Computational biochemistry ferenc Bogár György Ferency


 Temperature dependence of rateconstant



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3. Temperature dependence of rateconstant

Arrhenius equation

Changing the temperature dramatically affects reaction between either ligands and receptors, or enzymes and substrates. On the one hand, all the equilibrium and rate constants in the mechanisms are temperature dependent. On the other hand, the structure of biomolecules may change with temperature, thus their binding and activity can get modified.



The temperature dependence of the equilibrium constant (K) is described by the van’t Hoff equation [16]:




(12.14)

where T denotes the temperature, R is the gas constant, ΔHθ indicates the standard enthalpy change for the process.

Over temperature intervals where the reaction enthalpy can be considered constant, the above equation can be integrated to yield:
(12.15)

The empirical formula for the temperature dependence of rate constants was also suggested by van’t Hoff (in 1884 [17]), and was given a physical interpretation based on the collisional theory of gases by Arrhenius (in 1889 [18]). According to the equation named after him:




(12.16)

where A is the pre-exponential factor, EA is the so-called activation energy. In a simplified picture, A gives the total number of collisions and the exp(-EA/RT) is the probability that any given collision will result in a reaction.

Extended Arrhenius formulas

The Arrhenius equation was modified by several authors [19]. The most important theoretical treatments of reaction rates are the so-called t(TST) − also known as activated complex theory − introduced by Eyring, Evans and Polanyi [20], and by Pelzer and Wigner [21] in the 1930s.



The principal result of TST is the formula for the rate constant k:




(12.17)

where ΔG‡ is the free enthalpy of activation (describing the activation complex characteristic for the reaction), kB, ħ and R are the Boltzmann, Planck and gas constants, respectively.

The traditional phenomenological handling of the Arrhenius equation plot, ln(k) vs. 1/T. The slope of this, usually (nearly) linear plot is the empirical activation energy EA. Applying the procedure to the TST formula shows that the slope is:
(12.18)

The temperature dependence of ΔH‡ and ΔS‡ is usually negligible over the range of T measured, so the above expression simplifies to the formula:




(12.19)

The use of the Arrhenius equation is widespread in the literature. In fact, if the overall rate constant follows the Arrhenius equation, then this is occasionally considered evidence that the structure of biomolecules (receptor or enzyme) remained unchanged throughout the temperature range investigated. This reasoning is faulty, however: without a detailed knowledge of the mechanism, the appearance of a single Arrhenius equation is insufficient to exclude variations of the structure in question. There are also cases when a complex mechanism cannot be described even with extended Arrhenius-type formulas. Although detailed discussion of these cases is beyond the scope, it is important to keep in mind that this is typical rather than exceptional behavior when more than a single reaction dominates. It should also be noted that state-of-the-art computational methods, such as high-level molecular dynamics simulations, have matured to the point where accurate theoretical predictions can be made for the transition states and their associated activation energies for some enzyme catalyzed reactions - see, e.g., [22] for a recent example.

4. General remarks on parameter estimation

Traditionally kinetic models used to be linearized whenever possible, in order to estimate their parameters. Severe shortcomings of this approach have long been recognized [23]. It is important to calculate the statistical uncertainty of the parameters obtained, based on the experimental errors inherent in the input data. This becomes complicated when linearization is involved, and the extra effort needed for proper calculation negates the apparent simplicity of using linear regression. With advanced nonlinear methods readily available today these transformation are unnecessary. Direct parameter estimation is preferred. A recent theoretical paper by Tasi and Barna [24] elaborates this on the example of a Michaelis-Menten mechanisms evaluated according to the Woolf-Lineweaver-Burk form. They showed that non-linear least-squares fitting can be adequately handled with their method, which is based on the simplex optimization technique with error estimation.

There is a large variety of software available. For relatively simple systems, even using built-in nonlinear solvers for spreadsheet programs is feasible. General-purpose mathematical suites, such as Mathematica, MATLAB or their freeware equivalents, can also be used. For example, there are several specialized kinetics packages available in the “R” free software environment - for a current listing browse

http://cran.r-project.org/web/packages/available_packages_by_name.html.

Turányi et al. [25] reported developing in-house MATLAB code for encoding sophisticated statistical evaluation of large-scale kinetics models. There is also a number of special-purpose programs aimed at kinetics. One of the most comprehensive such suites is ZiTa [26]. This incorporates ODE-solver capability with parameter estimation, and utilizes flexible model definition that accommodates equilibrium equations besides kinetic ones.

