- Research article
- Open Access
On the curvature in logarithmic plots of rate coefficients for chemical reactions
© Canepa et al 2011
Received: 17 December 2010
Accepted: 6 May 2011
Published: 6 May 2011
In terms of the reduced potential energy barrier ζ = Δu TS /kT, the rate coefficients for chemical reactions are usually expressed as proportional to e-ζ. The coupling between vibrational modes of the medium to the reaction coordinate leads to a proportionality of the regularized gamma function of Euler Q(a,ζ) = Γ(a,ζ)/Γ(a), with a being the number of modes coupled to the reaction coordinate. In this work, the experimental rate coefficients at various temperatures for several chemical reactions were fitted to the theoretical expression in terms of Q(a,ζ) to determine the extent of its validity and generality. The new expression affords lower deviations from the experimental points in 29 cases out of 38 and it accounts for the curvature in the logarithmic plots of rate coefficients versus inverse temperature. In the absence of tunneling, conventional theories predict the curvature of these plots to be identically zero.
Curvatures in logarithmic plots of enzyme-catalyzed reactions were detected and discussed as early as 1949 by Kistiakowsky and Lumry  in a paper on the hydrolysis of urea by urease, and described as widespread by Maier and Tappel  in 1955. Among the many examples of curved Arrhenius plots known for enzyme-catalyzed processes are the cleaving of RNA by deoxyribozyme , the activation of spinach chloroplast fructose-1,6-biphosphatase , the catalytic activity of urocanase from Pseudomonas putida , adenylate cyclase from Saccharomyces cerevisiae , turnip peroxidase, bovine intestinal phosphatase , the kinetics of Shiff base formation between a cholesterol ozonolysis product and dimyristoyl phosphatidyl ethanolamine , the activity of cyanobacterial ADP-glucose pyrophosphorylase from Anabaena , and the hydrolysis of various amides catalyzed by α-chymotrypsin . Among the explanations put forward to rationalize the observed curvature are the temperature-induced conformational change in the reactant , the assumption that the measured rate coefficient actually is a combination of elementary rate coefficients for a multiple-step reaction , or the notion that the derivative of ΔH TS with respect to temperature is negative, i. e. . The curvature of lnk r versus 1/T is thus viewed as a result of a more complex microscopic mechanism in turn composed of individual elementary steps that do follow Transition State Theory (TST) behavior. The shortcomings of TST have been discussed by Pineda and Schwartz , who describe promoting vibrations coupled to the reaction coordinate. While their computational analysis spans a time scale of tens of picosecond , this work focuses on the low-frequency, long-range motions of the whole protein on a time scale of up to one second for α-chymotrypsin. The link between enzyme dynamics and hydrogen-transfer reactions was also discussed by Knapp and Klinman  in the context of the environmentally coupled hydrogen tunneling model. Their analysis rests on the relatively large deBroglie wavelength for hydrogen, and was tested on a thermophilic alcohol dehydrogenase and soybean lipoxygenase-1. Among the extended formulations of TST, the ensemble-averaged variational transition-state theory with multidimensional tunneling (EA-VTST/MT)  was successfully applied to many proton- or hydride-transfer reactions, as the interconversion of L-alanine and D-alanine catalyzed by alanine racemase , and the hydride transfer from nicotinamide adenine dinucleotide to flavin mononucleotide catalyzed by morphinone reductase . Another way of interpreting the curvature of Arrhenius plots has been described by Masgrau and González-Lafont , who postulate the temperature dependence of the activation energy. Their analysis shows that curvature is present even in the absence of variational or tunneling effects, and the calculated Arrhenius plots for two gas-phase reactions are convex at low temperature and concave at high temperatures. Quite interestingly, the experimental Arrhenius plots taken into account in our work, which is not by any means exhaustive, are, without exception, convex at any temperature, with the curvature being even more negative at higher temperatures. This difference could be a consequence of the condensed-phase environment of the reactions investigated in this work to evaluate the curvatures of Arrhenius plots.
with ζ = Δu TS /kT being the reduced potential energy barrier. In (1) the quantities
This work aims to establish whether the validity of (1) and (2) extends beyond the successful interpretations of the hydrolysis of amides by α-chymotrypsin  and the alkylation of 3-bromopyridine by iodomethane in acetonitrile . To this effect, the rate coefficient at different temperatures for various unimolecular, bimolecular, and enzyme-catalyzed reactions were examined, restricting the analysis to sets of data with at least four experimental points. By no means does the data discussed in this work cover the extensive literature reporting Arrhenius and Eyring plots for a wide variety of reacting systems. Whenever possible, a statistical analysis on the standard deviations of the activation parameters induced by the error bars of the rate coefficients was carried out.
with the simplex search method of Lagarias . The form (12) of σ was chosen in order to have the same response for x i and . A typical value of σ = 2.10-2 corresponds to a 15% average deviation. For comparison, constrained minimizations with a = 1 for all reactions (equivalent to Arrhenius plots) were also performed, and the results are given in the supporting information [Additional file 1].
