă, A. -M.; Putz, M. V.; Ostafe, V. New Enzymatic Kinetic Relating Michaelis-Menten Mechanisms. Annals of West University of Timisoara-Series of Chemistry
**2005**, 14(2), 179–190. [Google Scholar] - Mattick, J. S. The Hidden Genetic Program of Complex Organisms. Sci. Am.
**2004**, 291, 60–7. [Google Scholar] [CrossRef] - Silverman, P. H. Rethinking Genetic Determinism. The Scientist
**2004**, 18(10), 32–3. [Google Scholar] - Goodman, A. F.; Bellato, C. M.; Khidr, L. The Uncertain Future for Central Dogma. The Scientist
**2005**, 19(12), 20–1. [Google Scholar] - Cantor, C. R.; Schimmel, P. R. Biophysical Chemistry. Part III. The Behavior of Biological Macromolecules; W.H. Freeman and Company: San Francisco, 1980. [Google Scholar]
- Copeland, R. A. Enzymes-A Practical Introduction to Structure, Mechanism, and Data Analysis; Wiley-VCH: New York, 2000. [Google Scholar]
- Curran, J. M.; Gallagher, J. A.; Hunt, J. A. The Inflammatory Potential of Biphasic Calcium Phosphate Granules in Osteoblasts/Macrophage Co-Culture. Biomaterials
**2005**, 26, 5313–5320. [Google Scholar] [CrossRef] - Rubinow, S. I; Lebowitz, J. L. Time-Dependent Michaelis-Menten Kinetics for an Enzyme-Substrate-Inhibitor System. J. Am. Chem. Soc.
**1970**, 92, 3888–3893. [Google Scholar] [CrossRef] - Walsh, C. Enzymatic reaction mechanisms; W.H. Freeman and Company: San Francisco, 1979. [Google Scholar]
- Duggleby, R. G.; Wood, C. Analysis of Progress Curves for Enzyme-Catalysed Reactions. Automatic Construction of Computer Programs for Fitting Integrated Rate Equations. J. Biochem.
**1989**, 258, 397–402. [Google Scholar] - Duggleby, R. G. Analysis of Enzyme Progress Curves by Nonlinear Regression. Methods Enzymol.
**1995**, 249, 61–90. [Google Scholar] [CrossRef] - Duggleby, R. G. Progress Curves of Reactions Catalyzed by Unstable Enzymes. A Theoretical Approach. J. Theor. Biol.
**1986**, 123, 67–80. [Google Scholar] [CrossRef] - Ross, J.; Schreiber, I.; Vlad, M. O. Determination of Complex Reaction Mechanisms: Analysis of Chemical, Biological and Genetic Networks; Oxford University Press, 2006. [Google Scholar]

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**MDPI and ACS Style**

Putz, M.V.; Lacrama, A.-M.; Ostafe, V.
Full Analytic Progress Curves of Enzymic Reactions in Vitro. *Int. J. Mol. Sci.* **2006**, *7*, 469-484.
https://doi.org/10.3390/i7110469

**AMA Style**

Putz MV, Lacrama A-M, Ostafe V.
Full Analytic Progress Curves of Enzymic Reactions in Vitro. *International Journal of Molecular Sciences*. 2006; 7(11):469-484.
https://doi.org/10.3390/i7110469

**Chicago/Turabian Style**

Putz, Mihai V., Ana-Maria Lacrama, and Vasile Ostafe.
2006. "Full Analytic Progress Curves of Enzymic Reactions in Vitro" *International Journal of Molecular Sciences* 7, no. 11: 469-484.
https://doi.org/10.3390/i7110469