We have seen that even a relatively simple mechanism such as Michaelis-Menten leads to differential equations whose solution can be different, depending on the boundary conditions and on the approximations applied. The majority of enzyme reactions does not exactly follow Michaelis-Menten kinetics [27], and many reaction systems are more complicated. These types of problems can only be treated via simulation: with numerical solution of the system of differential equations written according to an assumed mechanism, and comparing the modeled results with experiments. Numerically solving the system of ordinary differential equations (ODEs) is a standard computational problem [28]. A particular problem that frequently occurs in kinetics, when there are unstable intermediates or other fast reacting species, is the so-called stiffness: the solution involves terms that may vary exceedingly rapidly, along with slower steps. This cause numerical instabilities, poor convergence, and undue restrictions on the step size applicable. There are many well-tested algorithms [29] and program implementations [30] available for overcoming these difficulties and routinely solving stiff ODEs in kinetic systems, however. While historically these tools were mostly developed by, and distributed to, users focusing on gas-phase kinetic applications, in recent years there have been growing awareness for their utility in the fields of biochemistry and systems biology, as well. It is now well recognized in enzyme kinetics, for example, that the co-existence of fast processes (like enzyme-substrate interaction) with slow steps (such as typical product formation) causes stiffness of the mechanism. There are now programs specifically targeted for biochemical research – and many of these tools are free for academic purposes. For a sampling of their continuously expanding range see the references [30e], and citations therein.

5. Parameter estimation in pharmacokinetics



The scientific discipline investigating rates of processes in pharmacology is called pharmacokinetics. Much of the mathematical formalism developed for chemical kinetics finds application here, even though biological transformations as well as pure physical processes also play important roles besides chemistry. Many effects of interest, typically depicted as dose-response curves, follow sigmoidal shape. One frequently used form is the Hill equation [32]:




(12.20)

where θ is the fraction of maximum (either for bound ligand or response). Application of Eq. 12.20 marked the start of quantitative treatment in pharmacology [33].

It is often desirable to describe these curves with a single parameter, for which the mid-point is the most frequently used practical choice. For active compounds (agonists), the term EC50 is defined as the concentration that produces 50% of the maximal possible effect. For inhibitors, the 50% inhibition concentration, IC50, is used: that is the concentration of the inhibitor which reduces an effect (such as some response, or binding of the agonist) to half its maximum value. When the curve is symmetrical then the half-maximum level coincides with the inflection point. There are cases, however, when asymmetrical curves are encountered where the inflection point is distinct from the mid-point, so care should be taken not to confuse the two.



Note that Eq. 12.20 assumes a zero baseline. Switching to a generic dependent variable y instead of θ, allowing for a non-specific y0 effect present at [L]=0, and rearranging for a traditional linearized plot yields




(12.21)

(where KS=KDn was introduced, and yM is the top asymptote). This equation (or one of its equivalent forms) is, mathematically, a four-parameter logistic curve. nH is called Hill slope (or coefficient). The mid-point is at [L]=KS.

A simulated dataset is used to illustrate fitting to Eqs. 12.20-21. Plotted below, the theoretical curve is characterized by IC50=10.0 µM, nH=1.00, KD=KS=10.0 µM. The data points shown are generated with 10% relative error. Non-linear parameter estimation based on Eq. 12.20 yields nH=1.00 and KD=9.93 µM (dashed curve), i.e. IC50=9.93 µM. Linearized fitting via Eq. 12.21 yields nH=1.01 and KS=9.72 µM (dotted curve), i.e. IC50=9.72 µM.



Figure 12.3. A simulated dataset is used to illustrate fitting to Eqs. 12.20-21

Due to the importance of the EC50 (or IC50) parameter in pharmaceutical research, many specialized software tools are available for its determination. Just as mentioned in the section on kinetic parameter estimation, it is crucial to use proper statistical treatment of the experimental errors, and for this reason direct non-linear algorithms should be preferred over deprecated linearization methods [34].

6. References



  1. IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). XML on-line corrected version: http://goldbook.iupac.org (2006-) created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. ISBN 0-9678550-9-8. doi:10.1351/goldbook

  2. a) M. C. Sauer Jr and B. Ward. Reactions of hydrogen atoms with benzene and toluene studied by pulsed radiolysis: reaction rate constants and transient spectra in the gas phase and aqueous solution. J. Phys. Chem. 71, 3971-3983 (1967).

b) V. D. Parker, W. Hao, Z. Li and R. Scow, Nonconventional versus conventional application of pseudo‐first‐order kinetics to fundamental organic reactions. Internat. J. Chem. Kinet., 44, 2-12 (2011).