3. Results and Discussion
Activation parameters for the thermolyses of N-benzyl-N-nitrosoamides
Δu TS / kcal mol-1
Activation parameters for the hydrolysis of trans-dinitrobis(ethylenediamine) cobalt III ion
Activation parameters for the rearrangement of bis-(4-chlorophenyl) thioncarbonate to bis-(4-chlorophenyl) thiolcarbonate (without solvent)
Activation parameters for the solvolysis of methyldiphenylsulfonium ion in H2O and EtOH
Activation parameters for the neutral hydrolysis of methyl trifluoroacetate in H2O/DMSO
Activation parameters for the oxidation of xanthine by xanthine oxidase
Activation parameters for hydrolyses catalyzed by α-chymotrypsin
Δu TS / kcal mol-1
Activation parameters for the hydride transfer catalyzed by Escherichia coli dihydrofolate reductase in various solvents
Δu TS / kcal mol-1
Calculated (Q(a,ζ H )/Q(a,ζ D )) and experimental kinetic deuterium isotope effects at 20 C for the hydride transfer catalyzed by Escherichia coli dihydrofolate reductase in various solvents
Activation parameters for the atom transfer radical polymerization of alkyl halides catalyzed by Cu(I)Br(PMDETAa) in acetonitrile
Δu TS / kcal mol-1
Activation parameters for the hydration of carbon dioxide in water
The high potential energy barriers obtained with the minimization of σ are a consequence of the augmented size of the reacting system with respect to the substrate molecule. Although the barrier of 143.96 kcal mol -1 obtained with (5) (second entry in Table 1) might be considered unusual, its high value is due to the participation of the solvent to the reaction coordinate. The gas-phase barriers obtained computationally may thus be regarded as lower limits to a much higher barrier in the condensed phase, augmented by the increase in potential energy due to change in coordinates of the medium to reach the transition structure of a large substrate-solvent cluster.
In order to estimate the error bars on the parameters obtained by nonlinear regression to (5) and (6), we reoptimized all the parameters in 10000 runs after allowing the rate constants to vary according to a normal distribution with the experimentally determined variance (the averaged variances for the rate coefficients of the reactions under consideration are: 16.9% for the thermolyses N-alkyl-N-nitrosopivalamides, 5% for the hydrolysis of dinitrobis(ethylenediamine) cobalt III ion, 4.0% for the solvolysis of methyldiphenylsulfonium ion, 0.04% for the hydrolysis of methyl trifluoroacetate, 8.7% for the hydrolyses catalyzed by α-chymotrypsin, 4.7% for the hydride transfer catalyzed by Escherichia coli dihydrofolate reductase, 7.1% for the hydration of CO2, and 1.9% for the solvolysis of 4-methylbenzhydryl 4-nitrobenzoate). The standard deviations for each activation parameter were determined and are given in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 in parentheses. In all cases where the standard deviations are large, the parameters must be regarded as mere estimates. The Arrhenius-type plots give much narrower error bars with respect to the plots obtained with (5). For example, the standard deviation on reaction barriers with a = 1 lie all below 2 kcal mol -1 , while, with the available experimental precision, some values of a exhibit a 50% error bar. Thus the accuracy required in order to obtain small error bars on the activation parameters with (5) is demanding. This degree of accuracy is matched by the excellent set of rate coefficients for the neutral hydrolysis of methyl trifluoroacetate in H2O/DMSO, affording the paramenters in Table 5, which exhibit standard deviations in stark contrast with their counterparts in the remaining Tables .
Although the high values of the standard deviations for the activation parameters obtained with (5) demand caution in their interpretation, we may qualitatively take the average curvature to measure the amount of deviation from TST caused by the cooperative mechanism, and outline four types of reactions: (a) systems with a limited coupling between the medium and the substrate that follow Arrhenius behavior. In this group the conventional mechanism of exclusive energy transfer to the reaction coordinate is the fastest, with a resulting linear logarithmic plot (Figure 2c); (b) systems with moderate to good coupling ( ) and low response frequencies (10-3 ÷ 10-2 Hz). As the ability of the medium to effectively couple modes to the reaction coordinate grows, the coupled mechanism prevails for sufficiently large clusters (Figure 2d); (c) systems with moderate coupling ( ) and high response frequencies (10-1 ÷ 10-2 Hz); (d) lastly, we may have the predominance of the cooperative mechanism with high response frequencies (10-1 ÷ 103 Hz), and optimal coupling with in enzyme-catalyzed reactions.
1. When the vibrational coupling between the substrate and the medium is taken into account, the rate coefficients of unimolecular and bimolecular reactions exhibit a dependence of the reduced potential energy barrier ζ = Δu TS /kT proportional to the regularized gamma function of Euler Q(a,ζ).
2. The deviations obtained from the experimental rate constants as a function of temperature using equation (5) are in many cases lower with respect to the standard TST equation.
3. The corresponding activation energies exhibit considerably higher values with respect to the values obtained with TST, a fact ascribed to the much larger size of the system under consideration, which adds the contribution of the reaction medium to the intrinsic gas-phase barrier. Response frequencies are correspondingly lower.
4. The standard deviations on the activation parameters obtained using equation (5) are of the same order of magnitude as the parameter themselves, a fact that puts more stringent requirements on the accuracy of the experimentally determined rate coefficients.
5. The reactions subjected to the above analysis may be divided into four groups according to the extent and the features of the substrate-medium coupling: (a) no coupling, linear logarithmic plots of k r versus ζ; (b) coupling with low response frequencies; (c) coupling with high response frequencies; (d) optimal coupling (a/ζ ≈ 1) with very high response frequencies (enzyme catalysis).
6. The proposed expression for the rate constants of reactions in the condensed phase does not yet allow direct calculations of rate coefficients of chemical reactions, in that the physical parameters in the expression (the reduced potential energy barrier ζ, the intrinsic response frequency ν of the reacting system, and the number a of active modes energetically coupled to the reaction coordinate) remain computationally elusive. However, its statistical nature is able to account for the negative curvatures of experimental Arrhenius plots through the intrinsically non-exponential dependence of the rate constants from the reduced potential energy barrier.
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