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  2. G. E. Briggs, J. B. S. Haldane, A note on the kinetics of enzyme action. Biochem. J. 18, 338-339 (1925).

  3. A. Fersht, Structure and Mechanism in protein science, W. H. Freeman and Company, New York (2000).

  4. C. S. Hanes, Studies on plant amylases: The effect of starch concentration upon the velocity of hydrolysis by the amylase of germinated barley. Biochem. J. 26, 1406 (1932).

  5. H. Lineweaver and D. Burk, The Determination of Enzyme Dissociation Constants. J. Am. Chem. Soc. 56, 658-666 (1934).

  6. R. Eisenthal and A. Cornish-Bowden, The direct linear plot. A new graphical procedure for estimating enzyme kinetic parameters. Biochem. J. 139, 715-720 (1974).

  7. M. Sakoda and K. Hiromi, Determination of the best-fit values of kinetic parameters of the Michaelis-Menten equation by the method of least squares with the Taylor expansion. J. Biochem. 80, 547-555 (1976).

  8. J.G.W. Raaijmakers, Statistical analysis of the Michaelis-Menten equation. Biometrics, 43, 793-803 (1987).

  9. J. A. Borghans, R. J. de Boer, L. A. Segel, Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58, 43-63 (1996).

  10. S. Garneau-Tsodikova, I. A. Shkel, O. V. Tsodikov, Exact and user-friendly kinetic analysis of the two-step rapid equilibrium Michaelis-Menten mechanism. Anal. Biochem. 387, 276-279 (2009).

  11. J. J. McGuire, Anticancer antifolates: current status and future directions. Curr. Pharm. Des. 9, 2593-613 (2003).

  12. S. E. Szedlacsek and R. G. Duggleby, Kinetics of slow and tight-binding inhibitors. Meth. Enzymol. 249, 144-180 (1995).

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  14. P. W. Atkins, Physical Chemistry 6th edition, Oxford University Press.

  15. J. H. van‘t Hoff, Études de dynamique chimique. F. Muller & Co., Amsterdam (1884). Studies of Chemical Dynamics, revised edition with additions by E. Cohen, translated by T. Ewan (1896); reprinted in 2010 by BiblioBazaar, LLC.

  16. S. Arrhenius, Über die Reaktionsgeschwindigkeit bei der Inversion von Rohzucker durch Säuren. Zeits. Phys. Chem. 4, 226-248 (1889). On the reaction velocity of the inversion of cane sugar by acids, translated and published in Margaret H. Back & Keith J. Laidler (Eds.): Selected Readings in Chemical Kinetics, Pergamon Press, Oxford (1967).

  17. J. N. Murrell, Understanding the Rates of Chemical Reactions, in: Fundamental World of Quantum Chemistry, Vol. 2, E. J. Brändas, E. S. Kryachko (Eds.), Kluwer Academic Publishers, p155-180 (2003).

  18. a) H. Eyring and M. Polanyi, Über einfache gasreaktionen (On Simple Gas Reactions). Z. Physik. Chemie, B12, 279 (1931). Part of this is translated in: M. H. Back, and K. J. Laidler, Selected Readings in Chemical Kinetics, Pergamon Press (1967).

b) M. G. Evans, M. Polanyi, Some applications of the transition state method to the calculation of reaction velocities, especially in solution. Trans. Faraday Soc. 31, 875 (1935).

  1. H. Pelzer and E.Wigner, Über die Geschwindigkeitkonstante von Austausch Reaktionen. Z. Physik. Chem., B15, 445-453 (1932).

  2. B. Kormányos, A. K. Horváth, G. Peintler and I. Nagypál. Inherent pitfalls in the simplified evaluation of kinetic curves. J. Phys. Chem. A 111, 8104-8109 (2007).

  3. M. W. Lee and M. Meuwly, "Molecular dynamics simulation of nitric oxide in myoglobin." J. Phys. Chem. B 116, 4154-4162 (2012).

  4. Gy. Tasi and D. Barna, "Analytical and numerical computation of error propagation of model parameters" J. Math. Chem. 49, 1322-1329 (2011).

  5. T. Turányi, T. Nagy, I. Gy. Zsély, M. Cserháti, T. Varga, B. T. Szabó, I. Sedyó, P. T. Kiss, A. Zempléni and H. J. Curran, "Determination of rate parameters based on both direct and indirect measurements" Internat. J. Chem. Kinet. 44, 284-302, (2012).

  6. G. Peintler “ZiTa, Version 5.0, a Comprehensive Program Package for Fitting Parameters of Chemical Reaction Mechanism”, University of Szeged, Hungary, 1989-1998.

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